latexindent (23feb17)

git-svn-id: svn://tug.org/texlive/trunk@43326 c570f23f-e606-0410-a88d-b1316a301751

111 files changed, 9286 insertions, 17959 deletions

diff --git a/Master/texmf-dist/doc/support/latexindent/README b/Master/texmf-dist/doc/support/latexindent/README
index 0fc5cf558ed..84048efae10 100644
--- a/Master/texmf-dist/doc/support/latexindent/README
+++ b/Master/texmf-dist/doc/support/latexindent/README
@@ -20,7 +20,7 @@
 
 *+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+*+
 
-FOR COMPLETE DETAILS, PLEASE SEE documentation/manual.pdf
+FOR COMPLETE DETAILS, PLEASE SEE documentation/latexindent.pdf
 
 Note: latexindent.exe was created using 
 
@@ -34,6 +34,7 @@ USAGE
     You'll need
 
         latexindent.pl
+        LatexIndent/*.pm
         defaultSettings.yaml
 
     in the same directory. Windows users might prefer to grab latexindent.exe
@@ -49,5 +50,6 @@ I recommend both using the following:
     - a visual check, at the very least, make sure that 
       each file has the same number of lines
     - a check using latexdiff inputfile.tex outputfile.tex
+    - git status myfile.tex
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/Master/texmf-dist/doc/support/latexindent/appendices.tex b/Master/texmf-dist/doc/support/latexindent/appendices.tex
new file mode 100644
index 00000000000..cc77004e45b
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/appendices.tex
@@ -0,0 +1,192 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\appendix
+	\section{Required \texttt{Perl} modules}\label{sec:requiredmodules}
+	 If you intend to use \texttt{latexindent.pl} and \emph{not} one of the supplied standalone executable files, then you will need a few standard Perl modules -- if you can run the
+	 minimum code in \cref{lst:helloworld} (\texttt{perl helloworld.pl}) then you will be able to run \texttt{latexindent.pl}, otherwise you may
+	 need to install the missing modules.
+
+	 \begin{cmhlistings}[language=Perl]{\texttt{helloworld.pl}}{lst:helloworld}
+	#!/usr/bin/perl
+
+	use strict;
+	use warnings;
+	use FindBin;
+	use YAML::Tiny;
+	use File::Copy;
+	use File::Basename;
+	use File::HomeDir;
+	use Getopt::Long;
+    use Data::Dumper;
+
+	print "hello world";
+	exit;
+\end{cmhlistings}
+	 My default installation on Ubuntu 12.04 did \emph{not} come
+	 with all of these modules as standard, but Strawberry Perl for Windows \cite{strawberryperl}
+	 did.
+
+	 Installing the modules given in \cref{lst:helloworld} will vary depending on your
+	 operating system and \texttt{Perl} distribution. For example, Ubuntu users
+	 might visit the software center, or else run
+	 \begin{commandshell}
+sudo perl -MCPAN -e 'install "File::HomeDir"'
+ \end{commandshell}
+
+	 Linux users may be interested in exploring Perlbrew \cite{perlbrew}; possible installation and setup
+	 options follow for Ubuntu (other distributions will need slightly different commands).
+	 \begin{commandshell}
+sudo apt-get install perlbrew
+perlbrew install perl-5.20.1
+perlbrew switch perl-5.20.1
+sudo apt-get install curl
+curl -L http://cpanmin.us | perl - App::cpanminus
+cpanm YAML::Tiny
+cpanm File::HomeDir
+\end{commandshell}
+
+	 Strawberry Perl users on Windows might use
+	 \texttt{CPAN client}. All of the modules are readily available on CPAN \cite{cpan}.
+
+	 \texttt{indent.log} will contain details of the location
+	 of the Perl modules on your system.  \texttt{latexindent.exe} is a standalone
+	 executable for Windows (and therefore does not require a Perl distribution) and caches copies of the Perl modules onto your system; if you
+	 wish to see where they are cached, use the  \texttt{trace} option, e.g
+	 \begin{dosprompt}
+latexindent.exe -t myfile.tex
+ \end{dosprompt}
+
+	\section{Updating the \texttt{path} variable}\label{sec:updating-path}
+	 \texttt{latexindent.pl} has a few scripts (available at \cite{latexindent-home}) that can update the \texttt{path} variables\footnote{Thanks to \cite{jasjuang} for this feature!}. If you're
+	 on a Linux or Mac machine, then you'll want \texttt{CMakeLists.txt} from \cite{latexindent-home}.
+	\subsection{Add to path for Linux}
+		To add \texttt{latexindent.pl} to the path for Linux, follow these steps:
+		\begin{enumerate}
+			\item download  \texttt{latexindent.pl} and its associated modules, \texttt{defaultSettings.yaml},
+			      to your chosen directory from \cite{latexindent-home} ;
+			\item within your directory, create a directory called \texttt{path-helper-files} and
+			      download \texttt{CMakeLists.txt} and \lstinline!cmake_uninstall.cmake.in!
+			      from \cite{latexindent-home}/path-helper-files to this directory;
+			\item run
+			      \begin{commandshell}
+ls /usr/local/bin
+          \end{commandshell}
+			      to see what is \emph{currently} in there;
+			\item run the following commands
+			      \begin{commandshell}
+sudo apt-get install cmake
+sudo apt-get update && sudo apt-get install build-essential
+mkdir build && cd build
+cmake ../path-helper-files
+sudo make install
+\end{commandshell}
+			\item run
+			      \begin{commandshell}
+ls /usr/local/bin
+          \end{commandshell}
+			      again to check that \texttt{latexindent.pl}, its modules and \texttt{defaultSettings.yaml} have been added.
+		\end{enumerate}
+		To \emph{remove} the files, run
+		\begin{commandshell}
+sudo make uninstall}.
+    \end{commandshell}
+	\subsection{Add to path for Windows}
+		To add \texttt{latexindent.exe} to the path for Windows, follow these steps:
+		\begin{enumerate}
+			\item download  \texttt{latexindent.exe}, \texttt{defaultSettings.yaml},  \texttt{add-to-path.bat}
+			      from \cite{latexindent-home} to your chosen directory;
+			\item open a command prompt and run the following command to see what is \emph{currently} in your \lstinline!%path%! variable;
+			      \begin{dosprompt}
+echo %path%
+          \end{dosprompt}
+			\item right click on \texttt{add-to-path.bat} and \emph{Run as administrator};
+			\item log out, and log back in;
+			\item open a command prompt and run
+			      \begin{dosprompt}
+echo %path%
+          \end{dosprompt}
+			      to check that the appropriate directory has been added to your \lstinline!%path%!.
+		\end{enumerate}
+		To \emph{remove} the directory from your \lstinline!%path%!, run \texttt{remove-from-path.bat} as administrator.
+
+	\section{Differences from Version 2.2 to 3.0}\label{app:differences}
+	 There are a few (small) changes to the interface when comparing Version 2.2 to Version 3.0.
+	 Explicitly, in previous versions you might have run, for example,
+	 \begin{commandshell}
+latexindent.pl -o myfile.tex outputfile.tex
+ \end{commandshell}
+	 whereas in Version 3.0 you would run any of the following, for example,
+	 \begin{commandshell}
+latexindent.pl -o=outputfile.tex myfile.tex
+latexindent.pl -o outputfile.tex myfile.tex
+latexindent.pl myfile.tex -o outputfile.tex 
+latexindent.pl myfile.tex -o=outputfile.tex 
+latexindent.pl myfile.tex -outputfile=outputfile.tex 
+latexindent.pl myfile.tex -outputfile outputfile.tex 
+ \end{commandshell}
+	 noting that the \emph{output} file is given \emph{next to} the \texttt{-o} switch.
+
+	 The fields given in \cref{lst:obsoleteYaml} are \emph{obsolete} from Version 3.0
+	 onwards.
+	 \begin{yaml}[style=yaml-LST,numbers=none]{Obsolete YAML fields from Version 3.0}[colframe=white!25!red,colbacktitle=white!75!red,colback=white!90!red,]{lst:obsoleteYaml}
+alwaysLookforSplitBrackets
+alwaysLookforSplitBrackets
+checkunmatched
+checkunmatchedELSE
+checkunmatchedbracket
+constructIfElseFi
+\end{yaml}
+
+	 There is a slight difference when specifying indentation after headings; specifically,
+	 we now write \texttt{indentAfterThisHeading} instead of \texttt{indent}. See \cref{lst:indentAfterThisHeadingOld,lst:indentAfterThisHeadingNew}
+
+	 \begin{minipage}{.45\textwidth}
+		 \begin{yaml}[style=yaml-LST,numbers=none]{\texttt{indentAfterThisHeading} in Version 2.2}{lst:indentAfterThisHeadingOld}
+indentAfterHeadings:
+    part:
+       indent: 0
+       level: 1
+\end{yaml}
+	 \end{minipage}%
+	 \hfill
+	 \begin{minipage}{.45\textwidth}
+		 \begin{yaml}[style=yaml-LST,numbers=none]{\texttt{indentAfterThisHeading} in Version 3.0}{lst:indentAfterThisHeadingNew}
+indentAfterHeadings:
+    part:
+       indentAfterThisHeading: 0
+       level: 1
+\end{yaml}
+	 \end{minipage}%
+
+	 To specify \texttt{noAdditionalIndent} for display-math environments in Version 2.2, you would write YAML
+	 as in \cref{lst:noAdditionalIndentOld}; as of Version 3.0, you would write YAML as in \cref{lst:indentAfterThisHeadingNew1}
+	 or, if you're using \texttt{-m} switch, \cref{lst:indentAfterThisHeadingNew2}.
+
+	 \begin{minipage}{.45\textwidth}
+		 \begin{yaml}[style=yaml-LST,numbers=none]{\texttt{noAdditionalIndent} in Version 2.2}{lst:noAdditionalIndentOld}
+noAdditionalIndent:
+    \[: 0
+    \]: 0
+\end{yaml}
+	 \end{minipage}%
+	 \hfill
+	 \begin{minipage}{.45\textwidth}
+		 \begin{yaml}[style=yaml-LST,numbers=none]{\texttt{noAdditionalIndent} for \texttt{displayMath} in Version 3.0}{lst:indentAfterThisHeadingNew1}
+specialBeginEnd:
+    displayMath:
+        begin: '\\\['
+        end: '\\\]'
+        lookForThis: 0
+\end{yaml}
+
+		 \begin{yaml}[style=yaml-LST,numbers=none]{\texttt{noAdditionalIndent} for \texttt{displayMath}  in Version 3.0}{lst:indentAfterThisHeadingNew2}
+noAdditionalIndent:
+    displayMath: 1
+\end{yaml}
+	 \end{minipage}%
+
+	 \mbox{}\hfill     \begin{minipage}{.25\textwidth}
+		 \hrule
+
+		 \hfill\itshape End
+
+	 \end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.pdf b/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.pdf
deleted file mode 100644
index 4ac36f075da..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.pdf
+++ /dev/null
diff --git a/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.tex b/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.tex
deleted file mode 100644
index c39b89e627e..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/documentation/latexindent.tex
+++ /dev/null
@@ -1,1512 +0,0 @@
-% arara: pdflatex
-% arara: bibtex
-% arara: pdflatex
-% arara: pdflatex
-% !arara: pdflatex
-% !arara: indent: {overwrite: yes, trace: yes, localSettings: yes, silent: yes}
-\documentclass[11pt]{article}
-%   This program is free software: you can redistribute it and/or modify
-%   it under the terms of the GNU General Public License as published by
-%   the Free Software Foundation, either version 3 of the License, or
-%   (at your option) any later version.
-%
-%   This program is distributed in the hope that it will be useful,
-%   but WITHOUT ANY WARRANTY; without even the implied warranty of
-%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-%   GNU General Public License for more details.
-%
-%   See <http://www.gnu.org/licenses/>.
-\usepackage[left=4.5cm,right=2.5cm,showframe=false,
-top=2cm,bottom=1.5cm]{geometry}                      % page setup
-\usepackage{parskip}                                 % paragraph skips
-\usepackage{booktabs}                                % beautiful tables
-\usepackage{listings}                                % nice verbatim environments
-\usepackage{titlesec}                                % customize headings
-\usepackage{changepage}                              % adjust width of page
-\usepackage{fancyhdr}                                % headers & footers
-\usepackage[sc,format=hang,font=small]{caption}      % captions
-\usepackage[backend=bibtex]{biblatex}                % bibliography
-\usepackage{tcolorbox}                                % framed environments
-\usepackage[charter]{mathdesign}                     % changes font
-\usepackage[expansion=false,kerning=true]{microtype} % better kerning
-\usepackage{enumitem}                                % custom lists
-% setup gitinfo2, as in the manual, details just above begin{document}
-\usepackage[mark,grumpy]{gitinfo2}
-\usepackage{examplep}
-% tcolorbox libraries
-\tcbuselibrary{breakable,skins,listings}
-% tikz libraries
-\usetikzlibrary{positioning}
-\usetikzlibrary{decorations.pathmorphing}
-\usetikzlibrary{decorations,shapes}
-\usepackage{varioref}                                % clever referencing
-\usepackage{hyperref}
-\hypersetup{
-   pdfauthor={Chris Hughes},
-   pdftitle={latexindent.pl package},
-   pdfkeywords={perl;beautify;indentation},
-	bookmarksnumbered,
-	bookmarksopen,
-	linktocpage,
-    colorlinks=true,
-    linkcolor=blue,
-    citecolor=black,
-}
-\usepackage{cleveref}
-
-\addbibresource{latex-indent}
-\addbibresource{contributors}
-
-
-% http://tex.stackexchange.com/questions/122135/how-to-add-a-png-icon-on-the-right-side-of-a-tcolorbox-title
-\newtcolorbox{warning}{parbox=false,breakable,enhanced,arc=0mm,colback=red!5,colframe=red,leftrule=12mm,%
-	overlay={\node[anchor=north west,outer sep=2pt] at (frame.north west) {\includegraphics[width=8mm]{warning}}; }}
-
-\makeatletter
-\tcbset{
-	addtolol/.style={list entry={\kvtcb@title},add to list={lol}{lstlisting}},
-}
-
-\newtcblisting[use counter=lstlisting]{cmhlistings}[3][]{%
-	width=\linewidth,
-	breakable,
-	colback=blue!5!white,
-	colframe=white!85!black,
-	top=0cm,
-	bottom=0cm,
-	left=6mm,
-	listing only,
-	listing options={#1},
-	center title,
-	title={\color{black}{\scshape Listing \thetcbcounter}: ~#2},label={#3},
-	addtolol,
-}
-
-\lstset{%
-	basicstyle=\small\ttfamily,language={[LaTeX]TeX},
-	%	numbers=left,
-	numberstyle=\ttfamily%\small,
-	breaklines=true,
-	%   frame=single,framexleftmargin=8mm, xleftmargin=8mm,
-	%	prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookrightarrow}},
-	%	backgroundcolor=\color{green!5},frameround=fttt,
-	%	rulecolor=\color{blue!70!black},
-	keywordstyle=\color{blue},                    % keywords
-	commentstyle=\color{purple},    % comments
-	tabsize=3,
-	%xleftmargin=1.5em,
-}%
-\lstdefinestyle{yaml}{%
-	numbers=left,
-	numberstyle=\color{black},}
-\lstdefinestyle{demo}{
-	numbers=none,
-	linewidth=1.25\textwidth,
-	columns=fullflexible,
-}
-
-% stars around contributors
-\pgfdeclaredecoration{stars}{initial}{
-	\state{initial}[width=15pt]
-	{
-		\pgfmathparse{round(rnd*100)}
-		\pgfsetfillcolor{yellow!\pgfmathresult!orange}
-		\pgfsetstrokecolor{yellow!\pgfmathresult!red}
-		\pgfnode{star}{center}{}{}{\pgfusepath{stroke,fill}}
-	}
-	\state{final}
-	{
-		\pgfpathmoveto{\pgfpointdecoratedpathlast}
-	}
-}
-
-\newtcolorbox{stars}{%
-	enhanced jigsaw,
-	breakable, % allow page breaks
-	left=0cm,
-	top=0cm,
-	before skip=0.2cm,
-	boxsep=0cm,
-	frame style={draw=none,fill=none}, % hide the default frame
-	colback=white,
-	overlay={
-		\draw[inner sep=0,minimum size=rnd*15pt+2pt]
-		decorate[decoration={stars,segment length=2cm}] {
-			decorate[decoration={zigzag,segment length=2cm,amplitude=0.3cm}] {
-				([xshift=-.5cm,yshift=0.1cm]frame.south west) --  ([xshift=-.5cm,yshift=0.4cm]frame.north west)
-		}};
-		\draw[inner sep=0,minimum size=rnd*15pt+2pt]
-		decorate[decoration={stars,segment length=2cm}] {
-			decorate[decoration={zigzag,segment length=2cm,amplitude=0.3cm}] {
-				([xshift=.75cm,yshift=0.1cm]frame.south east) --  ([xshift=.75cm,yshift=0.6cm]frame.north east)
-		}};
-	},
-	% paragraph skips obeyed within tcolorbox
-	parbox=false,
-}
-
-% copied from /usr/local/texlive/2013/texmf-dist/tex/latex/biblatex/bbx/numeric.bbx
-% the only modification is the \stars and \endstars
-\defbibenvironment{specialbib}
-{\stars\list
-	{\printtext[labelnumberwidth]{%
-		\printfield{prefixnumber}%
-		\printfield{labelnumber}}}
-	{\setlength{\labelwidth}{\labelnumberwidth}%
-		\setlength{\leftmargin}{\labelwidth}%
-		\setlength{\labelsep}{\biblabelsep}%
-		\addtolength{\leftmargin}{\labelsep}%
-		\setlength{\itemsep}{\bibitemsep}%
-		\setlength{\parsep}{\bibparsep}}%
-	\renewcommand*{\makelabel}[1]{\hss##1}}
-{\endlist\endstars}
-{\item}
-
-% see: http://tex.stackexchange.com/questions/2245/verbatim-description-list-item
-\newcommand{\verbitem}[1]{\small\PVerb{#1}}
-% stolen from arara.sty http://mirrors.med.harvard.edu/ctan/support/arara/doc/arara.sty
-%\lstnewenvironment{yaml}[1][]{\lstset{%
-%	basicstyle=\ttfamily,
-%	numbers=left,
-%	xleftmargin=1.5em,
-%	breaklines=true,
-%	numberstyle=\ttfamily\small,
-%	columns=flexible,
-%	mathescape=false,
-%	#1,
-%}}
-%{}
-
-\newcommand{\fixthis}[1]
-{%
-	\marginpar{\huge \color{red} \framebox{FIX}}%
-	\typeout{FIXTHIS: p\thepage : #1^^J}%
-}
-% custom section
-\titleformat{\section}
-{\normalfont\Large\bfseries}
-{\llap{\thesection\hskip.5cm}}
-{0pt}
-{}
-% custom subsection
-\titleformat{\subsection}
-{\normalfont\bfseries}
-{\llap{\thesubsection\hskip.5cm}}
-{0pt}
-{}
-% custom subsubsection
-\titleformat{\subsubsection}
-{\normalfont\bfseries}
-{\llap{\thesubsubsection\hskip.5cm}}
-{0pt}
-{}
-
-
-\titlespacing\section{0pt}{12pt plus 4pt minus 2pt}{-5pt plus 2pt minus 2pt}
-\titlespacing\subsection{0pt}{12pt plus 4pt minus 2pt}{-6pt plus 2pt minus 2pt}
-\titlespacing\subsubsection{0pt}{12pt plus 4pt minus 2pt}{-6pt plus 2pt minus 2pt}
-
-
-% cleveref settings
-\crefname{table}{Table}{Tables}
-\Crefname{table}{Table}{Tables}
-\crefname{figure}{Figure}{Figures}
-\Crefname{figure}{Figure}{Figures}
-\crefname{section}{Section}{Sections}
-\Crefname{section}{Section}{Sections}
-\crefname{listing}{Listing}{Listings}
-\Crefname{listing}{Listing}{Listings}
-
-% headers and footers
-\fancyhf{} % delete current header and footer
-\fancyhead[R]{\bfseries\thepage}
-\fancyheadoffset[L]{3cm}
-\pagestyle{fancy}
-
-% renew plain style
-\fancypagestyle{plain}{%
-\fancyhf{} % clear all header and footer fields
-\renewcommand{\headrulewidth}{0pt}
-\renewcommand{\footrulewidth}{0pt}}
-
-% sidebyside environment
-\newenvironment{sidebyside}{\begin{adjustwidth}{-3cm}{1cm}}{\end{adjustwidth}}
-
-% gitinfo2 settings
-\renewcommand{\gitMark}{\gitBranch\,@\,\gitAbbrevHash{}\,\textbullet{}\,\gitAuthorDate }
-
-% setting up gitinfo2:
-%   copy the file post-xxx-sample.txt from https://www.ctan.org/tex-archive/macros/latex/contrib/gitinfo2
-%   and put it in .git/hooks/post-checkout
-% then
-%   cd .git/hooks
-%   chmod g+x post-checkout
-%   chmod +x post-checkout
-%   cp post-checkout post-commit
-%   cp post-checkout post-merge
-%   cd ../..
-%   git checkout master
-%   git checkout develop
-%   ls .git
-% and you should see gitHeadInfo.gin
-\begin{document}
-% \begin{noindent}
-	\title{\lstinline[basicstyle=\huge\ttfamily]!latexindent.pl!\\[1cm]
-		Version 2.2}
-% \end{noindent}
-\author{Chris Hughes \footnote{and contributors! (See \vref{sec:contributors}.)}}
-\maketitle
-\begin{abstract}
-	\lstinline!latexindent.pl! is a \lstinline!Perl! script that indents \lstinline!.tex! (and other)
-	files according to an indentation scheme that the user can modify to suit their
-	taste. Environments, including those with alignment delimiters (such as \lstinline!tabular!),
-	and commands, including those that can split braces and brackets across lines,
-	are \emph{usually} handled correctly by the script. Options for \lstinline!verbatim!-like
-	environments and indentation after headings (such as \lstinline!\chapter!, \lstinline!\section!, etc)
-	are also available.
-\end{abstract}
-
-\tableofcontents
-\lstlistoflistings
-
-\section{Before we begin}
-\subsection{Thanks}
-I first created \lstinline!latexindent.pl! to help me format chapter files
-in a big project. After I blogged about it on the
-\TeX{} stack exchange \cite{cmhblog} I received some positive feedback and
-follow-up feature requests. A big thank you to Harish Kumar who has really
-helped to drive the script forward and has put it through a number of challenging
-tests--I look forward to more challenges in the future Harish!
-
-The \lstinline!yaml!-based interface of \lstinline!latexindent.pl! was inspired
-by the wonderful \lstinline!arara! tool; any similarities are deliberate, and
-I hope that it is perceived as the compliment that it is. Thank you to Paulo Cereda and the
-team for releasing this awesome tool; I initially worried that I was going to
-have to make a GUI for \lstinline!latexindent.pl!, but the release of \lstinline!arara!
-has meant there is no need. Thank you to Paulo for all of your advice and
-encouragement.
-
-There have been several contributors to the project so far (and hopefully more in
-the future!); thank you very much to the people detailed in \vref{sec:contributors}
-for their valued contributions.
-
-\subsection{License}
-\lstinline!latexindent.pl! is free and open source, and it always will be.
-Before you start using it on any important files, bear in mind that \lstinline!latexindent.pl! has the option to overwrite your \lstinline!.tex! files.
-It will always make at least one backup (you can choose how many it makes, see \cpageref{page:onlyonebackup})
-but you should still be careful when using it. The script has been tested on many
-files, but there are some known limitations (see \cref{sec:knownlimitations}).
-You, the user, are responsible for ensuring that you maintain backups of your files
-before running \lstinline!latexindent.pl! on them. I think it is important at this
-stage to restate an important part of the license here:
-\begin{quote}\itshape
-	This program is distributed in the hope that it will be useful,
-	but WITHOUT ANY WARRANTY; without even the implied warranty of
-	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-	GNU General Public License for more details.
-\end{quote}
-There is certainly no malicious intent in releasing this script, and I do hope
-that it works as you expect it to--if it does not, please first of all
-make sure that you have the correct settings, and then feel free to let me know with a
-complete minimum working example as I would like to improve the code as much as possible.
-
-\begin{warning}
-	Before you try the script on anything important (like your thesis), test it
-	out on the sample files that come with it in the \lstinline!success! directory.
-\end{warning}
-
-
-\section{Demonstration: before and after}
-Let's give a demonstration of some before and after code--after all, you probably
-won't want to try the script if you don't much like the results. You might also
-like to watch the video demonstration I made on youtube \cite{cmh:videodemo}
-
-As you look at \crefrange{lst:filecontentsbefore}{lst:pstricksafter}, remember
-that \lstinline!latexindent.pl! is just following its rules--there is nothing
-particular about these code snippets. All of the rules can be modified
-so that each user can personalize their indentation scheme.
-
-In each of the samples given in \crefrange{lst:filecontentsbefore}{lst:pstricksafter}
-the `before' case is a `worst case scenario' with no effort to make indentation. The `after'
-result would be the same, regardless of the leading white space at the beginning of
-each line which is stripped by \lstinline!latexindent.pl! (unless a \lstinline!verbatim!-like
-environment or \lstinline!noIndentBlock! is specified--more on this in \cref{sec:defuseloc}).
-
-  \begin{sidebyside}
-	\noindent
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!filecontents! before}{lst:filecontentsbefore}
-\begin{filecontents}{mybib.bib}
-@online{strawberryperl,
-title="Strawberry Perl",
-url="http://strawberryperl.com/"}
-@online{cmhblog,
-title="A Perl script ...
-url="...
-\end{filecontents}
-		\end{cmhlistings}
-	\end{minipage}%
-	\noindent
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!filecontents! after}{lst:filecontentsafter}
-\begin{filecontents}{mybib.bib}
-	@online{strawberryperl,
-		title="Strawberry Perl",
-		url="http://strawberryperl.com/"}
-	@online{cmhblog,
-		title="A Perl script for ...
-		url="...
-\end{filecontents}
-		\end{cmhlistings}
-	\end{minipage}
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!tikzset! before}{lst:tikzsetbefore}
-\tikzset{
-shrink inner sep/.code={
-\pgfkeysgetvalue...
-\pgfkeysgetvalue...
-}
-}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!tikzset! after}{lst:tikzsetafter}
-\tikzset{
-	shrink inner sep/.code={
-		\pgfkeysgetvalue...
-		\pgfkeysgetvalue...
-	}
-}
-		\end{cmhlistings}
-	\end{minipage}
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!pstricks! before}{lst:pstricksbefore}
-\def\Picture#1{%
-\def\stripH{#1}%
-\begin{pspicture}[showgrid...
-\psforeach{\row}{%
-{{3,2.8,2.7,3,3.1}},% <=== Only this
-{2.8,1,1.2,2,3},%
-...
-}{%
-\expandafter...
-}
-\end{pspicture}}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.6\textwidth}
-		\begin{cmhlistings}[style=demo]{\lstinline!pstricks! after}{lst:pstricksafter}
-\def\Picture#1{%
-	\def\stripH{#1}%
-	\begin{pspicture}[showgrid...
-		\psforeach{\row}{%
-			{{3,2.8,2.7,3,3.1}},% <===
-			{2.8,1,1.2,2,3},%
-            ...
-			}{%
-			\expandafter...
-		}
-	\end{pspicture}}
-		\end{cmhlistings}
-	\end{minipage}
-  \end{sidebyside}
-
-\section{How to use the script}
-\lstinline!latexindent.pl! ships as part of the \TeX Live distribution for
-Linux and Mac users; \lstinline!latexindent.exe! ships as part of the \TeX Live
-and MiK\TeX distributions for Windows users. These files are also available
-from github \cite{latexindent-home} should you wish to use them without
-a \TeX{} distribution; in this case, you may like to read \vref{sec:updating-path}
-which details how the \lstinline!path! variable can be updated.
-
-In what follows, we will always refer to \lstinline!latexindent.pl!, but depending on
-your operating system and preference, you might substitute \lstinline!latexindent.exe! or
-simply \lstinline!latexindent!.
-
-There are two ways to use \lstinline!latexindent.pl!: from the command line,
-and using \lstinline!arara!; we discuss these in \cref{sec:commandline} and
-\cref{sec:arara} respectively. We will discuss how to change the settings and 
-behaviour of the script in \vref{sec:defuseloc}.
-
-\lstinline!latexindent.pl! ships with \lstinline!latexindent.exe! for Windows
-users, so that you can use the script with or without a Perl distribution.
-If you plan to use \lstinline!latexindent.pl! (i.e, the original Perl script) then you will
-need a few standard Perl modules--see \vref{sec:requiredmodules} for details.
-
-\subsection{From the command line}\label{sec:commandline}
-\lstinline!latexindent.pl! has a number of different switches/flags/options, which
-can be combined in any way that you like, either in short or long form as detailed below. 
-\lstinline!latexindent.pl!  produces a \lstinline!.log! file, \lstinline!indent.log! every time it
-is run. There is a base of information that is written to \lstinline!indent.log!,
-but other additional information will be written depending
-on which of the following options are used.
-
-\begin{itemize}[labelsep=.25cm]
-	\item[] \lstinline!latexindent.pl!
-
-	      This will output a welcome message to the terminal, including the version number
-	      and available options.
-	\item[\verbitem{-h, --help}] \lstinline!latexindent.pl -h!
-
-	      As above this will output a welcome message to the terminal, including the version number
-	      and available options.
-	\item[] \lstinline!latexindent.pl myfile.tex!
-
-	      This will operate on \lstinline!myfile.tex!, but will simply output to your terminal; \lstinline!myfile.tex! will	not be changed in any way using this command.
-	\item[\verbitem{-w, --overwrite}] \lstinline!latexindent.pl -w myfile.tex!
-
-	      This \emph{will} overwrite \lstinline!myfile.tex!, but it will
-	      make a copy of \lstinline!myfile.tex! first. You can control the name of
-	      the extension (default is \lstinline!.bak!), and how many different backups are made--
-	      more on this in \cref{sec:defuseloc}; see \lstinline!backupExtension! and \lstinline!onlyOneBackUp!.
-
-	      Note that if \lstinline!latexindent.pl! can not create the backup, then it
-	      will exit without touching your original file; an error message will be given
-	      asking you to check the permissions of the backup file.
-	\item[\verbitem{-o,--outputfile}] \lstinline!latexindent.pl -o myfile.tex outputfile.tex!
-
-	      This will indent \lstinline!myfile.tex! and output it to \lstinline!outputfile.tex!,
-	      overwriting it (\lstinline!outputfile.tex!) if it already exists. Note that if \lstinline!latexindent.pl! is called with both
-	      the \lstinline!-w! and \lstinline!-o! switches, then \lstinline!-w! will
-	      be ignored and \lstinline!-o! will take priority (this seems safer than the
-	      other way round).
-
-	      Note that using \lstinline!-o! is equivalent to using \lstinline!latexindent.pl myfile.tex > outputfile.tex!
-	\item[\verbitem{-s, --silent}] \lstinline!latexindent.pl -s myfile.tex!
-
-	      Silent mode: no output will be given to the terminal.
-	\item[\verbitem{-t, --trace}] \lstinline!latexindent.pl -t myfile.tex!
-
-	      \label{page:traceswitch}
-	      Tracing mode: verbose output will be given to \lstinline!indent.log!. This
-	      is useful if \lstinline!latexindent.pl! has made a mistake and you're
-	      trying to find out where and why. You might also be interested in learning
-	      about \lstinline!latexindent.pl!'s thought process--if so, this
-	      switch is for you.
-	\item[\verbitem{-tt, --ttrace}] \lstinline!latexindent.pl -tt myfile.tex!
-
-      \emph{More detailed} tracing mode: this option gives more details to \lstinline!indent.log!
-      than the standard \lstinline!trace! option.
-	\item[\verbitem{-l, --local[=myyaml.yaml]}] \lstinline!latexindent.pl -l myfile.tex!
-
-\lstinline!latexindent.pl -l=myyaml.yaml myfile.tex!
-
-\lstinline!latexindent.pl -l myyaml.yaml myfile.tex!
-
-
-	      \label{page:localswitch}
-	      Local settings: you might like to read \cref{sec:defuseloc} before
-	      using this switch. \lstinline!latexindent.pl! will always load \lstinline!defaultSettings.yaml!
-	      and if it is called with the \lstinline!-l! switch and it finds \lstinline!localSettings.yaml!
-	      in the same directory as \lstinline!myfile.tex! then these settings will be
-	      added to the indentation scheme. Information will be given in \lstinline!indent.log! on
-	      the success or failure of loading \lstinline!localSettings.yaml!.
-
-The \lstinline!-l! flag can take an \emph{optional} parameter which details the name of a \lstinline!yaml! file
-that resides in the same directory as \lstinline!myfile.tex!; you can use this option if you would 
-like to load a settings file in the current working directory that is \emph{not} called \lstinline!localSettings.yaml!.
-
-	\item[\verbitem{-d, --onlydefault}] \lstinline!latexindent.pl -d myfile.tex!
-
-	      Only \lstinline!defaultSettings.yaml!: you might like to read \cref{sec:defuseloc} before
-	      using this switch. By default, \lstinline!latexindent.pl! will always search for
-	      \lstinline!indentconfig.yaml! or \lstinline!.indentconfig.yaml!  in your home directory. If you would prefer it not to do so
-	      then (instead of deleting or renaming \lstinline!indentconfig.yaml!/\lstinline!.indentconfig.yaml!) you can simply
-	      call the script with the \lstinline!-d! switch; note that this will also tell
-	      the script to ignore \lstinline!localSettings.yaml! even if it has been called with the
-	      \lstinline!-l! switch.
-
-	\item[\verbitem{-c, --cruft=<directory>}]\lstinline!latexindent.pl -c=/path/to/directory/ myfile.tex!
-
-	      If you wish to have backup files and \lstinline!indent.log! written to a directory
-	      other than the current working directory, then you can send these `cruft' files
-	      to another directory.
-	      % this switch was made as a result of http://tex.stackexchange.com/questions/142652/output-latexindent-auxiliary-files-to-a-different-directory
-\end{itemize}
-
-\lstinline!latexindent.pl! can also be called on a file without the file extension, for
-example \lstinline[breaklines=true,breakatwhitespace=true,]!latexindent.pl myfile! and in which case, you can specify
-the order in which extensions are searched for; see \vref{lst:fileExtensionPreference}
-for full details.
-
-\subsection{From \lstinline!arara!}\label{sec:arara}
-Using \lstinline!latexindent.pl! from the command line is fine for some folks, but
-others may find it easier to use from \lstinline!arara!. \lstinline!latexindent.pl!
-ships with an \lstinline!arara! rule, \lstinline!indent.yaml!, which can be copied
-to the directory of
-your other \lstinline!arara! rules; otherwise  you can add the directory in which \lstinline!latexindent.pl!
-resides to your \lstinline!araraconfig.yaml! file.
-
-Once you have told \lstinline!arara! where to find your \lstinline!indent! rule,
-you can use it any of the ways described in \cref{lst:arara} (or combinations thereof).
-In fact, \lstinline!arara! allows yet greater flexibility--you can use \lstinline!yes/no!, \lstinline!true/false!, or \lstinline!on/off! to toggle the various options.
-
-\begin{cmhlistings}[style=demo,escapeinside={(*@}{@*)}]{\lstinline!arara! sample usage}{lst:arara}
-%(*@@*) arara: indent
-%(*@@*) arara: indent: {overwrite: yes}
-%(*@@*) arara: indent: {output: myfile.tex}
-%(*@@*) arara: indent: {silent: yes}
-%(*@@*) arara: indent: {trace: yes}
-%(*@@*) arara: indent: {localSettings: yes}
-%(*@@*) arara: indent: {onlyDefault: on}
-%(*@@*) arara: indent: { cruft: /home/cmhughes/Desktop }
-\documentclass{article}
-...
-\end{cmhlistings}
-
-Hopefully the use of these rules is fairly self-explanatory, but for completeness
-\cref{tab:orbsandswitches} shows the relationship between \lstinline!arara! directive arguments and the
-switches given in \cref{sec:commandline}.
-
-\begin{table}[!ht]
-	\centering
-	\caption{\lstinline!arara! directive arguments and corresponding switches}
-	\label{tab:orbsandswitches}
-	\begin{tabular}{lc}
-		\toprule
-		\lstinline!arara! directive argument & switch         \\
-		\midrule
-		\lstinline!overwrite!                & \lstinline!-w! \\
-		\lstinline!output!                   & \lstinline!-o! \\
-		\lstinline!silent!                   & \lstinline!-s! \\
-		\lstinline!trace!                    & \lstinline!-t! \\
-		\lstinline!localSettings!            & \lstinline!-l! \\
-		\lstinline!onlyDefault!              & \lstinline!-d! \\
-		\lstinline!cruft!                    & \lstinline!-c! \\
-		\bottomrule
-	\end{tabular}
-\end{table}
-
-The \lstinline!cruft! directive does not work well when used with
-directories that contain spaces.
-
-\section{default, user, and local settings}\label{sec:defuseloc}
-\lstinline!latexindent.pl! loads its settings from \lstinline!defaultSettings.yaml!
-(rhymes with camel). The idea is to separate the behaviour of the script
-from the internal working--this is very similar to the way that we separate content
-from form when writing our documents in \LaTeX.
-
-\subsection{\lstinline!defaultSettings.yaml!}
-If you look in \lstinline!defaultSettings.yaml! you'll find the switches
-that govern the behaviour of \lstinline!latexindent.pl!. If you're not sure where
-\lstinline!defaultSettings.yaml! resides on your computer, don't worry as \lstinline!indent.log!
-will tell you where to find it.
-\lstinline!defaultSettings.yaml! is commented,
-but here is a description of what each switch is designed to do. The default
-value is given in each case.
-
-You can certainly feel free to edit \lstinline!defaultSettings.yaml!, but
-this is not ideal as it may be overwritten when you update your \TeX{} distribution --
-all of your hard work tweaking the script would be undone! Don't worry,
-there's a solution, feel free to peek ahead to \cref{sec:indentconfig} if you like.
-\begin{itemize}
-	\item[\verbitem{defaultIndent}] \lstinline!"\t"!
-
-	      This is the default indentation (\lstinline!\t! means a tab) used in the absence of other details
-	      for the command or environment we are working with--see \lstinline!indentRules!
-	      for more details (\cpageref{page:indentRules}).
-
-	      If you're interested in experimenting with \lstinline!latexindent.pl! then you
-	      can \emph{remove} all indentation by setting \lstinline!defaultIndent: ""!
-	\item[\verbitem{backupExtension}] \lstinline!.bak!
-
-	      If you call \lstinline!latexindent.pl! with the \lstinline!-w! switch (to overwrite
-	      \lstinline!myfile.tex!) then it will create a backup file before doing
-	      any indentation: \lstinline!myfile.bak0!
-
-	      By default, every time you call \lstinline!latexindent.pl! after this with
-	      the \lstinline!-w! switch it will create \lstinline!myfile.bak1!, \lstinline!myfile.bak2!,
-	      etc.
-	\item[\verbitem{onlyOneBackUp}] \lstinline!0!
-
-	      \label{page:onlyonebackup}
-	      If you don't want a backup for every time that you call \lstinline!latexindent.pl! (so
-	      you don't want \lstinline!myfile.bak1!, \lstinline!myfile.bak2!, etc) and you simply
-	      want \lstinline!myfile.bak! (or whatever you chose \lstinline!backupExtension! to be)
-	      then change \lstinline!onlyOneBackUp! to \lstinline!1!.
-
-	\item[\verbitem{maxNumberOfBackUps}]\lstinline!0!
-
-	      Some users may only want a finite number of backup files,
-	      say at most $3$, in which case, they can change this switch.
-	      The smallest value of \lstinline!maxNumberOfBackUps! is $0$ which will \emph{not}
-	      prevent backup files being made--in this case, the behaviour will be dictated
-	      entirely by \lstinline!onlyOneBackUp!.
-	      %\footnote{This was a feature request made on \href{https://github.com/cmhughes/latexindent.plx}{github}}
-	\item[\verbitem{cycleThroughBackUps}]\lstinline!0!
-
-	      Some users may wish to cycle through backup files, by deleting the
-	      oldest backup file and keeping only the most recent; for example,
-	      with \lstinline!maxNumberOfBackUps: 4!, and \lstinline!cycleThroughBackUps!
-	      set to \lstinline!1!  then the \lstinline!copy! procedure given in \cref{lst:cycleThroughBackUps}
-	      would be obeyed.
-
-	      \begin{cmhlistings}[language=Perl]{\lstinline!cycleThroughBackUps!}{lst:cycleThroughBackUps}
-copy myfile.bak1 to myfile.bak0
-copy myfile.bak2 to myfile.bak1
-copy myfile.bak3 to myfile.bak2
-copy myfile.bak4 to myfile.bak3
-	\end{cmhlistings}
-
-	\item[\verbitem{indentPreamble}] \lstinline!0!
-
-	      The preamble of a document can sometimes contain some trickier code
-	      for \lstinline!latexindent.pl! to work with. By default, \lstinline!latexindent.pl!
-	      won't try to operate on the preamble, but if you'd like it to try then
-	      change \lstinline!indentPreamble! to \lstinline!1!.
-	\item[\verbitem{alwaysLookforSplitBraces}] \lstinline!1!
-
-	      This switch tells \lstinline!latexindent.pl! to look for commands that
-	      can split \emph{braces} across lines, such as \lstinline!parbox!, \lstinline!tikzset!, etc. In older
-	      versions of \lstinline!latexindent.pl! you had to specify each one in \lstinline!checkunmatched!--this
-	      clearly became tedious, hence the introduction of \lstinline!alwaysLookforSplitBraces!.
-
-	      \emph{As long as you leave this switch on (set to 1) you don't need to specify which
-	      	commands can split braces across lines--you can ignore the
-	      	fields \lstinline!checkunmatched! and \lstinline!checkunmatchedELSE! described
-	      	later on \cpageref{lst:checkunmatched}}.
-	\item[\verbitem{alwaysLookforSplitBrackets}] \lstinline!1!
-
-	      This switch tells \lstinline!latexindent.pl! to look for commands that
-	      can split \emph{brackets} across lines, such as \lstinline!psSolid!, \lstinline!pgfplotstabletypeset!,
-	      etc. In older versions of \lstinline!latexindent.pl! you had to specify each one in \lstinline!checkunmatchedbracket!--
-	      this clearly became tedious, hence the introduction of \lstinline!alwaysLookforSplitBraces!.
-
-	      \emph{As long as you leave this switch on (set to 1) you don't need to specify which
-	      	commands can split brackets across lines--you can ignore \lstinline!checkunmatchedbracket! described later on
-	      	\cpageref{lst:checkunmatched}}.
-
-	\item[\verbitem{removeTrailingWhitespace}] \lstinline!0!
-
-	      By default \lstinline!latexindent.pl! indents every line (including empty lines)
-	      which creates `trailing white space' feared by most version control systems. If
-	      this option is set to \lstinline!1!, trailing white space is removed from all
-	      lines, also non-empty ones. In general this should not create any problems, but
-	      by precaution this option is turned off by default. Thanks to \cite{vosskuhle} for
-	      providing this feature.
-
-	\item[\verbitem{lookForAlignDelims}] This is the first example of a field
-	      in \lstinline!defaultSettings.yaml! that has more than one line; \cref{lst:aligndelims:basic}
-	      shows more details. In fact, the fields in \verbitem{lookForAlignDelims} can actually
-          take two different forms: the \emph{basic} version is shown in \cref{lst:aligndelims:basic} 
-          and the \emph{advanced} version in \cref{lst:aligndelims:advanced}; we will discuss each in turn.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!lookForAlignDelims! (basic)}{lst:aligndelims:basic}
-lookForAlignDelims:
-   tabular: 1
-   tabularx: 1
-   longtable: 1
-   array: 1
-   matrix: 1
-   bmatrix: 1
-   pmatrix: 1
-   align: 1
-   align*: 1
-   alignat: 1
-   alignat*: 1
-   aligned: 1
-   cases: 1
-   dcases: 1
-   pmatrix: 1
-   listabla: 1
-	\end{cmhlistings}
-
-	The environments specified in this field will be operated on in a special way  by \lstinline!latexindent.pl!. In particular, it will try and align each column by its alignment
-	tabs. It does have some limitations (discussed further in \cref{sec:knownlimitations}),
-	but in many cases it will produce results such as those in \cref{lst:tabularbefore:basic,lst:tabularafter:basic}.
-
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! before}{lst:tabularbefore:basic}
-\begin{tabular}{cccc}
-1&	2 &3       &4\\
-5& &6       &\\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! after (basic)}{lst:tabularafter:basic}
-\begin{tabular}{cccc}
- 1 & 2 & 3 & 4 \\
- 5 &   & 6 &   \\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}
-
-	If you find that \lstinline!latexindent.pl! does not perform satisfactorily on such
-	environments then you can either remove them from \lstinline!lookForAlignDelims! altogether, or set the relevant key to \lstinline!0!, for example \lstinline!tabular: 0!, or if you just want to ignore \emph{specific}
-	instances of the environment, you could wrap them in something from \lstinline!noIndentBlock! (see \cref{lst:noIndentBlock}).
-
-    If you wish to remove the alignment of the \lstinline!\\! within a delimiter-aligned block, then the
-    advanced form of \lstinline!lookForAlignDelims! shown in \cref{lst:aligndelims:advanced} is for you. 
-	      \begin{cmhlistings}[style=yaml]{\lstinline!lookForAlignDelims! (advanced)}{lst:aligndelims:advanced}
-lookForAlignDelims:
-   tabular: 
-      delims: 1
-      alignDoubleBackSlash: 0
-      spacesBeforeDoubleBackSlash: 0
-   tabularx:
-      delims: 1
-   longtable: 1
-	\end{cmhlistings}
-
-    Note that you can use a mixture of the basic and advanced form: in \cref{lst:aligndelims:advanced} \lstinline!tabular! and \lstinline!tabularx! 
-    are advanced and \lstinline!longtable! is basic. When using the advanced form, each field should receive at least 1 sub-field, and \emph{can} (but does not have to) receive up to 3 fields: 
-    \begin{itemize}
-      \item \lstinline!delims!: switch equivalent to simply specifying, for example, \lstinline!tabular: 1! in
-        the basic version shown in \cref{lst:aligndelims:basic} (default: 1);
-      \item \lstinline!alignDoubleBackSlash!: switch to determine if \lstinline!\\! should be aligned (default: 1);
-      \item \lstinline!spacesBeforeDoubleBackSlash!: optionally, specifies the number of spaces to be inserted
-        before (non-aligned) \lstinline!\\!. In order to use this field, \lstinline!alignDoubleBackSlash! needs
-        to be set to 0 (default: 0).
-    \end{itemize}
-    
-    With the settings shown in \cref{lst:aligndelims:advanced} we receive the before-and-after results shown in
-    \cref{lst:tabularbefore:advanced,lst:tabularafter:advanced}; note that the ampersands have been aligned, but 
-    the \lstinline!\\! have not (compare the alignment of \lstinline!\\! in \cref{lst:tabularafter:basic,lst:tabularafter:advanced}).
-
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! before }{lst:tabularbefore:advanced}
-\begin{tabular}{cccc}
-1&	2 &3       &4\\
-5& &6       &\\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! after (advanced)}{lst:tabularafter:advanced}
-\begin{tabular}{cccc}
- 1 & 2 & 3 & 4\\
- 5 &   & 6 &\\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}
-
-    Using  \lstinline!spacesBeforeDoubleBackSlash: 3! gives \cref{lst:tabularbefore:spacing,lst:tabularafter:spacing}, 
-    note the spacing before the \lstinline!\\! in \cref{lst:tabularafter:spacing}.
-
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! before}{lst:tabularbefore:spacing}
-\begin{tabular}{cccc}
-1&	2 &3       &4\\
-5& &6       &\\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,columns=fixed]{\lstinline!tabular! after (spacing)}{lst:tabularafter:spacing}
-\begin{tabular}{cccc}
- 1 & 2 & 3 & 4   \\
- 5 &   & 6 &   \\
-\end{tabular}
-		\end{cmhlistings}
-	\end{minipage}
-
-	
-
-
-	If you have blocks of code that you wish to align at the \&  character that
-	      are \emph{not} wrapped in, for example, \lstinline!\begin{tabular}...\end{tabular}!, then you use the mark up
-	      illustrated in \cref{lst:alignmentmarkup}. Note that the \lstinline!%*! must be next to
-	      each other, but that there can be any number of spaces (possibly none) between the
-	      \lstinline!*! and \lstinline!\begin{tabular}!; note also that you may use any
-	      environment name that you have specified in \lstinline!lookForAlignDelims!.
-	      \begin{cmhlistings}[style=demo,columns=fixed]{Mark up for aligning delimiters outside of environments}{lst:alignmentmarkup}
-\matrix{%
-%* \begin{tabular}
- 1 & 2 & 3 & 4 \\
- 5 &   & 6 &   \\
-%* \end{tabular}
-}
-	\end{cmhlistings}
-
-	\item[\verbitem{verbatimEnvironments}] A field that contains a list of environments
-	      that you would like left completely alone--no indentation will be done
-	      to environments that you have specified in this field--see \cref{lst:verbatimEnvironments}.
-
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!verbatimEnvironments!}{lst:verbatimEnvironments}
-verbatimEnvironments:
-    verbatim: 1
-    lstlisting: 1
-	\end{cmhlistings}
-	Note that if  you put an environment in \lstinline!verbatimEnvironments!
-	and in other fields such as \lstinline!lookForAlignDelims! or \lstinline!noAdditionalIndent!
-	then \lstinline!latexindent.pl! will \emph{always} prioritize \lstinline!verbatimEnvironments!.
-
-	\item[\verbitem{noIndentBlock}] If you have a block of code that you don't
-	      want \lstinline!latexindent.pl! to touch (even if it is \emph{not} a verbatim-like
-	      environment) then you can wrap it in an environment from \lstinline!noIndentBlock!;
-	      you can use any name you like for this, provided you populate it as demonstrate in
-	      \cref{lst:noIndentBlock}.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!noIndentBlock!}{lst:noIndentBlock}
-noIndentBlock:
-    noindent: 1
-    cmhtest: 1
-	\end{cmhlistings}
-
-	Of course, you don't want to have to specify these as null environments
-	in your code, so you use them with a comment symbol, \lstinline!%!, followed
-	by as many spaces (possibly none) as you like; see \cref{lst:noIndentBlockdemo} for
-	example.
-	\begin{cmhlistings}[style=demo,escapeinside={(*@}{@*)}]{\lstinline!noIndentBlock! demonstration}{lst:noIndentBlockdemo}
-%(*@@*) \begin{noindent}
-        this code
-                won't
-     be touched
-                    by
-             latexindent.pl!
-%(*@@*)\end{noindent}
-	\end{cmhlistings}
-
-	\item[\verbitem{noAdditionalIndent}] If you would prefer some of your
-	      environments or commands not to receive any additional indent, then
-	      populate \lstinline!noAdditionalIndent!; see \cref{lst:noAdditionalIndent}.
-	      Note that these environments will still receive the \emph{current} level
-	      of indentation unless they belong to \lstinline!verbatimEnvironments!, or \lstinline!noIndentBlock!.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!noAdditionalIndent!}{lst:noAdditionalIndent}
-noAdditionalIndent:
-    document: 1
-    myexample: 1
-    mydefinition: 1
-    problem: 1
-    exercises: 1
-    mysolution: 1
-    foreach: 0
-    widepage: 1
-    comment: 1
-    \[: 1
-    \]: 1
-    frame: 0
-	\end{cmhlistings}
-	Note in particular from \cref{lst:noAdditionalIndent} that if you wish content within
-	\lstinline!\[!  and \lstinline!\]! to receive no additional indentation then
-	you have to specify \emph{both} as \lstinline!1! (the default is \lstinline!0!).
-	If you do not specify both as the same value you may get some interesting results!
-	\item[\verbitem{indentRules}] If\label{page:indentRules} you would prefer to specify
-	      individual rules for certain environments or commands, just
-	      populate \lstinline!indentRules!; see \cref{lst:indentRules}
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!indentRules!}{lst:indentRules}
-indentRules:
-   myenvironment: "\t\t"
-   anotherenvironment: "\t\t\t\t"
-   \[: "\t"
-	\end{cmhlistings}      %%%%%\] just here to stop vim from colouring the rest of my code
-	Note that in contrast to \lstinline!noAdditionalIndent! you do \emph{not}
-	need to specify both \lstinline!\[! and \lstinline!\]! in this field.
-
-	If you put an environment in both \lstinline!noAdditionalIndent! and in
-	\lstinline!indentRules! then \lstinline!latexindent.pl! will resolve the conflict
-	by ignoring \lstinline!indentRules! and prioritizing \lstinline!noAdditionalIndent!.
-	You will get a warning message in \lstinline!indent.log!; note that you will only
-	get one warning message per command or environment. Further discussion
-	is given in \cref{sec:fieldhierachy}.
-
-	\item[\verbitem{indentAfterHeadings}] This field enables the user to specify
-	      indentation rules that take effect after heading commands such as \lstinline!\part!, \lstinline!\chapter!,
-	      \lstinline!\section!, \lstinline!\subsection*! etc. This field is slightly different from most
-	      of the fields that we have considered previously, because each element is
-	      itself a field which has two elements: \lstinline!indent! and \lstinline!level!. (Similar 
-          in structure to the advanced form of  \lstinline!lookForAlignDelims! in \cref{lst:aligndelims:advanced}.)
-	      \begin{cmhlistings}[style=yaml]{\lstinline!indentAfterHeadings!}{lst:indentAfterHeadings}
-indentAfterHeadings:
-    part:
-       indent: 0
-       level: 1
-    chapter:
-       indent: 0
-       level: 2
-    section:
-       indent: 0
-       level: 3
-    ...
-	\end{cmhlistings}
-	The default settings do \emph{not} place indentation after a heading--you
-	can easily switch them on by changing \lstinline!indent: 0! to \lstinline!indent: 1!.
-	The \lstinline!level! field tells \lstinline!latexindent.pl! the hierarchy of the heading
-	structure in your document. You might, for example, like to have both \lstinline!section!
-	and \lstinline!subsection! set with \lstinline!level: 3! because you do not want the indentation to go too deep.
-
-	You can add any of your own custom heading commands to this field, specifying the \lstinline!level!
-	as appropriate.  You can also specify your own indentation in \lstinline!indentRules!--
-	you will find the default \lstinline!indentRules! contains \lstinline!chapter: " "! which
-	tells \lstinline!latexindent.pl! simply to use a space character after \lstinline!\chapter! headings
-	(once \lstinline!indent! is set to \lstinline!1! for \lstinline!chapter!).
-
-	\item[\verbitem{indentAfterItems}] The environments specified in \lstinline!indentAfterItems!  tell
-	      \lstinline!latexindent.pl! to look for \lstinline!\item! commands; if these switches are set to \lstinline!1!
-	      then indentation will be performed so as indent the code after each \lstinline!item!.
-	      \begin{cmhlistings}{\lstinline!indentAfterItems!}{lst:indentafteritems}
-indentAfterItems:
-    itemize: 1
-    enumerate: 1
-	\end{cmhlistings}
-	A demonstration is given in \cref{lst:itemsbefore,lst:itemsafter}
-
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,xleftmargin=-3mm,columns=fixed]{\lstinline!items! before}{lst:itemsbefore}
-\begin{itemize}
-\item some text here
-some more text here
-some more text here
-\item another item
-\end{itemize}
-		\end{cmhlistings}
-	\end{minipage}%
-	\begin{minipage}{.5\textwidth}
-		\begin{cmhlistings}[style=demo,xleftmargin=-3mm,columns=fixed]{\lstinline!items! after}{lst:itemsafter}
-\begin{itemize}
-	\item some text here
-		   some more text here
-		   some more text here
-	\item another item
-\end{itemize}
-		\end{cmhlistings}
-	\end{minipage}
-
-	\item[\verbitem{itemNames}] If you have your own \lstinline!item! commands (perhaps you
-	      prefer to use \lstinline!myitem!, for example)
-	      then you can put populate them in \lstinline!itemNames!.
-	      For example, users of the \lstinline!exam! document class might like to add
-	      \lstinline!parts! to \lstinline!indentAfterItems! and \lstinline!part! to \lstinline!itemNames!
-	      to their user settings--see \vref{sec:indentconfig} for details of how to configure user settings,
-	      and \vref{lst:mysettings} in particular.\label{page:examsettings}
-
-	\item[\verbitem{constructIfElseFi}] The commands specified in this field
-	      will tell \lstinline!latexindent.pl! to look for constructs that
-	      have the form \lstinline!\if...! \lstinline!\else...! \lstinline!\fi!, such as,
-	      for example, \lstinline!\ifnum!; see \cref{lst:iffibefore,lst:iffiafter} for
-	      a before-and-after demonstration.
-
-	      \begin{minipage}{.5\textwidth}
-	      	\begin{cmhlistings}[style=demo,xleftmargin=-3mm,columns=fixed]{\lstinline!if-else-fi! construct before}{lst:iffibefore}
-\ifnum\radius>5
-\ifnum\radius<16
-\draw[decorate,...
-\fi
-\fi
-	      	\end{cmhlistings}
-	      \end{minipage}%
-	      \begin{minipage}{.5\textwidth}
-	      	\begin{cmhlistings}[style=demo,xleftmargin=-3mm,columns=fixed]{\lstinline!if-else-fi! construct after}{lst:iffiafter}
-\ifnum\radius>5
-    \ifnum\radius<16
-    	\draw[decorate,...
-	\fi
-\fi
-	      	\end{cmhlistings}
-	      \end{minipage}
-
-	\item[\verbitem{fileExtensionPreference}] \lstinline!latexindent.pl! can be called to
-	      act on a file without
-	      specifying the file extension.  For example we can call \lstinline!latexindent.pl myfile!
-	      in which case the script will look for \lstinline!myfile! with the extensions
-	      specified in \lstinline!fileExtensionPreference! in their numeric order. If
-	      no match is found, the script will exit. As with all of the fields, you should
-	      change and/or add to this as necessary.
-	      \begin{cmhlistings}[style=yaml]{\lstinline!fileExtensionPreference!}{lst:fileExtensionPreference}
-fileExtensionPreference:
-    .tex: 1
-    .sty: 2
-    .cls: 3
-    .bib: 4
-	\end{cmhlistings}
-	Calling \lstinline!latexindent.pl myfile! with the details specified in \cref{lst:fileExtensionPreference}
-	means that the script will first look for \lstinline!myfile.tex!, then \lstinline!myfile.sty!, \lstinline!myfile.cls!, 
-    and finally \lstinline!myfile.bib! in order.
-\item[\verbitem{logFilePreferences}] 
-  \lstinline!latexindent.pl! writes information to \lstinline!indent.log!, some 
-  of which can be customised by changing \lstinline!logFilePreferences!; see \cref{lst:logFilePreferences}.
-\begin{cmhlistings}[style=yaml]{\lstinline!logFilePreferences!}{lst:logFilePreferences}
-logFilePreferences:
-    showEveryYamlRead: 1
-    showAlmagamatedSettings: 0
-    endLogFileWith: '--------------' 
-    traceModeIncreaseIndent: '>>'
-    traceModeAddCurrentIndent: '||'
-    traceModeDecreaseIndent: '<<'
-    traceModeBetweenLines: "\n"
-  \end{cmhlistings}
-If you load your own user settings (see \vref{sec:indentconfig}) then \lstinline!latexindent.pl! will
-detail them in \lstinline!indent.log!; you can choose not to have the details logged by switching
-\lstinline!showEveryYamlRead! to \lstinline!0!. Once all of your settings have 
-been loaded, you can see the amalgamated settings by switching \lstinline!showAlmagamatedSettings! 
-to \lstinline!1!, if you wish. The log file will end with the characters 
-given in \lstinline!endLogFileWith!.
-
-When \lstinline!trace! mode is active (see \cpageref{page:traceswitch}) verbose information is written 
-to \lstinline!indent.log!. The decoration of this information can be customised through the remaining
-fields given in \cref{lst:logFilePreferences}; note, in particular, the use of \lstinline!"\n"! for 
-escaped characters (using single quotes will not produce the same results).
-
-\item[\verbitem{fileContentsEnvironments}] 
-  \lstinline!latexindent.pl! determines when the main document begins by looking for \lstinline!\begin{document}!; 
-  it will not do so when inside any of the environments specified in \lstinline!fileContentsEnvironments!, see
-  \cref{lst:fileContentsEnvironments}.
-\begin{cmhlistings}[style=yaml]{\lstinline!fileContentsEnvironments!}{lst:fileContentsEnvironments}
-fileContentsEnvironments:
-    filecontents: 1
-    filecontents*: 1
-  \end{cmhlistings}
-
-	\begin{warning}
-		\emph{The following fields are marked in red, as they are not necessary
-			unless you wish to micro-manage your indentation scheme.
-			Note that in each case, you should \emph{not} use the backslash.}
-	\end{warning}
-
-	% to anyone reading the source code- I know the next line isn't the
-	% correct way to do it :)
-	\item[\color{red}\verbitem{checkunmatched}] Assuming you keep \lstinline!alwaysLookforSplitBraces! set to \lstinline!1! (which
-	      is the default) then you don't need to worry about \lstinline!checkunmatched!.
-
-	      Should you wish to deactivate \lstinline!alwaysLookforSplitBraces! by setting it to \lstinline!0!, then
-	      you can populate \lstinline!checkunmatched! with commands that can split braces across
-	      lines--see \cref{lst:checkunmatched}.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!checkunmatched!}{lst:checkunmatched}
-checkunmatched:
-    parbox: 1
-    vbox: 1
-	\end{cmhlistings}
-	\item[\color{red}\verbitem{checkunmatchedELSE}] Similarly, assuming you keep \lstinline!alwaysLookforSplitBraces! set to \lstinline!1! (which
-	      is the default) then you don't need to worry about \lstinline!checkunmatchedELSE!.
-
-	      As in \lstinline!checkunmatched!, should you wish to deactivate \lstinline!alwaysLookforSplitBraces! by setting it to \lstinline!0!, then
-	      you can populate \lstinline!checkunmatchedELSE! with commands that can split braces across
-	      lines \emph{and} have an `else' statement--see \cref{lst:checkunmatchedELSE}.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!checkunmatchedELSE!}{lst:checkunmatchedELSE}
-checkunmatchedELSE:
-    pgfkeysifdefined: 1
-    DTLforeach: 1
-    ifthenelse: 1
-	\end{cmhlistings}
-	\item[\color{red}\verbitem{checkunmatchedbracket}] Assuming you keep \lstinline!alwaysLookforSplitBrackets!
-	      set to \lstinline!1! (which is the default) then you don't need to worry about \lstinline!checkunmatchedbracket!.
-
-	      Should you wish to deactivate \lstinline!alwaysLookforSplitBrackets! by setting it
-	      to \lstinline!0!, then you can populate \lstinline!checkunmatchedbracket! with commands that can
-	      split \emph{brackets} across lines--see \cref{lst:checkunmatchedbracket}.
-
-	      \begin{cmhlistings}[style=yaml]{\lstinline!checkunmatchedbracket!}{lst:checkunmatchedbracket}
-checkunmatchedbracket:
-    psSolid: 1
-    pgfplotstablecreatecol: 1
-    pgfplotstablesave: 1
-    pgfplotstabletypeset: 1
-    mycommand: 1
-	\end{cmhlistings}
-\end{itemize}
-
-\subsubsection{Hierarchy of fields}\label{sec:fieldhierachy}
-After reading the previous section, it should sound reasonable that
-\lstinline!noAdditionalIndent!, \lstinline!indentRules!, and
-\lstinline!verbatim! all serve mutually exclusive tasks. Naturally, you may
-well wonder what happens if you choose to ask \lstinline!latexindent.pl! to
-prioritize one above the other.
-
-For example, let's say that (after reading \cref{sec:indentconfig}) you put the fields in \cref{lst:conflict} into
-one of your settings files.
-\begin{cmhlistings}[style=yaml]{Conflicting ideas}{lst:conflict}
-indentRules:
-   myenvironment: "\t\t"
-noAdditionalIndent:
-   myenvironment: 1
-\end{cmhlistings}
-
-Clearly these fields conflict: first of all
-you are telling \lstinline!latexindent.pl! that \lstinline!myenvironment! should
-receive two tabs of indentation, and then you are telling it
-not to put any indentation in the environment. \lstinline!latexindent.pl!
-will always make the decision to prioritize \lstinline!noAdditionalIndent! above
-\lstinline!indentRules! regardless of the order that you load them in
-your settings file. The first
-time it encounters \lstinline!myenvironment! it will put a warning in \lstinline!indent.log!
-and delete the offending key from \lstinline!indentRules! so that any future
-conflicts will not have to be addressed.
-
-Let's consider another conflicting example in \cref{lst:bigconflict}
-\begin{cmhlistings}[style=yaml]{More conflicting ideas}{lst:bigconflict}
-lookForAlignDelims:
-   myenvironment: 1
-verbatimEnvironments:
-   myenvironment: 1
-\end{cmhlistings}
-This is quite a significant conflict--we are first of all telling \lstinline!latexindent.pl!
-to look for alignment delimiters in \lstinline!myenvironment! and then
-telling it that actually we would like \lstinline!myenvironment! to be considered
-as a \lstinline!verbatim!-like environment. Regardless of the order that we
-state \cref{lst:bigconflict} the \lstinline!verbatim! instruction will always win.
-As in \cref{lst:conflict} you will only receive a warning in \lstinline!indent.log! the
-first time \lstinline!latexindent.pl! encounters \lstinline!myenvironment! as the
-offending key is deleted from \lstinline!lookForAlignDelims!.
-
-To summarize, \lstinline!latexindent.pl! will prioritize the various fields in the
-following order:
-\begin{enumerate}
-	\item \lstinline!verbatimEnvironments!
-	\item \lstinline!noAdditionalIndent!
-	\item \lstinline!indentRules!
-\end{enumerate}
-\subsection{\lstinline!indentconfig.yaml! and \lstinline!.indentconfig.yaml! (for user settings)}\label{sec:indentconfig}
-Editing \lstinline!defaultSettings.yaml! is not ideal as it may be overwritten when
-updating your distribution--a better way to customize the settings to your liking
-is to set up your own settings file,
-\lstinline!mysettings.yaml! (or any name you like, provided it ends with \lstinline!.yaml!).
-The only thing you have to do is tell \lstinline!latexindent.pl! where to find it.
-
-\lstinline!latexindent.pl! will always check your home directory for \lstinline!indentconfig.yaml!
-and  \lstinline!.indentconfig.yaml! (unless
-it is called with the \lstinline!-d! switch),
-which is a plain text file you can create that contains the \emph{absolute}
-paths for any settings files that you wish \lstinline!latexindent.pl! to load. There is no difference
-between \lstinline!indentconfig.yaml! and \lstinline!.indentconfig.yaml!, other than the
-fact that \lstinline!.indentconfig.yaml! is a `hidden' file; thank you to \cite{jacobo-diaz-hidden-config}
-for providing this feature. In what follows, we will use \lstinline!indentconfig.yaml!, but it
-is understood that this equally represents \lstinline!.indentconfig.yaml! as well. If you
-have both files in existence,  \lstinline!indentconfig.yaml! takes priority.
-
-For Mac and Linux users, their home directory is \lstinline!~/username! while
-Windows (Vista onwards) is \lstinline!C:\Users\username! \footnote{If you're not sure
-	where to put \lstinline!indentconfig.yaml!, don't
-	worry \lstinline!latexindent.pl! will tell you in the log file exactly where to
-put it assuming it doesn't exist already.}
-\Cref{lst:indentconfig} shows a sample \lstinline!indentconfig.yaml! file.
-
-\begin{cmhlistings}[style=yaml]{\lstinline!indentconfig.yaml! (sample)}{lst:indentconfig}
-# Paths to user settings for latexindent.pl
-#
-# Note that the settings will be read in the order you
-# specify here- each successive settings file will overwrite
-# the variables that you specify
-
-paths:
-- /home/cmhughes/Documents/yamlfiles/mysettings.yaml
-- /home/cmhughes/folder/othersettings.yaml
-- /some/other/folder/anynameyouwant.yaml
-- C:\Users\chughes\Documents\mysettings.yaml
-- C:\Users\chughes\Desktop\test spaces\more spaces.yaml
-\end{cmhlistings}
-
-Note that the \lstinline!.yaml! files you specify in \lstinline!indentconfig.yaml!
-will be loaded in the order that you write them in. Each file doesn't have
-to have every switch from \lstinline!defaultSettings.yaml!; in fact, I recommend
-that you only keep the switches that you want to \emph{change} in these
-settings files.
-
-To get started with your own settings file, you might like to save a copy of
-\lstinline!defaultSettings.yaml! in another directory and call it, for
-example, \lstinline!mysettings.yaml!. Once you have added the path to \lstinline!indentconfig.yaml!
-feel free to start changing the switches and adding more environments to it
-as you see fit--have a look at \cref{lst:mysettings} for an example
-that uses four tabs for the default indent, adds the \lstinline!tabbing!
-environment to the list of environments that contains alignment delimiters,
-and adds the changes we described on \cpageref{page:examsettings}.
-
-\begin{cmhlistings}[style=yaml]{\lstinline!mysettings.yaml! (example)}{lst:mysettings}
-# Default value of indentation
-defaultIndent: "\t\t\t\t"
-
-# environments that have tab delimiters, add more
-# as needed
-lookForAlignDelims:
-   tabbing: 1
-
-# If you use the exam documentclass, you might 
-# like the following settings
-# environments that have \item commands
-indentAfterItems:
-    parts: 1
-
-# commands to be treated like \item
-itemNames:
-    part: 1
-\end{cmhlistings}
-
-You can make sure that your settings are loaded by checking \lstinline!indent.log!
-for details--if you have specified a path that \lstinline!latexindent.pl! doesn't
-recognize then you'll get a warning, otherwise you'll get confirmation that
-\lstinline!latexindent.pl! has read your settings file \footnote{Windows users
-	may find that they have to end \lstinline!.yaml! files with a blank line}.
-
-\begin{warning}
-	When editing \lstinline!.yaml! files it is \emph{extremely} important
-	to remember how sensitive they are to spaces. I highly recommend copying
-	and pasting from \lstinline!defaultSettings.yaml! when you create your
-	first \lstinline!whatevernameyoulike.yaml! file.
-
-	If \lstinline!latexindent.pl! can not read your \lstinline!.yaml! file it
-	will tell you so in \lstinline!indent.log!.
-\end{warning}
-
-\subsection{\lstinline!localSettings.yaml!}\label{sec:localsettings}
-You may remember on \cpageref{page:localswitch} we discussed the \lstinline!-l! switch
-that tells \lstinline!latexindent.pl! to look for \lstinline!localSettings.yaml! in the
-\emph{same directory} as \lstinline!myfile.tex!. This settings file will
-be read \emph{after} \lstinline!defaultSettings.yaml! and, assuming they exist,
-user settings.
-
-The \emph{local} settings file may be called \lstinline!localSettings.yaml!, and
-it can contain any switches that you'd
-like to change--a sample is shown in \cref{lst:localSettings}.
-
-\begin{cmhlistings}[style=yaml]{\lstinline!localSettings.yaml! (example)}{lst:localSettings}
-# Default value of indentation
-defaultIndent: " "
-
-# environments that have tab delimiters, add more
-# as needed
-lookForAlignDelims:
-   tabbing: 0
-
-#  verbatim environments- environments specified
-#  in this hash table will not be changed at all!
-verbatimEnvironments:
-    cmhenvironment: 0
-\end{cmhlistings}
-
-You can make sure that your local settings are loaded by checking \lstinline!indent.log!
-for details--if \lstinline!localSettings.yaml! can not be read then you will
-get a warning, otherwise you'll get confirmation that
-\lstinline!latexindent.pl! has read \lstinline!localSettings.yaml!.
-
-If you'd prefer to name your \lstinline!localSettings.yaml! file something different, (say, \lstinline!myyaml.yaml!) then
-you can call \lstinline!latexindent.pl! using, for example, \lstinline[breaklines=true]!latexindent.pl -l=myyaml.yaml myfile.tex!.
-
-\subsection{Settings load order}\label{sec:loadorder}
-\lstinline!latexindent.pl! loads the settings files in the following order:
-\begin{enumerate}
-	\item \lstinline!defaultSettings.yaml! is always loaded, and can not be renamed;
-	\item \lstinline!anyUserSettings.yaml! and any other arbitrarily-named files specified in \lstinline!indentconfig.yaml!;
-	\item \lstinline!localSettings.yaml! but only if found in the same directory as \lstinline!myfile.tex! and called
-	      with \lstinline!-l! switch; this file can be renamed, provided that the call to \lstinline!latexindent.pl! is adjusted 
-          accordingly (see \cref{sec:localsettings}).
-\end{enumerate}
-A visual representation of this is given in \cref{fig:loadorder}.
-
-\begin{figure}
-	\centering
-	\begin{tikzpicture}[
-			needed/.style={very thick, draw=blue,fill=blue!20,
-				text centered, minimum height=2.5em,rounded corners=1ex},
-			optional/.style={draw=black, very thick,scale=0.8,
-				text centered, minimum height=2.5em,rounded corners=1ex},
-			optionalfill/.style={fill=black!10},
-			connections/.style={draw=black!30,dotted,line width=3pt,text=red},
-		]
-		% Draw diagram elements
-		\node (latexindent) [needed,circle]  {\lstinline!latexindent.pl!};
-		\node (default) [needed,above right=.5cm of latexindent]  {\lstinline!defaultSettings.yaml!};
-		\node (indentconfig) [optional,right=of latexindent]  {\lstinline!indentconfig.yaml!};
-		\node (any) [optional,optionalfill,above right=of indentconfig]  {\lstinline!any.yaml!};
-		\node (name) [optional,optionalfill,right=of indentconfig]  {\lstinline!name.yaml!};
-		\node (you) [optional,optionalfill,below right=of indentconfig]  {\lstinline!you.yaml!};
-		\node (want) [optional,optionalfill,below=of indentconfig]  {\lstinline!want.yaml!};
-		\node (local) [optional,below=of latexindent]  {\lstinline!localSettings.yaml!};
-		% Draw arrows between elements
-		\draw[connections,solid] (latexindent) to[in=-90]node[pos=0.5,anchor=north]{1} (default.south) ;
-		\draw[connections,optional] (latexindent) -- node[pos=0.5,anchor=north]{2} (indentconfig) ;
-		\draw[connections,optional] (indentconfig) to[in=-90] (any.south) ;
-		\draw[connections,optional] (indentconfig) -- (name) ;
-		\draw[connections,optional] (indentconfig) to[out=-45,in=90] (you) ;
-		\draw[connections,optional] (indentconfig) -- (want) ;
-		\draw[connections,optional] (latexindent) -- node[pos=0.5,anchor=west]{3} (local) ;
-	\end{tikzpicture}
-	\caption{Schematic of the load order described in \cref{sec:loadorder}; solid lines represent
-		mandatory files, dotted lines represent optional files. \lstinline!indentconfig.yaml! can
-		contain as many files as you like--the files will be loaded in order; if you specify
-	settings for the same field in more than one file, the most recent takes priority. }
-	\label{fig:loadorder}
-\end{figure}
-
-\subsection{An important example}
-I was working on a document that had the text shown in \cref{lst:casestudy}.
-\begin{cmhlistings}[style=demo,escapeinside={(*@}{@*)}]{When to set \lstinline!alwaysLookforSplitBrackets=0!}{lst:casestudy}
-Hence determine how many zeros the function $h(x)=f(x)-g(x)$
-has on the interval $[0,9)$.(*@\label{line:interval1}@*)
-\begin{shortsolution}
-	The function $h$ has $10$ zeros on the interval $[0,9)$.(*@\label{line:interval2}@*)
-\end{shortsolution}
-\end{cmhlistings}
-I had allowed \lstinline!alwaysLookforSplitBrackets=1!, which is the default setting.
-Unfortunately, this caused undesired results, as \lstinline!latexindent.pl! thought that the opening
-\lstinline![!  in the interval notation (\cref{line:interval1,line:interval2})
-was an opening brace that needed to be closed (with a corresponding \lstinline!]!). Clearly
-this was inappropriate, but also expected since \lstinline!latexindent.pl! was simply
-following its matching rules.
-
-In this particular instance, I set up \lstinline!localSettings.yaml!
-to contain \lstinline!alwaysLookforSplitBrackets: 0! and then specified the commands
-that could split brackets across lines (such as \lstinline!begin{axis}!) individually
-in \lstinline!checkunmatchedbracket!.  Another option would have been to wrap the
-the line in an environment from \lstinline!noIndentBlock! which treats its contents
-as a verbatim environment.
-
-
-\section{Known limitations}\label{sec:knownlimitations}
-There are a number of known limitations of the script, and almost certainly quite a
-few that are \emph{unknown}!
-
-The main limitation is to do with the alignment routine of environments that contain
-delimiters--in other words, environments that are entered in \lstinline!lookForAlignDelims!.
-Indeed, this is the only part of the script that can \emph{potentially} remove
-lines from \lstinline!myfile.tex!. Note that \lstinline!indent.log! will always
-finish with a comparison of line counts before and after.
-
-The routine works well for `standard' blocks of code that have the same number of \lstinline!&!
-per line, but it will not do anything for lines that do not--such examples
-include \lstinline!tabular! environments that use \lstinline!\multicolumn! or
-perhaps spread cell contents across multiple lines.  For each alignment block (\lstinline!tabular!,
-\lstinline!align!, etc) \lstinline!latexindent.pl! first of all makes a record
-of the maximum number of \lstinline!&!; if each row does not have that
-number of \lstinline!&! then it will not try to format that row. Details
-will be given in \lstinline!indent.log! assuming that \lstinline!trace! mode
-is active.
-
-If you have a \lstinline!verbatim!-like environment inside a \lstinline!tabular!-like
-environment, the \lstinline!verbatim! environment \emph{will} be formatted, which
-is probably not what you want. I hope to address this in future versions, but for the
-moment wrap it in a \lstinline!noIndentBlock! (see \cpageref{lst:noIndentBlockdemo}).
-
-You can run \lstinline!latexindent! on \lstinline!.sty!, \lstinline!.cls! and any filetypes
-that you specify in \lstinline[breaklines=true]!fileExtensionPreference! (see \vref{lst:fileExtensionPreference}); 
-if you find a case in which the script struggles, please feel free
-to report it at \cite{latexindent-home}, and
-in the meantime, consider using a \lstinline!noIndentBlock! (see \cpageref{lst:noIndentBlockdemo}).
-
-I hope that this script is useful to some; if you find an example where the
-script does not behave as you think it should, the best way to contact me is to
-report an issue on \cite{latexindent-home}; otherwise, feel free to find me on 
-the \url{http://tex.stackexchange.com} site; I'm often around
-and in the chat room.
-
-\nocite{*}
-\section{References}
-\printbibliography[heading=subbibnumbered,title={External links},notkeyword=contributor]
-\printbibliography[env=specialbib,heading=subbibnumbered,title={Contributors\label{sec:contributors}},keyword=contributor]
-
-\appendix
-\section{Required \lstinline!Perl! modules}\label{sec:requiredmodules}
-If you intend to use \lstinline!latexindent.pl! and \emph{not} one of the supplied standalone executable files, then you will need a few standard Perl modules--if you can run the
-minimum code in \cref{lst:helloworld} (\lstinline!perl helloworld.pl!) then you will be able to run \lstinline!latexindent.pl!, otherwise you may
-need to install the missing modules.
-
-\begin{cmhlistings}[language=Perl]{\lstinline!helloworld.pl!}{lst:helloworld}
-#!/usr/bin/perl
-
-use strict;
-use warnings;
-use FindBin;
-use YAML::Tiny;
-use File::Copy;
-use File::Basename;
-use Getopt::Long;
-use File::HomeDir;
-
-print "hello world";
-exit;
-\end{cmhlistings}
-My default installation on Ubuntu 12.04 did \emph{not} come
-with all of these modules as standard, but Strawberry Perl for Windows \cite{strawberryperl}
-did.
-
-Installing the modules given in \cref{lst:helloworld} will vary depending on your
-operating system and \lstinline!Perl! distribution. For example, Ubuntu users
-might visit the software center, or else run
-\begin{lstlisting}[numbers=none]
-sudo perl -MCPAN -e 'install "File::HomeDir"'
-\end{lstlisting}
-
-Linux users may be interested in exploring Perlbrew \cite{perlbrew}; possible installation and setup 
-options follow for Ubuntu (other distributions will need slightly different commands).
-\begin{lstlisting}[numbers=none]
-sudo apt-get install perlbrew
-perlbrew install perl-5.20.1
-perlbrew switch perl-5.20.1
-sudo apt-get install curl
-curl -L http://cpanmin.us | perl - App::cpanminus
-cpanm YAML::Tiny
-cpanm File::HomeDir
-\end{lstlisting}
-
-Strawberry Perl users on Windows might use
-\lstinline!CPAN client!. All of the modules are readily available on CPAN \cite{cpan}.
-
-As of Version 2.1,  \lstinline!indent.log! will contain details of the location
-of the Perl modules on your system.  \lstinline!latexindent.exe! is a standalone
-executable for Windows (and therefore does not require a Perl distribution) and caches copies of the Perl modules onto your system; if you
-wish to see where they are cached, use the  \lstinline!trace! option, e.g  \lstinline!latexindent.exe -t myfile.tex!.
-
-\section{The \lstinline!arara! rule}
-The \lstinline!arara! rule (\lstinline!indent.yaml!) contains lines such as those
-given in \cref{lst:arararule}. With this setup, the user \emph{always} has
-to specify whether or not they want (in this example) to use the \lstinline!trace!
-identifier.
-\begin{cmhlistings}[style=yaml,numbers=none]{The \lstinline!arara! rule}{lst:arararule}
-...
-arguments:
-- identifier: trace
-  flag: <arara> @{ isTrue( parameters.trace, "-t" ) }
-...
-\end{cmhlistings}
-
-If you would like to have the \lstinline!trace! option on by default every time you
-call \lstinline!latexindent.pl! from \lstinline!arara! (without having to write \lstinline!% arara: indent: {trace: yes}!), then simply
-amend \cref{lst:arararule} so that it looks like \cref{lst:arararulemod}.
-\begin{cmhlistings}[style=yaml,numbers=none]{The \lstinline!arara! rule (modified)}{lst:arararulemod}
-...
-arguments:
-- identifier: trace
-  flag: <arara> @{ isTrue( parameters.trace, "-t" ) }
-  default: "-t"
-...
-\end{cmhlistings}
-
-With this modification in place, you now simply to write \lstinline!% arara: indent! and
-\lstinline!trace! mode will be activated by default. If you wish to turn off \lstinline!trace!
-mode then you can write \lstinline!% arara: indent: {trace: off}!.
-
-Of course, you can apply these types of modifications to \emph{any} of the identifiers,
-but proceed with caution if you intend to do this for \lstinline!overwrite!.
-
-\section{Updating the \lstinline!path! variable}\label{sec:updating-path}
-\lstinline!latexindent.pl! ships with a few scripts that can update the \lstinline!path! variables
-\footnote{Thanks to \cite{jasjuang} for this feature!}. If you're
-on a Linux or Mac machine, then you'll want \lstinline!CMakeLists.txt! from \cite{latexindent-home}.
-\subsection{Add to path for Linux}
-To add \lstinline!latexindent.pl! to the path for Linux, follow these steps:
-\begin{enumerate}
-	\item download  \lstinline!latexindent.pl!, \lstinline!defaultSettings.yaml!,  to your
-      chosen directory from \cite{latexindent-home} ; 
-    \item within your directory, create a directory called \lstinline!path-helper-files! and 
-      download \lstinline!CMakeLists.txt! and \lstinline!cmake_uninstall.cmake.in!
-	      from \cite{latexindent-home}/path-helper-files to this directory;
-	\item run \lstinline!ls /usr/local/bin! to see what is \emph{currently} in there;
-	\item run the commands given in \cref{linux-add-to-path};
-	\item run \lstinline!ls /usr/local/bin! again to check that \lstinline!latexindent.pl! and \lstinline!defaultSettings.yaml!
-	      have been added.
-\end{enumerate}
-\begin{cmhlistings}[style=yaml,numbers=none]{Add to path from a Linux terminal}{linux-add-to-path}
-sudo apt-get install cmake
-sudo apt-get update && sudo apt-get install build-essential
-mkdir build && cd build
-cmake ../path-helper-files
-sudo make install
-\end{cmhlistings}
-To \emph{remove} the files, run \lstinline!sudo make uninstall!.
-\subsection{Add to path for Windows}
-To add \lstinline!latexindent.exe! to the path for Windows, follow these steps:
-\begin{enumerate}
-	\item download  \lstinline!latexindent.exe!, \lstinline!defaultSettings.yaml!,  \lstinline!add-to-path.bat!
-	      from \cite{latexindent-home} to your chosen directory;
-	\item open a command prompt and run \lstinline!echo %path%! to see what is \emph{currently} in your \lstinline!%path%! variable;
-	\item right click on \lstinline!add-to-path.bat! and \emph{Run as administrator};
-	\item log out, and log back in;
-	\item open a command prompt and run \lstinline!echo %path%! to check that the appropriate directory has been added to your
-	      \lstinline!%path%!.
-\end{enumerate}
-To \emph{remove} the directory from your \lstinline!%path%!, run \lstinline!remove-from-path.bat! as administrator.
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/indent.yaml b/Master/texmf-dist/doc/support/latexindent/indent.yaml
deleted file mode 100644
index 0db81a066f2..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/indent.yaml
+++ /dev/null
@@ -1,38 +0,0 @@
-!config
-# indent rule for arara
-# author: Paulo Cereda, Chris Hughes
-# last updated: 11/9/2013
-# requires arara 3.0+
-#
-# Sample usage:
-#
-# % arara: indent
-# % arara: indent: {overwrite: yes}
-# % arara: indent: {output: myfile.tex, silent: no}
-# % arara: indent: {output: myfile.tex, silent: yes, overwrite: yes}
-# % arara: indent: {trace: true}
-# % arara: indent: {localSettings: true}
-# % arara: indent: {onlyDefault: on}
-# % arara: indent: { cruft: /home/cmhughes/Desktop }
-#
-# Directories with spaces will cause issues in the cruft call.
-# 
-# Note: output will take priority above overwrite
-identifier: indent
-name: Indent
-command: <arara> @{ isWindows( "cmd /c latexindent.exe", "latexindent.pl" ) } @{silent} @{trace} @{localSettings} @{cruft}@{ isNotEmpty( cruft, '="'.concat(parameters.cruft).concat('"') ) } @{overwrite}  @{onlyDefault} @{output} "@{file}" @{ isNotEmpty( output, '"'.concat(parameters.output).concat('"') ) }
-arguments:
-- identifier: overwrite
-  flag: <arara> @{ isTrue( parameters.overwrite, "-w" ) }
-- identifier: silent
-  flag: <arara> @{ isTrue( parameters.silent, "-s" ) }
-- identifier: trace
-  flag: <arara> @{ isTrue( parameters.trace, "-t" ) }
-- identifier: localSettings
-  flag: <arara> @{ isTrue( parameters.localSettings, "-l" ) }
-- identifier: output
-  flag: <arara> @{ isNotEmpty( parameters.output, "-o" ) }
-- identifier: onlyDefault
-  flag: <arara> @{ isTrue( parameters.onlyDefault, "-d" ) }
-- identifier: cruft
-  flag: <arara> @{ isNotEmpty( parameters.cruft, "-c" ) }
diff --git a/Master/texmf-dist/doc/support/latexindent/latexindent.pdf b/Master/texmf-dist/doc/support/latexindent/latexindent.pdf
new file mode 100644
index 00000000000..7c912e4b0fa
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/latexindent.pdf
diff --git a/Master/texmf-dist/doc/support/latexindent/latexindent.tex b/Master/texmf-dist/doc/support/latexindent/latexindent.tex
new file mode 100644
index 00000000000..b31a22b7141
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/latexindent.tex
@@ -0,0 +1,467 @@
+% arara: pdflatex: {shell: yes}
+% arara: bibtex
+% arara: pdflatex: {shell: yes}
+% arara: pdflatex: {shell: yes}
+% arara: pdflatex: {shell: yes}
+% !arara: indent: {overwrite: yes, trace: yes, localSettings: yes, silent: yes}
+\documentclass[10pt]{article}
+%   This program is free software: you can redistribute it and/or modify
+%   it under the terms of the GNU General Public License as published by
+%   the Free Software Foundation, either version 3 of the License, or
+%   (at your option) any later version.
+%
+%   This program is distributed in the hope that it will be useful,
+%   but WITHOUT ANY WARRANTY; without even the implied warranty of
+%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+%   GNU General Public License for more details.
+%
+%   See <http://www.gnu.org/licenses/>.
+\usepackage[left=4.5cm,right=2.5cm,showframe=false,
+	top=2cm,bottom=1.5cm]{geometry}                      % page setup
+\usepackage{lmodern}
+\usepackage{parskip}                                 % paragraph skips
+\usepackage{booktabs}                                % beautiful tables
+\usepackage{listings}                                % nice verbatim environments
+\usepackage{titlesec}                                % customize headings
+\usepackage{titletoc}                                % customize headings
+\usepackage{multicol}
+\usepackage{changepage}                              % adjust width of page
+\usepackage{fancyhdr}                                % headers & footers
+\usepackage{wrapfig}
+\usepackage{fontawesome}
+\usepackage[sc,format=hang,font=small]{caption}      % captions
+\usepackage[backend=bibtex]{biblatex}                % bibliography
+\usepackage{tcolorbox}                                % framed environments
+\usepackage{xparse}
+\usepackage[charter]{mathdesign}                     % changes font
+\usepackage[expansion=false,kerning=true]{microtype} % better kerning
+\usepackage{enumitem}                                % custom lists
+\usepackage{longtable}
+\usepackage{array}
+% setup gitinfo2, as in the manual, details just above begin{document}
+\usepackage[mark,grumpy]{gitinfo2}
+% tcolorbox libraries
+\tcbuselibrary{breakable,skins,listings,minted,xparse}
+%\usepackage{varioref}                                % clever referencing
+%\tcbuselibrary{documentation,breakable,skins,minted}
+% tikz libraries
+\usetikzlibrary{positioning}
+\usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{decorations,shapes}
+\usepackage{varioref}                                % clever referencing
+\usepackage{hyperref}
+\hypersetup{
+	pdfauthor={Chris Hughes},
+	pdftitle={latexindent.pl package},
+	pdfkeywords={perl;beautify;indentation},
+	bookmarksnumbered,
+	bookmarksopen,
+	linktocpage,
+	colorlinks=true,
+	linkcolor=blue,
+	citecolor=black,
+}
+\usepackage{cleveref}
+
+\addbibresource{latex-indent}
+\addbibresource{contributors}
+
+% http://tex.stackexchange.com/questions/122135/how-to-add-a-png-icon-on-the-right-side-of-a-tcolorbox-title
+\newtcolorbox{warning}{parbox=false,breakable,enhanced,arc=0mm,colback=red!5,colframe=red,leftrule=12mm,%
+	overlay={\node[anchor=north west,outer sep=2pt] at (frame.north west) {\includegraphics[width=8mm]{warning}}; }
+}
+
+\definecolor{harvestgold}{cmyk}{0.00, 0.05, 0.51, 0.07}  %EDE275
+\definecolor{cmhgold}{cmyk}{0,0.178,0.909,0.008}         %FDD017
+\makeatletter
+\tcbset{
+	addtolol/.style={list entry={\kvtcb@title},add to list={lol}{lstlisting}},
+	cmhlistings/.style={
+			%	width=\linewidth,
+			%breakable,
+			colback=blue!5!white,
+			colframe=white!25!black,colback=white,
+			top=0cm,
+			bottom=0cm,
+			left=0mm,
+			listing only,
+			center title,
+			listing engine=minted,
+			minted style=colorful,
+			minted options={obeytabs=true,showtabs=true,tabsize=4,showspaces=true},
+			addtolol,
+			boxrule=0pt,
+			toprule=1pt,bottomrule=1pt,
+			titlerule=1pt,
+			colframe=white!40!black,
+			colback=white,
+			sharp corners,
+			colbacktitle=white!75!black
+		},
+	yaml-TCB/.style={
+			listing only,
+			listing engine=listings,
+			left=0cm,
+			boxrule=0pt,
+			%leftrule=3pt,
+			sharp corners,
+			center title,
+			%colbacktitle=white!75!black,
+			colbacktitle=white!75!blue,
+			colframe=white!25!blue,
+			colback=white!90!blue,
+			toprule=2pt,
+			titlerule=2pt,
+			%bottomrule=1pt,
+		},
+	MLB-TCB/.style={
+			yaml-TCB,
+			center title,
+			colframe=cmhgold,
+			colbacktitle=harvestgold,
+			colback=white!60!cmhgold,
+			width=0.9\linewidth,
+			before=\centering,
+			%bottomrule=1pt,
+			enhanced,
+			overlay={\node[anchor=north east,outer sep=2pt,draw=cmhgold,very thick,double,fill=harvestgold,font =\small] at ([yshift=-3mm]frame.north east) {\texttt{-m}}; }
+		}
+}
+
+\newtcblisting[use counter=lstlisting]{cmhlistings}[3][]{%
+	cmhlistings,
+	center title,
+	title={\color{black}{\scshape Listing \thetcbcounter}: ~#2},label={#3},
+	listing engine=listings,
+	listing options={#1},
+}
+
+\DeclareTCBInputListing[use counter=lstlisting]{\cmhlistingsfromfile}{O{} m O{} m m}{%
+	cmhlistings,
+	listing file={#2},
+	listing options={#1},
+	title={\color{black}{\scshape Listing \thetcbcounter}: ~#4},label={#5},
+	#3,
+}
+
+% command shell
+\newtcblisting{commandshell}{colback=black,colupper=white,colframe=yellow!75!black,
+	listing only,listing options={style=tcblatex,language=sh,
+			morekeywords={latexindent,pl},
+			keywordstyle=\color{blue!35!white}\bfseries,
+		},
+	listing engine=listings,
+	left=0cm,
+	every listing line={\textcolor{red}{\small\ttfamily\fontseries{b}\selectfont cmh:$\sim$\$ }}}
+
+% dosprompt
+\newtcblisting{dosprompt}{
+	colback=black,
+	colupper=white,
+	colframe=yellow!75!black,
+	listing only,
+	listing options={
+			language=command.com,
+			morekeywords={latexindent,pl},
+			keywordstyle=\color{blue!35!white}\bfseries,
+			basicstyle=\small\color{white}\ttfamily
+		},
+	listing engine=listings,
+	left=0cm,
+	every listing line={\textcolor{white}{\small\ttfamily\fontseries{b}\selectfont C:\textbackslash Users\textbackslash cmh$>$}}}
+
+\lstset{%
+	basicstyle=\small\ttfamily,language={[LaTeX]TeX},
+	%	numbers=left,
+	numberstyle=\ttfamily%\small,
+	breaklines=true,
+	%   frame=single,framexleftmargin=8mm, xleftmargin=8mm,
+	%	prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookrightarrow}},
+	%	backgroundcolor=\color{green!5},frameround=fttt,
+	%	rulecolor=\color{blue!70!black},
+	keywordstyle=\color{blue},                    % keywords
+	commentstyle=\color{purple},    % comments
+	tabsize=3,
+	%xleftmargin=1.5em,
+}%
+\DeclareTCBListing[use counter=lstlisting]{yaml}{O{} m O{} m}{
+	yaml-TCB,
+	listing options={
+			style=tcblatex,
+			numbers=none,
+			numberstyle=\color{red},
+			#1,
+		},
+	title={\color{black}{\scshape Listing \thetcbcounter}: ~#2},label={#4},
+	#3,
+}
+
+\lstdefinestyle{yaml-LST}{
+	style=tcblatex,
+	numbers=none,
+	%numbers=left,
+	numberstyle=\color{red},
+}
+
+\lstdefinestyle{demo}{
+	numbers=none,
+	linewidth=1.25\textwidth,
+	columns=fullflexible,
+}
+
+% stars around contributors
+\pgfdeclaredecoration{stars}{initial}{
+	\state{initial}[width=15pt]
+	{
+		\pgfmathparse{round(rnd*100)}
+		\pgfsetfillcolor{yellow!\pgfmathresult!orange}
+		\pgfsetstrokecolor{yellow!\pgfmathresult!red}
+		\pgfnode{star}{center}{}{}{\pgfusepath{stroke,fill}}
+	}
+	\state{final}
+	{
+		\pgfpathmoveto{\pgfpointdecoratedpathlast}
+	}
+}
+
+\newtcolorbox{stars}{%
+	enhanced jigsaw,
+	breakable, % allow page breaks
+	left=0cm,
+	top=0cm,
+	before skip=0.2cm,
+	boxsep=0cm,
+	frame style={draw=none,fill=none}, % hide the default frame
+	colback=white,
+	overlay={
+			\draw[inner sep=0,minimum size=rnd*15pt+2pt]
+			decorate[decoration={stars,segment length=2cm}] {
+					decorate[decoration={zigzag,segment length=2cm,amplitude=0.3cm}] {
+							([xshift=-.5cm,yshift=0.1cm]frame.south west) --  ([xshift=-.5cm,yshift=0.4cm]frame.north west)
+						}};
+			\draw[inner sep=0,minimum size=rnd*15pt+2pt]
+			decorate[decoration={stars,segment length=2cm}] {
+					decorate[decoration={zigzag,segment length=2cm,amplitude=0.3cm}] {
+							([xshift=.75cm,yshift=0.1cm]frame.south east) --  ([xshift=.75cm,yshift=0.6cm]frame.north east)
+						}};
+			\node[anchor=north west,outer sep=2pt,opacity=0.25] at ([xshift=-4.25cm]frame.north west) {\resizebox{3cm}{!}{\faGithub}};
+		},
+	% paragraph skips obeyed within tcolorbox
+	parbox=false,
+}
+
+% copied from /usr/local/texlive/2013/texmf-dist/tex/latex/biblatex/bbx/numeric.bbx
+% the only modification is the \stars and \endstars
+\defbibenvironment{specialbib}
+{\stars\list
+	{\printtext[labelnumberwidth]{%
+			\printfield{prefixnumber}%
+			\printfield{labelnumber}}}
+	{\setlength{\labelwidth}{\labelnumberwidth}%
+		\setlength{\leftmargin}{\labelwidth}%
+		\setlength{\labelsep}{\biblabelsep}%
+		\addtolength{\leftmargin}{\labelsep}%
+		\setlength{\itemsep}{\bibitemsep}%
+		\setlength{\parsep}{\bibparsep}}%
+	\renewcommand*{\makelabel}[1]{\hss##1}}
+{\endlist\endstars}
+{\item}
+
+\newtcbox{yamltitlebox}[1][]{colframe=black!50!white,boxrule=.5pt,sharp corners,#1}
+
+\newcommand{\flagbox}[1]{%
+	\par
+	\makebox[30pt][l]{%
+		\hspace{-2cm}%
+		\ttfamily\fontseries{b}\selectfont #1
+	}%
+}
+
+\NewDocumentCommand{\yamltitle}{O{} m s m}{%
+	\par
+	\makebox[30pt][l]{%
+		\hspace{-2cm}%
+		\yamltitlebox[#1]{%
+			{\ttfamily\fontseries{b}\selectfont{\color{blue!80!white}#2}}: %
+			\IfBooleanTF{#3}
+			{$\langle$\itshape #4$\rangle$}
+			{{\bfseries #4}}
+		}}
+	\par\nobreak%
+}
+
+\newcommand{\fixthis}[1]
+{%
+	\marginpar{\huge \color{red} \framebox{FIX}}%
+	\typeout{FIXTHIS: p\thepage : #1^^J}%
+}
+% custom section
+\titleformat{\section}
+{\normalfont\Large\bfseries}
+{\llap{\thesection\hskip.5cm}}
+{0pt}
+{}
+% custom subsection
+\titleformat{\subsection}
+{\normalfont\bfseries}
+{\llap{\thesubsection\hskip.5cm}}
+{0pt}
+{}
+% custom subsubsection
+\titleformat{\subsubsection}
+{\normalfont\bfseries}
+{\llap{\thesubsubsection\hskip.5cm}}
+{0pt}
+{}
+
+\titlespacing\section{0pt}{12pt plus 4pt minus 2pt}{-5pt plus 2pt minus 2pt}
+\titlespacing\subsection{0pt}{12pt plus 4pt minus 2pt}{-6pt plus 2pt minus 2pt}
+\titlespacing\subsubsection{0pt}{12pt plus 4pt minus 2pt}{-6pt plus 2pt minus 2pt}
+
+% list of listings
+\contentsuse{lstlisting}{lol}
+\titlecontents{lstlisting}[2em]
+  {\addvspace{0.25pc}}
+  {\textbf{Code \thecontentslabel} }
+  {}
+  {\titlerule*[0.5em]{$\cdot$}\contentspage}
+  []
+\AtBeginDocument{\addtocontents{lol}{\protect\begin{widepage}\protect\begin{multicols}{2}}}
+\AtEndDocument{\addtocontents{lol}{\protect\end{multicols}\protect\end{widepage}}}
+
+% cleveref settings
+\crefname{table}{Table}{Tables}
+\Crefname{table}{Table}{Tables}
+\crefname{figure}{Figure}{Figures}
+\Crefname{figure}{Figure}{Figures}
+\crefname{section}{Section}{Sections}
+\Crefname{section}{Section}{Sections}
+\crefname{listing}{Listing}{Listings}
+\Crefname{listing}{Listing}{Listings}
+
+% headers and footers
+\fancyhf{} % delete current header and footer
+\fancyhead[R]{\bfseries\thepage%
+	\tikz[remember picture,overlay] {
+		\node at (1,0){\includegraphics{logo-bw}}; }
+}
+\fancyhead[L]{\rightmark}
+\fancyheadoffset[L]{3cm}
+\pagestyle{fancy}
+
+% renew plain style
+\fancypagestyle{plain}{%
+	\fancyhf{} % clear all header and footer fields
+	\renewcommand{\headrulewidth}{0pt}
+	\renewcommand{\footrulewidth}{0pt}}
+
+% widepage environment
+\newenvironment{widepage}{\begin{adjustwidth}{-3cm}{0cm}}{\end{adjustwidth}}
+
+% symbols for the m switch
+\newcommand{\BeginStartsOnOwnLine}{\color{red}\spadesuit}
+\newcommand{\BodyStartsOnOwnLine}{\color{red}\heartsuit}
+\newcommand{\EndStartsOnOwnLine}{\color{red}\diamondsuit}
+\newcommand{\EndFinishesWithLineBreak}{\color{red}\clubsuit}
+\newcommand{\ElseStartsOnOwnLine}{\color{red}\bigstar}
+\newcommand{\ElseFinishesWithLineBreak}{\color{red}\square}
+\newcommand{\EqualsStartsOnOwnLine}{\color{red}\bullet}
+
+% table rules
+\setlength\heavyrulewidth{0.25ex}
+% gitinfo2 settings
+\renewcommand{\gitMark}{\gitBranch\,@\,\gitAbbrevHash{}\,\textbullet{}\,\gitAuthorDate\,\textbullet{}\faGithub}
+
+% setting up gitinfo2:
+%   copy the file post-xxx-sample.txt from http://mirror.ctan.org/macros/latex/contrib/gitinfo2
+%   and put it in .git/hooks/post-checkout
+% then
+%   cd .git/hooks
+%   chmod g+x post-checkout
+%   chmod +x post-checkout
+%   cp post-checkout post-commit
+%   cp post-checkout post-merge
+%   cd ../..
+%   git checkout master
+%   git checkout develop
+%   ls .git
+% and you should see gitHeadInfo.gin
+
+% http://tex.stackexchange.com/questions/233843/textasteriskcentered-invisible-with-garamondmathdesign
+% remove the definition of \textasteriskcentered for TS1 encoding
+\UndeclareTextCommand{\textasteriskcentered}{TS1}
+% reinstate a default encoding
+\DeclareTextSymbolDefault{\textasteriskcentered}{OT1}
+% suitably define the command
+\DeclareTextCommand{\textasteriskcentered}{OT1}{\raisebox{-.7ex}[1ex][0pt]{*}}
+
+\begin{document}
+\renewcommand*{\thefootnote}{\arabic{footnote}}
+\title{%
+	\begin{tcolorbox}[
+			width=5.2cm,
+			boxrule=0pt,
+			colframe=white!40!black,
+			colback=white,
+			rightrule=2pt,
+			sharp corners,
+			enhanced,
+			overlay={\node[anchor=north east,outer sep=2pt] at ([xshift=3cm,yshift=4mm]frame.north east) {\includegraphics[width=3cm]{logo}}; }]
+		\centering\ttfamily\bfseries latexindent.pl\\[1cm] Version 3.0
+	\end{tcolorbox}
+}
+\author{Chris Hughes \thanks{and contributors! See \vref{sec:contributors}. For
+		all communication, please visit \cite{latexindent-home}.}}
+\maketitle
+\begin{adjustwidth}{1cm}{1cm}
+	\small
+	\texttt{latexindent.pl} is a \texttt{Perl} script that indents \texttt{.tex} (and other)
+	files according to an indentation scheme that the user can modify to suit their
+	taste. Environments, including those with alignment delimiters (such as \texttt{tabular}),
+	and commands, including those that can split braces and brackets across lines,
+	are \emph{usually} handled correctly by the script. Options for \texttt{verbatim}-like
+	environments and commands, together with indentation after headings (such as \lstinline!chapter!, \lstinline!section!, etc)
+	are also available. The script also has the ability to modifiy line breaks, and add
+	comment symbols. All user options are customisable via the switches in the YAML interface.
+\end{adjustwidth}
+\tableofcontents
+{\small
+	\lstlistoflistings
+}
+
+\input{sec-introduction}
+\input{sec-demonstration}
+\input{sec-how-to-use}
+\input{sec-indent-config-and-settings.tex}
+\input{sec-default-user-local}
+\input{subsec-noAdditionalIndent-indentRules}
+\input{subsubsec-environments-and-their-arguments}
+\input{subsubsec-environments-with-items}
+\input{subsubsec-commands-with-arguments}
+\input{subsubsec-ifelsefi}
+\input{subsubsec-special}
+\input{subsubsec-headings}
+\input{subsubsec-no-add-remaining-code-blocks}
+\stopcontents[noAdditionalIndent]
+\input{subsec-commands-and-their-options}
+\input{sec-the-m-switch}
+\input{subsec-partnering-poly-switches}
+\input{subsec-conflicting-poly-switches}
+\input{sec-conclusions-know-limitations}
+\input{references}
+\input{appendices}
+\end{document}
+
+\subsection{The phases of \texttt{latexindent.pl}}
+	With these rules in mind, let's study a few test cases:
+
+	latexindent.pl environments-line-break-conflict.tex -s -t -m -o environments-line-break-conflict-mod1.tex -l=env-conflicts-mod1.yaml
+	latexindent.pl environments-line-break-conflict-nested.tex -s -t -m -o environments-line-break-conflict-nested-mod-2.tex -l=env-conflicts-mod2.yaml
+	latexindent.pl environments-line-break-conflict-nested.tex -s -t -m -o environments-line-break-conflict-nested-mod-3.tex -l=env-conflicts-mod3.yaml
+	environments-first-opt-args.tex, see all of the different examples in test-cases.sh
+	environments-second-opt-args.tex provides some interesting cases too
+
+	The \lstinline!\fi! command knows to insert a space, so as to give, for example, \lstinline!\fi! text, and avoid things such as \lstinline!\fitext!
+
+	from yaml
+
diff --git a/Master/texmf-dist/doc/support/latexindent/logo.tex b/Master/texmf-dist/doc/support/latexindent/logo.tex
new file mode 100644
index 00000000000..3b0841fcae5
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/logo.tex
@@ -0,0 +1,20 @@
+% arara: pdflatex
+%
+% to create the black and white logo:
+%       convert -density 1000 -colorspace GRAY logo.pdf logo-bw.pdf
+\documentclass[border=1mm]{standalone}
+
+\usepackage{tikz}
+\usetikzlibrary{matrix}
+\definecolor{harvestgold}{cmyk}{0.00, 0.05, 0.51, 0.07}  %EDE275
+\definecolor{cmhgold}{cmyk}{0,0.178,0.909,0.008}         %FDD017
+\definecolor{bakeoffblue}{cmyk}{0.24, 0.00, 0.02, 0.18}  %9fd2cd
+\definecolor{bakeoffgreen}{cmyk}{0.30, 0.00, 0.20, 0.29} %80b692
+\definecolor{burntorange}{cmyk}{0.00, 0.41, 1.00, 0.04}
+\begin{document}
+\begin{tikzpicture}[logo/.style={draw=blue,circle,fill=white}]
+	\matrix{
+		\node[logo,dash pattern=on .5pt off 1.0pt,thick,draw=purple!75!white]{};  & \node[logo,draw=burntorange]{};                         \\
+		\node[logo,fill,draw=bakeoffgreen,fill=bakeoffblue]{}; & \node[logo,double,draw=cmhgold,fill=harvestgold]{};\\};
+\end{tikzpicture}
+\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/references.tex b/Master/texmf-dist/doc/support/latexindent/references.tex
new file mode 100644
index 00000000000..548b3945784
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/references.tex
@@ -0,0 +1,4 @@
+\nocite{*}
+\section{References}
+ \printbibliography[heading=subbibnumbered,title={External links},notkeyword=contributor]
+ \printbibliography[env=specialbib,heading=subbibnumbered,title={Contributors\label{sec:contributors}},keyword=contributor]
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-conclusions-know-limitations.tex b/Master/texmf-dist/doc/support/latexindent/sec-conclusions-know-limitations.tex
new file mode 100644
index 00000000000..4f9053b9717
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-conclusions-know-limitations.tex
@@ -0,0 +1,38 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\section{Conclusions and known limitations}\label{sec:knownlimitations}
+ There are a number of known limitations of the script, and almost certainly quite a
+ few that are \emph{unknown}!
+
+ For example, \texttt{latexindent.pl} will not indent the following code correctly,
+ because of the unmatched \lstinline![!. I'm hopeful to be able to resolve this
+ issue in a future version.
+
+ \begin{lstlisting}[,nolol=true,]
+\parbox{
+\@ifnextchar[{\@assignmentwithcutoff}{\@assignmentnocutoff}
+}
+\end{lstlisting}
+
+ The main other limitation is to do with the alignment routine of environments/commands that contain
+ delimiters which are specified in \texttt{lookForAlignDelims}.
+
+ The routine works well for `standard' blocks of code that have the same number of \lstinline!&!
+ per line, but it will not do anything for lines that do not -- such examples
+ include \texttt{tabular} environments that use \lstinline!\multicolumn! or
+ perhaps spread cell contents across multiple lines.  For each alignment block (\texttt{tabular},
+ \texttt{align}, etc) \texttt{latexindent.pl} first of all makes a record
+ of the maximum number of \lstinline!&!; if each row does not have that
+ number of \lstinline!&! then it will not try to format that row. Details
+ will be given in \texttt{indent.log} assuming that \texttt{trace} mode
+ is active.
+
+ You can run \texttt{latexindent} on \texttt{.sty}, \texttt{.cls} and any file types
+ that you specify in \lstinline[breaklines=true]!fileExtensionPreference! (see \vref{lst:fileExtensionPreference});
+ if you find a case in which the script struggles, please feel free
+ to report it at \cite{latexindent-home}, and
+ in the meantime, consider using a \texttt{noIndentBlock} (see \cpageref{lst:noIndentBlockdemo}).
+
+ I hope that this script is useful to some; if you find an example where the
+ script does not behave as you think it should, the best way to contact me is to
+ report an issue on \cite{latexindent-home}; otherwise, feel free to find me on
+ the \url{http://tex.stackexchange.com/users/6621/cmhughes}.
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-default-user-local.tex b/Master/texmf-dist/doc/support/latexindent/sec-default-user-local.tex
new file mode 100644
index 00000000000..5dc0e8a77b4
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-default-user-local.tex
@@ -0,0 +1,492 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\section{defaultSettings.yaml}\label{sec:defuseloc}
+ \texttt{latexindent.pl} loads its settings from \texttt{defaultSettings.yaml}. The idea is to separate the behaviour of the script
+ from the internal working -- this is very similar to the way that we separate content
+ from form when writing our documents in \LaTeX.
+
+ If you look in \texttt{defaultSettings.yaml} you'll find the switches
+ that govern the behaviour of \texttt{latexindent.pl}. If you're not sure where
+ \texttt{defaultSettings.yaml} resides on your computer, don't worry as \texttt{indent.log}
+ will tell you where to find it.
+ \texttt{defaultSettings.yaml} is commented,
+ but here is a description of what each switch is designed to do. The default
+ value is given in each case; whenever you see \emph{integer} in \emph{this}
+ section, assume that it must be greater than or equal to \texttt{0} unless
+ otherwise stated.
+
+\yamltitle{fileExtensionPreference}*{fields}
+	\texttt{latexindent.pl} can be called to
+	act on a file without
+	specifying the file extension.  For example we can call
+	\begin{commandshell}
+latexindent.pl myfile
+\end{commandshell}
+	\begin{wrapfigure}[8]{r}[0pt]{6cm}
+		\cmhlistingsfromfile[firstnumber=22,linerange={22-26},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{fileExtensionPreference}}{lst:fileExtensionPreference}
+	\end{wrapfigure}
+
+	in which case the script will look for \texttt{myfile} with the extensions
+	specified in \texttt{fileExtensionPreference} in their numeric order. If
+	no match is found, the script will exit. As with all of the fields, you should
+	change and/or add to this as necessary.
+
+	Calling \texttt{latexindent.pl myfile} with the (default) settings specified in \cref{lst:fileExtensionPreference}
+	means that the script will first look for \texttt{myfile.tex}, then \texttt{myfile.sty}, \texttt{myfile.cls},
+	and finally \texttt{myfile.bib} in order\footnote{Throughout this manual, listings with line numbers represent code
+		taken directly from \texttt{defaultSettings.yaml}.}.
+
+\yamltitle{backupExtension}*{extension name}
+
+	If you call \texttt{latexindent.pl} with the \texttt{-w} switch (to overwrite
+	\texttt{myfile.tex}) then it will create a backup file before doing
+	any indentation; the default extension is \texttt{.bak}, so, for example, \texttt{myfile.bak0}
+	would be created when calling \texttt{latexindent.pl myfile.tex}  for the first time.
+
+	By default, every time you subsequently call \texttt{latexindent.pl} with
+	the \texttt{-w} to act upon \texttt{myfile.tex}, it will create successive back up files: \texttt{myfile.bak1}, \texttt{myfile.bak2},
+	etc.
+
+\yamltitle{onlyOneBackUp}*{integer}
+	\label{page:onlyonebackup}
+	If you don't want a backup for every time that you call \texttt{latexindent.pl} (so
+	you don't want \texttt{myfile.bak1}, \texttt{myfile.bak2}, etc) and you simply
+	want \texttt{myfile.bak} (or whatever you chose \texttt{backupExtension} to be)
+	then change \texttt{onlyOneBackUp} to \texttt{1}; the default value of
+	\texttt{onlyOneBackUp} is \texttt{0}.
+
+\yamltitle{maxNumberOfBackUps}*{integer}
+	Some users may only want a finite number of backup files,
+	say at most $3$, in which case, they can change this switch.
+	The smallest value of \texttt{maxNumberOfBackUps} is $0$ which will \emph{not}
+	prevent backup files being made; in this case, the behaviour will be dictated
+	entirely by \texttt{onlyOneBackUp}. The default value of \texttt{maxNumberOfBackUps}
+	is \texttt{0}.
+
+\yamltitle{cycleThroughBackUps}*{integer}
+	Some users may wish to cycle through backup files, by deleting the
+	oldest backup file and keeping only the most recent; for example,
+	with \texttt{maxNumberOfBackUps: 4}, and \texttt{cycleThroughBackUps}
+	set to \texttt{1}  then the \texttt{copy} procedure given below
+	would be obeyed.
+
+	\begin{commandshell}
+copy myfile.bak1 to myfile.bak0
+copy myfile.bak2 to myfile.bak1
+copy myfile.bak3 to myfile.bak2
+copy myfile.bak4 to myfile.bak3
+	\end{commandshell}
+	The default value of \texttt{cycleThroughBackUps} is \texttt{0}.
+
+\yamltitle{logFilePreferences}*{fields}
+	\begin{wrapfigure}[10]{r}[0pt]{9cm}
+		\cmhlistingsfromfile[firstnumber=63,linerange={63-67},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.85\linewidth,before=\centering,yaml-TCB]{\texttt{logFilePreferences}}{lst:logFilePreferences}
+	\end{wrapfigure}
+	\texttt{latexindent.pl} writes information to \texttt{indent.log}, some
+	of which can be customised by changing \texttt{logFilePreferences}; see \cref{lst:logFilePreferences}.
+	If you load your own user settings (see \vref{sec:indentconfig}) then \texttt{latexindent.pl} will
+	detail them in \texttt{indent.log}; you can choose not to have the details logged by switching
+	\texttt{showEveryYamlRead} to \texttt{0}. Once all of your settings have
+	been loaded, you can see the amalgamated settings in the log file by switching \texttt{showAmalgamatedSettings}
+	to \texttt{1}, if you wish. The log file will end with the characters
+	given in \texttt{endLogFileWith}, and will report the \texttt{GitHub} address
+	of \texttt{latexindent.pl} to the log file if \texttt{showGitHubInfoFooter} is set to \texttt{1}.
+
+\yamltitle{verbatimEnvironments}*{fields}
+
+	\begin{wrapfigure}[14]{r}[0pt]{6cm}
+		\cmhlistingsfromfile[firstnumber=71,linerange={71-73},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{verbatimEnvironments}}{lst:verbatimEnvironments}
+
+		\vspace{.2cm}
+		\cmhlistingsfromfile[firstnumber=76,linerange={76-78},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{verbatimCommands}}{lst:verbatimCommands}
+	\end{wrapfigure}
+	A field that contains a list of environments
+	that you would like left completely alone -- no indentation will be performed
+	on environments that you have specified in this field, see \cref{lst:verbatimEnvironments}.
+
+	Note that if  you put an environment in \\ \texttt{verbatimEnvironments}
+	and in other fields such as \texttt{lookForAlignDelims} or \texttt{noAdditionalIndent}
+	then \texttt{latexindent.pl} will \emph{always} prioritize \\ \texttt{verbatimEnvironments}.
+
+\yamltitle{verbatimCommands}*{fields}
+	A field that contains a list of commands that are verbatim commands, for example
+	\lstinline|\lstinline|; any commands populated in this field are protected from line breaking
+	routines (only relevant if the \texttt{-m} is active, see \vref{sec:modifylinebreaks}).
+
+\yamltitle{noIndentBlock}*{fields}
+
+	\begin{wrapfigure}[8]{r}[0pt]{6cm}
+		\cmhlistingsfromfile[firstnumber=84,linerange={84-86},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{noIndentBlock}}{lst:noIndentBlock}
+	\end{wrapfigure}
+	If you have a block of code that you don't want \texttt{latexindent.pl} to touch (even if it is \emph{not} a verbatim-like
+	environment) then you can wrap it in an environment from \texttt{noIndentBlock};
+	you can use any name you like for this, provided you populate it as demonstrate in
+	\cref{lst:noIndentBlock}.
+
+	Of course, you don't want to have to specify these as null environments
+	in your code, so you use them with a comment symbol, \lstinline!%!, followed
+	by as many spaces (possibly none) as you like; see \cref{lst:noIndentBlockdemo} for
+	example.
+
+	\begin{cmhlistings}[style=demo,escapeinside={(*@}{@*)}]{\texttt{noIndentBlock} demonstration}{lst:noIndentBlockdemo}
+%(*@@*) \begin{noindent}
+        this code
+                won't
+     be touched
+                    by
+             latexindent.pl!
+%(*@@*)\end{noindent}
+	\end{cmhlistings}
+
+\yamltitle{removeTrailingWhitespace}*{fields}\label{yaml:removeTrailingWhitespace}
+
+	\begin{wrapfigure}[12]{r}[0pt]{6cm}
+		\cmhlistingsfromfile[firstnumber=89,linerange={89-91},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{removeTrailingWhitespace}{lst:removeTrailingWhitespace}
+
+		\vspace{.2cm}
+		\cmhlistingsfromfile[firstnumber=95,linerange={95-98},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{fileContentsEnvironments}}{lst:fileContentsEnvironments}
+	\end{wrapfigure}
+	Trailing white space can be removed both \emph{before} and \emph{after} processing
+	the document, as detailed in \cref{lst:removeTrailingWhitespace}; each of the fields
+	can take the values \texttt{0} or \texttt{1}. See \vref{lst:removeTWS-before,lst:env-mlb5-modAll,lst:env-mlb5-modAll-remove-WS}
+	for before and after results.  Thanks to \cite{vosskuhle} for providing this feature.
+
+\yamltitle{fileContentsEnvironments}*{field}
+
+	Before \texttt{latexindent.pl} determines the difference between preamble (if any) and the main document,
+	it first searches for any of the environments specified in \texttt{fileContentsEnvironments}, see
+	\cref{lst:fileContentsEnvironments}.
+	The behaviour of \texttt{latexindent.pl} on these environments is determined by their location (preamble or not), and
+	the value \texttt{indentPreamble}, discussed next.
+
+\yamltitle{indentPreamble}{0|1}
+
+	The preamble of a document can sometimes contain some trickier code
+	for \texttt{latexindent.pl} to operate upon. By default, \texttt{latexindent.pl}
+	won't try to operate on the preamble (as \texttt{indentPreamble} is set to \texttt{0},
+	by default), but if you'd like \texttt{latexindent.pl} to try then change \texttt{indentPreamble} to \texttt{1}.
+
+\yamltitle{lookForPreamble}*{fields}
+
+	\begin{wrapfigure}[8]{r}[0pt]{5cm}
+		\cmhlistingsfromfile[firstnumber=103,linerange={103-107},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{lookForPreamble}{lst:lookForPreamble}
+	\end{wrapfigure}
+	Not all files contain preamble; for example, \texttt{sty}, \texttt{cls} and \texttt{bib} files typically do \emph{not}. Referencing
+	\cref{lst:lookForPreamble}, if you set, for example, \texttt{.tex} to \texttt{0}, then regardless of the setting of the value of \texttt{indentPreamble}, preamble
+	will not be assumed when operating upon \texttt{.tex} files.
+\yamltitle{preambleCommandsBeforeEnvironments}{0|1}
+	Assuming that \texttt{latexindent.pl} is asked to operate upon the preamble of a document,
+	when this switch is set to \texttt{0} then environment code blocks will be sought first,
+	and then command code blocks. When this switch is set to \texttt{1}, commands
+	will be sought first. The example that first motivated this switch contained the code given in \cref{lst:motivatepreambleCommandsBeforeEnvironments}.
+
+	\begin{cmhlistings}{Motivating \texttt{preambleCommandsBeforeEnvironments}}{lst:motivatepreambleCommandsBeforeEnvironments}
+...
+preheadhook={\begin{mdframed}[style=myframedstyle]},
+postfoothook=\end{mdframed},
+...
+\end{cmhlistings}
+
+\yamltitle{defaultIndent}*{horizontal space}
+	This is the default indentation (\lstinline!\t! means a tab, and is the default value) used in the absence of other details
+	for the command or environment we are working with; see \texttt{indentRules} in \vref{sec:noadd-indent-rules}
+	for more details.
+
+	If you're interested in experimenting with \texttt{latexindent.pl} then you
+	can \emph{remove} all indentation by setting \texttt{defaultIndent: ""}.
+
+\yamltitle{lookForAlignDelims}*{fields}
+	\begin{wrapfigure}[12]{r}[0pt]{5cm}
+		\begin{yaml}[numbers=none]{\texttt{lookForAlignDelims} (basic)}[width=.8\linewidth,before=\centering]{lst:aligndelims:basic}
+lookForAlignDelims:
+   tabular: 1
+   tabularx: 1
+   longtable: 1
+   array: 1
+   matrix: 1
+   ...
+	\end{yaml}
+	\end{wrapfigure}
+	This contains a list of environments and/or commands that
+	are operated upon in a special way by \texttt{latexindent.pl} (see \cref{lst:aligndelims:basic}).
+	In fact, the fields in \texttt{lookForAlignDelims} can actually
+	take two different forms: the \emph{basic} version is shown in \cref{lst:aligndelims:basic}
+	and the \emph{advanced} version in \cref{lst:aligndelims:advanced}; we will discuss each in turn.
+
+	The environments specified in this field will be operated on in a special way  by \texttt{latexindent.pl}. In particular, it will try and align each column by its alignment
+	tabs. It does have some limitations (discussed further in \cref{sec:knownlimitations}),
+	but in many cases it will produce results such as those in \cref{lst:tabularbefore:basic,lst:tabularafter:basic}.
+
+	If you find that \texttt{latexindent.pl} does not perform satisfactorily on such
+	environments then you can set the relevant key to \texttt{0}, for example \texttt{tabular: 0}; alternatively, if you just want to ignore \emph{specific}
+	instances of the environment, you could wrap them in something from \texttt{noIndentBlock} (see \cref{lst:noIndentBlock}).
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tabular1.tex}{\texttt{tabular1.tex}}{lst:tabularbefore:basic}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tabular1-default.tex}{\texttt{tabular1.tex} default output}{lst:tabularafter:basic}
+	\end{minipage}%
+
+	If you wish to remove the alignment of the \lstinline!\\! within a delimiter-aligned block, then the
+	advanced form of \texttt{lookForAlignDelims} shown in \cref{lst:aligndelims:advanced} is for you.
+
+	\cmhlistingsfromfile[style=yaml-LST]{demonstrations/tabular.yaml}[yaml-TCB]{\texttt{tabular.yaml}}{lst:aligndelims:advanced}
+
+	Note that you can use a mixture of the basic and advanced form: in \cref{lst:aligndelims:advanced} \texttt{tabular} and \texttt{tabularx}
+	are advanced and \texttt{longtable} is basic. When using the advanced form, each field should receive at least 1 sub-field, and \emph{can} (but does not have to) receive up to 3 fields:
+	\begin{itemize}
+		\item \texttt{delims}: switch equivalent to simply specifying, for example, \texttt{tabular: 1} in
+		      the basic version shown in \cref{lst:aligndelims:basic} (default: 1);
+		\item \texttt{alignDoubleBackSlash}: switch to determine if \lstinline!\\! should be aligned (default: 1);
+		\item \texttt{spacesBeforeDoubleBackSlash}: optionally, specifies the number of spaces to be inserted
+		      before (non-aligned) \lstinline!\\!. In order to use this field, \texttt{alignDoubleBackSlash} needs
+		      to be set to 0 (default: 0).
+	\end{itemize}
+
+	Assuming that you have the settings in \cref{lst:aligndelims:advanced} saved in \texttt{tabular.yaml}, and the code
+	from \cref{lst:tabularbefore:basic} in \texttt{tabular1.tex} and you run
+	\begin{commandshell}
+latexindent.pl -l tabular.yaml tabular1.tex 
+\end{commandshell}
+	then you should receive the before-and-after results shown in
+	\cref{lst:tabularbefore:advanced,lst:tabularafter:advanced}; note that the ampersands have been aligned, but
+	the \lstinline!\\! have not (compare the alignment of \lstinline!\\! in \cref{lst:tabularafter:basic,lst:tabularafter:advanced}).
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tabular1.tex}{\texttt{tabular1.tex}}{lst:tabularbefore:advanced}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tabular1-advanced.tex}{\texttt{tabular1.tex} using \cref{lst:aligndelims:advanced}}{lst:tabularafter:advanced}
+	\end{minipage}%
+
+	Saving \cref{lst:aligndelims:advanced} into \texttt{tabular1.yaml} as in \cref{lst:tabular1YAML}, and running the command
+	\begin{commandshell}
+latexindent.pl -l tabular1.yaml tabular1.tex 
+\end{commandshell}
+	gives \cref{lst:tabularafter:spacing}; note the spacing before the \lstinline!\\!.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tabular1-advanced-3spaces.tex}{\texttt{tabular1.tex} using \cref{lst:tabular1YAML}}{lst:tabularafter:spacing}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.54\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/tabular1.yaml}[yaml-TCB]{\texttt{tabular1.yaml}}{lst:tabular1YAML}
+	\end{minipage}%
+
+	As of Version 3.0, the alignment routine works on mandatory and optional arguments within commands, and also within `special' code blocks
+	(see \texttt{specialBeginEnd} on \cpageref{yaml:specialBeginEnd}); for example, assuming that you have a command called \lstinline!\matrix!
+	and that it is populated within \texttt{lookForAlignDelims} (which it is, by default), and that you
+	run the command
+	\begin{commandshell}
+latexindent.pl matrix1.tex 
+    \end{commandshell}
+	then the before-and-after results
+	shown in \cref{lst:matrixbefore,lst:matrixafter} are achievable by default.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/matrix1.tex}{\texttt{matrix1.tex}}{lst:matrixbefore}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/matrix1-default.tex}{\texttt{matrix1.tex} default output}{lst:matrixafter}
+	\end{minipage}%
+
+	If you have blocks of code that you wish to align at the \&  character that
+	are \emph{not} wrapped in, for example, \lstinline!\begin{tabular}! \ldots \lstinline!\end{tabular}!, then you can use the mark up
+	illustrated in \cref{lst:alignmentmarkup}; the default output is shown in \cref{lst:alignmentmarkup-default}. Note that the \lstinline!%*! must be next to
+	each other, but that there can be any number of spaces (possibly none) between the
+	\lstinline!*! and \lstinline!\begin{tabular}!; note also that you may use any
+	environment name that you have specified in \texttt{lookForAlignDelims}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/align-block.tex}{\texttt{align-block.tex}}{lst:alignmentmarkup}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/align-block-default.tex}{\texttt{align-block.tex} default output}{lst:alignmentmarkup-default}
+	\end{minipage}%
+
+	With reference to \vref{tab:code-blocks} and the, yet undiscussed, fields of \texttt{noAdditionalIndent} and \texttt{indentRules}
+	(see \vref{sec:noadd-indent-rules}), these comment-marked blocks are considered \texttt{environments}.
+
+\yamltitle{indentAfterItems}*{fields}
+	\begin{wrapfigure}[5]{r}[0pt]{7cm}
+		\cmhlistingsfromfile[firstnumber=155,linerange={155-158},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{indentAfterItems}}{lst:indentafteritems}
+	\end{wrapfigure}
+	The environment names specified in \texttt{indentAfterItems}  tell
+	\texttt{latexindent.pl} to look for \lstinline!\item! commands; if these switches are set to \texttt{1}
+	then indentation will be performed so as indent the code after each \texttt{item}.
+	A demonstration is given in \cref{lst:itemsbefore,lst:itemsafter}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/items1.tex}{\texttt{items1.tex}}{lst:itemsbefore}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/items1-default.tex}{\texttt{items1.tex} default output}{lst:itemsafter}
+	\end{minipage}
+
+\yamltitle{itemNames}*{fields}
+	\begin{wrapfigure}[5]{r}[0pt]{5cm}
+		\cmhlistingsfromfile[firstnumber=164,linerange={164-166},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{itemNames}}{lst:itemNames}
+	\end{wrapfigure}
+	If you have your own \texttt{item} commands (perhaps you
+	prefer to use \texttt{myitem}, for example)
+	then you can put populate them in \texttt{itemNames}.
+	For example, users of the \texttt{exam} document class might like to add
+	\texttt{parts} to \texttt{indentAfterItems} and \texttt{part} to \texttt{itemNames}
+	to their user settings (see \vref{sec:indentconfig} for details of how to configure user settings,
+	and \vref{lst:mysettings} \\ in particular \label{page:examsettings}.)
+
+\yamltitle{specialBeginEnd}*{fields}\label{yaml:specialBeginEnd}
+	The fields specified in \texttt{specialBeginEnd} are, in their default state, focused on math mode begin and end statements, but
+	there is no requirement for this to be the case; \cref{lst:specialBeginEnd} shows the
+	default settings of \texttt{specialBeginEnd}.
+
+	\cmhlistingsfromfile[firstnumber=170,linerange={170-182},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{specialBeginEnd}}{lst:specialBeginEnd}
+
+	The field \texttt{displayMath} represents \lstinline!\[...\]!, \texttt{inlineMath} represents
+	\lstinline!$...$! and \texttt{displayMathTex} represents \lstinline!$$...$$!. You can, of course,
+	rename these in your own YAML files (see \vref{sec:localsettings}); indeed, you
+	might like to set up your own specil begin and end statements.
+
+	A demonstration of the before-and-after results are shown in \cref{lst:specialbefore,lst:specialafter}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1.tex}{\texttt{special1.tex} before}{lst:specialbefore}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1-default.tex}{\texttt{special1.tex} default output}{lst:specialafter}
+	\end{minipage}
+
+	For each field, \texttt{lookForThis} is set to \texttt{1} by default, which means that \texttt{latexindent.pl}
+	will look for this pattern; you can tell \texttt{latexindent.pl} not to look for the pattern, by setting
+	\texttt{lookForThis} to \texttt{0}.
+
+\yamltitle{indentAfterHeadings}*{fields}
+	\begin{wrapfigure}[17]{r}[0pt]{8cm}
+		\cmhlistingsfromfile[firstnumber=192,linerange={192-201},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{indentAfterHeadings}}{lst:indentAfterHeadings}
+	\end{wrapfigure}
+	This field enables the user to specify
+	indentation rules that take effect after heading commands such as \lstinline!\part!, \lstinline!\chapter!,
+	\lstinline!\section!, \lstinline!\subsection*!, or indeed any user-specified command written in this field.\footnote{There is a slight
+		difference in interface for this field when comparing Version 2.2 to Version 3.0; see \vref{app:differences} for details.}
+
+	The default settings do \emph{not} place indentation after a heading, but you
+	can easily switch them on by changing \\ \texttt{indentAfterThisHeading: 0} to \\ \texttt{indentAfterThisHeading: 1}.
+	The \texttt{level} field tells \texttt{latexindent.pl} the hierarchy of the heading
+	structure in your document. You might, for example, like to have both \texttt{section}
+	and \texttt{subsection} set with \texttt{level: 3} because you do not want the indentation to go too deep.
+
+	You can add any of your own custom heading commands to this field, specifying the \texttt{level}
+	as appropriate.  You can also specify your own indentation in \texttt{indentRules} (see \vref{sec:noadd-indent-rules});
+	you will find the default \texttt{indentRules} contains \lstinline!chapter: " "! which
+	tells \texttt{latexindent.pl} simply to use a space character after \texttt{\chapter} headings
+	(once \texttt{indent} is set to \texttt{1} for \texttt{chapter}).
+
+	For example, assuming that you have the code in \cref{lst:headings1yaml} saved into \texttt{headings1.yaml},
+	and that you have the text from \cref{lst:headings1} saved into \texttt{headings1.tex}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings1.yaml}[yaml-TCB]{\texttt{headings1.yaml}}{lst:headings1yaml}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings1.tex}{\texttt{headings1.tex}}{lst:headings1}
+	\end{minipage}
+
+	If you run the command
+	\begin{commandshell}
+latexindent.pl headings1.tex -l=headings1.yaml
+\end{commandshell}
+	then you should receive the output given in \cref{lst:headings1-mod1}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings1-mod1.tex}{\texttt{headings1.tex} using \cref{lst:headings1yaml}}{lst:headings1-mod1}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings1-mod2.tex}{\texttt{headings1.tex} second modification}{lst:headings1-mod2}
+	\end{minipage}
+
+	Now say that you modify the \texttt{YAML} from \cref{lst:headings1yaml} so that the \texttt{paragraph} \texttt{level} is \texttt{1}; after
+	running
+	\begin{commandshell}
+latexindent.pl headings1.tex -l=headings1.yaml
+\end{commandshell}
+	you should receive the code given in \cref{lst:headings1-mod2}; notice that
+	the \texttt{paragraph} and \texttt{subsection} are at the same indentation level.
+
+\subsection{The code blocks known \texttt{latexindent.pl}}\label{subsubsec:code-blocks}
+	As of Version 3.0, \texttt{latexindent.pl} processes documents using code blocks; each
+	of these are shown in \cref{tab:code-blocks}.
+
+	\begin{longtable}{m{.3\linewidth}@{\hspace{.25cm}}m{.4\linewidth}@{}m{.2\linewidth}}
+		\caption{Code blocks known to \texttt{latexindent.pl}}\label{tab:code-blocks}\\
+		\toprule
+		Code block                    & characters allowed in name                                                                                     & example                                                                                                                                                                                                                                     \\
+		\midrule
+		environments                  & \lstinline!a-zA-Z@\*0-9_\\!                                                                                        &
+		\begin{lstlisting}[,nolol=true,]
+\begin{myenv}
+body of myenv
+\end{myenv}
+  \end{lstlisting}
+		\\\cmidrule{2-3}
+		optionalArguments             & \emph{inherits} name from parent (e.g environment name)                                                        &
+		\begin{lstlisting}[,nolol=true,]
+[
+opt arg text
+]
+  \end{lstlisting}
+		\\\cmidrule{2-3}
+		mandatoryArguments            & \emph{inherits} name from parent (e.g environment name)                                                        &
+		\begin{lstlisting}[,nolol=true,]
+{
+mand arg text
+}
+  \end{lstlisting}
+		\\\cmidrule{2-3}
+		commands                      & \lstinline!+a-zA-Z@\*0-9_\:!                                                                                        & \lstinline!\mycommand!$\langle$\itshape{arguments}$\rangle$                                                                                                                                                                                \\\cmidrule{2-3}
+		keyEqualsValuesBracesBrackets & \lstinline!a-zA-Z@\*0-9_\/.\h\{\}:\#-!                                                                                        & \lstinline!my key/.style=!$\langle$\itshape{arguments}$\rangle$                                                                                                                                                                                \\\cmidrule{2-3}
+		namedGroupingBracesBrackets   & \lstinline!a-zA-Z@\*><!                                                                                        & \lstinline!in!$\langle$\itshape{arguments}$\rangle$                                                                                                                                                                                \\\cmidrule{2-3}
+		UnNamedGroupingBracesBrackets & \centering\emph{No name!}                                                                                      & \lstinline!{! or \lstinline![! or \lstinline!,! or \lstinline!&! or \lstinline!)! or \lstinline!(! or \lstinline!$! followed by $\langle$\itshape{arguments}$\rangle$ \\\cmidrule{2-3}
+		ifElseFi                      & \lstinline!@a-zA-Z! but must begin with either \newline \lstinline!\if! of \lstinline!\@if! &
+		\begin{lstlisting}[,nolol=true,]
+\ifnum...
+...
+\else
+...
+\fi
+  \end{lstlisting}\\\cmidrule{2-3}
+		items                         & User specified, see \vref{lst:indentafteritems,lst:itemNames}                                                  &
+		\begin{lstlisting}[,nolol=true,]
+\begin{enumerate}
+  \item ...
+\end{enumerate}
+  \end{lstlisting}\\\cmidrule{2-3}
+		specialBeginEnd               & User specified, see \vref{lst:specialBeginEnd}                                                                 &
+		\begin{lstlisting}[,nolol=true,]
+\[
+  ...
+\]
+  \end{lstlisting}\\\cmidrule{2-3}
+		afterHeading                  & User specified, see \vref{lst:indentAfterHeadings}                                                             &
+		\begin{lstlisting}[,morekeywords={chapter},nolol=true,]
+\chapter{title}
+  ...
+\section{title}
+  \end{lstlisting}\\\cmidrule{2-3}
+		filecontents                  & User specified, see \vref{lst:fileContentsEnvironments}                                                        &
+		\begin{lstlisting}[,nolol=true,]
+\begin{filecontents}
+...
+\end{filecontents}
+  \end{lstlisting}\\
+		\bottomrule
+	\end{longtable}
+
+	We will refer to these code blocks in what follows.
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-demonstration.tex b/Master/texmf-dist/doc/support/latexindent/sec-demonstration.tex
new file mode 100644
index 00000000000..656712e38fa
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-demonstration.tex
@@ -0,0 +1,40 @@
+\section{Demonstration: before and after}
+ Let's give a demonstration of some before and after code -- after all, you probably
+ won't want to try the script if you don't much like the results. You might also
+ like to watch the video demonstration I made on youtube \cite{cmh:videodemo}
+
+ As you look at \crefrange{lst:filecontentsbefore}{lst:pstricksafter}, remember
+ that \texttt{latexindent.pl} is just following its rules, and there is nothing
+ particular about these code snippets. All of the rules can be modified
+ so that you can personalize your indentation scheme.
+
+ In each of the samples given in \crefrange{lst:filecontentsbefore}{lst:pstricksafter}
+ the `before' case is a `worst case scenario' with no effort to make indentation. The `after'
+ result would be the same, regardless of the leading white space at the beginning of
+ each line which is stripped by \texttt{latexindent.pl} (unless a \texttt{verbatim}-like
+ environment or \texttt{noIndentBlock} is specified -- more on this in \cref{sec:defuseloc}).
+
+ \begin{widepage}
+	 \centering
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/filecontents1.tex}{\texttt{filecontents1.tex}}{lst:filecontentsbefore}
+	 \end{minipage}\hfill
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/filecontents1-default.tex}{\texttt{filecontents1.tex} default output}{lst:filecontentsafter}
+	 \end{minipage}%
+
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/tikzset.tex}{\texttt{tikzset.tex}}{lst:tikzsetbefore}
+	 \end{minipage}\hfill
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/tikzset-default.tex}{\texttt{tikzset.tex} default output}{lst:tikzsetafter}
+	 \end{minipage}%
+
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/pstricks.tex}{\texttt{pstricks.tex}}{lst:pstricksbefore}
+	 \end{minipage}\hfill
+	 \begin{minipage}{.45\linewidth}
+		 \cmhlistingsfromfile{demonstrations/pstricks-default.tex}{\texttt{pstricks.tex} default output}{lst:pstricksafter}
+	 \end{minipage}%
+ \end{widepage}
+
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-how-to-use.tex b/Master/texmf-dist/doc/support/latexindent/sec-how-to-use.tex
new file mode 100644
index 00000000000..8e76e5d8fbf
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-how-to-use.tex
@@ -0,0 +1,248 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\section{How to use the script}
+ \texttt{latexindent.pl} ships as part of the \TeX Live distribution for
+ Linux and Mac users; \texttt{latexindent.exe} ships as part of the \TeX Live
+ and MiK\TeX{} distributions for Windows users. These files are also available
+ from github \cite{latexindent-home} should you wish to use them without
+ a \TeX{} distribution; in this case, you may like to read \vref{sec:updating-path}
+ which details how the \texttt{path} variable can be updated.
+
+ In what follows, we will always refer to \texttt{latexindent.pl}, but depending on
+ your operating system and preference, you might substitute \texttt{latexindent.exe} or
+ simply \texttt{latexindent}.
+
+ There are two ways to use \texttt{latexindent.pl}: from the command line,
+ and using \texttt{arara}; we discuss these in \cref{sec:commandline} and
+ \cref{sec:arara} respectively. We will discuss how to change the settings and
+ behaviour of the script in \vref{sec:defuseloc}.
+
+ \texttt{latexindent.pl} ships with \texttt{latexindent.exe} for Windows
+ users, so that you can use the script with or without a Perl distribution.
+ If you plan to use \texttt{latexindent.pl} (i.e, the original Perl script) then you will
+ need a few standard Perl modules -- see \vref{sec:requiredmodules} for details.
+
+\subsection{From the command line}\label{sec:commandline}
+	\texttt{latexindent.pl} has a number of different switches/flags/options, which
+	can be combined in any way that you like, either in short or long form as detailed below.
+	\texttt{latexindent.pl}  produces a \texttt{.log} file, \texttt{indent.log}, every time it
+	is run; the name of the log file can be customised, but we will
+	refer to the log file as \texttt{indent.log} throughout this document.
+	There is a base of information that is written to \texttt{indent.log},
+	but other additional information will be written depending
+	on which of the following options are used.
+
+	\begin{commandshell}
+latexindent.pl
+      \end{commandshell}
+
+	This will output a welcome message to the terminal, including the version number
+	and available options.
+
+\flagbox{-h, --help}
+
+	\begin{commandshell}
+latexindent.pl -h
+      \end{commandshell}
+
+	As above this will output a welcome message to the terminal, including the version number
+	and available options.
+	\begin{commandshell}
+latexindent.pl myfile.tex
+      \end{commandshell}
+
+	This will operate on \texttt{myfile.tex}, but will simply output to your terminal; \texttt{myfile.tex} will	not be changed
+	by \texttt{latexindent.pl} in any way using this command.
+
+\flagbox{-w, --overwrite}
+	\begin{commandshell}
+latexindent.pl -w myfile.tex
+latexindent.pl --overwrite myfile.tex
+latexindent.pl myfile.tex --overwrite 
+      \end{commandshell}
+
+	This \emph{will} overwrite \texttt{myfile.tex}, but it will
+	make a copy of \texttt{myfile.tex} first. You can control the name of
+	the extension (default is \texttt{.bak}), and how many different backups are made --
+	more on this in \cref{sec:defuseloc}, and in particular see \texttt{backupExtension} and \texttt{onlyOneBackUp}.
+
+	Note that if \texttt{latexindent.pl} can not create the backup, then it
+	will exit without touching your original file; an error message will be given
+	asking you to check the permissions of the backup file.
+
+\flagbox{-o=output.tex,--outputfile=output.tex}
+	\begin{commandshell} 
+latexindent.pl -o=output.tex myfile.tex
+latexindent.pl myfile.tex -o=output.tex 
+latexindent.pl --outputfile=output.tex myfile.tex
+latexindent.pl --outputfile output.tex myfile.tex
+      \end{commandshell}
+
+	This will indent \texttt{myfile.tex} and output it to \texttt{output.tex},
+	overwriting it (\texttt{output.tex}) if it already exists\footnote{Users of version 2.* should
+		note the subtle change in syntax}. Note that if \texttt{latexindent.pl} is called with both
+	the \texttt{-w} and \texttt{-o} switches, then \texttt{-w} will
+	be ignored and \texttt{-o} will take priority (this seems safer than the
+	other way round).
+
+	Note that using \texttt{-o} is equivalent to using
+	\begin{commandshell}
+latexindent.pl myfile.tex > output.tex
+\end{commandshell}
+	See \vref{app:differences} for details of how the interface has changed
+	from Version 2.2 to Version 3.0 for this flag.
+\flagbox{-s, --silent}
+	\begin{commandshell}
+latexindent.pl -s myfile.tex
+latexindent.pl myfile.tex -s
+      \end{commandshell}
+
+	Silent mode: no output will be given to the terminal.
+
+\flagbox{-t, --trace}
+	\begin{commandshell}
+latexindent.pl -t myfile.tex
+latexindent.pl myfile.tex -t
+      \end{commandshell}
+
+	\label{page:traceswitch}
+	Tracing mode: verbose output will be given to \texttt{indent.log}. This
+	is useful if \texttt{latexindent.pl} has made a mistake and you're
+	trying to find out where and why. You might also be interested in learning
+	about \texttt{latexindent.pl}'s thought process -- if so, this
+	switch is for you, although it should be noted that, especially for large files, this does affect
+	performance of the script.
+
+\flagbox{-tt, --ttrace}
+	\begin{commandshell}
+latexindent.pl -tt myfile.tex
+latexindent.pl myfile.tex -tt
+      \end{commandshell}
+
+	\emph{More detailed} tracing mode: this option gives more details to \texttt{indent.log}
+	than the standard \texttt{trace} option (note that, even more so than with \texttt{-t},
+	especially for large files, performance of the script will be affected).
+
+\flagbox{-l, --local[=myyaml.yaml,other.yaml,...]}
+	\begin{commandshell}
+latexindent.pl -l myfile.tex
+latexindent.pl -l=myyaml.yaml myfile.tex
+latexindent.pl -l myyaml.yaml myfile.tex
+latexindent.pl -l first.yaml,second.yaml,third.yaml myfile.tex
+latexindent.pl -l=first.yaml,second.yaml,third.yaml myfile.tex
+latexindent.pl myfile.tex -l=first.yaml,second.yaml,third.yaml 
+      \end{commandshell}
+
+	\label{page:localswitch}
+	\texttt{latexindent.pl} will always load \texttt{defaultSettings.yaml} (rhymes with camel)
+	and if it is called with the \texttt{-l} switch and it finds \texttt{localSettings.yaml}
+	in the same directory as \texttt{myfile.tex} then these settings will be
+	added to the indentation scheme. Information will be given in \texttt{indent.log} on
+	the success or failure of loading \texttt{localSettings.yaml}.
+
+	The \texttt{-l} flag can take an \emph{optional} parameter which details the name (or names separated by commas) of a YAML file(s)
+	that resides in the same directory as \texttt{myfile.tex}; you can use this option if you would
+	like to load a settings file in the current working directory that is \emph{not} called \texttt{localSettings.yaml}.
+	In fact, you can specify \emph{relative} path names to the current directory, but \emph{not}
+	absolute paths -- for absolute paths, see \vref{sec:indentconfig}.
+	Explicit demonstrations of how to use the \texttt{-l} switch are given throughout this documentation.
+
+\flagbox{-d, --onlydefault}
+	\begin{commandshell}
+latexindent.pl -d myfile.tex
+      \end{commandshell}
+
+	Only \texttt{defaultSettings.yaml}: you might like to read \cref{sec:defuseloc} before
+	using this switch. By default, \texttt{latexindent.pl} will always search for
+	\texttt{indentconfig.yaml} or \texttt{.indentconfig.yaml}  in your home directory. If you would prefer it not to do so
+	then (instead of deleting or renaming \texttt{indentconfig.yaml}/\texttt{.indentconfig.yaml}) you can simply
+	call the script with the \texttt{-d} switch; note that this will also tell
+	the script to ignore \texttt{localSettings.yaml} even if it has been called with the
+	\texttt{-l} switch.
+
+\flagbox{-c, --cruft=<directory>}
+	\begin{commandshell}
+latexindent.pl -c=/path/to/directory/ myfile.tex
+      \end{commandshell}
+
+	If you wish to have backup files and \texttt{indent.log} written to a directory
+	other than the current working directory, then you can send these `cruft' files
+	to another directory.
+	% this switch was made as a result of http://tex.stackexchange.com/questions/142652/output-latexindent-auxiliary-files-to-a-different-directory
+
+\flagbox{-g, --logfile}
+	\begin{commandshell}
+latexindent.pl -g=other.log myfile.tex
+latexindent.pl -g other.log myfile.tex
+latexindent.pl --logfile other.log myfile.tex
+latexindent.pl myfile.tex -g other.log 
+      \end{commandshell}
+
+	By default, \texttt{latexindent.pl} reports information to \texttt{indent.log}, but if you wish to change the
+	name of this file, simply call the script with your chosen name after the \texttt{-g} switch as demonstrated above.
+
+\flagbox{-m, --modifylinebreaks}
+	\begin{commandshell}
+latexindent.pl -m myfile.tex
+latexindent.pl -modifylinebreaks myfile.tex
+      \end{commandshell}
+
+	One of the most exciting developments in Version~3.0 is the ability to modify line breaks; for full details
+	see \vref{sec:modifylinebreaks}
+
+	\texttt{latexindent.pl} can also be called on a file without the file extension, for
+	example
+	\begin{commandshell}
+latexindent.pl myfile
+    \end{commandshell}
+	and in which case, you can specify
+	the order in which extensions are searched for; see \vref{lst:fileExtensionPreference}
+	for full details.
+
+\subsection{From \texttt{arara}}\label{sec:arara}
+	Using \texttt{latexindent.pl} from the command line is fine for some folks, but
+	others may find it easier to use from \texttt{arara}. \texttt{arara} ships with
+	a rule, \texttt{indent.yaml}, but in case you do not have this rule, you can find it at \cite{paulo}.
+
+	You can use the rule in any of the ways described in \cref{lst:arara} (or combinations thereof).
+	In fact, \texttt{arara} allows yet greater flexibility -- you can use \texttt{yes/no}, \texttt{true/false}, or \texttt{on/off} to toggle the various options.
+	\begin{cmhlistings}[style=demo,escapeinside={(*@}{@*)}]{\texttt{arara} sample usage}{lst:arara}
+%(*@@*) arara: indent
+%(*@@*) arara: indent: {overwrite: yes}
+%(*@@*) arara: indent: {output: myfile.tex}
+%(*@@*) arara: indent: {silent: yes}
+%(*@@*) arara: indent: {trace: yes}
+%(*@@*) arara: indent: {localSettings: yes}
+%(*@@*) arara: indent: {onlyDefault: on}
+%(*@@*) arara: indent: { cruft: /home/cmhughes/Desktop }
+\documentclass{article}
+...
+\end{cmhlistings}
+%(*@@*) arara: indent: { modifylinebreaks: yes }
+
+	Hopefully the use of these rules is fairly self-explanatory, but for completeness
+	\cref{tab:orbsandswitches} shows the relationship between \texttt{arara} directive arguments and the
+	switches given in \cref{sec:commandline}.
+
+	\begin{table}[!ht]
+		\centering
+		\caption{\texttt{arara} directive arguments and corresponding switches}
+		\label{tab:orbsandswitches}
+		\begin{tabular}{lc}
+			\toprule
+			\texttt{arara} directive argument & switch      \\
+			\midrule
+			\texttt{overwrite}                & \texttt{-w} \\
+			\texttt{output}                   & \texttt{-o} \\
+			\texttt{silent}                   & \texttt{-s} \\
+			\texttt{trace}                    & \texttt{-t} \\
+			\texttt{localSettings}            & \texttt{-l} \\
+			\texttt{onlyDefault}              & \texttt{-d} \\
+			\texttt{cruft}                    & \texttt{-c} \\
+			\texttt{modifylinebreaks}         & \texttt{-m} \\
+			\bottomrule
+		\end{tabular}
+	\end{table}
+
+	The \texttt{cruft} directive does not work well when used with
+	directories that contain spaces.
+
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-indent-config-and-settings.tex b/Master/texmf-dist/doc/support/latexindent/sec-indent-config-and-settings.tex
new file mode 100644
index 00000000000..a3cffa0921a
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-indent-config-and-settings.tex
@@ -0,0 +1,154 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\section{User, local settings, \texttt{indentconfig.yaml} and \texttt{.indentconfig.yaml}}\label{sec:indentconfig}
+ Editing \texttt{defaultSettings.yaml} is not ideal as it may be overwritten when
+ updating your distribution--a better way to customize the settings to your liking
+ is to set up your own settings file,
+ \texttt{mysettings.yaml} (or any name you like, provided it ends with \texttt{.yaml}).
+ The only thing you have to do is tell \texttt{latexindent.pl} where to find it.
+
+ \texttt{latexindent.pl} will always check your home directory for \texttt{indentconfig.yaml}
+ and  \texttt{.indentconfig.yaml} (unless
+ it is called with the \texttt{-d} switch),
+ which is a plain text file you can create that contains the \emph{absolute}
+ paths for any settings files that you wish \texttt{latexindent.pl} to load. There is no difference
+ between \texttt{indentconfig.yaml} and \texttt{.indentconfig.yaml}, other than the
+ fact that \texttt{.indentconfig.yaml} is a `hidden' file; thank you to \cite{jacobo-diaz-hidden-config}
+ for providing this feature. In what follows, we will use \texttt{indentconfig.yaml}, but it
+ is understood that this equally represents \texttt{.indentconfig.yaml} as well. If you
+ have both files in existence,  \texttt{indentconfig.yaml} takes priority.
+
+ For Mac and Linux users, their home directory is \texttt{~/username} while
+ Windows (Vista onwards) is \lstinline!C:\Users\username!\footnote{If you're not sure
+	 where to put \texttt{indentconfig.yaml}, don't
+	 worry \texttt{latexindent.pl} will tell you in the log file exactly where to
+	 put it assuming it doesn't exist already.}
+ \Cref{lst:indentconfig} shows a sample \texttt{indentconfig.yaml} file.
+
+ \begin{yaml}{\texttt{indentconfig.yaml} (sample)}{lst:indentconfig}
+	# Paths to user settings for latexindent.pl
+	#
+	# Note that the settings will be read in the order you
+	# specify here- each successive settings file will overwrite
+	# the variables that you specify
+
+	paths:
+	- /home/cmhughes/Documents/yamlfiles/mysettings.yaml
+	- /home/cmhughes/folder/othersettings.yaml
+	- /some/other/folder/anynameyouwant.yaml
+	- C:\Users\chughes\Documents\mysettings.yaml
+	- C:\Users\chughes\Desktop\test spaces\more spaces.yaml
+\end{yaml}
+
+ Note that the \texttt{.yaml} files you specify in \texttt{indentconfig.yaml}
+ will be loaded in the order that you write them in. Each file doesn't have
+ to have every switch from \texttt{defaultSettings.yaml}; in fact, I recommend
+ that you only keep the switches that you want to \emph{change} in these
+ settings files.
+
+ To get started with your own settings file, you might like to save a copy of
+ \texttt{defaultSettings.yaml} in another directory and call it, for
+ example, \texttt{mysettings.yaml}. Once you have added the path to \texttt{indentconfig.yaml}
+ you can change the switches and add more code-block names to it
+ as you see fit -- have a look at \cref{lst:mysettings} for an example
+ that uses four tabs for the default indent, adds the \texttt{tabbing}
+ environment/command to the list of environments that contains alignment delimiters; you might also like to
+ refer to the many YAML files detailed throughout the rest of this documentation.
+
+ \begin{yaml}{\texttt{mysettings.yaml} (example)}{lst:mysettings}
+# Default value of indentation
+defaultIndent: "\t\t\t\t"
+
+# environments that have tab delimiters, add more
+# as needed
+lookForAlignDelims:
+    tabbing: 1
+\end{yaml}
+
+ You can make sure that your settings are loaded by checking \texttt{indent.log}
+ for details -- if you have specified a path that \texttt{latexindent.pl} doesn't
+ recognize then you'll get a warning, otherwise you'll get confirmation that
+ \texttt{latexindent.pl} has read your settings file \footnote{Windows users
+	 may find that they have to end \texttt{.yaml} files with a blank line}.
+
+ \begin{warning}
+	 When editing \texttt{.yaml} files it is \emph{extremely} important
+	 to remember how sensitive they are to spaces. I highly recommend copying
+	 and pasting from \texttt{defaultSettings.yaml} when you create your
+	 first \texttt{whatevernameyoulike.yaml} file.
+
+	 If \texttt{latexindent.pl} can not read your \texttt{.yaml} file it
+	 will tell you so in \texttt{indent.log}.
+ \end{warning}
+
+\subsection{\texttt{localSettings.yaml}}\label{sec:localsettings}
+	The \texttt{-l} switch tells \texttt{latexindent.pl} to look for \texttt{localSettings.yaml} in the
+	\emph{same directory} as \texttt{myfile.tex}.  If you'd prefer to name your \texttt{localSettings.yaml} file something
+	different, (say, \texttt{myyaml.yaml}) then
+	you can call \texttt{latexindent.pl} using, for example,
+	\begin{commandshell}
+latexindent.pl -l=myyaml.yaml myfile.tex
+\end{commandshell}
+
+	Any settings file(s) specified using the \texttt{-l} switch will be read \emph{after} \texttt{defaultSettings.yaml} and, assuming they exist,
+	user settings from \texttt{indentconfig.yaml}.
+
+	Your settings file can contain any switches that you'd
+	like to change; a sample is shown in \cref{lst:localSettings}, and you'll find plenty of further examples throughout this manual.
+
+	\begin{yaml}{\texttt{localSettings.yaml} (example)}{lst:localSettings}
+#  verbatim environments- environments specified
+#  in this hash table will not be changed at all!
+verbatimEnvironments:
+    cmhenvironment: 0
+\end{yaml}
+
+	You can make sure that your settings file has been loaded by checking \texttt{indent.log}
+	for details; if it can not be read then you receive a warning, otherwise you'll get confirmation that
+	\texttt{latexindent.pl} has read your settings file.
+
+\subsection{Settings load order}\label{sec:loadorder}
+	\texttt{latexindent.pl} loads the settings files in the following order:
+	\begin{enumerate}
+		\item \texttt{defaultSettings.yaml} is always loaded, and can not be renamed;
+		\item \texttt{anyUserSettings.yaml} and any other arbitrarily-named files specified in \texttt{indentconfig.yaml};
+		\item \texttt{localSettings.yaml} but only if found in the same directory as \texttt{myfile.tex} and called
+		      with \texttt{-l} switch; this file can be renamed, provided that the call to \texttt{latexindent.pl} is adjusted
+		      accordingly (see \cref{sec:localsettings}). You may specify relative  paths to other
+		      YAML files using the \texttt{-l} switch, separating multiple files using commas.
+	\end{enumerate}
+	A visual representation of this is given in \cref{fig:loadorder}.
+
+	\begin{figure}
+		\centering
+		\begin{tikzpicture}[
+				needed/.style={very thick, draw=blue,fill=blue!20,
+						text centered, minimum height=2.5em,rounded corners=1ex},
+				optional/.style={draw=black, very thick,scale=0.8,
+						text centered, minimum height=2.5em,rounded corners=1ex},
+				optionalfill/.style={fill=black!10},
+				connections/.style={draw=black!30,dotted,line width=3pt,text=red},
+			]
+			% Draw diagram elements
+			\node (latexindent) [needed,circle]  {\texttt{latexindent.pl}};
+			\node (default) [needed,above right=.5cm of latexindent]  {\texttt{defaultSettings.yaml}};
+			\node (indentconfig) [optional,right=of latexindent]  {\texttt{indentconfig.yaml}};
+			\node (any) [optional,optionalfill,above right=of indentconfig]  {\texttt{any.yaml}};
+			\node (name) [optional,optionalfill,right=of indentconfig]  {\texttt{name.yaml}};
+			\node (you) [optional,optionalfill,below right=of indentconfig]  {\texttt{you.yaml}};
+			\node (want) [optional,optionalfill,below=of indentconfig]  {\texttt{want.yaml}};
+			\node (local) [optional,below=of latexindent]  {\texttt{localSettings.yaml}};
+			% Draw arrows between elements
+			\draw[connections,solid] (latexindent) to[in=-90]node[pos=0.5,anchor=north]{1} (default.south) ;
+			\draw[connections,optional] (latexindent) -- node[pos=0.5,anchor=north]{2} (indentconfig) ;
+			\draw[connections,optional] (indentconfig) to[in=-90] (any.south) ;
+			\draw[connections,optional] (indentconfig) -- (name) ;
+			\draw[connections,optional] (indentconfig) to[out=-45,in=90] (you) ;
+			\draw[connections,optional] (indentconfig) -- (want) ;
+			\draw[connections,optional] (latexindent) -- node[pos=0.5,anchor=west]{3} (local) ;
+		\end{tikzpicture}
+		\caption{Schematic of the load order described in \cref{sec:loadorder}; solid lines represent
+			mandatory files, dotted lines represent optional files. \texttt{indentconfig.yaml} can
+			contain as many files as you like. The files will be loaded in order; if you specify
+			settings for the same field in more than one file, the most recent takes priority. }
+		\label{fig:loadorder}
+	\end{figure}
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-introduction.tex b/Master/texmf-dist/doc/support/latexindent/sec-introduction.tex
new file mode 100644
index 00000000000..ffe81105a01
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-introduction.tex
@@ -0,0 +1,47 @@
+\section{Introduction}
+\subsection{Thanks}
+	I first created \texttt{latexindent.pl} to help me format chapter files
+	in a big project. After I blogged about it on the
+	\TeX{} stack exchange \cite{cmhblog} I received some positive feedback and
+	follow-up feature requests. A big thank you to Harish Kumar who
+	helped to develop and test the initial versions of the script.
+
+	The \texttt{YAML}-based interface of \texttt{latexindent.pl} was inspired
+	by the wonderful \texttt{arara} tool; any similarities are deliberate, and
+	I hope that it is perceived as the compliment that it is. Thank you to Paulo Cereda and the
+	team for releasing this awesome tool; I initially worried that I was going to
+	have to make a GUI for \texttt{latexindent.pl}, but the release of \texttt{arara}
+	has meant there is no need.
+
+	There have been several contributors to the project so far (and hopefully more in
+	the future!); thank you very much to the people detailed in \vref{sec:contributors}
+	for their valued contributions, and thank you to those who report bugs and request features
+	at \cite{latexindent-home}.
+
+\subsection{License}
+	\texttt{latexindent.pl} is free and open source, and it always will be.
+	Before you start using it on any important files, bear in mind that \texttt{latexindent.pl} has the option to overwrite your \texttt{.tex} files.
+	It will always make at least one backup (you can choose how many it makes, see \cpageref{page:onlyonebackup})
+	but you should still be careful when using it. The script has been tested on many
+	files, but there are some known limitations (see \cref{sec:knownlimitations}).
+	You, the user, are responsible for ensuring that you maintain backups of your files
+	before running \texttt{latexindent.pl} on them. I think it is important at this
+	stage to restate an important part of the license here:
+	\begin{quote}\itshape
+		This program is distributed in the hope that it will be useful,
+		but WITHOUT ANY WARRANTY; without even the implied warranty of
+		MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+		GNU General Public License for more details.
+	\end{quote}
+	There is certainly no malicious intent in releasing this script, and I do hope
+	that it works as you expect it to; if it does not, please first of all
+	make sure that you have the correct settings, and then feel free to let me know at \cite{latexindent-home} with a
+	complete minimum working example as I would like to improve the code as much as possible.
+	\begin{warning}
+		Before you try the script on anything important (like your thesis), test it
+		out on the sample files in the \texttt{test-case} directory \cite{latexindent-home}.
+	\end{warning}
+
+	\emph{If you have used any version 2.* of \texttt{latexindent.pl}, there
+		are a few changes to the interface; see \vref{app:differences} and the comments
+		throughout this document for details}.
diff --git a/Master/texmf-dist/doc/support/latexindent/sec-the-m-switch.tex b/Master/texmf-dist/doc/support/latexindent/sec-the-m-switch.tex
new file mode 100644
index 00000000000..541a66572d2
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/sec-the-m-switch.tex
@@ -0,0 +1,555 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+% the -m switch
+% the -m switch
+% the -m switch
+
+\fancyhead[R]{\bfseries\thepage%
+	\tikz[remember picture,overlay] {
+		\node at (1,0){\includegraphics{logo}};
+	}}
+\section{The \texttt{-m} (\texttt{modifylinebreaks}) switch}\label{sec:modifylinebreaks}
+ All features described in this section will only be relevant if the \texttt{-m} switch
+ is used.
+
+\yamltitle{modifylinebreaks}*{fields}
+	\begin{wrapfigure}[7]{r}[0pt]{8cm}
+		\cmhlistingsfromfile[firstnumber=356,linerange={356-358},style=yaml-LST,numbers=left,]{../defaultSettings.yaml}[MLB-TCB,width=.85\linewidth,before=\centering]{\texttt{modifyLineBreaks}}{lst:modifylinebreaks}
+	\end{wrapfigure}
+	\makebox[0pt][r]{%
+		\raisebox{-\totalheight}[0pt][0pt]{%
+			\tikz\node[opacity=1] at (0,0) {\includegraphics[width=4cm]{logo}};}}%	
+	One of the most exciting features of Version 3.0 is the \texttt{-m} switch, which
+	permits \texttt{latexindent.pl} to modify line breaks, according to the
+	specifications in the \texttt{modifyLineBreaks} field. \emph{The settings
+		in this field will only be considered if the \texttt{-m} switch has been used}.
+	A snippet of the default settings of this field is shown in \cref{lst:modifylinebreaks}.
+
+	Having read the previous paragraph, it should sound reasonable that, if you call \texttt{latexindent.pl}
+	using the \texttt{-m} switch, then you give it permission to modify line breaks in your file,
+	but let's be clear:
+
+	\begin{warning}
+		If you call \texttt{latexindent.pl} with the \texttt{-m} switch, then you
+		are giving it permission to modify line breaks. By default, the only
+		thing that will happen is that multiple blank lines will be condensed into
+		one blank line; many other settings are possible, discussed next.
+	\end{warning}
+
+\yamltitle{preserveBlankLines}{0|1}
+	This field is directly related to \emph{poly-switches}, discussed below.
+	By default, it is set to \texttt{1}, which means that blank lines will
+	be protected from removal; however, regardless of this setting, multiple
+	blank lines can be condensed if \texttt{condenseMultipleBlankLinesInto} is
+	greater than \texttt{0}, discussed next.
+
+\yamltitle{condenseMultipleBlankLinesInto}*{integer $\geq 0$}
+	Assuming that this switch takes an integer value greater than \texttt{0}, \texttt{latexindent.pl} will condense multiple blank lines into
+	the number of blank lines illustrated by this switch. As an example, \cref{lst:mlb-bl} shows a sample file
+	with blank lines; upon running
+	\begin{commandshell}
+latexindent.pl myfile.tex -m  
+\end{commandshell}
+	the output is shown in \cref{lst:mlb-bl-out}; note that the multiple blank lines have been
+	condensed into one blank line, and note also that we have used the \texttt{-m} switch!
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mlb1.tex}{\texttt{mlb1.tex}}{lst:mlb-bl}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mlb1-out.tex}{\texttt{mlb1.tex} out output}{lst:mlb-bl-out}
+	\end{minipage}
+
+\subsection{Poly-switches}
+	Every other field in the \texttt{modifyLineBreaks} field uses poly-switches, and can take
+	one of four integer values\footnote{You might like to associate one of the four circles in the logo with one of the four given values}:
+	\begin{itemize}[font=\bfseries]
+		\item[$-1$] \emph{remove mode}: line breaks before or after the \emph{<part of thing>} can be removed (assuming that \texttt{preserveBlankLines} is set to \texttt{0});
+		\item[0] \emph{off mode}: line breaks will not be modified for the \emph{<part of thing>} under consideration;
+		\item[1] \emph{add mode}: a line break will be added before or after the \emph{<part of thing>} under consideration, assuming that
+		      there is not already a line break before or after the \emph{<part of thing>};
+		\item[2] \emph{comment then add mode}: a comment symbol will be added, followed by a line break before or after the \emph{<part of thing>} under consideration, assuming that
+		      there is not already a comment and line break before or after the \emph{<part of thing>}.
+	\end{itemize}
+	All poly-switches are \emph{off} by default; \texttt{latexindent.pl} searches first of all for per-name settings, and then followed by global per-thing settings.
+
+\subsection{modifyLineBreaks for environments}\label{sec:modifylinebreaks-environments}
+	We start by viewing a snippet of \texttt{defaultSettings.yaml} in \cref{lst:environments-mlb}; note that it contains \emph{global} settings (immediately
+	after the \texttt{environments} field) and that \emph{per-name} settings are also allowed -- in the case of \cref{lst:environments-mlb}, settings
+	for \texttt{equation*} have been specified. Note that all poly-switches are \emph{off} by default.
+
+	\cmhlistingsfromfile[firstnumber=359,linerange={359-368},style=yaml-LST,numbers=left,]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,MLB-TCB]{\texttt{environments}}{lst:environments-mlb}
+
+\subsubsection{Adding line breaks (poly-switches set to $1$ or $2$)}
+	Let's begin with the simple example given in \cref{lst:env-mlb1-tex}; note that we have annotated key parts of the file using $\BeginStartsOnOwnLine$,
+	$\BodyStartsOnOwnLine$, $\EndStartsOnOwnLine$ and $\EndFinishesWithLineBreak$, these will be related to fields specified in \cref{lst:environments-mlb}.
+
+	\begin{cmhlistings}[escapeinside={(*@}{@*)}]{\texttt{env-mlb1.tex}}{lst:env-mlb1-tex}
+before words(*@$\BeginStartsOnOwnLine$@*) \begin{myenv}(*@$\BodyStartsOnOwnLine$@*)body of myenv(*@$\EndStartsOnOwnLine$@*)\end{myenv}(*@$\EndFinishesWithLineBreak$@*) after words
+\end{cmhlistings}
+
+	Let's explore \texttt{BeginStartsOnOwnLine} and \texttt{BodyStartsOnOwnLine} in \cref{lst:env-mlb1,lst:env-mlb2}, and in particular,
+	let's allow each of them in turn to take a value of $1$.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb1.yaml}[MLB-TCB]{\texttt{env-mlb1.yaml}}{lst:env-mlb1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb2.yaml}[MLB-TCB]{\texttt{env-mlb2.yaml}}{lst:env-mlb2}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl -m env-mlb.tex -l env-mlb1.yaml
+latexindent.pl -m env-mlb.tex -l env-mlb2.yaml
+\end{commandshell}
+	the output is as in \cref{lst:env-mlb-mod1,lst:env-mlb-mod2} respectively.
+
+	\begin{widepage}
+		\begin{minipage}{.57\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod1.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb1}}{lst:env-mlb-mod1}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{.42\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod2.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb2}}{lst:env-mlb-mod2}
+		\end{minipage}
+	\end{widepage}
+
+	There are a couple of points to note:
+	\begin{itemize}
+		\item in \cref{lst:env-mlb-mod1} a line break has been added at the point denoted by $\BeginStartsOnOwnLine$ in \cref{lst:env-mlb1-tex}; no
+		      other line breaks have been changed;
+		\item in \cref{lst:env-mlb-mod2} a line break has been added at the point denoted by $\BodyStartsOnOwnLine$ in \cref{lst:env-mlb1-tex};
+		      furthermore, note that the \emph{body} of \texttt{myenv} has received the appropriate (default) indentation.
+	\end{itemize}
+
+	Let's now change each of the \texttt{1} values in \cref{lst:env-mlb1,lst:env-mlb2} so that they are $2$ and
+	save them into \texttt{env-mlb3.yaml} and \texttt{env-mlb4.yaml} respectively (see \cref{lst:env-mlb3,lst:env-mlb4}).
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb3.yaml}[MLB-TCB]{\texttt{env-mlb3.yaml}}{lst:env-mlb3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb4.yaml}[MLB-TCB]{\texttt{env-mlb4.yaml}}{lst:env-mlb4}
+	\end{minipage}
+
+	Upon running  commands analogous to the above, we obtain \cref{lst:env-mlb-mod3,lst:env-mlb-mod4}.
+
+	\begin{widepage}
+		\begin{minipage}{.57\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod3.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb3}}{lst:env-mlb-mod3}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{.42\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod4.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb4}}{lst:env-mlb-mod4}
+		\end{minipage}
+	\end{widepage}
+
+	Note that line breaks have been added as in \cref{lst:env-mlb-mod1,lst:env-mlb-mod2}, but this time a comment symbol
+	has been added before adding the line break; in both cases, trailing horizontal
+	space has been stripped before doing so.
+
+	Let's explore \texttt{EndStartsOnOwnLine} and \texttt{EndFinishesWithLineBreak} in \cref{lst:env-mlb5,lst:env-mlb6},
+	and in particular, let's allow each of them in turn to take a value of $1$.
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb5.yaml}[MLB-TCB]{\texttt{env-mlb5.yaml}}{lst:env-mlb5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb6.yaml}[MLB-TCB]{\texttt{env-mlb6.yaml}}{lst:env-mlb6}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl -m env-mlb.tex -l env-mlb5.yaml
+latexindent.pl -m env-mlb.tex -l env-mlb6.yaml
+\end{commandshell}
+	the output is as in \cref{lst:env-mlb-mod5,lst:env-mlb-mod6}.
+
+	\begin{widepage}
+		\begin{minipage}{.42\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod5.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb5}}{lst:env-mlb-mod5}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{.57\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod6.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb6}}{lst:env-mlb-mod6}
+		\end{minipage}
+	\end{widepage}
+
+	There are a couple of points to note:
+	\begin{itemize}
+		\item in \cref{lst:env-mlb-mod5} a line break has been added at the point denoted by $\EndStartsOnOwnLine$ in \vref{lst:env-mlb1-tex}; no
+		      other line breaks have been changed and the \lstinline!\end{myenv}! statement has \emph{not} received indentation (as intended);
+		\item in \cref{lst:env-mlb-mod6} a line break has been added at the point denoted by $\EndFinishesWithLineBreak$ in \vref{lst:env-mlb1-tex}.
+	\end{itemize}
+
+	Let's now change each of the \texttt{1} values in \cref{lst:env-mlb5,lst:env-mlb6} so that they are $2$ and
+	save them into \texttt{env-mlb7.yaml} and \texttt{env-mlb8.yaml} respectively (see \cref{lst:env-mlb7,lst:env-mlb8}).
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb7.yaml}[MLB-TCB]{\texttt{env-mlb7.yaml}}{lst:env-mlb7}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb8.yaml}[MLB-TCB]{\texttt{env-mlb8.yaml}}{lst:env-mlb8}
+	\end{minipage}
+
+	Upon running  commands analogous to the above, we obtain \cref{lst:env-mlb-mod7,lst:env-mlb-mod8}.
+
+	\begin{widepage}
+		\begin{minipage}{.42\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod7.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb7}}{lst:env-mlb-mod7}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{.57\linewidth}
+			\cmhlistingsfromfile{demonstrations/env-mlb-mod8.tex}{\texttt{env-mlb.tex} using \cref{lst:env-mlb8}}{lst:env-mlb-mod8}
+		\end{minipage}
+	\end{widepage}
+
+	Note that line breaks have been added as in \cref{lst:env-mlb-mod5,lst:env-mlb-mod6}, but this time a comment symbol
+	has been added before adding the line break; in both cases, trailing horizontal
+	space has been stripped before doing so.
+
+	If you ask \texttt{latexindent.pl} to add a line break (possibly with a comment) using a poly-switch value of $1$ (or $2$),
+	it will only do so if necessary. For example, if you process the file in \vref{lst:mlb2} using any of the YAML
+	files presented so far in this section, it will be left unchanged.
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb2.tex}{\texttt{env-mlb2.tex}}{lst:mlb2}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb3.tex}{\texttt{env-mlb3.tex}}{lst:mlb3}
+	\end{minipage}
+
+	In contrast, the output from processing the file in \cref{lst:mlb3} will vary depending
+	on the poly-switches used; in \cref{lst:env-mlb3-mod2} you'll see that the comment symbol after
+	the \lstinline!\begin{myenv}! has been moved to the next line, as \texttt{BodyStartsOnOwnLine}
+	is set to \texttt{1}. In \cref{lst:env-mlb3-mod4} you'll see that the comment has been accounted
+	for correctly because \texttt{BodyStartsOnOwnLine} has been set to \texttt{2},
+	and the comment symbol has \emph{not} been moved to its own line. You're encouraged to experiment
+	with \cref{lst:mlb3} and by setting the other poly-switches considered so far to \texttt{2} in turn.
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb3-mod2.tex}{\texttt{env-mlb3.tex} using \vref{lst:env-mlb2}}{lst:env-mlb3-mod2}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb3-mod4.tex}{\texttt{env-mlb3.tex} using \vref{lst:env-mlb4}}{lst:env-mlb3-mod4}
+	\end{minipage}
+
+	The details of the discussion in this section have concerned \emph{global} poly-switches in the \texttt{environments} field;
+	each switch can also be specified on a \emph{per-name} basis, which would take priority over the global values; with
+	reference to \vref{lst:environments-mlb}, an example is shown for the \texttt{equation*} environment.
+
+\subsubsection{Removing line breaks (poly-switches set to $-1$)}
+	Setting poly-switches to $-1$ tells \texttt{latexindent.pl} to remove line breaks of the \emph{<part of the thing>}, if necessary. We will consider the
+	example code given in \cref{lst:mlb4}, noting in particular the positions of
+	the line break highlighters, $\BeginStartsOnOwnLine$, $\BodyStartsOnOwnLine$, $\EndStartsOnOwnLine$
+	and $\EndFinishesWithLineBreak$, together with the associated YAML files in \crefrange{lst:env-mlb9}{lst:env-mlb12}.
+
+	\begin{minipage}{.45\linewidth}
+		\begin{cmhlistings}[escapeinside={(*@}{@*)}]{\texttt{env-mlb4.tex}}{lst:mlb4}
+before words(*@$\BeginStartsOnOwnLine$@*)
+\begin{myenv}(*@$\BodyStartsOnOwnLine$@*)
+body of myenv(*@$\EndStartsOnOwnLine$@*)
+\end{myenv}(*@$\EndFinishesWithLineBreak$@*)
+after words
+\end{cmhlistings}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.51\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb9.yaml}[MLB-TCB]{\texttt{env-mlb9.yaml}}{lst:env-mlb9}
+
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb10.yaml}[MLB-TCB]{\texttt{env-mlb10.yaml}}{lst:env-mlb10}
+
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb11.yaml}[MLB-TCB]{\texttt{env-mlb11.yaml}}{lst:env-mlb11}
+
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/env-mlb12.yaml}[MLB-TCB]{\texttt{env-mlb12.yaml}}{lst:env-mlb12}
+	\end{minipage}
+
+	After running the commands
+	\begin{commandshell}
+latexindent.pl -m env-mlb4.tex -l env-mlb9.yaml
+latexindent.pl -m env-mlb4.tex -l env-mlb10.yaml
+latexindent.pl -m env-mlb4.tex -l env-mlb11.yaml
+latexindent.pl -m env-mlb4.tex -l env-mlb12.yaml
+\end{commandshell}
+
+	we obtain the respective output in \crefrange{lst:env-mlb4-mod9}{lst:env-mlb4-mod12}.
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb4-mod9.tex}{\texttt{env-mlb4.tex} using \cref{lst:env-mlb9}}{lst:env-mlb4-mod9}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb4-mod10.tex}{\texttt{env-mlb4.tex} using \cref{lst:env-mlb10}}{lst:env-mlb4-mod10}
+	\end{minipage}
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb4-mod11.tex}{\texttt{env-mlb4.tex} using \cref{lst:env-mlb11}}{lst:env-mlb4-mod11}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/env-mlb4-mod12.tex}{\texttt{env-mlb4.tex} using \cref{lst:env-mlb12}}{lst:env-mlb4-mod12}
+	\end{minipage}
+
+	Notice that in
+	\begin{itemize}
+		\item \cref{lst:env-mlb4-mod9} the line break denoted by $\BeginStartsOnOwnLine$ in \cref{lst:mlb4} has been removed;
+		\item \cref{lst:env-mlb4-mod10} the line break denoted by $\BodyStartsOnOwnLine$ in \cref{lst:mlb4} has been removed;
+		\item \cref{lst:env-mlb4-mod11} the line break denoted by $\EndStartsOnOwnLine$ in \cref{lst:mlb4} has been removed;
+		\item \cref{lst:env-mlb4-mod12} the line break denoted by $\EndFinishesWithLineBreak$ in \cref{lst:mlb4} has been removed.
+	\end{itemize}
+	We examined each of these cases separately for clarity of explanation, but you can combine all of the YAML
+	settings in \crefrange{lst:env-mlb9}{lst:env-mlb12} into one file; alternatively, you could tell \texttt{latexindent.pl}
+	to load them all by using the following command, for example
+	\begin{widepage}
+		\begin{commandshell}
+latexindent.pl -m env-mlb4.tex -l env-mlb9.yaml,env-mlb10.yaml,env-mlb11.yaml,env-mlb12.yaml
+\end{commandshell}
+	\end{widepage}
+	which gives the output in \vref{lst:env-mlb1-tex}.
+
+	\paragraph{About trailing horizontal space}
+		Recall that on \cpageref{yaml:removeTrailingWhitespace} we discussed the YAML field \texttt{removeTrailingWhitespace},
+		and that it has two (binary) switches to determine if horizontal space should be removed \texttt{beforeProcessing} and \texttt{afterProcessing}.
+		The \texttt{beforeProcessing} is particularly relevant when considering the \texttt{-m} switch; let's consider the
+		file shown in \cref{lst:mlb5}, which highlights trailing spaces.
+
+		\begin{minipage}{.45\linewidth}
+			\begin{cmhlistings}[showspaces=true,escapeinside={(*@}{@*)}]{\texttt{env-mlb5.tex}}{lst:mlb5}
+before words   (*@$\BeginStartsOnOwnLine$@*) 
+\begin{myenv}           (*@$\BodyStartsOnOwnLine$@*)
+body of myenv      (*@$\EndStartsOnOwnLine$@*) 
+\end{myenv}     (*@$\EndFinishesWithLineBreak$@*)
+after words
+\end{cmhlistings}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{.45\linewidth}
+			\cmhlistingsfromfile[style=yaml-LST]{demonstrations/removeTWS-before.yaml}[yaml-TCB]{\texttt{removeTWS-before.yaml}}{lst:removeTWS-before}
+		\end{minipage}
+
+		The output from the following commands
+		\begin{widepage}
+			\begin{commandshell}
+latexindent.pl -m env-mlb5.tex -l env-mlb9.yaml,env-mlb10.yaml,env-mlb11.yaml,env-mlb12.yaml
+latexindent.pl -m env-mlb5.tex -l env-mlb9.yaml,env-mlb10.yaml,env-mlb11.yaml,env-mlb12.yaml,removeTWS-before.yaml
+\end{commandshell}
+		\end{widepage}
+		is shown, respectively, in \cref{lst:env-mlb5-modAll,lst:env-mlb5-modAll-remove-WS}; note that
+		the trailing horizontal white space has been preserved (by default) in \cref{lst:env-mlb5-modAll}, while
+		in \cref{lst:env-mlb5-modAll-remove-WS}, it has been removed using the switch specified in \cref{lst:removeTWS-before}.
+
+		\begin{widepage}
+			\cmhlistingsfromfile{demonstrations/env-mlb5-modAll.tex}{\texttt{env-mlb5.tex} using \crefrange{lst:env-mlb4-mod9}{lst:env-mlb4-mod12}}{lst:env-mlb5-modAll}
+
+			\cmhlistingsfromfile{demonstrations/env-mlb5-modAll-remove-WS.tex}{\texttt{env-mlb5.tex} using \crefrange{lst:env-mlb4-mod9}{lst:env-mlb4-mod12} \emph{and} \cref{lst:removeTWS-before}}{lst:env-mlb5-modAll-remove-WS}
+		\end{widepage}
+
+
+	\paragraph{Blank lines}
+		Now let's consider the file in \cref{lst:mlb6}, which contains blank lines.
+
+		\begin{minipage}{.45\linewidth}
+			\begin{cmhlistings}[escapeinside={(*@}{@*)}]{\texttt{env-mlb6.tex}}{lst:mlb6}
+before words(*@$\BeginStartsOnOwnLine$@*)
+
+
+\begin{myenv}(*@$\BodyStartsOnOwnLine$@*)
+
+
+body of myenv(*@$\EndStartsOnOwnLine$@*)
+
+
+\end{myenv}(*@$\EndFinishesWithLineBreak$@*)
+
+after words
+\end{cmhlistings}
+		\end{minipage}%
+		\hfill
+		\begin{minipage}{.45\linewidth}
+			\cmhlistingsfromfile[style=yaml-LST]{demonstrations/UnpreserveBlankLines.yaml}[MLB-TCB]{\texttt{UnpreserveBlankLines.yaml}}{lst:UnpreserveBlankLines}
+		\end{minipage}
+
+		Upon running the following commands
+		\begin{widepage}
+			\begin{commandshell}
+latexindent.pl -m env-mlb6.tex -l env-mlb9.yaml,env-mlb10.yaml,env-mlb11.yaml,env-mlb12.yaml
+latexindent.pl -m env-mlb6.tex -l env-mlb9.yaml,env-mlb10.yaml,env-mlb11.yaml,env-mlb12.yaml,UnpreserveBlankLines.yaml
+\end{commandshell}
+		\end{widepage}
+		we receive the respective outputs in \cref{lst:env-mlb6-modAll,lst:env-mlb6-modAll-un-Preserve-Blank-Lines}. In
+		\cref{lst:env-mlb6-modAll} we see that the multiple blank lines have each been condensed into one blank line,
+		but that blank lines have \emph{not} been removed by the poly-switches -- this is because, by default, \texttt{preserveBlankLines}
+		is set to \texttt{1}. By contrast, in \cref{lst:env-mlb6-modAll-un-Preserve-Blank-Lines}, we have allowed
+		the poly-switches to remove blank lines because, in \cref{lst:UnpreserveBlankLines}, we have set \texttt{preserveBlankLines} to \texttt{0}.
+
+		\begin{widepage}
+			\begin{minipage}{.30\linewidth}
+				\cmhlistingsfromfile{demonstrations/env-mlb6-modAll.tex}{\texttt{env-mlb6.tex} using \crefrange{lst:env-mlb4-mod9}{lst:env-mlb4-mod12}}{lst:env-mlb6-modAll}
+			\end{minipage}
+			\hfill
+			\begin{minipage}{.65\linewidth}
+				\cmhlistingsfromfile{demonstrations/env-mlb6-modAll-un-Preserve-Blank-Lines.tex}{\texttt{env-mlb6.tex} using \crefrange{lst:env-mlb4-mod9}{lst:env-mlb4-mod12} \emph{and} \cref{lst:UnpreserveBlankLines}}{lst:env-mlb6-modAll-un-Preserve-Blank-Lines}
+			\end{minipage}
+		\end{widepage}
+
+\subsection{Poly-switches for other code blocks}
+	Rather than repeat the examples shown for the environment code blocks (in \vref{sec:modifylinebreaks-environments}), we choose to detail the poly-switches for
+	all other code blocks in \cref{tab:poly-switch-mapping}; note that each and every one of these poly-switches is \emph{off by default}, i.e, set to \texttt{0}. Note also that,
+	by design, line breaks involving \texttt{verbatim}, \texttt{filecontents} and `comment-marked' code blocks (\vref{lst:alignmentmarkup}) can \emph{not} be
+	modified using \texttt{latexindent.pl}.
+
+	\begin{longtable}{m{.2\textwidth}@{\hspace{.75cm}}m{.35\textwidth}@{}m{.4\textwidth}}
+		\caption{Poly-switch mappings for all code-block types}\label{tab:poly-switch-mapping}\\
+		\toprule
+		Code block & Sample & Poly-switch mapping \\
+		\midrule
+		environment &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+before words(*@$\BeginStartsOnOwnLine$@*)
+\begin{myenv}(*@$\BodyStartsOnOwnLine$@*)
+body of myenv(*@$\EndStartsOnOwnLine$@*)
+\end{myenv}(*@$\EndFinishesWithLineBreak$@*)
+after words
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$     & BeginStartsOnOwnLine     \\
+			$\BodyStartsOnOwnLine$      & BodyStartsOnOwnLine      \\
+			$\EndStartsOnOwnLine$       & EndStartsOnOwnLine       \\
+			$\EndFinishesWithLineBreak$ & EndFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		ifelsefi &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+before words(*@$\BeginStartsOnOwnLine$@*)
+\if...(*@$\BodyStartsOnOwnLine$@*)
+body of if statement(*@$\ElseStartsOnOwnLine$@*)
+\else(*@$\ElseFinishesWithLineBreak$@*)
+body of else statement(*@$\EndStartsOnOwnLine$@*)
+\fi(*@$\EndFinishesWithLineBreak$@*)
+after words
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$      & IfStartsOnOwnLine         \\
+			$\BodyStartsOnOwnLine$       & BodyStartsOnOwnLine       \\
+			$\ElseStartsOnOwnLine$       & ElseStartsOnOwnLine       \\
+			$\ElseFinishesWithLineBreak$ & ElseFinishesWithLineBreak \\
+			$\EndStartsOnOwnLine$        & FiStartsOnOwnLine         \\
+			$\EndFinishesWithLineBreak$  & FiFinishesWithLineBreak   \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		optionalArguments &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+...(*@$\BeginStartsOnOwnLine$@*)
+[(*@$\BodyStartsOnOwnLine$@*)
+body of opt arg(*@$\EndStartsOnOwnLine$@*)
+](*@$\EndFinishesWithLineBreak$@*)
+...
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$     & LSqBStartsOnOwnLine\footnote{LSqB stands for Left Square Bracket} \\
+			$\BodyStartsOnOwnLine$      & OptArgBodyStartsOnOwnLine                                         \\
+			$\EndStartsOnOwnLine$       & RSqBStartsOnOwnLine                                               \\
+			$\EndFinishesWithLineBreak$ & RSqBFinishesWithLineBreak                                         \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		mandatoryArguments &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+...(*@$\BeginStartsOnOwnLine$@*)
+{(*@$\BodyStartsOnOwnLine$@*)
+body of mand arg(*@$\EndStartsOnOwnLine$@*)
+}(*@$\EndFinishesWithLineBreak$@*)
+...
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$     & LCuBStartsOnOwnLine\footnote{LCuB stands for Left Curly Brace} \\
+			$\BodyStartsOnOwnLine$      & MandArgBodyStartsOnOwnLine                                     \\
+			$\EndStartsOnOwnLine$       & RCuBStartsOnOwnLine                                            \\
+			$\EndFinishesWithLineBreak$ & RCuBFinishesWithLineBreak                                      \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		commands &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},morekeywords={mycommand},nolol=true,]
+before words(*@$\BeginStartsOnOwnLine$@*)
+\mycommand(*@$\BodyStartsOnOwnLine$@*)
+(*@$\langle$\itshape{arguments}$\rangle$@*)
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$ & CommandStartsOnOwnLine           \\
+			$\BodyStartsOnOwnLine$  & CommandNameFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		namedGroupingBraces Brackets &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},morekeywords={myname},nolol=true,]
+before words(*@$\BeginStartsOnOwnLine$@*)
+myname(*@$\BodyStartsOnOwnLine$@*)
+(*@$\langle$\itshape{braces/brackets}$\rangle$@*)
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$ & NameStartsOnOwnLine       \\
+			$\BodyStartsOnOwnLine$  & NameFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		keyEqualsValuesBraces\newline Brackets &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},morekeywords={key},nolol=true,]
+before words(*@$\BeginStartsOnOwnLine$@*)
+key(*@$\EqualsStartsOnOwnLine$@*)=(*@$\BodyStartsOnOwnLine$@*)
+(*@$\langle$\itshape{braces/brackets}$\rangle$@*)
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$  & KeyStartsOnOwnLine          \\
+			$\EqualsStartsOnOwnLine$ & EqualsStartsOnOwnLine       \\
+			$\BodyStartsOnOwnLine$   & EqualsFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		items &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+before words(*@$\BeginStartsOnOwnLine$@*)
+\item(*@$\BodyStartsOnOwnLine$@*)
+...
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$ & ItemStartsOnOwnLine       \\
+			$\BodyStartsOnOwnLine$  & ItemFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\cmidrule{2-3}
+		specialBeginEnd &
+		\begin{lstlisting}[escapeinside={(*@}{@*)},nolol=true]
+before words(*@$\BeginStartsOnOwnLine$@*)
+\[(*@$\BodyStartsOnOwnLine$@*)
+body of special(*@$\EndStartsOnOwnLine$@*)
+\](*@$\EndFinishesWithLineBreak$@*)
+after words
+  \end{lstlisting}
+		&
+		\begin{tabular}[t]{c@{~}l@{}}
+			$\BeginStartsOnOwnLine$     & SpecialBeginStartsOnOwnLine     \\
+			$\BodyStartsOnOwnLine$      & SpecialBodyStartsOnOwnLine      \\
+			$\EndStartsOnOwnLine$       & SpecialEndStartsOnOwnLine       \\
+			$\EndFinishesWithLineBreak$ & SpecialEndFinishesWithLineBreak \\
+		\end{tabular}
+		\\
+		\bottomrule
+	\end{longtable}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsec-commands-and-their-options.tex b/Master/texmf-dist/doc/support/latexindent/subsec-commands-and-their-options.tex
new file mode 100644
index 00000000000..8ef571bc574
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsec-commands-and-their-options.tex
@@ -0,0 +1,139 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsection{Commands and the strings between their arguments}\label{subsec:commands-string-between}
+	The \texttt{command} code blocks will always look for optional (square bracketed) and
+	mandatory (curly braced) arguments which can contain comments, line breaks and
+	`beamer' commands \lstinline!<.*?>!  between them. There are switches that can allow them to contain
+	other strings, which we discuss next.
+
+\yamltitle{commandCodeBlocks}*{fields}
+
+	The \texttt{commandCodeBlocks} field contains a few switches detailed in \cref{lst:commandCodeBlocks}.
+
+	\cmhlistingsfromfile[firstnumber=278,linerange={278-286},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{commandCodeBlocks}}{lst:commandCodeBlocks}
+
+\yamltitle{roundParenthesesAllowed}{0|1}
+
+	The need for this field was mostly motivated by commands found in code used to generate images in \texttt{PSTricks} and \texttt{tikz}; for example,
+	let's consider the code given in \cref{lst:pstricks1}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/pstricks1.tex}{\texttt{pstricks1.tex}}{lst:pstricks1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/pstricks1-default.tex}{\texttt{pstricks1} default output}{lst:pstricks1-default}
+	\end{minipage}
+
+	Notice that the \lstinline!\defFunction! command has an optional argument, followed by a
+	mandatory argument, followed by a round-parenthesis argument, $(u,v)$.
+
+	By default, because \texttt{roundParenthesesAllowed} is set to $1$ in \cref{lst:commandCodeBlocks}, then \texttt{latexindent.pl}
+	will allow round parenthesis between optional and mandatory arguments. In the case of the code in \cref{lst:pstricks1},
+	\texttt{latexindent.pl} finds \emph{all} the arguments of \lstinline!defFunction!, both before and after \lstinline!(u,v)!.
+
+	The default output from running \texttt{latexindent.pl} on \cref{lst:pstricks1} actually leaves it unchanged (see \cref{lst:pstricks1-default});
+	note in particular, this is because of \texttt{noAdditionalIndentGlobal} as discussed on \cpageref{page:command:noAddGlobal}.
+
+	Upon using the YAML settings in \cref{lst:noRoundParentheses}, and running the command
+	\begin{commandshell}
+latexindent.pl pstricks1.tex -l noRoundParentheses.yaml
+        \end{commandshell}
+	we obtain the output given in  \cref{lst:pstricks1-nrp}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/pstricks1-nrp.tex}{\texttt{pstricks1.tex} using \cref{lst:noRoundParentheses}}{lst:pstricks1-nrp}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/noRoundParentheses.yaml}[yaml-TCB]{\texttt{noRoundParentheses.yaml}}{lst:noRoundParentheses}
+	\end{minipage}
+
+	Notice the difference between \cref{lst:pstricks1-default} and \cref{lst:pstricks1-nrp}; in particular, in \cref{lst:pstricks1-nrp}, because
+	round parentheses are \emph{not} allowed, \texttt{latexindent.pl} finds that the \lstinline!\defFunction! command finishes at the first opening
+	round parenthesis. As such, the remaining braced, mandatory, arguments are found to be \texttt{UnNamedGroupingBracesBrackets} (see \vref{tab:code-blocks})
+	which, by default, assume indentation for their body, and hence the tabbed indentation in \cref{lst:pstricks1-nrp}.
+
+	Let's explore this using the YAML given in \cref{lst:defFunction} and run the command
+	\begin{commandshell}
+latexindent.pl pstricks1.tex -l defFunction.yaml
+        \end{commandshell}
+	then the output is as in \cref{lst:pstricks1-indent-rules}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/pstricks1-indent-rules.tex}{\texttt{pstricks1.tex} using \cref{lst:defFunction}}{lst:pstricks1-indent-rules}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/defFunction.yaml}[yaml-TCB]{\texttt{defFunction.yaml}}{lst:defFunction}
+	\end{minipage}
+
+	Notice in \cref{lst:pstricks1-indent-rules} that the \emph{body} of the \lstinline!defFunction! command i.e, the subsequent lines
+	containing arguments after the command name, have received the single space of indentation specified by \cref{lst:defFunction}.
+
+\yamltitle{stringsAllowedBetweenArguments}*{fields}
+	\texttt{tikz} users may well specify code such as that given in \cref{lst:tikz-node1}; processing this code using
+	\texttt{latexindent.pl} gives the default output in \cref{lst:tikz-node1-default}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tikz-node1.tex}{\texttt{tikz-node1.tex}}{lst:tikz-node1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[columns=fixed]{demonstrations/tikz-node1-default.tex}{\texttt{tikz-node1} default output}{lst:tikz-node1-default}
+	\end{minipage}
+
+	With reference to \vref{lst:commandCodeBlocks}, we see that the strings
+	\begin{quote}
+		to, node, ++
+	\end{quote}
+	are all allowed to appear between arguments, as they are each set to $1$; importantly, you are encouraged to add further names
+	to this field as necessary. This means that when \texttt{latexindent.pl}
+	processes \cref{lst:tikz-node1}, it consumes:
+	\begin{itemize}
+		\item the optional argument \lstinline![thin]!
+		\item the round-bracketed argument \lstinline!(c)! because \texttt{roundParenthesesAllowed} is $1$ by default
+		\item the string \lstinline!to! (specified in \texttt{stringsAllowedBetweenArguments})
+		\item the optional argument \lstinline![in=110,out=-90]!
+		\item the string \lstinline!++! (specified in \texttt{stringsAllowedBetweenArguments})
+		\item the round-bracketed argument \lstinline!(0,-0.5cm)! because \texttt{roundParenthesesAllowed} is $1$ by default
+		\item the string \lstinline!node! (specified in \texttt{stringsAllowedBetweenArguments})
+		\item the optional argument \lstinline![below,align=left,scale=0.5]!
+	\end{itemize}
+
+	We can explore this further, for example using \cref{lst:draw} and running the command
+	\begin{commandshell}
+latexindent.pl tikz-node1.tex -l draw.yaml  
+\end{commandshell}
+	we receive the output given in \cref{lst:tikz-node1-draw}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/tikz-node1-draw.tex}{\texttt{tikz-node1.tex} using \cref{lst:draw}}{lst:tikz-node1-draw}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/draw.yaml}[yaml-TCB]{\texttt{draw.yaml}}{lst:draw}
+	\end{minipage}
+
+	Notice that each line after the \lstinline!\draw! command (its `body') in \cref{lst:tikz-node1-draw} has been given the
+	appropriate two-spaces worth of indentation specified in \cref{lst:draw}.
+
+	Let's compare this with the output from using the YAML settings in \cref{lst:no-to}, and running the command
+	\begin{commandshell}
+latexindent.pl tikz-node1.tex -l no-to.yaml  
+\end{commandshell}
+	given in \cref{lst:tikz-node1-no-to}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/tikz-node1-no-to.tex}{\texttt{tikz-node1.tex} using \cref{lst:no-to}}{lst:tikz-node1-no-to}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/no-to.yaml}[yaml-TCB]{\texttt{no-to.yaml}}{lst:no-to}
+	\end{minipage}
+
+	In this case, \texttt{latexindent.pl} sees that:
+	\begin{itemize}
+		\item the \lstinline!\draw! command finishes after the \lstinline!(c)! as (\texttt{stringsAllowedBetweenArguments} has \texttt{to} set to $0$)
+		\item it finds a \texttt{namedGroupingBracesBrackets} called \texttt{to} (see \vref{tab:code-blocks}) \emph{with} argument \lstinline![in=110,out=-90]!
+		\item it finds another \texttt{namedGroupingBracesBrackets} but this time called \texttt{node} with argument \lstinline![below,align=left,scale=0.5]!
+	\end{itemize}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsec-conflicting-poly-switches.tex b/Master/texmf-dist/doc/support/latexindent/subsec-conflicting-poly-switches.tex
new file mode 100644
index 00000000000..61216c96c7c
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsec-conflicting-poly-switches.tex
@@ -0,0 +1,137 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsection{Conflicting poly-switches: sequential code blocks}
+	It is very easy to have conflicting poly-switches; if we use the example from \vref{lst:mycommand1},
+	and consider the YAML settings given in \cref{lst:mycom-mlb4}. The output from running
+	\begin{commandshell}
+latexindent.pl -m -l=mycom-mlb4.yaml mycommand1.tex
+\end{commandshell}
+	is given in \cref{lst:mycom-mlb4}.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb4.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb4}}{lst:mycommand1-mlb4}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb4.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb4.yaml}}{lst:mycom-mlb4}
+	\end{minipage}
+
+	Studying \cref{lst:mycom-mlb4}, we see that the two poly-switches are at opposition with one another:
+	\begin{itemize}
+		\item on the one hand, \texttt{LCuBStartsOnOwnLine} should \emph{not} start on its own line (as poly-switch is set to $-1$);
+		\item on the other hand, \texttt{RCuBFinishesWithLineBreak} \emph{should} finish with a line break.
+	\end{itemize}
+	So, which should win the conflict? As demonstrated in \cref{lst:mycommand1-mlb4}, it is clear that \texttt{LCuBStartsOnOwnLine} won
+	this conflict, and the reason is that \emph{the second argument was processed after the first} -- in general, the most recently-processed
+	code block and associated poly-switch takes priority.
+
+	We can explore this further by considering the YAML settings in \cref{lst:mycom-mlb5}; upon running the command
+	\begin{commandshell}
+latexindent.pl -m -l=mycom-mlb5.yaml mycommand1.tex
+\end{commandshell}
+	we obtain the output given in \cref{lst:mycommand1-mlb5}.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb5.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb5}}{lst:mycommand1-mlb5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb5.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb5.yaml}}{lst:mycom-mlb5}
+	\end{minipage}
+
+	As previously, the most-recently-processed code block takes priority -- as before, the second (i.e, \emph{last}) argument. Exploring this
+	further, we consider the YAML settings in \cref{lst:mycom-mlb6}, which give associated output in \cref{lst:mycommand1-mlb6}.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb6.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb6}}{lst:mycommand1-mlb6}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb6.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb6.yaml}}{lst:mycom-mlb6}
+	\end{minipage}
+
+	Note that a \lstinline!%! \emph{has} been added to the trailing first \lstinline!}!; this is because:
+	\begin{itemize}
+		\item while processing the \emph{first} argument, the trailing line break has been removed (\texttt{RCuBFinishesWithLineBreak} set to $-1$);
+		\item while processing the \emph{second} argument, \texttt{latexindent.pl} finds that it does \emph{not} begin on its own line, and so
+		      because \texttt{LCuBStartsOnOwnLine} is set to $2$, it adds a comment, followed by a line break.
+	\end{itemize}
+
+\subsection{Conflicting poly-switches: nested code blocks}
+	Now let's consider an example when nested code blocks have conflicting poly-switches; we'll use the code in \cref{lst:nested-env},
+	noting that it contains nested environments.
+
+	\cmhlistingsfromfile{demonstrations/nested-env.tex}{\texttt{nested-env.tex}}{lst:nested-env}
+
+	Let's use the YAML settings given in \cref{lst:nested-env-mlb1-yaml}, which upon running the command
+	\begin{commandshell}
+latexindent.pl -m -l=nested-env-mlb1.yaml nested-env.tex
+        \end{commandshell}
+	gives the output in \cref{lst:nested-env-mlb1}.
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/nested-env-mlb1.tex}{\texttt{nested-env.tex} using \cref{lst:nested-env-mlb1}}{lst:nested-env-mlb1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/nested-env-mlb1.yaml}[MLB-TCB,width=\linewidth]{\texttt{nested-env-mlb1.yaml}}{lst:nested-env-mlb1-yaml}
+	\end{minipage}
+
+	In \cref{lst:nested-env-mlb1}, let's first of all note that both environments have received the appropriate (default) indentation; secondly,
+	note that the poly-switch \texttt{EndStartsOnOwnLine} appears to have won the conflict, as \lstinline!\end{one}! has had its leading line break removed.
+
+	To understand it, let's talk about the three basic phases of \texttt{latexindent.pl}:
+	\begin{enumerate}
+		\item Phase 1: packing, in which code blocks are replaced with unique ids, working from \emph{the inside to the outside}, and then sequentially -- for example, in
+		      \cref{lst:nested-env}, the \texttt{two} environment is found \emph{before} the \texttt{one} environment; if the -m switch is active, then during this phase:
+		      \begin{itemize}
+			      \item line breaks at the beginning of the \texttt{body} can be added (if \texttt{BodyStartsOnOwnLine} is $1$ or $2$) or removed (if \texttt{BodyStartsOnOwnLine} is $-1$);
+			      \item line breaks at the end of the body can be added (if \texttt{EndStartsOnOwnLine} is $1$ or $2$) or removed (if \texttt{EndStartsOnOwnLine} is $-1$);
+			      \item line breaks after the end statement can be added (if \texttt{EndFinishesWithLineBreak} is $1$ or $2$).
+		      \end{itemize}
+		\item Phase 2: indentation, in which white space is added to the begin, body, and end statements;
+		\item Phase 3: unpacking, in which unique ids are replaced by their \emph{indented} code blocks; if the -m switch is active, then during this phase,
+		      \begin{itemize}
+			      \item line breaks before \texttt{begin} statements can be added or removed (depending upon \texttt{BeginStartsOnOwnLine});
+			      \item line breaks after \emph{end} statements can be removed but \emph{NOT} added (see \texttt{EndFinishesWithLineBreak}).
+		      \end{itemize}
+	\end{enumerate}
+
+	With reference to \cref{lst:nested-env-mlb1}, this means that during Phase 1:
+	\begin{itemize}
+		\item the \texttt{two} environment is found first, and
+		      the line break ahead of the \lstinline!\end{two}! statement is removed because \texttt{EndStartsOnOwnLine} is set to $-1$. Importantly,
+		      because, \emph{at this stage}, \lstinline!\end{two}! \emph{does} finish with a line break, \texttt{EndFinishesWithLineBreak} causes
+		      no action.
+		\item next, the \texttt{one} environment is found; the line break ahead of \lstinline!\end{one}! is removed because \texttt{EndStartsOnOwnLine}
+		      is set to $-1$.
+	\end{itemize}
+	The indentation is done in Phase 2, and then in Phase 3, \emph{there is no option to add a line break after the \lstinline!end! statements}.
+	We can justify this by remembering that during Phase 3, the \texttt{one} environment will be found and processed first, followed
+	by the \texttt{two} environment. If the \texttt{two} environment were to add a line break after the \lstinline!\end{two}! statement, then
+	\texttt{latexindent.pl} would have no way of knowing how much indentation to add to the subsequent text (in this case, \lstinline!\end{one}!).
+
+	We can explore this further using the poly-switches in \cref{lst:nested-env-mlb2}; upon running the command
+	\begin{commandshell}
+latexindent.pl -m -l=nested-env-mlb2.yaml nested-env.tex
+        \end{commandshell}
+	we obtain the output given in \cref{lst:nested-env-mlb2-output}.
+
+	\begin{minipage}{.45\linewidth}
+		\cmhlistingsfromfile{demonstrations/nested-env-mlb2.tex}{\texttt{nested-env.tex} using \cref{lst:nested-env-mlb2}}{lst:nested-env-mlb2-output}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/nested-env-mlb2.yaml}[MLB-TCB,width=\linewidth]{\texttt{nested-env-mlb2.yaml}}{lst:nested-env-mlb2}
+	\end{minipage}
+
+	During Phase 1:
+	\begin{itemize}
+		\item the \texttt{two} environment is found first, and
+		      the line break ahead of the \lstinline!\end{two}! statement is not changed because \texttt{EndStartsOnOwnLine} is set to $1$.
+		      Importantly, because, \emph{at this stage}, \lstinline!\end{two}! \emph{does} finish with a line break, \texttt{EndFinishesWithLineBreak} causes
+		      no action.
+		\item next, the \texttt{one} environment is found; the line break ahead of \lstinline!\end{one}! is already present, and no action is needed.
+	\end{itemize}
+	The indentation is done in Phase 2, and then in Phase 3, the \texttt{one} environment is found and processed first, followed by
+	the \texttt{two} environment. \emph{At this stage}, the \texttt{two} environment finds \texttt{EndFinishesWithLineBreak} is $-1$, so it removes
+	the trailing line break; remember, at this point, \texttt{latexindent.pl} has completely finished with the \texttt{one} environment.
diff --git a/Master/texmf-dist/doc/support/latexindent/subsec-noAdditionalIndent-indentRules.tex b/Master/texmf-dist/doc/support/latexindent/subsec-noAdditionalIndent-indentRules.tex
new file mode 100644
index 00000000000..e0a20542877
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsec-noAdditionalIndent-indentRules.tex
@@ -0,0 +1,24 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsection{\texttt{noAdditionalIndent} and \texttt{indentRules}}\label{sec:noadd-indent-rules}
+	\texttt{latexindent.pl} operates on files by looking for code blocks, as detailed in \vref{subsubsec:code-blocks};
+	for each type of code block  in \vref{tab:code-blocks} (which we will call a \emph{$\langle$thing$\rangle$} in what follows)
+	it searches YAML fields for information in the following order:
+	\begin{enumerate}
+		\item \texttt{noAdditionalIndent} for the \emph{name} of the current \emph{$\langle$thing$\rangle$};
+		\item \texttt{indentRules} for the \emph{name} of the current \emph{$\langle$thing$\rangle$};
+		\item \texttt{noAdditionalIndentGlobal} for the \emph{type} of the current \emph{$\langle$thing$\rangle$};
+		\item \texttt{indentRulesGlobal} for the \emph{type} of the current \emph{$\langle$thing$\rangle$}.
+	\end{enumerate}
+
+	Using the above list, the first piece of information to be found will be used; failing that,
+	the value of \texttt{defaultIndent} is used.
+	If information is found in multiple fields, the first one according to the list above will be used; for example,
+	if information is present in both \texttt{indentRules} and in \texttt{noAdditionalIndentGlobal}, then the information from \texttt{indentRules}
+	takes priority.
+
+	We now present details for the different type of code blocks known to \texttt{latexindent.pl}, as detailed in \vref{tab:code-blocks}; for
+	reference, there follows a list of the code blocks covered.
+
+	\startcontents[noAdditionalIndent]
+	\printcontents[noAdditionalIndent]{}{0}{}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsec-partnering-poly-switches.tex b/Master/texmf-dist/doc/support/latexindent/subsec-partnering-poly-switches.tex
new file mode 100644
index 00000000000..d906fe84e9e
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsec-partnering-poly-switches.tex
@@ -0,0 +1,47 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsection{Partnering \texttt{BodyStartsOnOwnLine} with argument-based poly-switches}
+	Some poly-switches need to be partnered together; in particular, when line breaks involving the \emph{first} argument
+	of a code block need to be accounted for using both \texttt{BodyStartsOnOwnLine} (or its equivalent, see \vref{tab:poly-switch-mapping})
+	and \texttt{LCuBStartsOnOwnLine} for mandatory arguments, and \texttt{LSqBStartsOnOwnLine} for optional arguments.
+
+	Let's begin with the code in \cref{lst:mycommand1} and the YAML settings in \cref{lst:mycom-mlb1}; with reference
+	to \vref{tab:poly-switch-mapping}, the key \texttt{CommandNameFinishesWithLineBreak} is an alias for \texttt{BodyStartsOnOwnLine}.
+
+	\cmhlistingsfromfile{demonstrations/mycommand1.tex}{\texttt{mycommand1.tex}}{lst:mycommand1}
+
+	Upon running the command
+	\begin{commandshell}
+latexindent.pl -m -l=mycom-mlb1.yaml mycommand1.tex
+\end{commandshell}
+	we obtain \cref{lst:mycommand1-mlb1}; note that the \emph{second} mandatory argument beginning brace \lstinline!{! has had
+	its leading line break removed, but that the \emph{first} brace has not.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb1.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb1}}{lst:mycommand1-mlb1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb1.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb1.yaml}}{lst:mycom-mlb1}
+	\end{minipage}
+
+	Now let's change the YAML file so that it is as in \cref{lst:mycom-mlb2}; upon running the analogous command to that given above,
+	we obtain \cref{lst:mycommand1-mlb2}; both beginning braces \lstinline!{! have had their leading line breaks removed.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb2.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb2}}{lst:mycommand1-mlb2}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb2.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb2.yaml}}{lst:mycom-mlb2}
+	\end{minipage}
+
+	Now let's change the YAML file so that it is as in \cref{lst:mycom-mlb3}; upon running the analogous command to that given above,
+	we obtain \cref{lst:mycommand1-mlb3}.
+
+	\begin{minipage}{.4\linewidth}
+		\cmhlistingsfromfile{demonstrations/mycommand1-mlb3.tex}{\texttt{mycommand1.tex} using \cref{lst:mycom-mlb3}}{lst:mycommand1-mlb3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.55\linewidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycom-mlb3.yaml}[MLB-TCB,width=\linewidth]{\texttt{mycom-mlb3.yaml}}{lst:mycom-mlb3}
+	\end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-commands-with-arguments.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-commands-with-arguments.tex
new file mode 100644
index 00000000000..592e7fe4243
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-commands-with-arguments.tex
@@ -0,0 +1,100 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{Commands with arguments}\label{subsubsec:commands-arguments}
+	Let's begin with the simple example in \cref{lst:mycommand}; when \texttt{latexindent.pl} operates
+	on this file, the default output is shown in \cref{lst:mycommand-default}. \footnote{The command code blocks
+		have quite a few subtleties, described in \vref{subsec:commands-string-between}.}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand.tex}{\texttt{mycommand.tex}}{lst:mycommand}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-default.tex}{\texttt{mycommand.tex} default output}{lst:mycommand-default}
+	\end{minipage}
+
+	As in the environment-based case (see \vref{lst:myenv-noAdd1,lst:myenv-noAdd2}) we may specify \texttt{noAdditionalIndent}
+	either in `scalar' form, or in `field' form, as shown in \cref{lst:mycommand-noAdd1,lst:mycommand-noAdd2}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd1.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd1.yaml}}{lst:mycommand-noAdd1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd2.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd2.yaml}}{lst:mycommand-noAdd2}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl mycommand.tex -l mycommand-noAdd1.yaml  
+latexindent.pl mycommand.tex -l mycommand-noAdd2.yaml  
+\end{commandshell}
+	we receive the respective output given in \cref{lst:mycommand-output-noAdd1,lst:mycommand-output-noAdd2}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd1.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd1}}{lst:mycommand-output-noAdd1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd2.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd2}}{lst:mycommand-output-noAdd2}
+	\end{minipage}
+
+	Note that in \cref{lst:mycommand-output-noAdd1} that the `body', optional argument \emph{and} mandatory argument have \emph{all} received
+	no additional indentation, while in \cref{lst:mycommand-output-noAdd2}, only the `body' has not received any additional indentation. We define
+	the `body' of a command as any lines following the command name that include its optional or mandatory arguments.
+
+	We may further customise \texttt{noAdditionalIndent} for \texttt{mycommand} as we did in \vref{lst:myenv-noAdd5,lst:myenv-noAdd6}; explicit examples
+	are given in \cref{lst:mycommand-noAdd3,lst:mycommand-noAdd4}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd3.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd3.yaml}}{lst:mycommand-noAdd3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd4.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd4.yaml}}{lst:mycommand-noAdd4}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl mycommand.tex -l mycommand-noAdd3.yaml  
+latexindent.pl mycommand.tex -l mycommand-noAdd4.yaml  
+\end{commandshell}
+	we receive the respective output given in \cref{lst:mycommand-output-noAdd3,lst:mycommand-output-noAdd4}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd3.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd3}}{lst:mycommand-output-noAdd3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd4.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd4}}{lst:mycommand-output-noAdd4}
+	\end{minipage}
+
+	Attentive readers will note that the body of \texttt{mycommand} in both \cref{lst:mycommand-output-noAdd3,lst:mycommand-output-noAdd4}
+	has received no additional indent, even though \texttt{body} is explicitly set to \texttt{0} in both \cref{lst:mycommand-noAdd3,lst:mycommand-noAdd4}.
+	This is because, by default, \texttt{noAdditionalIndentGlobal} for \texttt{commands} is set to \texttt{1} by default; this can be easily
+	fixed as in \cref{lst:mycommand-noAdd5,lst:mycommand-noAdd6}.\label{page:command:noAddGlobal}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd5.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd5.yaml}}{lst:mycommand-noAdd5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mycommand-noAdd6.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{mycommand-noAdd6.yaml}}{lst:mycommand-noAdd6}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl mycommand.tex -l mycommand-noAdd5.yaml  
+latexindent.pl mycommand.tex -l mycommand-noAdd6.yaml  
+\end{commandshell}
+	we receive the respective output given in \cref{lst:mycommand-output-noAdd5,lst:mycommand-output-noAdd6}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd5.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd5}}{lst:mycommand-output-noAdd5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/mycommand-noAdd6.tex}{\texttt{mycommand.tex} using \cref{lst:mycommand-noAdd6}}{lst:mycommand-output-noAdd6}
+	\end{minipage}
+
+	Both \texttt{indentRules} and \texttt{indentRulesGlobal} can be adjusted as they were for \emph{environment} code blocks, as in
+	\vref{lst:myenv-rules3,lst:myenv-rules4} and \vref{lst:indentRulesGlobal:environments,lst:opt-args-indent-rules-glob,lst:mand-args-indent-rules-glob}.
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-and-their-arguments.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-and-their-arguments.tex
new file mode 100644
index 00000000000..908c2607b78
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-and-their-arguments.tex
@@ -0,0 +1,255 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{Environments and their arguments}\label{subsubsec:env-and-their-args}
+	There are a few different YAML switches governing the indentation of environments; let's start
+	with the code shown in \cref{lst:myenvtex}.
+
+	\cmhlistingsfromfile{demonstrations/myenvironment-simple.tex}{\texttt{myenv.tex}}{lst:myenvtex}
+
+\yamltitle{noAdditionalIndent}*{fields}
+	If we do not wish \texttt{myenv} to receive any additional indentation, we have a few choices available to us,
+	as demonstrated in \cref{lst:myenv-noAdd1,lst:myenv-noAdd2}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd1.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd1.yaml}}{lst:myenv-noAdd1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd2.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd2.yaml}}{lst:myenv-noAdd2}
+	\end{minipage}
+
+	On applying either of the following commands,
+	\begin{commandshell}
+latexindent.pl myenv.tex -l myenv-noAdd1.yaml  
+latexindent.pl myenv.tex -l myenv-noAdd2.yaml  
+\end{commandshell}
+	we obtain the output given in \cref{lst:myenv-output}; note in particular that the environment \texttt{myenv}
+	has not received any \emph{additional} indentation, but that the \texttt{outer} environment \emph{has} still
+	received indentation.
+
+	\cmhlistingsfromfile{demonstrations/myenvironment-simple-noAdd-body1.tex}{\texttt{myenv.tex output (using either \cref{lst:myenv-noAdd1} or \cref{lst:myenv-noAdd2})}}{lst:myenv-output}
+
+	Upon changing the YAML files to those shown in \cref{lst:myenv-noAdd3,lst:myenv-noAdd4}, and running either
+	\begin{commandshell}
+latexindent.pl myenv.tex -l myenv-noAdd3.yaml  
+latexindent.pl myenv.tex -l myenv-noAdd4.yaml  
+\end{commandshell}
+	we obtain the output given in \cref{lst:myenv-output-4}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd3.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd3.yaml}}{lst:myenv-noAdd3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd4.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd4.yaml}}{lst:myenv-noAdd4}
+	\end{minipage}
+
+	\cmhlistingsfromfile{demonstrations/myenvironment-simple-noAdd-body4.tex}{\texttt{myenv.tex output} (using either \cref{lst:myenv-noAdd3} or \cref{lst:myenv-noAdd4})}{lst:myenv-output-4}
+
+	Let's now allow \texttt{myenv} to have some optional and mandatory arguments, as in \cref{lst:myenv-args}.
+	\cmhlistingsfromfile{demonstrations/myenvironment-args.tex}{\texttt{myenv-args.tex}}{lst:myenv-args}
+	Upon running
+	\begin{commandshell}
+latexindent.pl -l=myenv-noAdd1.yaml myenv-args.tex  
+\end{commandshell}
+	we obtain the output shown in \cref{lst:myenv-args-noAdd1}; note that the optional argument, mandatory argument and body \emph{all}
+	have received no additional indent. This is because, when \texttt{noAdditionalIndent} is specified in `scalar' form (as in \cref{lst:myenv-noAdd1}),
+	then \emph{all} parts of the environment (body, optional and mandatory arguments) are assumed to want no additional indent.
+	\cmhlistingsfromfile{demonstrations/myenvironment-args-noAdd-body1.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-noAdd1}}{lst:myenv-args-noAdd1}
+
+	We may customise \texttt{noAdditionalIndent} for optional and mandatory arguments of the \texttt{myenv} environment, as shown in, for example, \cref{lst:myenv-noAdd5,lst:myenv-noAdd6}.
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd5.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd5.yaml}}{lst:myenv-noAdd5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-noAdd6.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-noAdd6.yaml}}{lst:myenv-noAdd6}
+	\end{minipage}
+
+	Upon running
+	\begin{commandshell}
+latexindent.pl myenv.tex -l myenv-noAdd5.yaml  
+latexindent.pl myenv.tex -l myenv-noAdd6.yaml  
+\end{commandshell}
+	we obtain the respective outputs given in \cref{lst:myenv-args-noAdd5,lst:myenv-args-noAdd6}. Note that in \cref{lst:myenv-args-noAdd5}
+	the text for the \emph{optional} argument has not received any additional indentation, and that in \cref{lst:myenv-args-noAdd6} the
+	\emph{mandatory} argument has not received any additional indentation; in both cases, the \emph{body} has not received any additional indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-noAdd5.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-noAdd5}}{lst:myenv-args-noAdd5}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-noAdd6.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-noAdd6}}{lst:myenv-args-noAdd6}
+	\end{minipage}
+
+\yamltitle{indentRules}*{fields}
+	We may also specify indentation rules for environment code blocks using the \texttt{indentRules} field; see, for example,
+	\cref{lst:myenv-rules1,lst:myenv-rules2}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-rules1.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-rules1.yaml}}{lst:myenv-rules1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-rules2.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-rules2.yaml}}{lst:myenv-rules2}
+	\end{minipage}
+
+	On applying either of the following commands,
+	\begin{commandshell}
+latexindent.pl myenv.tex -l myenv-rules1.yaml  
+latexindent.pl myenv.tex -l myenv-rules2.yaml  
+\end{commandshell}
+	we obtain the output given in \cref{lst:myenv-rules-output}; note in particular that the environment \texttt{myenv}
+	has received one tab (from the \texttt{outer} environment) plus three spaces from \cref{lst:myenv-rules1} or \ref{lst:myenv-rules2}.
+
+	\cmhlistingsfromfile{demonstrations/myenv-rules1.tex}{\texttt{myenv.tex output (using either \cref{lst:myenv-rules1} or \cref{lst:myenv-rules2})}}{lst:myenv-rules-output}
+
+	If you specify a field in \texttt{indentRules} using anything other than horizontal space, it will be ignored.
+
+	Returning to the example in \cref{lst:myenv-args} that contains optional and mandatory arguments. Upon using \cref{lst:myenv-rules1} as in
+	\begin{commandshell}
+latexindent.pl myenv-args.tex -l=myenv-rules1.yaml  
+\end{commandshell}
+	we obtain the output in \cref{lst:myenv-args-rules1}; note that the body, optional argument and mandatory argument have \emph{all}
+	received the same customised indentation.
+	\cmhlistingsfromfile{demonstrations/myenvironment-args-rules1.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-rules1}}{lst:myenv-args-rules1}
+
+	You can specify different indentation rules for the different features using, for example, \cref{lst:myenv-rules3,lst:myenv-rules4}
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-rules3.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-rules3.yaml}}{lst:myenv-rules3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/myenv-rules4.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{myenv-rules4.yaml}}{lst:myenv-rules4}
+	\end{minipage}
+
+	After running
+	\begin{commandshell}
+latexindent.pl myenv-args.tex -l myenv-rules3.yaml  
+latexindent.pl myenv-args.tex -l myenv-rules4.yaml  
+\end{commandshell}
+	then we obtain the respective outputs given in \cref{lst:myenv-args-rules3,lst:myenv-args-rules4}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules3.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-rules3}}{lst:myenv-args-rules3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules4.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-rules4}}{lst:myenv-args-rules4}
+	\end{minipage}
+
+	Note that in \cref{lst:myenv-args-rules3}, the optional argument has only received a single space of indentation, while the mandatory argument
+	has received the default (tab) indentation; the environment body has received three spaces of indentation.
+
+	In \cref{lst:myenv-args-rules4}, the optional argument has received the default (tab) indentation, the mandatory argument has received two tabs
+	of indentation, and the body has received three spaces of indentation.
+
+\yamltitle{noAdditionalIndentGlobal}*{fields}
+	\begin{wrapfigure}[6]{r}[0pt]{7cm}
+		\cmhlistingsfromfile[firstnumber=247,linerange={247-248},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{env-noAdditionalGlobal.yaml}}{lst:noAdditionalIndentGlobal:environments}
+	\end{wrapfigure}
+	Assuming that your environment name is not found within neither \texttt{noAdditionalIndent} nor \texttt{indentRules}, the next
+	place that \texttt{latexindent.pl} will look is \texttt{noAdditionalIndentGlobal}, and in particular \emph{for the environments} key
+	(see \cref{lst:noAdditionalIndentGlobal:environments}). Let's say that you change
+	the value of \texttt{environments} to \texttt{1} in \cref{lst:noAdditionalIndentGlobal:environments}, and that you run
+
+	\begin{widepage}
+		\begin{commandshell}
+latexindent.pl myenv-args.tex -l env-noAdditionalGlobal.yaml
+latexindent.pl myenv-args.tex -l myenv-rules1.yaml,env-noAdditionalGlobal.yaml
+\end{commandshell}
+	\end{widepage}
+
+	The respective output from these two commands are in \cref{lst:myenv-args-no-add-global1,lst:myenv-args-no-add-global2}; in \cref{lst:myenv-args-no-add-global1} notice that \emph{both}
+	environments receive no additional indentation but that the arguments of \texttt{myenv} still \emph{do} receive indentation. In \cref{lst:myenv-args-no-add-global2}
+	notice that the \emph{outer} environment does not receive additional indentation, but because of the settings from \texttt{myenv-rules1.yaml} (in \vref{lst:myenv-rules1}), the \texttt{myenv}
+	environment still \emph{does} receive indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules1-noAddGlobal1.tex}{\texttt{myenv-args.tex} using \cref{lst:noAdditionalIndentGlobal:environments}}{lst:myenv-args-no-add-global1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules1-noAddGlobal2.tex}{\texttt{myenv-args.tex} using \cref{lst:noAdditionalIndentGlobal:environments,lst:myenv-rules1}}{lst:myenv-args-no-add-global2}
+	\end{minipage}
+
+	In fact, \texttt{noAdditionalIndentGlobal} also contains keys that control the indentation of optional and mandatory
+	arguments; on referencing \cref{lst:opt-args-no-add-glob,lst:mand-args-no-add-glob}
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/opt-args-no-add-glob.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{opt-args-no-add-glob.yaml}}{lst:opt-args-no-add-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mand-args-no-add-glob.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{mand-args-no-add-glob.yaml}}{lst:mand-args-no-add-glob}
+	\end{minipage}
+
+	we may run the commands
+	\begin{commandshell}
+latexindent.pl  myenv-args.tex -local opt-args-no-add-glob.yaml
+latexindent.pl  myenv-args.tex -local mand-args-no-add-glob.yaml
+\end{commandshell}
+	which produces the respective outputs given in \cref{lst:myenv-args-no-add-opt,lst:myenv-args-no-add-mand}. Notice that in \cref{lst:myenv-args-no-add-opt}
+	the \emph{optional} argument has not received any additional indentation, and in \cref{lst:myenv-args-no-add-mand} the \emph{mandatory} argument
+	has not received any additional indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules1-noAddGlobal3.tex}{\texttt{myenv-args.tex} using \cref{lst:opt-args-no-add-glob}}{lst:myenv-args-no-add-opt}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-rules1-noAddGlobal4.tex}{\texttt{myenv-args.tex} using \cref{lst:mand-args-no-add-glob}}{lst:myenv-args-no-add-mand}
+	\end{minipage}
+
+\yamltitle{indentRulesGlobal}*{fields}
+	\begin{wrapfigure}[4]{r}[0pt]{7cm}
+		\cmhlistingsfromfile[firstnumber=263,linerange={263-264},style=yaml-LST]{../defaultSettings.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{env-indentRulesGlobal.yaml}}{lst:indentRulesGlobal:environments}
+	\end{wrapfigure}
+	The final check that \texttt{latexindent.pl} will make is to look for \texttt{indentRulesGlobal} as detailed in \cref{lst:indentRulesGlobal:environments}; if you change the \texttt{environments}
+	field to anything involving horizontal space, say \lstinline!" "!, and then run the following commands
+
+	\begin{commandshell}
+latexindent.pl  myenv-args.tex -l env-indentRules.yaml
+latexindent.pl  myenv-args.tex -l myenv-rules1.yaml,env-indentRules.yaml
+\end{commandshell}
+	then the respective output is shown in \cref{lst:myenv-args-indent-rules-global1,lst:myenv-args-indent-rules-global2}. Note that
+	in \cref{lst:myenv-args-indent-rules-global1}, both the environment blocks have received a single-space indentation, whereas in
+	\cref{lst:myenv-args-indent-rules-global2} the \texttt{outer} environment has received single-space indentation (specified by \texttt{indentRulesGlobal}),
+	but \texttt{myenv} has received \lstinline!"   "!, as specified by the particular \texttt{indentRules} for \texttt{myenv} \vref{lst:myenv-rules1}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-global-rules1.tex}{\texttt{myenv-args.tex} using \cref{lst:indentRulesGlobal:environments}}{lst:myenv-args-indent-rules-global1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-global-rules2.tex}{\texttt{myenv-args.tex} using \cref{lst:myenv-rules1,lst:indentRulesGlobal:environments}}{lst:myenv-args-indent-rules-global2}
+	\end{minipage}
+
+	You can specify \texttt{indentRulesGlobal} for both optional and mandatory arguments, as detailed in \cref{lst:opt-args-indent-rules-glob,lst:mand-args-indent-rules-glob}
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/opt-args-indent-rules-glob.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{opt-args-indent-rules-glob.yaml}}{lst:opt-args-indent-rules-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/mand-args-indent-rules-glob.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{mand-args-indent-rules-glob.yaml}}{lst:mand-args-indent-rules-glob}
+	\end{minipage}
+
+	Upon running the following commands
+	\begin{commandshell}
+latexindent.pl  myenv-args.tex -local opt-args-indent-rules-glob.yaml
+latexindent.pl  myenv-args.tex -local mand-args-indent-rules-glob.yaml
+\end{commandshell}
+	we obtain the respective outputs in \cref{lst:myenv-args-indent-rules-global3,lst:myenv-args-indent-rules-global4}. Note that the \emph{optional}
+	argument in \cref{lst:myenv-args-indent-rules-global3} has received two tabs worth of indentation, while the \emph{mandatory} argument has
+	done so in \cref{lst:myenv-args-indent-rules-global4}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-global-rules3.tex}{\texttt{myenv-args.tex} using \cref{lst:opt-args-indent-rules-glob}}{lst:myenv-args-indent-rules-global3}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/myenvironment-args-global-rules4.tex}{\texttt{myenv-args.tex} using \cref{lst:mand-args-indent-rules-glob}}{lst:myenv-args-indent-rules-global4}
+	\end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-with-items.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-with-items.tex
new file mode 100644
index 00000000000..82bcc9044f2
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-environments-with-items.tex
@@ -0,0 +1,59 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{Environments with items}
+	With reference to \vref{lst:indentafteritems,lst:itemNames}, some commands
+	may contain \texttt{item} commands; for the purposes of this discussion,
+	we will use the code from \vref{lst:itemsbefore}.
+
+	Assuming that you've populated \texttt{itemNames} with the name of your
+	\texttt{item}, you can put the item name into \texttt{noAdditionalIndent}
+	as in \cref{lst:item-noAdd1}, although a more efficient approach may be
+	to change the relevant field in \texttt{itemNames} to \texttt{0}. Similarly,
+	you can customise the indentation that your \texttt{item} receives using
+	\texttt{indentRules}, as in \cref{lst:item-rules1}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/item-noAdd1.yaml}[yaml-TCB]{\texttt{item-noAdd1.yaml}}{lst:item-noAdd1}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/item-rules1.yaml}[yaml-TCB]{\texttt{item-rules1.yaml}}{lst:item-rules1}
+	\end{minipage}
+
+	Upon running the following commands
+	\begin{commandshell}
+latexindent.pl items1.tex -local item-noAdd1.yaml  
+latexindent.pl items1.tex -local item-rules1.yaml  
+\end{commandshell}
+	the respective outputs are given in \cref{lst:items1-noAdd1,lst:items1-rules1}; note that in \cref{lst:items1-noAdd1}
+	that the text after each \texttt{item} has not received any additional indentation, and in \cref{lst:items1-rules1},
+	the text after each \texttt{item} has received a single space of indentation, specified by \cref{lst:item-rules1}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/items1-noAdd1.tex}{\texttt{items1.tex} using \cref{lst:item-noAdd1}}{lst:items1-noAdd1}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/items1-rules1.tex}{\texttt{items1.tex} using \cref{lst:item-rules1}}{lst:items1-rules1}
+	\end{minipage}
+
+	Alternatively, you might like to populate \texttt{noAdditionalIndentGlobal} or \texttt{indentRulesGlobal} using the \texttt{items}
+	key, as demonstrated in \cref{lst:items-noAdditionalGlobal,lst:items-indentRulesGlobal}. Note that there is a need to
+	`reset/remove' the \texttt{item} field from \texttt{indentRules} in both cases (see the hierarchy description given on \cpageref{sec:noadd-indent-rules})
+	as the \texttt{item} command is a member of \texttt{indentRules} by default.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/items-noAdditionalGlobal.yaml}[yaml-TCB]{\texttt{items-noAdditionalGlobal.yaml}}{lst:items-noAdditionalGlobal}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/items-indentRulesGlobal.yaml}[yaml-TCB]{\texttt{items-indentRulesGlobal.yaml}}{lst:items-indentRulesGlobal}
+	\end{minipage}
+
+	Upon running the following commands,
+	\begin{commandshell}
+latexindent.pl items1.tex -local items-noAdditionalGlobal.yaml
+latexindent.pl items1.tex -local items-indentRulesGlobal.yaml
+\end{commandshell}
+	the respective outputs from \cref{lst:items1-noAdd1,lst:items1-rules1} are obtained; note, however, that
+	\emph{all} such \texttt{item} commands without their own individual \texttt{noAdditionalIndent} or \texttt{indentRules}
+	settings would behave as in these listings.
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-headings.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-headings.tex
new file mode 100644
index 00000000000..7c062666978
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-headings.tex
@@ -0,0 +1,93 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{\texttt{afterHeading} code blocks}\label{subsubsec-headings-no-add-indent-rules}
+	Let's use the example \cref{lst:headings2} for demonstration throughout this \namecref{subsubsec-headings-no-add-indent-rules}.
+	As discussed on \cpageref{lst:headings1}, by default \texttt{latexindent.pl} will not add indentation after headings.
+
+	\cmhlistingsfromfile{demonstrations/headings2.tex}{\texttt{headings2.tex}}{lst:headings2}
+
+	On using the YAML file in \cref{lst:headings3yaml} by running the command
+	\begin{commandshell}
+latexindent.pl headings2.tex -l headings3.yaml      
+    \end{commandshell}
+	we obtain the output in \cref{lst:headings2-mod3}. Note that the argument of \texttt{paragraph} has received (default) indentation,
+	and that the body after the heading statement has received (default) indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod3.tex}{\texttt{headings2.tex} using \cref{lst:headings3yaml}}{lst:headings2-mod3}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings3.yaml}[yaml-TCB]{\texttt{headings3.yaml}}{lst:headings3yaml}
+	\end{minipage}
+
+	If we specify \texttt{noAdditionalIndent} as in \cref{lst:headings4yaml} and run the command
+	\begin{commandshell}
+latexindent.pl headings2.tex -l headings4.yaml      
+    \end{commandshell}
+	then we receive the output in \cref{lst:headings2-mod4}. Note that the arguments \emph{and} the body after the heading
+	of \texttt{paragraph} has received no additional indentation, because we have specified \texttt{noAdditionalIndent} in scalar form.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod4.tex}{\texttt{headings2.tex} using \cref{lst:headings4yaml}}{lst:headings2-mod4}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings4.yaml}[yaml-TCB]{\texttt{headings4.yaml}}{lst:headings4yaml}
+	\end{minipage}
+
+	Similarly, if we specify \texttt{indentRules} as in \cref{lst:headings5yaml} and run analogous commands to those above,
+	we receive the output in \cref{lst:headings2-mod5}; note that the \emph{body}, \emph{mandatory argument} and content
+	\emph{after the heading} of \texttt{paragraph} have \emph{all} received three tabs worth of indentation.
+
+	\begin{minipage}{.55\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod5.tex}{\texttt{headings2.tex} using \cref{lst:headings5yaml}}{lst:headings2-mod5}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.42\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings5.yaml}[yaml-TCB]{\texttt{headings5.yaml}}{lst:headings5yaml}
+	\end{minipage}
+
+	We may, instead, specify \texttt{noAdditionalIndent} in `field' form, as in \cref{lst:headings6yaml} which gives the output in \cref{lst:headings2-mod6}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod6.tex}{\texttt{headings2.tex} using \cref{lst:headings6yaml}}{lst:headings2-mod6}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings6.yaml}[yaml-TCB]{\texttt{headings6.yaml}}{lst:headings6yaml}
+	\end{minipage}
+
+	Analogously, we may specify \texttt{indentRules} as in \cref{lst:headings7yaml} which gives the output in \cref{lst:headings2-mod7};
+	note that mandatory argument text has only received a single space of indentation, while the body after the heading has
+	received three tabs worth of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod7.tex}{\texttt{headings2.tex} using \cref{lst:headings7yaml}}{lst:headings2-mod7}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings7.yaml}[yaml-TCB]{\texttt{headings7.yaml}}{lst:headings7yaml}
+	\end{minipage}
+
+	Finally, let's consider \texttt{noAdditionalIndentGlobal} and \texttt{indentRulesGlobal} shown in \cref{lst:headings8yaml,lst:headings9yaml}
+	respectively, with respective output in \cref{lst:headings2-mod8,lst:headings2-mod9}. Note that in \cref{lst:headings8yaml} the
+	\emph{mandatory argument} of \texttt{paragraph} has received a (default) tab's worth of indentation, while the body after the
+	heading has received \emph{no additional indentation}. Similarly, in \cref{lst:headings2-mod9}, the \emph{argument} has received both a
+	(default) tab plus two spaces of indentation (from the global rule specified in \cref{lst:headings9yaml}), and the remaining body
+	after \texttt{paragraph} has received just two spaces of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod8.tex}{\texttt{headings2.tex} using \cref{lst:headings8yaml}}{lst:headings2-mod8}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings8.yaml}[yaml-TCB]{\texttt{headings8.yaml}}{lst:headings8yaml}
+	\end{minipage}
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/headings2-mod9.tex}{\texttt{headings2.tex} using \cref{lst:headings9yaml}}{lst:headings2-mod9}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/headings9.yaml}[yaml-TCB]{\texttt{headings9.yaml}}{lst:headings9yaml}
+	\end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-ifelsefi.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-ifelsefi.tex
new file mode 100644
index 00000000000..5c9a37dbb22
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-ifelsefi.tex
@@ -0,0 +1,70 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{ifelsefi code blocks}
+	Let's use the simple example shown in \cref{lst:ifelsefi1}; when
+	\texttt{latexindent.pl} operates on this file, the output as in \cref{lst:ifelsefi1-default};
+	note that the body of each of the \lstinline!\if! statements have been indented,
+	and that the \lstinline!\else! statement has been accounted for correctly.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1.tex}{\texttt{ifelsefi1.tex}}{lst:ifelsefi1}
+	\end{minipage}%
+	\hfill
+	\begin{minipage}{.54\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1-default.tex}{\texttt{ifelsefi1.tex} default output}{lst:ifelsefi1-default}
+	\end{minipage}
+
+	It is recommended to specify \texttt{noAdditionalIndent} and \texttt{indentRules} in the `scalar' form only
+	for these type of code blocks, although the `field' form would work, assuming that \texttt{body} was specified.
+	Examples are shown in \cref{lst:ifnum-noAdd,lst:ifnum-indent-rules}.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/ifnum-noAdd.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{ifnum-noAdd.yaml}}{lst:ifnum-noAdd}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/ifnum-indent-rules.yaml}[width=.8\linewidth,before=\centering,yaml-TCB]{\texttt{ifnum-indent-rules.yaml}}{lst:ifnum-indent-rules}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl ifelsefi1.tex -local ifnum-noAdd.yaml  
+latexindent.pl ifelsefi1.tex -l ifnum-indent-rules.yaml  
+\end{commandshell}
+	we receive the respective output given in \cref{lst:ifelsefi1-output-noAdd,lst:ifelsefi1-output-indent-rules}; note that
+	in \cref{lst:ifelsefi1-output-noAdd}, the \texttt{ifnum} code block has \emph{not} received any additional indentation,
+	while in \cref{lst:ifelsefi1-output-indent-rules}, the \texttt{ifnum} code block has received one tab and two spaces of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1-noAdd.tex}{\texttt{ifelsefi1.tex} using \cref{lst:ifnum-noAdd}}{lst:ifelsefi1-output-noAdd}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.5\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1-indent-rules.tex}{\texttt{ifelsefi1.tex} using \cref{lst:ifnum-indent-rules}}{lst:ifelsefi1-output-indent-rules}
+	\end{minipage}
+
+	We may specify \texttt{noAdditionalIndentGlobal} and \texttt{indentRulesGlobal} as in \cref{lst:ifelsefi-noAdd-glob,lst:ifelsefi-indent-rules-global}.
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/ifelsefi-noAdd-glob.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{ifelsefi-noAdd-glob.yaml}}{lst:ifelsefi-noAdd-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/ifelsefi-indent-rules-global.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{ifelsefi-indent-rules-global.yaml}}{lst:ifelsefi-indent-rules-global}
+	\end{minipage}
+
+	Upon running the following commands
+	\begin{commandshell}
+latexindent.pl ifelsefi1.tex -local ifelsefi-noAdd-glob.yaml  
+latexindent.pl ifelsefi1.tex -l ifelsefi-indent-rules-global.yaml  
+\end{commandshell}
+	we receive the outputs in \cref{lst:ifelsefi1-output-noAdd-glob,lst:ifelsefi1-output-indent-rules-global}; notice that  in
+	\cref{lst:ifelsefi1-output-noAdd-glob} neither of the \texttt{ifelsefi} code blocks have received indentation, while in
+	\cref{lst:ifelsefi1-output-indent-rules-global} both code blocks have received a single space of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1-noAdd-glob.tex}{\texttt{ifelsefi1.tex} using \cref{lst:ifelsefi-noAdd-glob}}{lst:ifelsefi1-output-noAdd-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/ifelsefi1-indent-rules-global.tex}{\texttt{ifelsefi1.tex} using \cref{lst:ifelsefi-indent-rules-global}}{lst:ifelsefi1-output-indent-rules-global}
+	\end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-no-add-remaining-code-blocks.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-no-add-remaining-code-blocks.tex
new file mode 100644
index 00000000000..60063fbb2b7
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-no-add-remaining-code-blocks.tex
@@ -0,0 +1,109 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{The remaining code blocks}
+	Referencing the different types of code blocks in \vref{tab:code-blocks}, we have a few
+	code blocks yet to cover; these are very similar to the \texttt{commands} code block type
+	covered comprehensively in \vref{subsubsec:commands-arguments}, but a small discussion
+	defining these remaining code blocks is necessary.
+
+	\paragraph{\texttt{keyEqualsValuesBracesBrackets}}
+		\texttt{latexindent.pl} defines this type of code block by the following criteria:
+		\begin{itemize}
+			\item it must immediately follow either \lstinline!{! OR \lstinline![! OR \lstinline!,! with comments
+			      and blank lines allowed;
+			\item then it has a name made up of the characters detailed in \vref{tab:code-blocks};
+			\item then an $=$ symbol;
+			\item then at least one set of curly braces or square brackets (comments and line breaks allowed throughout).
+		\end{itemize}
+
+		An example is shown in \cref{lst:pgfkeysbefore}, with the default output given in \cref{lst:pgfkeys1:default}.
+
+		\begin{minipage}{.45\textwidth}
+			\cmhlistingsfromfile{demonstrations/pgfkeys1.tex}{\texttt{pgfkeys1.tex}}{lst:pgfkeysbefore}
+		\end{minipage}%
+		\hfill
+		\begin{minipage}{.5\textwidth}
+			\cmhlistingsfromfile{demonstrations/pgfkeys1-default.tex}{\texttt{pgfkeys1.tex} default output}{lst:pgfkeys1:default}
+		\end{minipage}%
+
+		In \cref{lst:pgfkeys1:default}, note that the maximum indentation is three tabs, and these come from:
+		\begin{itemize}
+			\item the \lstinline!\pgfkeys! command's mandatory argument;
+			\item the \lstinline!start coordinate/.initial! key's mandatory argument;
+			\item the \lstinline!start coordinate/.initial! key's body, which is defined as any lines following the name of the
+			      key that include its arguments.  This is the part controlled by the \emph{body} field for \texttt{noAdditionalIndent}
+			      and friends from \cpageref{sec:noadd-indent-rules}.
+		\end{itemize}
+	\paragraph{\texttt{namedGroupingBracesBrackets}}
+		This type of code block is mostly motivated by tikz-based code; we define this code block as follows:
+		\begin{itemize}
+			\item it must immediately follow either \emph{horizontal space} OR \emph{one or more line breaks} OR \lstinline!{! OR \lstinline![!
+			      OR \lstinline!$!;
+			\item the name may contain the characters detailed in \vref{tab:code-blocks};
+			\item then at least one set of curly braces or square brackets (comments and line breaks allowed throughout).
+		\end{itemize}
+		A simple example is given in \cref{lst:child1}, with default output in \cref{lst:child1:default}.
+
+		\begin{minipage}{.45\textwidth}
+			\cmhlistingsfromfile{demonstrations/child1.tex}{\texttt{child1.tex}}{lst:child1}
+		\end{minipage}%
+		\hfill
+		\begin{minipage}{.5\textwidth}
+			\cmhlistingsfromfile{demonstrations/child1-default.tex}{\texttt{child1.tex} default output}{lst:child1:default}
+		\end{minipage}%
+
+		In particular, \texttt{latexindent.pl} considers \texttt{child}, \texttt{parent} and \texttt{node} all to be \texttt{namedGroupingBracesBrackets}\footnote{
+			You may like to verify this by using the \texttt{-tt} option and checking \texttt{indent.log}! }.
+		Referencing \cref{lst:child1:default},
+		note that the maximum indentation is two tabs, and these come from:
+		\begin{itemize}
+			\item the \lstinline!child!'s mandatory argument;
+			\item the \lstinline!child!'s body, which is defined as any lines following the name of the \texttt{namedGroupingBracesBrackets}
+			      that include its arguments. This is the part controlled by the \emph{body} field for \texttt{noAdditionalIndent}
+			      and friends from \cpageref{sec:noadd-indent-rules}.
+		\end{itemize}
+
+	\paragraph{\texttt{UnNamedGroupingBracesBrackets}} occur in a variety of situations; specifically, we define
+		this type of code block as satisfying the following criteria:
+		\begin{itemize}
+			\item it must immediately follow either \lstinline!{! OR \lstinline![! OR \lstinline!,! OR \lstinline!&! OR \lstinline!)! OR \lstinline!(!
+			      OR \lstinline!$!;
+			\item then at least one set of curly braces or square brackets (comments and line breaks allowed throughout).
+		\end{itemize}
+
+		An example is shown in \cref{lst:psforeach1} with default output give in \cref{lst:psforeach:default}.
+
+		\begin{minipage}{.45\textwidth}
+			\cmhlistingsfromfile{demonstrations/psforeach1.tex}{\texttt{psforeach1.tex}}{lst:psforeach1}
+		\end{minipage}%
+		\hfill
+		\begin{minipage}{.5\textwidth}
+			\cmhlistingsfromfile{demonstrations/psforeach1-default.tex}{\texttt{psforeach1.tex} default output}{lst:psforeach:default}
+		\end{minipage}%
+
+		Referencing \cref{lst:psforeach:default}, there are \emph{three} sets of unnamed braces. Note also that the maximum value
+		of indentation is three tabs, and these come from:
+		\begin{itemize}
+			\item the \lstinline!\psforeach! command's mandatory argument;
+			\item the \emph{first} un-named braces mandatory argument;
+			\item the \emph{first} un-named braces \emph{body}, which we define as any lines following the first opening \lstinline!{! or \lstinline![!
+			      that defined the code block.  This is the part controlled by the \emph{body} field for \texttt{noAdditionalIndent}
+			      and friends from \cpageref{sec:noadd-indent-rules}.
+		\end{itemize}
+		Users wishing to customise the mandatory and/or optional arguments on a \emph{per-name} basis for the \texttt{UnNamedGroupingBracesBrackets}
+		should use \texttt{always-un-named}.
+
+	\paragraph{\texttt{filecontents}} code blocks behave just as \texttt{environments}, except that neither arguments nor items are sought.
+
+\subsubsection{Summary}
+	Having considered all of the different types of code blocks, the functions of the fields given in
+	\cref{lst:noAdditionalIndentGlobal,lst:indentRulesGlobal} should now make sense.
+
+	\begin{widepage}
+		\begin{minipage}{.47\linewidth}
+			\cmhlistingsfromfile[firstnumber=247,linerange={247-259},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[before=\centering,yaml-TCB]{\texttt{noAdditionalIndentGlobal}}{lst:noAdditionalIndentGlobal}
+		\end{minipage}%
+		\hfill
+		\begin{minipage}{.47\linewidth}
+			\cmhlistingsfromfile[firstnumber=263,linerange={263-275},style=yaml-LST,numbers=left]{../defaultSettings.yaml}[before=\centering,yaml-TCB]{\texttt{indentRulesGlobal}}{lst:indentRulesGlobal}
+		\end{minipage}%
+	\end{widepage}
diff --git a/Master/texmf-dist/doc/support/latexindent/subsubsec-special.tex b/Master/texmf-dist/doc/support/latexindent/subsubsec-special.tex
new file mode 100644
index 00000000000..8deca581f0d
--- /dev/null
+++ b/Master/texmf-dist/doc/support/latexindent/subsubsec-special.tex
@@ -0,0 +1,60 @@
+% arara: pdflatex: {shell: yes, files: [latexindent]}
+\subsubsection{\texttt{specialBeginEnd} code blocks}
+	Let's use the example from \vref{lst:specialbefore} which has default output shown in
+	\vref{lst:specialafter}.
+
+	It is recommended to specify \texttt{noAdditionalIndent} and \texttt{indentRules} in the `scalar' form
+	for these type of code blocks, although the `field' form would work, assuming that \texttt{body} was specified.
+	Examples are shown in \cref{lst:displayMath-noAdd,lst:displayMath-indent-rules}.
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/displayMath-noAdd.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{displayMath-noAdd.yaml}}{lst:displayMath-noAdd}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/displayMath-indent-rules.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{displayMath-indent-rules.yaml}}{lst:displayMath-indent-rules}
+	\end{minipage}
+
+	After running the following commands,
+	\begin{commandshell}
+latexindent.pl special1.tex -local displayMath-noAdd.yaml  
+latexindent.pl special1.tex -l displayMath-indent-rules.yaml  
+\end{commandshell}
+	we receive the respective output given in \cref{lst:special1-output-noAdd,lst:special1-output-indent-rules}; note that
+	in \cref{lst:special1-output-noAdd}, the \texttt{displayMath} code block has \emph{not} received any additional indentation,
+	while in \cref{lst:special1-output-indent-rules}, the \texttt{displayMath} code block has received three tabs worth of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1-noAdd.tex}{\texttt{special1.tex} using \cref{lst:displayMath-noAdd}}{lst:special1-output-noAdd}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1-indent-rules.tex}{\texttt{special1.tex} using \cref{lst:displayMath-indent-rules}}{lst:special1-output-indent-rules}
+	\end{minipage}
+
+	We may specify \texttt{noAdditionalIndentGlobal} and \texttt{indentRulesGlobal} as in \cref{lst:special-noAdd-glob,lst:special-indent-rules-global}.
+
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/special-noAdd-glob.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{special-noAdd-glob.yaml}}{lst:special-noAdd-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.49\textwidth}
+		\cmhlistingsfromfile[style=yaml-LST]{demonstrations/special-indent-rules-global.yaml}[width=.9\linewidth,before=\centering,yaml-TCB]{\texttt{special-indent-rules-global.yaml}}{lst:special-indent-rules-global}
+	\end{minipage}
+
+	Upon running the following commands
+	\begin{commandshell}
+latexindent.pl special1.tex -local special-noAdd-glob.yaml  
+latexindent.pl special1.tex -l special-indent-rules-global.yaml  
+\end{commandshell}
+	we receive the outputs in \cref{lst:special1-output-noAdd-glob,lst:special1-output-indent-rules-global}; notice that  in
+	\cref{lst:special1-output-noAdd-glob} neither of the \texttt{special} code blocks have received indentation, while in
+	\cref{lst:special1-output-indent-rules-global} both code blocks have received a single space of indentation.
+
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1-noAdd-glob.tex}{\texttt{special1.tex} using \cref{lst:special-noAdd-glob}}{lst:special1-output-noAdd-glob}
+	\end{minipage}
+	\hfill
+	\begin{minipage}{.45\textwidth}
+		\cmhlistingsfromfile{demonstrations/special1-indent-rules-global.tex}{\texttt{special1.tex} using \cref{lst:special-indent-rules-global}}{lst:special1-output-indent-rules-global}
+	\end{minipage}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/alignmentoutsideEnvironments.tex b/Master/texmf-dist/doc/support/latexindent/success/alignmentoutsideEnvironments.tex
deleted file mode 100644
index 04053c027a9..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/alignmentoutsideEnvironments.tex
+++ /dev/null
@@ -1,29 +0,0 @@
-% arara: indent: {overwrite: true, trace: yes, silent: yes}
-
-\matrix{%
-	%* \begin{tabular}
-		    & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & A_9 \\\hline
-		A_1 & 0   &     &     &     &     &     &     &     &     \\
-		A_2 &     & 0   &     &     &     &     &     &     &     \\
-		A_3 &     &     & 0   &     &     &     &     &     &     \\
-		A_4 &     &     &     & 0   &     &     &     &     &     \\
-		A_5 &     &     &     &     & 0   &     &     &     &     \\
-		A_6 &     &     &     &     &     & 0   &     &     &     \\
-		A_7 &     &     &     &     &     &     & 0   &     &     \\
-		A_8 &     &     &     &     &     &     &     & 0   &     \\
-		A_9 &     &     &     &     &     &     &     &     & 0   \\
-	%* \end{tabular}
-}
-\begin{tabular}
-	    & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & A_9 \\\hline
-	&
-	A_1 & 0   &     &     &     &     &     &     &     &     \\
-	A_2 &     & 0   &     &     &     &     &     &     &     \\
-	A_3 &     &     & 0   &     &     &     &     &     &     \\
-	A_4 &     &     &     & 0   &     &     &     &     &     \\
-	A_5 &     &     &     &     & 0   &     &     &     &     \\
-	A_6 &     &     &     &     &     & 0   &     &     &     \\
-	A_7 &     &     &     &     &     &     & 0   &     &     \\
-	A_8 &     &     &     &     &     &     &     & 0   &     \\
-	A_9 &     &     &     &     &     &     &     &     & 0   \\
-\end{tabular}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/bigTest.tex b/Master/texmf-dist/doc/support/latexindent/success/bigTest.tex
deleted file mode 100644
index e38913da9b3..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/bigTest.tex
+++ /dev/null
@@ -1,293 +0,0 @@
-% arara: indent: {trace: true, overwrite: yes}
-\documentclass[10pt,twoside]{report}
-\begin{document}
-
-% needed for the mini-tableofcontents
-\dominitoc
-\faketableofcontents
-
-\fancyhf{} % delete current header and footer
-\fancyhead[LE,RO]{\bfseries\thepage}
-\fancyhead[LO,RE]{\tiny\rightmark}
-\fancyheadoffset[LE,LO]{4cm}
-
-\pagestyle{fancy}
-%\include{coverpage}
-\include{functions}
-%\include{exponentialfunctions}
-%\include{functions2}
-%\include{logarithms}
-%\include{polyrat}
-%\include{ideas}
-
-%=======================
-%   BEGIN SOLUTIONS
-%=======================
-
-% change the page geometry using \newgeometry
-%\cleardoublepage
-\clearpage
-%\setbool{@twoside}{false}
-\fancyheadoffset[RE,RO]{2cm}
-\fancyheadoffset[LE,LO]{1cm}
-\renewcommand{\rightmark}{Solutions to Section \thesection}
-\fancyhead[LO,RE]{\rightmark}
-\newgeometry{left=4cm,right=4cm,showframe=true,
-	marginratio=1:1,
-	top=1.5cm,bottom=1.5cm,bindingoffset=0cm}
-
-% finish the php file
-\Writetofile{crossrefsWEB}{?>}
-
-% close the solutions files
-\Closesolutionfile{shortsolutions}
-\Closesolutionfile{longsolutions}
-%\Closesolutionfile{hints}
-\Closesolutionfile{crossrefsWEB}
-
-% when itemized lists are used in the solutions, they
-% are actually at 2nd depth because the solution environment
-% uses an \itemize environment to get the indendation correct
-\setlist[itemize,2]{label=\textbullet}
-
-% SHORT solution to problem (show only odd, even, all)
-% Note: this renewenvironment needs to go here
-%       so that the answers package can still
-%       display correctly to the page if needed
-\newbool{oddproblemnumber}
-\renewenvironment{shortSoln}[1]{%
-	\exploregroups % needed to ignore {}
-	% before the environment starts - this is a stretchable space
-	\vskip 0.1cm plus 2cm minus 0.1cm%
-	\fullexpandarg % need this line so that '.' are counted
-	%
-	% either problems, or subproblems, e.g: 3.1 or 3.1.4 respectively
-	% determine which one by counting the '.'
-	\StrCount{#1}{.}[\numberofdecimals]
-	%
-	% find the problem number by splitting the string
-	\ifnumequal{\numberofdecimals}{0}%
-	{%
-		% problems, such as 4, 5, 6, ...
-		\def\problemnumber{#1}%
-	}%
-	{}%
-	\ifnumequal{\numberofdecimals}{1}%
-	{%
-		% subproblems, such as 4.3, 1.2, 10.5
-		\StrBefore{#1}{.}[\problemnumber]%
-	}%
-	{}%
-	\ifnumequal{\numberofdecimals}{2}%
-	{%
-		% subproblems such as 1.3.1, 1.2.4, 7.5.6
-		% note that these aren't currently used, but maybe someday
-		\StrBehind{#1}{.}[\newbit]%
-		\StrBefore{\newbit}{.}[\problemnumber]%
-	}%
-	{}%
-	%
-	% determine if the problem number is odd or even
-	% and depending on our choices above, display or not
-	\ifnumodd{\problemnumber}%
-	{%
-		% set a boolean that says the problem number is odd (used later)
-		\setbool{oddproblemnumber}{true}%
-		% display or not
-		\ifbool{showoddsolns}%
-		{%
-			% if we want to show the odd problems
-			\ifbool{coreproblemYesNo}%
-			{%  Core problem
-				\expandafter\itemize[label=\llap{$\bigstar$ }\bfseries \hyperlink{prob:#1:\thechapter:\thesection}{#1}.]\item%
-			}%
-			{%  NOT Core problem
-				\expandafter\itemize[label=\bfseries \hyperlink{prob:#1:\thechapter:\thesection}{#1}.]\item%
-			}%
-		}%
-		{%
-			% otherwise don't show them!
-			\expandafter\comment%
-		}%
-	}%
-	{%
-		% even numbered problem, set the boolean (used later)
-		\setbool{oddproblemnumber}{false}%
-		\ifbool{showevensolns}%
-		{%
-			% if we want to show the even problems
-			\ifbool{coreproblemYesNo}%
-			{%  Core problem
-				\expandafter\itemize[label=\llap{$\bigstar$ }\itshape \hyperlink{prob:#1:\thechapter:\thesection}{#1}.]\item%
-			}%
-			{%  NOT Core problem
-				\expandafter\itemize[label=\itshape \hyperlink{prob:#1:\thechapter:\thesection}{#1}.]\item%
-			}%
-		}%
-		{%
-			% otherwise don't show them!
-			\expandafter\comment%
-		}%
-	}%
-}%
-{%
-	% after the environment finishes
-	\ifbool{oddproblemnumber}%
-	{%
-		% odd numbered problems
-		\ifbool{showoddsolns}%
-		{%
-			% if we want to show the odd problems
-			% then the environment is finished
-			\enditemize%
-		}%
-		{%
-			% otherwise we need to finish the comment
-			\expandafter\endcomment%
-		}%
-	}%
-	{%
-		% even numbered problems
-		\ifbool{showevensolns}%
-		{%
-			% if we want to show the even problems
-			% then the environment is finished
-			\enditemize%
-		}%
-		{%
-			% otherwise we need to finish the comment
-			\expandafter\endcomment%
-		}%
-	}%
-}
-
-% LONG solution to problem (show only odd, even, all)
-% Note: this renewenvironment needs to go here
-%       so that the answers package can still
-%       display correctly to the page if needed
-\renewenvironment{longSoln}[1]{%
-	\exploregroups % needed to ignore {}
-	% before the environment starts - this is a stretchable space
-	\vskip 0.1cm plus 2cm minus 0.1cm%
-	\fullexpandarg % need this line so that '.' are counted
-	%
-	% either problems, or subproblems, e.g: 3.1 or 3.1.4 respectively
-	% determine which one by counting the '.'
-	\StrCount{#1}{.}[\numberofdecimals]
-	%
-	% find the problem number by splitting the string
-	\ifnumequal{\numberofdecimals}{0}%
-	{%
-		% problems, such as 4, 5, 6, ...
-		\def\problemnumber{#1}%
-	}%
-	{}%
-	\ifnumequal{\numberofdecimals}{1}%
-	{%
-		% problems, such as 4.3, 1.2, 10.5
-		\StrBefore{#1}{.}[\problemnumber]%
-	}%
-	{}%
-	\ifnumequal{\numberofdecimals}{2}%
-	{%
-		% subproblems such as 1.3.1, 1.2.4, 7.5.6
-		\StrBehind{#1}{.}[\newbit]%
-		\StrBefore{\newbit}{.}[\problemnumber]%
-	}%
-	{}%
-	%
-	% determine if the problem number is odd or even
-	% and depending on our choices above, display or not
-	\ifnumodd{\problemnumber}%
-	{%
-		% set a boolean that says the problem number is odd (used later)
-		\setbool{oddproblemnumber}{true}%
-		% display or not
-		\ifbool{showoddsolns}%
-		{%
-			% if we want to show the odd problems
-			{\bfseries \hyperlink{prob:#1:\thechapter:\thesection}{#1}.}%
-		}%
-		{%
-			% otherwise don't show them!
-			\expandafter\comment%
-		}%
-	}%
-	{%
-		% even numbered problem, set the boolean (used later)
-		\setbool{oddproblemnumber}{false}%
-		\ifbool{showevensolns}%
-		{%
-			% if we want to show the even problems
-			{\itshape \hyperlink{prob:#1:\thechapter:\thesection}{#1}.}%
-		}%
-		{%
-			% otherwise don't show them!
-			\expandafter\comment%
-		}%
-	}%
-}%
-{%
-	% after the environment finishes
-	\ifbool{oddproblemnumber}%
-	{%
-		% odd numbered problems
-		\ifbool{showoddsolns}%
-		{%
-			% if we want to show the odd problems
-			% then the environment is finished
-		}%
-		{%
-			% otherwise we need to finish the comment
-			\expandafter\endcomment%
-		}%
-	}%
-	{%
-		% even numbered problems
-		\ifbool{showevensolns}%
-		{%
-			% if we want to show the even problems
-			% then the environment is finished
-		}%
-		{%
-			% otherwise we need to finish the comment
-			\expandafter\endcomment%
-		}%
-	}%
-}
-
-% renew tikzpicture environment to make it use valign=t
-% on every one, which fixes vertical alignment of tikzpicture
-% with the solution label: http://tex.stackexchange.com/questions/30367/aligning-enumerate-labels-to-top-of-image
-\BeforeBeginEnvironment{tikzpicture}{\begin{adjustbox}{valign=t}}
-\AfterEndEnvironment{tikzpicture}{\end{adjustbox}}
-
-% do the same for the tabular environment
-\BeforeBeginEnvironment{tabular}{\begin{adjustbox}{valign=t}}
-\AfterEndEnvironment{tabular}{\end{adjustbox}}
-
-% set every picture in the solutions to have \solutionfigurewidth
-\pgfplotsset{
-	every axis/.append style={%
-		width=\solutionfigurewidth}}
-
-% input the SHORT solutions file
-\IfFileExists{shortsolutions.tex}{\input{shortsolutions.tex}}{}
-
-\clearpage
-% input the LONG solutions file
-%\IfFileExists{longsolutions.tex}{\input{longsolutions.tex}}{}
-
-\clearpage
-% input the HINTS file
-%\IfFileExists{hints.tex}{\input{hints.tex}}{}
-%=======================
-%   END SOLUTIONS
-%=======================
-
-%=======================
-%   INDEX
-%=======================
-\printindex
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/braceTest.tex b/Master/texmf-dist/doc/support/latexindent/success/braceTest.tex
deleted file mode 100644
index b389b436602..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/braceTest.tex
+++ /dev/null
@@ -1,57 +0,0 @@
-% arara: indent: { overwrite: on}
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-		}
-	}%
-	%---------------------------------------------------------------------------
-}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-		}
-}}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-}}}
-
-some other text
-some other text
-some other text
-
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-	}}
-}
-
-some other text
-some other text
-some other text
-
-\parbox{
-	\begin{something}
-						
-	\end{something}
-}
-
-some other text
-some other text
-some other text
diff --git a/Master/texmf-dist/doc/support/latexindent/success/braceTestsmall.tex b/Master/texmf-dist/doc/support/latexindent/success/braceTestsmall.tex
deleted file mode 100644
index 38b7f495db6..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/braceTestsmall.tex
+++ /dev/null
@@ -1,31 +0,0 @@
-% arara: indent: {trace: on, overwrite: on}
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep }%
-	}%
-	%---------------------------------------------------------------------------
-}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep }}%
-}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={\someothercommand{here}%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep }}}
-
-some other text
-some other text
-some other text
diff --git a/Master/texmf-dist/doc/support/latexindent/success/bracketTest.tex b/Master/texmf-dist/doc/support/latexindent/success/bracketTest.tex
deleted file mode 100644
index 34cf9df16fb..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/bracketTest.tex
+++ /dev/null
@@ -1,6 +0,0 @@
-\cmh[
-	here is some text
-]
-\cmh{
-	here is some text
-}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/conditional.tex b/Master/texmf-dist/doc/support/latexindent/success/conditional.tex
deleted file mode 100644
index 9a857e0af64..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/conditional.tex
+++ /dev/null
@@ -1,12 +0,0 @@
-% arara: indent: {overwrite: yes}
-% add these lines to a localSettings.yaml file
-%constructIfElseFi:
-%    if@isu@draftcls@: 1
-\newif\if@isu@draftcls@
-\if@isu@draftcls@
-	\RequirePackage[firstpage]{draftwatermark}
-	\RequirePackage{datetime}
-	\newcommand{\isu@draftfooter}{DRAFT --- rendered \today\ at \currenttime}
-\else
-	\newcommand{\isu@draftfooter}{}
-\fi
diff --git a/Master/texmf-dist/doc/support/latexindent/success/environments.tex b/Master/texmf-dist/doc/support/latexindent/success/environments.tex
deleted file mode 100644
index 986474dca79..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/environments.tex
+++ /dev/null
@@ -1,39 +0,0 @@
-% arara: indent: { cruft: /home/cmhughes/Desktop/tmp/, overwrite: on, trace: yes, silent: yes, onlyDefault: no}
-some
-other
-text
-\begin{enumerate}
-	\item $3x^3y(-4x^3+y) = -12x^6y+3x^3y^2$
-	\item $3x^3+y(-4x^3+y) = 3x^3-4x^3y+y^2$
-	\item
-	      \begin{align*}
-	      	{\color{red}(3x^3+y)}(-4x^3+y) & =	{\color{red}(3x^3+y)}(-4x^3)+{\color{red}(3x^3+y)}(y) \\
-	      	                               & =	-12x^6-4x^3y+3x^3y+y^2                                \\
-	      	                               & =	-12x^6-x^3y+y^2                                       
-	      \end{align*}
-\end{enumerate}
-\begin{enumerate}
-	\item $3x^3y(-4x^3+y) = -12x^6y+3x^3y^2$
-	\item $3x^3+y(-4x^3+y) = 3x^3-4x^3y+y^2$
-	\item
-	      \begin{align*}
-	      	{\color{red}(3x^3+y)}(-4x^3+y) & =	{\color{red}(3x^3+y)}(-4x^3)+{\color{red}(3x^3+y)}(y) \\
-	      	                               & =	-12x^6-4x^3y+3x^3y+y^2                                \\
-	      	                               & =	-12x^6-x^3y+y^2                                       
-	      \end{align*}
-\end{enumerate}
-
-\begin{something}
-	\begin{else}
-		again
-	\end{else}
-\end{something}
-no
-environments
-here
-\[ x^2+ 3x\]
-other text
-\[
-	x^2+ 3x
-\]
-other text
diff --git a/Master/texmf-dist/doc/support/latexindent/success/figureValign.tex b/Master/texmf-dist/doc/support/latexindent/success/figureValign.tex
deleted file mode 100644
index 8c2bc33281a..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/figureValign.tex
+++ /dev/null
@@ -1,124 +0,0 @@
-% arara: pdflatex
-% !arara: indent: {overwrite: yes}
-\documentclass{article}
-\usepackage{lipsum}
-\usepackage{graphicx}
-\usepackage{environ}
-\usepackage[showframe=true]{geometry}
-
-\newsavebox{\vabox}
-
-\NewEnviron{verticallyaligned}{%
-	% temporarily set \vamaxheight to nothing during the measurements
-	\let\vamaxheight\relax
-	% measure the height of the body
-	\begin{lrbox}{\vabox}
-		\BODY%
-	\end{lrbox}%
-	% set the height of the minipage box
-	\newlength{\vamaxheight}
-	\setlength{\vamaxheight}{\ht\vabox}
-	% output the body, which now contains the new height :)
-	\noindent\makebox[\linewidth][c]{\mbox{}\hfill\BODY\hfill\mbox{}}%
-}
-
-\begin{document}
-
-\sbox{\vabox}{%
-	\begin{minipage}[b][][b]{.10\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image-a}
-	\end{minipage}%
-	\begin{minipage}[b][][t]{.15\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image-b}
-	\end{minipage}%
-	\begin{minipage}[b][][t]{.20\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image-c}
-	\end{minipage}%
-	\begin{minipage}[b][][c]{.10\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image}
-	\end{minipage}%
-	\begin{minipage}[b][][t]{.20\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image}
-	\end{minipage}%
-}%
-
-\noindent
-\begin{minipage}[b][\ht\vabox][b]{.10\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image-a}
-\end{minipage}%
-\hfill
-\begin{minipage}[b][\ht\vabox][t]{.15\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image-b}
-\end{minipage}%
-\hfill
-\begin{minipage}[b][\ht\vabox][t]{.20\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image-c}
-\end{minipage}%
-\hfill
-\begin{minipage}[b][\ht\vabox][c]{.10\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image}
-\end{minipage}%
-\hfill
-\begin{minipage}[b][\ht\vabox][t]{.20\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image}
-\end{minipage}%
-\par\lipsum
-
-
-\begin{verticallyaligned}
-	\noindent\begin{minipage}[b][\vamaxheight][b]{.30\textwidth}
-	\centering
-	\includegraphics[width=\textwidth]{example-image-a}
-	\end{minipage}%
-	\hfill
-	\begin{minipage}[b][\vamaxheight][t]{.10\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image-b}
-	\end{minipage}%
-	\hfill
-	\begin{minipage}[b][\vamaxheight][t]{.20\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image-c}
-	\end{minipage}%
-	\hfill
-	\begin{minipage}[b][\vamaxheight][c]{.20\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image}
-	\end{minipage}%
-	\hfill
-	\begin{minipage}[b][\vamaxheight][t]{.20\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image}
-	\end{minipage}%
-\end{verticallyaligned}
-
-\lipsum[1]
-
-\begin{figure}
-	\begin{verticallyaligned}
-		\noindent\begin{minipage}[b][\vamaxheight][c]{.50\textwidth}
-		\centering
-		\includegraphics[width=\textwidth]{example-image}
-		\caption{}
-		\end{minipage}%
-		\hfill
-		\begin{minipage}[b][\vamaxheight][t]{.40\textwidth}
-			\centering
-			\includegraphics[width=\textwidth]{example-image}
-			\caption{}
-		\end{minipage}%
-	\end{verticallyaligned}
-	\caption{}
-\end{figure}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/filecontents.tex b/Master/texmf-dist/doc/support/latexindent/success/filecontents.tex
deleted file mode 100644
index ccc35c32684..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/filecontents.tex
+++ /dev/null
@@ -1,73 +0,0 @@
-% arara: indent: {overwrite: true, trace: false, localSettings: yes}
-
-% used with localSettings.yaml as follows
-%indentPreamble: 1
-%indentRules:
-%   @online: "\t\t\t\t"
-%   #myenvironment: "\t\t"
-%   myotherenvironment: "\t\t"
-%   \[: "\t\t"
-%   tabular: "\t\t\t"
-%noAdditionalIndent:
-%   @online: 0
-%   myotherenvironment: 1
-%   \[: 0
-%   \]: 0
-%   tabular: 0
-%   something: 0
-%   parbox: 1
-%verbatimEnvironments:
-%   myotherenvironment: 1
-%   tabular: 0
-%   someothername: 0
-
-
-%        \begin{noindent}
-here we are in a block
-% \end{noindent}
-some more
-\begin{tabular}{cccc}
-	1 & 2 & 3 & 4 \\
-	5 &   & 6 &   \\
-\end{tabular}
-
-another test
-\begin{tabular}{cccc}
-	1 & 2 & 3 & 4 \\
-	5 &   & 6 &   \\
-\end{tabular}
-
-\begin{something}
-	\parbox{something
-		else
-		goes
-		here
-	}
-	some text some text
-	some text some text
-	some text some text
-	\[
-		x^2+2x
-	\]
-	some text some text
-	some text some text
-	some text some text
-	some text some text
-	some text some text
-\end{something}
-\begin{filecontents}{mybib.bib}
-	@online{strawberryperl,
-		title="Strawberry Perl",
-		url="http://strawberryperl.com/"}
-	@online{cmhblog,
-		title="A Perl script for indenting tex files",
-		url="http://tex.blogoverflow.com/2012/08/a-perl-script-for-indenting-tex-files/"}
-\end{filecontents}
-
-\begin{myotherenvironment}
-	some text goes here
-	some text goes here
-	some text goes here
-	some text goes here
-\end{myotherenvironment}
-
diff --git a/Master/texmf-dist/doc/support/latexindent/success/filecontents1.tex b/Master/texmf-dist/doc/support/latexindent/success/filecontents1.tex
deleted file mode 100644
index 46e501ebaa0..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/filecontents1.tex
+++ /dev/null
@@ -1,18 +0,0 @@
-% arara: indent: {overwrite: true, trace: false, localSettings: yes}
-
-\documentclass{article}
-\begin{filecontents}{mybib.bib}
-	\begin{document}
-	here is some text
-	\end{document}
-\end{filecontents}
-
-\begin{document}
-\begin{myotherenvironment}
-	some text goes here
-	some text goes here
-	some text goes here
-	some text goes here
-\end{myotherenvironment}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/filecontents2.tex b/Master/texmf-dist/doc/support/latexindent/success/filecontents2.tex
deleted file mode 100644
index 54307023361..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/filecontents2.tex
+++ /dev/null
@@ -1,21 +0,0 @@
-% arara: indent: {overwrite: true, trace: false, localSettings: yes}
-
-\documentclass{article}
-\usepackage{verbatim}
-\begin{filecontents}{mybib.bib}
-	\begin{document}
-	here is some text
-	\end{document}
-\end{filecontents}
-
-\begin{document}
-
-\begin{verbatim}
-\begin{filecontents}{mybib.bib}
-	\begin{document}
-	here is some text
-	\end{document}
-        \end{filecontents}
-\end{verbatim}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/ifelsefi.tex b/Master/texmf-dist/doc/support/latexindent/success/ifelsefi.tex
deleted file mode 100644
index 389985acaed..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/ifelsefi.tex
+++ /dev/null
@@ -1,103 +0,0 @@
-% !arara: pdflatex
-% arara: indent: {overwrite: yes, trace: yes, localSettings: yes}
-\documentclass{report}
-\usepackage{etoolbox}
-\usepackage{pgfplots}
-\usepackage{lipsum}
-
-\makeatletter
-\newcommand{\totalchapters}[1]{%
-	\@ifundefined{c@totalchapters}
-	{%
-		\newcounter{totalchapters}
-		\setcounter{totalchapters}{#1}
-		\typeout{Defining a new counter: totalchapters (#1)}
-	}%
-	{%
-		\ifodd\value{totalchapters}=#1
-			\typeout{Total Chapters match auxilary file (#1)}
-		\else
-			\typeout{Warning: total Chapter count updated from \the\value{totalchapters} to #1-- recompile to fix}
-		\fi
-		\setcounter{totalchapters}{#1}
-	}%
-}
-\newcommand{\definetotalpagecount}[2]{%
-	\@ifundefined{c@totalpages\@roman{#1}}%
-	{%
-		\newcounter{totalpages\@roman{#1}}
-		\setcounter{totalpages\@roman{#1}}{#2}
-		\typeout{Defining a new counter: totalpages\@roman{#1}}
-	}%
-	{%
-		\ifnum\value{totalpages\@roman{#1}}=#2
-			\typeout{Total pages for Chapter #1 match auxilary file (#2)}
-		\else
-			\typeout{Warning: total pages for Chapter #1 updated from \the\value{totalpages\@roman{#1}} to #2-- recompile to fix}
-		\fi
-		\setcounter{totalpages\@roman{#1}}{#2}
-	}%
-}
-
-
-\preto\chapter{%
-	\ifnum\value{chapter}>0
-		\immediate\write\@auxout{%
-			\string\definetotalpagecount\string{\thechapter\string}\string{\the\value{page}\string}
-		}
-	\fi
-}
-
-\AtEndDocument{%
-	\immediate\write\@auxout{%
-		\string\definetotalpagecount\string{\thechapter\string}\string{\the\value{page}\string}
-		\string\totalchapters\string{\thechapter\string}%
-	}
-}
-
-\newcommand{\drawPageChart}{%
-	\begin{tikzpicture}
-		\begin{axis}[
-				xbar stacked,
-				xmin=-0.1,
-				%ymin=0,ymax=1,
-				bar width=40pt,
-				nodes near coords,
-				axis lines=none,
-				nodes near coords align={horizontal},
-				visualization depends on=x \as \myxcoord,
-				nodes near coords={\pgfmathprintnumber\myxcoord},
-				every node near coord/.append style={
-					anchor=east},
-			]
-			\@ifundefined{c@totalchapters}
-			{}
-			{%
-				\foreach \i in {1,...,\thetotalchapters}{%
-					\addplot coordinates {(\the\value{totalpages\@roman{\i}},0)};
-				}
-			}
-		\end{axis}
-	\end{tikzpicture}
-}
-\begin{document}
-
-\begin{figure}[!htb]
-	\centering
-	\drawPageChart
-	\caption{Blueprint of my thesis}
-\end{figure}
-
-
-\chapter{}
- \lipsum
-\chapter{}
- \lipsum
- \lipsum
-\chapter{}
- \lipsum
- \lipsum
- \lipsum
- \lipsum
- \lipsum
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/ifelsefiONE.tex b/Master/texmf-dist/doc/support/latexindent/success/ifelsefiONE.tex
deleted file mode 100644
index 0d3c4f39322..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/ifelsefiONE.tex
+++ /dev/null
@@ -1,43 +0,0 @@
-% arara: indent: {overwrite: yes, trace: yes}
-\documentclass[tikz]{standalone}
-\usepackage{tikz}
-\usetikzlibrary{decorations.pathmorphing,decorations.shapes}
-
-\begin{document}
-
-\foreach \radius in {1,2,...,20}
-{
-	\begin{tikzpicture}
-		% background rectangle
-		\filldraw[black] (-3,-3) rectangle (5,3);
-		% skyline
-		\filldraw[black!80!blue](-3,-3)--(-3,-2)--(-2.5,-2)--(-2.5,-1)--(-2.25,-1)--(-2.25,-2)--(-2,-2)
-		--(-2,-1)--(-1.75,-0.75)--(-1.5,-1)
-		--(-1.5,-2)--(-1.1,-2)--(-1.1,0)--(-0.5,0)--(-0.5,-2)
-		--(0,-2)--(0,-1.5)--(1,-1.5)--(1.25,-0.5)--(1.5,-1.5)--(1.5,-2)
-		--(2,-2)--(2,0)--(2.5,0)--(2.5,-2)
-		--(3,-2)--(3,-1)--(4,-1)--(4,-2)--(5,-2)--(5,-3)--cycle;
-		% moon- what a hack!
-		\filldraw[white] (4,2.5) arc (90:-90:20pt);
-		\filldraw[black] (3.8,2.5) arc (90:-90:20pt);
-		% fireworks
-		\pgfmathparse{100-(\radius-1)*10};
-		% red firework
-		\ifnum\radius<11
-			\draw[decorate,decoration={crosses},red!\pgfmathresult!black] (0,0) circle (\radius ex);
-		\fi
-		% orange firework
-		\pgfmathparse{100-(\radius-6)*10};
-		\ifnum\radius>5
-			\ifnum\radius<16
-				\draw[decorate,decoration={crosses},orange!\pgfmathresult!black] (1,1) circle ( \radius ex-5ex);
-			\fi
-		\fi
-		% yellow firework
-		\pgfmathparse{100-(\radius-11)*10};
-		\ifnum\radius>10
-			\draw[decorate,decoration={crosses},yellow!\pgfmathresult!black] (2.5,1) circle (\radius ex-10ex);
-		\fi
-	\end{tikzpicture}
-}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/ifelsefiSmall.tex b/Master/texmf-dist/doc/support/latexindent/success/ifelsefiSmall.tex
deleted file mode 100644
index f4b4bad3e64..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/ifelsefiSmall.tex
+++ /dev/null
@@ -1,10 +0,0 @@
-\ifnum\radius>5
-	\ifnum\radius<16
-		\draw[decorate,decoration={crosses},orange!\pgfmathresult!black] (1,1) circle ( \radius ex-5ex);
-	\fi
-\fi
-\ifnum
-	1
-\else
-	2
-\fi
diff --git a/Master/texmf-dist/doc/support/latexindent/success/items1.tex b/Master/texmf-dist/doc/support/latexindent/success/items1.tex
deleted file mode 100644
index 29be75d5fdc..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/items1.tex
+++ /dev/null
@@ -1,10 +0,0 @@
-\begin{itemize}
-	\item one
-	      here is some text
-	      here is some text
-	      here is some text
-	\item two
-	      here is some text
-	      here is some text
-	      here is some text
-\end{itemize}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/items2.tex b/Master/texmf-dist/doc/support/latexindent/success/items2.tex
deleted file mode 100644
index 76d70b95729..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/items2.tex
+++ /dev/null
@@ -1,21 +0,0 @@
-\begin{enumerate}
-	\item one
-	      here is some text
-	      here is some text
-	      here is some text
-	\item two
-	      here is some text
-	      here is some text
-	      here is some text
-	      \begin{itemize}
-	      	\item bullet
-	      	      here is some text
-	      	      here is some text
-	      	      here is some text
-	      	\item bullet
-	      	      here is some text
-	      	      here is some text
-	      	      here is some text
-	      \end{itemize}
-	      some text
-\end{enumerate}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/items3.tex b/Master/texmf-dist/doc/support/latexindent/success/items3.tex
deleted file mode 100644
index 30460f40e72..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/items3.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-\begin{myenv}
-	\begin{itemize}
-	\item one
-	      here is some text
-	      here is some text
-	      here is some text
-	\item two
-	      here is some text
-	      here is some text
-	      here is some text
-	\end{itemize}
-\end{myenv}
-\begin{itemize}
-\item one
-      here is some text
-      here is some text
-      here is some text
-\item two
-      here is some text
-      here is some text
-      here is some text
-\end{itemize}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/items4.tex b/Master/texmf-dist/doc/support/latexindent/success/items4.tex
deleted file mode 100644
index 49383c47e1a..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/items4.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-\begin{myenv}
-\begin{itemize}
-\item one
-      here is some text
-      here is some text
-      here is some text
-\item two
-      here is some text
-      here is some text
-      here is some text
-\end{itemize}
-\end{myenv}
-\begin{itemize}
-\item one
-      here is some text
-      here is some text
-      here is some text
-\item two
-      here is some text
-      here is some text
-      here is some text
-\end{itemize}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/matrix.tex b/Master/texmf-dist/doc/support/latexindent/success/matrix.tex
deleted file mode 100644
index 79c959e1ba2..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/matrix.tex
+++ /dev/null
@@ -1,14 +0,0 @@
-% arara: indent: {overwrite: true, trace: true}
-\[
-	\begin{matrix}[cc|cccccc|c]
-		                &   &                       &                       &           &                       & {\color{blue}\downarrow} & {\color{blue}\downarrow} & S                       \\\hline
-		                & 6 & {\color{red}\newmoon} &                       &           &                       &                          &                          & {\color{red}\leftarrow} \\
-		{\color{blue}*} & 5 &                       &                       &           &                       &                          & {\color{blue}\newmoon}   &                         \\
-		                & 4 &                       &                       &           & {\color{red}\newmoon} &                          &                          & {\color{red}\leftarrow} \\
-		                & 3 &                       & {\color{red}\newmoon} &           &                       &                          &                          & {\color{red}\leftarrow} \\
-		                & 2 &                       &                       & \fullmoon &                       &                          &                          &                         \\
-		{\color{blue}*} & 1 &                       &                       &           &                       & {\color{blue}\newmoon}   &                          &                         \\\hline
-		                &   & 1                     & 2                     & 3         & 4                     & 5                        & 6                        &                         \\
-		\%              &   & {\color{red}*}        & {\color{red}*}        &           & {\color{red}*}        &                          &                          &                         
-	\end{matrix}
-\]
diff --git a/Master/texmf-dist/doc/support/latexindent/success/multipleBraces.tex b/Master/texmf-dist/doc/support/latexindent/success/multipleBraces.tex
deleted file mode 100644
index 63a2967dab3..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/multipleBraces.tex
+++ /dev/null
@@ -1,15 +0,0 @@
-% arara: indent: {overwrite: yes, trace: on}
-\xapptocmd{\tableofcontents}{%
-	\end{singlespace}%
-	\pagestyle{plain}%
-	\clearpage}{}{}
-
-\xapptocmd{\tableofcontents}{%
-	\end{singlespace}%
-	\pagestyle{plain}%
-	\clearpage}{}{}
-
-\xapptocmd{\tableofcontents}{%
-	\end{singlespace}%
-	\pagestyle{plain}%
-	\clearpage}{}{}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/nestedalignment.tex b/Master/texmf-dist/doc/support/latexindent/success/nestedalignment.tex
deleted file mode 100644
index 6578b4f4e00..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/nestedalignment.tex
+++ /dev/null
@@ -1,33 +0,0 @@
-% arara: indent: { overwrite: on, trace: yes}
-\begin{tabular}{p{3cm}|c|p{8cm}}
-	Example & Choice                         & Why                                                                      \\ \hline \hline
-	\(
-	\left\{
-	\begin{array}{rl}
-	x+y & =6	\\
-	2x+y & =8
-	\end{array}
-	\right.
-	\)
-	        & Substitution {\em or }Addition & Because it is easy to solve for $x$ in the 1st equation                  
-	{\em or}
-	Because it is easy to multiply the first equation by -1 \\ \hline
-	\(
-	\left\{
-	\begin{array}{rl}
-	3x-7y & =13 \\
-	6x+5y & =7
-	\end{array}
-	\right.
-	\)
-	        & Addition                       & Because there is no obvious way to use substitution                      \\ \hline
-	\(
-	\left\{
-	\begin{array}{rl}
-	x-7y  & =13 \\
-	6x+5y & =7
-	\end{array}
-	\right.
-	\)
-	        & Substitution                   & Because the first equation can easily be solved for one of the variables 
-\end{tabular}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/nestedalignment1.tex b/Master/texmf-dist/doc/support/latexindent/success/nestedalignment1.tex
deleted file mode 100644
index 9a19ca8da43..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/nestedalignment1.tex
+++ /dev/null
@@ -1,25 +0,0 @@
-% arara: indent: { overwrite: on, silent: no, trace: yes}
-\begin{tabular}{p{3cm}|c|p{8cm}}
-	x+y       & = & 6     \\
-	2x+y & \&\%=8 % \% &
-	\%\&\%\%  &   &       \\ % &   & 2x+y & =8
-	x+y       & = & 6     \\
-	2x+y & =8
-	2x+y \&\& & = & 8 % trailine comment   
-\end{tabular}
-
-here's another line $\{ x^2 + 5x \}$
-\begin{minipage}{\textwidth}
-	content
-	content
-	content
-	content
-	content
-\end{minipage}\\[3cm]
-\begin{minipage}{\textwidth}
-	content
-	content
-	content
-	content
-	content
-\end{minipage}\\[3cm]
diff --git a/Master/texmf-dist/doc/support/latexindent/success/outputfile.tex b/Master/texmf-dist/doc/support/latexindent/success/outputfile.tex
deleted file mode 100644
index aac040a48c7..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/outputfile.tex
+++ /dev/null
@@ -1,57 +0,0 @@
-% arara: indent: { overwrite: yes}
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-		}
-	}%
-	%---------------------------------------------------------------------------
-}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-		}
-}}
-
-some other text
-some other text
-some other text
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-}}}
-
-some other text
-some other text
-some other text
-
-
-\hypersetup{%
-	pdfstartview={%
-		FitH \hypercalcbp{\paperheight-\topmargin-0in
-			-\headheight-\headsep
-	}}
-}
-
-some other text
-some other text
-some other text
-
-\parbox{
-	\begin{something}
-						
-	\end{something}
-}
-
-some other text
-some other text
-some other text
diff --git a/Master/texmf-dist/doc/support/latexindent/success/preamble.tex b/Master/texmf-dist/doc/support/latexindent/success/preamble.tex
deleted file mode 100644
index 749e9819c20..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/preamble.tex
+++ /dev/null
@@ -1,29 +0,0 @@
-% arara: indent: {trace: on, overwrite: yes}
-\documentclass[10pt,twoside]{report}
-\newenvironment{widepage}{\begin{adjustwidth}{-\offsetpage}{}%
-	\addtolength{\textwidth}{\offsetpage}}%
-{\end{adjustwidth}}
-
-% Define fix command
-% 	- it puts a comment in the margin
-% 	- it writes to a file with a list of things that need fixing
-\newcommand{\fixthis}[1]
-{%
-	\marginpar{\huge \color{red} \framebox{FIX}}%
-	\typeout{FIXTHIS: p\thepage : #1^^J}%
-}
-
-% Define pccname command
-% 	- it writes to the log file with a detail of the name-
-%     this is useful for tracking names for diversity purposes
-\newcommand{\pccname}[1]
-{%
-	#1%
-	\typeout{PCCNAME: p\thepage : #1}%
-}
-
-\begin{document}
-
-some text
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/pstricks1.tex b/Master/texmf-dist/doc/support/latexindent/success/pstricks1.tex
deleted file mode 100644
index ff55548b6c7..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/pstricks1.tex
+++ /dev/null
@@ -1,26 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass[pstricks]{standalone}
-\usepackage{pstricks,multido}
-
-\def\Bottle#1{{\pscustom[linewidth=2pt]{%
-	\rotate{#1}
-	\psline(-1,3.5)(-1,4)(1,4)(1,3.5)
-	\pscurve(3,2)(1,0)\psline(-1,0)
-	\pscurve(-3,2)(-1,3.5)}}}
-
-\def\BottleWithWater(#1)#2{%
-	\rput[c]{#2}(#1){%
-		\rput{*0}(0,0){%
-			\psclip{\Bottle{#2}}
-			\psframe*[linecolor=gray](-6,-2)(6,2)
-			\endpsclip}\rput{*0}(0,0){\Bottle{#2}}}}
-
-\begin{document}
-
-\multido{\iA=-45+5}{19}{%
-	\begin{pspicture}(-2.5,-0.5)(6,5.5)
-		\BottleWithWater(1.5,1){\iA}
-	\end{pspicture}
-}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/pstricks2.tex b/Master/texmf-dist/doc/support/latexindent/success/pstricks2.tex
deleted file mode 100644
index 7690da820e7..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/pstricks2.tex
+++ /dev/null
@@ -1,68 +0,0 @@
-% arara: indent: {overwrite: true, trace: true, localSettings: off}
-\documentclass[pstricks,border=12pt]{standalone}
-\usepackage{pst-eucl}
-\usepackage[nomessages]{fp}
-
-\definecolor{myblue}{RGB}{37,111,197}
-\definecolor{mybrown}{RGB}{211,200,134}
-
-\def\rOne{0.75}
-\def\rTwo{0.75}
-\def\tOne{20}
-\def\tTwo{-45}
-\FPset\RowMaxIndex{4}% because zero based index
-\FPset\ColMaxIndex{4}% because zero based index
-
-
-\psset
-{
-	PointName=none,
-	PointSymbol=none,
-	fillstyle=solid,
-	linejoin=1,
-}
-
-\def\Bar(#1,#2)#3{%
-	\rput(!#2 \rTwo\space mul \tTwo\space PtoC){%
-		\rput(!#1 \rOne\space mul \tOne\space 180 add PtoC){%
-			\pstGeonode{O}(\rOne;\tOne){One}(\rTwo;\tTwo){Two}
-			\pstTranslation{O}{Two}{One}[Three]
-			\pnode(0,\stripH){O'}
-			\pstTranslation{O}{O'}{One,Two,Three}
-			\pnode(0,#3){O''}
-			\pstTranslation{O}{O''}{One,Two,Three}[One'',Two'',Three'']
-			\psset{fillcolor=mybrown}
-			\pspolygon(O'')(O)(Two)(Two'')
-			\pspolygon(Two'')(Two)(Three)(Three'')
-			\pspolygon(One'')(O'')(Two'')(Three'')
-			\psset{fillcolor=myblue,opacity=0.75,linestyle=none,linewidth=0}
-			\FPifeq{#1}{\RowMaxIndex}\pspolygon(O')(O)(Two)(Two')\fi
-			\FPifeq{#2}{\ColMaxIndex}\pspolygon(Two')(Two)(Three)(Three')\fi
-			\FPiflt{#3}{\stripH}\pspolygon(One')(O')(Two')(Three')\fi
-		}%
-	}%
-}
-
-\newcount\OuterIndex
-\def\SaveListContents#1\relax{\def\Contents{#1}}
-
-\def\Picture#1{%
-	\def\stripH{#1}%
-	\begin{pspicture}[showgrid=false](-2.5,-3.35)(3.05,3.05)
-		\psforeach{\row}{%
-			{{3,2.8,2.7,3,3.1}},% <=== Only this row must use double curly braces. It is a feature!
-			{2.8,1,1.2,2,3},%
-			{2.8,1,1.2,2,2.8},%
-			{2.6,1.6,1.8,1.9,1.8},%
-			{2.4,1.5,1.7,1.9,1.5}%
-			}{%
-			\expandafter\SaveListContents\row\relax
-			\OuterIndex=\psLoopIndex\relax
-			\psforeach{\col}{\Contents}{\Bar(\the\OuterIndex,\the\psLoopIndex){\col}}%
-			\psLoopIndex=\OuterIndex\relax
-		}
-	\end{pspicture}}
-
-\begin{document}
-\multido{\n=0.0+0.2}{17}{\Picture{\n}}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/pstricks3.tex b/Master/texmf-dist/doc/support/latexindent/success/pstricks3.tex
deleted file mode 100644
index 09c132f5c73..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/pstricks3.tex
+++ /dev/null
@@ -1,49 +0,0 @@
-% arara: indent: {overwrite: yes}
-\documentclass[pstricks,border=12pt]{standalone}
-\usepackage{pst-node}
-\addtopsstyle{gridstyle}{gridlabels=0pt,strokeopacity=.25}
-
-\begin{document}
-\begin{pspicture}[showgrid=top](8,8)
-	\multips(0,.5)(0,1){8}{%
-		\multips(.5,0)(1,0){8}{%
-			\psline[linecolor=red](6pt;-135)(6pt;45)
-			\psline[linecolor=red](6pt;135)(6pt;-45)}}
-	\pscustom
-	[
-		dimen=middle,
-		fillstyle=eovlines*,
-		fillcolor=white,
-		hatchcolor=blue,
-		linecolor=blue,
-	]
-	{
-		\psframe(8,8)
-		\pspolygon
-		(3,1)
-		(3,4)
-		(1,4)
-		(1,6)
-		(2,6)
-		(2,7)
-		(7,7)
-		(7,4)
-		(6,4)
-		(6,6)
-		(5,6)
-		(5,4)
-		(6,4)
-		(6,3)
-		(7,3)
-		(7,1)
-		(6,1)
-		(6,2)
-		(5,2)
-		(5,1)
-	}
-	\psset{linecolor=blue,nodesep=7pt}
-	\pscircle*(4,2){2pt}
-	\pscircle*(4,3){2pt}
-	\pcline(4,2)(4,3)
-\end{pspicture}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/sampleAFTER.tex b/Master/texmf-dist/doc/support/latexindent/success/sampleAFTER.tex
deleted file mode 100644
index b1f37474fe0..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/sampleAFTER.tex
+++ /dev/null
@@ -1,5819 +0,0 @@
-% arara: indent: {overwrite: yes}
-% A sample chapter file- it contains a lot of
-% environments, including tabulars, align, etc
-%
-% Don't try and compile this file using pdflatex etc, just
-% compare the *format* of it to the format of the
-% sampleAFTER.tex
-%
-% In particular, compare the tabular and align-type
-% environments before and after running the script
-
-\section{Polynomial functions}
- \reformatstepslist{P} % the steps list should be P1, P2, \ldots
- In your previous mathematics classes you have studied \emph{linear} and
- \emph{quadratic} functions. The most general forms of these types of
- functions can be represented (respectively) by  the functions $f$
- and $g$ that have formulas
- \begin{equation}\label{poly:eq:linquad}
- 	f(x)=mx+b, \qquad g(x)=ax^2+bx+c
- \end{equation}
- We know that $m$ is the slope of $f$, and that $a$ is the \emph{leading coefficient}
- of $g$. We also know that the \emph{signs} of $m$ and $a$ completely
- determine the behavior of the functions $f$ and $g$. For example, if $m>0$
- then $f$ is an \emph{increasing} function, and if $m<0$ then $f$ is
- a \emph{decreasing} function.  Similarly, if $a>0$ then $g$ is
- \emph{concave up} and if $a<0$ then $g$ is \emph{concave down}. Graphical
- representations of these statements are given in \cref{poly:fig:linquad}.
-   
- \begin{figure}[!htb]
- 	\setlength{\figurewidth}{.2\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-10:8]{(x+2)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$m>0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-10:8]{-(x+2)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$m<0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-4:4]{(x^2-6)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$a>0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-4:4]{-(x^2-6)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$a<0$}
- 	\end{subfigure}
- 	\caption{Typical graphs of linear and quadratic functions.}
- 	\label{poly:fig:linquad}
- \end{figure}
-   
- Let's look a little more closely at the formulas for $f$ and $g$ in
- \cref{poly:eq:linquad}. Note that the \emph{degree}
- of $f$ is $1$ since the highest power of $x$ that is present in the
- formula for $f(x)$ is $1$. Similarly, the degree of $g$ is $2$ since
- the highest power of $x$ that is present in the formula for $g(x)$
- is $2$.
-   
- In this section we will build upon our knowledge of these elementary
- functions. In particular, we will generalize the functions $f$ and $g$ to a function $p$ that has
- any degree that we wish.
-   
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{essentialskills}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Quadratic functions]
- 	Every quadratic function has the form $y=ax^2+bx+c$; state the value
- 	of $a$ for each of the following functions, and hence decide if the
- 	parabola that represents the function opens upward or downward.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			$F(x)=x^2+3$
- 			\begin{shortsolution}
- 				$a=1$; the parabola opens upward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$G(t)=4-5t^2$
- 			\begin{shortsolution}
- 				$a=-5$; the parabola opens downward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$H(y)=4y^2-96y+8$
- 			\begin{shortsolution}
- 				$a=4$; the parabola opens upward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$K(z)=-19z^2$
- 			\begin{shortsolution}
- 				$m=-19$; the parabola opens downward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	Now let's generalize our findings for the most general quadratic function $g$
- 	that has formula $g(x)=a_2x^2+a_1x+a_0$. Complete the following sentences.
- 	\begin{subproblem}
- 		When $a_2>0$, the parabola that represents $y=g(x)$ opens $\ldots$
- 		\begin{shortsolution}
- 			When $a_2>0$, the parabola that represents the function opens upward.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		When $a_2<0$, the parabola that represents $y=g(x)$ opens $\ldots$
- 		\begin{shortsolution}
- 			When $a_2<0$, the parabola that represents the function opens downward.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- \end{essentialskills}
-   
- \subsection*{Power functions with positive exponents}
- 	The study of polynomials will rely upon a good knowledge
- 	of power functions| you may reasonably ask, what is a power function?
- 	\begin{pccdefinition}[Power functions]
- 		Power functions have the form
- 		\[
- 			f(x) = a_n x^n
- 		\]
- 		where $n$ can be any real number.
- 		 		 		
- 		Note that for this section we will only be concerned with the
- 		case when $n$ is a positive integer.
- 	\end{pccdefinition}
- 	 	 	
- 	You may find assurance in the fact that you are already very comfortable
- 	with power functions that have $n=1$ (linear) and $n=2$ (quadratic). Let's
- 	explore some power functions that you might not be so familiar with.
- 	As you read \cref{poly:ex:oddpow,poly:ex:evenpow}, try and spot
- 	as many patterns and similarities as you can.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Power functions with odd positive exponents]
- 		\label{poly:ex:oddpow}
- 		Graph each of the following functions, state their domain, and their
- 		long-run behavior as $x\rightarrow\pm\infty$
- 		\[
- 			f(x)=x^3,   \qquad  g(x)=x^5, \qquad h(x)=x^7
- 		\]
- 		\begin{pccsolution}
- 			The functions $f$, $g$, and $h$ are plotted in \cref{poly:fig:oddpow}.
- 			The domain of each of the functions $f$, $g$, and $h$ is $(-\infty,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				f(x)\rightarrow\infty                           & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}   f(x)\rightarrow-\infty & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same results hold for $g$ and $h$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-1.5,xmax=1.5,
- 						ymin=-5,ymax=5,
- 						xtick={-1.0,-0.5,...,1.0},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=north west,
- 					]
- 					\addplot expression[domain=-1.5:1.5]{x^3};
- 					\addplot expression[domain=-1.379:1.379]{x^5};
- 					\addplot expression[domain=-1.258:1.258]{x^7};
- 					\addplot[soldot]coordinates{(-1,-1)} node[axisnode,anchor=north west]{$(-1,-1)$};
- 					\addplot[soldot]coordinates{(1,1)} node[axisnode,anchor=south east]{$(1,1)$};
- 					\legend{$f$,$g$,$h$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{Odd power functions}
- 			\label{poly:fig:oddpow}
- 		\end{minipage}%
- 		\hfill
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-2.5,xmax=2.5,
- 						ymin=-5,ymax=5,
- 						xtick={-2.0,-1.5,...,2.0},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=south east,
- 					]
- 					\addplot expression[domain=-2.236:2.236]{x^2};
- 					\addplot expression[domain=-1.495:1.495]{x^4};
- 					\addplot expression[domain=-1.307:1.307]{x^6};
- 					\addplot[soldot]coordinates{(-1,1)} node[axisnode,anchor=east]{$(-1,1)$};
- 					\addplot[soldot]coordinates{(1,1)} node[axisnode,anchor=west]{$(1,1)$};
- 					\legend{$F$,$G$,$H$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{Even power functions}
- 			\label{poly:fig:evenpow}
- 		\end{minipage}%
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Power functions with even positive exponents]\label{poly:ex:evenpow}%
- 		Graph each of the following functions, state their domain, and their
- 		long-run behavior as $x\rightarrow\pm\infty$
- 		\[
- 			F(x)=x^2, \qquad G(x)=x^4, \qquad H(x)=x^6
- 		\]
- 		\begin{pccsolution}
- 			The functions $F$, $G$, and $H$ are plotted in \cref{poly:fig:evenpow}. The domain
- 			of each of the functions is $(-\infty,\infty)$. Note that the long-run behavior
- 			of each of the functions is the same, and in particular
- 			\begin{align*}
- 				F(x)\rightarrow\infty                          & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}   F(x)\rightarrow\infty & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same result holds for $G$ and $H$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Repeat \cref{poly:ex:oddpow,poly:ex:evenpow} using (respectively)
- 		\begin{subproblem}
- 			$f(x)=-x^3,   \qquad  g(x)=-x^5, \qquad h(x)=-x^7$
- 			\begin{shortsolution}
- 				The functions $f$, $g$, and $h$ have domain $(-\infty,\infty)$ and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-1.5,xmax=1.5,
- 							ymin=-5,ymax=5,
- 							xtick={-1.0,-0.5,...,0.5},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-1.5:1.5]{-x^3};
- 						\addplot expression[domain=-1.379:1.379]{-x^5};
- 						\addplot expression[domain=-1.258:1.258]{-x^7};
- 						\legend{$f$,$g$,$h$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					f(x)\rightarrow-\infty                         & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}   f(x)\rightarrow\infty & \text{ as } x\rightarrow-\infty 
- 				\end{align*}
- 				The same is true for $g$ and $h$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$F(x)=-x^2,   \qquad  G(x)=-x^4, \qquad H(x)=-x^6$
- 			\begin{shortsolution}
- 				The functions $F$, $G$, and $H$ have domain $(-\infty,\infty)$ and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-2.5,xmax=2.5,
- 							ymin=-5,ymax=5,
- 							xtick={-1.0,-0.5,...,0.5},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-2.236:2.236]{-x^2};
- 						\addplot expression[domain=-1.495:1.495]{-x^4};
- 						\addplot expression[domain=-1.307:1.307]{-x^6};
- 						\legend{$F$,$G$,$H$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					F(x)\rightarrow-\infty                          & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}   F(x)\rightarrow-\infty & \text{ as } x\rightarrow-\infty 
- 				\end{align*}
- 				The same is true for $G$ and $H$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- \subsection*{Polynomial functions}
- 	Now that we have a little more familiarity with power functions,
- 	we can define polynomial functions. Provided that you were comfortable
- 	with our opening discussion about linear and quadratic functions (see
- 	$f$ and $g$ in \cref{poly:eq:linquad}) then there is every chance
- 	that you'll be able to master polynomial functions as well; just remember
- 	that polynomial functions are a natural generalization of linear
- 	and quadratic functions. Once you've studied the examples and problems
- 	in this section, you'll hopefully agree that polynomial functions
- 	are remarkably predictable.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccdefinition}[Polynomial functions]
- 		Polynomial functions have the form
- 		\[
- 			p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0
- 		\]
- 		where $a_n$, $a_{n-1}$, $a_{n-2}$, \ldots, $a_0$ are real numbers.
- 		\begin{itemize}
- 			\item We call $n$ the degree of the polynomial, and require that $n$
- 			      is a non-negative integer;
- 			\item $a_n$, $a_{n-1}$, $a_{n-2}$, \ldots, $a_0$ are called the coefficients;
- 			\item We typically write polynomial functions in descending powers of $x$.
- 		\end{itemize}
- 		In particular, we call $a_n$ the \emph{leading} coefficient, and $a_nx^n$ the
- 		\emph{leading term}.
- 		 		 		
- 		Note that if a polynomial is given in factored form, then the degree can be found
- 		by counting the number of linear factors.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Polynomial or not]
- 		Identify the following functions as polynomial or not; if the function
- 		is a polynomial, state its degree.
- 		\begin{multicols}{3}
- 			\begin{enumerate}
- 				\item $p(x)=x^2-3$
- 				\item $q(x)=-4x^{\nicefrac{1}{2}}+10$
- 				\item $r(x)=10x^5$
- 				\item $s(x)=x^{-2}+x^{23}$
- 				\item $f(x)=-8$
- 				\item $g(x)=3^x$
- 				\item $h(x)=\sqrt[3]{x^7}-x^2+x$
- 				\item $k(x)=4x(x+2)(x-3)$
- 				\item $j(x)=x^2(x-4)(5-x)$
- 			\end{enumerate}
- 		\end{multicols}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item $p$ is a polynomial, and its degree is $2$.
- 				\item $q$ is \emph{not} a polynomial, because $\frac{1}{2}$ is not an integer.
- 				\item $r$ is a polynomial, and its degree is $5$.
- 				\item $s$ is \emph{not} a polynomial, because $-2$ is not a positive integer.
- 				\item $f$ is a polynomial, and its degree is $0$.
- 				\item $g$ is \emph{not} a polynomial, because the independent
- 				      variable, $x$, is in the exponent.
- 				\item $h$ is \emph{not} a polynomial, because $\frac{7}{3}$ is not an integer.
- 				\item $k$ is a polynomial, and its degree is $3$.
- 				\item $j$ is a polynomial, and its degree is $4$.
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Typical graphs]\label{poly:ex:typical}
- 		\Cref{poly:fig:typical} shows graphs of some polynomial functions;
- 		the ticks have deliberately been left off the axis to allow us to concentrate
- 		on the features of each graph. Note in particular that:
- 		\begin{itemize}
- 			\item \cref{poly:fig:typical1} shows a degree-$1$ polynomial (you might also
- 			      classify the function as linear) whose leading coefficient, $a_1$, is positive.
- 			\item \cref{poly:fig:typical2} shows a degree-$2$ polynomial (you might also
- 			      classify the function as quadratic) whose leading coefficient, $a_2$, is positive.
- 			\item \cref{poly:fig:typical3} shows a degree-$3$ polynomial whose leading coefficient, $a_3$,
- 			      is positive| compare its overall
- 			      shape and long-run behavior to the functions described in \cref{poly:ex:oddpow}.
- 			\item \cref{poly:fig:typical4} shows a degree-$4$ polynomial whose leading coefficient, $a_4$,
- 			      is positive|compare its overall shape and long-run behavior to the functions described in \cref{poly:ex:evenpow}.
- 			\item \cref{poly:fig:typical5} shows a degree-$5$ polynomial whose leading coefficient, $a_5$,
- 			      is positive| compare its overall
- 			      shape and long-run behavior to the functions described in \cref{poly:ex:oddpow}.
- 		\end{itemize}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{\textwidth/6}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-10:8]{(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_1>0$}
- 			\label{poly:fig:typical1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-4:4]{(x^2-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_2>0$}
- 			\label{poly:fig:typical2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-7.5:7.5]{0.05*(x+6)*x*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_3>0$}
- 			\label{poly:fig:typical3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-2.35:5.35,samples=100]{0.2*(x-5)*x*(x-3)*(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_4>0$}
- 			\label{poly:fig:typical4}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-5.5:6.3,samples=100]{0.01*(x+2)*x*(x-3)*(x+5)*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_5>0$}
- 			\label{poly:fig:typical5}
- 		\end{subfigure}
- 		\end{widepage}
- 		\caption{Graphs to illustrate typical curves of polynomial functions.}
- 		\label{poly:fig:typical}
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Use \cref{poly:ex:typical} and \cref{poly:fig:typical} to help you sketch
- 		the graphs of polynomial functions that have negative leading coefficients| note
- 		that there are many ways to do this! The intention with this problem
- 		is to use your knowledge of transformations- in particular, \emph{reflections}-
- 		to guide you.
- 		\begin{shortsolution}
- 			$a_1<0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-10:8]{-(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_2<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-4:4]{-(x^2-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_3<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-7.5:7.5]{-0.05*(x+6)*x*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_4<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-2.35:5.35,samples=100]{-0.2*(x-5)*x*(x-3)*(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_5<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-5.5:6.3,samples=100]{-0.01*(x+2)*x*(x-3)*(x+5)*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- 	\fixthis{poly: Need a more basic example here- it can have a similar
- 		format to the multiple zeros example, but just keep it simple; it should
- 	be halfway between the 2 examples surrounding it}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Multiple zeros]
- 		Consider the polynomial functions $p$, $q$, and $r$ which are
- 		graphed in \cref{poly:fig:moremultiple}.
- 		The formulas for $p$, $q$, and $r$ are as follows
- 		\begin{align*}
- 			p(x) & =(x-3)^2(x+4)^2       \\
- 			q(x) & =x(x+2)^2(x-1)^2(x-3) \\
- 			r(x) & =x(x-3)^3(x+1)^2      
- 		\end{align*}
- 		Find the degree of $p$, $q$, and $r$, and decide if the functions bounce off or cut
- 		through the horizontal axis at each of their zeros.
- 		\begin{pccsolution}
- 			The degree of $p$ is 4. Referring to \cref{poly:fig:bouncep},
- 			the curve bounces off the horizontal axis at both zeros, $3$ and $4$.
- 			 			 			
- 			The degree of $q$ is 6. Referring to \cref{poly:fig:bounceq},
- 			the curve bounces off the horizontal axis at $-2$ and $1$, and cuts
- 			through the horizontal axis at $0$ and $3$.
- 			 			 			
- 			The degree of $r$ is 6. Referring to \cref{poly:fig:bouncer},
- 			the curve bounces off the horizontal axis at $-1$, and cuts through
- 			the horizontal axis at $0$ and at $3$, although is flattened immediately to the left and right of $3$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\setlength{\figurewidth}{0.25\textwidth}
- 	\begin{figure}[!htb]
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-6,xmax=5,
- 						ymin=-30,ymax=200,
- 						xtick={-4,-2,...,4},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-5.63733:4.63733,samples=50]{(x-3)^2*(x+4)^2};
- 					\addplot[soldot]coordinates{(3,0)(-4,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=p(x)$}
- 			\label{poly:fig:bouncep}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-3,xmax=4,
- 						xtick={-2,...,3},
- 						ymin=-60,ymax=40,
- 						width=\textwidth,
- 					]
- 					\addplot+[samples=50] expression[domain=-2.49011:3.11054]{x*(x+2)^2*(x-1)^2*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(0,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=q(x)$}
- 			\label{poly:fig:bounceq}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-2,xmax=4,
- 						xtick={-1,...,3},
- 						ymin=-40,ymax=40,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-1.53024:3.77464,samples=50]{x*(x-3)^3*(x+1)^2};
- 					\addplot[soldot]coordinates{(-1,0)(0,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=r(x)$}
- 			\label{poly:fig:bouncer}
- 		\end{subfigure}
- 		\caption{}
- 		\label{poly:fig:moremultiple}
- 	\end{figure}
- 	 	 	
- 	\begin{pccdefinition}[Multiple zeros]\label{poly:def:multzero}
- 		Let $p$ be a polynomial that has a repeated linear factor $(x-a)^n$. Then we say
- 		that $p$ has a multiple zero at $a$ of multiplicity $n$ and
- 		\begin{itemize}
- 			\item if the factor $(x-a)$ is repeated an even number of times, the graph of $y=p(x)$ does not
- 			      cross the $x$ axis at $a$, but `bounces' off the horizontal axis at $a$.
- 			\item if the factor $(x-a)$ is repeated an odd number of times, the graph of $y=p(x)$ crosses the
- 			      horizontal axis at $a$, but it looks `flattened' there
- 		\end{itemize}
- 		If $n=1$, then we say that $p$ has a \emph{simple} zero at $a$.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Find a formula]
- 		Find formulas for the polynomial functions, $p$ and $q$, graphed in \cref{poly:fig:findformulademoboth}.
- 		\begin{figure}[!htb]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-5,xmax=5,
- 							ymin=-10,ymax=10,
- 							xtick={-4,-2,...,4},
- 							minor xtick={-3,-1,...,3},
- 							ytick={-8,-6,...,8},
- 							width=\textwidth,
- 						grid=both]
- 						\addplot expression[domain=-3.25842:2.25842,samples=50]{-x*(x-2)*(x+3)*(x+1)};
- 						\addplot[soldot]coordinates{(1,8)}node[axisnode,inner sep=.35cm,anchor=west]{$(1,8)$};
- 						\addplot[soldot]coordinates{(-3,0)(-1,0)(0,0)(2,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$p$}
- 				\label{poly:fig:findformulademo}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-5,xmax=5,
- 							ymin=-10,ymax=10,
- 							xtick={-4,-2,...,4},
- 							minor xtick={-3,-1,...,3},
- 							ytick={-8,-6,...,8},
- 							width=\textwidth,
- 						grid=both]
- 						\addplot expression[domain=-4.33:4.08152]{-.25*(x+2)^2*(x-3)};
- 						\addplot[soldot]coordinates{(2,4)}node[axisnode,anchor=south west]{$(2,4)$};
- 						\addplot[soldot]coordinates{(-2,0)(3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$q$}
- 				\label{poly:fig:findformulademo1}
- 			\end{subfigure}
- 			\caption{}
- 			\label{poly:fig:findformulademoboth}
- 		\end{figure}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item We begin by noting that the horizontal intercepts of $p$ are $(-3,0)$, $(-1,0)$, $(0,0)$ and $(2,0)$.
- 				      We also note that each zero is simple (multiplicity $1$).
- 				      If we assume that $p$ has no other zeros, then we can start by writing
- 				      \begin{align*}
- 				      	p(x) & =(x+3)(x+1)(x-0)(x-2) \\
- 				      	     & =x(x+3)(x+1)(x-2)     \\
- 				      \end{align*}
- 				      According to \cref{poly:fig:findformulademo}, the point $(1,8)$ lies
- 				      on the curve $y=p(x)$.
- 				      Let's check if the formula we have written satisfies this requirement
- 				      \begin{align*}
- 				      	p(1) & = (1)(4)(2)(-1) \\
- 				      	     & = -8            
- 				      \end{align*}
- 				      which is clearly not correct| it is close though. We can correct this by
- 				      multiplying $p$ by a constant $k$; so let's assume that
- 				      \[
- 				      	p(x)=kx(x+3)(x+1)(x-2)
- 				      \]
- 				      Then $p(1)=-8k$, and if this is to equal $8$, then $k=-1$. Therefore
- 				      the formula for $p(x)$ is
- 				      \[
- 				      	p(x)=-x(x+3)(x+1)(x-2)
- 				      \]
- 				\item The function $q$ has a zero at $-2$ of multiplicity $2$, and zero of
- 				      multiplicity $1$ at $3$ (so $3$ is a simple zero of $q$); we can therefore assume that $q$ has the form
- 				      \[
- 				      	q(x)=k(x+2)^2(x-3)
- 				      \]
- 				      where $k$ is some real number. In order to find $k$, we use the given ordered pair, $(2,4)$, and
- 				      evaluate $p(2)$
- 				      \begin{align*}
- 				      	p(2) & =k(4)^2(-1) \\
- 				      	     & =-16k       
- 				      \end{align*}
- 				      We solve the equation $4=-8k$ and obtain $k=-\frac{1}{4}$ and conclude that the
- 				      formula for $q(x)$ is
- 				      \[
- 				      	q(x)=-\frac{1}{4}(x+2)^2(x-3)
- 				      \]
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	\fixthis{Chris: need sketching polynomial problems}
- 	\begin{pccspecialcomment}[Steps to follow when sketching polynomial functions]
- 		\begin{steps}
- 			\item \label{poly:step:first} Determine the degree of the polynomial,
- 			its leading term and leading coefficient, and hence determine
- 			the long-run behavior of the polynomial| does it behave like $\pm x^2$ or $\pm x^3$
- 			as $x\rightarrow\pm\infty$?
- 			\item Determine the zeros and their multiplicity. Mark all zeros
- 			and the vertical intercept on the graph using solid circles $\bullet$.
- 			\item \label{poly:step:last}  Deduce the overall shape of the curve, and sketch it. If there isn't
- 			enough information from the previous steps, then construct a table of values.
- 		\end{steps}
- 		Remember that until we have the tools of calculus, we won't be able to
- 		find the exact coordinates of local minimums, local maximums, and points
- 		of inflection.
- 	\end{pccspecialcomment}
- 	Before we demonstrate some examples, it is important to remember the following:
- 	\begin{itemize}
- 		\item our sketches will give a good representation of the overall
- 		      shape of the graph, but until we have the tools of calculus (from MTH 251)
- 		      we can not find local minimums, local maximums, and inflection points algebraically. This
- 		      means that we will make our best guess as to where these points are.
- 		\item we will not concern ourselves too much with the vertical scale (because of
- 		      our previous point)| we will, however, mark the vertical intercept (assuming there is one),
- 		      and any horizontal asymptotes.
- 	\end{itemize}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:simplecubic}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $p$
- 		that has formula
- 		\[
- 			p(x)=\frac{1}{2}(x-4)(x-1)(x+3)
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $p$ has degree $3$. The leading term of $p$ is $\frac{1}{2}x^3$, so the leading coefficient of $p$
- 				is $\frac{1}{2}$. The long-run behavior of $p$ is therefore similar to that of $x^3$.
- 				\item The zeros of $p$ are $-3$, $1$, and $4$; each zero is simple (i.e, it has multiplicity $1$).
- 				This means that the curve of $p$ cuts the horizontal axis at each zero. The vertical
- 				intercept of $p$ is $(0,6)$.
- 				\item We draw the details we have obtained so far on \cref{poly:fig:simplecubicp1}. Given
- 				that the curve of $p$ looks like the curve of $x^3$ in the long-run, we are able to complete a sketch of the
- 				graph of $p$ in \cref{poly:fig:simplecubicp2}.
- 				 				 				
- 				Note that we can not find the coordinates of the local minimums, local maximums, and inflection
- 				points| for the moment we make reasonable guesses as to where these points are (you'll find how
- 				to do this in calculus).
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=15,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-3,0)(1,0)(4,0)(0,6)}node[axisnode,anchor=south west]{$(0,6)$};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:simplecubicp1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=15,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-3,0)(1,0)(4,0)(0,6)}node[axisnode,anchor=south west]{$(0,6)$};
- 							\addplot[pccplot] expression[domain=-3.57675:4.95392,samples=100]{.5*(x-4)*(x-1)*(x+3)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:simplecubicp2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{2}(x-4)(x-1)(x+3)$}
- 				\label{poly:fig:simplecubic}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:degree5}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $q$
- 		that has formula
- 		\[
- 			q(x)=\frac{1}{200}(x+7)^2(2-x)(x-6)^2
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $q$ has degree $4$. The leading term of $q$ is
- 				\[
- 					-\frac{1}{200}x^5
- 				\]
- 				so the leading coefficient of $q$ is $-\frac{1}{200}$. The long-run behavior of $q$
- 				is therefore similar to that of $-x^5$.
- 				\item The zeros of $q$ are $-7$ (multiplicity 2), $2$ (simple), and $6$ (multiplicity $2$).
- 				The curve of $q$ bounces off the horizontal axis at the zeros with multiplicity $2$ and
- 				cuts the horizontal axis at the simple zeros. The vertical intercept of $q$ is $\left( 0,\frac{441}{25} \right)$.
- 				\item We mark the details we have found so far on \cref{poly:fig:degree5p1}. Given that
- 				the curve of $q$ looks like the curve of $-x^5$ in the long-run, we can complete \cref{poly:fig:degree5p2}.
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=40,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,0,...,35},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-7,0)(2,0)(6,0)(0,441/25)}node[axisnode,anchor=south west]{$\left( 0, \frac{441}{25} \right)$};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree5p1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=40,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,0,...,35},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-7,0)(2,0)(6,0)(0,441/25)}node[axisnode,anchor=south west]{$\left( 0, \frac{441}{25} \right)$};
- 							\addplot[pccplot] expression[domain=-8.83223:7.34784,samples=50]{1/200*(x+7)^2*(2-x)*(x-6)^2};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree5p2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{200}(x+7)^2(2-x)(x-6)^2$}
- 				\label{poly:fig:degree5}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $r$
- 		that has formula
- 		\[
- 			r(x)=\frac{1}{100}x^3(x+4)(x-4)(x-6)
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $r$ has degree $6$. The leading term of $r$ is
- 				\[
- 					\frac{1}{100}x^6
- 				\]
- 				so the leading coefficient of $r$ is $\frac{1}{100}$. The long-run behavior of $r$
- 				is therefore similar to that of $x^6$.
- 				\item The zeros of $r$ are $-4$ (simple), $0$ (multiplicity $3$), $4$ (simple),
- 				and $6$ (simple). The vertical intercept of $r$ is $(0,0)$. The curve of $r$
- 				cuts the horizontal axis at the simple zeros, and goes through the axis
- 				at $(0,0)$, but does so in a flattened way.
- 				\item We mark the zeros and vertical intercept on \cref{poly:fig:degree6p1}. Given that
- 				the curve of $r$ looks like the curve of $x^6$ in the long-run, we complete the graph
- 				of $r$ in \cref{poly:fig:degree6p2}.
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-5,xmax=10,
- 								ymin=-20,ymax=10,
- 								xtick={-4,-2,...,8},
- 								ytick={-15,-10,...,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-4,0)(0,0)(4,0)(6,0)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree6p1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-5,xmax=10,
- 								ymin=-20,ymax=10,
- 								xtick={-4,-2,...,8},
- 								ytick={-15,-10,...,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-4,0)(0,0)(4,0)(6,0)};
- 							\addplot[pccplot] expression[domain=-4.16652:6.18911,samples=100]{1/100*(x+4)*x^3*(x-4)*(x-6)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree6p2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{100}(x+4)x^3(x-4)(x-6)$}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[An open-topped box]
- 		A cardboard company makes open-topped boxes for their clients. The specifications
- 		dictate that the box must have a square base, and that it must be open-topped.
- 		The company uses sheets of cardboard that are $\unit[1200]{cm^2}$. Assuming that
- 		the base of each box has side $x$ (measured in cm), it can be shown that the volume of each box, $V(x)$,
- 		has formula
- 		\[
- 			V(x)=\frac{x}{4}(1200-x^2)
- 		\]
- 		Find the dimensions of the box that maximize the volume.
- 		\begin{pccsolution}
- 			We graph $y=V(x)$ in \cref{poly:fig:opentoppedbox}. Note that because
- 			$x$ represents the length of a side, and $V(x)$ represents the volume
- 			of the box, we necessarily require both values to be positive; we illustrate
- 			the part of the curve that applies to this problem using a solid line.
- 			 			 			
- 			\begin{figure}[!htb]
- 				\centering
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-50,xmax=50,
- 							ymin=-5000,ymax=5000,
- 							xtick={-40,-30,...,40},
- 							minor xtick={-45,-35,...,45},
- 							minor ytick={-3000,-1000,1000,3000},
- 							width=.75\textwidth,
- 							height=.5\textwidth,
- 						grid=both]
- 						\addplot[pccplot,dashed,<-] expression[domain=-40:0,samples=50]{x/4*(1200-x^2)};
- 						\addplot[pccplot,-] expression[domain=0:34.64,samples=50]{x/4*(1200-x^2)};
- 						\addplot[pccplot,dashed,->] expression[domain=34.64:40,samples=50]{x/4*(1200-x^2)};
- 						\addplot[soldot] coordinates{(20,4000)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=V(x)$}
- 				\label{poly:fig:opentoppedbox}
- 			\end{figure}
- 			 			 			
- 			According to \cref{poly:fig:opentoppedbox}, the maximum volume of such a box is
- 			approximately $\unit[4000]{cm^2}$, and we achieve it using a base of length
- 			approximately $\unit[20]{cm}$. Since the base is square and each sheet of cardboard
- 			is $\unit[1200]{cm^2}$, we conclude that the dimensions of each box are $\unit[20]{cm}\times\unit[20]{cm}\times\unit[30]{cm}$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- \subsection*{Complex zeros}
- 	There has been a pattern to all of the examples that we have seen so far|
- 	the degree of the polynomial has dictated the number of \emph{real} zeros that the
- 	polynomial has. For example, the function $p$ in \cref{poly:ex:simplecubic}
- 	has degree $3$, and $p$ has $3$ real zeros; the function $q$ in \cref{poly:ex:degree5}
- 	has degree $5$ and $q$ has $5$ real zeros.
- 	 	 	
- 	You may wonder if this result can be generalized| does every polynomial that
- 	has degree $n$ have $n$ real zeros? Before we tackle the general result,
- 	let's consider an example that may help motivate it.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:complx}
- 		Consider the polynomial function $c$ that has formula
- 		\[
- 			c(x)=x(x^2+1)
- 		\]
- 		It is clear that $c$ has degree $3$, and that $c$ has a (simple) zero at $0$. Does
- 		$c$ have any other zeros, i.e, can we find any values of $x$ that satisfy the equation
- 		\begin{equation}\label{poly:eq:complx}
- 			x^2+1=0
- 		\end{equation}
- 		The solutions to \cref{poly:eq:complx} are $\pm i$.
- 		 		 		
- 		We conclude that $c$ has $3$ zeros: $0$ and $\pm i$; we note that \emph{not
- 		all of them are real}.
- 	\end{pccexample}
- 	\Cref{poly:ex:complx} shows that not every degree-$3$ polynomial has $3$
- 	\emph{real} zeros; however, if we are prepared to venture into the complex numbers,
- 	then we can state the following theorem.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccspecialcomment}[The fundamental theorem of algebra]
- 		Every polynomial function of degree $n$ has $n$ roots, some of which may
- 		be complex, and some may be repeated.
- 	\end{pccspecialcomment}
- 	\fixthis{Fundamental theorem of algebra: is this wording ok? do we want
- 	it as a theorem?}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}
- 		Find all the zeros of the polynomial function $p$ that has formula
- 		\[
- 			p(x)=x^4-2x^3+5x^2
- 		\]
- 		\begin{pccsolution}
- 			We begin by factoring $p$
- 			\begin{align*}
- 				p(x) & =x^4-2x^3+5x^2 \\
- 				     & =x^2(x^2-2x+5) 
- 			\end{align*}
- 			We note that $0$ is a zero of $p$ with multiplicity $2$. The other zeros of $p$
- 			can be found by solving the equation
- 			\[
- 				x^2-2x+5=0
- 			\]
- 			This equation can not be factored, so we use the quadratic formula
- 			\begin{align*}
- 				x & =\frac{2\pm\sqrt{(-2)^2}-20}{2(1)} \\
- 				  & =\frac{2\pm\sqrt{-16}}{2}          \\
- 				  & =1\pm 2i                           
- 			\end{align*}
- 			We conclude that $p$ has $4$ zeros: $0$ (multiplicity $2$), and $1\pm 2i$ (simple).
- 		\end{pccsolution}
- 	\end{pccexample}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}
- 		Find a polynomial that has zeros at $2\pm i\sqrt{2}$.
- 		\begin{pccsolution}
- 			We know that the zeros of a polynomial can be found by analyzing the linear
- 			factors. We are given the zeros, and have to work backwards to find the
- 			linear factors.
- 			 			 			
- 			We begin by assuming that $p$ has the form
- 			\begin{align*}
- 				p(x) & =(x-(2-i\sqrt{2}))(x-(2+i\sqrt{2}))                           \\
- 				     & =x^2-x(2+i\sqrt{2})-x(2-i\sqrt{2})+(2-i\sqrt{2})(2+i\sqrt{2}) \\
- 				     & =x^2-4x+(4-2i^2)                                              \\
- 				     & =x^2-4x+6                                                     
- 			\end{align*}
- 			We conclude that a possible formula for a polynomial function, $p$,
- 			that has zeros at $2\pm i\sqrt{2}$ is
- 			\[
- 				p(x)=x^2-4x+6
- 			\]
- 			Note that we could multiply $p$ by any real number and still ensure
- 			that $p$ has the same zeros.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	\investigation*{}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a graph]
- 	For each of the polynomials in \cref{poly:fig:findformula}
- 	\begin{enumerate}
- 		\item count the number of times the curve turns round, and cuts/bounces off the $x$ axis;
- 		\item approximate the degree of the polynomial;
- 		\item use your information to find the linear factors of each polynomial, and therefore write a possible formula for each;
- 		\item make sure your polynomial goes through the given ordered pair.
- 	\end{enumerate}
- 	\begin{shortsolution}
- 		\Vref{poly:fig:findformdeg2}:
- 		\begin{enumerate}
- 			\item the curve turns round once;
- 			\item the degree could be 2;
- 			\item based on the zeros, the linear factors are $(x+5)$ and $(x-3)$; since the
- 			      graph opens downwards, we will assume the leading coefficient is negative: $p(x)=-k(x+5)(x-3)$;
- 			\item $p$ goes through $(2,2)$, so we need to solve $2=-k(7)(-1)$ and therefore $k=\nicefrac{2}{7}$, so
- 			      \[
- 			      	p(x)=-\frac{2}{7}(x+5)(x-3)
- 			      \]
- 		\end{enumerate}
- 		\Vref{poly:fig:findformdeg3}:
- 		\begin{enumerate}
- 			\item the curve turns around twice;
- 			\item the degree could be 3;
- 			\item based on the zeros, the linear factors are $(x+2)^2$, and $(x-1)$;
- 			      based on the behavior of $p$, we assume that the leading coefficient is positive, and try $p(x)=k(x+2)^2(x-1)$;
- 			\item $p$ goes through $(0,-2)$, so we need to solve $-2=k(4)(-1)$ and therefore $k=\nicefrac{1}{2}$, so
- 			      \[
- 			      	p(x)=\frac{1}{2}(x+2)^2(x-1)
- 			      \]
- 		\end{enumerate}
- 		\Vref{poly:fig:findformdeg5}:
- 		\begin{enumerate}
- 			\item the curve turns around 4 times;
- 			\item the degree could be 5;
- 			\item based on the zeros, the linear factors are $(x+5)^2$, $(x+1)$, $(x-2)$, $(x-3)$;
- 			      based on the behavior of $p$, we assume that the leading coefficient is positive, and try $p(x)=k(x+5)^2(x+1)(x-2)(x-3)$;
- 			\item $p$ goes through $(-3,-50)$, so we need to solve $-50=k(64)(-2)(-5)(-6)$ and therefore $k=\nicefrac{5}{384}$, so
- 			      \[
- 			      	p(x)=\frac{5}{384}(x+5)^2(x+1)(x-2)(x-3)
- 			      \]
- 		\end{enumerate}
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-2,ymax=5,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.5:3.75]{-1/3*(x+4)*(x-3)};
- 					\addplot[soldot] coordinates{(-4,0)(3,0)(2,2)} node[axisnode,above right]{$(2,2)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-3,xmax=2,
- 						ymin=-2,ymax=4,
- 						xtick={-2,...,1},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.95:1.75]{1/3*(x+2)^2*(x-1)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(0,-1.33)}node[axisnode,anchor=north west]{$(0,-2)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-100,ymax=150,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.5:3.4,samples=50]{(x+4)^2*(x+1)*(x-2)*(x-3)};
- 					\addplot[soldot]coordinates{(-4,0)(-1,0)(2,0)(3,0)(-3,-60)}node[axisnode,anchor=north]{$(-3,-50)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg5}
- 		\end{subfigure}
- 		\caption{}
- 		\label{poly:fig:findformula}
- 	\end{figure}
- 	 	 	
- 	 	 	
- 	 	 	
- 	 	 	
- 	\begin{exercises}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Prerequisite classifacation skills]
- 	Decide if each of the following functions are linear or quadratic.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x)=2x+3$
- 			\begin{shortsolution}
- 				$f$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x)=10-7x$
- 			\begin{shortsolution}
- 				$g$ is linear
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x)=-x^2+3x-9$
- 			\begin{shortsolution}
- 				$h$ is quadratic.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(x)=-17$
- 			\begin{shortsolution}
- 				$k$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(x)=-82x^2-4$
- 			\begin{shortsolution}
- 				$l$ is quadratic
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(x)=6^2x-8$
- 			\begin{shortsolution}
- 				$m$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Prerequisite slope identification]
- 	State the slope of each of the following linear functions, and
- 	hence decide if each function is increasing or decreasing.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$\alpha(x)=4x+1$
- 			\begin{shortsolution}
- 				$m=4$; $\alpha$ is increasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\beta(x)=-9x$
- 			\begin{shortsolution}
- 				$m=-9$; $\beta$ is decreasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\gamma(t)=18t+100$
- 			\begin{shortsolution}
- 				$m=18$; $\gamma$ is increasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\delta(y)=23-y$
- 			\begin{shortsolution}
- 				$m=-1$; $\delta$ is decreasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	Now let's generalize our findings for the most general linear function $f$
- 	that has formula $f(x)=mx+b$. Complete the following sentences.
- 	\begin{subproblem}
- 		When $m>0$, the function $f$ is $\ldots$
- 		\begin{shortsolution}
- 			When $m>0$, the function $f$ is $\ldots$  \emph{increasing}.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		When $m<0$, the function $f$ is $\ldots$
- 		\begin{shortsolution}
- 			When $m<0$, the function $f$ is $\ldots$  \emph{decreasing}.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Polynomial or not?]
- 	Identify whether each of the following functions is a polynomial or not.
- 	If the function is a polynomial, state its degree.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$p(x)=2x+1$
- 			\begin{shortsolution}
- 				$p$ is a polynomial (you might also describe $p$ as linear). The degree of $p$ is 1.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=7x^2+4x$
- 			\begin{shortsolution}
- 				$p$ is a polynomial (you might also describe $p$ as quadratic). The degree of $p$ is 2.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=\sqrt{x}+2x+1$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial; we require the powers of $x$ to be integer values.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=2^x-45$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial; the $2^x$ term is exponential.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=6x^4-5x^3+9$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $6$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=-5x^{17}+9x+2$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is 17.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=4x(x+7)^2(x-3)^3$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $6$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=4x^{-5}-x^2+x$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial because $-5$ is not a positive integer.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=-x^6(x^2+1)(x^3-2)$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $11$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Polynomial graphs]
- 	Three polynomial functions $p$, $m$, and $n$ are shown in \crefrange{poly:fig:functionp}{poly:fig:functionn}.
- 	The functions have the following formulas
- 	\begin{align*}
- 		p(x) & = (x-1)(x+2)(x-3)           \\
- 		m(x) & = -(x-1)(x+2)(x-3)          \\
- 		n(x) & = (x-1)(x+2)(x-3)(x+1)(x+4) 
- 	\end{align*}
- 	Note that for our present purposes we are not concerned with the vertical scale of the graphs.
- 	\begin{subproblem}
- 		Identify both on the graph {\em and} algebraically, the zeros of each polynomial.
- 		\begin{shortsolution}
- 			$y=p(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot] coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$y=m(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{-1*(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot] coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$y=n(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-90,ymax=70,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-4.15:3.15,samples=50]{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)};
- 					\addplot[soldot] coordinates{(-4,0)(-2,0)(-1,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			The zeros of $p$ are $-2$, $1$, and $3$; the zeros of $m$ are $-2$, $1$, and $3$; the zeros of $n$ are
- 			$-4$, $-2$, $-1$, and $3$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Write down the degree, how many times the curve of each function `turns around',
- 		and how many zeros it has
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item The degree of $p$ is 3, and the curve $y=p(x)$ turns around twice.
- 				\item The degree of $q$ is also 3, and the curve $y=q(x)$ turns around twice.
- 				\item The degree of $n$ is $5$, and the curve $y=n(x)$ turns around 4 times.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						ytick={-5,5},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=p(x)$}
- 			\label{poly:fig:functionp}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						ytick={-5,5},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{-1*(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=m(x)$}
- 			\label{poly:fig:functionm}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-90,ymax=70,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.15:3.15,samples=100]{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)};
- 					\addplot[soldot]coordinates{(-4,0)(-2,0)(-1,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=n(x)$}
- 			\label{poly:fig:functionn}
- 		\end{subfigure}
- 		\caption{}
- 		\end{widepage}
- 	\end{figure}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Horizontal intercepts]\label{poly:prob:matchpolys}%
- 	State the horizontal intercepts (as ordered pairs) of the following polynomials.
- 	\begin{multicols}{2}
- 		\begin{subproblem}\label{poly:prob:degree5}
- 			$p(x)=(x-1)(x+2)(x-3)(x+1)(x+4)$
- 			\begin{shortsolution}
- 				$(-4,0)$, $(-2,0)$, $(-1,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(x)=-(x-1)(x+2)(x-3)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(x)=(x-1)(x+2)(x-3)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}\label{poly:prob:degree2}
- 			$s(x)=(x-2)(x+2)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(2,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Minimums, maximums, and concavity]\label{poly:prob:incdec}
- 	Four polynomial functions are graphed in \cref{poly:fig:incdec}. The formulas
- 	for these functions are (not respectively)
- 	\begin{gather*}
- 		p(x)=\frac{x^3}{6}-\frac{x^2}{4}-3x, \qquad q(x)=\frac{x^4}{20}+\frac{x^3}{15}-\frac{6}{5}x^2+1\\
- 		r(x)=-\frac{x^5}{50}-\frac{x^4}{40}+\frac{2x^3}{5}+6, \qquad s(x)=-\frac{x^6}{6000}-\frac{x^5}{2500}+\frac{67x^4}{4000}+\frac{17x^3}{750}-\frac{42x^2}{125}
- 	\end{gather*}
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{.23\textwidth}
- 		\centering
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=major,
- 					]
- 					\addplot expression[domain=-5.28:4.68,samples=50]{-x^5/50-x^4/40+2*x^3/5+6};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=major,
- 					]
- 					\addplot expression[domain=-6.08:4.967,samples=50]{x^4/20+x^3/15-6/5*x^2+1};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-6,xmax=8,ymin=-10,ymax=10,
- 						xtick={-4,-2,...,6},
- 						ytick={-8,-4,4,8},
- 						minor ytick={-6,-2,...,6},
- 						grid=both,
- 					]
- 					\addplot expression[domain=-4.818:6.081,samples=50]{x^3/6-x^2/4-3*x};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,ymin=-10,ymax=10,
- 						xtick={-8,-4,4,8},
- 						ytick={-8,-4,4,8},
- 						minor xtick={-6,-2,...,6},
- 						minor ytick={-6,-2,...,6},
- 						grid=both,
- 					]
- 					\addplot expression[domain=-9.77:8.866,samples=50]{-x^6/6000-x^5/2500+67*x^4/4000+17/750*x^3-42/125*x^2};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec4}
- 		\end{subfigure}
- 		\caption{Graphs for \cref{poly:prob:incdec}.}
- 		\label{poly:fig:incdec}
- 		\end{widepage}
- 	\end{figure}
- 	\begin{subproblem}
- 		Match each of the formulas with one of the given graphs.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is graphed in \vref{poly:fig:incdec1};
- 				\item $q$ is graphed in \vref{poly:fig:incdec2};
- 				\item $r$ is graphed in \vref{poly:fig:incdec3};
- 				\item $s$ is graphed in \vref{poly:fig:incdec4}.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the zeros of each function using the appropriate graph.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ has simple zeros at about $-3.8$, $0$, and $5$.
- 				\item $q$ has simple zeros at about $-5.9$, $-1$, $1$, and $4$.
- 				\item $r$ has simple zeros at about $-5$, $-2.9$, and $4.1$.
- 				\item $s$ has simple zeros at about $-9$, $-6$, $4.2$, $8.1$, and a zero of multiplicity $2$ at $0$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the local maximums and minimums of each of the functions.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ has a local maximum of approximately $3.9$ at $-2$, and a local minimum of approximately $-6.5$ at $3$.
- 				\item $q$ has a local minimum of approximately $-10$ at $-4$, and $-4$ at $3$; $q$ has a local maximum of approximately $1$ at $0$.
- 				\item $r$ has a local minimum of approximately $-5.5$ at $-4$, and a local maximum of approximately $10$ at $3$.
- 				\item $s$ has a local maximum of approximately $5$ at $-8$, $0$ at $0$, and $5$ at  $7$; $s$ has local minimums
- 				      of approximately $-3$ at $-4$, and $-1$ at $3$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the global maximums and minimums of each of the functions.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ does not have a global maximum, nor a global minimum.
- 				\item $q$ has a global minimum of approximately $-10$; it does not have a global maximum.
- 				\item $r$ does not have a global maximum, nor a global minimum.
- 				\item $s$ has a global maximum of approximately $5$; it does not have a global minimum.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the intervals on which each function is increasing and decreasing.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is increasing on $(-\infty,-2)\cup (3,\infty)$, and decreasing on $(-2,3)$.
- 				\item $q$ is increasing on $(-4,0)\cup (3,\infty)$, and decreasing on $(-\infty,-4)\cup (0,3)$.
- 				\item $r$ is increasing on $(-4,3)$, and decreasing on $(-\infty,-4)\cup (3,\infty)$.
- 				\item $s$ is increasing on $(-\infty,-8)\cup (-4,0)\cup (3,5)$, and decreasing on $(-8,-4)\cup (0,3)\cup (5,\infty)$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the intervals on which each function is concave up and concave down.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is concave up on  $(1,\infty)$, and concave down on  $(-\infty,1)$.
- 				\item $q$ is concave up on $(-\infty,-1)\cup (1,\infty)$, and concave down on $(-1,1)$.
- 				\item $r$ is concave up on $(-\infty,-3)\cup (0,2)$, and concave down on $(-3,0)\cup (2,\infty)$.
- 				\item $s$ is concave up on $(-6,-2)\cup (2,5)$, and concave down on $(-\infty,-6)\cup (-2,2)\cup (5,\infty)$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		The degree of $q$ is $5$. Assuming that all of the real zeros of $q$ are
- 		shown in its graph, how many complex zeros does $q$ have?
- 		\begin{shortsolution}
- 			\Vref{poly:fig:incdec2} shows that $q$ has $3$ real zeros
- 			since the curve of $q$ cuts the horizontal axis $3$ times.
- 			Since $q$ has degree $5$, $q$ must have $2$ complex zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Long-run behaviour of polynomials]
- 	Describe the long-run behavior of each of polynomial functions in
- 	\crefrange{poly:prob:degree5}{poly:prob:degree2}.
- 	\begin{shortsolution}
- 		$\dd\lim_{x\rightarrow-\infty}p(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow\infty}p(x)=\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}q(x)=\infty$,
- 		$\dd\lim_{x\rightarrow\infty}q(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}r(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow\infty}r(x)=\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}s(x)=\infty$,
- 		$\dd\lim_{x\rightarrow\infty}s(x)=\infty$,
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[True of false?]
- 	Let $p$ be a polynomial function.
- 	Label each of the following statements as true (T) or false (F); if they are false,
- 	provide an example that supports your answer.
- 	\begin{subproblem}
- 		If $p$ has degree $3$, then $p$ has $3$ distinct zeros.
- 		\begin{shortsolution}
- 			False. Consider $p(x)=x^2(x+1)$ which has only 2 distinct zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has degree $4$, then $\dd\lim_{x\rightarrow-\infty}p(x)=\infty$ and $\dd\lim_{x\rightarrow\infty}p(x)=\infty$.
- 		\begin{shortsolution}
- 			False. Consider $p(x)=-x^4$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has even degree, then it is possible that $p$ can have no real zeros.
- 		\begin{shortsolution}
- 			True.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has odd degree, then it is possible that $p$ can have no real zeros.
- 		\begin{shortsolution}
- 			False. All odd degree polynomials will cut the horizontal axis at least once.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a possible formula for a polynomial
- 	function that has the specified properties.
- 	\begin{subproblem}
- 		Degree 2 and has zeros at $4$ and $5$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-4)(x-5)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 3 and has zeros at $4$,$5$ and $-3$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-4)(x-5)(x+3)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 4 and has zeros at $0$, $4$, $5$, $-3$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=x(x-4)(x-5)(x+3)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 4, with zeros that make the graph cut at $2$, $-5$, and a zero that makes the graph touch at $-2$;
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-2)(x+5)(x+2)^2$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 3, with only one zero at $-1$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x+1)^3$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[\Cref{poly:step:last}]
- 	\pccname{Saheed} is graphing a polynomial function, $p$.
- 	He is following \crefrange{poly:step:first}{poly:step:last} and has so far
- 	marked the zeros of $p$ on \cref{poly:fig:optionsp1}. Saheed tells you that
- 	$p$ has degree $3$, but does \emph{not} say if the leading coefficient
- 	of $p$ is positive or negative.
- 	\begin{figure}[!htbp]
- 		\begin{widepage}
- 		\begin{subfigure}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\textwidth,
- 						height=.5\textwidth,
- 					]
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:optionsp1}
- 		\end{subfigure}%
- 		\hfill
- 		\begin{subfigure}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\textwidth,
- 						height=.5\textwidth,
- 					]
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:optionsp2}
- 		\end{subfigure}%
- 		\caption{}
- 		\end{widepage}
- 	\end{figure}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp1} to help sketch $p$, assuming that the leading coefficient
- 		is positive.
- 		\begin{shortsolution}
- 			Assuming that $a_3>0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-6.78179:8.35598,samples=50]{1/20*(x+5)*(x-2)*(x-6)};
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp1} to help sketch $p$, assuming that the leading coefficient
- 		is negative.
- 		\begin{shortsolution}
- 			Assuming that $a_3<0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-6.78179:8.35598,samples=50]{-1/20*(x+5)*(x-2)*(x-6)};
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	Saheed now turns his attention to another polynomial function, $q$. He finds
- 	the zeros of $q$ (there are only $2$) and marks them on \cref{poly:fig:optionsp2}.
- 	Saheed knows that $q$ has degree $3$, but doesn't know if the leading
- 	coefficient is positive or negative.
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp2} to help sketch $q$, assuming that the leading
- 		coefficient of $q$ is positive. Hint: only one of the zeros is simple.
- 		\begin{shortsolution}
- 			Assuming that $a_4>0$ there are $2$ different options:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-8.68983:7.31809,samples=50]{1/20*(x+5)^2*(x-6)};
- 					\addplot expression[domain=-6.31809:9.68893,samples=50]{1/20*(x+5)*(x-6)^2};
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp2} to help sketch $q$, assuming that the leading
- 		coefficient of $q$ is negative.
- 		\begin{shortsolution}
- 			Assuming that $a_4<0$ there are $2$ different options:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-8.68983:7.31809,samples=50]{-1/20*(x+5)^2*(x-6)};
- 					\addplot expression[domain=-6.31809:9.68893,samples=50]{-1/20*(x+5)*(x-6)^2};
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Zeros]
- 	Find all zeros of each of the following polynomial functions, making
- 	sure to detail their multiplicity. Note that
- 	you may need to use factoring, or the quadratic formula, or both! Also note
- 	that some zeros may be repeated, and some may be complex.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$p(x)=x^2+1$
- 			\begin{shortsolution}
- 				$\pm i$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(y)=(y^2-9)(y^2-7)$
- 			\begin{shortsolution}
- 				$\pm 3$, $\pm \sqrt{7}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(z)=-4z^3(z^2+3)(z^2+64)$
- 			\begin{shortsolution}
- 				$0$ (multiplicity $3$), $\pm\sqrt{3}$ (simple), $\pm\sqrt{8}$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$a(x)=x^4-81$
- 			\begin{shortsolution}
- 				$\pm 3$, $\pm 3i$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$b(y)=y^3-8$
- 			\begin{shortsolution}
- 				$2$, $-1\pm i\sqrt{3}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$c(m)=m^3-m^2$
- 			\begin{shortsolution}
- 				$0$ (multiplicity $2$), $1$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(n)=(n+1)(n^2+4)$
- 			\begin{shortsolution}
- 				$-1$, $\pm 2i$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$f(\alpha)=(\alpha^2-16)(\alpha^2-5\alpha+4)$
- 			\begin{shortsolution}
- 				$-4$ (simple), $4$ (multiplicity $2$), $1$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(\beta)=(\beta^2-25)(\beta^2-5\beta-4)$
- 			\begin{shortsolution}
- 				$\pm 5$, $\dfrac{5\pm\sqrt{41}}{2}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Given zeros, find a formula]
- 	In each of the following problems you are given the zeros of a polynomial.
- 	Write a possible formula for each polynomial| you may leave your
- 	answer in factored form, but it may not contain complex numbers. Unless
- 	otherwise stated, assume that the zeros are simple.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$1$, $2$
- 			\begin{shortsolution}
- 				$p(x)=(x-1)(x-2)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$0$, $5$, $13$
- 			\begin{shortsolution}
- 				$p(x)=x(x-5)(x-13)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$-7$, $2$ (multiplicity $3$), $5$
- 			\begin{shortsolution}
- 				$p(x)=(x+7)(x-2)^3(x-5)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$0$, $\pm i$
- 			\begin{shortsolution}
- 				$p(x)=x(x^2+1)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\pm 2i$, $\pm 7$
- 			\begin{shortsolution}
- 				$p(x)=(x^2+4)(x^2-49)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$-2\pm i\sqrt{6}$
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Composition of polynomials]
- 	Let $p$ and $q$ be polynomial functions that have formulas
- 	\[
- 		p(x)=(x+1)(x+2)(x+5), \qquad q(x)=3-x^4
- 	\]
- 	Evaluate each of the following.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(p\circ q)(0)$
- 			\begin{shortsolution}
- 				$160$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(q\circ p)(0)$
- 			\begin{shortsolution}
- 				$-9997$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\circ q)(1)$
- 			\begin{shortsolution}
- 				$84$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\circ p)(0)$
- 			\begin{shortsolution}
- 				$1980$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Piecewise polynomial functions]
- 	Let $P$ be the piecewise-defined function with formula
- 	\[
- 		P(x)=\begin{cases}
- 		(1-x)(2x+5)(x^2+1), &   x\leq -3\\
- 		4-x^2,              &   -3<x < 4\\
- 		x^3                 &   x\geq 4
- 		\end{cases}
- 	\]
- 	Evaluate each of the following
- 	\begin{multicols}{5}
- 		\begin{subproblem}
- 			$P(-4)$
- 			\begin{shortsolution}
- 				$-255$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(0)$
- 			\begin{shortsolution}
- 				$4$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(4)$
- 			\begin{shortsolution}
- 				$64$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(-3)$
- 			\begin{shortsolution}
- 				$-40$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(P\circ P)(0)$
- 			\begin{shortsolution}
- 				$64$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Function algebra]
- 	Let $p$ and $q$ be the polynomial functions that have formulas
- 	\[
- 		p(x)=x(x+1)(x-3)^2, \qquad q(x)=7-x^2
- 	\]
- 	Evaluate each of the following (if possible).
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(p+q)(1)$
- 			\begin{shortsolution}
- 				$14$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p-q)(0)$
- 			\begin{shortsolution}
- 				$7$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\cdot q)(\sqrt{7})$
- 			\begin{shortsolution}
- 				$0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\left( \frac{q}{p} \right)(1)$
- 			\begin{shortsolution}
- 				$\frac{3}{4}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\begin{subproblem}
- 		What is the domain of the function $\frac{q}{p}$?
- 		\begin{shortsolution}
- 			$(-\infty,-1)\cup (-1,0)\cup (0,3)\cup (3,\infty)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Transformations: given the transformation, find the formula]
- 	Let $p$ be the polynomial function that has formula.
- 	\[
- 		p(x)=4x(x^2-1)(x+3)
- 	\]
- 	In each of the following
- 	problems apply the given transformation to the function $p$ and
- 	write a formula for the transformed version of $p$.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			Shift $p$ to the right by $5$ units.
- 			\begin{shortsolution}
- 				$p(x-5)=4(x-5)(x-2)(x^2-10x+24)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ to the left by $6$ units.
- 			\begin{shortsolution}
- 				$p(x+6)=4(x+6)(x+9)(x^2+12x+35)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ up by $12$ units.
- 			\begin{shortsolution}
- 				$p(x)+12=4x(x^2-1)(x+3)+12$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ down by $2$ units.
- 			\begin{shortsolution}
- 				$p(x)-2=4x(x^2-1)(x+3)-2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $p$ over the horizontal axis.
- 			\begin{shortsolution}
- 				$-p(x)=-4x(x^2-1)(x+3)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $p$ over the vertical axis.
- 			\begin{shortsolution}
- 				$p(-x)=-4x(x^2-1)(3-x)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a table]\label{poly:prob:findformula}
- 	\Crefrange{poly:tab:findformulap}{poly:tab:findformulas} show values of polynomial functions, $p$, $q$,
- 	$r$, and $s$.
- 	 	 	
- 	\begin{table}[!htb]
- 		\centering
- 		\begin{widepage}
- 		\caption{Tables for \cref{poly:prob:findformula}}
- 		\label{poly:tab:findformula}
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=p(x)$}
- 			\label{poly:tab:findformulap}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				$-4$          & $-56$         \\\normalline
- 				$-3$          & $-18$         \\\normalline
- 				$-2$          & $0$           \\\normalline
- 				$-1$          & $4$           \\\normalline
- 				$0$           & $0$           \\\normalline
- 				$1$           & $-6$          \\\normalline
- 				$2$           & $-8$          \\\normalline
- 				$3$           & $0$           \\\normalline
- 				$4$           & $24$          \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=q(x)$}
- 			\label{poly:tab:findformulaq}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $-16$         \\\normalline
- 				$-3$          & $-3$          \\\normalline
- 				$-2$          & $0$           \\\normalline
- 				$-1$          & $-1$          \\\normalline
- 				$0$           & $0$           \\\normalline
- 				$1$           & $9$           \\\normalline
- 				$2$           & $32$          \\\normalline
- 				$3$           & $75$          \\\normalline
- 				$4$           & $144$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=r(x)$}
- 			\label{poly:tab:findformular}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $105$         \\\normalline
- 				$-3$          & $0$           \\\normalline
- 				$-2$          & $-15$         \\\normalline
- 				$-1$          & $0$           \\\normalline
- 				$0$           & $9$           \\\normalline
- 				$1$           & $0$           \\\normalline
- 				$2$           & $-15$         \\\normalline
- 				$3$           & $0$           \\\normalline
- 				$4$           & $105$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=s(x)$}
- 			\label{poly:tab:findformulas}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $75$          \\\normalline
- 				$-3$          & $0$           \\\normalline
- 				$-2$          & $-9$          \\\normalline
- 				$-1$          & $0$           \\\normalline
- 				$0$           & $3$           \\\normalline
- 				$1$           & $0$           \\\normalline
- 				$2$           & $15$          \\\normalline
- 				$3$           & $96$          \\\normalline
- 				$4$           & $760$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\end{widepage}
- 	\end{table}
- 	 	 	
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $p$ are shown (in \cref{poly:tab:findformulap}), how many zeros does $p$ have?
- 		\begin{shortsolution}
- 			$p$ has 3 zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What is the degree of $p$?
- 		\begin{shortsolution}
- 			$p$ is degree 3.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Write a formula for $p(x)$.
- 		\begin{shortsolution}
- 			$p(x)=x(x+2)(x-3)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $q$ are shown (in \cref{poly:tab:findformulaq}), how many zeros does $q$ have?
- 		\begin{shortsolution}
- 			$q$ has 2 zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Describe the difference in behavior of $p$ and $q$ at $-2$.
- 		\begin{shortsolution}
- 			$p$ changes sign at $-2$, and $q$ does not change sign at $-2$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that $q$ is a degree-$3$ polynomial, write a formula for $q(x)$.
- 		\begin{shortsolution}
- 			$q(x)=x(x+2)^2$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $r$ are shown (in \cref{poly:tab:findformular}), find a formula for $r(x)$.
- 		\begin{shortsolution}
- 			$r(x)=(x+3)(x+1)(x-1)(x-3)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $s$ are shown (in \cref{poly:tab:findformulas}), find a formula for $s(x)$.
- 		\begin{shortsolution}
- 			$s(x)=(x+3)(x+1)(x-1)^2$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	\end{exercises}
- 	 	 	
-\section{Rational functions}
- \subsection*{Power functions with negative exponents}
- 	The study of rational functions will rely upon a good knowledge
- 	of power functions with negative exponents. \Cref{rat:ex:oddpow,rat:ex:evenpow} are
- 	simple but fundamental to understanding the behavior of rational functions.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Power functions with odd negative exponents]\label{rat:ex:oddpow}
- 		Graph each of the following functions on your calculator, state their domain in interval notation, and their
- 		behavior as $x\rightarrow 0^-$ and $x\rightarrow 0^+$.
- 		\[
- 			f(x)=\frac{1}{x},\qquad g(x)=\dfrac{1}{x^3},\qquad h(x)=\dfrac{1}{x^5}
- 		\]
- 		\begin{pccsolution}
- 			The functions $f$, $g$, and $k$ are plotted in \cref{rat:fig:oddpow}.
- 			The domain of each of the functions $f$, $g$, and $h$ is $(-\infty,0)\cup (0,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				f(x)\rightarrow 0                         & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}  f(x)\rightarrow 0 & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same results hold for $g$ and $h$. Note also that each of the functions
- 			has a \emph{vertical asymptote} at $0$. We see that
- 			\begin{align*}
- 				f(x)\rightarrow -\infty                         & \text{ as } x\rightarrow 0^- \\
- 				\mathllap{\text{and }}   f(x)\rightarrow \infty & \text{ as } x\rightarrow 0^+ 
- 			\end{align*}
- 			The same results hold for $g$ and $h$.
- 			 			 			
- 			The curve of a function that has a vertical asymptote is necessarily separated
- 			into \emph{branches}| each of the functions $f$, $g$, and $h$ have $2$ branches.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-3,xmax=3,
- 						ymin=-5,ymax=5,
- 						xtick={-2,-1,...,2},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=north west,
- 					]
- 					\addplot expression[domain=-3:-0.2]{1/x};
- 					\addplot expression[domain=-3:-0.584]{1/x^3};
- 					\addplot expression[domain=-3:-0.724]{1/x^5};
- 					\addplot expression[domain=0.2:3]{1/x};
- 					\addplot expression[domain=0.584:3]{1/x^3};
- 					\addplot expression[domain=0.724:3]{1/x^5};
- 					\addplot[soldot]coordinates{(-1,-1)}node[axisnode,anchor=north east]{$(-1,-1)$};
- 					\addplot[soldot]coordinates{(1,1)}node[axisnode,anchor=south west]{$(1,1)$};
- 					\legend{$f$,$g$,$h$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:oddpow}
- 		\end{minipage}%
- 		\hfill
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-3,xmax=3,
- 						ymin=-5,ymax=5,
- 						xtick={-2,-1,...,2},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=south east,
- 					]
- 					\addplot expression[domain=-3:-0.447]{1/x^2};
- 					\addplot expression[domain=-3:-0.668]{1/x^4};
- 					\addplot expression[domain=-3:-0.764]{1/x^6};
- 					\addplot expression[domain=0.447:3]{1/x^2};
- 					\addplot expression[domain=0.668:3]{1/x^4};
- 					\addplot expression[domain=0.764:3]{1/x^6};
- 					\addplot[soldot]coordinates{(-1,1)}node[axisnode,anchor=south east]{$(-1,1)$};
- 					\addplot[soldot]coordinates{(1,1)}node[axisnode,anchor=south west]{$(1,1)$};
- 					\legend{$F$,$G$,$H$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:evenpow}
- 		\end{minipage}%
- 	\end{figure}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Power functions with even negative exponents]\label{rat:ex:evenpow}%
- 		Graph each of the following functions, state their domain, and their
- 		behavior as $x\rightarrow 0^-$ and $x\rightarrow 0^+$.
- 		\[
- 			f(x)=\frac{1}{x^2},\qquad g(x)=\frac{1}{x^4},\qquad h(x)=\frac{1}{x^6}
- 		\]
- 		\begin{pccsolution}
- 			The functions $F$, $G$, and $H$ are plotted in \cref{rat:fig:evenpow}.
- 			The domain of each of the functions $F$, $G$, and $H$ is $(-\infty,0)\cup (0,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				F(x)\rightarrow 0                           & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}    f(x)\rightarrow 0 & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			As in \cref{rat:ex:oddpow}, $F$ has a horizontal asymptote  that
- 			has equation $y=0$.
- 			The same results hold for $G$ and $H$. Note also that each of the functions
- 			has a \emph{vertical asymptote} at $0$. We see that
- 			\begin{align*}
- 				F(x)\rightarrow \infty                          & \text{ as } x\rightarrow 0^- \\
- 				\mathllap{\text{and }}   F(x)\rightarrow \infty & \text{ as } x\rightarrow 0^+ 
- 			\end{align*}
- 			The same results hold for $G$ and $H$. Each of the functions $F$, $G$, and $H$
- 			have $2$ branches.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Repeat \cref{rat:ex:oddpow,rat:ex:evenpow} using (respectively)
- 		\begin{subproblem}
- 			$k(x)=-\dfrac{1}{x}$, $ m(x)=-\dfrac{1}{x^3}$, $ n(x)=-\dfrac{1}{x^5}$
- 			\begin{shortsolution}
- 				The functions $k$, $m$, and $n$ have domain $(-\infty,0)\cup (0,\infty)$, and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-3,xmax=3,
- 							ymin=-5,ymax=5,
- 							xtick={-2,-1,...,2},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-3:-0.2]{-1/x};
- 						\addplot expression[domain=-3:-0.584]{-1/x^3};
- 						\addplot expression[domain=-3:-0.724]{-1/x^5};
- 						\addplot expression[domain=0.2:3]{-1/x};
- 						\addplot expression[domain=0.584:3]{-1/x^3};
- 						\addplot expression[domain=0.724:3]{-1/x^5};
- 						\legend{$k$,$m$,$n$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					k(x)\rightarrow 0                                & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}    k(x)\rightarrow 0      & \text{ as } x\rightarrow-\infty \\
- 					\intertext{and also}
- 					k(x)\rightarrow \infty                           & \text{ as } x\rightarrow 0^-    \\
- 					\mathllap{\text{and }}   k(x)\rightarrow -\infty & \text{ as } x\rightarrow 0^+    
- 				\end{align*}
- 				The same are true for $m$ and $n$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$ K(x)=-\dfrac{1}{x^2}$, $ M(x)=-\dfrac{1}{x^4}$, $ N(x)=-\dfrac{1}{x^6}$
- 			\begin{shortsolution}
- 				The functions $K$, $M$, and $N$ have domain $(-\infty,0)\cup (0,\infty)$, and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-3,xmax=3,
- 							ymin=-5,ymax=5,
- 							xtick={-2,-1,...,2},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-3:-0.447]{-1/x^2};
- 						\addplot expression[domain=-3:-0.668]{-1/x^4};
- 						\addplot expression[domain=-3:-0.764]{-1/x^6};
- 						\addplot expression[domain=0.447:3]{-1/x^2};
- 						\addplot expression[domain=0.668:3]{-1/x^4};
- 						\addplot expression[domain=0.764:3]{-1/x^6};
- 						\legend{$K$,$M$,$N$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					K(x)\rightarrow 0                                & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}    K(x)\rightarrow 0      & \text{ as } x\rightarrow-\infty \\
- 					\intertext{and also}
- 					K(x)\rightarrow -\infty                          & \text{ as } x\rightarrow 0^-    \\
- 					\mathllap{\text{and }}   K(x)\rightarrow -\infty & \text{ as } x\rightarrow 0^+    
- 				\end{align*}
- 				The same are true for $M$ and $N$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- \subsection*{Rational functions}
- 	\begin{pccdefinition}[Rational functions]\label{rat:def:function}
- 		Rational functions have the form
- 		\[
- 			r(x) = \frac{p(x)}{q(x)}
- 		\]
- 		where both $p$ and $q$ are polynomials.
- 		 		 		
- 		Note that
- 		\begin{itemize}
- 			\item the domain or $r$ will be all real numbers, except those that
- 			      make the \emph{denominator}, $q(x)$, equal to $0$;
- 			\item the zeros of $r$ are the zeros of $p$, i.e the real numbers
- 			      that make the \emph{numerator}, $p(x)$, equal to $0$.
- 		\end{itemize}
- 		 		 		
- 		\Cref{rat:ex:oddpow,rat:ex:evenpow} are particularly important because $r$
- 		will behave like $\frac{1}{x}$, or $\frac{1}{x^2}$ around its vertical asymptotes,
- 		depending on the power that the relevant term is raised to| we will demonstrate
- 		this in what follows.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Rational or not]
- 		Identify whether each of the following functions is a rational or not. If
- 		the function is rational, state the domain.
- 		\begin{multicols}{3}
- 			\begin{enumerate}
- 				\item $r(x)=\dfrac{1}{x}$
- 				\item $f(x)=2^x+3$
- 				\item $g(x)=19$
- 				\item $h(x)=\dfrac{3+x}{4-x}$
- 				\item $k(x)=\dfrac{x^3+2x}{x-15}$
- 				\item $l(x)=9-4x$
- 				\item $m(x)=\dfrac{x+5}{(x-7)(x+9)}$
- 				\item $n(x)=x^2+6x+7$
- 				\item $q(x)=1-\dfrac{3}{x+1}$
- 			\end{enumerate}
- 		\end{multicols}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item $r$ is rational; the domain of $r$ is $(-\infty,0)\cup(0,\infty)$.
- 				\item $f$ is not rational.
- 				\item $g$ is not rational; $g$ is constant.
- 				\item $h$ is rational; the domain of $h$ is $(-\infty,4)\cup(4,\infty)$.
- 				\item $k$ is rational; the domain of $k$ is $(-\infty,15)\cup(15,\infty)$.
- 				\item $l$ is not rational; $l$ is linear.
- 				\item $m$ is rational; the domain of $m$ is $(-\infty,-9)\cup(-9,7)\cup(7,\infty)$.
- 				\item $n$ is not rational; $n$ is quadratic (or you might describe $n$ as a polynomial).
- 				\item $q$ is rational; the domain of $q$ is $(-\infty,-1)\cup (-1,\infty)$.
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Match formula to graph]
- 		Each of the following functions is graphed in \cref{rat:fig:whichiswhich}.
- 		Which is which?
- 		\[
- 			r(x)=\frac{1}{x-3}, \qquad q(x)=\frac{x-2}{x+5}, \qquad k(x)=\frac{1}{(x+2)(x-3)}
- 		\]
- 		\begin{figure}[!htb]
- 			\setlength{\figurewidth}{0.3\textwidth}
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=(x-2)/(x+5);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-6,ymax=6,
- 							xtick={-8,-6,...,8},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-6.37]{f};
- 						\addplot[pccplot] expression[domain=-3.97:10]{f};
- 						\addplot[soldot] coordinates{(2,0)};
- 						\addplot[asymptote,domain=-6:6]({-5},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which1}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=1/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-5,ymax=6,
- 							xtick={-8,-6,...,8},
- 							ytick={-4,4},
- 							minor ytick={-3,...,5},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:2.8]{f};
- 						\addplot[pccplot] expression[domain=3.17:10]{f};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which2}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=1/((x-3)*(x+2));}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-5,ymax=5,
- 							xtick={-8,-6,...,8},
- 							ytick={-4,4},
- 							minor ytick={-3,...,3},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-2.03969]{f};
- 						\addplot[pccplot] expression[domain=-1.95967:2.95967]{f};
- 						\addplot[pccplot] expression[domain=3.03969:10]{f};
- 						\addplot[asymptote,domain=-5:5]({-2},{x});
- 						\addplot[asymptote,domain=-5:5]({3},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which3}
- 			\end{subfigure}
- 			\caption{}
- 			\label{rat:fig:whichiswhich}
- 		\end{figure}
- 		 		 		
- 		\begin{pccsolution}
- 			Let's start with the function $r$. Note that domain of $r$ is $(-\infty,3)\cup(0,3)$, so
- 			we search for a function that has a vertical asymptote at $3$. There
- 			are two possible choices: the functions graphed in \cref{rat:fig:which2,rat:fig:which3},
- 			but note that the function in \cref{rat:fig:which3} also has a vertical asymptote at $-2$
- 			which is not consistent with the formula for $r(x)$. Therefore, $y=r(x)$
- 			is graphed in \cref{rat:fig:which2}.
- 			 			 			
- 			The function $q$ has domain $(-\infty,-5)\cup(-5,\infty)$, so we search
- 			for a function that has a vertical asymptote at $-5$. The only candidate
- 			is the curve shown in \cref{rat:fig:which1}; note that the curve also goes through $(2,0)$,
- 			which is consistent with the formula for $q(x)$, since $q(2)=0$, i.e $q$
- 			has a zero at $2$.
- 			 			 			
- 			The function $k$ has domain $(-\infty,-2)\cup(-2,3)\cup(3,\infty)$, and
- 			has vertical asymptotes at $-2$ and $3$. This is consistent with
- 			the graph in \cref{rat:fig:which3} (and is the only curve that
- 			has $3$ branches).
- 			 			 			
- 			We note that each function behaves like $\frac{1}{x}$ around its vertical asymptotes,
- 			because each linear factor in each denominator is raised to the power $1$; if (for example)
- 			the definition of $r$ was instead
- 			\[
- 				r(x)=\frac{1}{(x-3)^2}
- 			\]
- 			then we would see that $r$ behaves like $\frac{1}{x^2}$ around its vertical asymptote, and
- 			the graph of $r$ would be very different. We will deal with these cases in the examples that follow.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Repeated factors in the denominator]
- 		Consider the functions $f$, $g$, and $h$ that have formulas
- 		\[
- 			f(x)=\frac{x-2}{(x-3)(x+2)}, \qquad g(x)=\frac{x-2}{(x-3)^2(x+2)}, \qquad h(x)=\frac{x-2}{(x-3)(x+2)^2}
- 		\]
- 		which are graphed in \cref{rat:fig:repfactd}. Note that each function has $2$
- 		vertical asymptotes, and the domain of each function is
- 		\[
- 			(-\infty,-2)\cup(-2,3)\cup(3,\infty)
- 		\]
- 		so we are not surprised to see that each curve has $3$ branches. We also note that
- 		the numerator of each function is the same, which tells us that each function has
- 		only $1$ zero at $2$.
- 		 		 		
- 		The functions $g$ and $h$ are different from those that we have considered previously,
- 		because they have a repeated factor in the denominator. Notice in particular
- 		the way that the functions behave around their asymptotes:
- 		\begin{itemize}
- 			\item $f$ behaves like $\frac{1}{x}$ around both of its asymptotes;
- 			\item $g$ behaves like $\frac{1}{x}$ around $-2$, and like $\frac{1}{x^2}$ around $3$;
- 			\item $h$ behaves like $\frac{1}{x^2}$ around $-2$, and like $\frac{1}{x}$ around $3$.
- 		\end{itemize}
- 	\end{pccexample}
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,4},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.201]{f};
- 					\addplot[pccplot] expression[domain=-1.802:2.951]{f};
- 					\addplot[pccplot] expression[domain=3.052:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-2},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)(x-3)}$}
- 			\label{rat:fig:repfactd1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)*(x-3)^2);}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,4},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.039]{f};
- 					\addplot[pccplot] expression[domain=-1.959:2.796]{f};
- 					\addplot[pccplot] expression[domain=3.243:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-4:4]({-2},{x});
- 					%                 \addplot[asymptote,domain=-4:4]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)(x-3)^2}$}
- 			\label{rat:fig:repfactd2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)^2*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,2},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.451]{f};
- 					\addplot[pccplot] expression[domain=-1.558:2.990]{f};
- 					\addplot[pccplot] expression[domain=3.010:6]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-4:4]({-2},{x});
- 					%                 \addplot[asymptote,domain=-4:4]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)^2(x-3)}$}
- 			\label{rat:fig:repfactd3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:repfactd}
- 	\end{figure}
- 	 	 	
- 	\Cref{rat:def:function} says that the zeros of
- 	the rational function $r$ that has formula $r(x)=\frac{p(x)}{q(x)}$ are
- 	the zeros of $p$. Let's explore this a little more.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Zeros] Find the zeros of each of the following functions
- 		\[
- 			\alpha(x)=\frac{x+5}{3x-7}, \qquad \beta(x)=\frac{9-x}{x+1}, \qquad \gamma(x)=\frac{17x^2-10}{2x+1}
- 		\]
- 		\begin{pccsolution}
- 			We find the zeros of each function in turn by setting the numerator equal to $0$. The zeros of
- 			$\alpha$ are found by solving
- 			\[
- 				x+5=0
- 			\]
- 			The zero of $\alpha$ is $-5$.
- 			 			 			
- 			Similarly, we may solve $9-x=0$ to find the zero of $\beta$, which is clearly $9$.
- 			 			 			
- 			The zeros of $\gamma$ satisfy the equation
- 			\[
- 				17x^2-10=0
- 			\]
- 			which we can solve using the square root property to obtain
- 			\[
- 				x=\pm\frac{10}{17}
- 			\]
- 			The zeros of $\gamma$ are $\pm\frac{10}{17}$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- \subsection*{Long-run behavior}
- 	Our focus so far has been on the behavior of rational functions around
- 	their \emph{vertical} asymptotes. In fact, rational functions also
- 	have interesting long-run behavior around their \emph{horizontal} or
- 	\emph{oblique} asymptotes. A rational function will always have either
- 	a horizontal or an oblique asymptote| the case is determined by the degree
- 	of the numerator and the degree of the denominator.
- 	\begin{pccdefinition}[Long-run behavior]\label{rat:def:longrun}
- 		Let $r$ be the rational function that has formula
- 		\[
- 			r(x) = \frac{a_n x^n + a_{n-1}x^{n-1}+\ldots + a_0}{b_m x^m + b_{m-1}x^{m-1}+\ldots+b_0}
- 		\]
- 		We can classify the long-run behavior of the rational function $r$
- 		according to the following criteria:
- 		\begin{itemize}
- 			\item if $n<m$ then  $r$ has a horizontal asymptote with equation $y=0$;
- 			\item if $n=m$ then $r$ has a horizontal asymptote with equation $y=\dfrac{a_n}{b_m}$;
- 			\item if $n>m$ then $r$ will have an oblique asymptote as $x\rightarrow\pm\infty$ (more on this in \cref{rat:sec:oblique})
- 		\end{itemize}
- 	\end{pccdefinition}
- 	We will concentrate on functions that have horizontal asymptotes until
- 	we reach \cref{rat:sec:oblique}.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Long-run behavior graphically]\label{rat:ex:horizasymp}
- 		\pccname{Kebede} has graphed the following functions in his graphing calculator
- 		\[
- 			r(x)=\frac{x+1}{x-3}, \qquad s(x)=\frac{2(x+1)}{x-3}, \qquad t(x)=\frac{3(x+1)}{x-3}
- 		\]
- 		and obtained the curves shown in \cref{rat:fig:horizasymp}. Kebede decides
- 		to test his knowledgeable friend \pccname{Oscar}, and asks him
- 		to match the formulas to the graphs.
- 		 		 		
- 		\begin{figure}[!htb]
- 			\setlength{\figurewidth}{0.3\textwidth}
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=2*(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:2]{f};
- 						\addplot[pccplot] expression[domain=5:15]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{2});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp1}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:2.42857,samples=50]{f};
- 						\addplot[pccplot] expression[domain=3.8:15,samples=50]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{1});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp2}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=3*(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:1.6666,samples=50]{f};
- 						\addplot[pccplot] expression[domain=7:15]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{3});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp3}
- 			\end{subfigure}
- 			\caption{Horizontal asymptotes}
- 			\label{rat:fig:horizasymp}
- 		\end{figure}
- 		 		 		
- 		Oscar notices that each function has a vertical asymptote at $3$ and a zero at $-1$.
- 		The main thing that catches Oscar's eye is that each function has a different
- 		coefficient in the numerator, and that each curve has a different horizontal asymptote.
- 		In particular, Oscar notes that
- 		\begin{itemize}
- 			\item the curve shown in \cref{rat:fig:horizasymp1} has a horizontal asymptote with equation $y=2$;
- 			\item the curve shown in \cref{rat:fig:horizasymp2} has a horizontal asymptote with equation $y=1$;
- 			\item the curve shown in \cref{rat:fig:horizasymp3} has a horizontal asymptote with equation $y=3$.
- 		\end{itemize}
- 		Oscar is able to tie it all together for Kebede by referencing \cref{rat:def:longrun}. He says
- 		that since the degree of the numerator and the degree of the denominator is the same
- 		for each of the functions $r$, $s$, and $t$, the horizontal asymptote will be determined
- 		by evaluating the ratio of their leading coefficients.
- 		 		 		
- 		Oscar therefore says that $r$ should have a horizontal asymptote $y=\frac{1}{1}=1$, $s$ should
- 		have a horizontal asymptote $y=\frac{2}{1}=2$, and $t$ should have a horizontal asymptote
- 		$y=\frac{3}{1}=3$. Kebede is able to finish the problem from here, and says that $r$ is
- 		shown in \cref{rat:fig:horizasymp2}, $s$ is shown in \cref{rat:fig:horizasymp1}, and
- 		$t$ is shown in \cref{rat:fig:horizasymp3}.
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Long-run behavior numerically]
- 		\pccname{Xiao} and \pccname{Dwayne} saw \cref{rat:ex:horizasymp} but are a little confused
- 		about horizontal asymptotes. What does it mean to say that a function $r$ has a horizontal
- 		asymptote?
- 		 		 		
- 		They decide to explore the concept by
- 		constructing a table of values for the rational functions $R$ and  $S$ that have formulas
- 		\[
- 			R(x)=\frac{-5(x+1)}{x-3}, \qquad S(x)=\frac{7(x-5)}{2(x+1)}
- 		\]
- 		In \cref{rat:tab:plusinfty} they model the behavior of $R$ and $S$ as $x\rightarrow\infty$,
- 		and in \cref{rat:tab:minusinfty} they model the behavior of $R$ and $S$ as $x\rightarrow-\infty$
- 		by substituting very large values of $|x|$ into each function.
- 		\begin{table}[!htb]
- 			\begin{minipage}{.5\textwidth}
- 				\centering
- 				\caption{$R$ and $S$ as $x\rightarrow\infty$}
- 				\label{rat:tab:plusinfty}
- 				\begin{tabular}{crr}
- 					\beforeheading
- 					$x$            & $R(x)$     & $S(x)$    \\ \afterheading
- 					$1\times 10^2$ & $-5.20619$ & $3.29208$ \\\normalline
- 					$1\times 10^3$ & $-5.02006$ & $3.47902$ \\\normalline
- 					$1\times 10^4$ & $-5.00200$ & $3.49790$ \\\normalline
- 					$1\times 10^5$ & $-5.00020$ & $3.49979$ \\\normalline
- 					$1\times 10^6$ & $-5.00002$ & $3.49998$ \\\lastline
- 				\end{tabular}
- 			\end{minipage}%
- 			\begin{minipage}{.5\textwidth}
- 				\centering
- 				\caption{$R$ and $S$ as $x\rightarrow-\infty$}
- 				\label{rat:tab:minusinfty}
- 				\begin{tabular}{crr}
- 					\beforeheading
- 					$x$             & $R(x)$     & $S(x)$    \\ \afterheading
- 					$-1\times 10^2$ & $-4.80583$ & $3.71212$ \\\normalline
- 					$-1\times 10^3$ & $-4.98006$ & $3.52102$ \\\normalline
- 					$-1\times 10^4$ & $-4.99800$ & $3.50210$ \\\normalline
- 					$-1\times 10^5$ & $-4.99980$ & $3.50021$ \\\normalline
- 					$-1\times 10^6$ & $-4.99998$ & $3.50002$ \\\lastline
- 				\end{tabular}
- 			\end{minipage}
- 		\end{table}
- 		 		 		
- 		Xiao and Dwayne study \cref{rat:tab:plusinfty,rat:tab:minusinfty} and decide that
- 		the functions $R$ and $S$ never actually touch their horizontal asymptotes, but they
- 		do get infinitely close. They also feel as if they have a better understanding of
- 		what it means to study the behavior of a function as $x\rightarrow\pm\infty$.
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Repeated factors in the numerator]
- 		Consider the functions $f$, $g$, and $h$ that have formulas
- 		\[
- 			f(x)=\frac{(x-2)^2}{(x-3)(x+1)}, \qquad g(x)=\frac{x-2}{(x-3)(x+1)}, \qquad h(x)=\frac{(x-2)^3}{(x-3)(x+1)}
- 		\]
- 		which are graphed in \cref{rat:fig:repfactn}. We note that each function has vertical
- 		asymptotes at $-1$ and $3$, and so the domain of each function is
- 		\[
- 			(-\infty,-1)\cup(-1,3)\cup(3,\infty)
- 		\]
- 		We also notice that the numerators of each function are quite similar| indeed, each
- 		function has a zero at $2$, but how does each function behave around their zero?
- 		 		 		
- 		Using \cref{rat:fig:repfactn} to guide us, we note that
- 		\begin{itemize}
- 			\item $f$ has a horizontal intercept $(2,0)$, but the curve of
- 			      $f$ does not cut the horizontal axis| it bounces off it;
- 			\item $g$ also has a horizontal intercept $(2,0)$, and the curve
- 			      of $g$ \emph{does} cut the horizontal axis;
- 			\item $h$ has a horizontal intercept $(2,0)$, and the curve of $h$
- 			      also cuts the axis, but appears flattened as it does so.
- 		\end{itemize}
- 		 		 		
- 		We can further enrich our study by discussing the long-run behavior of each function.
- 		Using the tools of \cref{rat:def:longrun}, we can deduce that
- 		\begin{itemize}
- 			\item $f$ has a horizontal asymptote with equation $y=1$;
- 			\item $g$ has a horizontal asymptote with equation $y=0$;
- 			\item $h$ does \emph{not} have a horizontal asymptote| it has an oblique asymptote (we'll
- 			      study this more in \cref{rat:sec:oblique}).
- 		\end{itemize}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)^2/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						xtick={-4,-2,...,4},
- 						ytick={-8,-4,...,8},
- 						%                    grid=both,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.248,samples=50]{f};
- 					\addplot[pccplot] expression[domain=-0.794:2.976,samples=50]{f};
- 					\addplot[pccplot] expression[domain=3.026:5,samples=50]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-1},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{(x-2)^2}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						xtick={-4,-2,...,4},
- 						ytick={-8,-4,...,8},
- 						%                    grid=both,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.075]{f};
- 					\addplot[pccplot] expression[domain=-0.925:2.975]{f};
- 					\addplot[pccplot] expression[domain=3.025:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-1},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)^3/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						xtick={-8,-6,...,8},
- 						%                    grid=both,
- 						ymin=-30,ymax=30,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.27]{f};
- 					\addplot[pccplot] expression[domain=-0.806:2.99185]{f};
- 					\addplot[pccplot] expression[domain=3.0085:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-30:30]({-1},{x});
- 					%                 \addplot[asymptote,domain=-30:30]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{(x-2)^3}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:repfactn}
- 	\end{figure}
- 	 	 	
- \subsection*{Holes}
- 	Rational functions have a vertical asymptote at $a$ if the denominator is $0$ at $a$.
- 	What happens if the numerator is $0$ at the same place? In this case, we say that the rational
- 	function has a \emph{hole} at $a$.
- 	\begin{pccdefinition}[Holes]
- 		The rational function
- 		\[
- 			r(x)=\frac{p(x)}{q(x)}
- 		\]
- 		has a hole at $a$ if $p(a)=q(a)=0$. Note that holes are different from
- 		a vertical asymptotes. We represent that $r$ has a hole at the point
- 		$(a,r(a))$ on the curve $y=r(x)$ by
- 		using a hollow circle, $\circ$.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}
- 		\pccname{Mohammed} and \pccname{Sue} have graphed the function $r$ that has formula
- 		\[
- 			r(x)=\frac{x^2+x-6}{(x-2)}
- 		\]
- 		in their calculators, and can not decide if the correct graph
- 		is \cref{rat:fig:hole} or \cref{rat:fig:hole1}.
- 		 		 		
- 		Luckily for them, Oscar is nearby, and can help them settle the debate.
- 		Oscar demonstrates that
- 		\begin{align*}
- 			r(x) & =\frac{(x+3)(x-2)}{(x-2)} \\
- 			     & = x+3                     
- 		\end{align*}
- 		but only when $x\ne 2$, because the function is undefined at $2$. Oscar
- 		says that this necessarily means that the domain or $r$ is
- 		\[
- 			(-\infty,2)\cup(2,\infty)
- 		\]
- 		and that $r$ must have a hole at $2$.
- 		 		 		
- 		Mohammed and Sue are very grateful for the clarification, and conclude that
- 		the graph of $r$ is shown in \cref{rat:fig:hole1}.
- 		\begin{figure}[!htb]
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-4,...,8},
- 							ytick={-8,-4,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot expression[domain=-10:7]{x+3};
- 						\addplot[soldot] coordinates{(-3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:hole}
- 			\end{minipage}%
- 			\hfill
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-4,...,8},
- 							ytick={-8,-4,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot expression[domain=-10:7]{x+3};
- 						\addplot[holdot] coordinates{(2,5)};
- 						\addplot[soldot] coordinates{(-3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:hole1}
- 			\end{minipage}%
- 		\end{figure}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}
- 		Consider the function $f$ that has formula
- 		\[
- 			f(x)=\frac{x(x+3)}{x^2-4x}
- 		\]
- 		The domain of $f$ is $(-\infty,0)\cup(0,4)\cup(4,\infty)$ because both $0$ and $4$
- 		make the denominator equal to $0$. Notice that
- 		\begin{align*}
- 			f(x) & = \frac{x(x+3)}{x(x-4)} \\
- 			     & = \frac{x+3}{x-4}       
- 		\end{align*}
- 		provided that $x\ne 0$. Since $0$ makes the numerator
- 		and the denominator 0 at the same time, we say that $f$ has a hole at $(0,-\nicefrac{3}{4})$.
- 		Note that this necessarily means that $f$ does not have a vertical intercept.
- 		 		 		
- 		We also note $f$ has a vertical asymptote at $4$; the function is graphed in \cref{rat:fig:holeex}.
- 		\begin{figure}[!htb]
- 			\centering
- 			\begin{tikzpicture}[/pgf/declare function={f=(x+3)/(x-4);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=both,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.36364,samples=50]{f};
- 					\addplot[pccplot] expression[domain=4.77:10]{f};
- 					\addplot[asymptote,domain=-10:10]({4},{x});
- 					\addplot[holdot]coordinates{(0,-0.75)};
- 					\addplot[soldot] coordinates{(-3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x(x+3)}{x^2-4x}$}
- 			\label{rat:fig:holeex}
- 		\end{figure}
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Minimums and maximums]
- 		\pccname{Seamus} and \pccname{Trang} are discussing rational functions. Seamus says that
- 		if a rational function has a vertical asymptote, then it can
- 		not possibly have local minimums and maximums, nor can it have
- 		global minimums and maximums.
- 		 		 		
- 		Trang says this statement is not always true. She plots the functions
- 		$f$ and $g$ that have formulas
- 		\[
- 			f(x)=-\frac{32(x-1)(x+1)}{(x-2)^2(x+2)^2}, \qquad g(x)=\frac{32(x-1)(x+1)}{(x-2)^2(x+2)^2}
- 		\]
- 		in \cref{rat:fig:minmax1,rat:fig:minmax2} and shows them to Seamus. On seeing the graphs,
- 		Seamus quickly corrects himself, and says  that $f$ has a local (and global)
- 		maximum of $2$ at $0$, and that $g$ has a local (and global) minimum of $-2$ at $0$.
- 		 		 		
- 		\begin{figure}[!htb]
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=-32*(x-1)*(x+1)/(( x-2)^2*(x+2)^2);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-6,...,8},
- 							ytick={-8,-6,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-3.01]{f};
- 						\addplot[pccplot] expression[domain=-1.45:1.45]{f};
- 						\addplot[pccplot] expression[domain=3.01:10]{f};
- 						\addplot[soldot] coordinates{(-1,0)(1,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=f(x)$}
- 				\label{rat:fig:minmax1}
- 			\end{minipage}%
- 			\hfill
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=32*(x-1)*(x+1)/(( x-2)^2*(x+2)^2);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-6,...,8},
- 							ytick={-8,-6,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-3.01]{f};
- 						\addplot[pccplot] expression[domain=-1.45:1.45]{f};
- 						\addplot[pccplot] expression[domain=3.01:10]{f};
- 						\addplot[soldot] coordinates{(-1,0)(1,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=g(x)$}
- 				\label{rat:fig:minmax2}
- 			\end{minipage}%
- 		\end{figure}
- 		 		 		
- 		Seamus also notes that (in its domain) the function $f$ is always concave down, and
- 		that (in its domain) the function $g$ is always concave up. Furthermore, Trang
- 		observes that each function behaves like $\frac{1}{x^2}$ around each of its vertical
- 		asymptotes, because each linear factor in the denominator is raised to the power $2$.
- 		 		 		
- 		\pccname{Oscar} stops by and reminds both students about the long-run behavior; according
- 		to \cref{rat:def:longrun} since the degree of the denominator is greater than the
- 		degree of the numerator (in both functions), each function has a horizontal asymptote
- 		at $y=0$.
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	\investigation*{}
- 	%===================================
- 	%   Author: Pettit/Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[The spaghetti incident]
- 	The same Queen from \vref{exp:prob:queenschessboard} has recovered from
- 	the rice experiments, and has called her loyal jester for another challenge.
- 	 	 	
- 	The jester has an $11-$inch piece of uncooked spaghetti that he puts on a table;
- 	he uses a book to cover $\unit[1]{inch}$ of it so that
- 	$\unit[10]{inches}$ hang over the edge. The jester then produces a box of $\unit{mg}$
- 	weights that can be hung from the spaghetti.
- 	 	 	
- 	The jester says it will take $\unit[y]{mg}$ to break the spaghetti when hung
- 	$\unit[x]{inches}$ from the edge, according to the rule $y=\frac{100}{x}$.
- 	\begin{margintable}
- 		\centering
- 		\captionof{table}{}
- 		\label{rat:tab:spaghetti}
- 		\begin{tabular}{cc}
- 			\beforeheading
- 			\heading{$x$} & \heading{$y$} \\
- 			\afterheading
- 			$1$           &               \\\normalline
- 			$2$           &               \\\normalline
- 			$3$           &               \\\normalline
- 			$4$           &               \\\normalline
- 			$5$           &               \\\normalline
- 			$6$           &               \\\normalline
- 			$7$           &               \\\normalline
- 			$8$           &               \\\normalline
- 			$9$           &               \\\normalline
- 			$10$          &               \\\lastline
- 		\end{tabular}
- 	\end{margintable}
- 	\begin{subproblem}\label{rat:prob:spaggt1}
- 		Help the Queen complete \cref{rat:tab:spaghetti}, and use $2$ digits after the decimal
- 		where appropriate.
- 		\begin{shortsolution}
- 			\begin{tabular}[t]{ld{2}}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				$1$           & 100           \\\normalline
- 				$2$           & 50            \\\normalline
- 				$3$           & 33.33         \\\normalline
- 				$4$           & 25            \\\normalline
- 				$5$           & 20            \\\normalline
- 				$6$           & 16.67         \\\normalline
- 				$7$           & 14.29         \\\normalline
- 				$8$           & 12.50         \\\normalline
- 				$9$           & 11.11         \\\normalline
- 				$10$          & 10            \\\lastline
- 			\end{tabular}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What do you notice about the number of $\unit{mg}$ that it takes to break
- 		the spaghetti as $x$ increases?
- 		\begin{shortsolution}
- 			It seems that the number of $\unit{mg}$ that it takes to break the spaghetti decreases
- 			as $x$ increases.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:spaglt1}
- 		The Queen wonders what happens when $x$ gets very small| help the Queen construct
- 		a table of values for $x$ and $y$ when $x=0.0001, 0.001, 0.01, 0.1, 0.5, 1$.
- 		\begin{shortsolution}
- 			\begin{tabular}[t]{d{2}l}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				0.0001        & $1000000$     \\\normalline
- 				0.001         & $100000$      \\\normalline
- 				0.01          & $10000$       \\\normalline
- 				0.1           & $1000$        \\\normalline
- 				0.5           & $200$         \\\normalline
- 				1             & $100$         \\\lastline
- 			\end{tabular}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What do you notice about the number of $\unit{mg}$ that it takes to break the spaghetti
- 		as $x\rightarrow 0$? Would it ever make sense to let $x=0$?
- 		\begin{shortsolution}
- 			The number of $\unit{mg}$ required to break the spaghetti increases as $x\rightarrow 0$.
- 			We can not allow $x$ to be $0$, as we can not divide by $0$, and we can not
- 			be $0$ inches from the edge of the table.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Plot your results from \cref{rat:prob:spaggt1,rat:prob:spaglt1} on the same graph,
- 		and join the points using a smooth curve| set the maximum value of $y$ as $200$, and
- 		note that this necessarily means that you will not be able to plot all of the points.
- 		\begin{shortsolution}
- 			The graph of $y=\frac{100}{x}$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-2,xmax=11,
- 						ymin=-20,ymax=200,
- 						xtick={2,4,...,10},
- 						ytick={20,40,...,180},
- 						grid=major,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot+[-] expression[domain=0.5:10]{100/x};
- 					\addplot[soldot] coordinates{(0.5,200)(1,100)(2,50)(3,33.33)
- 					(4,25)(5,20)(16.67)(7,14.29)(8,12.50)(9,11.11)(10,10)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Using your graph, observe what happens to $y$ as $x$ increases. If we could somehow
- 		construct a piece of uncooked spaghetti that was $\unit[101]{inches}$ long, how many
- 		$\unit{mg}$ would it take to break the spaghetti?
- 		\begin{shortsolution}
- 			As $x$ increases, $y\rightarrow 0$. If we could construct a piece of spaghetti
- 			$\unit[101]{inches}$ long, it would only take $\unit[1]{mg}$ to break it $\left(\frac{100}{100}=1\right)$. Of course,
- 			the weight of spaghetti would probably cause it to break without the weight.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	The Queen looks forward to more food-related investigations from her jester.
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Adams (Hughes)
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Debt Amortization]
- 	To amortize a debt means to pay it off in a given length of time using
- 	equal periodic payments. The payments include interest on the unpaid
- 	balance. The following formula gives the monthly payment, $M$, in dollars
- 	that is necessary to amortize a debt of $P$ dollars in $n$ months
- 	at a monthly interest rate of $i$
- 	\[
- 		M=\frac{P\cdot i}{1-(1+i)^{-n}}
- 	\]
- 	Use this formula in each of the following problems.
- 	\begin{subproblem}
- 		What monthly payments are necessary on a credit card debt of \$2000 at
- 		$\unit[1.5]{\%}$ monthly if you want to pay off the debt in $2$ years?
- 		In one year? How much money will you save by paying off the debt in the
- 		shorter amount of time?
- 		\begin{shortsolution}
- 			Paying off the debt in $2$ years, we use
- 			\begin{align*}
- 				M & = \frac{2000\cdot 0.015}{1-(1+0.015)^{-24}} \\
- 				  & \approx 99.85                               
- 			\end{align*}
- 			The monthly payments are \$99.85.
- 			 			 			
- 			Paying off the debt in $1$ year, we use
- 			\begin{align*}
- 				M & = \frac{2000\cdot 0.015}{1-(1+0.015)^{-12}} \\
- 				  & \approx 183.36                              
- 			\end{align*}
- 			The monthly payments are \$183.36
- 			 			 			
- 			In the $2$-year model we would pay a total of $\$99.85\cdot 12=\$2396.40$. In the
- 			$1$-year model we would pay a total of $\$183.36\cdot 12=\$2200.32$. We would therefore
- 			save $\$196.08$ if we went with the $1$-year model instead of the $2$-year model.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		To purchase a home, a family needs a loan of \$300,000 at $\unit[5.2]{\%}$
- 		annual interest.  Compare a $20$ year loan to a $30$ year loan and make
- 		a recommendation for the family.
- 		(Note: when given an annual interest rate, it is a common business practice to divide by
- 		$12$ to get a monthly rate.)
- 		\begin{shortsolution}
- 			For the $20$-year loan we use
- 			\begin{align*}
- 				M & = \frac{300000\cdot \frac{0.052}{12}}{1-\left( 1+\frac{0.052}{12} \right)^{-12\cdot 20}} \\
- 				  & \approx 2013.16                                                                          
- 			\end{align*}
- 			The monthly payments are \$2013.16.
- 			 			 			
- 			For the $30$-year loan we use
- 			\begin{align*}
- 				M & = \frac{300000\cdot \frac{0.052}{12}}{1-\left( 1+\frac{0.052}{12} \right)^{-12\cdot 30}} \\
- 				  & \approx 1647.33                                                                          
- 			\end{align*}
- 			The monthly payments are \$1647.33.
- 			 			 			
- 			The total amount paid during the $20$-year loan is $\$2013.16\cdot 12\cdot 20=\$483,158.40$.
- 			The total amount paid during the $30$-year loan is $\$1647.33\cdot 12\cdot 30=\$593,038.80$.
- 			 			 			
- 			Recommendation: if you can afford the payments, choose the $20$-year loan.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		\pccname{Ellen} wants to make monthly payments of \$100 to pay off a debt of \$3000
- 		at \unit[12]{\%} annual interest. How long will it take her to pay off the
- 		debt?
- 		\begin{shortsolution}
- 			We are given $M=100$, $P=3000$, $i=0.01$, and we need to find $n$
- 			in the equation
- 			\[
- 				100 = \frac{3000\cdot 0.01}{1-(1+0.01)^{-n}}
- 			\]
- 			Using logarithms, we find that $n\approx 36$. It will take
- 			Ellen about $3$ years to pay off the debt.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		\pccname{Jake} is going to buy a new car. He puts \$2000 down and wants to finance the
- 		remaining \$14,000. The dealer will offer him \unit[4]{\%} annual interest for
- 		$5$ years, or a \$2000
- 		rebate which he can use to reduce the amount of the loan and \unit[8]{\%}
- 		annual interest for 5 years. Which should he choose?
- 		\begin{shortsolution}
- 			\begin{description}
- 				\item[Option 1:] $\unit[4]{\%}$ annual interest for $5$ years on \$14,000.
- 				This means that the monthly payments will be calculated using
- 				\begin{align*}
- 					M & = \frac{14000\cdot \frac{0.04}{12}}{1-\left( 1+\frac{0.04}{12} \right)^{-12\cdot 5}} \\
- 					  & \approx 257.83                                                                       
- 				\end{align*}
- 				The monthly payments will be $\$257.83$. The total amount paid will be
- 				$\$257.83\cdot 5\cdot 12=\$15,469.80$, of which $\$1469.80$ is interest.
- 				\item[Option 2:] $\unit[8]{\%}$ annual interest for $5$ years on \$12,000.
- 				This means that the monthly payments will be calculated using
- 				\begin{align*}
- 					M & = \frac{12000\cdot \frac{0.08}{12}}{1-\left( 1+\frac{0.08}{12} \right)^{-12\cdot 5}} \\
- 					  & \approx 243.32                                                                       
- 				\end{align*}
- 				The monthly payments will be $\$243.32$. The total amount paid
- 				will be $\$243.32\cdot 5\cdot 12 =\$14,599.20$, of which $\$2599.2$ is
- 				interest.
- 			\end{description}
- 			Jake should choose option 1 to minimize the amount of interest
- 			he has to pay.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	\begin{exercises}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Rational or not]
- 	Decide if each of the following functions are rational or not. If
- 	they are rational, state their domain.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$r(x)=\dfrac{3}{x}$
- 			\begin{shortsolution}
- 				$r$ is rational; the domain of $r$ is $(-\infty,0)\cup (0,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$s(y)=\dfrac{y}{6}$
- 			\begin{shortsolution}
- 				$s$ is not rational ($s$ is linear).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$t(z)=\dfrac{4-x}{7-8z}$
- 			\begin{shortsolution}
- 				$t$ is rational; the domain of $t$ is $\left( -\infty,\dfrac{7}{8} \right)\cup \left( \dfrac{7}{8},\infty \right)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$u(w)=\dfrac{w^2}{(w-3)(w+4)}$
- 			\begin{shortsolution}
- 				$u$ is rational; the domain of $w$ is $(-\infty,-4)\cup(-4,3)\cup(3,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$v(x)=\dfrac{4}{(x-2)^2}$
- 			\begin{shortsolution}
- 				$v$ is rational; the domain of $v$ is $(-\infty,2)\cup(2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$w(x)=\dfrac{9-x}{x+17}$
- 			\begin{shortsolution}
- 				$w$ is rational; the domain of $w$ is $(-\infty,-17)\cup(-17,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$a(x)=x^2+4$
- 			\begin{shortsolution}
- 				$a$ is not rational ($a$ is quadratic, or a polynomial of degree $2$).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$b(y)=3^y$
- 			\begin{shortsolution}
- 				$b$ is not rational ($b$ is exponential).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$c(z)=\dfrac{z^2}{z^3}$
- 			\begin{shortsolution}
- 				$c$ is rational; the domain of $c$ is $(-\infty,0)\cup (0,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$d(x)=x^2(x+3)(5x-7)$
- 			\begin{shortsolution}
- 				$d$ is not rational ($d$ is a polynomial).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$e(\alpha)=\dfrac{\alpha^2}{\alpha^2-1}$
- 			\begin{shortsolution}
- 				$e$ is rational; the domain of $e$ is $(-\infty,-1)\cup(-1,1)\cup(1,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$f(\beta)=\dfrac{3}{4}$
- 			\begin{shortsolution}
- 				$f$ is not rational ($f$ is constant).
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Function evaluation]
- 	Let $r$ be the function that has formula
- 	\[
- 		r(x)=\frac{(x-2)(x+3)}{(x+5)(x-7)}
- 	\]
- 	Evaluate each of the following (if possible); if the value is undefined,
- 	then state so.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$r(0)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(0) & =\frac{(0-2)(0+3)}{(0+5)(0-7)} \\
- 					     & =\frac{-6}{-35}                \\
- 					     & =\frac{6}{35}                  
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(1)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(1) & =\frac{(1-2)(1+3)}{(1+5)(1-7)} \\
- 					     & =\frac{-4}{-36}                \\
- 					     & =\frac{1}{9}                   
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(2)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(2) & =\frac{(2-2)(2+3)}{(2+5)(2-7)} \\
- 					     & = \frac{0}{-50}                \\
- 					     & =0                             
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(4)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(4) & =\frac{(4-2)(4+3)}{(4+5)(4-7)} \\
- 					     & =\frac{14}{-27}                \\
- 					     & =-\frac{14}{27}                
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(7)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(7) & =\frac{(7-2)(7+3)}{(7+5)(7-7)} \\
- 					     & =\frac{50}{0}                  
- 				\end{aligned}$
- 				 				 				
- 				$r(7)$ is undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(-3)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(-3) & =\frac{(-3-2)(-3+3)}{(-3+5)(-3-7)} \\
- 					      & =\frac{0}{-20}                     \\
- 					      & =0                                 
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(-5)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(-5) & =\frac{(-5-2)(-5+3)}{(-5+5)(-5-7)} \\
- 					      & =\frac{14}{0}                      
- 				\end{aligned}$
- 				 				 				
- 				$r(-5)$ is undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r\left( \frac{1}{2} \right)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r\left( \frac{1}{2} \right) & = \frac{\left( \frac{1}{2}-2 \right)\left( \frac{1}{2}+3 \right)}{\left( \frac{1}{2}+5 \right)\left( \frac{1}{2}-7 \right)} \\
- 					                            & =\frac{-\frac{3}{2}\cdot\frac{7}{2}}{\frac{11}{2}\left( -\frac{13}{2} \right)}                                              \\
- 					                            & =\frac{-\frac{21}{4}}{-\frac{143}{4}}                                                                                       \\
- 					                            & =\frac{37}{143}                                                                                                             
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Holes or asymptotes?]
- 	State the domain of each of the following rational functions. Identify
- 	any holes or asymptotes.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x)=\dfrac{12}{x-2}$
- 			\begin{shortsolution}
- 				$f$ has a vertical asymptote at $2$; the domain of $f$ is $(-\infty,2)\cup (2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x)=\dfrac{x^2+x}{(x+1)(x-2)}$
- 			\begin{shortsolution}
- 				$g$ has a vertical asymptote at $2$, and a hole at $-1$; the domain of $g$ is $(-\infty,-1)\cup(-1,2)\cup(2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x)=\dfrac{x^2+5x+4}{x^2+x-12}$
- 			\begin{shortsolution}
- 				$h$ has a vertical asymptote at $3$, and a whole at $-4$; the domain of $h$ is $(-\infty,-4)\cup(-4,3)\cup(3,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(z)=\dfrac{z+2}{2z-3}$
- 			\begin{shortsolution}
- 				$k$ has a vertical asymptote at $\dfrac{3}{2}$; the domain of $k$ is $\left( -\infty,\dfrac{3}{2} \right)\cup\left( \dfrac{3}{2},\infty \right)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(w)=\dfrac{w}{w^2+1}$
- 			\begin{shortsolution}
- 				$l$ does not have any vertical asymptotes nor holes; the domain of $w$ is $(-\infty,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(t)=\dfrac{14}{13-t^2}$
- 			\begin{shortsolution}
- 				$m$ has vertical asymptotes at $\pm\sqrt{13}$; the domain of $m$ is $(-\infty,\sqrt{13})\cup(-\sqrt{13},\sqrt{13})\cup(\sqrt{13},\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a graph]
- 	Consider the rational functions graphed in \cref{rat:fig:findformula}. Find
- 	the vertical asymptotes for each function, together with any zeros, and
- 	give a possible formula for each.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item \Vref{rat:fig:formula1}: possible formula is $r(x)=\dfrac{1}{x+5}$
- 			\item \Vref{rat:fig:formula2}: possible formula is $r(x)=\dfrac{(x+3)}{(x-5)}$
- 			\item \Vref{rat:fig:formula3}: possible formula is $r(x)=\dfrac{1}{(x-4)(x+3)}$.
- 		\end{itemize}
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=1/(x+4);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-6,ymax=6,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-4.16667,samples=50]{f};
- 					\addplot[pccplot] expression[domain=-3.83333:10,samples=50]{f};
- 					\addplot[asymptote,domain=-6:6]({-4},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x+3)/(x-5);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-6,ymax=6,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.85714]{f};
- 					\addplot[pccplot] expression[domain=6.6:10]{f};
- 					\addplot[soldot] coordinates{(-3,0)};
- 					\addplot[asymptote,domain=-6:6]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{1});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=1/((x-4)*(x+3));}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-3,ymax=3,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-3.0473]{f};
- 					\addplot[pccplot] expression[domain=-2.95205:3.95205]{f};
- 					\addplot[pccplot] expression[domain=4.0473:10]{f};
- 					\addplot[asymptote,domain=-3:3]({-3},{x});
- 					\addplot[asymptote,domain=-3:3]({4},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:findformula}
- 		\end{widepage}
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a formula of a rational
- 	function that has the listed properties.
- 	\begin{subproblem}
- 		Vertical asymptote at $2$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{1}{x-2}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Vertical asymptote at $5$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{1}{x-5}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Vertical asymptote at $-2$, and zero at $6$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{x-6}{x+2}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Zeros at $2$ and $-5$ and vertical asymptotes at $1$ and $-7$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{(x-2)(x+5)}{(x-1)(x+7)}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Given formula, find horizontal asymptotes]
- 	Each of the following functions has a horizontal asymptote. Write the equation
- 	of the horizontal asymptote for each function.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x) = \dfrac{1}{x}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x) = \dfrac{2x+3}{x}$
- 			\begin{shortsolution}
- 				$y=2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x) = \dfrac{x^2+2x}{x^2+3}$
- 			\begin{shortsolution}
- 				$y=1$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(x) = \dfrac{x^2+7}{x}$
- 			\begin{shortsolution}
- 				$y=1$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(x)=\dfrac{3x-2}{5x+8}$
- 			\begin{shortsolution}
- 				$y=\dfrac{3}{5}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(x)=\dfrac{3x-2}{5x^2+8}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$n(x)=\dfrac{(6x+1)(x-7)}{(11x-8)(x-5)}$
- 			\begin{shortsolution}
- 				$y=\dfrac{6}{11}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=\dfrac{19x^3}{5-x^4}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(x)=\dfrac{14x^2+x}{1-7x^2}$
- 			\begin{shortsolution}
- 				$y=-2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{problem}[Given horizontal asymptotes, find formula]
- 	In each of the following problems, give a formula for a function that
- 	has the given horizontal asymptote. Note that there may be more than one option.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$y=7$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{7(x-2)}{x+1}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $7$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-1$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{5-x^2}{x^2+10}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $10$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=53$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{53x^3}{x^3+4x^2-7}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $53$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-17$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{34(x+2)}{7-2x}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $-17$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=\dfrac{3}{2}$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{3x+4}{2(x+1)}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $\dfrac{3}{2}$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=0$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{4}{x}$. Note that there
- 				are other options, provided that the degree of the numerator is less than the degree
- 				of the denominator.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-1$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{10x}{5-10x}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $-1$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=2$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{8x-3}{4x+1}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $2$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a formula for a function that
- 	has the prescribed properties. Note that there may be more than one option.
- 	\begin{subproblem}
- 		$f(x)\rightarrow 3$ as $x\rightarrow\pm\infty$.
- 		\begin{shortsolution}
- 			Possible option: $f(x)=\dfrac{3(x-2)}{x+7}$. Note that
- 			the zero and asymptote of $f$ could be changed, and $f$ would still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$r(x)\rightarrow -4$ as $x\rightarrow\pm\infty$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{-4(x-2)}{x+7}$. Note that
- 			the zero and asymptote of $r$ could be changed, and $r$ would still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$k(x)\rightarrow 2$ as $x\rightarrow\pm\infty$, and $k$ has vertical asymptotes at $-3$ and $5$.
- 		\begin{shortsolution}
- 			Possible option: $k(x)=\dfrac{2x^2}{(x+3)(x-5)}$. Note that the denominator
- 			must have the given factors; the numerator could be any degree $2$ polynomial, provided the
- 			leading coefficient is $2$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   Feb 2011
- 	%===================================
- 	\begin{problem}
- 	Let $r$ be the rational function that has
- 	\[
- 		r(x) = \frac{(x+2)(x-1)}{(x+3)(x-4)}
- 	\]
- 	Each of the following questions are in relation to this function.
- 	\begin{subproblem}
- 		What is the vertical intercept of this function? State your answer as an
- 		ordered pair. \index{rational functions!vertical intercept}
- 		\begin{shortsolution}
- 			$\left(0,\frac{1}{6}\right)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:rational}
- 		What values of $x$ make the denominator equal to $0$?
- 		\begin{shortsolution}
- 			$-3,4$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use your answer to \cref{rat:prob:rational} to write the domain of the function in
- 		both interval, and set builder notation. %\index{rational functions!domain}\index{domain!rational functions}
- 		\begin{shortsolution}
- 			Interval notation: $(-\infty,-3)\cup (-3,4)\cup (4,\infty)$.
- 			Set builder: $\{x|x\ne -3, \mathrm{and}\, x\ne 4\}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What are the vertical asymptotes of the function? State your answers in
- 		the form $x=$
- 		\begin{shortsolution}
- 			$x=-3$ and $x=4$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:zeroes}
- 		What values of $x$ make the numerator equal to $0$?
- 		\begin{shortsolution}
- 			$-2,1$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use your answer to \cref{rat:prob:zeroes} to write the horizontal intercepts of
- 		$r$ as ordered pairs.
- 		\begin{shortsolution}
- 			$(-2,0)$ and $(1,0)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Holes]
- 	\pccname{Josh} and \pccname{Pedro} are discussing the function
- 	\[
- 		r(x)=\frac{x^2-1}{(x+3)(x-1)}
- 	\]
- 	\begin{subproblem}
- 		What is the domain of $r$?
- 		\begin{shortsolution}
- 			The domain of $r$ is $(-\infty,-3)\cup(-3,1)\cup(1,\infty)$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Josh notices that the numerator can be factored- can you see how?
- 		\begin{shortsolution}
- 			$(x^2-1)=(x-1)(x+1)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Pedro asks, `Doesn't that just mean that
- 		\[
- 			r(x)=\frac{x+1}{x+3}
- 		\]
- 		for all values of $x$?' Josh says, `Nearly\ldots but not for all values of $x$'.
- 		What does Josh mean?
- 		\begin{shortsolution}
- 			$r(x)=\dfrac{x+1}{x+3}$ provided that $x\ne -1$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Where does $r$ have vertical asymptotes, and where does it have holes?
- 		\begin{shortsolution}
- 			The function $r$ has a vertical asymptote at $-3$, and a hole at $1$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Sketch a graph of $r$.
- 		\begin{shortsolution}
- 			A graph of $r$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-3.25]{(x+1)/(x+3)};
- 					\addplot[pccplot] expression[domain=-2.75:10]{(x+1)/(x+3)};
- 					\addplot[asymptote,domain=-10:10]({-3},{x});
- 					\addplot[holdot]coordinates{(1,0.5)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Function algebra]
- 	Let $r$ and $s$ be the rational functions that have formulas
- 	\[
- 		r(x)=\frac{2-x}{x+3}, \qquad s(x)=\frac{x^2}{x-4}
- 	\]
- 	Evaluate each of the following (if possible).
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(r+s)(5)$
- 			\begin{shortsolution}
- 				$\frac{197}{8}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(r-s)(3)$
- 			\begin{shortsolution}
- 				$\frac{53}{6}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(r\cdot s)(4)$
- 			\begin{shortsolution}
- 				Undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\left( \frac{r}{s} \right)(1)$
- 			\begin{shortsolution}
- 				$-\frac{3}{4}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Transformations: given the transformation, find the formula]
- 	Let $r$ be the rational function that has formula.
- 	\[
- 		r(x)=\frac{x+5}{2x-3}
- 	\]
- 	In each of the following problems apply the given transformation to the function $r$ and
- 	write a formula for the transformed version of $r$.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			Shift $r$ to the right by $3$ units.
- 			\begin{shortsolution}
- 				$r(x-3)=\frac{x+2}{2x-9}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ to the left by $4$ units.
- 			\begin{shortsolution}
- 				$r(x+4)=\frac{x+9}{2x+5}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ up by $\pi$ units.
- 			\begin{shortsolution}
- 				$r(x)+\pi=\frac{x+5}{2x-3}+\pi$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ down by $17$ units.
- 			\begin{shortsolution}
- 				$r(x)-17=\frac{x+5}{2x-3}-17$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $r$ over the horizontal axis.
- 			\begin{shortsolution}
- 				$-r(x)=-\frac{x+5}{2x-3}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $r$ over the vertical axis.
- 			\begin{shortsolution}
- 				$r(-x)=\frac{x-5}{2x+3}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a table]\label{rat:prob:findformula}
- 	\Crefrange{rat:tab:findformular}{rat:tab:findformulau} show values of rational functions $r$, $q$, $s$,
- 	and $t$. Assume that any values marked with an X are undefined.
- 	 	 	
- 	\begin{table}[!htb]
- 		\begin{widepage}
- 		\centering
- 		\caption{Tables for \cref{rat:prob:findformula}}
- 		\label{rat:tab:findformula}
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=r(x)$}
- 			\label{rat:tab:findformular}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                \\ \afterheading
- 				$-4$ & $\nicefrac{7}{2}$  \\\normalline
- 				$-3$ & $-18$              \\\normalline
- 				$-2$ & X                  \\\normalline
- 				$-1$ & $-4$               \\\normalline
- 				$0$  & $\nicefrac{-3}{2}$ \\\normalline
- 				$1$  & $\nicefrac{-2}{3}$ \\\normalline
- 				$2$  & $\nicefrac{-1}{4}$ \\\normalline
- 				$3$  & $0$                \\\normalline
- 				$4$  & $\nicefrac{1}{6}$  \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=s(x)$}
- 			\label{rat:tab:findformulas}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                 \\ \afterheading
- 				$-4$ & $\nicefrac{-2}{21}$ \\\normalline
- 				$-3$ & $\nicefrac{-1}{12}$ \\\normalline
- 				$-2$ & $0$                 \\\normalline
- 				$-1$ & X                   \\\normalline
- 				$0$  & $\nicefrac{-2}{3}$  \\\normalline
- 				$1$  & $\nicefrac{-3}{4}$  \\\normalline
- 				$2$  & $\nicefrac{-4}{3}$  \\\normalline
- 				$3$  & X                   \\\normalline
- 				$4$  & $\nicefrac{6}{5}$   \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=t(x)$}
- 			\label{rat:tab:findformulat}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$               \\ \afterheading
- 				$-4$ & $\nicefrac{3}{5}$ \\\normalline
- 				$-3$ & $0$               \\\normalline
- 				$-2$ & X                 \\\normalline
- 				$-1$ & $3$               \\\normalline
- 				$0$  & $3$               \\\normalline
- 				$1$  & X                 \\\normalline
- 				$2$  & $0$               \\\normalline
- 				$3$  & $\nicefrac{3}{5}$ \\\normalline
- 				$4$  & $\nicefrac{7}{9}$ \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=u(x)$}
- 			\label{rat:tab:findformulau}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                \\ \afterheading
- 				$-4$ & $\nicefrac{16}{7}$ \\\normalline
- 				$-3$ & X                  \\\normalline
- 				$-2$ & $-\nicefrac{4}{5}$ \\\normalline
- 				$-1$ & $-\nicefrac{1}{8}$ \\\normalline
- 				$0$  & $0$                \\\normalline
- 				$1$  & $-\nicefrac{1}{8}$ \\\normalline
- 				$2$  & $-\nicefrac{4}{5}$ \\\normalline
- 				$3$  & X                  \\\normalline
- 				$4$  & $\nicefrac{16}{7}$ \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\end{widepage}
- 	\end{table}
- 	\begin{subproblem}
- 		Given that the formula for $r(x)$ has the form $r(x)=\dfrac{x-A}{x-B}$, use \cref{rat:tab:findformular}
- 		to find values of $A$ and $B$.
- 		\begin{shortsolution}
- 			$A=3$ and $B=-2$, so $r(x)=\dfrac{x-3}{x+2}$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Check your formula by computing $r(x)$ at the values specified in the table.
- 		\begin{shortsolution}
- 			$\begin{aligned}[t]
- 				r(-4) & = \frac{-4-3}{-4+2} \\
- 				      & = \frac{7}{2}       \\
- 			\end{aligned}$
- 			 			 			
- 			$r(-3)=\ldots$ etc
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		The function $s$ in \cref{rat:tab:findformulas} has two vertical asymptotes and one zero.
- 		Can you find a formula for $s(x)$?
- 		\begin{shortsolution}
- 			$s(x)=\dfrac{x+2}{(x-3)(x+1)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Check your formula by computing $s(x)$ at the values specified in the table.
- 		\begin{shortsolution}
- 			$\begin{aligned}[t]
- 				s(-4) & =\frac{-4+2}{(-4-3)(-4+1)} \\
- 				      & =-\frac{2}{21}             
- 			\end{aligned}$
- 			 			 			
- 			$s(-3)=\ldots$ etc
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that the formula for $t(x)$ has  the form $t(x)=\dfrac{(x-A)(x-B)}{(x-C)(x-D)}$, use \cref{rat:tab:findformulat} to find the
- 		values of $A$, $B$, $C$, and $D$; hence write a formula for $t(x)$.
- 		\begin{shortsolution}
- 			$t(x)=\dfrac{(x+3)(x-2)}{(x+2)(x+1)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that the formula for $u(x)$ has  the form $u(x)=\dfrac{(x-A)^2}{(x-B)(x-C)}$, use \cref{rat:tab:findformulau} to find the
- 		values of $A$, $B$, and $C$; hence write a formula for $u(x)$.
- 		\begin{shortsolution}
- 			$u(x)=\dfrac{x^2}{(x+3)(x-3)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	\end{exercises}
- 	 	 	
-\section{Graphing rational functions (horizontal asymptotes)}
- \reformatstepslist{R} % the steps list should be R1, R2, \ldots
- We studied rational functions in the previous section, but were
- not asked to graph them; in this section we will demonstrate the
- steps to be followed in order to sketch graphs of the functions.
-   
- Remember from \vref{rat:def:function} that rational functions have
- the form
- \[
- 	r(x)=\frac{p(x)}{q(x)}
- \]
- In this section we will restrict attention to the case when
- \[
- 	\text{degree of }p\leq \text{degree of }q
- \]
- Note that this necessarily means that each function that we consider
- in this section \emph{will have a horizontal asymptote} (see \vref{rat:def:longrun}).
- The cases in which the degree of $p$ is greater than the degree of $q$
- is covered in the next section.
-   
- Before we begin, it is important to remember the following:
- \begin{itemize}
- 	\item Our sketches will give a good representation of the overall
- 	      shape of the graph, but until we have the tools of calculus (from MTH 251)
- 	      we can not find local minimums, local maximums, and inflection points algebraically. This
- 	      means that we will make our best guess as to where these points are.
- 	\item We will not concern ourselves too much with the vertical scale (because of
- 	      our previous point)| we will, however, mark the vertical intercept (assuming there is one),
- 	      and any horizontal asymptotes.
- \end{itemize}
- \begin{pccspecialcomment}[Steps to follow when sketching rational functions]\label{rat:def:stepsforsketch}
- 	\begin{steps}
- 		\item \label{rat:step:first} Find all vertical asymptotes and holes, and mark them on the
- 		graph using dashed vertical lines and open circles $\circ$ respectively.
- 		\item Find any intercepts, and mark them using solid circles $\bullet$;
- 		determine if the curve cuts the axis, or bounces off it at each zero.
- 		\item Determine the behavior of the function around each asymptote| does
- 		it behave like $\frac{1}{x}$ or $\frac{1}{x^2}$?
- 		\item \label{rat:step:penultimate} Determine the long-run behavior of the function, and mark the horizontal
- 		asymptote using a dashed horizontal line.
- 		\item \label{rat:step:last}  Deduce the overall shape of the curve, and sketch it. If there isn't
- 		enough information from the previous steps, then construct a table of values
- 		including sample points from each branch.
- 	\end{steps}
- 	Remember that until we have the tools of calculus, we won't be able to
- 	find the exact coordinates of local minimums, local maximums, and points
- 	of inflection.
- \end{pccspecialcomment}
-   
- The examples that follow show how \crefrange{rat:step:first}{rat:step:last} can be
- applied to a variety of different rational functions.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:1overxminus2p2}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $r$
- 	that has formula
- 	\[
- 		r(x)=\frac{1}{x-2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $r$ has a vertical asymptote at $2$; $r$ does not have any holes. The curve of
- 			$r$ will have $2$ branches.
- 			\item $r$ does not have any zeros since the numerator is never equal to $0$. The
- 			vertical intercept of $r$ is $\left( 0,-\frac{1}{2} \right)$.
- 			\item $r$ behaves like $\frac{1}{x}$ around its vertical asymptote since $(x-2)$
- 			is raised to the power $1$.
- 			\item Since the degree of the numerator is less than the degree of the denominator,
- 			according to \vref{rat:def:longrun} the horizontal asymptote of $r$ has equation $y=0$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:1overxminus2p1}. Notice
- 			that there is only one way to complete the graph, which we have done in \cref{rat:fig:1overxminus2p2}.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-5,xmax=5,
- 					ymin=-5,ymax=5,
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-5:5]({2},{x});
- 				\addplot[asymptote,domain=-5:5]({x},{0});
- 				\addplot[soldot] coordinates{(0,-0.5)}node[axisnode,anchor=north east]{$\left( 0,-\frac{1}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:1overxminus2p1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=1/(x-2);}]
- 			\begin{axis}[
- 					xmin=-5,xmax=5,
- 					ymin=-5,ymax=5,
- 					width=\textwidth,
- 				]
- 				\addplot[pccplot] expression[domain=-5:1.8,samples=50]{f};
- 				\addplot[pccplot] expression[domain=2.2:5]{f};
- 				\addplot[asymptote,domain=-5:5]({2},{x});
- 				\addplot[asymptote,domain=-5:5]({x},{0});
- 				\addplot[soldot] coordinates{(0,-0.5)}node[axisnode,anchor=north east]{$\left( 0,-\frac{1}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:1overxminus2p2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{1}{x-2}$}
- \end{figure}
-   
- The function $r$ in \cref{rat:ex:1overxminus2p2} has a horizontal asymptote which has equation $y=0$.
- This asymptote lies on the horizontal axis, and you might (understandably) find it hard
- to distinguish between the two lines (\cref{rat:fig:1overxminus2p2}). When faced
- with such a situation, it is perfectly acceptable to draw the horizontal axis
- as a dashed line| just make sure to label it correctly. We will demonstrate this
- in the next example.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:1overxp1}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $v$
- 	that has formula
- 	\[
- 		v(x)=\frac{10}{x}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $v$ has a vertical asymptote at $0$. $v$ does not have
- 			any holes. The curve of $v$ will have $2$ branches.
- 			\item $v$ does not have any zeros (since $10\ne 0$). Furthermore, $v$
- 			does not have a vertical intercept since $v(0)$ is undefined.
- 			\item $v$ behaves like $\frac{1}{x}$ around its vertical asymptote.
- 			\item $v$ has a horizontal asymptote with equation $y=0$.
- 			\item We put the details we have obtained so far in \cref{rat:fig:1overxp1}.
- 			We do not have enough information to sketch $v$ yet (because $v$ does
- 			not have any intercepts), so let's pick a sample
- 			point in either of the $2$ branches| it doesn't matter where our sample point
- 			is, because we know what the overall shape will be. Let's compute $v(2)$
- 			\begin{align*}
- 				v(2) & =\dfrac{10}{2} \\
- 				     & = 5            
- 			\end{align*}
- 			We therefore mark the point $(2,5)$ on \cref{rat:fig:1overxp2}, and then complete the sketch using
- 			the details we found in the previous steps.
- 		\end{steps}
- 		 		 		
- 		\begin{figure}[!htbp]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-5,5},
- 							ytick={-5,5},
- 							axis line style={color=white},
- 							width=\textwidth,
- 						]
- 						\addplot[asymptote,<->,domain=-10:10]({0},{x});
- 						\addplot[asymptote,<->,domain=-10:10]({x},{0});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:1overxp1}
- 			\end{subfigure}%
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=10/x;}]
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-5,5},
- 							ytick={-5,5},
- 							axis line style={color=white},
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-1]{f};
- 						\addplot[pccplot] expression[domain=1:10]{f};
- 						\addplot[soldot] coordinates{(2,5)}node[axisnode,anchor=south west]{$(2,5)$};
- 						\addplot[asymptote,<->,domain=-10:10]({0},{x});
- 						\addplot[asymptote,<->,domain=-10:10]({x},{0});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:1overxp2}
- 			\end{subfigure}%
- 			\caption{$y=\dfrac{10}{x}$}
- 		\end{figure}
- 	\end{pccsolution}
- \end{pccexample}
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:asympandholep1}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $u$
- 	that has formula
- 	\[
- 		u(x)=\frac{-4(x^2-9)}{x^2-8x+15}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item We begin by factoring both the numerator and denominator of $u$ to help
- 			us find any vertical asymptotes or holes
- 			\begin{align*}
- 				u(x) & =\frac{-4(x^2-9)}{x^2-8x+15}     \\
- 				     & =\frac{-4(x+3)(x-3)}{(x-5)(x-3)} \\
- 				     & =\frac{-4(x+3)}{x-5}             
- 			\end{align*}
- 			provided that $x\ne 3$. Therefore $u$ has a vertical asymptote at $5$ and
- 			a hole at $3$. The curve of $u$ has $2$ branches.
- 			\item $u$ has a simple zero at $-3$. The vertical intercept of $u$ is $\left( 0,\frac{12}{5} \right)$.
- 			\item $u$ behaves like $\frac{1}{x}$ around its vertical asymptote at $4$.
- 			\item Using \vref{rat:def:longrun} the equation of the horizontal asymptote of $u$ is $y=-4$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:1overxminus2p1}. Notice
- 			that there is only one way to complete the graph, which we have done in \cref{rat:fig:1overxminus2p2}.
- 		\end{steps}
- 		 		 		
- 		\begin{figure}[!htbp]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-20,ymax=20,
- 							xtick={-8,-6,...,8},
- 							ytick={-10,10},
- 							width=\textwidth,
- 						]
- 						\addplot[asymptote,domain=-20:20]({4},{x});
- 						\addplot[asymptote,domain=-10:10]({x},{-4});
- 						\addplot[soldot] coordinates{(-3,0)(0,2.4)}node[axisnode,anchor=south east]{$\left( 0,\frac{12}{5} \right)$};
- 						\addplot[holdot] coordinates{(3,12)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:asympandholep1}
- 			\end{subfigure}%
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=-4*(x+3)/(x-5);}]
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-20,ymax=20,
- 							xtick={-8,-6,...,8},
- 							ytick={-10,10},
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:3.6666,samples=50]{f};
- 						\addplot[pccplot] expression[domain=7:10]{f};
- 						\addplot[asymptote,domain=-20:20]({5},{x});
- 						\addplot[asymptote,domain=-10:10]({x},{-4});
- 						\addplot[soldot] coordinates{(-3,0)(0,2.4)}node[axisnode,anchor=south east]{$\left( 0,\frac{12}{5} \right)$};
- 						\addplot[holdot] coordinates{(3,12)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:asympandholep2}
- 			\end{subfigure}%
- 			\caption{$y=\dfrac{-4(x+3)}{x-5}$}
- 		\end{figure}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \Cref{rat:ex:1overxminus2p2,rat:ex:1overxp1,rat:ex:asympandholep1} have focused on functions
- that only have one vertical asymptote; the remaining examples in this section
- concern functions that have more than one vertical asymptote. We will demonstrate
- that \crefrange{rat:step:first}{rat:step:last} still apply.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:sketchtwoasymp}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $w$
- 	that has formula
- 	\[
- 		w(x)=\frac{2(x+3)(x-5)}{(x+5)(x-4)}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $w$ has vertical asymptotes at $-5$ and $4$. $w$ does not have
- 			any holes. The curve of $w$ will have $3$ branches.
- 			\item $w$ has simple zeros at $-3$ and $5$. The vertical intercept of $w$
- 			is $\left( 0,\frac{3}{2} \right)$.
- 			\item $w$ behaves like $\frac{1}{x}$ around both of its vertical
- 			asymptotes.
- 			\item The degree of the numerator of $w$ is $2$ and the degree of the
- 			denominator of $w$ is also $2$. Using the ratio of the leading coefficients
- 			of the numerator and denominator, we say that $w$ has a horizontal
- 			asymptote with equation $y=\frac{2}{1}=2$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:sketchtwoasymptp1}.
- 			 			 			
- 			The function $w$ is a little more complicated than the functions that
- 			we have considered in the previous examples because the curve has $3$
- 			branches. When graphing such functions, it is generally a good idea to start with the branch
- 			for which you have the most information| in this case, that is the \emph{middle} branch
- 			on the interval $(-5,4)$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			(because of our observations about the behavior of $w$ around its vertical asymptotes),
- 			which we have done in \cref{rat:fig:sketchtwoasymptp2}.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[soldot] coordinates{(-3,0)(5,0)};
- 				\addplot[soldot] coordinates{(0,1.5)}node[axisnode,anchor=north west]{$\left( 0,\frac{3}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:sketchtwoasymptp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=2*(x+3)*(x-5)/( (x+5)*(x-4));}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[soldot] coordinates{(-3,0)(5,0)};
- 				\addplot[soldot] coordinates{(0,1.5)}node[axisnode,anchor=north west]{$\left( 0,\frac{3}{2} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.56708]{f};
- 				\addplot[pccplot] expression[domain=-4.63511:3.81708]{f};
- 				\addplot[pccplot] expression[domain=4.13511:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:sketchtwoasymptp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{2(x+3)(x-5)}{(x+5)(x-4)}$}
- \end{figure}
-   
- The rational functions that we have considered so far have had simple
- factors in the denominator; each function has behaved like $\frac{1}{x}$
- around each of its vertical asymptotes. \Cref{rat:ex:2asympnozeros,rat:ex:2squaredasymp}
- consider functions that have a repeated factor in the denominator.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:2asympnozeros}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $f$
- 	that has formula
- 	\[
- 		f(x)=\frac{100}{(x+5)(x-4)^2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $f$ has vertical asymptotes at $-5$ and $4$. $f$ does not have
- 			any holes. The curve of $f$ will have $3$ branches.
- 			\item $f$ does not have any zeros (since $100\ne 0$). The vertical intercept of $f$
- 			is $\left( 0,\frac{5}{4} \right)$.
- 			\item $f$ behaves like $\frac{1}{x}$ around $-5$ and behaves like $\frac{1}{x^2}$
- 			around $4$.
- 			\item The degree of the numerator of $f$ is $0$ and the degree of the
- 			denominator of $f$ is $2$. $f$ has a horizontal asymptote with
- 			equation $y=0$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:2asympnozerosp1}.
- 			 			 			
- 			The function $f$ is similar to the function $w$ that we considered in \cref{rat:ex:sketchtwoasymp}|
- 			it has two vertical asymptotes and $3$ branches, but in contrast to $w$ it does not have any zeros.
- 			 			 			
- 			We sketch $f$ in \cref{rat:fig:2asympnozerosp2}, using the middle branch as our guide
- 			because we have the most information about the function on the interval $(-5,4)$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $f$ around its vertical asymptotes (it behaves like $\frac{1}{x}$),
- 			which we have done in \cref{rat:fig:2asympnozerosp2}.
- 			 			 			
- 			Note that we are not yet able to find the local minimum of $f$ algebraically on the interval $(-5,4)$,
- 			so we make a reasonable guess as to where it is| we can be confident that it is above the horizontal axis
- 			since $f$ has no zeros. You may think that this is unsatisfactory, but once we have the tools of calculus, we will
- 			be able to find local minimums more precisely.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(0,1.25)}node[axisnode,anchor=south east]{$\left( 0,\frac{5}{4} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2asympnozerosp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=100/( (x+5)*(x-4)^2);}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(0,1.25)}node[axisnode,anchor=south east]{$\left( 0,\frac{5}{4} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.12022]{f};
- 				\addplot[pccplot] expression[domain=-4.87298:2.87298,samples=50]{f};
- 				\addplot[pccplot] expression[domain=5:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2asympnozerosp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{100}{(x+5)(x-4)^2}$}
- \end{figure}
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:2squaredasymp}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $g$
- 	that has formula
- 	\[
- 		g(x)=\frac{50(2-x)}{(x+3)^2(x-5)^2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $g$ has vertical asymptotes at $-3$ and $5$. $g$ does
- 			not have any holes. The curve of $g$ will have $3$ branches.
- 			\item $g$ has a simple zero at $2$. The vertical intercept of $g$ is
- 			$\left( 0,\frac{4}{9} \right)$.
- 			\item $g$ behaves like $\frac{1}{x^2}$ around both of its
- 			vertical asymptotes.
- 			\item The degree of the numerator of $g$ is $1$ and the degree of the denominator
- 			of $g$ is $4$. Using \vref{rat:def:longrun}, we calculate that
- 			the horizontal asymptote of $g$ has equation $y=0$.
- 			\item The details that we have found so far have been drawn in
- 			\cref{rat:fig:2squaredasymp1}. The function $g$ is similar to the functions
- 			we considered in \cref{rat:ex:sketchtwoasymp,rat:ex:2asympnozeros} because
- 			it has $2$ vertical asymptotes and $3$ branches.
- 			 			 			
- 			We sketch $g$ using the middle branch as our guide because we have the most information
- 			about $g$ on the interval $(-3,5)$. Note that there is no other way to draw this branch
- 			without introducing other zeros which $g$ does not have.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $g$ around its vertical asymptotes| it
- 			behaves like $\frac{1}{x^2}$.
- 			 			 			
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-3},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(2,0)(0,4/9)}node[axisnode,anchor=south west]{$\left( 0,\frac{4}{9} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2squaredasymp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=50*(2-x)/( (x+3)^2*(x-5)^2);}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-3},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(2,0)(0,4/9)}node[axisnode,anchor=south west]{$\left( 0,\frac{4}{9} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-3.61504]{f};
- 				\addplot[pccplot] expression[domain=-2.3657:4.52773]{f};
- 				\addplot[pccplot] expression[domain=5.49205:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2squaredasymp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{50(2-x)}{(x+3)^2(x-5)^2}$}
- \end{figure}
-   
- Each of the rational functions that we have considered so far has had either
- a \emph{simple} zero, or no zeros at all. Remember from our work on polynomial
- functions, and particularly \vref{poly:def:multzero}, that a \emph{repeated} zero
- corresponds to the curve of the function behaving differently at the zero
- when compared to how the curve behaves at a simple zero. \Cref{rat:ex:doublezero} details a
- function that has a non-simple zero.
-   
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{pccexample}\label{rat:ex:doublezero}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $g$
- 	that has formula
- 	\[
- 		h(x)=\frac{(x-3)^2}{(x+4)(x-6)}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $h$ has vertical asymptotes at $-4$ and $6$. $h$ does
- 			not have any holes. The curve of $h$ will have $3$ branches.
- 			\item $h$ has a zero at $3$ that has \emph{multiplicity $2$}.
- 			The vertical intercept of $h$ is
- 			$\left( 0,-\frac{3}{8} \right)$.
- 			\item $h$ behaves like $\frac{1}{x}$ around both of its
- 			vertical asymptotes.
- 			\item The degree of the numerator of $h$ is $2$ and the degree of the denominator
- 			of $h$ is $2$. Using \vref{rat:def:longrun}, we calculate that
- 			the horizontal asymptote of $h$ has equation $y=1$.
- 			\item The details that we have found so far have been drawn in
- 			\cref{rat:fig:doublezerop1}. The function $h$ is different
- 			from the functions that we have considered in previous examples because
- 			of the multiplicity of the zero at $3$.
- 			 			 			
- 			We sketch $h$ using the middle branch as our guide because we have the most information
- 			about $h$ on the interval $(-4,6)$. Note that there is no other way to draw this branch
- 			without introducing other zeros which $h$ does not have| also note how
- 			the curve bounces off the horizontal axis at $3$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $h$ around its vertical asymptotes| it
- 			behaves like $\frac{1}{x}$.
- 			 			 			
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-5,ymax=5,
- 					xtick={-8,-6,...,8},
- 					ytick={-3,3},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-4},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{1});
- 				\addplot[soldot] coordinates{(3,0)(0,-3/8)}node[axisnode,anchor=north west]{$\left( 0,-\frac{3}{8} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:doublezerop1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=(x-3)^2/((x+4)*(x-6));}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-5,ymax=5,
- 					xtick={-8,-6,...,8},
- 					ytick={-3,3},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-4},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{1});
- 				\addplot[soldot] coordinates{(3,0)(0,-3/8)}node[axisnode,anchor=north west]{$\left( 0,-\frac{3}{8} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.20088]{f};
- 				\addplot[pccplot] expression[domain=-3.16975:5.83642,samples=50]{f};
- 				\addplot[pccplot] expression[domain=6.20088:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:doublezerop2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{(x-3)^2}{(x+4)(x-6)}$}
- \end{figure}
- \begin{exercises}
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{problem}[\Cref{rat:step:last}]\label{rat:prob:deduce}
- \pccname{Katie} is working on graphing rational functions. She
- has been concentrating on functions that have the form
- \begin{equation}\label{rat:eq:deducecurve}
- 	f(x)=\frac{a(x-b)}{x-c}
- \end{equation}
- Katie notes that functions with this type of formula have a zero
- at $b$, and a vertical asymptote at $c$. Furthermore, these functions
- behave like $\frac{1}{x}$ around their vertical asymptote, and the
- curve of each function will have $2$ branches.
-   
- Katie has been working with $3$ functions that have the form given
- in \cref{rat:eq:deducecurve}, and has followed \crefrange{rat:step:first}{rat:step:penultimate};
- her results are shown in \cref{rat:fig:deducecurve}. There is just one
- more thing to do to complete the graphs| follow \cref{rat:step:last}.
- Help Katie finish each graph by deducing the curve of each function.
- \begin{shortsolution}
- 	\Vref{rat:fig:deducecurve1}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=3*(x+4)/(x+5);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-4,0)(0,12/5)};
- 			\addplot[asymptote,domain=-10:10]({-5},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{3});
- 			\addplot[pccplot] expression[domain=-10:-5.42857]{f};
- 			\addplot[pccplot] expression[domain=-4.76923:10,samples=50]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducecurve2}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=-3*(x-2)/(x-4);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(2,0)(0,-3/2)};
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{-3});
- 			\addplot[pccplot] expression[domain=-10:3.53846,samples=50]{f};
- 			\addplot[pccplot] expression[domain=4.85714:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducecurve4}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=2*(x-6)/(x-4);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(6,0)(0,3)};
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[pccplot] expression[domain=-10:3.5,samples=50]{f};
- 			\addplot[pccplot] expression[domain=4.3333:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- \end{shortsolution}
- \end{problem}
-   
- \begin{figure}[!htb]
- 	\begin{widepage}
- 	\setlength{\figurewidth}{0.3\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-4,0)(0,12/5)};
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(2,0)(0,-3/2)};
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{-3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve2}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(6,0)(0,3)};
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve4}
- 	\end{subfigure}
- 	\caption{Graphs for \cref{rat:prob:deduce}}
- 	\label{rat:fig:deducecurve}
- 	\end{widepage}
- \end{figure}
-   
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{problem}[\Cref{rat:step:last} for  more complicated rational functions]\label{rat:prob:deducehard}
- \pccname{David} is also working on graphing rational functions, and
- has been concentrating on functions that have the form
- \[
- 	r(x)=\frac{a(x-b)(x-c)}{(x-d)(x-e)}
- \]
- David notices that functions with this type of formula have simple zeros
- at $b$ and $c$, and vertical asymptotes at $d$ and $e$. Furthermore,
- these functions behave like $\frac{1}{x}$ around both vertical asymptotes,
- and the curve of the function will have $3$ branches.
-   
- David has followed \crefrange{rat:step:first}{rat:step:penultimate} for
- $3$ separate functions, and drawn the results in \cref{rat:fig:deducehard}.
- Help David finish each graph by deducing the curve of each function.
- \begin{shortsolution}
- 	\Vref{rat:fig:deducehard1}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=(x-6)*(x+3)/( (x-4)*(x+1));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(6,0)(0,9/2)};
- 			\addplot[asymptote,domain=-10:10]({-1},{x});
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[pccplot] expression[domain=-10:-1.24276]{f};
- 			\addplot[pccplot] expression[domain=-0.6666:3.66667]{f};
- 			\addplot[pccplot] expression[domain=4.24276:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducehard2}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=3*(x-2)*(x+3)/( (x-6)*(x+5));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(2,0)(0,3/5)};
- 			\addplot[asymptote,domain=-10:10]({-5},{x});
- 			\addplot[asymptote,domain=-10:10]({6},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{3});
- 			\addplot[pccplot] expression[domain=-10:-5.4861]{f};
- 			\addplot[pccplot] expression[domain=-4.68395:5.22241]{f};
- 			\addplot[pccplot] expression[domain=7.34324:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducehard3}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=2*(x-7)*(x+3)/( (x+6)*(x-5));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(7,0)(0,1.4)};
- 			\addplot[asymptote,domain=-10:10]({-6},{x});
- 			\addplot[asymptote,domain=-10:10]({5},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[pccplot] expression[domain=-10:-6.91427]{f};
- 			\addplot[pccplot] expression[domain=-5.42252:4.66427]{f};
- 			\addplot[pccplot] expression[domain=5.25586:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- \end{shortsolution}
- \end{problem}
-   
- \begin{figure}[!htb]
- 	\begin{widepage}
- 	\setlength{\figurewidth}{0.3\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(6,0)(0,9/2)};
- 				\addplot[asymptote,domain=-10:10]({-1},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(2,0)(0,3/5)};
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard2}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(7,0)(0,1.4)};
- 				\addplot[asymptote,domain=-10:10]({-6},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard3}
- 	\end{subfigure}%
- 	\hfill
- 	\caption{Graphs for \cref{rat:prob:deducehard}}
- 	\label{rat:fig:deducehard}
- 	\end{widepage}
- \end{figure}
- %===================================
- %   Author: Adams (Hughes)
- %   Date:   March 2012
- %===================================
- \begin{problem}[\Crefrange{rat:step:first}{rat:step:last}]
- Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of
- each of the following functions
- \fixthis{need 2 more subproblems here}
- \begin{multicols}{4}
- 	\begin{subproblem}
- 		$y=\dfrac{4}{x+2}$
- 		\begin{shortsolution}
- 			Vertical intercept: $(0,2)$; vertical asymptote: $x=-2$, horizontal asymptote: $y=0$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.8]{4/(x+2)};
- 					\addplot[pccplot] expression[domain=-1.2:5]{4/(x+2)};
- 					\addplot[soldot]coordinates{(0,2)};
- 					\addplot[asymptote,domain=-5:5]({-2},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{2x-1}{x^2-9}$
- 		\begin{shortsolution}
- 			Vertical intercept:$\left( 0,\frac{1}{9} \right)$;
- 			horizontal intercept: $\left( \frac{1}{2},0 \right)$;
- 			vertical asymptotes: $x=-3$, $x=3$, horizontal asymptote: $y=0$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-3.23974]{(2*x-1)/(x^2-9)};
- 					\addplot[pccplot,samples=50] expression[domain=-2.77321:2.83974]{(2*x-1)/(x^2-9)};
- 					\addplot[pccplot] expression[domain=3.17321:5]{(2*x-1)/(x^2-9)};
- 					\addplot[soldot]coordinates{(0,1/9)(1/2,0)};
- 					\addplot[asymptote,domain=-5:5]({-3},{x});
- 					\addplot[asymptote,domain=-5:5]({3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{x+3}{x-5}$
- 		\begin{shortsolution}
- 			Vertical intercept $\left( 0,-\frac{3}{5} \right)$; horizontal
- 			intercept: $(-3,0)$; vertical asymptote: $x=5$; horizontal asymptote: $y=1$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-5,ymax=5,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.666]{(x+3)/(x-5)};
- 					\addplot[pccplot] expression[domain=7:10]{(x+3)/(x-5)};
- 					\addplot[asymptote,domain=-5:5]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{1});
- 					\addplot[soldot]coordinates{(0,-3/5)(-3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{2x+3}{3x-1}$
- 		\begin{shortsolution}
- 			Vertical intercept: $(0,-3)$; horizontal intercept: $\left( -\frac{3}{2},0 \right)$;
- 			vertical asymptote: $x=\frac{1}{3}$, horizontal asymptote: $y=\frac{2}{3}$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(2*x+3)/(3*x-1);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:0.1176]{f};
- 					\addplot[pccplot] expression[domain=0.6153:5]{f};
- 					\addplot[asymptote,domain=-5:5]({1/3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{2/3});
- 					\addplot[soldot]coordinates{(0,-3)(-3/2,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{4-x^2}{x^2-9}$
- 		\begin{shortsolution}
- 			Vertical intercept: $\left( 0,-\frac{4}{9} \right)$;
- 			horizontal intercepts: $(2,0)$, $(-2,0)$;
- 			vertical asymptotes: $x=-3$, $x=3$; horizontal asymptote: $y=-1$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(4-x^2)/(x^2-9);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-3.20156]{f};
- 					\addplot[pccplot,samples=50] expression[domain=-2.85774:2.85774]{f};
- 					\addplot[pccplot] expression[domain=3.20156:5]{f};
- 					\addplot[asymptote,domain=-5:5]({-3},{x});
- 					\addplot[asymptote,domain=-5:5]({3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{-1});
- 					\addplot[soldot] coordinates{(-2,0)(2,0)(0,-4/9)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{(4x+5)(3x-4)}{(2x+5)(x-5)}$
- 		\begin{shortsolution}
- 			Vertical intercept: $\left( 0,\frac{4}{5} \right)$;
- 			horizontal intercepts: $\left( -\frac{5}{4},0 \right)$, $\left( \frac{4}{3},0 \right)$;
- 			vertical asymptotes: $x=-\frac{5}{2}$, $x=5$; horizontal asymptote: $y=6$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(4*x+5)*(3*x-4)/((2*x+5)*(x-5));}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-20,ymax=20,
- 						xtick={-8,-6,...,8},
- 						ytick={-10,0,...,10},
- 						minor ytick={-15,-5,...,15},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-2.73416]{f};
- 					\addplot[pccplot] expression[domain=-2.33689:4.2792]{f};
- 					\addplot[pccplot] expression[domain=6.26988:10]{f};
- 					\addplot[asymptote,domain=-20:20]({-5/2},{x});
- 					\addplot[asymptote,domain=-20:20]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{6});
- 					\addplot[soldot]coordinates{(0,4/5)(-5/4,0)(4/3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Inverse functions]
- Each of the following rational functions are invertible
- \[
- 	F(x)=\frac{2x+1}{x-3}, \qquad G(x)= \frac{1-4x}{x+3}
- \]
- \begin{subproblem}
- 	State the domain of each function.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The domain of $F$ is $(-\infty,3)\cup(3,\infty)$.
- 			\item The domain of $G$ is $(-\infty,-3)\cup(-3,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	Find the inverse of each function, and state its domain.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item $F^{-1}(x)=\frac{3x+1}{x-2}$; the domain of $F^{-1}$ is $(-\infty,2)\cup(2,\infty)$.
- 			\item $G^{-1}(x)=\frac{3x+1}{x+4}$; the domain of $G^{-1}$ is $(-\infty,-4)\cup(-4,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	Hence state the range of the original functions.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The range of $F$ is the domain of $F^{-1}$, which is $(-\infty,2)\cup(2,\infty)$.
- 			\item The range of $G$ is the domain of $G^{-1}$, which is $(-\infty,-4)\cup(-4,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	State the range of each inverse function.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The range of $F^{-1}$ is the domain of $F$, which is $(-\infty,3)\cup(3,\infty)$.
- 			\item The range of $G^{-1}$ is the domain of $G$, which is $(-\infty,-3)\cup(-3,\infty)$.
- 		\end{itemize}<++>
- 	\end{shortsolution}
- \end{subproblem}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Composition]
- Let $r$ and $s$ be the rational functions that have formulas
- \[
- 	r(x)=\frac{3}{x^2},\qquad s(x)=\frac{4-x}{x+5}
- \]
- Evaluate each of the following.
- \begin{multicols}{3}
- 	\begin{subproblem}
- 		$(r\circ s)(0)$
- 		\begin{shortsolution}
- 			$\frac{75}{16}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(0)$
- 		\begin{shortsolution}
- 			$(s\circ r)(0)$ is undefined.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(r\circ s)(2)$
- 		\begin{shortsolution}
- 			$\frac{147}{4}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(3)$
- 		\begin{shortsolution}
- 			$192$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(4)$
- 		\begin{shortsolution}
- 			$(s\circ r)(4)$ is undefined.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(x)$
- 		\begin{shortsolution}
- 			$\dfrac{4x^2-3}{1+5x^2}$
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Piecewise rational functions]
- The function $R$ has formula
- \[
- 	R(x)=
- 	\begin{dcases}
- 		\frac{2}{x+3},    & x<-5     \\
- 		\frac{x-4}{x-10}, & x\geq -5 
- 	\end{dcases}
- \]
- Evaluate each of the following.
- \begin{multicols}{4}
- 	\begin{subproblem}
- 		$R(-6)$
- 		\begin{shortsolution}
- 			$-\frac{2}{3}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(-5)$
- 		\begin{shortsolution}
- 			$\frac{3}{5}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(-3)$
- 		\begin{shortsolution}
- 			$\frac{7}{13}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(5)$
- 		\begin{shortsolution}
- 			$-\frac{1}{5}$
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \begin{subproblem}
- 	What is the domain of $R$?
- 	\begin{shortsolution}
- 		$(-\infty,10)\cup(10,\infty)$
- 	\end{shortsolution}
- \end{subproblem}
- \end{problem}
- \end{exercises}
-   
-\section{Graphing rational functions (oblique asymptotes)}\label{rat:sec:oblique}
- \begin{subproblem}
- 	$y=\dfrac{x^2+1}{x-4}$
- 	\begin{shortsolution}
- 		\begin{enumerate}
- 			\item $\left( 0,-\frac{1}{4} \right)$
- 			\item Vertical asymptote: $x=4$.
- 			\item A graph of the function is shown below
- 			       			       			      
- 			      \begin{tikzpicture}[/pgf/declare function={f=(x^2+1)/(x-4);}]
- 			      	\begin{axis}[
- 			      			framed,
- 			      			xmin=-20,xmax=20,
- 			      			ymin=-30,ymax=30,
- 			      			xtick={-10,10},
- 			      			minor xtick={-15,-5,...,15},
- 			      			minor ytick={-10,10},
- 			      			grid=both,
- 			      			width=\solutionfigurewidth,
- 			      		]
- 			      		\addplot[pccplot,samples=50] expression[domain=-20:3.54724]{f};
- 			      		\addplot[pccplot,samples=50] expression[domain=4.80196:20]{f};
- 			      		\addplot[asymptote,domain=-30:30]({4},{x});
- 			      	\end{axis}
- 			      \end{tikzpicture}
- 		\end{enumerate}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	$y=\dfrac{x^3(x+3)}{x-5}$
- 	\begin{shortsolution}
- 		\begin{enumerate}
- 			\item $(0,0)$, $(-3,0)$
- 			\item Vertical asymptote: $x=5$, horizontal asymptote: none.
- 			\item A graph of the function is shown below
- 			       			       			      
- 			      \begin{tikzpicture}[/pgf/declare function={f=x^3*(x+3)/(x-5);}]
- 			      	\begin{axis}[
- 			      			framed,
- 			      			xmin=-10,xmax=10,
- 			      			ymin=-500,ymax=2500,
- 			      			xtick={-8,-6,...,8},
- 			      			ytick={500,1000,1500,2000},
- 			      			grid=both,
- 			      			width=\solutionfigurewidth,
- 			      		]
- 			      		\addplot[pccplot,samples=50] expression[domain=-10:4]{f};
- 			      		\addplot[pccplot] expression[domain=5.6068:9.777]{f};
- 			      		\addplot[asymptote,domain=-500:2500]({5},{x});
- 			      	\end{axis}
- 			      \end{tikzpicture}
- 		\end{enumerate}
- 	\end{shortsolution}
- \end{subproblem}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/sampleBEFORE.tex b/Master/texmf-dist/doc/support/latexindent/success/sampleBEFORE.tex
deleted file mode 100644
index ae704bd4bc1..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/sampleBEFORE.tex
+++ /dev/null
@@ -1,5819 +0,0 @@
-% arara: indent: {overwrite: true, trace: on}
-% A sample chapter file- it contains a lot of
-% environments, including tabulars, align, etc
-%
-% Don't try and compile this file using pdflatex etc, just
-% compare the *format* of it to the format of the
-% sampleAFTER.tex
-%
-% In particular, compare the tabular and align-type
-% environments before and after running the script
-
-\section{Polynomial functions}
- \reformatstepslist{P} % the steps list should be P1, P2, \ldots
- In your previous mathematics classes you have studied \emph{linear} and
- \emph{quadratic} functions. The most general forms of these types of
- functions can be represented (respectively) by  the functions $f$
- and $g$ that have formulas
- \begin{equation}\label{poly:eq:linquad}
- 	f(x)=mx+b, \qquad g(x)=ax^2+bx+c
- \end{equation}
- We know that $m$ is the slope of $f$, and that $a$ is the \emph{leading coefficient}
- of $g$. We also know that the \emph{signs} of $m$ and $a$ completely
- determine the behavior of the functions $f$ and $g$. For example, if $m>0$
- then $f$ is an \emph{increasing} function, and if $m<0$ then $f$ is
- a \emph{decreasing} function.  Similarly, if $a>0$ then $g$ is
- \emph{concave up} and if $a<0$ then $g$ is \emph{concave down}. Graphical
- representations of these statements are given in \cref{poly:fig:linquad}.
-   
- \begin{figure}[!htb]
- 	\setlength{\figurewidth}{.2\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-10:8]{(x+2)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$m>0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-10:8]{-(x+2)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$m<0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-4:4]{(x^2-6)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$a>0$}
- 	\end{subfigure}
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					framed,
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					width=\textwidth,
- 					xtick={-11},
- 					ytick={-11},
- 				]
- 				\addplot expression[domain=-4:4]{-(x^2-6)};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{$a<0$}
- 	\end{subfigure}
- 	\caption{Typical graphs of linear and quadratic functions.}
- 	\label{poly:fig:linquad}
- \end{figure}
-   
- Let's look a little more closely at the formulas for $f$ and $g$ in
- \cref{poly:eq:linquad}. Note that the \emph{degree}
- of $f$ is $1$ since the highest power of $x$ that is present in the
- formula for $f(x)$ is $1$. Similarly, the degree of $g$ is $2$ since
- the highest power of $x$ that is present in the formula for $g(x)$
- is $2$.
-   
- In this section we will build upon our knowledge of these elementary
- functions. In particular, we will generalize the functions $f$ and $g$ to a function $p$ that has
- any degree that we wish.
-   
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{essentialskills}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Quadratic functions]
- 	Every quadratic function has the form $y=ax^2+bx+c$; state the value
- 	of $a$ for each of the following functions, and hence decide if the
- 	parabola that represents the function opens upward or downward.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			$F(x)=x^2+3$
- 			\begin{shortsolution}
- 				$a=1$; the parabola opens upward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$G(t)=4-5t^2$
- 			\begin{shortsolution}
- 				$a=-5$; the parabola opens downward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$H(y)=4y^2-96y+8$
- 			\begin{shortsolution}
- 				$a=4$; the parabola opens upward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$K(z)=-19z^2$
- 			\begin{shortsolution}
- 				$m=-19$; the parabola opens downward.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	Now let's generalize our findings for the most general quadratic function $g$
- 	that has formula $g(x)=a_2x^2+a_1x+a_0$. Complete the following sentences.
- 	\begin{subproblem}
- 		When $a_2>0$, the parabola that represents $y=g(x)$ opens $\ldots$
- 		\begin{shortsolution}
- 			When $a_2>0$, the parabola that represents the function opens upward.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		When $a_2<0$, the parabola that represents $y=g(x)$ opens $\ldots$
- 		\begin{shortsolution}
- 			When $a_2<0$, the parabola that represents the function opens downward.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- \end{essentialskills}
-   
- \subsection*{Power functions with positive exponents}
- 	The study of polynomials will rely upon a good knowledge
- 	of power functions| you may reasonably ask, what is a power function?
- 	\begin{pccdefinition}[Power functions]
- 		Power functions have the form
- 		\[
- 			f(x) = a_n x^n
- 		\]
- 		where $n$ can be any real number.
- 		 		 		
- 		Note that for this section we will only be concerned with the
- 		case when $n$ is a positive integer.
- 	\end{pccdefinition}
- 	 	 	
- 	You may find assurance in the fact that you are already very comfortable
- 	with power functions that have $n=1$ (linear) and $n=2$ (quadratic). Let's
- 	explore some power functions that you might not be so familiar with.
- 	As you read \cref{poly:ex:oddpow,poly:ex:evenpow}, try and spot
- 	as many patterns and similarities as you can.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Power functions with odd positive exponents]
- 		\label{poly:ex:oddpow}
- 		Graph each of the following functions, state their domain, and their
- 		long-run behavior as $x\rightarrow\pm\infty$
- 		\[
- 			f(x)=x^3,   \qquad  g(x)=x^5, \qquad h(x)=x^7
- 		\]
- 		\begin{pccsolution}
- 			The functions $f$, $g$, and $h$ are plotted in \cref{poly:fig:oddpow}.
- 			The domain of each of the functions $f$, $g$, and $h$ is $(-\infty,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				f(x)\rightarrow\infty                           & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}   f(x)\rightarrow-\infty & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same results hold for $g$ and $h$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-1.5,xmax=1.5,
- 						ymin=-5,ymax=5,
- 						xtick={-1.0,-0.5,...,1.0},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=north west,
- 					]
- 					\addplot expression[domain=-1.5:1.5]{x^3};
- 					\addplot expression[domain=-1.379:1.379]{x^5};
- 					\addplot expression[domain=-1.258:1.258]{x^7};
- 					\addplot[soldot]coordinates{(-1,-1)} node[axisnode,anchor=north west]{$(-1,-1)$};
- 					\addplot[soldot]coordinates{(1,1)} node[axisnode,anchor=south east]{$(1,1)$};
- 					\legend{$f$,$g$,$h$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{Odd power functions}
- 			\label{poly:fig:oddpow}
- 		\end{minipage}%
- 		\hfill
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-2.5,xmax=2.5,
- 						ymin=-5,ymax=5,
- 						xtick={-2.0,-1.5,...,2.0},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=south east,
- 					]
- 					\addplot expression[domain=-2.236:2.236]{x^2};
- 					\addplot expression[domain=-1.495:1.495]{x^4};
- 					\addplot expression[domain=-1.307:1.307]{x^6};
- 					\addplot[soldot]coordinates{(-1,1)} node[axisnode,anchor=east]{$(-1,1)$};
- 					\addplot[soldot]coordinates{(1,1)} node[axisnode,anchor=west]{$(1,1)$};
- 					\legend{$F$,$G$,$H$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{Even power functions}
- 			\label{poly:fig:evenpow}
- 		\end{minipage}%
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Power functions with even positive exponents]\label{poly:ex:evenpow}%
- 		Graph each of the following functions, state their domain, and their
- 		long-run behavior as $x\rightarrow\pm\infty$
- 		\[
- 			F(x)=x^2, \qquad G(x)=x^4, \qquad H(x)=x^6
- 		\]
- 		\begin{pccsolution}
- 			The functions $F$, $G$, and $H$ are plotted in \cref{poly:fig:evenpow}. The domain
- 			of each of the functions is $(-\infty,\infty)$. Note that the long-run behavior
- 			of each of the functions is the same, and in particular
- 			\begin{align*}
- 				F(x)\rightarrow\infty                          & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}   F(x)\rightarrow\infty & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same result holds for $G$ and $H$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Repeat \cref{poly:ex:oddpow,poly:ex:evenpow} using (respectively)
- 		\begin{subproblem}
- 			$f(x)=-x^3,   \qquad  g(x)=-x^5, \qquad h(x)=-x^7$
- 			\begin{shortsolution}
- 				The functions $f$, $g$, and $h$ have domain $(-\infty,\infty)$ and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-1.5,xmax=1.5,
- 							ymin=-5,ymax=5,
- 							xtick={-1.0,-0.5,...,0.5},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-1.5:1.5]{-x^3};
- 						\addplot expression[domain=-1.379:1.379]{-x^5};
- 						\addplot expression[domain=-1.258:1.258]{-x^7};
- 						\legend{$f$,$g$,$h$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					f(x)\rightarrow-\infty                         & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}   f(x)\rightarrow\infty & \text{ as } x\rightarrow-\infty 
- 				\end{align*}
- 				The same is true for $g$ and $h$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$F(x)=-x^2,   \qquad  G(x)=-x^4, \qquad H(x)=-x^6$
- 			\begin{shortsolution}
- 				The functions $F$, $G$, and $H$ have domain $(-\infty,\infty)$ and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-2.5,xmax=2.5,
- 							ymin=-5,ymax=5,
- 							xtick={-1.0,-0.5,...,0.5},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-2.236:2.236]{-x^2};
- 						\addplot expression[domain=-1.495:1.495]{-x^4};
- 						\addplot expression[domain=-1.307:1.307]{-x^6};
- 						\legend{$F$,$G$,$H$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					F(x)\rightarrow-\infty                          & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}   F(x)\rightarrow-\infty & \text{ as } x\rightarrow-\infty 
- 				\end{align*}
- 				The same is true for $G$ and $H$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- \subsection*{Polynomial functions}
- 	Now that we have a little more familiarity with power functions,
- 	we can define polynomial functions. Provided that you were comfortable
- 	with our opening discussion about linear and quadratic functions (see
- 	$f$ and $g$ in \cref{poly:eq:linquad}) then there is every chance
- 	that you'll be able to master polynomial functions as well; just remember
- 	that polynomial functions are a natural generalization of linear
- 	and quadratic functions. Once you've studied the examples and problems
- 	in this section, you'll hopefully agree that polynomial functions
- 	are remarkably predictable.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccdefinition}[Polynomial functions]
- 		Polynomial functions have the form
- 		\[
- 			p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0
- 		\]
- 		where $a_n$, $a_{n-1}$, $a_{n-2}$, \ldots, $a_0$ are real numbers.
- 		\begin{itemize}
- 			\item We call $n$ the degree of the polynomial, and require that $n$
- 			      is a non-negative integer;
- 			\item $a_n$, $a_{n-1}$, $a_{n-2}$, \ldots, $a_0$ are called the coefficients;
- 			\item We typically write polynomial functions in descending powers of $x$.
- 		\end{itemize}
- 		In particular, we call $a_n$ the \emph{leading} coefficient, and $a_nx^n$ the
- 		\emph{leading term}.
- 		 		 		
- 		Note that if a polynomial is given in factored form, then the degree can be found
- 		by counting the number of linear factors.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Polynomial or not]
- 		Identify the following functions as polynomial or not; if the function
- 		is a polynomial, state its degree.
- 		\begin{multicols}{3}
- 			\begin{enumerate}
- 				\item $p(x)=x^2-3$
- 				\item $q(x)=-4x^{\nicefrac{1}{2}}+10$
- 				\item $r(x)=10x^5$
- 				\item $s(x)=x^{-2}+x^{23}$
- 				\item $f(x)=-8$
- 				\item $g(x)=3^x$
- 				\item $h(x)=\sqrt[3]{x^7}-x^2+x$
- 				\item $k(x)=4x(x+2)(x-3)$
- 				\item $j(x)=x^2(x-4)(5-x)$
- 			\end{enumerate}
- 		\end{multicols}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item $p$ is a polynomial, and its degree is $2$.
- 				\item $q$ is \emph{not} a polynomial, because $\frac{1}{2}$ is not an integer.
- 				\item $r$ is a polynomial, and its degree is $5$.
- 				\item $s$ is \emph{not} a polynomial, because $-2$ is not a positive integer.
- 				\item $f$ is a polynomial, and its degree is $0$.
- 				\item $g$ is \emph{not} a polynomial, because the independent
- 				      variable, $x$, is in the exponent.
- 				\item $h$ is \emph{not} a polynomial, because $\frac{7}{3}$ is not an integer.
- 				\item $k$ is a polynomial, and its degree is $3$.
- 				\item $j$ is a polynomial, and its degree is $4$.
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Typical graphs]\label{poly:ex:typical}
- 		\Cref{poly:fig:typical} shows graphs of some polynomial functions;
- 		the ticks have deliberately been left off the axis to allow us to concentrate
- 		on the features of each graph. Note in particular that:
- 		\begin{itemize}
- 			\item \cref{poly:fig:typical1} shows a degree-$1$ polynomial (you might also
- 			      classify the function as linear) whose leading coefficient, $a_1$, is positive.
- 			\item \cref{poly:fig:typical2} shows a degree-$2$ polynomial (you might also
- 			      classify the function as quadratic) whose leading coefficient, $a_2$, is positive.
- 			\item \cref{poly:fig:typical3} shows a degree-$3$ polynomial whose leading coefficient, $a_3$,
- 			      is positive| compare its overall
- 			      shape and long-run behavior to the functions described in \cref{poly:ex:oddpow}.
- 			\item \cref{poly:fig:typical4} shows a degree-$4$ polynomial whose leading coefficient, $a_4$,
- 			      is positive|compare its overall shape and long-run behavior to the functions described in \cref{poly:ex:evenpow}.
- 			\item \cref{poly:fig:typical5} shows a degree-$5$ polynomial whose leading coefficient, $a_5$,
- 			      is positive| compare its overall
- 			      shape and long-run behavior to the functions described in \cref{poly:ex:oddpow}.
- 		\end{itemize}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{\textwidth/6}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-10:8]{(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_1>0$}
- 			\label{poly:fig:typical1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-4:4]{(x^2-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_2>0$}
- 			\label{poly:fig:typical2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-7.5:7.5]{0.05*(x+6)*x*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_3>0$}
- 			\label{poly:fig:typical3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-2.35:5.35,samples=100]{0.2*(x-5)*x*(x-3)*(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_4>0$}
- 			\label{poly:fig:typical4}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\textwidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-5.5:6.3,samples=100]{0.01*(x+2)*x*(x-3)*(x+5)*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$a_5>0$}
- 			\label{poly:fig:typical5}
- 		\end{subfigure}
- 		\end{widepage}
- 		\caption{Graphs to illustrate typical curves of polynomial functions.}
- 		\label{poly:fig:typical}
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Use \cref{poly:ex:typical} and \cref{poly:fig:typical} to help you sketch
- 		the graphs of polynomial functions that have negative leading coefficients| note
- 		that there are many ways to do this! The intention with this problem
- 		is to use your knowledge of transformations- in particular, \emph{reflections}-
- 		to guide you.
- 		\begin{shortsolution}
- 			$a_1<0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-10:8]{-(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_2<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-4:4]{-(x^2-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_3<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-7.5:7.5]{-0.05*(x+6)*x*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_4<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-2.35:5.35,samples=100]{-0.2*(x-5)*x*(x-3)*(x+2)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$a_5<0$
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 						xtick={-11},
- 						ytick={-11},
- 					]
- 					\addplot expression[domain=-5.5:6.3,samples=100]{-0.01*(x+2)*x*(x-3)*(x+5)*(x-6)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- 	\fixthis{poly: Need a more basic example here- it can have a similar
- 		format to the multiple zeros example, but just keep it simple; it should
- 	be halfway between the 2 examples surrounding it}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Multiple zeros]
- 		Consider the polynomial functions $p$, $q$, and $r$ which are
- 		graphed in \cref{poly:fig:moremultiple}.
- 		The formulas for $p$, $q$, and $r$ are as follows
- 		\begin{align*}
- 			p(x) & =(x-3)^2(x+4)^2       \\
- 			q(x) & =x(x+2)^2(x-1)^2(x-3) \\
- 			r(x) & =x(x-3)^3(x+1)^2      
- 		\end{align*}
- 		Find the degree of $p$, $q$, and $r$, and decide if the functions bounce off or cut
- 		through the horizontal axis at each of their zeros.
- 		\begin{pccsolution}
- 			The degree of $p$ is 4. Referring to \cref{poly:fig:bouncep},
- 			the curve bounces off the horizontal axis at both zeros, $3$ and $4$.
- 			 			 			
- 			The degree of $q$ is 6. Referring to \cref{poly:fig:bounceq},
- 			the curve bounces off the horizontal axis at $-2$ and $1$, and cuts
- 			through the horizontal axis at $0$ and $3$.
- 			 			 			
- 			The degree of $r$ is 6. Referring to \cref{poly:fig:bouncer},
- 			the curve bounces off the horizontal axis at $-1$, and cuts through
- 			the horizontal axis at $0$ and at $3$, although is flattened immediately to the left and right of $3$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\setlength{\figurewidth}{0.25\textwidth}
- 	\begin{figure}[!htb]
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-6,xmax=5,
- 						ymin=-30,ymax=200,
- 						xtick={-4,-2,...,4},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-5.63733:4.63733,samples=50]{(x-3)^2*(x+4)^2};
- 					\addplot[soldot]coordinates{(3,0)(-4,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=p(x)$}
- 			\label{poly:fig:bouncep}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-3,xmax=4,
- 						xtick={-2,...,3},
- 						ymin=-60,ymax=40,
- 						width=\textwidth,
- 					]
- 					\addplot+[samples=50] expression[domain=-2.49011:3.11054]{x*(x+2)^2*(x-1)^2*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(0,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=q(x)$}
- 			\label{poly:fig:bounceq}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-2,xmax=4,
- 						xtick={-1,...,3},
- 						ymin=-40,ymax=40,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-1.53024:3.77464,samples=50]{x*(x-3)^3*(x+1)^2};
- 					\addplot[soldot]coordinates{(-1,0)(0,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=r(x)$}
- 			\label{poly:fig:bouncer}
- 		\end{subfigure}
- 		\caption{}
- 		\label{poly:fig:moremultiple}
- 	\end{figure}
- 	 	 	
- 	\begin{pccdefinition}[Multiple zeros]\label{poly:def:multzero}
- 		Let $p$ be a polynomial that has a repeated linear factor $(x-a)^n$. Then we say
- 		that $p$ has a multiple zero at $a$ of multiplicity $n$ and
- 		\begin{itemize}
- 			\item if the factor $(x-a)$ is repeated an even number of times, the graph of $y=p(x)$ does not
- 			      cross the $x$ axis at $a$, but `bounces' off the horizontal axis at $a$.
- 			\item if the factor $(x-a)$ is repeated an odd number of times, the graph of $y=p(x)$ crosses the
- 			      horizontal axis at $a$, but it looks `flattened' there
- 		\end{itemize}
- 		If $n=1$, then we say that $p$ has a \emph{simple} zero at $a$.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Find a formula]
- 		Find formulas for the polynomial functions, $p$ and $q$, graphed in \cref{poly:fig:findformulademoboth}.
- 		\begin{figure}[!htb]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-5,xmax=5,
- 							ymin=-10,ymax=10,
- 							xtick={-4,-2,...,4},
- 							minor xtick={-3,-1,...,3},
- 							ytick={-8,-6,...,8},
- 							width=\textwidth,
- 						grid=both]
- 						\addplot expression[domain=-3.25842:2.25842,samples=50]{-x*(x-2)*(x+3)*(x+1)};
- 						\addplot[soldot]coordinates{(1,8)}node[axisnode,inner sep=.35cm,anchor=west]{$(1,8)$};
- 						\addplot[soldot]coordinates{(-3,0)(-1,0)(0,0)(2,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$p$}
- 				\label{poly:fig:findformulademo}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-5,xmax=5,
- 							ymin=-10,ymax=10,
- 							xtick={-4,-2,...,4},
- 							minor xtick={-3,-1,...,3},
- 							ytick={-8,-6,...,8},
- 							width=\textwidth,
- 						grid=both]
- 						\addplot expression[domain=-4.33:4.08152]{-.25*(x+2)^2*(x-3)};
- 						\addplot[soldot]coordinates{(2,4)}node[axisnode,anchor=south west]{$(2,4)$};
- 						\addplot[soldot]coordinates{(-2,0)(3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$q$}
- 				\label{poly:fig:findformulademo1}
- 			\end{subfigure}
- 			\caption{}
- 			\label{poly:fig:findformulademoboth}
- 		\end{figure}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item We begin by noting that the horizontal intercepts of $p$ are $(-3,0)$, $(-1,0)$, $(0,0)$ and $(2,0)$.
- 				      We also note that each zero is simple (multiplicity $1$).
- 				      If we assume that $p$ has no other zeros, then we can start by writing
- 				      \begin{align*}
- 				      	p(x) & =(x+3)(x+1)(x-0)(x-2) \\
- 				      	     & =x(x+3)(x+1)(x-2)     \\
- 				      \end{align*}
- 				      According to \cref{poly:fig:findformulademo}, the point $(1,8)$ lies
- 				      on the curve $y=p(x)$.
- 				      Let's check if the formula we have written satisfies this requirement
- 				      \begin{align*}
- 				      	p(1) & = (1)(4)(2)(-1) \\
- 				      	     & = -8            
- 				      \end{align*}
- 				      which is clearly not correct| it is close though. We can correct this by
- 				      multiplying $p$ by a constant $k$; so let's assume that
- 				      \[
- 				      	p(x)=kx(x+3)(x+1)(x-2)
- 				      \]
- 				      Then $p(1)=-8k$, and if this is to equal $8$, then $k=-1$. Therefore
- 				      the formula for $p(x)$ is
- 				      \[
- 				      	p(x)=-x(x+3)(x+1)(x-2)
- 				      \]
- 				\item The function $q$ has a zero at $-2$ of multiplicity $2$, and zero of
- 				      multiplicity $1$ at $3$ (so $3$ is a simple zero of $q$); we can therefore assume that $q$ has the form
- 				      \[
- 				      	q(x)=k(x+2)^2(x-3)
- 				      \]
- 				      where $k$ is some real number. In order to find $k$, we use the given ordered pair, $(2,4)$, and
- 				      evaluate $p(2)$
- 				      \begin{align*}
- 				      	p(2) & =k(4)^2(-1) \\
- 				      	     & =-16k       
- 				      \end{align*}
- 				      We solve the equation $4=-8k$ and obtain $k=-\frac{1}{4}$ and conclude that the
- 				      formula for $q(x)$ is
- 				      \[
- 				      	q(x)=-\frac{1}{4}(x+2)^2(x-3)
- 				      \]
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	\fixthis{Chris: need sketching polynomial problems}
- 	\begin{pccspecialcomment}[Steps to follow when sketching polynomial functions]
- 		\begin{steps}
- 			\item \label{poly:step:first} Determine the degree of the polynomial,
- 			its leading term and leading coefficient, and hence determine
- 			the long-run behavior of the polynomial| does it behave like $\pm x^2$ or $\pm x^3$
- 			as $x\rightarrow\pm\infty$?
- 			\item Determine the zeros and their multiplicity. Mark all zeros
- 			and the vertical intercept on the graph using solid circles $\bullet$.
- 			\item \label{poly:step:last}  Deduce the overall shape of the curve, and sketch it. If there isn't
- 			enough information from the previous steps, then construct a table of values.
- 		\end{steps}
- 		Remember that until we have the tools of calculus, we won't be able to
- 		find the exact coordinates of local minimums, local maximums, and points
- 		of inflection.
- 	\end{pccspecialcomment}
- 	Before we demonstrate some examples, it is important to remember the following:
- 	\begin{itemize}
- 		\item our sketches will give a good representation of the overall
- 		      shape of the graph, but until we have the tools of calculus (from MTH 251)
- 		      we can not find local minimums, local maximums, and inflection points algebraically. This
- 		      means that we will make our best guess as to where these points are.
- 		\item we will not concern ourselves too much with the vertical scale (because of
- 		      our previous point)| we will, however, mark the vertical intercept (assuming there is one),
- 		      and any horizontal asymptotes.
- 	\end{itemize}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:simplecubic}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $p$
- 		that has formula
- 		\[
- 			p(x)=\frac{1}{2}(x-4)(x-1)(x+3)
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $p$ has degree $3$. The leading term of $p$ is $\frac{1}{2}x^3$, so the leading coefficient of $p$
- 				is $\frac{1}{2}$. The long-run behavior of $p$ is therefore similar to that of $x^3$.
- 				\item The zeros of $p$ are $-3$, $1$, and $4$; each zero is simple (i.e, it has multiplicity $1$).
- 				This means that the curve of $p$ cuts the horizontal axis at each zero. The vertical
- 				intercept of $p$ is $(0,6)$.
- 				\item We draw the details we have obtained so far on \cref{poly:fig:simplecubicp1}. Given
- 				that the curve of $p$ looks like the curve of $x^3$ in the long-run, we are able to complete a sketch of the
- 				graph of $p$ in \cref{poly:fig:simplecubicp2}.
- 				 				 				
- 				Note that we can not find the coordinates of the local minimums, local maximums, and inflection
- 				points| for the moment we make reasonable guesses as to where these points are (you'll find how
- 				to do this in calculus).
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=15,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-3,0)(1,0)(4,0)(0,6)}node[axisnode,anchor=south west]{$(0,6)$};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:simplecubicp1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=15,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-3,0)(1,0)(4,0)(0,6)}node[axisnode,anchor=south west]{$(0,6)$};
- 							\addplot[pccplot] expression[domain=-3.57675:4.95392,samples=100]{.5*(x-4)*(x-1)*(x+3)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:simplecubicp2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{2}(x-4)(x-1)(x+3)$}
- 				\label{poly:fig:simplecubic}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:degree5}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $q$
- 		that has formula
- 		\[
- 			q(x)=\frac{1}{200}(x+7)^2(2-x)(x-6)^2
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $q$ has degree $4$. The leading term of $q$ is
- 				\[
- 					-\frac{1}{200}x^5
- 				\]
- 				so the leading coefficient of $q$ is $-\frac{1}{200}$. The long-run behavior of $q$
- 				is therefore similar to that of $-x^5$.
- 				\item The zeros of $q$ are $-7$ (multiplicity 2), $2$ (simple), and $6$ (multiplicity $2$).
- 				The curve of $q$ bounces off the horizontal axis at the zeros with multiplicity $2$ and
- 				cuts the horizontal axis at the simple zeros. The vertical intercept of $q$ is $\left( 0,\frac{441}{25} \right)$.
- 				\item We mark the details we have found so far on \cref{poly:fig:degree5p1}. Given that
- 				the curve of $q$ looks like the curve of $-x^5$ in the long-run, we can complete \cref{poly:fig:degree5p2}.
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=40,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,0,...,35},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-7,0)(2,0)(6,0)(0,441/25)}node[axisnode,anchor=south west]{$\left( 0, \frac{441}{25} \right)$};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree5p1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-10,xmax=10,
- 								ymin=-10,ymax=40,
- 								xtick={-8,-6,...,8},
- 								ytick={-5,0,...,35},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-7,0)(2,0)(6,0)(0,441/25)}node[axisnode,anchor=south west]{$\left( 0, \frac{441}{25} \right)$};
- 							\addplot[pccplot] expression[domain=-8.83223:7.34784,samples=50]{1/200*(x+7)^2*(2-x)*(x-6)^2};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree5p2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{200}(x+7)^2(2-x)(x-6)^2$}
- 				\label{poly:fig:degree5}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}
- 		Use \crefrange{poly:step:first}{poly:step:last} to sketch a graph of the function $r$
- 		that has formula
- 		\[
- 			r(x)=\frac{1}{100}x^3(x+4)(x-4)(x-6)
- 		\]
- 		\begin{pccsolution}
- 			\begin{steps}
- 				\item $r$ has degree $6$. The leading term of $r$ is
- 				\[
- 					\frac{1}{100}x^6
- 				\]
- 				so the leading coefficient of $r$ is $\frac{1}{100}$. The long-run behavior of $r$
- 				is therefore similar to that of $x^6$.
- 				\item The zeros of $r$ are $-4$ (simple), $0$ (multiplicity $3$), $4$ (simple),
- 				and $6$ (simple). The vertical intercept of $r$ is $(0,0)$. The curve of $r$
- 				cuts the horizontal axis at the simple zeros, and goes through the axis
- 				at $(0,0)$, but does so in a flattened way.
- 				\item We mark the zeros and vertical intercept on \cref{poly:fig:degree6p1}. Given that
- 				the curve of $r$ looks like the curve of $x^6$ in the long-run, we complete the graph
- 				of $r$ in \cref{poly:fig:degree6p2}.
- 			\end{steps}
- 			 			 			
- 			\begin{figure}[!htbp]
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-5,xmax=10,
- 								ymin=-20,ymax=10,
- 								xtick={-4,-2,...,8},
- 								ytick={-15,-10,...,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-4,0)(0,0)(4,0)(6,0)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree6p1}
- 				\end{subfigure}%
- 				\hfill
- 				\begin{subfigure}{.45\textwidth}
- 					\begin{tikzpicture}
- 						\begin{axis}[
- 								xmin=-5,xmax=10,
- 								ymin=-20,ymax=10,
- 								xtick={-4,-2,...,8},
- 								ytick={-15,-10,...,5},
- 								width=\textwidth,
- 							]
- 							\addplot[soldot] coordinates{(-4,0)(0,0)(4,0)(6,0)};
- 							\addplot[pccplot] expression[domain=-4.16652:6.18911,samples=100]{1/100*(x+4)*x^3*(x-4)*(x-6)};
- 						\end{axis}
- 					\end{tikzpicture}
- 					\caption{}
- 					\label{poly:fig:degree6p2}
- 				\end{subfigure}%
- 				\caption{$y=\dfrac{1}{100}(x+4)x^3(x-4)(x-6)$}
- 			\end{figure}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[An open-topped box]
- 		A cardboard company makes open-topped boxes for their clients. The specifications
- 		dictate that the box must have a square base, and that it must be open-topped.
- 		The company uses sheets of cardboard that are $\unit[1200]{cm^2}$. Assuming that
- 		the base of each box has side $x$ (measured in cm), it can be shown that the volume of each box, $V(x)$,
- 		has formula
- 		\[
- 			V(x)=\frac{x}{4}(1200-x^2)
- 		\]
- 		Find the dimensions of the box that maximize the volume.
- 		\begin{pccsolution}
- 			We graph $y=V(x)$ in \cref{poly:fig:opentoppedbox}. Note that because
- 			$x$ represents the length of a side, and $V(x)$ represents the volume
- 			of the box, we necessarily require both values to be positive; we illustrate
- 			the part of the curve that applies to this problem using a solid line.
- 			 			 			
- 			\begin{figure}[!htb]
- 				\centering
- 				\begin{tikzpicture}
- 					\begin{axis}[framed,
- 							xmin=-50,xmax=50,
- 							ymin=-5000,ymax=5000,
- 							xtick={-40,-30,...,40},
- 							minor xtick={-45,-35,...,45},
- 							minor ytick={-3000,-1000,1000,3000},
- 							width=.75\textwidth,
- 							height=.5\textwidth,
- 						grid=both]
- 						\addplot[pccplot,dashed,<-] expression[domain=-40:0,samples=50]{x/4*(1200-x^2)};
- 						\addplot[pccplot,-] expression[domain=0:34.64,samples=50]{x/4*(1200-x^2)};
- 						\addplot[pccplot,dashed,->] expression[domain=34.64:40,samples=50]{x/4*(1200-x^2)};
- 						\addplot[soldot] coordinates{(20,4000)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=V(x)$}
- 				\label{poly:fig:opentoppedbox}
- 			\end{figure}
- 			 			 			
- 			According to \cref{poly:fig:opentoppedbox}, the maximum volume of such a box is
- 			approximately $\unit[4000]{cm^2}$, and we achieve it using a base of length
- 			approximately $\unit[20]{cm}$. Since the base is square and each sheet of cardboard
- 			is $\unit[1200]{cm^2}$, we conclude that the dimensions of each box are $\unit[20]{cm}\times\unit[20]{cm}\times\unit[30]{cm}$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- \subsection*{Complex zeros}
- 	There has been a pattern to all of the examples that we have seen so far|
- 	the degree of the polynomial has dictated the number of \emph{real} zeros that the
- 	polynomial has. For example, the function $p$ in \cref{poly:ex:simplecubic}
- 	has degree $3$, and $p$ has $3$ real zeros; the function $q$ in \cref{poly:ex:degree5}
- 	has degree $5$ and $q$ has $5$ real zeros.
- 	 	 	
- 	You may wonder if this result can be generalized| does every polynomial that
- 	has degree $n$ have $n$ real zeros? Before we tackle the general result,
- 	let's consider an example that may help motivate it.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}\label{poly:ex:complx}
- 		Consider the polynomial function $c$ that has formula
- 		\[
- 			c(x)=x(x^2+1)
- 		\]
- 		It is clear that $c$ has degree $3$, and that $c$ has a (simple) zero at $0$. Does
- 		$c$ have any other zeros, i.e, can we find any values of $x$ that satisfy the equation
- 		\begin{equation}\label{poly:eq:complx}
- 			x^2+1=0
- 		\end{equation}
- 		The solutions to \cref{poly:eq:complx} are $\pm i$.
- 		 		 		
- 		We conclude that $c$ has $3$ zeros: $0$ and $\pm i$; we note that \emph{not
- 		all of them are real}.
- 	\end{pccexample}
- 	\Cref{poly:ex:complx} shows that not every degree-$3$ polynomial has $3$
- 	\emph{real} zeros; however, if we are prepared to venture into the complex numbers,
- 	then we can state the following theorem.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccspecialcomment}[The fundamental theorem of algebra]
- 		Every polynomial function of degree $n$ has $n$ roots, some of which may
- 		be complex, and some may be repeated.
- 	\end{pccspecialcomment}
- 	\fixthis{Fundamental theorem of algebra: is this wording ok? do we want
- 	it as a theorem?}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}
- 		Find all the zeros of the polynomial function $p$ that has formula
- 		\[
- 			p(x)=x^4-2x^3+5x^2
- 		\]
- 		\begin{pccsolution}
- 			We begin by factoring $p$
- 			\begin{align*}
- 				p(x) & =x^4-2x^3+5x^2 \\
- 				     & =x^2(x^2-2x+5) 
- 			\end{align*}
- 			We note that $0$ is a zero of $p$ with multiplicity $2$. The other zeros of $p$
- 			can be found by solving the equation
- 			\[
- 				x^2-2x+5=0
- 			\]
- 			This equation can not be factored, so we use the quadratic formula
- 			\begin{align*}
- 				x & =\frac{2\pm\sqrt{(-2)^2}-20}{2(1)} \\
- 				  & =\frac{2\pm\sqrt{-16}}{2}          \\
- 				  & =1\pm 2i                           
- 			\end{align*}
- 			We conclude that $p$ has $4$ zeros: $0$ (multiplicity $2$), and $1\pm 2i$ (simple).
- 		\end{pccsolution}
- 	\end{pccexample}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{pccexample}
- 		Find a polynomial that has zeros at $2\pm i\sqrt{2}$.
- 		\begin{pccsolution}
- 			We know that the zeros of a polynomial can be found by analyzing the linear
- 			factors. We are given the zeros, and have to work backwards to find the
- 			linear factors.
- 			 			 			
- 			We begin by assuming that $p$ has the form
- 			\begin{align*}
- 				p(x) & =(x-(2-i\sqrt{2}))(x-(2+i\sqrt{2}))                           \\
- 				     & =x^2-x(2+i\sqrt{2})-x(2-i\sqrt{2})+(2-i\sqrt{2})(2+i\sqrt{2}) \\
- 				     & =x^2-4x+(4-2i^2)                                              \\
- 				     & =x^2-4x+6                                                     
- 			\end{align*}
- 			We conclude that a possible formula for a polynomial function, $p$,
- 			that has zeros at $2\pm i\sqrt{2}$ is
- 			\[
- 				p(x)=x^2-4x+6
- 			\]
- 			Note that we could multiply $p$ by any real number and still ensure
- 			that $p$ has the same zeros.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	\investigation*{}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a graph]
- 	For each of the polynomials in \cref{poly:fig:findformula}
- 	\begin{enumerate}
- 		\item count the number of times the curve turns round, and cuts/bounces off the $x$ axis;
- 		\item approximate the degree of the polynomial;
- 		\item use your information to find the linear factors of each polynomial, and therefore write a possible formula for each;
- 		\item make sure your polynomial goes through the given ordered pair.
- 	\end{enumerate}
- 	\begin{shortsolution}
- 		\Vref{poly:fig:findformdeg2}:
- 		\begin{enumerate}
- 			\item the curve turns round once;
- 			\item the degree could be 2;
- 			\item based on the zeros, the linear factors are $(x+5)$ and $(x-3)$; since the
- 			      graph opens downwards, we will assume the leading coefficient is negative: $p(x)=-k(x+5)(x-3)$;
- 			\item $p$ goes through $(2,2)$, so we need to solve $2=-k(7)(-1)$ and therefore $k=\nicefrac{2}{7}$, so
- 			      \[
- 			      	p(x)=-\frac{2}{7}(x+5)(x-3)
- 			      \]
- 		\end{enumerate}
- 		\Vref{poly:fig:findformdeg3}:
- 		\begin{enumerate}
- 			\item the curve turns around twice;
- 			\item the degree could be 3;
- 			\item based on the zeros, the linear factors are $(x+2)^2$, and $(x-1)$;
- 			      based on the behavior of $p$, we assume that the leading coefficient is positive, and try $p(x)=k(x+2)^2(x-1)$;
- 			\item $p$ goes through $(0,-2)$, so we need to solve $-2=k(4)(-1)$ and therefore $k=\nicefrac{1}{2}$, so
- 			      \[
- 			      	p(x)=\frac{1}{2}(x+2)^2(x-1)
- 			      \]
- 		\end{enumerate}
- 		\Vref{poly:fig:findformdeg5}:
- 		\begin{enumerate}
- 			\item the curve turns around 4 times;
- 			\item the degree could be 5;
- 			\item based on the zeros, the linear factors are $(x+5)^2$, $(x+1)$, $(x-2)$, $(x-3)$;
- 			      based on the behavior of $p$, we assume that the leading coefficient is positive, and try $p(x)=k(x+5)^2(x+1)(x-2)(x-3)$;
- 			\item $p$ goes through $(-3,-50)$, so we need to solve $-50=k(64)(-2)(-5)(-6)$ and therefore $k=\nicefrac{5}{384}$, so
- 			      \[
- 			      	p(x)=\frac{5}{384}(x+5)^2(x+1)(x-2)(x-3)
- 			      \]
- 		\end{enumerate}
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-2,ymax=5,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.5:3.75]{-1/3*(x+4)*(x-3)};
- 					\addplot[soldot] coordinates{(-4,0)(3,0)(2,2)} node[axisnode,above right]{$(2,2)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-3,xmax=2,
- 						ymin=-2,ymax=4,
- 						xtick={-2,...,1},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.95:1.75]{1/3*(x+2)^2*(x-1)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(0,-1.33)}node[axisnode,anchor=north west]{$(0,-2)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-100,ymax=150,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.5:3.4,samples=50]{(x+4)^2*(x+1)*(x-2)*(x-3)};
- 					\addplot[soldot]coordinates{(-4,0)(-1,0)(2,0)(3,0)(-3,-60)}node[axisnode,anchor=north]{$(-3,-50)$};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:findformdeg5}
- 		\end{subfigure}
- 		\caption{}
- 		\label{poly:fig:findformula}
- 	\end{figure}
- 	 	 	
- 	 	 	
- 	 	 	
- 	 	 	
- 	\begin{exercises}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Prerequisite classifacation skills]
- 	Decide if each of the following functions are linear or quadratic.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x)=2x+3$
- 			\begin{shortsolution}
- 				$f$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x)=10-7x$
- 			\begin{shortsolution}
- 				$g$ is linear
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x)=-x^2+3x-9$
- 			\begin{shortsolution}
- 				$h$ is quadratic.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(x)=-17$
- 			\begin{shortsolution}
- 				$k$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(x)=-82x^2-4$
- 			\begin{shortsolution}
- 				$l$ is quadratic
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(x)=6^2x-8$
- 			\begin{shortsolution}
- 				$m$ is linear.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Prerequisite slope identification]
- 	State the slope of each of the following linear functions, and
- 	hence decide if each function is increasing or decreasing.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$\alpha(x)=4x+1$
- 			\begin{shortsolution}
- 				$m=4$; $\alpha$ is increasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\beta(x)=-9x$
- 			\begin{shortsolution}
- 				$m=-9$; $\beta$ is decreasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\gamma(t)=18t+100$
- 			\begin{shortsolution}
- 				$m=18$; $\gamma$ is increasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\delta(y)=23-y$
- 			\begin{shortsolution}
- 				$m=-1$; $\delta$ is decreasing.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	Now let's generalize our findings for the most general linear function $f$
- 	that has formula $f(x)=mx+b$. Complete the following sentences.
- 	\begin{subproblem}
- 		When $m>0$, the function $f$ is $\ldots$
- 		\begin{shortsolution}
- 			When $m>0$, the function $f$ is $\ldots$  \emph{increasing}.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		When $m<0$, the function $f$ is $\ldots$
- 		\begin{shortsolution}
- 			When $m<0$, the function $f$ is $\ldots$  \emph{decreasing}.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Polynomial or not?]
- 	Identify whether each of the following functions is a polynomial or not.
- 	If the function is a polynomial, state its degree.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$p(x)=2x+1$
- 			\begin{shortsolution}
- 				$p$ is a polynomial (you might also describe $p$ as linear). The degree of $p$ is 1.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=7x^2+4x$
- 			\begin{shortsolution}
- 				$p$ is a polynomial (you might also describe $p$ as quadratic). The degree of $p$ is 2.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=\sqrt{x}+2x+1$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial; we require the powers of $x$ to be integer values.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=2^x-45$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial; the $2^x$ term is exponential.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=6x^4-5x^3+9$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $6$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=-5x^{17}+9x+2$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is 17.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=4x(x+7)^2(x-3)^3$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $6$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=4x^{-5}-x^2+x$
- 			\begin{shortsolution}
- 				$p$ is not a polynomial because $-5$ is not a positive integer.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=-x^6(x^2+1)(x^3-2)$
- 			\begin{shortsolution}
- 				$p$ is a polynomial, and the degree of $p$ is $11$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Polynomial graphs]
- 	Three polynomial functions $p$, $m$, and $n$ are shown in \crefrange{poly:fig:functionp}{poly:fig:functionn}.
- 	The functions have the following formulas
- 	\begin{align*}
- 		p(x) & = (x-1)(x+2)(x-3)           \\
- 		m(x) & = -(x-1)(x+2)(x-3)          \\
- 		n(x) & = (x-1)(x+2)(x-3)(x+1)(x+4) 
- 	\end{align*}
- 	Note that for our present purposes we are not concerned with the vertical scale of the graphs.
- 	\begin{subproblem}
- 		Identify both on the graph {\em and} algebraically, the zeros of each polynomial.
- 		\begin{shortsolution}
- 			$y=p(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot] coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$y=m(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{-1*(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot] coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			$y=n(x)$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-90,ymax=70,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-4.15:3.15,samples=50]{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)};
- 					\addplot[soldot] coordinates{(-4,0)(-2,0)(-1,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			 			 			
- 			The zeros of $p$ are $-2$, $1$, and $3$; the zeros of $m$ are $-2$, $1$, and $3$; the zeros of $n$ are
- 			$-4$, $-2$, $-1$, and $3$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Write down the degree, how many times the curve of each function `turns around',
- 		and how many zeros it has
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item The degree of $p$ is 3, and the curve $y=p(x)$ turns around twice.
- 				\item The degree of $q$ is also 3, and the curve $y=q(x)$ turns around twice.
- 				\item The degree of $n$ is $5$, and the curve $y=n(x)$ turns around 4 times.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						ytick={-5,5},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=p(x)$}
- 			\label{poly:fig:functionp}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						ytick={-5,5},
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-2.5:3.5,samples=50]{-1*(x-1)*(x+2)*(x-3)};
- 					\addplot[soldot]coordinates{(-2,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=m(x)$}
- 			\label{poly:fig:functionm}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-5,xmax=5,
- 						ymin=-90,ymax=70,
- 						width=\textwidth,
- 					]
- 					\addplot expression[domain=-4.15:3.15,samples=100]{(x-1)*(x+2)*(x-3)*(x+1)*(x+4)};
- 					\addplot[soldot]coordinates{(-4,0)(-2,0)(-1,0)(1,0)(3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=n(x)$}
- 			\label{poly:fig:functionn}
- 		\end{subfigure}
- 		\caption{}
- 		\end{widepage}
- 	\end{figure}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Horizontal intercepts]\label{poly:prob:matchpolys}%
- 	State the horizontal intercepts (as ordered pairs) of the following polynomials.
- 	\begin{multicols}{2}
- 		\begin{subproblem}\label{poly:prob:degree5}
- 			$p(x)=(x-1)(x+2)(x-3)(x+1)(x+4)$
- 			\begin{shortsolution}
- 				$(-4,0)$, $(-2,0)$, $(-1,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(x)=-(x-1)(x+2)(x-3)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(x)=(x-1)(x+2)(x-3)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(1,0)$, $(3,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}\label{poly:prob:degree2}
- 			$s(x)=(x-2)(x+2)$
- 			\begin{shortsolution}
- 				$(-2,0)$, $(2,0)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Minimums, maximums, and concavity]\label{poly:prob:incdec}
- 	Four polynomial functions are graphed in \cref{poly:fig:incdec}. The formulas
- 	for these functions are (not respectively)
- 	\begin{gather*}
- 		p(x)=\frac{x^3}{6}-\frac{x^2}{4}-3x, \qquad q(x)=\frac{x^4}{20}+\frac{x^3}{15}-\frac{6}{5}x^2+1\\
- 		r(x)=-\frac{x^5}{50}-\frac{x^4}{40}+\frac{2x^3}{5}+6, \qquad s(x)=-\frac{x^6}{6000}-\frac{x^5}{2500}+\frac{67x^4}{4000}+\frac{17x^3}{750}-\frac{42x^2}{125}
- 	\end{gather*}
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{.23\textwidth}
- 		\centering
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=major,
- 					]
- 					\addplot expression[domain=-5.28:4.68,samples=50]{-x^5/50-x^4/40+2*x^3/5+6};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec3}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=major,
- 					]
- 					\addplot expression[domain=-6.08:4.967,samples=50]{x^4/20+x^3/15-6/5*x^2+1};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-6,xmax=8,ymin=-10,ymax=10,
- 						xtick={-4,-2,...,6},
- 						ytick={-8,-4,4,8},
- 						minor ytick={-6,-2,...,6},
- 						grid=both,
- 					]
- 					\addplot expression[domain=-4.818:6.081,samples=50]{x^3/6-x^2/4-3*x};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						width=\textwidth,
- 						xmin=-10,xmax=10,ymin=-10,ymax=10,
- 						xtick={-8,-4,4,8},
- 						ytick={-8,-4,4,8},
- 						minor xtick={-6,-2,...,6},
- 						minor ytick={-6,-2,...,6},
- 						grid=both,
- 					]
- 					\addplot expression[domain=-9.77:8.866,samples=50]{-x^6/6000-x^5/2500+67*x^4/4000+17/750*x^3-42/125*x^2};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:incdec4}
- 		\end{subfigure}
- 		\caption{Graphs for \cref{poly:prob:incdec}.}
- 		\label{poly:fig:incdec}
- 		\end{widepage}
- 	\end{figure}
- 	\begin{subproblem}
- 		Match each of the formulas with one of the given graphs.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is graphed in \vref{poly:fig:incdec1};
- 				\item $q$ is graphed in \vref{poly:fig:incdec2};
- 				\item $r$ is graphed in \vref{poly:fig:incdec3};
- 				\item $s$ is graphed in \vref{poly:fig:incdec4}.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the zeros of each function using the appropriate graph.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ has simple zeros at about $-3.8$, $0$, and $5$.
- 				\item $q$ has simple zeros at about $-5.9$, $-1$, $1$, and $4$.
- 				\item $r$ has simple zeros at about $-5$, $-2.9$, and $4.1$.
- 				\item $s$ has simple zeros at about $-9$, $-6$, $4.2$, $8.1$, and a zero of multiplicity $2$ at $0$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the local maximums and minimums of each of the functions.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ has a local maximum of approximately $3.9$ at $-2$, and a local minimum of approximately $-6.5$ at $3$.
- 				\item $q$ has a local minimum of approximately $-10$ at $-4$, and $-4$ at $3$; $q$ has a local maximum of approximately $1$ at $0$.
- 				\item $r$ has a local minimum of approximately $-5.5$ at $-4$, and a local maximum of approximately $10$ at $3$.
- 				\item $s$ has a local maximum of approximately $5$ at $-8$, $0$ at $0$, and $5$ at  $7$; $s$ has local minimums
- 				      of approximately $-3$ at $-4$, and $-1$ at $3$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the global maximums and minimums of each of the functions.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ does not have a global maximum, nor a global minimum.
- 				\item $q$ has a global minimum of approximately $-10$; it does not have a global maximum.
- 				\item $r$ does not have a global maximum, nor a global minimum.
- 				\item $s$ has a global maximum of approximately $5$; it does not have a global minimum.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the intervals on which each function is increasing and decreasing.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is increasing on $(-\infty,-2)\cup (3,\infty)$, and decreasing on $(-2,3)$.
- 				\item $q$ is increasing on $(-4,0)\cup (3,\infty)$, and decreasing on $(-\infty,-4)\cup (0,3)$.
- 				\item $r$ is increasing on $(-4,3)$, and decreasing on $(-\infty,-4)\cup (3,\infty)$.
- 				\item $s$ is increasing on $(-\infty,-8)\cup (-4,0)\cup (3,5)$, and decreasing on $(-8,-4)\cup (0,3)\cup (5,\infty)$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Approximate the intervals on which each function is concave up and concave down.
- 		\begin{shortsolution}
- 			\begin{itemize}
- 				\item $p$ is concave up on  $(1,\infty)$, and concave down on  $(-\infty,1)$.
- 				\item $q$ is concave up on $(-\infty,-1)\cup (1,\infty)$, and concave down on $(-1,1)$.
- 				\item $r$ is concave up on $(-\infty,-3)\cup (0,2)$, and concave down on $(-3,0)\cup (2,\infty)$.
- 				\item $s$ is concave up on $(-6,-2)\cup (2,5)$, and concave down on $(-\infty,-6)\cup (-2,2)\cup (5,\infty)$.
- 			\end{itemize}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		The degree of $q$ is $5$. Assuming that all of the real zeros of $q$ are
- 		shown in its graph, how many complex zeros does $q$ have?
- 		\begin{shortsolution}
- 			\Vref{poly:fig:incdec2} shows that $q$ has $3$ real zeros
- 			since the curve of $q$ cuts the horizontal axis $3$ times.
- 			Since $q$ has degree $5$, $q$ must have $2$ complex zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Long-run behaviour of polynomials]
- 	Describe the long-run behavior of each of polynomial functions in
- 	\crefrange{poly:prob:degree5}{poly:prob:degree2}.
- 	\begin{shortsolution}
- 		$\dd\lim_{x\rightarrow-\infty}p(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow\infty}p(x)=\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}q(x)=\infty$,
- 		$\dd\lim_{x\rightarrow\infty}q(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}r(x)=-\infty$,
- 		$\dd\lim_{x\rightarrow\infty}r(x)=\infty$,
- 		$\dd\lim_{x\rightarrow-\infty}s(x)=\infty$,
- 		$\dd\lim_{x\rightarrow\infty}s(x)=\infty$,
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[True of false?]
- 	Let $p$ be a polynomial function.
- 	Label each of the following statements as true (T) or false (F); if they are false,
- 	provide an example that supports your answer.
- 	\begin{subproblem}
- 		If $p$ has degree $3$, then $p$ has $3$ distinct zeros.
- 		\begin{shortsolution}
- 			False. Consider $p(x)=x^2(x+1)$ which has only 2 distinct zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has degree $4$, then $\dd\lim_{x\rightarrow-\infty}p(x)=\infty$ and $\dd\lim_{x\rightarrow\infty}p(x)=\infty$.
- 		\begin{shortsolution}
- 			False. Consider $p(x)=-x^4$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has even degree, then it is possible that $p$ can have no real zeros.
- 		\begin{shortsolution}
- 			True.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		If $p$ has odd degree, then it is possible that $p$ can have no real zeros.
- 		\begin{shortsolution}
- 			False. All odd degree polynomials will cut the horizontal axis at least once.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a possible formula for a polynomial
- 	function that has the specified properties.
- 	\begin{subproblem}
- 		Degree 2 and has zeros at $4$ and $5$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-4)(x-5)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 3 and has zeros at $4$,$5$ and $-3$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-4)(x-5)(x+3)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 4 and has zeros at $0$, $4$, $5$, $-3$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=x(x-4)(x-5)(x+3)$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 4, with zeros that make the graph cut at $2$, $-5$, and a zero that makes the graph touch at $-2$;
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x-2)(x+5)(x+2)^2$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Degree 3, with only one zero at $-1$.
- 		\begin{shortsolution}
- 			Possible option: $p(x)=(x+1)^3$. Note we could multiply $p$ by any real number, and still meet the requirements.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[\Cref{poly:step:last}]
- 	\pccname{Saheed} is graphing a polynomial function, $p$.
- 	He is following \crefrange{poly:step:first}{poly:step:last} and has so far
- 	marked the zeros of $p$ on \cref{poly:fig:optionsp1}. Saheed tells you that
- 	$p$ has degree $3$, but does \emph{not} say if the leading coefficient
- 	of $p$ is positive or negative.
- 	\begin{figure}[!htbp]
- 		\begin{widepage}
- 		\begin{subfigure}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\textwidth,
- 						height=.5\textwidth,
- 					]
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:optionsp1}
- 		\end{subfigure}%
- 		\hfill
- 		\begin{subfigure}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\textwidth,
- 						height=.5\textwidth,
- 					]
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{poly:fig:optionsp2}
- 		\end{subfigure}%
- 		\caption{}
- 		\end{widepage}
- 	\end{figure}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp1} to help sketch $p$, assuming that the leading coefficient
- 		is positive.
- 		\begin{shortsolution}
- 			Assuming that $a_3>0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-6.78179:8.35598,samples=50]{1/20*(x+5)*(x-2)*(x-6)};
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp1} to help sketch $p$, assuming that the leading coefficient
- 		is negative.
- 		\begin{shortsolution}
- 			Assuming that $a_3<0$:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-6.78179:8.35598,samples=50]{-1/20*(x+5)*(x-2)*(x-6)};
- 					\addplot[soldot] coordinates{(-5,0)(2,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	Saheed now turns his attention to another polynomial function, $q$. He finds
- 	the zeros of $q$ (there are only $2$) and marks them on \cref{poly:fig:optionsp2}.
- 	Saheed knows that $q$ has degree $3$, but doesn't know if the leading
- 	coefficient is positive or negative.
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp2} to help sketch $q$, assuming that the leading
- 		coefficient of $q$ is positive. Hint: only one of the zeros is simple.
- 		\begin{shortsolution}
- 			Assuming that $a_4>0$ there are $2$ different options:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-8.68983:7.31809,samples=50]{1/20*(x+5)^2*(x-6)};
- 					\addplot expression[domain=-6.31809:9.68893,samples=50]{1/20*(x+5)*(x-6)^2};
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use the information in \cref{poly:fig:optionsp2} to help sketch $q$, assuming that the leading
- 		coefficient of $q$ is negative.
- 		\begin{shortsolution}
- 			Assuming that $a_4<0$ there are $2$ different options:
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-15},
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot expression[domain=-8.68983:7.31809,samples=50]{-1/20*(x+5)^2*(x-6)};
- 					\addplot expression[domain=-6.31809:9.68893,samples=50]{-1/20*(x+5)*(x-6)^2};
- 					\addplot[soldot] coordinates{(-5,0)(6,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Zeros]
- 	Find all zeros of each of the following polynomial functions, making
- 	sure to detail their multiplicity. Note that
- 	you may need to use factoring, or the quadratic formula, or both! Also note
- 	that some zeros may be repeated, and some may be complex.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$p(x)=x^2+1$
- 			\begin{shortsolution}
- 				$\pm i$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(y)=(y^2-9)(y^2-7)$
- 			\begin{shortsolution}
- 				$\pm 3$, $\pm \sqrt{7}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(z)=-4z^3(z^2+3)(z^2+64)$
- 			\begin{shortsolution}
- 				$0$ (multiplicity $3$), $\pm\sqrt{3}$ (simple), $\pm\sqrt{8}$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$a(x)=x^4-81$
- 			\begin{shortsolution}
- 				$\pm 3$, $\pm 3i$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$b(y)=y^3-8$
- 			\begin{shortsolution}
- 				$2$, $-1\pm i\sqrt{3}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$c(m)=m^3-m^2$
- 			\begin{shortsolution}
- 				$0$ (multiplicity $2$), $1$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(n)=(n+1)(n^2+4)$
- 			\begin{shortsolution}
- 				$-1$, $\pm 2i$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$f(\alpha)=(\alpha^2-16)(\alpha^2-5\alpha+4)$
- 			\begin{shortsolution}
- 				$-4$ (simple), $4$ (multiplicity $2$), $1$ (simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(\beta)=(\beta^2-25)(\beta^2-5\beta-4)$
- 			\begin{shortsolution}
- 				$\pm 5$, $\dfrac{5\pm\sqrt{41}}{2}$ (all are simple).
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Given zeros, find a formula]
- 	In each of the following problems you are given the zeros of a polynomial.
- 	Write a possible formula for each polynomial| you may leave your
- 	answer in factored form, but it may not contain complex numbers. Unless
- 	otherwise stated, assume that the zeros are simple.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$1$, $2$
- 			\begin{shortsolution}
- 				$p(x)=(x-1)(x-2)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$0$, $5$, $13$
- 			\begin{shortsolution}
- 				$p(x)=x(x-5)(x-13)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$-7$, $2$ (multiplicity $3$), $5$
- 			\begin{shortsolution}
- 				$p(x)=(x+7)(x-2)^3(x-5)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$0$, $\pm i$
- 			\begin{shortsolution}
- 				$p(x)=x(x^2+1)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\pm 2i$, $\pm 7$
- 			\begin{shortsolution}
- 				$p(x)=(x^2+4)(x^2-49)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$-2\pm i\sqrt{6}$
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Composition of polynomials]
- 	Let $p$ and $q$ be polynomial functions that have formulas
- 	\[
- 		p(x)=(x+1)(x+2)(x+5), \qquad q(x)=3-x^4
- 	\]
- 	Evaluate each of the following.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(p\circ q)(0)$
- 			\begin{shortsolution}
- 				$160$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(q\circ p)(0)$
- 			\begin{shortsolution}
- 				$-9997$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\circ q)(1)$
- 			\begin{shortsolution}
- 				$84$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\circ p)(0)$
- 			\begin{shortsolution}
- 				$1980$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   June 2012
- 	%===================================
- 	\begin{problem}[Piecewise polynomial functions]
- 	Let $P$ be the piecewise-defined function with formula
- 	\[
- 		P(x)=\begin{cases}
- 		(1-x)(2x+5)(x^2+1), &   x\leq -3\\
- 		4-x^2,              &   -3<x < 4\\
- 		x^3                 &   x\geq 4
- 		\end{cases}
- 	\]
- 	Evaluate each of the following
- 	\begin{multicols}{5}
- 		\begin{subproblem}
- 			$P(-4)$
- 			\begin{shortsolution}
- 				$-255$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(0)$
- 			\begin{shortsolution}
- 				$4$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(4)$
- 			\begin{shortsolution}
- 				$64$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$P(-3)$
- 			\begin{shortsolution}
- 				$-40$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(P\circ P)(0)$
- 			\begin{shortsolution}
- 				$64$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Function algebra]
- 	Let $p$ and $q$ be the polynomial functions that have formulas
- 	\[
- 		p(x)=x(x+1)(x-3)^2, \qquad q(x)=7-x^2
- 	\]
- 	Evaluate each of the following (if possible).
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(p+q)(1)$
- 			\begin{shortsolution}
- 				$14$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p-q)(0)$
- 			\begin{shortsolution}
- 				$7$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(p\cdot q)(\sqrt{7})$
- 			\begin{shortsolution}
- 				$0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\left( \frac{q}{p} \right)(1)$
- 			\begin{shortsolution}
- 				$\frac{3}{4}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\begin{subproblem}
- 		What is the domain of the function $\frac{q}{p}$?
- 		\begin{shortsolution}
- 			$(-\infty,-1)\cup (-1,0)\cup (0,3)\cup (3,\infty)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Transformations: given the transformation, find the formula]
- 	Let $p$ be the polynomial function that has formula.
- 	\[
- 		p(x)=4x(x^2-1)(x+3)
- 	\]
- 	In each of the following
- 	problems apply the given transformation to the function $p$ and
- 	write a formula for the transformed version of $p$.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			Shift $p$ to the right by $5$ units.
- 			\begin{shortsolution}
- 				$p(x-5)=4(x-5)(x-2)(x^2-10x+24)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ to the left by $6$ units.
- 			\begin{shortsolution}
- 				$p(x+6)=4(x+6)(x+9)(x^2+12x+35)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ up by $12$ units.
- 			\begin{shortsolution}
- 				$p(x)+12=4x(x^2-1)(x+3)+12$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $p$ down by $2$ units.
- 			\begin{shortsolution}
- 				$p(x)-2=4x(x^2-1)(x+3)-2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $p$ over the horizontal axis.
- 			\begin{shortsolution}
- 				$-p(x)=-4x(x^2-1)(x+3)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $p$ over the vertical axis.
- 			\begin{shortsolution}
- 				$p(-x)=-4x(x^2-1)(3-x)$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a table]\label{poly:prob:findformula}
- 	\Crefrange{poly:tab:findformulap}{poly:tab:findformulas} show values of polynomial functions, $p$, $q$,
- 	$r$, and $s$.
- 	 	 	
- 	\begin{table}[!htb]
- 		\centering
- 		\begin{widepage}
- 		\caption{Tables for \cref{poly:prob:findformula}}
- 		\label{poly:tab:findformula}
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=p(x)$}
- 			\label{poly:tab:findformulap}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				$-4$          & $-56$         \\\normalline
- 				$-3$          & $-18$         \\\normalline
- 				$-2$          & $0$           \\\normalline
- 				$-1$          & $4$           \\\normalline
- 				$0$           & $0$           \\\normalline
- 				$1$           & $-6$          \\\normalline
- 				$2$           & $-8$          \\\normalline
- 				$3$           & $0$           \\\normalline
- 				$4$           & $24$          \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=q(x)$}
- 			\label{poly:tab:findformulaq}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $-16$         \\\normalline
- 				$-3$          & $-3$          \\\normalline
- 				$-2$          & $0$           \\\normalline
- 				$-1$          & $-1$          \\\normalline
- 				$0$           & $0$           \\\normalline
- 				$1$           & $9$           \\\normalline
- 				$2$           & $32$          \\\normalline
- 				$3$           & $75$          \\\normalline
- 				$4$           & $144$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=r(x)$}
- 			\label{poly:tab:findformular}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $105$         \\\normalline
- 				$-3$          & $0$           \\\normalline
- 				$-2$          & $-15$         \\\normalline
- 				$-1$          & $0$           \\\normalline
- 				$0$           & $9$           \\\normalline
- 				$1$           & $0$           \\\normalline
- 				$2$           & $-15$         \\\normalline
- 				$3$           & $0$           \\\normalline
- 				$4$           & $105$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=s(x)$}
- 			\label{poly:tab:findformulas}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\ \afterheading
- 				$-4$          & $75$          \\\normalline
- 				$-3$          & $0$           \\\normalline
- 				$-2$          & $-9$          \\\normalline
- 				$-1$          & $0$           \\\normalline
- 				$0$           & $3$           \\\normalline
- 				$1$           & $0$           \\\normalline
- 				$2$           & $15$          \\\normalline
- 				$3$           & $96$          \\\normalline
- 				$4$           & $760$         \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\end{widepage}
- 	\end{table}
- 	 	 	
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $p$ are shown (in \cref{poly:tab:findformulap}), how many zeros does $p$ have?
- 		\begin{shortsolution}
- 			$p$ has 3 zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What is the degree of $p$?
- 		\begin{shortsolution}
- 			$p$ is degree 3.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Write a formula for $p(x)$.
- 		\begin{shortsolution}
- 			$p(x)=x(x+2)(x-3)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $q$ are shown (in \cref{poly:tab:findformulaq}), how many zeros does $q$ have?
- 		\begin{shortsolution}
- 			$q$ has 2 zeros.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Describe the difference in behavior of $p$ and $q$ at $-2$.
- 		\begin{shortsolution}
- 			$p$ changes sign at $-2$, and $q$ does not change sign at $-2$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that $q$ is a degree-$3$ polynomial, write a formula for $q(x)$.
- 		\begin{shortsolution}
- 			$q(x)=x(x+2)^2$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $r$ are shown (in \cref{poly:tab:findformular}), find a formula for $r(x)$.
- 		\begin{shortsolution}
- 			$r(x)=(x+3)(x+1)(x-1)(x-3)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Assuming that all of the zeros of $s$ are shown (in \cref{poly:tab:findformulas}), find a formula for $s(x)$.
- 		\begin{shortsolution}
- 			$s(x)=(x+3)(x+1)(x-1)^2$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	\end{exercises}
- 	 	 	
-\section{Rational functions}
- \subsection*{Power functions with negative exponents}
- 	The study of rational functions will rely upon a good knowledge
- 	of power functions with negative exponents. \Cref{rat:ex:oddpow,rat:ex:evenpow} are
- 	simple but fundamental to understanding the behavior of rational functions.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Power functions with odd negative exponents]\label{rat:ex:oddpow}
- 		Graph each of the following functions on your calculator, state their domain in interval notation, and their
- 		behavior as $x\rightarrow 0^-$ and $x\rightarrow 0^+$.
- 		\[
- 			f(x)=\frac{1}{x},\qquad g(x)=\dfrac{1}{x^3},\qquad h(x)=\dfrac{1}{x^5}
- 		\]
- 		\begin{pccsolution}
- 			The functions $f$, $g$, and $k$ are plotted in \cref{rat:fig:oddpow}.
- 			The domain of each of the functions $f$, $g$, and $h$ is $(-\infty,0)\cup (0,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				f(x)\rightarrow 0                         & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}  f(x)\rightarrow 0 & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			The same results hold for $g$ and $h$. Note also that each of the functions
- 			has a \emph{vertical asymptote} at $0$. We see that
- 			\begin{align*}
- 				f(x)\rightarrow -\infty                         & \text{ as } x\rightarrow 0^- \\
- 				\mathllap{\text{and }}   f(x)\rightarrow \infty & \text{ as } x\rightarrow 0^+ 
- 			\end{align*}
- 			The same results hold for $g$ and $h$.
- 			 			 			
- 			The curve of a function that has a vertical asymptote is necessarily separated
- 			into \emph{branches}| each of the functions $f$, $g$, and $h$ have $2$ branches.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-3,xmax=3,
- 						ymin=-5,ymax=5,
- 						xtick={-2,-1,...,2},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=north west,
- 					]
- 					\addplot expression[domain=-3:-0.2]{1/x};
- 					\addplot expression[domain=-3:-0.584]{1/x^3};
- 					\addplot expression[domain=-3:-0.724]{1/x^5};
- 					\addplot expression[domain=0.2:3]{1/x};
- 					\addplot expression[domain=0.584:3]{1/x^3};
- 					\addplot expression[domain=0.724:3]{1/x^5};
- 					\addplot[soldot]coordinates{(-1,-1)}node[axisnode,anchor=north east]{$(-1,-1)$};
- 					\addplot[soldot]coordinates{(1,1)}node[axisnode,anchor=south west]{$(1,1)$};
- 					\legend{$f$,$g$,$h$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:oddpow}
- 		\end{minipage}%
- 		\hfill
- 		\begin{minipage}{.45\textwidth}
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-3,xmax=3,
- 						ymin=-5,ymax=5,
- 						xtick={-2,-1,...,2},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\textwidth,
- 						legend pos=south east,
- 					]
- 					\addplot expression[domain=-3:-0.447]{1/x^2};
- 					\addplot expression[domain=-3:-0.668]{1/x^4};
- 					\addplot expression[domain=-3:-0.764]{1/x^6};
- 					\addplot expression[domain=0.447:3]{1/x^2};
- 					\addplot expression[domain=0.668:3]{1/x^4};
- 					\addplot expression[domain=0.764:3]{1/x^6};
- 					\addplot[soldot]coordinates{(-1,1)}node[axisnode,anchor=south east]{$(-1,1)$};
- 					\addplot[soldot]coordinates{(1,1)}node[axisnode,anchor=south west]{$(1,1)$};
- 					\legend{$F$,$G$,$H$}
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:evenpow}
- 		\end{minipage}%
- 	\end{figure}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Power functions with even negative exponents]\label{rat:ex:evenpow}%
- 		Graph each of the following functions, state their domain, and their
- 		behavior as $x\rightarrow 0^-$ and $x\rightarrow 0^+$.
- 		\[
- 			f(x)=\frac{1}{x^2},\qquad g(x)=\frac{1}{x^4},\qquad h(x)=\frac{1}{x^6}
- 		\]
- 		\begin{pccsolution}
- 			The functions $F$, $G$, and $H$ are plotted in \cref{rat:fig:evenpow}.
- 			The domain of each of the functions $F$, $G$, and $H$ is $(-\infty,0)\cup (0,\infty)$. Note that
- 			the long-run behavior of each of the functions is the same, and in particular
- 			\begin{align*}
- 				F(x)\rightarrow 0                           & \text{ as } x\rightarrow\infty  \\
- 				\mathllap{\text{and }}    f(x)\rightarrow 0 & \text{ as } x\rightarrow-\infty 
- 			\end{align*}
- 			As in \cref{rat:ex:oddpow}, $F$ has a horizontal asymptote  that
- 			has equation $y=0$.
- 			The same results hold for $G$ and $H$. Note also that each of the functions
- 			has a \emph{vertical asymptote} at $0$. We see that
- 			\begin{align*}
- 				F(x)\rightarrow \infty                          & \text{ as } x\rightarrow 0^- \\
- 				\mathllap{\text{and }}   F(x)\rightarrow \infty & \text{ as } x\rightarrow 0^+ 
- 			\end{align*}
- 			The same results hold for $G$ and $H$. Each of the functions $F$, $G$, and $H$
- 			have $2$ branches.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{doyouunderstand}
- 		\begin{problem}
- 		Repeat \cref{rat:ex:oddpow,rat:ex:evenpow} using (respectively)
- 		\begin{subproblem}
- 			$k(x)=-\dfrac{1}{x}$, $ m(x)=-\dfrac{1}{x^3}$, $ n(x)=-\dfrac{1}{x^5}$
- 			\begin{shortsolution}
- 				The functions $k$, $m$, and $n$ have domain $(-\infty,0)\cup (0,\infty)$, and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-3,xmax=3,
- 							ymin=-5,ymax=5,
- 							xtick={-2,-1,...,2},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-3:-0.2]{-1/x};
- 						\addplot expression[domain=-3:-0.584]{-1/x^3};
- 						\addplot expression[domain=-3:-0.724]{-1/x^5};
- 						\addplot expression[domain=0.2:3]{-1/x};
- 						\addplot expression[domain=0.584:3]{-1/x^3};
- 						\addplot expression[domain=0.724:3]{-1/x^5};
- 						\legend{$k$,$m$,$n$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					k(x)\rightarrow 0                                & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}    k(x)\rightarrow 0      & \text{ as } x\rightarrow-\infty \\
- 					\intertext{and also}
- 					k(x)\rightarrow \infty                           & \text{ as } x\rightarrow 0^-    \\
- 					\mathllap{\text{and }}   k(x)\rightarrow -\infty & \text{ as } x\rightarrow 0^+    
- 				\end{align*}
- 				The same are true for $m$ and $n$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$ K(x)=-\dfrac{1}{x^2}$, $ M(x)=-\dfrac{1}{x^4}$, $ N(x)=-\dfrac{1}{x^6}$
- 			\begin{shortsolution}
- 				The functions $K$, $M$, and $N$ have domain $(-\infty,0)\cup (0,\infty)$, and
- 				are graphed below.
- 				 				 				
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-3,xmax=3,
- 							ymin=-5,ymax=5,
- 							xtick={-2,-1,...,2},
- 							minor ytick={-3,-1,...,3},
- 							grid=both,
- 							width=\solutionfigurewidth,
- 							legend pos=north east,
- 						]
- 						\addplot expression[domain=-3:-0.447]{-1/x^2};
- 						\addplot expression[domain=-3:-0.668]{-1/x^4};
- 						\addplot expression[domain=-3:-0.764]{-1/x^6};
- 						\addplot expression[domain=0.447:3]{-1/x^2};
- 						\addplot expression[domain=0.668:3]{-1/x^4};
- 						\addplot expression[domain=0.764:3]{-1/x^6};
- 						\legend{$K$,$M$,$N$}
- 					\end{axis}
- 				\end{tikzpicture}
- 				 				 				
- 				Note that
- 				\begin{align*}
- 					K(x)\rightarrow 0                                & \text{ as } x\rightarrow\infty  \\
- 					\mathllap{\text{and }}    K(x)\rightarrow 0      & \text{ as } x\rightarrow-\infty \\
- 					\intertext{and also}
- 					K(x)\rightarrow -\infty                          & \text{ as } x\rightarrow 0^-    \\
- 					\mathllap{\text{and }}   K(x)\rightarrow -\infty & \text{ as } x\rightarrow 0^+    
- 				\end{align*}
- 				The same are true for $M$ and $N$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\end{problem}
- 	\end{doyouunderstand}
- 	 	 	
- \subsection*{Rational functions}
- 	\begin{pccdefinition}[Rational functions]\label{rat:def:function}
- 		Rational functions have the form
- 		\[
- 			r(x) = \frac{p(x)}{q(x)}
- 		\]
- 		where both $p$ and $q$ are polynomials.
- 		 		 		
- 		Note that
- 		\begin{itemize}
- 			\item the domain or $r$ will be all real numbers, except those that
- 			      make the \emph{denominator}, $q(x)$, equal to $0$;
- 			\item the zeros of $r$ are the zeros of $p$, i.e the real numbers
- 			      that make the \emph{numerator}, $p(x)$, equal to $0$.
- 		\end{itemize}
- 		 		 		
- 		\Cref{rat:ex:oddpow,rat:ex:evenpow} are particularly important because $r$
- 		will behave like $\frac{1}{x}$, or $\frac{1}{x^2}$ around its vertical asymptotes,
- 		depending on the power that the relevant term is raised to| we will demonstrate
- 		this in what follows.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Rational or not]
- 		Identify whether each of the following functions is a rational or not. If
- 		the function is rational, state the domain.
- 		\begin{multicols}{3}
- 			\begin{enumerate}
- 				\item $r(x)=\dfrac{1}{x}$
- 				\item $f(x)=2^x+3$
- 				\item $g(x)=19$
- 				\item $h(x)=\dfrac{3+x}{4-x}$
- 				\item $k(x)=\dfrac{x^3+2x}{x-15}$
- 				\item $l(x)=9-4x$
- 				\item $m(x)=\dfrac{x+5}{(x-7)(x+9)}$
- 				\item $n(x)=x^2+6x+7$
- 				\item $q(x)=1-\dfrac{3}{x+1}$
- 			\end{enumerate}
- 		\end{multicols}
- 		\begin{pccsolution}
- 			\begin{enumerate}
- 				\item $r$ is rational; the domain of $r$ is $(-\infty,0)\cup(0,\infty)$.
- 				\item $f$ is not rational.
- 				\item $g$ is not rational; $g$ is constant.
- 				\item $h$ is rational; the domain of $h$ is $(-\infty,4)\cup(4,\infty)$.
- 				\item $k$ is rational; the domain of $k$ is $(-\infty,15)\cup(15,\infty)$.
- 				\item $l$ is not rational; $l$ is linear.
- 				\item $m$ is rational; the domain of $m$ is $(-\infty,-9)\cup(-9,7)\cup(7,\infty)$.
- 				\item $n$ is not rational; $n$ is quadratic (or you might describe $n$ as a polynomial).
- 				\item $q$ is rational; the domain of $q$ is $(-\infty,-1)\cup (-1,\infty)$.
- 			\end{enumerate}
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Match formula to graph]
- 		Each of the following functions is graphed in \cref{rat:fig:whichiswhich}.
- 		Which is which?
- 		\[
- 			r(x)=\frac{1}{x-3}, \qquad q(x)=\frac{x-2}{x+5}, \qquad k(x)=\frac{1}{(x+2)(x-3)}
- 		\]
- 		\begin{figure}[!htb]
- 			\setlength{\figurewidth}{0.3\textwidth}
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=(x-2)/(x+5);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-6,ymax=6,
- 							xtick={-8,-6,...,8},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-6.37]{f};
- 						\addplot[pccplot] expression[domain=-3.97:10]{f};
- 						\addplot[soldot] coordinates{(2,0)};
- 						\addplot[asymptote,domain=-6:6]({-5},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which1}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=1/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-5,ymax=6,
- 							xtick={-8,-6,...,8},
- 							ytick={-4,4},
- 							minor ytick={-3,...,5},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:2.8]{f};
- 						\addplot[pccplot] expression[domain=3.17:10]{f};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which2}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=1/((x-3)*(x+2));}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-5,ymax=5,
- 							xtick={-8,-6,...,8},
- 							ytick={-4,4},
- 							minor ytick={-3,...,3},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-2.03969]{f};
- 						\addplot[pccplot] expression[domain=-1.95967:2.95967]{f};
- 						\addplot[pccplot] expression[domain=3.03969:10]{f};
- 						\addplot[asymptote,domain=-5:5]({-2},{x});
- 						\addplot[asymptote,domain=-5:5]({3},{x});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:which3}
- 			\end{subfigure}
- 			\caption{}
- 			\label{rat:fig:whichiswhich}
- 		\end{figure}
- 		 		 		
- 		\begin{pccsolution}
- 			Let's start with the function $r$. Note that domain of $r$ is $(-\infty,3)\cup(0,3)$, so
- 			we search for a function that has a vertical asymptote at $3$. There
- 			are two possible choices: the functions graphed in \cref{rat:fig:which2,rat:fig:which3},
- 			but note that the function in \cref{rat:fig:which3} also has a vertical asymptote at $-2$
- 			which is not consistent with the formula for $r(x)$. Therefore, $y=r(x)$
- 			is graphed in \cref{rat:fig:which2}.
- 			 			 			
- 			The function $q$ has domain $(-\infty,-5)\cup(-5,\infty)$, so we search
- 			for a function that has a vertical asymptote at $-5$. The only candidate
- 			is the curve shown in \cref{rat:fig:which1}; note that the curve also goes through $(2,0)$,
- 			which is consistent with the formula for $q(x)$, since $q(2)=0$, i.e $q$
- 			has a zero at $2$.
- 			 			 			
- 			The function $k$ has domain $(-\infty,-2)\cup(-2,3)\cup(3,\infty)$, and
- 			has vertical asymptotes at $-2$ and $3$. This is consistent with
- 			the graph in \cref{rat:fig:which3} (and is the only curve that
- 			has $3$ branches).
- 			 			 			
- 			We note that each function behaves like $\frac{1}{x}$ around its vertical asymptotes,
- 			because each linear factor in each denominator is raised to the power $1$; if (for example)
- 			the definition of $r$ was instead
- 			\[
- 				r(x)=\frac{1}{(x-3)^2}
- 			\]
- 			then we would see that $r$ behaves like $\frac{1}{x^2}$ around its vertical asymptote, and
- 			the graph of $r$ would be very different. We will deal with these cases in the examples that follow.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Repeated factors in the denominator]
- 		Consider the functions $f$, $g$, and $h$ that have formulas
- 		\[
- 			f(x)=\frac{x-2}{(x-3)(x+2)}, \qquad g(x)=\frac{x-2}{(x-3)^2(x+2)}, \qquad h(x)=\frac{x-2}{(x-3)(x+2)^2}
- 		\]
- 		which are graphed in \cref{rat:fig:repfactd}. Note that each function has $2$
- 		vertical asymptotes, and the domain of each function is
- 		\[
- 			(-\infty,-2)\cup(-2,3)\cup(3,\infty)
- 		\]
- 		so we are not surprised to see that each curve has $3$ branches. We also note that
- 		the numerator of each function is the same, which tells us that each function has
- 		only $1$ zero at $2$.
- 		 		 		
- 		The functions $g$ and $h$ are different from those that we have considered previously,
- 		because they have a repeated factor in the denominator. Notice in particular
- 		the way that the functions behave around their asymptotes:
- 		\begin{itemize}
- 			\item $f$ behaves like $\frac{1}{x}$ around both of its asymptotes;
- 			\item $g$ behaves like $\frac{1}{x}$ around $-2$, and like $\frac{1}{x^2}$ around $3$;
- 			\item $h$ behaves like $\frac{1}{x^2}$ around $-2$, and like $\frac{1}{x}$ around $3$.
- 		\end{itemize}
- 	\end{pccexample}
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,4},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.201]{f};
- 					\addplot[pccplot] expression[domain=-1.802:2.951]{f};
- 					\addplot[pccplot] expression[domain=3.052:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-2},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)(x-3)}$}
- 			\label{rat:fig:repfactd1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)*(x-3)^2);}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,4},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.039]{f};
- 					\addplot[pccplot] expression[domain=-1.959:2.796]{f};
- 					\addplot[pccplot] expression[domain=3.243:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-4:4]({-2},{x});
- 					%                 \addplot[asymptote,domain=-4:4]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)(x-3)^2}$}
- 			\label{rat:fig:repfactd2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+2)^2*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-4,ymax=4,
- 						xtick={-4,-2,...,2},
- 						ytick={-2,2},
- 						%                    grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.451]{f};
- 					\addplot[pccplot] expression[domain=-1.558:2.990]{f};
- 					\addplot[pccplot] expression[domain=3.010:6]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-4:4]({-2},{x});
- 					%                 \addplot[asymptote,domain=-4:4]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+2)^2(x-3)}$}
- 			\label{rat:fig:repfactd3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:repfactd}
- 	\end{figure}
- 	 	 	
- 	\Cref{rat:def:function} says that the zeros of
- 	the rational function $r$ that has formula $r(x)=\frac{p(x)}{q(x)}$ are
- 	the zeros of $p$. Let's explore this a little more.
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Zeros] Find the zeros of each of the following functions
- 		\[
- 			\alpha(x)=\frac{x+5}{3x-7}, \qquad \beta(x)=\frac{9-x}{x+1}, \qquad \gamma(x)=\frac{17x^2-10}{2x+1}
- 		\]
- 		\begin{pccsolution}
- 			We find the zeros of each function in turn by setting the numerator equal to $0$. The zeros of
- 			$\alpha$ are found by solving
- 			\[
- 				x+5=0
- 			\]
- 			The zero of $\alpha$ is $-5$.
- 			 			 			
- 			Similarly, we may solve $9-x=0$ to find the zero of $\beta$, which is clearly $9$.
- 			 			 			
- 			The zeros of $\gamma$ satisfy the equation
- 			\[
- 				17x^2-10=0
- 			\]
- 			which we can solve using the square root property to obtain
- 			\[
- 				x=\pm\frac{10}{17}
- 			\]
- 			The zeros of $\gamma$ are $\pm\frac{10}{17}$.
- 		\end{pccsolution}
- 	\end{pccexample}
- 	 	 	
- \subsection*{Long-run behavior}
- 	Our focus so far has been on the behavior of rational functions around
- 	their \emph{vertical} asymptotes. In fact, rational functions also
- 	have interesting long-run behavior around their \emph{horizontal} or
- 	\emph{oblique} asymptotes. A rational function will always have either
- 	a horizontal or an oblique asymptote| the case is determined by the degree
- 	of the numerator and the degree of the denominator.
- 	\begin{pccdefinition}[Long-run behavior]\label{rat:def:longrun}
- 		Let $r$ be the rational function that has formula
- 		\[
- 			r(x) = \frac{a_n x^n + a_{n-1}x^{n-1}+\ldots + a_0}{b_m x^m + b_{m-1}x^{m-1}+\ldots+b_0}
- 		\]
- 		We can classify the long-run behavior of the rational function $r$
- 		according to the following criteria:
- 		\begin{itemize}
- 			\item if $n<m$ then  $r$ has a horizontal asymptote with equation $y=0$;
- 			\item if $n=m$ then $r$ has a horizontal asymptote with equation $y=\dfrac{a_n}{b_m}$;
- 			\item if $n>m$ then $r$ will have an oblique asymptote as $x\rightarrow\pm\infty$ (more on this in \cref{rat:sec:oblique})
- 		\end{itemize}
- 	\end{pccdefinition}
- 	We will concentrate on functions that have horizontal asymptotes until
- 	we reach \cref{rat:sec:oblique}.
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Long-run behavior graphically]\label{rat:ex:horizasymp}
- 		\pccname{Kebede} has graphed the following functions in his graphing calculator
- 		\[
- 			r(x)=\frac{x+1}{x-3}, \qquad s(x)=\frac{2(x+1)}{x-3}, \qquad t(x)=\frac{3(x+1)}{x-3}
- 		\]
- 		and obtained the curves shown in \cref{rat:fig:horizasymp}. Kebede decides
- 		to test his knowledgeable friend \pccname{Oscar}, and asks him
- 		to match the formulas to the graphs.
- 		 		 		
- 		\begin{figure}[!htb]
- 			\setlength{\figurewidth}{0.3\textwidth}
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=2*(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:2]{f};
- 						\addplot[pccplot] expression[domain=5:15]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{2});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp1}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:2.42857,samples=50]{f};
- 						\addplot[pccplot] expression[domain=3.8:15,samples=50]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{1});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp2}
- 			\end{subfigure}
- 			\hfill
- 			\begin{subfigure}{\figurewidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=3*(x+1)/(x-3);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-15,xmax=15,
- 							ymin=-6,ymax=6,
- 							xtick={-12,-8,...,12},
- 							minor ytick={-4,-3,...,4},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-15:1.6666,samples=50]{f};
- 						\addplot[pccplot] expression[domain=7:15]{f};
- 						\addplot[soldot] coordinates{(-1,0)};
- 						\addplot[asymptote,domain=-6:6]({3},{x});
- 						\addplot[asymptote,domain=-15:15]({x},{3});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:horizasymp3}
- 			\end{subfigure}
- 			\caption{Horizontal asymptotes}
- 			\label{rat:fig:horizasymp}
- 		\end{figure}
- 		 		 		
- 		Oscar notices that each function has a vertical asymptote at $3$ and a zero at $-1$.
- 		The main thing that catches Oscar's eye is that each function has a different
- 		coefficient in the numerator, and that each curve has a different horizontal asymptote.
- 		In particular, Oscar notes that
- 		\begin{itemize}
- 			\item the curve shown in \cref{rat:fig:horizasymp1} has a horizontal asymptote with equation $y=2$;
- 			\item the curve shown in \cref{rat:fig:horizasymp2} has a horizontal asymptote with equation $y=1$;
- 			\item the curve shown in \cref{rat:fig:horizasymp3} has a horizontal asymptote with equation $y=3$.
- 		\end{itemize}
- 		Oscar is able to tie it all together for Kebede by referencing \cref{rat:def:longrun}. He says
- 		that since the degree of the numerator and the degree of the denominator is the same
- 		for each of the functions $r$, $s$, and $t$, the horizontal asymptote will be determined
- 		by evaluating the ratio of their leading coefficients.
- 		 		 		
- 		Oscar therefore says that $r$ should have a horizontal asymptote $y=\frac{1}{1}=1$, $s$ should
- 		have a horizontal asymptote $y=\frac{2}{1}=2$, and $t$ should have a horizontal asymptote
- 		$y=\frac{3}{1}=3$. Kebede is able to finish the problem from here, and says that $r$ is
- 		shown in \cref{rat:fig:horizasymp2}, $s$ is shown in \cref{rat:fig:horizasymp1}, and
- 		$t$ is shown in \cref{rat:fig:horizasymp3}.
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{pccexample}[Long-run behavior numerically]
- 		\pccname{Xiao} and \pccname{Dwayne} saw \cref{rat:ex:horizasymp} but are a little confused
- 		about horizontal asymptotes. What does it mean to say that a function $r$ has a horizontal
- 		asymptote?
- 		 		 		
- 		They decide to explore the concept by
- 		constructing a table of values for the rational functions $R$ and  $S$ that have formulas
- 		\[
- 			R(x)=\frac{-5(x+1)}{x-3}, \qquad S(x)=\frac{7(x-5)}{2(x+1)}
- 		\]
- 		In \cref{rat:tab:plusinfty} they model the behavior of $R$ and $S$ as $x\rightarrow\infty$,
- 		and in \cref{rat:tab:minusinfty} they model the behavior of $R$ and $S$ as $x\rightarrow-\infty$
- 		by substituting very large values of $|x|$ into each function.
- 		\begin{table}[!htb]
- 			\begin{minipage}{.5\textwidth}
- 				\centering
- 				\caption{$R$ and $S$ as $x\rightarrow\infty$}
- 				\label{rat:tab:plusinfty}
- 				\begin{tabular}{crr}
- 					\beforeheading
- 					$x$            & $R(x)$     & $S(x)$    \\ \afterheading
- 					$1\times 10^2$ & $-5.20619$ & $3.29208$ \\\normalline
- 					$1\times 10^3$ & $-5.02006$ & $3.47902$ \\\normalline
- 					$1\times 10^4$ & $-5.00200$ & $3.49790$ \\\normalline
- 					$1\times 10^5$ & $-5.00020$ & $3.49979$ \\\normalline
- 					$1\times 10^6$ & $-5.00002$ & $3.49998$ \\\lastline
- 				\end{tabular}
- 			\end{minipage}%
- 			\begin{minipage}{.5\textwidth}
- 				\centering
- 				\caption{$R$ and $S$ as $x\rightarrow-\infty$}
- 				\label{rat:tab:minusinfty}
- 				\begin{tabular}{crr}
- 					\beforeheading
- 					$x$             & $R(x)$     & $S(x)$    \\ \afterheading
- 					$-1\times 10^2$ & $-4.80583$ & $3.71212$ \\\normalline
- 					$-1\times 10^3$ & $-4.98006$ & $3.52102$ \\\normalline
- 					$-1\times 10^4$ & $-4.99800$ & $3.50210$ \\\normalline
- 					$-1\times 10^5$ & $-4.99980$ & $3.50021$ \\\normalline
- 					$-1\times 10^6$ & $-4.99998$ & $3.50002$ \\\lastline
- 				\end{tabular}
- 			\end{minipage}
- 		\end{table}
- 		 		 		
- 		Xiao and Dwayne study \cref{rat:tab:plusinfty,rat:tab:minusinfty} and decide that
- 		the functions $R$ and $S$ never actually touch their horizontal asymptotes, but they
- 		do get infinitely close. They also feel as if they have a better understanding of
- 		what it means to study the behavior of a function as $x\rightarrow\pm\infty$.
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}[Repeated factors in the numerator]
- 		Consider the functions $f$, $g$, and $h$ that have formulas
- 		\[
- 			f(x)=\frac{(x-2)^2}{(x-3)(x+1)}, \qquad g(x)=\frac{x-2}{(x-3)(x+1)}, \qquad h(x)=\frac{(x-2)^3}{(x-3)(x+1)}
- 		\]
- 		which are graphed in \cref{rat:fig:repfactn}. We note that each function has vertical
- 		asymptotes at $-1$ and $3$, and so the domain of each function is
- 		\[
- 			(-\infty,-1)\cup(-1,3)\cup(3,\infty)
- 		\]
- 		We also notice that the numerators of each function are quite similar| indeed, each
- 		function has a zero at $2$, but how does each function behave around their zero?
- 		 		 		
- 		Using \cref{rat:fig:repfactn} to guide us, we note that
- 		\begin{itemize}
- 			\item $f$ has a horizontal intercept $(2,0)$, but the curve of
- 			      $f$ does not cut the horizontal axis| it bounces off it;
- 			\item $g$ also has a horizontal intercept $(2,0)$, and the curve
- 			      of $g$ \emph{does} cut the horizontal axis;
- 			\item $h$ has a horizontal intercept $(2,0)$, and the curve of $h$
- 			      also cuts the axis, but appears flattened as it does so.
- 		\end{itemize}
- 		 		 		
- 		We can further enrich our study by discussing the long-run behavior of each function.
- 		Using the tools of \cref{rat:def:longrun}, we can deduce that
- 		\begin{itemize}
- 			\item $f$ has a horizontal asymptote with equation $y=1$;
- 			\item $g$ has a horizontal asymptote with equation $y=0$;
- 			\item $h$ does \emph{not} have a horizontal asymptote| it has an oblique asymptote (we'll
- 			      study this more in \cref{rat:sec:oblique}).
- 		\end{itemize}
- 	\end{pccexample}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)^2/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						xtick={-4,-2,...,4},
- 						ytick={-8,-4,...,8},
- 						%                    grid=both,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.248,samples=50]{f};
- 					\addplot[pccplot] expression[domain=-0.794:2.976,samples=50]{f};
- 					\addplot[pccplot] expression[domain=3.026:5,samples=50]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-1},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{(x-2)^2}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						ymin=-10,ymax=10,
- 						xtick={-4,-2,...,4},
- 						ytick={-8,-4,...,8},
- 						%                    grid=both,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.075]{f};
- 					\addplot[pccplot] expression[domain=-0.925:2.975]{f};
- 					\addplot[pccplot] expression[domain=3.025:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-6:6]({-1},{x});
- 					%                 \addplot[asymptote,domain=-6:6]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x-2}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x-2)^3/((x+1)*(x-3));}]
- 				\begin{axis}[
- 						%                    framed,
- 						xmin=-5,xmax=5,
- 						xtick={-8,-6,...,8},
- 						%                    grid=both,
- 						ymin=-30,ymax=30,
- 						width=\figurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-1.27]{f};
- 					\addplot[pccplot] expression[domain=-0.806:2.99185]{f};
- 					\addplot[pccplot] expression[domain=3.0085:5]{f};
- 					\addplot[soldot] coordinates{(2,0)};
- 					%                 \addplot[asymptote,domain=-30:30]({-1},{x});
- 					%                 \addplot[asymptote,domain=-30:30]({3},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{(x-2)^3}{(x+1)(x-3)}$}
- 			\label{rat:fig:repfactn3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:repfactn}
- 	\end{figure}
- 	 	 	
- \subsection*{Holes}
- 	Rational functions have a vertical asymptote at $a$ if the denominator is $0$ at $a$.
- 	What happens if the numerator is $0$ at the same place? In this case, we say that the rational
- 	function has a \emph{hole} at $a$.
- 	\begin{pccdefinition}[Holes]
- 		The rational function
- 		\[
- 			r(x)=\frac{p(x)}{q(x)}
- 		\]
- 		has a hole at $a$ if $p(a)=q(a)=0$. Note that holes are different from
- 		a vertical asymptotes. We represent that $r$ has a hole at the point
- 		$(a,r(a))$ on the curve $y=r(x)$ by
- 		using a hollow circle, $\circ$.
- 	\end{pccdefinition}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}
- 		\pccname{Mohammed} and \pccname{Sue} have graphed the function $r$ that has formula
- 		\[
- 			r(x)=\frac{x^2+x-6}{(x-2)}
- 		\]
- 		in their calculators, and can not decide if the correct graph
- 		is \cref{rat:fig:hole} or \cref{rat:fig:hole1}.
- 		 		 		
- 		Luckily for them, Oscar is nearby, and can help them settle the debate.
- 		Oscar demonstrates that
- 		\begin{align*}
- 			r(x) & =\frac{(x+3)(x-2)}{(x-2)} \\
- 			     & = x+3                     
- 		\end{align*}
- 		but only when $x\ne 2$, because the function is undefined at $2$. Oscar
- 		says that this necessarily means that the domain or $r$ is
- 		\[
- 			(-\infty,2)\cup(2,\infty)
- 		\]
- 		and that $r$ must have a hole at $2$.
- 		 		 		
- 		Mohammed and Sue are very grateful for the clarification, and conclude that
- 		the graph of $r$ is shown in \cref{rat:fig:hole1}.
- 		\begin{figure}[!htb]
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-4,...,8},
- 							ytick={-8,-4,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot expression[domain=-10:7]{x+3};
- 						\addplot[soldot] coordinates{(-3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:hole}
- 			\end{minipage}%
- 			\hfill
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-4,...,8},
- 							ytick={-8,-4,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot expression[domain=-10:7]{x+3};
- 						\addplot[holdot] coordinates{(2,5)};
- 						\addplot[soldot] coordinates{(-3,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:hole1}
- 			\end{minipage}%
- 		\end{figure}
- 	\end{pccexample}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{pccexample}
- 		Consider the function $f$ that has formula
- 		\[
- 			f(x)=\frac{x(x+3)}{x^2-4x}
- 		\]
- 		The domain of $f$ is $(-\infty,0)\cup(0,4)\cup(4,\infty)$ because both $0$ and $4$
- 		make the denominator equal to $0$. Notice that
- 		\begin{align*}
- 			f(x) & = \frac{x(x+3)}{x(x-4)} \\
- 			     & = \frac{x+3}{x-4}       
- 		\end{align*}
- 		provided that $x\ne 0$. Since $0$ makes the numerator
- 		and the denominator 0 at the same time, we say that $f$ has a hole at $(0,-\nicefrac{3}{4})$.
- 		Note that this necessarily means that $f$ does not have a vertical intercept.
- 		 		 		
- 		We also note $f$ has a vertical asymptote at $4$; the function is graphed in \cref{rat:fig:holeex}.
- 		\begin{figure}[!htb]
- 			\centering
- 			\begin{tikzpicture}[/pgf/declare function={f=(x+3)/(x-4);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=both,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.36364,samples=50]{f};
- 					\addplot[pccplot] expression[domain=4.77:10]{f};
- 					\addplot[asymptote,domain=-10:10]({4},{x});
- 					\addplot[holdot]coordinates{(0,-0.75)};
- 					\addplot[soldot] coordinates{(-3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{$y=\dfrac{x(x+3)}{x^2-4x}$}
- 			\label{rat:fig:holeex}
- 		\end{figure}
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{pccexample}[Minimums and maximums]
- 		\pccname{Seamus} and \pccname{Trang} are discussing rational functions. Seamus says that
- 		if a rational function has a vertical asymptote, then it can
- 		not possibly have local minimums and maximums, nor can it have
- 		global minimums and maximums.
- 		 		 		
- 		Trang says this statement is not always true. She plots the functions
- 		$f$ and $g$ that have formulas
- 		\[
- 			f(x)=-\frac{32(x-1)(x+1)}{(x-2)^2(x+2)^2}, \qquad g(x)=\frac{32(x-1)(x+1)}{(x-2)^2(x+2)^2}
- 		\]
- 		in \cref{rat:fig:minmax1,rat:fig:minmax2} and shows them to Seamus. On seeing the graphs,
- 		Seamus quickly corrects himself, and says  that $f$ has a local (and global)
- 		maximum of $2$ at $0$, and that $g$ has a local (and global) minimum of $-2$ at $0$.
- 		 		 		
- 		\begin{figure}[!htb]
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=-32*(x-1)*(x+1)/(( x-2)^2*(x+2)^2);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-6,...,8},
- 							ytick={-8,-6,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-3.01]{f};
- 						\addplot[pccplot] expression[domain=-1.45:1.45]{f};
- 						\addplot[pccplot] expression[domain=3.01:10]{f};
- 						\addplot[soldot] coordinates{(-1,0)(1,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=f(x)$}
- 				\label{rat:fig:minmax1}
- 			\end{minipage}%
- 			\hfill
- 			\begin{minipage}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=32*(x-1)*(x+1)/(( x-2)^2*(x+2)^2);}]
- 					\begin{axis}[
- 							framed,
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-8,-6,...,8},
- 							ytick={-8,-6,...,8},
- 							grid=both,
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-3.01]{f};
- 						\addplot[pccplot] expression[domain=-1.45:1.45]{f};
- 						\addplot[pccplot] expression[domain=3.01:10]{f};
- 						\addplot[soldot] coordinates{(-1,0)(1,0)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{$y=g(x)$}
- 				\label{rat:fig:minmax2}
- 			\end{minipage}%
- 		\end{figure}
- 		 		 		
- 		Seamus also notes that (in its domain) the function $f$ is always concave down, and
- 		that (in its domain) the function $g$ is always concave up. Furthermore, Trang
- 		observes that each function behaves like $\frac{1}{x^2}$ around each of its vertical
- 		asymptotes, because each linear factor in the denominator is raised to the power $2$.
- 		 		 		
- 		\pccname{Oscar} stops by and reminds both students about the long-run behavior; according
- 		to \cref{rat:def:longrun} since the degree of the denominator is greater than the
- 		degree of the numerator (in both functions), each function has a horizontal asymptote
- 		at $y=0$.
- 	\end{pccexample}
- 	 	 	
- 	 	 	
- 	\investigation*{}
- 	%===================================
- 	%   Author: Pettit/Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[The spaghetti incident]
- 	The same Queen from \vref{exp:prob:queenschessboard} has recovered from
- 	the rice experiments, and has called her loyal jester for another challenge.
- 	 	 	
- 	The jester has an $11-$inch piece of uncooked spaghetti that he puts on a table;
- 	he uses a book to cover $\unit[1]{inch}$ of it so that
- 	$\unit[10]{inches}$ hang over the edge. The jester then produces a box of $\unit{mg}$
- 	weights that can be hung from the spaghetti.
- 	 	 	
- 	The jester says it will take $\unit[y]{mg}$ to break the spaghetti when hung
- 	$\unit[x]{inches}$ from the edge, according to the rule $y=\frac{100}{x}$.
- 	\begin{margintable}
- 		\centering
- 		\captionof{table}{}
- 		\label{rat:tab:spaghetti}
- 		\begin{tabular}{cc}
- 			\beforeheading
- 			\heading{$x$} & \heading{$y$} \\
- 			\afterheading
- 			$1$           &               \\\normalline
- 			$2$           &               \\\normalline
- 			$3$           &               \\\normalline
- 			$4$           &               \\\normalline
- 			$5$           &               \\\normalline
- 			$6$           &               \\\normalline
- 			$7$           &               \\\normalline
- 			$8$           &               \\\normalline
- 			$9$           &               \\\normalline
- 			$10$          &               \\\lastline
- 		\end{tabular}
- 	\end{margintable}
- 	\begin{subproblem}\label{rat:prob:spaggt1}
- 		Help the Queen complete \cref{rat:tab:spaghetti}, and use $2$ digits after the decimal
- 		where appropriate.
- 		\begin{shortsolution}
- 			\begin{tabular}[t]{ld{2}}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				$1$           & 100           \\\normalline
- 				$2$           & 50            \\\normalline
- 				$3$           & 33.33         \\\normalline
- 				$4$           & 25            \\\normalline
- 				$5$           & 20            \\\normalline
- 				$6$           & 16.67         \\\normalline
- 				$7$           & 14.29         \\\normalline
- 				$8$           & 12.50         \\\normalline
- 				$9$           & 11.11         \\\normalline
- 				$10$          & 10            \\\lastline
- 			\end{tabular}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What do you notice about the number of $\unit{mg}$ that it takes to break
- 		the spaghetti as $x$ increases?
- 		\begin{shortsolution}
- 			It seems that the number of $\unit{mg}$ that it takes to break the spaghetti decreases
- 			as $x$ increases.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:spaglt1}
- 		The Queen wonders what happens when $x$ gets very small| help the Queen construct
- 		a table of values for $x$ and $y$ when $x=0.0001, 0.001, 0.01, 0.1, 0.5, 1$.
- 		\begin{shortsolution}
- 			\begin{tabular}[t]{d{2}l}
- 				\beforeheading
- 				\heading{$x$} & \heading{$y$} \\
- 				\afterheading
- 				0.0001        & $1000000$     \\\normalline
- 				0.001         & $100000$      \\\normalline
- 				0.01          & $10000$       \\\normalline
- 				0.1           & $1000$        \\\normalline
- 				0.5           & $200$         \\\normalline
- 				1             & $100$         \\\lastline
- 			\end{tabular}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What do you notice about the number of $\unit{mg}$ that it takes to break the spaghetti
- 		as $x\rightarrow 0$? Would it ever make sense to let $x=0$?
- 		\begin{shortsolution}
- 			The number of $\unit{mg}$ required to break the spaghetti increases as $x\rightarrow 0$.
- 			We can not allow $x$ to be $0$, as we can not divide by $0$, and we can not
- 			be $0$ inches from the edge of the table.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Plot your results from \cref{rat:prob:spaggt1,rat:prob:spaglt1} on the same graph,
- 		and join the points using a smooth curve| set the maximum value of $y$ as $200$, and
- 		note that this necessarily means that you will not be able to plot all of the points.
- 		\begin{shortsolution}
- 			The graph of $y=\frac{100}{x}$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-2,xmax=11,
- 						ymin=-20,ymax=200,
- 						xtick={2,4,...,10},
- 						ytick={20,40,...,180},
- 						grid=major,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot+[-] expression[domain=0.5:10]{100/x};
- 					\addplot[soldot] coordinates{(0.5,200)(1,100)(2,50)(3,33.33)
- 					(4,25)(5,20)(16.67)(7,14.29)(8,12.50)(9,11.11)(10,10)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Using your graph, observe what happens to $y$ as $x$ increases. If we could somehow
- 		construct a piece of uncooked spaghetti that was $\unit[101]{inches}$ long, how many
- 		$\unit{mg}$ would it take to break the spaghetti?
- 		\begin{shortsolution}
- 			As $x$ increases, $y\rightarrow 0$. If we could construct a piece of spaghetti
- 			$\unit[101]{inches}$ long, it would only take $\unit[1]{mg}$ to break it $\left(\frac{100}{100}=1\right)$. Of course,
- 			the weight of spaghetti would probably cause it to break without the weight.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	The Queen looks forward to more food-related investigations from her jester.
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Adams (Hughes)
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Debt Amortization]
- 	To amortize a debt means to pay it off in a given length of time using
- 	equal periodic payments. The payments include interest on the unpaid
- 	balance. The following formula gives the monthly payment, $M$, in dollars
- 	that is necessary to amortize a debt of $P$ dollars in $n$ months
- 	at a monthly interest rate of $i$
- 	\[
- 		M=\frac{P\cdot i}{1-(1+i)^{-n}}
- 	\]
- 	Use this formula in each of the following problems.
- 	\begin{subproblem}
- 		What monthly payments are necessary on a credit card debt of \$2000 at
- 		$\unit[1.5]{\%}$ monthly if you want to pay off the debt in $2$ years?
- 		In one year? How much money will you save by paying off the debt in the
- 		shorter amount of time?
- 		\begin{shortsolution}
- 			Paying off the debt in $2$ years, we use
- 			\begin{align*}
- 				M & = \frac{2000\cdot 0.015}{1-(1+0.015)^{-24}} \\
- 				  & \approx 99.85                               
- 			\end{align*}
- 			The monthly payments are \$99.85.
- 			 			 			
- 			Paying off the debt in $1$ year, we use
- 			\begin{align*}
- 				M & = \frac{2000\cdot 0.015}{1-(1+0.015)^{-12}} \\
- 				  & \approx 183.36                              
- 			\end{align*}
- 			The monthly payments are \$183.36
- 			 			 			
- 			In the $2$-year model we would pay a total of $\$99.85\cdot 12=\$2396.40$. In the
- 			$1$-year model we would pay a total of $\$183.36\cdot 12=\$2200.32$. We would therefore
- 			save $\$196.08$ if we went with the $1$-year model instead of the $2$-year model.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		To purchase a home, a family needs a loan of \$300,000 at $\unit[5.2]{\%}$
- 		annual interest.  Compare a $20$ year loan to a $30$ year loan and make
- 		a recommendation for the family.
- 		(Note: when given an annual interest rate, it is a common business practice to divide by
- 		$12$ to get a monthly rate.)
- 		\begin{shortsolution}
- 			For the $20$-year loan we use
- 			\begin{align*}
- 				M & = \frac{300000\cdot \frac{0.052}{12}}{1-\left( 1+\frac{0.052}{12} \right)^{-12\cdot 20}} \\
- 				  & \approx 2013.16                                                                          
- 			\end{align*}
- 			The monthly payments are \$2013.16.
- 			 			 			
- 			For the $30$-year loan we use
- 			\begin{align*}
- 				M & = \frac{300000\cdot \frac{0.052}{12}}{1-\left( 1+\frac{0.052}{12} \right)^{-12\cdot 30}} \\
- 				  & \approx 1647.33                                                                          
- 			\end{align*}
- 			The monthly payments are \$1647.33.
- 			 			 			
- 			The total amount paid during the $20$-year loan is $\$2013.16\cdot 12\cdot 20=\$483,158.40$.
- 			The total amount paid during the $30$-year loan is $\$1647.33\cdot 12\cdot 30=\$593,038.80$.
- 			 			 			
- 			Recommendation: if you can afford the payments, choose the $20$-year loan.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		\pccname{Ellen} wants to make monthly payments of \$100 to pay off a debt of \$3000
- 		at \unit[12]{\%} annual interest. How long will it take her to pay off the
- 		debt?
- 		\begin{shortsolution}
- 			We are given $M=100$, $P=3000$, $i=0.01$, and we need to find $n$
- 			in the equation
- 			\[
- 				100 = \frac{3000\cdot 0.01}{1-(1+0.01)^{-n}}
- 			\]
- 			Using logarithms, we find that $n\approx 36$. It will take
- 			Ellen about $3$ years to pay off the debt.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		\pccname{Jake} is going to buy a new car. He puts \$2000 down and wants to finance the
- 		remaining \$14,000. The dealer will offer him \unit[4]{\%} annual interest for
- 		$5$ years, or a \$2000
- 		rebate which he can use to reduce the amount of the loan and \unit[8]{\%}
- 		annual interest for 5 years. Which should he choose?
- 		\begin{shortsolution}
- 			\begin{description}
- 				\item[Option 1:] $\unit[4]{\%}$ annual interest for $5$ years on \$14,000.
- 				This means that the monthly payments will be calculated using
- 				\begin{align*}
- 					M & = \frac{14000\cdot \frac{0.04}{12}}{1-\left( 1+\frac{0.04}{12} \right)^{-12\cdot 5}} \\
- 					  & \approx 257.83                                                                       
- 				\end{align*}
- 				The monthly payments will be $\$257.83$. The total amount paid will be
- 				$\$257.83\cdot 5\cdot 12=\$15,469.80$, of which $\$1469.80$ is interest.
- 				\item[Option 2:] $\unit[8]{\%}$ annual interest for $5$ years on \$12,000.
- 				This means that the monthly payments will be calculated using
- 				\begin{align*}
- 					M & = \frac{12000\cdot \frac{0.08}{12}}{1-\left( 1+\frac{0.08}{12} \right)^{-12\cdot 5}} \\
- 					  & \approx 243.32                                                                       
- 				\end{align*}
- 				The monthly payments will be $\$243.32$. The total amount paid
- 				will be $\$243.32\cdot 5\cdot 12 =\$14,599.20$, of which $\$2599.2$ is
- 				interest.
- 			\end{description}
- 			Jake should choose option 1 to minimize the amount of interest
- 			he has to pay.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	\begin{exercises}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Rational or not]
- 	Decide if each of the following functions are rational or not. If
- 	they are rational, state their domain.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$r(x)=\dfrac{3}{x}$
- 			\begin{shortsolution}
- 				$r$ is rational; the domain of $r$ is $(-\infty,0)\cup (0,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$s(y)=\dfrac{y}{6}$
- 			\begin{shortsolution}
- 				$s$ is not rational ($s$ is linear).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$t(z)=\dfrac{4-x}{7-8z}$
- 			\begin{shortsolution}
- 				$t$ is rational; the domain of $t$ is $\left( -\infty,\dfrac{7}{8} \right)\cup \left( \dfrac{7}{8},\infty \right)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$u(w)=\dfrac{w^2}{(w-3)(w+4)}$
- 			\begin{shortsolution}
- 				$u$ is rational; the domain of $w$ is $(-\infty,-4)\cup(-4,3)\cup(3,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$v(x)=\dfrac{4}{(x-2)^2}$
- 			\begin{shortsolution}
- 				$v$ is rational; the domain of $v$ is $(-\infty,2)\cup(2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$w(x)=\dfrac{9-x}{x+17}$
- 			\begin{shortsolution}
- 				$w$ is rational; the domain of $w$ is $(-\infty,-17)\cup(-17,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$a(x)=x^2+4$
- 			\begin{shortsolution}
- 				$a$ is not rational ($a$ is quadratic, or a polynomial of degree $2$).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$b(y)=3^y$
- 			\begin{shortsolution}
- 				$b$ is not rational ($b$ is exponential).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$c(z)=\dfrac{z^2}{z^3}$
- 			\begin{shortsolution}
- 				$c$ is rational; the domain of $c$ is $(-\infty,0)\cup (0,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$d(x)=x^2(x+3)(5x-7)$
- 			\begin{shortsolution}
- 				$d$ is not rational ($d$ is a polynomial).
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$e(\alpha)=\dfrac{\alpha^2}{\alpha^2-1}$
- 			\begin{shortsolution}
- 				$e$ is rational; the domain of $e$ is $(-\infty,-1)\cup(-1,1)\cup(1,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$f(\beta)=\dfrac{3}{4}$
- 			\begin{shortsolution}
- 				$f$ is not rational ($f$ is constant).
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Function evaluation]
- 	Let $r$ be the function that has formula
- 	\[
- 		r(x)=\frac{(x-2)(x+3)}{(x+5)(x-7)}
- 	\]
- 	Evaluate each of the following (if possible); if the value is undefined,
- 	then state so.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$r(0)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(0) & =\frac{(0-2)(0+3)}{(0+5)(0-7)} \\
- 					     & =\frac{-6}{-35}                \\
- 					     & =\frac{6}{35}                  
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(1)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(1) & =\frac{(1-2)(1+3)}{(1+5)(1-7)} \\
- 					     & =\frac{-4}{-36}                \\
- 					     & =\frac{1}{9}                   
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(2)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(2) & =\frac{(2-2)(2+3)}{(2+5)(2-7)} \\
- 					     & = \frac{0}{-50}                \\
- 					     & =0                             
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(4)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(4) & =\frac{(4-2)(4+3)}{(4+5)(4-7)} \\
- 					     & =\frac{14}{-27}                \\
- 					     & =-\frac{14}{27}                
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(7)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(7) & =\frac{(7-2)(7+3)}{(7+5)(7-7)} \\
- 					     & =\frac{50}{0}                  
- 				\end{aligned}$
- 				 				 				
- 				$r(7)$ is undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(-3)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(-3) & =\frac{(-3-2)(-3+3)}{(-3+5)(-3-7)} \\
- 					      & =\frac{0}{-20}                     \\
- 					      & =0                                 
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r(-5)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r(-5) & =\frac{(-5-2)(-5+3)}{(-5+5)(-5-7)} \\
- 					      & =\frac{14}{0}                      
- 				\end{aligned}$
- 				 				 				
- 				$r(-5)$ is undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$r\left( \frac{1}{2} \right)$
- 			\begin{shortsolution}
- 				$\begin{aligned}[t]
- 					r\left( \frac{1}{2} \right) & = \frac{\left( \frac{1}{2}-2 \right)\left( \frac{1}{2}+3 \right)}{\left( \frac{1}{2}+5 \right)\left( \frac{1}{2}-7 \right)} \\
- 					                            & =\frac{-\frac{3}{2}\cdot\frac{7}{2}}{\frac{11}{2}\left( -\frac{13}{2} \right)}                                              \\
- 					                            & =\frac{-\frac{21}{4}}{-\frac{143}{4}}                                                                                       \\
- 					                            & =\frac{37}{143}                                                                                                             
- 				\end{aligned}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   March 2012
- 	%===================================
- 	\begin{problem}[Holes or asymptotes?]
- 	State the domain of each of the following rational functions. Identify
- 	any holes or asymptotes.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x)=\dfrac{12}{x-2}$
- 			\begin{shortsolution}
- 				$f$ has a vertical asymptote at $2$; the domain of $f$ is $(-\infty,2)\cup (2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x)=\dfrac{x^2+x}{(x+1)(x-2)}$
- 			\begin{shortsolution}
- 				$g$ has a vertical asymptote at $2$, and a hole at $-1$; the domain of $g$ is $(-\infty,-1)\cup(-1,2)\cup(2,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x)=\dfrac{x^2+5x+4}{x^2+x-12}$
- 			\begin{shortsolution}
- 				$h$ has a vertical asymptote at $3$, and a whole at $-4$; the domain of $h$ is $(-\infty,-4)\cup(-4,3)\cup(3,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(z)=\dfrac{z+2}{2z-3}$
- 			\begin{shortsolution}
- 				$k$ has a vertical asymptote at $\dfrac{3}{2}$; the domain of $k$ is $\left( -\infty,\dfrac{3}{2} \right)\cup\left( \dfrac{3}{2},\infty \right)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(w)=\dfrac{w}{w^2+1}$
- 			\begin{shortsolution}
- 				$l$ does not have any vertical asymptotes nor holes; the domain of $w$ is $(-\infty,\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(t)=\dfrac{14}{13-t^2}$
- 			\begin{shortsolution}
- 				$m$ has vertical asymptotes at $\pm\sqrt{13}$; the domain of $m$ is $(-\infty,\sqrt{13})\cup(-\sqrt{13},\sqrt{13})\cup(\sqrt{13},\infty)$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a graph]
- 	Consider the rational functions graphed in \cref{rat:fig:findformula}. Find
- 	the vertical asymptotes for each function, together with any zeros, and
- 	give a possible formula for each.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item \Vref{rat:fig:formula1}: possible formula is $r(x)=\dfrac{1}{x+5}$
- 			\item \Vref{rat:fig:formula2}: possible formula is $r(x)=\dfrac{(x+3)}{(x-5)}$
- 			\item \Vref{rat:fig:formula3}: possible formula is $r(x)=\dfrac{1}{(x-4)(x+3)}$.
- 		\end{itemize}
- 	\end{shortsolution}
- 	\end{problem}
- 	 	 	
- 	\begin{figure}[!htb]
- 		\begin{widepage}
- 		\setlength{\figurewidth}{0.3\textwidth}
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=1/(x+4);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-6,ymax=6,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-4.16667,samples=50]{f};
- 					\addplot[pccplot] expression[domain=-3.83333:10,samples=50]{f};
- 					\addplot[asymptote,domain=-6:6]({-4},{x});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula1}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=(x+3)/(x-5);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-6,ymax=6,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.85714]{f};
- 					\addplot[pccplot] expression[domain=6.6:10]{f};
- 					\addplot[soldot] coordinates{(-3,0)};
- 					\addplot[asymptote,domain=-6:6]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{1});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula2}
- 		\end{subfigure}
- 		\hfill
- 		\begin{subfigure}{\figurewidth}
- 			\begin{tikzpicture}[/pgf/declare function={f=1/((x-4)*(x+3));}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-3,ymax=3,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-4,-3,...,4},
- 						grid=both,
- 						width=\textwidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-3.0473]{f};
- 					\addplot[pccplot] expression[domain=-2.95205:3.95205]{f};
- 					\addplot[pccplot] expression[domain=4.0473:10]{f};
- 					\addplot[asymptote,domain=-3:3]({-3},{x});
- 					\addplot[asymptote,domain=-3:3]({4},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 			\caption{}
- 			\label{rat:fig:formula3}
- 		\end{subfigure}
- 		\caption{}
- 		\label{rat:fig:findformula}
- 		\end{widepage}
- 	\end{figure}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a formula of a rational
- 	function that has the listed properties.
- 	\begin{subproblem}
- 		Vertical asymptote at $2$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{1}{x-2}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Vertical asymptote at $5$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{1}{x-5}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Vertical asymptote at $-2$, and zero at $6$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{x-6}{x+2}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Zeros at $2$ and $-5$ and vertical asymptotes at $1$ and $-7$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{(x-2)(x+5)}{(x-1)(x+7)}$. Note that we could multiply the
- 			numerator or denominator by any real number and still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Given formula, find horizontal asymptotes]
- 	Each of the following functions has a horizontal asymptote. Write the equation
- 	of the horizontal asymptote for each function.
- 	\begin{multicols}{3}
- 		\begin{subproblem}
- 			$f(x) = \dfrac{1}{x}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$g(x) = \dfrac{2x+3}{x}$
- 			\begin{shortsolution}
- 				$y=2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$h(x) = \dfrac{x^2+2x}{x^2+3}$
- 			\begin{shortsolution}
- 				$y=1$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$k(x) = \dfrac{x^2+7}{x}$
- 			\begin{shortsolution}
- 				$y=1$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$l(x)=\dfrac{3x-2}{5x+8}$
- 			\begin{shortsolution}
- 				$y=\dfrac{3}{5}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$m(x)=\dfrac{3x-2}{5x^2+8}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$n(x)=\dfrac{(6x+1)(x-7)}{(11x-8)(x-5)}$
- 			\begin{shortsolution}
- 				$y=\dfrac{6}{11}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$p(x)=\dfrac{19x^3}{5-x^4}$
- 			\begin{shortsolution}
- 				$y=0$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$q(x)=\dfrac{14x^2+x}{1-7x^2}$
- 			\begin{shortsolution}
- 				$y=-2$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2012
- 	%===================================
- 	\begin{problem}[Given horizontal asymptotes, find formula]
- 	In each of the following problems, give a formula for a function that
- 	has the given horizontal asymptote. Note that there may be more than one option.
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$y=7$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{7(x-2)}{x+1}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $7$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-1$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{5-x^2}{x^2+10}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $10$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=53$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{53x^3}{x^3+4x^2-7}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $53$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-17$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{34(x+2)}{7-2x}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $-17$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=\dfrac{3}{2}$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{3x+4}{2(x+1)}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $\dfrac{3}{2}$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=0$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{4}{x}$. Note that there
- 				are other options, provided that the degree of the numerator is less than the degree
- 				of the denominator.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=-1$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{10x}{5-10x}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $-1$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$y=2$
- 			\begin{shortsolution}
- 				Possible option: $f(x)=\dfrac{8x-3}{4x+1}$. Note that there
- 				are other options, provided that the degree of the numerator is the same as the degree
- 				of the denominator, and that the ratio of the leading
- 				coefficients is $2$.
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a description]
- 	In each of the following problems, give a formula for a function that
- 	has the prescribed properties. Note that there may be more than one option.
- 	\begin{subproblem}
- 		$f(x)\rightarrow 3$ as $x\rightarrow\pm\infty$.
- 		\begin{shortsolution}
- 			Possible option: $f(x)=\dfrac{3(x-2)}{x+7}$. Note that
- 			the zero and asymptote of $f$ could be changed, and $f$ would still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$r(x)\rightarrow -4$ as $x\rightarrow\pm\infty$.
- 		\begin{shortsolution}
- 			Possible option: $r(x)=\dfrac{-4(x-2)}{x+7}$. Note that
- 			the zero and asymptote of $r$ could be changed, and $r$ would still have the desired properties.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$k(x)\rightarrow 2$ as $x\rightarrow\pm\infty$, and $k$ has vertical asymptotes at $-3$ and $5$.
- 		\begin{shortsolution}
- 			Possible option: $k(x)=\dfrac{2x^2}{(x+3)(x-5)}$. Note that the denominator
- 			must have the given factors; the numerator could be any degree $2$ polynomial, provided the
- 			leading coefficient is $2$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   Feb 2011
- 	%===================================
- 	\begin{problem}
- 	Let $r$ be the rational function that has
- 	\[
- 		r(x) = \frac{(x+2)(x-1)}{(x+3)(x-4)}
- 	\]
- 	Each of the following questions are in relation to this function.
- 	\begin{subproblem}
- 		What is the vertical intercept of this function? State your answer as an
- 		ordered pair. \index{rational functions!vertical intercept}
- 		\begin{shortsolution}
- 			$\left(0,\frac{1}{6}\right)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:rational}
- 		What values of $x$ make the denominator equal to $0$?
- 		\begin{shortsolution}
- 			$-3,4$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use your answer to \cref{rat:prob:rational} to write the domain of the function in
- 		both interval, and set builder notation. %\index{rational functions!domain}\index{domain!rational functions}
- 		\begin{shortsolution}
- 			Interval notation: $(-\infty,-3)\cup (-3,4)\cup (4,\infty)$.
- 			Set builder: $\{x|x\ne -3, \mathrm{and}\, x\ne 4\}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		What are the vertical asymptotes of the function? State your answers in
- 		the form $x=$
- 		\begin{shortsolution}
- 			$x=-3$ and $x=4$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}\label{rat:prob:zeroes}
- 		What values of $x$ make the numerator equal to $0$?
- 		\begin{shortsolution}
- 			$-2,1$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Use your answer to \cref{rat:prob:zeroes} to write the horizontal intercepts of
- 		$r$ as ordered pairs.
- 		\begin{shortsolution}
- 			$(-2,0)$ and $(1,0)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Holes]
- 	\pccname{Josh} and \pccname{Pedro} are discussing the function
- 	\[
- 		r(x)=\frac{x^2-1}{(x+3)(x-1)}
- 	\]
- 	\begin{subproblem}
- 		What is the domain of $r$?
- 		\begin{shortsolution}
- 			The domain of $r$ is $(-\infty,-3)\cup(-3,1)\cup(1,\infty)$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Josh notices that the numerator can be factored- can you see how?
- 		\begin{shortsolution}
- 			$(x^2-1)=(x-1)(x+1)$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Pedro asks, `Doesn't that just mean that
- 		\[
- 			r(x)=\frac{x+1}{x+3}
- 		\]
- 		for all values of $x$?' Josh says, `Nearly\ldots but not for all values of $x$'.
- 		What does Josh mean?
- 		\begin{shortsolution}
- 			$r(x)=\dfrac{x+1}{x+3}$ provided that $x\ne -1$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Where does $r$ have vertical asymptotes, and where does it have holes?
- 		\begin{shortsolution}
- 			The function $r$ has a vertical asymptote at $-3$, and a hole at $1$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Sketch a graph of $r$.
- 		\begin{shortsolution}
- 			A graph of $r$ is shown below.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-10,ymax=10,
- 						xtick={-8,-6,...,8},
- 						ytick={-8,-6,...,8},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-3.25]{(x+1)/(x+3)};
- 					\addplot[pccplot] expression[domain=-2.75:10]{(x+1)/(x+3)};
- 					\addplot[asymptote,domain=-10:10]({-3},{x});
- 					\addplot[holdot]coordinates{(1,0.5)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Function algebra]
- 	Let $r$ and $s$ be the rational functions that have formulas
- 	\[
- 		r(x)=\frac{2-x}{x+3}, \qquad s(x)=\frac{x^2}{x-4}
- 	\]
- 	Evaluate each of the following (if possible).
- 	\begin{multicols}{4}
- 		\begin{subproblem}
- 			$(r+s)(5)$
- 			\begin{shortsolution}
- 				$\frac{197}{8}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(r-s)(3)$
- 			\begin{shortsolution}
- 				$\frac{53}{6}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$(r\cdot s)(4)$
- 			\begin{shortsolution}
- 				Undefined.
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			$\left( \frac{r}{s} \right)(1)$
- 			\begin{shortsolution}
- 				$-\frac{3}{4}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   July 2012
- 	%===================================
- 	\begin{problem}[Transformations: given the transformation, find the formula]
- 	Let $r$ be the rational function that has formula.
- 	\[
- 		r(x)=\frac{x+5}{2x-3}
- 	\]
- 	In each of the following problems apply the given transformation to the function $r$ and
- 	write a formula for the transformed version of $r$.
- 	\begin{multicols}{2}
- 		\begin{subproblem}
- 			Shift $r$ to the right by $3$ units.
- 			\begin{shortsolution}
- 				$r(x-3)=\frac{x+2}{2x-9}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ to the left by $4$ units.
- 			\begin{shortsolution}
- 				$r(x+4)=\frac{x+9}{2x+5}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ up by $\pi$ units.
- 			\begin{shortsolution}
- 				$r(x)+\pi=\frac{x+5}{2x-3}+\pi$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Shift $r$ down by $17$ units.
- 			\begin{shortsolution}
- 				$r(x)-17=\frac{x+5}{2x-3}-17$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $r$ over the horizontal axis.
- 			\begin{shortsolution}
- 				$-r(x)=-\frac{x+5}{2x-3}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 		\begin{subproblem}
- 			Reflect $r$ over the vertical axis.
- 			\begin{shortsolution}
- 				$r(-x)=\frac{x-5}{2x+3}$
- 			\end{shortsolution}
- 		\end{subproblem}
- 	\end{multicols}
- 	\end{problem}
- 	 	 	
- 	 	 	
- 	%===================================
- 	%   Author: Hughes
- 	%   Date:   May 2011
- 	%===================================
- 	\begin{problem}[Find a formula from a table]\label{rat:prob:findformula}
- 	\Crefrange{rat:tab:findformular}{rat:tab:findformulau} show values of rational functions $r$, $q$, $s$,
- 	and $t$. Assume that any values marked with an X are undefined.
- 	 	 	
- 	\begin{table}[!htb]
- 		\begin{widepage}
- 		\centering
- 		\caption{Tables for \cref{rat:prob:findformula}}
- 		\label{rat:tab:findformula}
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=r(x)$}
- 			\label{rat:tab:findformular}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                \\ \afterheading
- 				$-4$ & $\nicefrac{7}{2}$  \\\normalline
- 				$-3$ & $-18$              \\\normalline
- 				$-2$ & X                  \\\normalline
- 				$-1$ & $-4$               \\\normalline
- 				$0$  & $\nicefrac{-3}{2}$ \\\normalline
- 				$1$  & $\nicefrac{-2}{3}$ \\\normalline
- 				$2$  & $\nicefrac{-1}{4}$ \\\normalline
- 				$3$  & $0$                \\\normalline
- 				$4$  & $\nicefrac{1}{6}$  \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=s(x)$}
- 			\label{rat:tab:findformulas}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                 \\ \afterheading
- 				$-4$ & $\nicefrac{-2}{21}$ \\\normalline
- 				$-3$ & $\nicefrac{-1}{12}$ \\\normalline
- 				$-2$ & $0$                 \\\normalline
- 				$-1$ & X                   \\\normalline
- 				$0$  & $\nicefrac{-2}{3}$  \\\normalline
- 				$1$  & $\nicefrac{-3}{4}$  \\\normalline
- 				$2$  & $\nicefrac{-4}{3}$  \\\normalline
- 				$3$  & X                   \\\normalline
- 				$4$  & $\nicefrac{6}{5}$   \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=t(x)$}
- 			\label{rat:tab:findformulat}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$               \\ \afterheading
- 				$-4$ & $\nicefrac{3}{5}$ \\\normalline
- 				$-3$ & $0$               \\\normalline
- 				$-2$ & X                 \\\normalline
- 				$-1$ & $3$               \\\normalline
- 				$0$  & $3$               \\\normalline
- 				$1$  & X                 \\\normalline
- 				$2$  & $0$               \\\normalline
- 				$3$  & $\nicefrac{3}{5}$ \\\normalline
- 				$4$  & $\nicefrac{7}{9}$ \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\hfill
- 		\begin{subtable}{.2\textwidth}
- 			\centering
- 			\caption{$y=u(x)$}
- 			\label{rat:tab:findformulau}
- 			\begin{tabular}{rr}
- 				\beforeheading
- 				$x$  & $y$                \\ \afterheading
- 				$-4$ & $\nicefrac{16}{7}$ \\\normalline
- 				$-3$ & X                  \\\normalline
- 				$-2$ & $-\nicefrac{4}{5}$ \\\normalline
- 				$-1$ & $-\nicefrac{1}{8}$ \\\normalline
- 				$0$  & $0$                \\\normalline
- 				$1$  & $-\nicefrac{1}{8}$ \\\normalline
- 				$2$  & $-\nicefrac{4}{5}$ \\\normalline
- 				$3$  & X                  \\\normalline
- 				$4$  & $\nicefrac{16}{7}$ \\\lastline
- 			\end{tabular}
- 		\end{subtable}
- 		\end{widepage}
- 	\end{table}
- 	\begin{subproblem}
- 		Given that the formula for $r(x)$ has the form $r(x)=\dfrac{x-A}{x-B}$, use \cref{rat:tab:findformular}
- 		to find values of $A$ and $B$.
- 		\begin{shortsolution}
- 			$A=3$ and $B=-2$, so $r(x)=\dfrac{x-3}{x+2}$.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Check your formula by computing $r(x)$ at the values specified in the table.
- 		\begin{shortsolution}
- 			$\begin{aligned}[t]
- 				r(-4) & = \frac{-4-3}{-4+2} \\
- 				      & = \frac{7}{2}       \\
- 			\end{aligned}$
- 			 			 			
- 			$r(-3)=\ldots$ etc
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		The function $s$ in \cref{rat:tab:findformulas} has two vertical asymptotes and one zero.
- 		Can you find a formula for $s(x)$?
- 		\begin{shortsolution}
- 			$s(x)=\dfrac{x+2}{(x-3)(x+1)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Check your formula by computing $s(x)$ at the values specified in the table.
- 		\begin{shortsolution}
- 			$\begin{aligned}[t]
- 				s(-4) & =\frac{-4+2}{(-4-3)(-4+1)} \\
- 				      & =-\frac{2}{21}             
- 			\end{aligned}$
- 			 			 			
- 			$s(-3)=\ldots$ etc
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that the formula for $t(x)$ has  the form $t(x)=\dfrac{(x-A)(x-B)}{(x-C)(x-D)}$, use \cref{rat:tab:findformulat} to find the
- 		values of $A$, $B$, $C$, and $D$; hence write a formula for $t(x)$.
- 		\begin{shortsolution}
- 			$t(x)=\dfrac{(x+3)(x-2)}{(x+2)(x+1)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		Given that the formula for $u(x)$ has  the form $u(x)=\dfrac{(x-A)^2}{(x-B)(x-C)}$, use \cref{rat:tab:findformulau} to find the
- 		values of $A$, $B$, and $C$; hence write a formula for $u(x)$.
- 		\begin{shortsolution}
- 			$u(x)=\dfrac{x^2}{(x+3)(x-3)}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\end{problem}
- 	\end{exercises}
- 	 	 	
-\section{Graphing rational functions (horizontal asymptotes)}
- \reformatstepslist{R} % the steps list should be R1, R2, \ldots
- We studied rational functions in the previous section, but were
- not asked to graph them; in this section we will demonstrate the
- steps to be followed in order to sketch graphs of the functions.
-   
- Remember from \vref{rat:def:function} that rational functions have
- the form
- \[
- 	r(x)=\frac{p(x)}{q(x)}
- \]
- In this section we will restrict attention to the case when
- \[
- 	\text{degree of }p\leq \text{degree of }q
- \]
- Note that this necessarily means that each function that we consider
- in this section \emph{will have a horizontal asymptote} (see \vref{rat:def:longrun}).
- The cases in which the degree of $p$ is greater than the degree of $q$
- is covered in the next section.
-   
- Before we begin, it is important to remember the following:
- \begin{itemize}
- 	\item Our sketches will give a good representation of the overall
- 	      shape of the graph, but until we have the tools of calculus (from MTH 251)
- 	      we can not find local minimums, local maximums, and inflection points algebraically. This
- 	      means that we will make our best guess as to where these points are.
- 	\item We will not concern ourselves too much with the vertical scale (because of
- 	      our previous point)| we will, however, mark the vertical intercept (assuming there is one),
- 	      and any horizontal asymptotes.
- \end{itemize}
- \begin{pccspecialcomment}[Steps to follow when sketching rational functions]\label{rat:def:stepsforsketch}
- 	\begin{steps}
- 		\item \label{rat:step:first} Find all vertical asymptotes and holes, and mark them on the
- 		graph using dashed vertical lines and open circles $\circ$ respectively.
- 		\item Find any intercepts, and mark them using solid circles $\bullet$;
- 		determine if the curve cuts the axis, or bounces off it at each zero.
- 		\item Determine the behavior of the function around each asymptote| does
- 		it behave like $\frac{1}{x}$ or $\frac{1}{x^2}$?
- 		\item \label{rat:step:penultimate} Determine the long-run behavior of the function, and mark the horizontal
- 		asymptote using a dashed horizontal line.
- 		\item \label{rat:step:last}  Deduce the overall shape of the curve, and sketch it. If there isn't
- 		enough information from the previous steps, then construct a table of values
- 		including sample points from each branch.
- 	\end{steps}
- 	Remember that until we have the tools of calculus, we won't be able to
- 	find the exact coordinates of local minimums, local maximums, and points
- 	of inflection.
- \end{pccspecialcomment}
-   
- The examples that follow show how \crefrange{rat:step:first}{rat:step:last} can be
- applied to a variety of different rational functions.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:1overxminus2p2}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $r$
- 	that has formula
- 	\[
- 		r(x)=\frac{1}{x-2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $r$ has a vertical asymptote at $2$; $r$ does not have any holes. The curve of
- 			$r$ will have $2$ branches.
- 			\item $r$ does not have any zeros since the numerator is never equal to $0$. The
- 			vertical intercept of $r$ is $\left( 0,-\frac{1}{2} \right)$.
- 			\item $r$ behaves like $\frac{1}{x}$ around its vertical asymptote since $(x-2)$
- 			is raised to the power $1$.
- 			\item Since the degree of the numerator is less than the degree of the denominator,
- 			according to \vref{rat:def:longrun} the horizontal asymptote of $r$ has equation $y=0$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:1overxminus2p1}. Notice
- 			that there is only one way to complete the graph, which we have done in \cref{rat:fig:1overxminus2p2}.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-5,xmax=5,
- 					ymin=-5,ymax=5,
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-5:5]({2},{x});
- 				\addplot[asymptote,domain=-5:5]({x},{0});
- 				\addplot[soldot] coordinates{(0,-0.5)}node[axisnode,anchor=north east]{$\left( 0,-\frac{1}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:1overxminus2p1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=1/(x-2);}]
- 			\begin{axis}[
- 					xmin=-5,xmax=5,
- 					ymin=-5,ymax=5,
- 					width=\textwidth,
- 				]
- 				\addplot[pccplot] expression[domain=-5:1.8,samples=50]{f};
- 				\addplot[pccplot] expression[domain=2.2:5]{f};
- 				\addplot[asymptote,domain=-5:5]({2},{x});
- 				\addplot[asymptote,domain=-5:5]({x},{0});
- 				\addplot[soldot] coordinates{(0,-0.5)}node[axisnode,anchor=north east]{$\left( 0,-\frac{1}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:1overxminus2p2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{1}{x-2}$}
- \end{figure}
-   
- The function $r$ in \cref{rat:ex:1overxminus2p2} has a horizontal asymptote which has equation $y=0$.
- This asymptote lies on the horizontal axis, and you might (understandably) find it hard
- to distinguish between the two lines (\cref{rat:fig:1overxminus2p2}). When faced
- with such a situation, it is perfectly acceptable to draw the horizontal axis
- as a dashed line| just make sure to label it correctly. We will demonstrate this
- in the next example.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:1overxp1}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $v$
- 	that has formula
- 	\[
- 		v(x)=\frac{10}{x}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $v$ has a vertical asymptote at $0$. $v$ does not have
- 			any holes. The curve of $v$ will have $2$ branches.
- 			\item $v$ does not have any zeros (since $10\ne 0$). Furthermore, $v$
- 			does not have a vertical intercept since $v(0)$ is undefined.
- 			\item $v$ behaves like $\frac{1}{x}$ around its vertical asymptote.
- 			\item $v$ has a horizontal asymptote with equation $y=0$.
- 			\item We put the details we have obtained so far in \cref{rat:fig:1overxp1}.
- 			We do not have enough information to sketch $v$ yet (because $v$ does
- 			not have any intercepts), so let's pick a sample
- 			point in either of the $2$ branches| it doesn't matter where our sample point
- 			is, because we know what the overall shape will be. Let's compute $v(2)$
- 			\begin{align*}
- 				v(2) & =\dfrac{10}{2} \\
- 				     & = 5            
- 			\end{align*}
- 			We therefore mark the point $(2,5)$ on \cref{rat:fig:1overxp2}, and then complete the sketch using
- 			the details we found in the previous steps.
- 		\end{steps}
- 		 		 		
- 		\begin{figure}[!htbp]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-5,5},
- 							ytick={-5,5},
- 							axis line style={color=white},
- 							width=\textwidth,
- 						]
- 						\addplot[asymptote,<->,domain=-10:10]({0},{x});
- 						\addplot[asymptote,<->,domain=-10:10]({x},{0});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:1overxp1}
- 			\end{subfigure}%
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=10/x;}]
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-10,ymax=10,
- 							xtick={-5,5},
- 							ytick={-5,5},
- 							axis line style={color=white},
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:-1]{f};
- 						\addplot[pccplot] expression[domain=1:10]{f};
- 						\addplot[soldot] coordinates{(2,5)}node[axisnode,anchor=south west]{$(2,5)$};
- 						\addplot[asymptote,<->,domain=-10:10]({0},{x});
- 						\addplot[asymptote,<->,domain=-10:10]({x},{0});
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:1overxp2}
- 			\end{subfigure}%
- 			\caption{$y=\dfrac{10}{x}$}
- 		\end{figure}
- 	\end{pccsolution}
- \end{pccexample}
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:asympandholep1}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $u$
- 	that has formula
- 	\[
- 		u(x)=\frac{-4(x^2-9)}{x^2-8x+15}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item We begin by factoring both the numerator and denominator of $u$ to help
- 			us find any vertical asymptotes or holes
- 			\begin{align*}
- 				u(x) & =\frac{-4(x^2-9)}{x^2-8x+15}     \\
- 				     & =\frac{-4(x+3)(x-3)}{(x-5)(x-3)} \\
- 				     & =\frac{-4(x+3)}{x-5}             
- 			\end{align*}
- 			provided that $x\ne 3$. Therefore $u$ has a vertical asymptote at $5$ and
- 			a hole at $3$. The curve of $u$ has $2$ branches.
- 			\item $u$ has a simple zero at $-3$. The vertical intercept of $u$ is $\left( 0,\frac{12}{5} \right)$.
- 			\item $u$ behaves like $\frac{1}{x}$ around its vertical asymptote at $4$.
- 			\item Using \vref{rat:def:longrun} the equation of the horizontal asymptote of $u$ is $y=-4$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:1overxminus2p1}. Notice
- 			that there is only one way to complete the graph, which we have done in \cref{rat:fig:1overxminus2p2}.
- 		\end{steps}
- 		 		 		
- 		\begin{figure}[!htbp]
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-20,ymax=20,
- 							xtick={-8,-6,...,8},
- 							ytick={-10,10},
- 							width=\textwidth,
- 						]
- 						\addplot[asymptote,domain=-20:20]({4},{x});
- 						\addplot[asymptote,domain=-10:10]({x},{-4});
- 						\addplot[soldot] coordinates{(-3,0)(0,2.4)}node[axisnode,anchor=south east]{$\left( 0,\frac{12}{5} \right)$};
- 						\addplot[holdot] coordinates{(3,12)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:asympandholep1}
- 			\end{subfigure}%
- 			\hfill
- 			\begin{subfigure}{.45\textwidth}
- 				\begin{tikzpicture}[/pgf/declare function={f=-4*(x+3)/(x-5);}]
- 					\begin{axis}[
- 							xmin=-10,xmax=10,
- 							ymin=-20,ymax=20,
- 							xtick={-8,-6,...,8},
- 							ytick={-10,10},
- 							width=\textwidth,
- 						]
- 						\addplot[pccplot] expression[domain=-10:3.6666,samples=50]{f};
- 						\addplot[pccplot] expression[domain=7:10]{f};
- 						\addplot[asymptote,domain=-20:20]({5},{x});
- 						\addplot[asymptote,domain=-10:10]({x},{-4});
- 						\addplot[soldot] coordinates{(-3,0)(0,2.4)}node[axisnode,anchor=south east]{$\left( 0,\frac{12}{5} \right)$};
- 						\addplot[holdot] coordinates{(3,12)};
- 					\end{axis}
- 				\end{tikzpicture}
- 				\caption{}
- 				\label{rat:fig:asympandholep2}
- 			\end{subfigure}%
- 			\caption{$y=\dfrac{-4(x+3)}{x-5}$}
- 		\end{figure}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \Cref{rat:ex:1overxminus2p2,rat:ex:1overxp1,rat:ex:asympandholep1} have focused on functions
- that only have one vertical asymptote; the remaining examples in this section
- concern functions that have more than one vertical asymptote. We will demonstrate
- that \crefrange{rat:step:first}{rat:step:last} still apply.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:sketchtwoasymp}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $w$
- 	that has formula
- 	\[
- 		w(x)=\frac{2(x+3)(x-5)}{(x+5)(x-4)}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $w$ has vertical asymptotes at $-5$ and $4$. $w$ does not have
- 			any holes. The curve of $w$ will have $3$ branches.
- 			\item $w$ has simple zeros at $-3$ and $5$. The vertical intercept of $w$
- 			is $\left( 0,\frac{3}{2} \right)$.
- 			\item $w$ behaves like $\frac{1}{x}$ around both of its vertical
- 			asymptotes.
- 			\item The degree of the numerator of $w$ is $2$ and the degree of the
- 			denominator of $w$ is also $2$. Using the ratio of the leading coefficients
- 			of the numerator and denominator, we say that $w$ has a horizontal
- 			asymptote with equation $y=\frac{2}{1}=2$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:sketchtwoasymptp1}.
- 			 			 			
- 			The function $w$ is a little more complicated than the functions that
- 			we have considered in the previous examples because the curve has $3$
- 			branches. When graphing such functions, it is generally a good idea to start with the branch
- 			for which you have the most information| in this case, that is the \emph{middle} branch
- 			on the interval $(-5,4)$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			(because of our observations about the behavior of $w$ around its vertical asymptotes),
- 			which we have done in \cref{rat:fig:sketchtwoasymptp2}.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[soldot] coordinates{(-3,0)(5,0)};
- 				\addplot[soldot] coordinates{(0,1.5)}node[axisnode,anchor=north west]{$\left( 0,\frac{3}{2} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:sketchtwoasymptp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=2*(x+3)*(x-5)/( (x+5)*(x-4));}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[soldot] coordinates{(-3,0)(5,0)};
- 				\addplot[soldot] coordinates{(0,1.5)}node[axisnode,anchor=north west]{$\left( 0,\frac{3}{2} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.56708]{f};
- 				\addplot[pccplot] expression[domain=-4.63511:3.81708]{f};
- 				\addplot[pccplot] expression[domain=4.13511:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:sketchtwoasymptp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{2(x+3)(x-5)}{(x+5)(x-4)}$}
- \end{figure}
-   
- The rational functions that we have considered so far have had simple
- factors in the denominator; each function has behaved like $\frac{1}{x}$
- around each of its vertical asymptotes. \Cref{rat:ex:2asympnozeros,rat:ex:2squaredasymp}
- consider functions that have a repeated factor in the denominator.
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:2asympnozeros}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $f$
- 	that has formula
- 	\[
- 		f(x)=\frac{100}{(x+5)(x-4)^2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $f$ has vertical asymptotes at $-5$ and $4$. $f$ does not have
- 			any holes. The curve of $f$ will have $3$ branches.
- 			\item $f$ does not have any zeros (since $100\ne 0$). The vertical intercept of $f$
- 			is $\left( 0,\frac{5}{4} \right)$.
- 			\item $f$ behaves like $\frac{1}{x}$ around $-5$ and behaves like $\frac{1}{x^2}$
- 			around $4$.
- 			\item The degree of the numerator of $f$ is $0$ and the degree of the
- 			denominator of $f$ is $2$. $f$ has a horizontal asymptote with
- 			equation $y=0$.
- 			\item We put the details we have obtained so far on \cref{rat:fig:2asympnozerosp1}.
- 			 			 			
- 			The function $f$ is similar to the function $w$ that we considered in \cref{rat:ex:sketchtwoasymp}|
- 			it has two vertical asymptotes and $3$ branches, but in contrast to $w$ it does not have any zeros.
- 			 			 			
- 			We sketch $f$ in \cref{rat:fig:2asympnozerosp2}, using the middle branch as our guide
- 			because we have the most information about the function on the interval $(-5,4)$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $f$ around its vertical asymptotes (it behaves like $\frac{1}{x}$),
- 			which we have done in \cref{rat:fig:2asympnozerosp2}.
- 			 			 			
- 			Note that we are not yet able to find the local minimum of $f$ algebraically on the interval $(-5,4)$,
- 			so we make a reasonable guess as to where it is| we can be confident that it is above the horizontal axis
- 			since $f$ has no zeros. You may think that this is unsatisfactory, but once we have the tools of calculus, we will
- 			be able to find local minimums more precisely.
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(0,1.25)}node[axisnode,anchor=south east]{$\left( 0,\frac{5}{4} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2asympnozerosp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=100/( (x+5)*(x-4)^2);}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(0,1.25)}node[axisnode,anchor=south east]{$\left( 0,\frac{5}{4} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.12022]{f};
- 				\addplot[pccplot] expression[domain=-4.87298:2.87298,samples=50]{f};
- 				\addplot[pccplot] expression[domain=5:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2asympnozerosp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{100}{(x+5)(x-4)^2}$}
- \end{figure}
-   
- %===================================
- %   Author: Hughes
- %   Date:   May 2012
- %===================================
- \begin{pccexample}\label{rat:ex:2squaredasymp}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $g$
- 	that has formula
- 	\[
- 		g(x)=\frac{50(2-x)}{(x+3)^2(x-5)^2}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $g$ has vertical asymptotes at $-3$ and $5$. $g$ does
- 			not have any holes. The curve of $g$ will have $3$ branches.
- 			\item $g$ has a simple zero at $2$. The vertical intercept of $g$ is
- 			$\left( 0,\frac{4}{9} \right)$.
- 			\item $g$ behaves like $\frac{1}{x^2}$ around both of its
- 			vertical asymptotes.
- 			\item The degree of the numerator of $g$ is $1$ and the degree of the denominator
- 			of $g$ is $4$. Using \vref{rat:def:longrun}, we calculate that
- 			the horizontal asymptote of $g$ has equation $y=0$.
- 			\item The details that we have found so far have been drawn in
- 			\cref{rat:fig:2squaredasymp1}. The function $g$ is similar to the functions
- 			we considered in \cref{rat:ex:sketchtwoasymp,rat:ex:2asympnozeros} because
- 			it has $2$ vertical asymptotes and $3$ branches.
- 			 			 			
- 			We sketch $g$ using the middle branch as our guide because we have the most information
- 			about $g$ on the interval $(-3,5)$. Note that there is no other way to draw this branch
- 			without introducing other zeros which $g$ does not have.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $g$ around its vertical asymptotes| it
- 			behaves like $\frac{1}{x^2}$.
- 			 			 			
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-3},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(2,0)(0,4/9)}node[axisnode,anchor=south west]{$\left( 0,\frac{4}{9} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2squaredasymp1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=50*(2-x)/( (x+3)^2*(x-5)^2);}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					ytick={-5,5},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-3},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{0});
- 				\addplot[soldot] coordinates{(2,0)(0,4/9)}node[axisnode,anchor=south west]{$\left( 0,\frac{4}{9} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-3.61504]{f};
- 				\addplot[pccplot] expression[domain=-2.3657:4.52773]{f};
- 				\addplot[pccplot] expression[domain=5.49205:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:2squaredasymp2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{50(2-x)}{(x+3)^2(x-5)^2}$}
- \end{figure}
-   
- Each of the rational functions that we have considered so far has had either
- a \emph{simple} zero, or no zeros at all. Remember from our work on polynomial
- functions, and particularly \vref{poly:def:multzero}, that a \emph{repeated} zero
- corresponds to the curve of the function behaving differently at the zero
- when compared to how the curve behaves at a simple zero. \Cref{rat:ex:doublezero} details a
- function that has a non-simple zero.
-   
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{pccexample}\label{rat:ex:doublezero}
- 	Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of the function $g$
- 	that has formula
- 	\[
- 		h(x)=\frac{(x-3)^2}{(x+4)(x-6)}
- 	\]
- 	\begin{pccsolution}
- 		\begin{steps}
- 			\item $h$ has vertical asymptotes at $-4$ and $6$. $h$ does
- 			not have any holes. The curve of $h$ will have $3$ branches.
- 			\item $h$ has a zero at $3$ that has \emph{multiplicity $2$}.
- 			The vertical intercept of $h$ is
- 			$\left( 0,-\frac{3}{8} \right)$.
- 			\item $h$ behaves like $\frac{1}{x}$ around both of its
- 			vertical asymptotes.
- 			\item The degree of the numerator of $h$ is $2$ and the degree of the denominator
- 			of $h$ is $2$. Using \vref{rat:def:longrun}, we calculate that
- 			the horizontal asymptote of $h$ has equation $y=1$.
- 			\item The details that we have found so far have been drawn in
- 			\cref{rat:fig:doublezerop1}. The function $h$ is different
- 			from the functions that we have considered in previous examples because
- 			of the multiplicity of the zero at $3$.
- 			 			 			
- 			We sketch $h$ using the middle branch as our guide because we have the most information
- 			about $h$ on the interval $(-4,6)$. Note that there is no other way to draw this branch
- 			without introducing other zeros which $h$ does not have| also note how
- 			the curve bounces off the horizontal axis at $3$.
- 			 			 			
- 			Once we have drawn the middle branch, there is only one way to complete the graph
- 			because of our observations about the behavior of $h$ around its vertical asymptotes| it
- 			behaves like $\frac{1}{x}$.
- 			 			 			
- 		\end{steps}
- 	\end{pccsolution}
- \end{pccexample}
-   
- \begin{figure}[!htbp]
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-5,ymax=5,
- 					xtick={-8,-6,...,8},
- 					ytick={-3,3},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-4},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{1});
- 				\addplot[soldot] coordinates{(3,0)(0,-3/8)}node[axisnode,anchor=north west]{$\left( 0,-\frac{3}{8} \right)$};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:doublezerop1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{.45\textwidth}
- 		\begin{tikzpicture}[/pgf/declare function={f=(x-3)^2/((x+4)*(x-6));}]
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-5,ymax=5,
- 					xtick={-8,-6,...,8},
- 					ytick={-3,3},
- 					width=\textwidth,
- 				]
- 				\addplot[asymptote,domain=-10:10]({-4},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{1});
- 				\addplot[soldot] coordinates{(3,0)(0,-3/8)}node[axisnode,anchor=north west]{$\left( 0,-\frac{3}{8} \right)$};
- 				\addplot[pccplot] expression[domain=-10:-5.20088]{f};
- 				\addplot[pccplot] expression[domain=-3.16975:5.83642,samples=50]{f};
- 				\addplot[pccplot] expression[domain=6.20088:10]{f};
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:doublezerop2}
- 	\end{subfigure}%
- 	\caption{$y=\dfrac{(x-3)^2}{(x+4)(x-6)}$}
- \end{figure}
- \begin{exercises}
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{problem}[\Cref{rat:step:last}]\label{rat:prob:deduce}
- \pccname{Katie} is working on graphing rational functions. She
- has been concentrating on functions that have the form
- \begin{equation}\label{rat:eq:deducecurve}
- 	f(x)=\frac{a(x-b)}{x-c}
- \end{equation}
- Katie notes that functions with this type of formula have a zero
- at $b$, and a vertical asymptote at $c$. Furthermore, these functions
- behave like $\frac{1}{x}$ around their vertical asymptote, and the
- curve of each function will have $2$ branches.
-   
- Katie has been working with $3$ functions that have the form given
- in \cref{rat:eq:deducecurve}, and has followed \crefrange{rat:step:first}{rat:step:penultimate};
- her results are shown in \cref{rat:fig:deducecurve}. There is just one
- more thing to do to complete the graphs| follow \cref{rat:step:last}.
- Help Katie finish each graph by deducing the curve of each function.
- \begin{shortsolution}
- 	\Vref{rat:fig:deducecurve1}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=3*(x+4)/(x+5);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-4,0)(0,12/5)};
- 			\addplot[asymptote,domain=-10:10]({-5},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{3});
- 			\addplot[pccplot] expression[domain=-10:-5.42857]{f};
- 			\addplot[pccplot] expression[domain=-4.76923:10,samples=50]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducecurve2}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=-3*(x-2)/(x-4);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(2,0)(0,-3/2)};
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{-3});
- 			\addplot[pccplot] expression[domain=-10:3.53846,samples=50]{f};
- 			\addplot[pccplot] expression[domain=4.85714:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducecurve4}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=2*(x-6)/(x-4);}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(6,0)(0,3)};
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[pccplot] expression[domain=-10:3.5,samples=50]{f};
- 			\addplot[pccplot] expression[domain=4.3333:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- \end{shortsolution}
- \end{problem}
-   
- \begin{figure}[!htb]
- 	\begin{widepage}
- 	\setlength{\figurewidth}{0.3\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-4,0)(0,12/5)};
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(2,0)(0,-3/2)};
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{-3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve2}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(6,0)(0,3)};
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducecurve4}
- 	\end{subfigure}
- 	\caption{Graphs for \cref{rat:prob:deduce}}
- 	\label{rat:fig:deducecurve}
- 	\end{widepage}
- \end{figure}
-   
- %===================================
- %   Author: Hughes
- %   Date:   June 2012
- %===================================
- \begin{problem}[\Cref{rat:step:last} for  more complicated rational functions]\label{rat:prob:deducehard}
- \pccname{David} is also working on graphing rational functions, and
- has been concentrating on functions that have the form
- \[
- 	r(x)=\frac{a(x-b)(x-c)}{(x-d)(x-e)}
- \]
- David notices that functions with this type of formula have simple zeros
- at $b$ and $c$, and vertical asymptotes at $d$ and $e$. Furthermore,
- these functions behave like $\frac{1}{x}$ around both vertical asymptotes,
- and the curve of the function will have $3$ branches.
-   
- David has followed \crefrange{rat:step:first}{rat:step:penultimate} for
- $3$ separate functions, and drawn the results in \cref{rat:fig:deducehard}.
- Help David finish each graph by deducing the curve of each function.
- \begin{shortsolution}
- 	\Vref{rat:fig:deducehard1}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=(x-6)*(x+3)/( (x-4)*(x+1));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(6,0)(0,9/2)};
- 			\addplot[asymptote,domain=-10:10]({-1},{x});
- 			\addplot[asymptote,domain=-10:10]({4},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[pccplot] expression[domain=-10:-1.24276]{f};
- 			\addplot[pccplot] expression[domain=-0.6666:3.66667]{f};
- 			\addplot[pccplot] expression[domain=4.24276:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducehard2}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=3*(x-2)*(x+3)/( (x-6)*(x+5));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(2,0)(0,3/5)};
- 			\addplot[asymptote,domain=-10:10]({-5},{x});
- 			\addplot[asymptote,domain=-10:10]({6},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{3});
- 			\addplot[pccplot] expression[domain=-10:-5.4861]{f};
- 			\addplot[pccplot] expression[domain=-4.68395:5.22241]{f};
- 			\addplot[pccplot] expression[domain=7.34324:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- 	\Vref{rat:fig:deducehard3}
- 	 	 	
- 	\begin{tikzpicture}[/pgf/declare function={f=2*(x-7)*(x+3)/( (x+6)*(x-5));}]
- 		\begin{axis}[
- 				xmin=-10,xmax=10,
- 				ymin=-10,ymax=10,
- 				xtick={-8,-6,...,8},
- 				width=\solutionfigurewidth,
- 			]
- 			\addplot[soldot] coordinates{(-3,0)(7,0)(0,1.4)};
- 			\addplot[asymptote,domain=-10:10]({-6},{x});
- 			\addplot[asymptote,domain=-10:10]({5},{x});
- 			\addplot[asymptote,domain=-10:10]({x},{2});
- 			\addplot[pccplot] expression[domain=-10:-6.91427]{f};
- 			\addplot[pccplot] expression[domain=-5.42252:4.66427]{f};
- 			\addplot[pccplot] expression[domain=5.25586:10]{f};
- 		\end{axis}
- 	\end{tikzpicture}
- 	 	 	
- \end{shortsolution}
- \end{problem}
-   
- \begin{figure}[!htb]
- 	\begin{widepage}
- 	\setlength{\figurewidth}{0.3\textwidth}
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(6,0)(0,9/2)};
- 				\addplot[asymptote,domain=-10:10]({-1},{x});
- 				\addplot[asymptote,domain=-10:10]({4},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard1}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(2,0)(0,3/5)};
- 				\addplot[asymptote,domain=-10:10]({-5},{x});
- 				\addplot[asymptote,domain=-10:10]({6},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{3});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard2}
- 	\end{subfigure}%
- 	\hfill
- 	\begin{subfigure}{\figurewidth}
- 		\begin{tikzpicture}
- 			\begin{axis}[
- 					xmin=-10,xmax=10,
- 					ymin=-10,ymax=10,
- 					xtick={-8,-6,...,8},
- 					width=\textwidth,
- 				]
- 				\addplot[soldot] coordinates{(-3,0)(7,0)(0,1.4)};
- 				\addplot[asymptote,domain=-10:10]({-6},{x});
- 				\addplot[asymptote,domain=-10:10]({5},{x});
- 				\addplot[asymptote,domain=-10:10]({x},{2});
- 			\end{axis}
- 		\end{tikzpicture}
- 		\caption{}
- 		\label{rat:fig:deducehard3}
- 	\end{subfigure}%
- 	\hfill
- 	\caption{Graphs for \cref{rat:prob:deducehard}}
- 	\label{rat:fig:deducehard}
- 	\end{widepage}
- \end{figure}
- %===================================
- %   Author: Adams (Hughes)
- %   Date:   March 2012
- %===================================
- \begin{problem}[\Crefrange{rat:step:first}{rat:step:last}]
- Use \crefrange{rat:step:first}{rat:step:last} to sketch a graph of
- each of the following functions
- \fixthis{need 2 more subproblems here}
- \begin{multicols}{4}
- 	\begin{subproblem}
- 		$y=\dfrac{4}{x+2}$
- 		\begin{shortsolution}
- 			Vertical intercept: $(0,2)$; vertical asymptote: $x=-2$, horizontal asymptote: $y=0$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-2.8]{4/(x+2)};
- 					\addplot[pccplot] expression[domain=-1.2:5]{4/(x+2)};
- 					\addplot[soldot]coordinates{(0,2)};
- 					\addplot[asymptote,domain=-5:5]({-2},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{2x-1}{x^2-9}$
- 		\begin{shortsolution}
- 			Vertical intercept:$\left( 0,\frac{1}{9} \right)$;
- 			horizontal intercept: $\left( \frac{1}{2},0 \right)$;
- 			vertical asymptotes: $x=-3$, $x=3$, horizontal asymptote: $y=0$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-3.23974]{(2*x-1)/(x^2-9)};
- 					\addplot[pccplot,samples=50] expression[domain=-2.77321:2.83974]{(2*x-1)/(x^2-9)};
- 					\addplot[pccplot] expression[domain=3.17321:5]{(2*x-1)/(x^2-9)};
- 					\addplot[soldot]coordinates{(0,1/9)(1/2,0)};
- 					\addplot[asymptote,domain=-5:5]({-3},{x});
- 					\addplot[asymptote,domain=-5:5]({3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{0});
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{x+3}{x-5}$
- 		\begin{shortsolution}
- 			Vertical intercept $\left( 0,-\frac{3}{5} \right)$; horizontal
- 			intercept: $(-3,0)$; vertical asymptote: $x=5$; horizontal asymptote: $y=1$.
- 			 			 			
- 			\begin{tikzpicture}
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-5,ymax=5,
- 						xtick={-8,-6,...,8},
- 						minor ytick={-3,-1,...,3},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:3.666]{(x+3)/(x-5)};
- 					\addplot[pccplot] expression[domain=7:10]{(x+3)/(x-5)};
- 					\addplot[asymptote,domain=-5:5]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{1});
- 					\addplot[soldot]coordinates{(0,-3/5)(-3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{2x+3}{3x-1}$
- 		\begin{shortsolution}
- 			Vertical intercept: $(0,-3)$; horizontal intercept: $\left( -\frac{3}{2},0 \right)$;
- 			vertical asymptote: $x=\frac{1}{3}$, horizontal asymptote: $y=\frac{2}{3}$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(2*x+3)/(3*x-1);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:0.1176]{f};
- 					\addplot[pccplot] expression[domain=0.6153:5]{f};
- 					\addplot[asymptote,domain=-5:5]({1/3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{2/3});
- 					\addplot[soldot]coordinates{(0,-3)(-3/2,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{4-x^2}{x^2-9}$
- 		\begin{shortsolution}
- 			Vertical intercept: $\left( 0,-\frac{4}{9} \right)$;
- 			horizontal intercepts: $(2,0)$, $(-2,0)$;
- 			vertical asymptotes: $x=-3$, $x=3$; horizontal asymptote: $y=-1$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(4-x^2)/(x^2-9);}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-5,xmax=5,
- 						ymin=-5,ymax=5,
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-5:-3.20156]{f};
- 					\addplot[pccplot,samples=50] expression[domain=-2.85774:2.85774]{f};
- 					\addplot[pccplot] expression[domain=3.20156:5]{f};
- 					\addplot[asymptote,domain=-5:5]({-3},{x});
- 					\addplot[asymptote,domain=-5:5]({3},{x});
- 					\addplot[asymptote,domain=-5:5]({x},{-1});
- 					\addplot[soldot] coordinates{(-2,0)(2,0)(0,-4/9)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$y=\dfrac{(4x+5)(3x-4)}{(2x+5)(x-5)}$
- 		\begin{shortsolution}
- 			Vertical intercept: $\left( 0,\frac{4}{5} \right)$;
- 			horizontal intercepts: $\left( -\frac{5}{4},0 \right)$, $\left( \frac{4}{3},0 \right)$;
- 			vertical asymptotes: $x=-\frac{5}{2}$, $x=5$; horizontal asymptote: $y=6$.
- 			 			 			
- 			\begin{tikzpicture}[/pgf/declare function={f=(4*x+5)*(3*x-4)/((2*x+5)*(x-5));}]
- 				\begin{axis}[
- 						framed,
- 						xmin=-10,xmax=10,
- 						ymin=-20,ymax=20,
- 						xtick={-8,-6,...,8},
- 						ytick={-10,0,...,10},
- 						minor ytick={-15,-5,...,15},
- 						grid=both,
- 						width=\solutionfigurewidth,
- 					]
- 					\addplot[pccplot] expression[domain=-10:-2.73416]{f};
- 					\addplot[pccplot] expression[domain=-2.33689:4.2792]{f};
- 					\addplot[pccplot] expression[domain=6.26988:10]{f};
- 					\addplot[asymptote,domain=-20:20]({-5/2},{x});
- 					\addplot[asymptote,domain=-20:20]({5},{x});
- 					\addplot[asymptote,domain=-10:10]({x},{6});
- 					\addplot[soldot]coordinates{(0,4/5)(-5/4,0)(4/3,0)};
- 				\end{axis}
- 			\end{tikzpicture}
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Inverse functions]
- Each of the following rational functions are invertible
- \[
- 	F(x)=\frac{2x+1}{x-3}, \qquad G(x)= \frac{1-4x}{x+3}
- \]
- \begin{subproblem}
- 	State the domain of each function.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The domain of $F$ is $(-\infty,3)\cup(3,\infty)$.
- 			\item The domain of $G$ is $(-\infty,-3)\cup(-3,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	Find the inverse of each function, and state its domain.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item $F^{-1}(x)=\frac{3x+1}{x-2}$; the domain of $F^{-1}$ is $(-\infty,2)\cup(2,\infty)$.
- 			\item $G^{-1}(x)=\frac{3x+1}{x+4}$; the domain of $G^{-1}$ is $(-\infty,-4)\cup(-4,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	Hence state the range of the original functions.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The range of $F$ is the domain of $F^{-1}$, which is $(-\infty,2)\cup(2,\infty)$.
- 			\item The range of $G$ is the domain of $G^{-1}$, which is $(-\infty,-4)\cup(-4,\infty)$.
- 		\end{itemize}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	State the range of each inverse function.
- 	\begin{shortsolution}
- 		\begin{itemize}
- 			\item The range of $F^{-1}$ is the domain of $F$, which is $(-\infty,3)\cup(3,\infty)$.
- 			\item The range of $G^{-1}$ is the domain of $G$, which is $(-\infty,-3)\cup(-3,\infty)$.
- 		\end{itemize}<++>
- 	\end{shortsolution}
- \end{subproblem}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Composition]
- Let $r$ and $s$ be the rational functions that have formulas
- \[
- 	r(x)=\frac{3}{x^2},\qquad s(x)=\frac{4-x}{x+5}
- \]
- Evaluate each of the following.
- \begin{multicols}{3}
- 	\begin{subproblem}
- 		$(r\circ s)(0)$
- 		\begin{shortsolution}
- 			$\frac{75}{16}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(0)$
- 		\begin{shortsolution}
- 			$(s\circ r)(0)$ is undefined.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(r\circ s)(2)$
- 		\begin{shortsolution}
- 			$\frac{147}{4}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(3)$
- 		\begin{shortsolution}
- 			$192$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(4)$
- 		\begin{shortsolution}
- 			$(s\circ r)(4)$ is undefined.
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$(s\circ r)(x)$
- 		\begin{shortsolution}
- 			$\dfrac{4x^2-3}{1+5x^2}$
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \end{problem}
- %===================================
- %   Author: Hughes
- %   Date:   March 2012
- %===================================
- \begin{problem}[Piecewise rational functions]
- The function $R$ has formula
- \[
- 	R(x)=
- 	\begin{dcases}
- 		\frac{2}{x+3},    & x<-5     \\
- 		\frac{x-4}{x-10}, & x\geq -5 
- 	\end{dcases}
- \]
- Evaluate each of the following.
- \begin{multicols}{4}
- 	\begin{subproblem}
- 		$R(-6)$
- 		\begin{shortsolution}
- 			$-\frac{2}{3}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(-5)$
- 		\begin{shortsolution}
- 			$\frac{3}{5}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(-3)$
- 		\begin{shortsolution}
- 			$\frac{7}{13}$
- 		\end{shortsolution}
- 	\end{subproblem}
- 	\begin{subproblem}
- 		$R(5)$
- 		\begin{shortsolution}
- 			$-\frac{1}{5}$
- 		\end{shortsolution}
- 	\end{subproblem}
- \end{multicols}
- \begin{subproblem}
- 	What is the domain of $R$?
- 	\begin{shortsolution}
- 		$(-\infty,10)\cup(10,\infty)$
- 	\end{shortsolution}
- \end{subproblem}
- \end{problem}
- \end{exercises}
-   
-\section{Graphing rational functions (oblique asymptotes)}\label{rat:sec:oblique}
- \begin{subproblem}
- 	$y=\dfrac{x^2+1}{x-4}$
- 	\begin{shortsolution}
- 		\begin{enumerate}
- 			\item $\left( 0,-\frac{1}{4} \right)$
- 			\item Vertical asymptote: $x=4$.
- 			\item A graph of the function is shown below
- 			       			       			      
- 			      \begin{tikzpicture}[/pgf/declare function={f=(x^2+1)/(x-4);}]
- 			      	\begin{axis}[
- 			      			framed,
- 			      			xmin=-20,xmax=20,
- 			      			ymin=-30,ymax=30,
- 			      			xtick={-10,10},
- 			      			minor xtick={-15,-5,...,15},
- 			      			minor ytick={-10,10},
- 			      			grid=both,
- 			      			width=\solutionfigurewidth,
- 			      		]
- 			      		\addplot[pccplot,samples=50] expression[domain=-20:3.54724]{f};
- 			      		\addplot[pccplot,samples=50] expression[domain=4.80196:20]{f};
- 			      		\addplot[asymptote,domain=-30:30]({4},{x});
- 			      	\end{axis}
- 			      \end{tikzpicture}
- 		\end{enumerate}
- 	\end{shortsolution}
- \end{subproblem}
- \begin{subproblem}
- 	$y=\dfrac{x^3(x+3)}{x-5}$
- 	\begin{shortsolution}
- 		\begin{enumerate}
- 			\item $(0,0)$, $(-3,0)$
- 			\item Vertical asymptote: $x=5$, horizontal asymptote: none.
- 			\item A graph of the function is shown below
- 			       			       			      
- 			      \begin{tikzpicture}[/pgf/declare function={f=x^3*(x+3)/(x-5);}]
- 			      	\begin{axis}[
- 			      			framed,
- 			      			xmin=-10,xmax=10,
- 			      			ymin=-500,ymax=2500,
- 			      			xtick={-8,-6,...,8},
- 			      			ytick={500,1000,1500,2000},
- 			      			grid=both,
- 			      			width=\solutionfigurewidth,
- 			      		]
- 			      		\addplot[pccplot,samples=50] expression[domain=-10:4]{f};
- 			      		\addplot[pccplot] expression[domain=5.6068:9.777]{f};
- 			      		\addplot[asymptote,domain=-500:2500]({5},{x});
- 			      	\end{axis}
- 			      \end{tikzpicture}
- 		\end{enumerate}
- 	\end{shortsolution}
- \end{subproblem}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/stylefile.tex b/Master/texmf-dist/doc/support/latexindent/success/stylefile.tex
deleted file mode 100644
index c7e3cce3730..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/stylefile.tex
+++ /dev/null
@@ -1,135 +0,0 @@
-% arara: indent: {overwrite: yes}
-% http://tex.stackexchange.com/questions/106244/using-a-lot-of-marginpars
-\ProvidesPackage{tabto}[2013/03/25 \space v 1.3 \space
-Another tabbing mechanism]\relax
-
-\newdimen\CurrentLineWidth
-\let\TabPrevPos\z@
-
-\newcommand\tabto[1]{%
-	\leavevmode
-	\begingroup
-	\def\@tempa{*}\def\@tempb{#1}%
-	\ifx\@tempa\@tempb % \tab*
-		\endgroup
-		\TTo@overlaptrue % ... set a flag and re-issue \tabto to get argument
-		\expandafter\tabto
-	\else
-		\ifinner % in a \hbox, so ignore
-		\else % unrestricted horizontal mode
-			\null% \predisplaysize will tell the position of this box (must be box)
-			\parfillskip\fill
-			\everydisplay{}\everymath{}%
-			\predisplaypenalty\@M \postdisplaypenalty\@M
-			$$% math display so we can test \predisplaysize
-			\lineskiplimit=-999pt % so we get pure \baselineskip
-			\abovedisplayskip=-\baselineskip \abovedisplayshortskip=-\baselineskip
-			\belowdisplayskip\z@skip \belowdisplayshortskip\z@skip
-			\halign{##\cr\noalign{%
-				% get the width of the line above
-				%\message{>>> Line \the\inputlineno\space -- \predisplaydirection\the\predisplaydirection, \predisplaysize\the\predisplaysize, \displayindent\the\displayindent, \leftskip\the\leftskip, \linewidth\the\linewidth. }%
-				\ifdim\predisplaysize=\maxdimen % mixed R and L; call the line full
-					\message{Mixed R and L, so line is full. }%
-					\CurrentLineWidth\linewidth
-				\else
-					\ifdim\predisplaysize=-\maxdimen % impossible, in vmode; call the line empty
-						\message{Not in paragraph, so line is empty. }%
-						\CurrentLineWidth\z@
-					\else
-						\ifnum\TTo@Direction<\z@
-							\CurrentLineWidth\linewidth \advance\CurrentLineWidth\predisplaysize
-						\else
-							\CurrentLineWidth\predisplaysize
-						\fi
-						% Correct the 2em offset
-						\advance\CurrentLineWidth -2em
-						\advance\CurrentLineWidth -\displayindent
-						\advance\CurrentLineWidth -\leftskip
-					\fi
-				\fi
-				\ifdim\CurrentLineWidth<\z@ \CurrentLineWidth\z@\fi
-				% Enshrine the tab-to position; #1 might reference \CurrentLineWidth
-				\@tempdimb=#1\relax
-				\message{*** Tab to \the\@tempdimb, previous width is \the\CurrentLineWidth. ***}%
-				% Save width for possible return use
-				\xdef\TabPrevPos{\the\CurrentLineWidth}%
-				% Build the action to perform
-				\protected@xdef\TTo@action{%
-					\vrule\@width\z@\@depth\the\prevdepth
-					\ifdim\CurrentLineWidth>\@tempdimb
-						\ifTTo@overlap\else
-						\protect\newline \protect\null
-					\fi
-					\fi
-					\protect\nobreak
-					\protect\hskip\the\@tempdimb\relax
-				}%
-				%\message{\string\TTo@action: \meaning \TTo@action. }%
-				% get back to the baseline, regardless of its depth.
-				\vskip-\prevdepth
-				\prevdepth-99\p@
-				\vskip\prevdepth
-			}}%
-			$$
-			% Don't count the display as lines in the paragraph
-			\count@\prevgraf \advance\count@-4 \prevgraf\count@
-			\TTo@action
-			%%   \penalty\@m % to allow a penalized line break
-		\fi
-		\endgroup
-		\TTo@overlapfalse
-		\ignorespaces
-	\fi
-}
-
-% \tab -- to the next position
-% \hskip so \tab\tab moves two positions
-% Allow a (penalized but flexible) line-break right after the tab.
-%
-\newcommand\tab{\leavevmode\hskip2sp\tabto{\NextTabStop}%
-	\nobreak\hskip\z@\@plus 30\p@\penalty4000\hskip\z@\@plus-30\p@\relax}
-
-
-% Expandable macro to select the next tab position from the list
-
-\newcommand\NextTabStop{%
-	\expandafter \TTo@nexttabstop \TabStopList,\maxdimen,>%
-}
-
-\def\TTo@nexttabstop #1,{%
-	\ifdim#1<\CurrentLineWidth
-		\expandafter\TTo@nexttabstop
-	\else
-		\ifdim#1<0.9999\linewidth#1\else\z@\fi
-		\expandafter\strip@prefix
-	\fi
-}
-\def\TTo@foundtabstop#1>{}
-
-\newcommand\TabPositions[1]{\def\TabStopList{\z@,#1}}
-
-\newcommand\NumTabs[1]{%
-	\def\TabStopList{}%
-	\@tempdimb\linewidth
-	\divide\@tempdimb by#1\relax
-	\advance\@tempdimb 1sp % counteract rounding-down by \divide
-	\CurrentLineWidth\z@
-	\@whiledim\CurrentLineWidth<\linewidth\do {%
-		\edef\TabStopList{\TabStopList\the\CurrentLineWidth,}%
-		\advance\CurrentLineWidth\@tempdimb
-	}%
-	\edef\TabStopList{\TabStopList\linewidth}%
-}
-
-% default setting of tab positions:
-\TabPositions{\parindent,.5\linewidth}
-
-\newif\ifTTo@overlap \TTo@overlapfalse
-
-\@ifundefined{predisplaydirection}{
-	\let\TTo@Direction\predisplaysize
-	\let\predisplaydirection\@undefined
-}
-{
-	\let\TTo@Direction\predisplaydirection
-}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/table1.tex b/Master/texmf-dist/doc/support/latexindent/success/table1.tex
deleted file mode 100644
index 5002f8b81a6..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/table1.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass{article}
-\usepackage{multirow}
-\usepackage{booktabs}
-\begin{document}
-\begin{table}[h!]
-	\centering
-	\caption{mycaption}
-	\label{tab:test}
-	\begin{tabular}{llll}
-		\toprule
-		\textbf{headerone} & \textbf{headertwo} & \textbf{headerthree} & \textbf{headerfour}                           \\\midrule
-		r1c1               & r1c2               & r1c3                 & \multirow{4}{*}{norowlinesinthefirstfourrows} \\\cmidrule{1-3}
-		r2c1               & r2c2               & r2c3                 &                                               \\\cmidrule{1-3}
-		r3c1               & r3c2               & r3c3                 &                                               \\\cmidrule{1-3}
-		r4c1               & r4c2               & r4c3                 &                                               \\\midrule
-		r5c1               & r5c2               & r5c3                 &                                               \\\midrule
-		r6c1               & r6c2               & r6c3                 &                                               \\\midrule
-		r7c1               & r7c2               & r7c3                 &                                               \\\bottomrule
-	\end{tabular}
-\end{table}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/table2.tex b/Master/texmf-dist/doc/support/latexindent/success/table2.tex
deleted file mode 100644
index cc0c12cc763..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/table2.tex
+++ /dev/null
@@ -1,26 +0,0 @@
-% arara: indent: {overwrite: true, silent: yes}
-\documentclass{article}
-\usepackage{array} % Thanks to Heiko for catching the redundant package loading
-\newcolumntype{M}{>{$}c<{$}}
-
-\begin{document}
-
-\begin{table}%
-	\centering
-	\begin{tabular}{M|MMMMMMMMM}
-		    & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & A_9 \\\hline
-		A_1 & 0   &     &     &     &     &     &     &     &     \\
-		A_2 &     & 0   &     &     &     &     &     &     &     \\
-		A_3 &     &     & 0   &     &     &     &     &     &     \\
-		A_4 &     &     &     & 0   &     &     &     &     &     \\
-		A_5 &     &     &     &     & 0   &     &     &     &     \\
-		A_6 &     &     &     &     &     & 0   &     &     &     \\
-		A_7 &     &     &     &     &     &     & 0   &     &     \\
-		A_8 &     &     &     &     &     &     &     & 0   &     \\
-		A_9 &     &     &     &     &     &     &     &     & 0   \\
-	\end{tabular}
-	\caption{Some caption}
-	\label{table:mytable}
-\end{table}
-\end{document}
-
diff --git a/Master/texmf-dist/doc/support/latexindent/success/table3.tex b/Master/texmf-dist/doc/support/latexindent/success/table3.tex
deleted file mode 100644
index 19007908e01..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/table3.tex
+++ /dev/null
@@ -1,26 +0,0 @@
-% !arara: pdflatex
-% arara: indent: {overwrite: yes, trace: on}
-\documentclass{article}
-\usepackage{multirow}
-
-\begin{document}
-\begin{figure*}
-	\centering
-	\begin{tabular}{|c|c|c c c c|c|}
-		\hline
-		\multicolumn{2}{|c|}{\multirow{2}{*}{$V_{\rm rot}/{\sigma}$}}&\multicolumn{4}{c|}{W1}\\
-		\cline{3-6}
-		\multicolumn{2}{|c|}{}&3&6&9&12\\
-		\hline
-		\multirow{6}{*}{W2} & \multirow{3}{*}{3} & $0.090475\pm 0.011115$ & \multirow{3}{*}{21} & \multirow{3}{*}{6} & \multirow{3}{*}{3} \\
-		                    &                    & $0.14861\pm 0.03562$   &                     &                    &                    \\
-		                    &                    & $0.1861 \pm 0.01728$   &                     &                    &                    \\
-		                    & 6                  & 8                      & 14                  & 5                  & 2                  \\
-		                    & 9                  & 8                      & 14                  & 5                  & 2                  \\
-		                    & 12                 & 8                      & 14                  & 5                  & 2                  \\
-		\hline
-	\end{tabular}
-	\caption{Multirow in multirow}
-	\label{ta.Multirow}
-\end{figure*}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/table4.tex b/Master/texmf-dist/doc/support/latexindent/success/table4.tex
deleted file mode 100644
index 6f6e71323ce..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/table4.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-% arara: indent: {overwrite: true, silent: on, localSettings: on}
-\documentclass{article}
-\usepackage{siunitx}
-
-\begin{document}
-\begin{tabular}{%
-	l
-	S[table-format=3.0]
-	S[table-format=3.2]
-	S[table-format=2.2]
-	S[table-format=3.2]
-	S[table-format=-2.2]
-	S[table-format=3.2]
-	}
-	Latex              & 360 & 101.77 & 10.71 & 101.86 & 64.60  & 127.20 \\
-	Manufacturing      & 360 & -7.33  & 12.59 & -7.24  & -49.00 & 22.00  \\
-	Cons               & 360 & -17.19 & 23.4  & -17.22 & -79.00 & 43.00  \\
-	Apple and Orange   & 360 & 3.38   & 13.84 & 3.60   & -47.00 & 29.00  \\
-	Services and Harry & 104 & -4.96  & 20.8  & -4.81  & -57.00 & 30.00  \\
-	Manchester         & 360 & -9.29  & 8.64  & -9.26  & -35.00 & 8.00   \\
-\end{tabular}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/table5.tex b/Master/texmf-dist/doc/support/latexindent/success/table5.tex
deleted file mode 100644
index e2750f40d78..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/table5.tex
+++ /dev/null
@@ -1,35 +0,0 @@
-\documentclass{article}
-
-\begin{document}
-
-\begin{tabular}{M|MMMMMMMMM}
-	    & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & A_9  \\\hline
-	A_1 & 0   &     &     &     &     &     &     &     &  \\
-	A_2 &     & 0   &     &     &     &     &     &     &  \\
-	A_3 &     &     & 0   &     &     &     &     &     &  \\
-	A_4 &     &     &     & 0   &     &     &     &     &  \\
-	A_5 &     &     &     &     & 0   &     &     &     &  \\
-	A_6 &     &     &     &     &     & 0   &     &     &  \\
-	A_7 &     &     &     &     &     &     & 0   &     &  \\
-	A_8 &     &     &     &     &     &     &     & 0   &  \\
-	A_9 &     &     &     &     &     &     &     &     & 0  \\
-\end{tabular}
-\begin{tabularx}{M|MMMMMMMMM}
-	    & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & A_9 \\\hline
-	A_1 & 0   &     &     &     &     &     &     &     &     \\
-	A_2 &     & 0   &     &     &     &     &     &     &     \\
-	A_3 &     &     & 0   &     &     &     &     &     &     \\
-	A_4 &     &     &     & 0   &     &     &     &     &     \\
-	A_5 &     &     &     &     & 0   &     &     &     &     \\
-	A_6 &     &     &     &     &     & 0   &     &     &     \\
-	A_7 &     &     &     &     &     &     & 0   &     &     \\
-	A_8 &     &     &     &     &     &     &     & 0   &     \\
-	A_9 &     &     &     &     &     &     &     &     & 0   \\
-\end{tabularx}
-\begin{align*}
-	CCI_n          & = \frac{p_n-SMA(p_n)}{0.015 \cdot \sigma(p_n)}\\
-	\textrm{wobei} & n = \textrm{Perioden, i.\,d.\,R. 20};\p_n = \textrm{Typischer Preis/Kurs};\ SMA(p_n) = \textrm{SMA der typischen Preise};\\
-	               & \sigma(p_n) = \textrm{Standardabweichung}\\
-\end{align*}
-\end{document}
-
diff --git a/Master/texmf-dist/doc/support/latexindent/success/testHeadings-simple.tex b/Master/texmf-dist/doc/support/latexindent/success/testHeadings-simple.tex
deleted file mode 100644
index 565157dcaca..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/testHeadings-simple.tex
+++ /dev/null
@@ -1,56 +0,0 @@
-% arara: indent: {onlyDefault: no, overwrite: true, trace: on, silent: yes, localSettings: true}
-\part{part}
-         part text
-         part text
-         \chapter{chapter long title}
-          chapter text
-          chapter text
-          \section[for the toc]{section}
-           section text
-           section text
-          \section[for the toc]{section}
-           section text
-           section text
-          \subsection[for the toc]{subsection}
-          	subsection text
-          	subsection text
-          \subsection[for the toc]{subsection}
-          	subsection text
-          	subsection text
-          \section[for the toc]{section}
-           section text
-           section text
-         \chapter{chapter}
-          chapter text
-          chapter text
-\part{part}
-         part text
-         part text
-         \chapter[toc]{chapter title}
-          chapter text
-          chapter text
-          \section[for the toc]{section}
-           section text
-           section text
-           \subsubsection[for the toc]{subsubsection}
-           	subsubsection text
-           	subsubsection text
-           	\paragraph{paragraph}
-           		paragraph text
-           		paragraph text
-           		\subparagraph{subparagraph}
-           			subparagraph text
-           			subparagraph text
-          \section[for the toc]{section}
-           section text
-           section text
-           \subsubsection[for the toc]{subsubsection}
-           	subsubsection text
-           	subsubsection text
-           	\paragraph{paragraph}
-           		paragraph text
-           		paragraph text
-           		\subparagraph{subparagraph}
-           			subparagraph text
-           			subparagraph text
-         \chapter[somethingelse]{goes here}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/testHeadings.tex b/Master/texmf-dist/doc/support/latexindent/success/testHeadings.tex
deleted file mode 100644
index 2a8fa072e4e..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/testHeadings.tex
+++ /dev/null
@@ -1,59 +0,0 @@
-% arara: indent: {onlyDefault: no, overwrite: true, trace: on, silent: yes, localSettings: true}
-\part{part}
-         part text
-         part text
-         \chapter{chapter long title}
-          chapter text
-          chapter text
-          \[
-          	f(x)=x^2
-          \]
-          \section[for the toc]{section}
-           section text
-           section text
-          \section[for the toc]{section}
-           section text
-           section text
-          \subsection[for the toc]{subsection}
-          	subsection text
-          	subsection text
-          \subsection[for the toc]{subsection}
-          	subsection text
-          	subsection text
-          \section[for the toc]{section}
-           section text
-           section text
-         \chapter{chapter}
-          chapter text
-          chapter text
-\part{part}
-         part text
-         part text
-         \chapter[toc]{chapter title}
-          chapter text
-          chapter text
-          \section[for the toc]{section}
-           section text
-           section text
-           \subsubsection[for the toc]{subsubsection}
-           	subsubsection text
-           	subsubsection text
-           	\paragraph{paragraph}
-           		paragraph text
-           		paragraph text
-           		\subparagraph{subparagraph}
-           			subparagraph text
-           			subparagraph text
-          \section[for the toc]{section}
-           section text
-           section text
-           \subsubsection[for the toc]{subsubsection}
-           	subsubsection text
-           	subsubsection text
-           	\paragraph{paragraph}
-           		paragraph text
-           		paragraph text
-           		\subparagraph{subparagraph}
-           			subparagraph text
-           			subparagraph text
-         \chapter[somethingelse]{goes here}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/testHeadings1.tex b/Master/texmf-dist/doc/support/latexindent/success/testHeadings1.tex
deleted file mode 100644
index 164fcbe1057..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/testHeadings1.tex
+++ /dev/null
@@ -1,25 +0,0 @@
-% arara: indent: {overwrite: yes, localSettings: yes, trace: on}
-\documentclass[a4paper]{article}
-\usepackage{filecontents}
-\begin{filecontents}
-	\begin{document}
-	hello world
-	\end{document}
-\end{filecontents}
-\begin{document}
-\section{}
-\subsection{}
-	\subsubsection{}
-		some text goes here
-		some text goes here
-		some text goes here
-		\begin{verbatim}
-                \documentclass[<+options+>]{<+class+>}
-
-           \begin{document}
-        <++>
-                            \end{document}
-                     \end{document}
-                            more text here
-		\end{verbatim}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/testItems.tex b/Master/texmf-dist/doc/support/latexindent/success/testItems.tex
deleted file mode 100644
index fba97784548..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/testItems.tex
+++ /dev/null
@@ -1,94 +0,0 @@
-% arara: indent: {overwrite: yes, silent: off, trace: yes, localSettings: yes}
-
-\begin{itemize}
-	\item some text here
-	      some more text here
-	      some more text here
-	\item another item more text
-	      more text more text more text
-	      more text more text more text
-	\item
-	      more text more text more text
-	      more text more text more text
-\end{itemize}
-
-regular text
-regular text
-regular text
-\begin{itemize}
-	\item some other stuff
-	\item[1] some other stuff
-	\myitem another item
-	\myitem[2] another item
-	\item some other stuff
-	\item some other stuff
-	\item some other stuff
-	\item some text some text
-	      some text some text
-	      some text some text
-	      % here's a comment
-	\item some text some text
-	      some text some text
-	      some text some text
-	      \[
-	      	x^2
-	      \]
-	\item some text some text
-	      some text some text
-	      some text some text
-	      \begin{list}
-	      	\item some other stuff
-	      	      some other stuff
-	      	      some other stuff
-	      	      % here's a comment
-	      \end{list}
-	\item some text some text
-	      some text some text
-	      some text some text
-	      \begin{myenv}
-	      	some other stuff
-	      	some other stuff
-	      	some other stuff
-	      \end{myenv}
-	      \begin{itemize}
-	      	\item some text some text
-	      	      some text some text
-	      	      some text some text
-	      	      % here's a comment
-	      	\item some text some text
-	      	      some text some text
-	      	      some text some text
-	      	      \[
-	      	      	x^2
-	      	      \]
-	      	\item some text some text
-	      	      some text some text
-	      	      some text some text
-	      	      \begin{list}
-	      	      	\item some other stuff
-	      	      	      some other stuff
-	      	      	      some other stuff
-	      	      	      % here's a comment
-	      	      \end{list}
-	      	      	      	      	      	      
-	      	\item some text some text
-	      	      some text some text
-	      	      some text some text
-	      	      \begin{myenv}
-	      	      	some other stuff
-	      	      	some other stuff
-	      	      	some other stuff
-	      	      \end{myenv}
-	      	\item just one line
-	      	      some more text
-	      	\item something here
-	      \end{itemize}
-	\item this one belongs here
-	      more text more text more text
-\end{itemize}
-
-\begin{myenv}
-	some other stuff
-	some other stuff
-	some other stuff
-\end{myenv}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/testcls.cls b/Master/texmf-dist/doc/support/latexindent/success/testcls.cls
deleted file mode 100644
index d4c23660c5a..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/testcls.cls
+++ /dev/null
@@ -1,375 +0,0 @@
-% arara: indent: {overwrite: yes}
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesClass{rpisudiss}[2014/07/06 Ryan Pavlik's ISU Thesis]
-
-\newcommand{\@isu@classname}{rpisudiss}
-\RequirePackage{setspace}
-
-% option doublespacing: Double-space where we would have 1.5 spaced.
-\newcommand\isu@spacing{\onehalfspacing}
-\newcommand\isu@space{onehalfspace}
-\DeclareOption{doublespacing}{%
-	\renewcommand\isu@spacing{\doublespacing}
-	\renewcommand\isu@space{doublespace}
-}
-
-% option capstoc: Capitalize chapters in the table of contents
-% Technically required if you go by the text of the grad college site,
-% but rule inconsistently applied and results painful to read.
-\newif\if@isu@capstoc	\@isu@capstocfalse
-\DeclareOption{capstoc}{\@isu@capstoctrue}
-
-% option capschap: Capitalize chapters everywhere else
-% Technically required if you go by the text of the grad college site,
-% but rule inconsistently applied and results painful to read.
-\newif\if@isu@capschap	\@isu@capschapfalse
-\DeclareOption{capschap}{\@isu@capschaptrue}
-
-% option print: Optimize for print rather than on-screen (hide links, etc.)
-% Apparently doesn't work right now. Sad.
-\newif\if@isu@print	\@isu@printfalse
-\DeclareOption{print}{\@isu@printtrue}
-
-% option tocnumbersections: Number sections in the ToC.
-\newif\if@isu@tocnumbersections@	\@isu@tocnumbersections@false
-\DeclareOption{tocnumbersections}{\@isu@tocnumbersections@true}
-
-% option tocnumbersubsections: Number subsections in the ToC. Implies tocnumbersections.
-\newif\if@isu@tocnumbersubsections@	\@isu@tocnumbersubsections@false
-\DeclareOption{tocnumbersubsections}{	\@isu@tocnumbersubsections@true\@isu@tocnumbersections@true}
-
-% option ignoremissingmainmatter: Don't error if we've seen a titlepage but no \mainmatter by the end of the document.
-\newif\if@isu@errormissingmainmatter@ \@isu@errormissingmainmatter@true
-\DeclareOption{ignoremissingmainmatter}{\@isu@errormissingmainmatter@false}
-
-% option draftcls: adds "DRAFT" and a date/time stamp on the footer.
-\newif\if@isu@draftcls@	\@isu@draftcls@false
-\DeclareOption{draftcls}{\@isu@draftcls@true}
-
-% option draft: draftcls plus whatever anyone else thinks about draft.
-\DeclareOption{draft}{
-	\@isu@draftcls@true
-	\PassOptionsToClass{draft}{report}
-}
-
-% Forward everything not recognized
-\DeclareOption*{\PassOptionsToClass{\CurrentOption}{report}}
-
-% Process options
-\ProcessOptions\relax
-
-% Based on report
-\LoadClass[12pt]{report}
-
-% had to choose to put this here for tocloft
-\RequirePackage{subfig}
-
-% Put lot, lof, and bibliography (but not the contents itself) in the ToC
-\RequirePackage[nottoc]{tocbibind}
-
-% Basic hyperref - note that backreferences are incompatible with bibtopic (for per-chapter bibliographies)
-\RequirePackage[pdftex,pdfusetitle,hypertexnames=false,linktocpage=true]{hyperref}
-\hypersetup{bookmarksnumbered=true,bookmarksopen=true,pdfpagemode=UseOutlines,pdfview=FitB}
-
-\if@isu@print
-% Hide links for print
-% Apparently doesn't work right now. Sad.
-\hypersetup{hidelinks}
-\else
-% Nice blue links.
-\hypersetup{colorlinks=true,linkcolor=blue,anchorcolor=blue,citecolor=blue,filecolor=blue,urlcolor=blue}
-\fi
-
-% Indent first paragraph after sectioning things.
-\RequirePackage{indentfirst}
-
-% Setup page layout
-\RequirePackage{geometry}
-\geometry{left=1in, top=1in, headheight=0.25in, headsep=0.5in, right=1in, bottom=1in, includehead=false}
-
-% Handle draftcls option placing timestamp in footer and watermarking first page.
-\if@isu@draftcls@
-	\RequirePackage[firstpage]{draftwatermark}
-	\RequirePackage{datetime}
-	\newcommand{\isu@draftfooter}{DRAFT --- rendered \today\ at \currenttime}
-\else
-	\newcommand{\isu@draftfooter}{}
-\fi
-
-% Setup headers/footers - override the plain page style.
-\RequirePackage{fancyhdr}
-\fancypagestyle{plain}{%
-	\fancyhf{} % clear all header and footer fields
-	\fancyhead[C]{\thepage} % Always put the page in the center header
-	\fancyfoot[C]{\isu@draftfooter} % In draft mode, put stuff in the center footer.
-	\renewcommand{\headrulewidth}{0pt}
-	\renewcommand{\footrulewidth}{0pt}}
-
-\AtBeginDocument{\pagestyle{plain}}
-
-% Set up a bool for mainmatter or not.
-\newif\if@mainmatter \@mainmatterfalse
-
-% Title page:
-% - empty style (no numbering shown)
-% - starts the preface/roman numerals
-% - gets a PDF bookmark, just because we can.
-\renewcommand{\titlepage}{
-	\thispagestyle{empty}
-	\pagenumbering{roman}
-	\pdfbookmark[0]{\@title}{toc}
-	% backup and set secnumdepth
-	\newcounter{isu@secnumdepth}
-	\setcounter{isu@secnumdepth}{\value{secnumdepth}}
-	\setcounter{secnumdepth}{-1}
-	\@mainmatterfalse
-}
-\renewcommand{\endtitlepage}{\newpage\pagestyle{plain}}
-
-%%%
-% Set up sectioning
-\RequirePackage{titlesec}
-% \titleformat{command}[shape]{format}{label}{sep}{before}[after]
-
-% Chapter titles:
-% - Caps (optional)
-% - large
-% - bold
-% - center
-\titleformat{\chapter}[block]
-{\normalfont\large\bfseries\centering}
-{\if@mainmatter%
-	\if@isu@capschap\MakeUppercase{\chaptertitlename}%
-	\else\chaptertitlename\fi
-	\thechapter.\quad\fi}
-{0pt}
-{\if@isu@capschap\MakeUppercase\fi}{}
-
-% Section:
-% - bold
-% - center
-\titleformat{\section}[block]
-{\normalfont\normalsize\bfseries\centering}
-{\thesection.}{1em}{}{}
-
-% Subsection:
-% - bold
-% - left-justified
-\titleformat{\subsection}[block]
-{\normalfont\normalsize\bfseries}
-{\thesubsection.}{1em}{}{}
-
-% Subsubsection:
-% - bold
-% - left-justified with indent
-\titleformat{\subsubsection}[block]
-{\normalfont\normalsize\bfseries}
-{\quad\thesubsubsection.}{1em}{}{}
-
-% Apply default spacing
-\isu@spacing
-
-
-% Needed for toc/lof/lot spacing and headfoot tweaking
-\RequirePackage{xpatch}
-
-%%%
-% ToC:
-
-% Rename the ToC
-\RequirePackage[subfigure,titles]{tocloft}
-\renewcommand{\contentsname}{Table of Contents}
-
-% - Remove parskips from toc (and lof/lot)
-\setlength{\cftparskip}{0pt}
-
-% - Single space
-% - Page break after
-% - TODO: couldn't get pdfbookmark to point to this page instead of the
-%   first page so it was removed.
-%\begin{noindent}
-\xpretocmd{\tableofcontents}{%
-	\begin{singlespace}}{}{}
-\xapptocmd{\tableofcontents}{%
-\end{singlespace}%
-\pagestyle{plain}%
-\clearpage}{}{}
-%\end{noindent}
-
-% - Add dot leader for chapter levels
-\renewcommand\cftchapdotsep{\cftdotsep}
-
-% - Prefix "Chapter " to chapter number
-% - Adjust indentation of levels
-% - Capitalize title entries, if requested
-\if@isu@capstoc
-	\renewcommand\cftchappresnum{\MakeUppercase{\chaptertitlename} }
-	\cftsetindents{chapter}{0em}{8em}
-	\cftsetindents{section}{2em}{0em}
-	\cftsetindents{subsection}{3em}{0em}
-	\renewcommand{\cftchapfont}{\MakeUppercase}
-\else
-	\renewcommand\cftchappresnum{\chaptertitlename\ }
-	\cftsetindents{chapter}{0em}{6em}
-	\cftsetindents{section}{1em}{0em}
-	\cftsetindents{subsection}{2em}{0em}
-\fi
-
-% - Remove section/subsection numbers from ToC by capturing
-%   see idea at  http://tex.stackexchange.com/questions/71123/remove-section-number-toc-entries-with-tocloft
-\if@isu@tocnumbersections@\else
-	\renewcommand{\cftsecpresnum}{\begin{lrbox}{\@tempboxa}}
-	\renewcommand{\cftsecaftersnum}{\end{lrbox}}
-\fi
-
-\if@isu@tocnumbersubsections@\else
-\renewcommand{\cftsubsecpresnum}{\begin{lrbox}{\@tempboxa}}
-\renewcommand{\cftsubsecaftersnum}{\end{lrbox}}
-\fi
-
-%%%
-% List of Figures:
-% - Single space
-% - Page break after
-%\begin{noindent}
-\xpretocmd{\listoffigures}{%
-\begin{singlespace}}{}{}
-\xapptocmd{\listoffigures}{%
-\end{singlespace}\clearpage}{}{}
-%\end{noindent}
-
-% - Prepend the word "Figure" to the number
-\renewcommand\cftfigpresnum{Figure }
-\cftsetindents{figure}{0em}{6em}
-
-%%%
-% List of Tables:
-% - Single space
-% - Page break after
-%\begin{noindent}
-\xpretocmd{\listoftables}{%
-	\begin{singlespace}}{}{}
-\xapptocmd{\listoftables}{%
-    \end{singlespace}\clearpage}{}{}
-%\end{noindent}
-
-% - Prepend the word "Table" to the number
-\renewcommand\cfttabpresnum{Table }
-\cftsetindents{table}{0em}{6em}
-
-%%%
-% Document division commands
-
-% Command to indicate when we're done
-% with preface content - must be called!
-% (If we're actually typesetting a full thesis...)
-\newcommand{\mainmatter}{%
-	\clearpage
-	\pagenumbering{arabic}
-	\pagestyle{plain}
-	\@mainmattertrue
-	\newcommand{\@isu@gotmainmatter@}{}
-	\setcounter{chapter}{0}
-	% restore secnumdepth
-	\setcounter{secnumdepth}{\value{isu@secnumdepth}}
-}
-
-% Command to indicate we're done with main content
-\newcommand{\backmatter}{
-	\setcounter{isu@secnumdepth}{\value{secnumdepth}}
-	\setcounter{secnumdepth}{-1}
-	\@mainmatterfalse
-}
-
-\RequirePackage{etoolbox}
-% Verify that we actually got some main matter
-\AfterEndDocument{
-	\if@isu@errormissingmainmatter@
-		\ifdefined\@isu@gottitle
-		\ClassInfo{\@isu@classname}{Full dissertation mode}
-		\unless\ifdefined\@isu@gotmainmatter@
-		\ClassError{\@isu@classname}{Missing \protect\mainmatter\space before your first real chapter!}{Missing mainmatter}
-		%             \ClassError{\@isu@classname}{%
-		%                 \protect\mainmatter\space not called in your document expected before your first real chapter}{%
-		%                 You need to put \protect\mainmatter\space before your first real numbered chapter, typically your introduction.}
-	\fi
-	\fi
-	\fi
-}
-
-%%%
-% Title Page
-
-% Temporary: hardcode these values in.
-\newcommand\isu@degree{Doctor of Philosophy}
-\newcommand\isu@gradyear{2014}
-\newcommand\isu@submissiontype{dissertation}
-\newcommand\isu@majorline{Co-majors: Human-Computer Interaction; Computer Science}
-\newcommand\isu@committee{%
-	Judy M. Vance, Co-major Professor\\%
-	Leslie Miller, Co-major Professor\\%
-	Debra Satterfield \\ Jonathan Kelly \\ David Weiss \\ Horea Ilies}
-\newcommand\isu@copyrightnotice{\\       % Unclear what the spacing between notice and the text above should be.
-	Copyright \copyright\ \@author, \isu@gradyear.
-All rights reserved.}
-
-% Spacing tools
-% - used for what the thesis office calls "two blank lines"
-\newcommand{\@isu@twoblanklines}{20pt}
-\newcommand{\@isu@maketwoblanklines}{\vspace{\@isu@twoblanklines}}
-
-% The actual title page layout.
-% Note that portions that appear double-spaced in the sample/annotated PDF
-% are given the same spacing as the body of the document (1.5 or 2)
-\renewcommand{\maketitle}{
-	\newcommand{\@isu@gottitle}{} % Assume that \maketitle implies typesetting a full thesis, not just a chapter.
-	\ClassInfo{\@isu@classname}{Generating title page -- assuming we are typesetting a full dissertation.}
-			
-	\begin{titlepage}
-		\setlength{\parindent}{0pt} % Don't you dare try to indent!
-		\vbox to \textheight{ % Full-page box to contain everything and stretch everything.
-			\begin{center}                      % Center this whole page
-				\vspace*{12pt}                      % Designated blank line at the top of the page.
-												
-				\begin{singlespace}                 % Single-space this section, we manually add spacing.
-					\textbf{\@title}\\                  % Title in bold
-					\@isu@maketwoblanklines             % "two blank lines"
-					by\\                                % the word 'by', not in bold
-					\@isu@maketwoblanklines             % "two blank lines"
-					\textbf{\@author}                   % Author in bold
-				\end{singlespace}
-												
-				\vfill{}                            % let LaTeX decide what "4-6 blank lines" should be.
-												
-				\begin{\isu@space}                  % Normal document spacing here.
-					A \isu@submissiontype\ submitted to the graduate faculty\\
-					in partial fulfillment of the requirements of the degree of\\
-					\MakeUppercase{\isu@degree}         % Doctor of Philosophy, etc. We enforce caps so they don't have to.
-				\end{\isu@space}
-												
-				\@isu@maketwoblanklines             % "two blank lines"
-												
-				\begin{singlespace}                 % Committee is single-spaced, looks like really long majors would be too.
-					\isu@majorline\\                    % "Major: MAJ (CONC)" or "Co-majors: MAJ; MAJ;
-					\medskip                            % Unspecified space here, looks like just "one blank line" on sample
-					Program of Study Committee:\\
-					\isu@committee                      % Committee, user-delimited with \\ and user-annoted with ", Major Professor" or "Co-major Professor"
-				\end{singlespace}
-												
-				\vfill{}                            % let LaTeX decide what "7-8 blank lines" should be.
-				\@isu@maketwoblanklines             % Suggest it's a little longer than the other vfill.
-												
-				\begin{\isu@space}                  % Normal document spacing here
-					Iowa State University\\
-					Ames, Iowa\\
-					\isu@gradyear                       % Newline after here, if needed, is in the copyright notice macro.
-					\isu@copyrightnotice                % Copyright line optional if copyright not formally filed.
-				\end{\isu@space}
-												
-			\end{center}
-		}
-	\end{titlepage}
-}
-
-\endinput
-
diff --git a/Master/texmf-dist/doc/support/latexindent/success/theorem.tex b/Master/texmf-dist/doc/support/latexindent/success/theorem.tex
deleted file mode 100644
index 6541ed31ceb..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/theorem.tex
+++ /dev/null
@@ -1,47 +0,0 @@
-% arara: indent: {overwrite: on}
-\documentclass[12pt,twoside]{report}
-\usepackage[margin=2cm]{geometry}
-\usepackage{amsmath,amsthm,amssymb}
-\usepackage{thmtools}
-\usepackage{tikz}
-\usepackage[framemethod=TikZ]{mdframed}
-
-\declaretheoremstyle
-[
-	spaceabove=0pt, spacebelow=0pt, headfont=\normalfont\bfseries,
-	notefont=\mdseries, notebraces={(}{)}, headpunct={\newline}, headindent={},
-	postheadspace={ }, postheadspace=4pt, bodyfont=\normalfont, qed=$\blacktriangle$,
-	preheadhook={\begin{mdframed}[style=myframedstyle]},
-	postfoothook=\end{mdframed},
-]{mystyle}
-
-\declaretheorem[style=mystyle,numberwithin=chapter,title=Exemplo]{example}
-\mdfdefinestyle{myframedstyle}{%
-	outermargin = 1.3cm , %
-	leftmargin = 0pt , rightmargin = 0pt , %
-	innerleftmargin = 5pt , innerrightmargin = 5pt , %
-	innertopmargin = 5pt, innerbottommargin = 5pt , %
-	backgroundcolor = blue!10 , %
-	align = center , % align the environment itself (left, center, rigth)
-	nobreak = true, % prevent a frame from splitting
-	hidealllines = true , %
-	topline = true , bottomline = true , %
-	splittopskip = \topskip , splitbottomskip = 0pt , %
-	skipabove = 0.5\baselineskip ,  skipbelow = 0.3\baselineskip}
-
-\begin{document}
-\section{Introduction}
- Lorem ipsum sed nulla id risus adipiscing vulputate.
-   
- \begin{example}
- 	Um consumidor financiou a compra de um veículo pagando 48 parcelas de \$800,00 mensais e a taxa de juros cobrada pela concessionária foi de 1,2\% a.m.. Qual era o valor à vista do automóvel adquirido?
- 	\newline
- 	\textbf{Solução:}
- 	\newline
- 	$PV = 800 \times \left[ \dfrac{1,012^{48}-1}{1,012^{48}\times 0,012} \right] \newline
- 	PV = 800 \times \left[ \dfrac{0,772820}{0,021274} \right] \newline
- 	PV = \$29.061,79$
- \end{example}
-   
- Lorem ipsum sed nulla id risus adipiscing vulputate.
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/tikz1.tex b/Master/texmf-dist/doc/support/latexindent/success/tikz1.tex
deleted file mode 100644
index ccef3d047f2..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/tikz1.tex
+++ /dev/null
@@ -1,88 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass[png,border=10pt,tikz]{standalone}
-\usepackage{xstring}
-\usepackage{tikz}
-\usetikzlibrary{calc}
-
-\pgfkeys{/tikz/.cd,
-	vertical factor/.initial=0.5,
-	vertical factor/.get=\vertfactor,
-	vertical factor/.store in=\vertfactor,
-	start coordinate/.initial={0,\vertfactor},
-	start coordinate/.get=\startcoord,
-	start coordinate/.store in=\startcoord,
-	sample color/.initial=black,
-	sample color/.get=\samplecol,
-	sample color/.store in=\samplecol,
-	sample size/.initial=1pt,
-	sample size/.get=\samplesize,
-	sample size/.store in=\samplesize,
-	sample line width/.initial=very thick,
-	sample line width/.get=\samplelinewidth,
-	sample line width/.store in=\samplelinewidth,
-}
-
-
-\newcommand{\samplepath}[1]{%
-	\coordinate (start) at (\startcoord) ;
-	\foreach \samples[count=\xi from 1] in {#1}{%
-		\StrCut{\samples}{|}{\vertdir}{\hordir}
-		\ifnum\xi=1
-			\draw[\samplelinewidth,\samplecol](start)
-			--++(\hordir,0) coordinate (start);
-		\else
-			\IfStrEq{\vertdir}{+}{%true
-				\draw[\samplelinewidth,\samplecol]($(start)+(0,\vertfactor)$)
-				--++(\hordir,0)coordinate(start);
-				}{%false
-				\relax
-			}
-			\IfStrEq{\vertdir}{-}{%true
-				\draw[\samplelinewidth,\samplecol]($(start)+(0,-\vertfactor)$)
-				--++(\hordir,0)coordinate(start);
-				}{%false
-				\relax
-			}
-		\fi
-	}
-}
-
-\tikzset{sample/.style={
-	circle,
-	inner sep=\samplesize,
-	fill=\samplecol,
-	}
-}
-
-\newcommand{\discretesamplepath}[1]{%
-	\coordinate (start) at (\startcoord) ;
-	\foreach \samples[count=\xi from 1] in {#1}{%
-		\StrCut{\samples}{|}{\vertdir}{\hordir}
-		\ifnum\xi=1
-			\path(start)node[sample]{}
-			--++(\hordir,0) coordinate (start);
-		\else
-			\IfStrEq{\vertdir}{+}{%true
-				\path($(start)+(0,\vertfactor)$)node[sample]{}
-				--++(\hordir,0)coordinate(start);
-				}{%false
-				\relax
-			}
-			\IfStrEq{\vertdir}{-}{%true
-				\path($(start)+(0,-\vertfactor)$)node[sample]{}
-				--++(\hordir,0)coordinate(start);
-				}{%false
-				\relax
-			}
-		\fi
-	}
-}
-
-\begin{document}
-\begin{tikzpicture}
-	% axis
-	\draw[-stealth] (0,-1)--(0,4) node[left]{$X(t)$};
-	\draw[-stealth] (-1,0)--(5,0) node[below]{$t$};
-	\samplepath{+|0.5,+|0.25,-|1.5,+|1,+|0.5,+|0.75}
-\end{tikzpicture}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/tikz2.tex b/Master/texmf-dist/doc/support/latexindent/success/tikz2.tex
deleted file mode 100644
index 5ba03622079..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/tikz2.tex
+++ /dev/null
@@ -1,61 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-% http://tex.stackexchange.com/questions/104528/tikz-shade-also-the-border-of-a-node
-\documentclass[tikz,border=10pt,png]{standalone}
-\usepackage{tikz}
-\usetikzlibrary{calc}
-\begin{document}
-\tikzset{
-	shrink inner sep/.code={
-		\pgfkeysgetvalue{/pgf/inner xsep}{\currentinnerxsep}
-		\pgfkeysgetvalue{/pgf/inner ysep}{\currentinnerysep}
-		\pgfkeyssetvalue{/pgf/inner xsep}{\currentinnerxsep - 0.5\pgflinewidth}
-		\pgfkeyssetvalue{/pgf/inner ysep}{\currentinnerysep - 0.5\pgflinewidth}
-	}
-}
-
-\tikzset{horizontal shaded border/.style args={#1 and #2}{
-	append after command={
-		\pgfextra{%
-			\begin{pgfinterruptpath}
-				\path[rounded corners,left color=#1,right color=#2]
-				($(\tikzlastnode.south west)+(-\pgflinewidth,-\pgflinewidth)$)
-				rectangle
-				($(\tikzlastnode.north east)+(\pgflinewidth,\pgflinewidth)$);
-			\end{pgfinterruptpath}
-		}
-	}
-	},
-	vertical shaded border/.style args={#1 and #2}{
-		append after command={
-			\pgfextra{%
-				\begin{pgfinterruptpath}
-					\path[rounded corners,top color=#1,bottom color=#2]
-					($(\tikzlastnode.south west)+(-\pgflinewidth,-\pgflinewidth)$)
-					rectangle
-					($(\tikzlastnode.north east)+(\pgflinewidth,\pgflinewidth)$);
-				\end{pgfinterruptpath}
-			}
-		}
-	}
-}
-\begin{tikzpicture}
-	\draw (0,0) node[rectangle,
-		rounded corners,
-		thick,
-		outer sep=0pt,
-		shrink inner sep,
-		left color=red!50!white,
-		right color=green!50!white,
-		horizontal shaded border=red and green
-	](A){abcabc abc};
-	\draw (2.5,0) node[rectangle,
-		rounded corners,
-		thick,
-		outer sep=0pt,
-		shrink inner sep,
-		top color=cyan!50,
-		bottom color=orange!50,
-		vertical shaded border=blue and orange
-	](A){abcabc abc};
-\end{tikzpicture}
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/tikz3.tex b/Master/texmf-dist/doc/support/latexindent/success/tikz3.tex
deleted file mode 100644
index f50646ac5c8..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/tikz3.tex
+++ /dev/null
@@ -1,53 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass[11pt]{article}
-\usepackage{tikz}
-\usetikzlibrary{trees}
-\usetikzlibrary{decorations.pathmorphing}
-\usetikzlibrary{decorations.markings}
-
-\begin{document}
-
-\tikzset{
-	photon/.style={decorate, decoration={snake}, draw=red},
-	particle/.style={draw=blue, postaction={decorate},
-		decoration={markings,mark=at position .5 with {\arrow[draw=blue]{>}}}},
-	antiparticle/.style={draw=blue, postaction={decorate},
-		decoration={markings,mark=at position .5 with {\arrow[draw=blue]{<}}}},
-	gluon/.style={decorate, draw=black,
-		decoration={coil,amplitude=4pt, segment length=5pt}}
-}
-
-\begin{tikzpicture}[
-		thick,
-		% Set the overall layout of the tree
-		level/.style={level distance=1.5cm},
-		level 2/.style={sibling distance=3.5cm},
-	]
-	\coordinate
-	child[grow=down]{
-		edge from parent [antiparticle]
-		child {
-			node{$E$}
-			edge from parent [particle]
-		}
-		child {
-			node{$D$}
-			edge from parent [gluon]
-		}
-		node [above=3pt] {$C$}
-	}
-	% I have to insert a dummy child to get the tree to grow
-	% correctly to the right.
-	child[grow=right, level distance=0pt] {
-		child {
-			node{$A$}
-			edge from parent [gluon]
-		}
-		child {
-			node{$B$}
-			edge from parent [particle]
-		}
-	};
-\end{tikzpicture}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/tikz4.tex b/Master/texmf-dist/doc/support/latexindent/success/tikz4.tex
deleted file mode 100644
index 97830271b3f..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/tikz4.tex
+++ /dev/null
@@ -1,27 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass{article}
-
-% in the preamble
-% nothing
-% should happend
-\foreach \x in {0,1,2,3,4}{
-	\foreach \y in {0,1,2,3,4}{
-		\foreach \z in {0,1,2,3,4}{
-			\fill[black] (\x, \y, \z) circle (0.1);
-		}
-	}
-};
-\usepackage{tikz}
-
-\begin{document}
-\begin{tikzpicture}
-	\foreach \x in {0,1,2,3,4}{
-		\foreach \y in {0,1,2,3,4}{
-			\foreach \z in {0,1,2,3,4}{
-				\fill[black] (\x, \y, \z) circle (0.1);
-			}
-		}
-	};
-\end{tikzpicture}
-\end{document}
-
diff --git a/Master/texmf-dist/doc/support/latexindent/success/tikz5.tex b/Master/texmf-dist/doc/support/latexindent/success/tikz5.tex
deleted file mode 100644
index b24b429e318..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/tikz5.tex
+++ /dev/null
@@ -1,34 +0,0 @@
-% arara: indent: {overwrite: yes}
-\documentclass[professionalfont, fleqn]{beamer}
-\mode<presentation>
-\usetheme{Warsaw}
-\usetheme{CambridgeUS}
-
-\usepackage{pgfplots}
-\usetikzlibrary{arrows,shapes,positioning}
-\graphicspath{{graphics/}}
-
-\begin{document}
-\frame
-{
-	\frametitle{Frame Title}
-	\begin{tikzpicture}
-		\begin{axis}
-			[
-				axis x line  = bottom,
-				axis y line  = left,
-				width        = 1.01\textwidth,
-				height       = .63\textwidth,          % Adjusted
-				ymax         = 93,
-				ymin         = 27,
-				ytick        = {30,40,...,90},
-				xmax         = 1993,
-				xmin         = 1967,
-				xtick        = {1970, 1980, ..., 1990},
-			]
-			\node[anchor=west] at (axis cs:1968.5,89.5){%
-				\textbullet\ Comment here about data
-			};
-		\end{axis}
-	\end{tikzpicture}
-}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/torusPGF.tex b/Master/texmf-dist/doc/support/latexindent/success/torusPGF.tex
deleted file mode 100644
index 0edf21a5e29..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/torusPGF.tex
+++ /dev/null
@@ -1,20 +0,0 @@
-% arara: indent: {overwrite: yes}
-\documentclass{article}
-\usepackage{pgfplots}
-
-\begin{document}
-
-\begin{tikzpicture}
-	\begin{axis}
-		\addplot3[surf,
-			colormap/cool,
-			samples=20,
-			domain=0:2*pi,y domain=0:2*pi,
-		z buffer=sort]
-		({(2+cos(deg(x)))*cos(deg(y+pi/2))},
-		{(2+cos(deg(x)))*sin(deg(y+pi/2))},
-		{sin(deg(x))});
-	\end{axis}
-\end{tikzpicture}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/torusPSTricks.tex b/Master/texmf-dist/doc/support/latexindent/success/torusPSTricks.tex
deleted file mode 100644
index cd8cc0db4ad..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/torusPSTricks.tex
+++ /dev/null
@@ -1,18 +0,0 @@
-% arara: indent: {overwrite: on, silent: yes}
-\documentclass{article}
-\usepackage{pst-solides3d}
-\begin{document}
-
-\begin{pspicture}(-3,-4)(3,6)
-	\psset{viewpoint=20 40 40 rtp2xyz,Decran=30,lightsrc=20 10 10}
-	\defFunction[algebraic]{torus}(u,v)
-	{(2+cos(u))*cos(v+\Pi)}
-	{(2+cos(u))*sin(v+\Pi)}
-	{sin(u)}
-	\psSolid[object=surfaceparametree,
-		base=-10 10 0 6.28,fillcolor=black!70,incolor=orange,
-		function=torus,ngrid=60 0.4,
-	opacity=0.25]
-\end{pspicture}
-
-\end{document}
diff --git a/Master/texmf-dist/doc/support/latexindent/success/trailingComments.tex b/Master/texmf-dist/doc/support/latexindent/success/trailingComments.tex
deleted file mode 100644
index 060e1888df7..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/trailingComments.tex
+++ /dev/null
@@ -1,7 +0,0 @@
-% arara: indent: {overwrite: yes, trace: true}
-\parbox{% more comments here
-	some stuff% comments go here }}}}
-	some \% stuff this is not a comment
-}
-some stuff
-some stuff
diff --git a/Master/texmf-dist/doc/support/latexindent/success/verbatim1.tex b/Master/texmf-dist/doc/support/latexindent/success/verbatim1.tex
deleted file mode 100644
index 635192051f2..00000000000
--- a/Master/texmf-dist/doc/support/latexindent/success/verbatim1.tex
+++ /dev/null
@@ -1,21 +0,0 @@
-% arara: indent: {overwrite: true, silent: on}
-\documentclass{article}
-
-\begin{document}
-\begin{verbatim}{%
-		l
-		S[table-format=3.0]
-		S[table-format=3.2]
-		S[table-format=2.2]
-		S[table-format=3.2]
-		S[table-format=-2.2]
-		S[table-format=3.2]
-		}
-		Latex              & 360 & 101.77 & 10.71 & 101.86 & 64.60  & 127.20 \\
-Manufacturing      & 360 & -7.33  & 12.59 & -7.24  & -49.00 & 22.00  \\
-		Cons           & 360 & -17.19 & 23.4  & -17.22 & -79.00 & 43.00  \\
-		Apple and Orange  & 360 &       3.38   & 13.84 & 3.60   & -47.00 & 29.00  \\
-		Services and Harry & 104 & -4.96  & 20.8  & -4.81  & -57.00 & 30.00  \\
-		Manchester         & 360 & -9.29  & 8.64  & -9.26  & -35.00 & 8.00   \\
-\end{verbatim}
-\end{document}
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/AlignmentAtAmpersand.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/AlignmentAtAmpersand.pm
new file mode 100644
index 00000000000..3f86cffd10f
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/AlignmentAtAmpersand.pm
@@ -0,0 +1,255 @@
+package LatexIndent::AlignmentAtAmpersand;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Tokens qw/%tokens/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/align_at_ampersand find_aligned_block/;
+our $alignmentBlockCounter;
+
+sub find_aligned_block{
+    my $self = shift;
+
+    return unless (${$self}{body} =~ m/(?!<\\)%\*\h*\\begin\{/s);
+
+    # aligned block
+	#      %* \begin{tabular}
+	#         1 & 2 & 3 & 4 \\
+	#         5 &   & 6 &   \\
+	#        %* \end{tabular}
+    $self->logger('looking for ALIGNED blocks marked by comments','heading')if($is_t_switch_active);
+    $self->logger(Dumper(\%{$masterSettings{lookForAlignDelims}})) if($is_t_switch_active);
+    while( my ($alignmentBlock,$yesno)= each %{$masterSettings{lookForAlignDelims}}){
+        if(ref $yesno eq "HASH"){
+              $yesno = (defined ${$yesno}{delims} ) ? ${$yesno}{delims} : 1;
+            }
+        if($yesno){
+            $self->logger("looking for $alignmentBlock:$yesno environments");
+
+            my $noIndentRegExp = qr/
+                            (
+                                (?!<\\)
+                                %
+                                \*
+                                \h*                     # possible horizontal spaces
+                                \\begin\{
+                                        $alignmentBlock  # environment name captured into $2
+                                       \}               # %* \begin{alignmentBlock} statement
+                            )
+                            (
+                                .*?
+                            )
+                            \R
+                            \h*
+                            (
+                                (?!<\\)
+                                %\*                     # %
+                                \h*                     # possible horizontal spaces
+                                \\end\{$alignmentBlock\} # \end{alignmentBlock}
+                            )                           # %* \end{<something>} statement
+                            #\R
+                        /sx;
+
+            while( ${$self}{body} =~ m/$noIndentRegExp/sx){
+
+              ${$self}{body} =~ s/
+                                    $noIndentRegExp
+                                /
+                                    # create a new Environment object
+                                    my $alignmentBlock = LatexIndent::AlignmentAtAmpersand->new( begin=>$1,
+                                                                          body=>$2,
+                                                                          end=>$3,
+                                                                          name=>$alignmentBlock,
+                                                                          modifyLineBreaksYamlName=>"environments",
+                                                                          linebreaksAtEnd=>{
+                                                                            begin=>1,
+                                                                            body=>1,
+                                                                            end=>0,
+                                                                          },
+                                                                          );
+            
+                                    # the settings and storage of most objects has a lot in common
+                                    $self->get_settings_and_store_new_object($alignmentBlock);
+                                    ${@{${$self}{children}}[-1]}{replacementText};
+                              /xseg;
+            } 
+      } else {
+            $self->logger("*not* looking for $alignmentBlock as $alignmentBlock:$yesno");
+      }
+    }
+    return;
+}
+
+sub modify_line_breaks_settings{
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+#    $self->remove_leading_space;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $alignmentBlockCounter++;
+    ${$self}{id} = "$tokens{alignmentBlock}$alignmentBlockCounter";
+    return;
+}
+
+sub align_at_ampersand{
+    my $self = shift;
+    return if(${$self}{bodyLineBreaks}==0);
+
+    # calculate the maximum number of ampersands in a row in the body
+    my $maximumNumberOfAmpersands = 0;
+    foreach(split("\n",${$self}{body})){
+        my $numberOfAmpersands = () = $_ =~ /(?<!\\)&/g;
+        $maximumNumberOfAmpersands = $numberOfAmpersands if($numberOfAmpersands>$maximumNumberOfAmpersands);
+    }
+
+    # create an array of zeros
+    my @columnSizes = (0) x ($maximumNumberOfAmpersands+1); 
+
+    # array for the new body
+    my @formattedBody;
+
+    # now loop back through the body, and store the maximum column size
+    foreach(split("\n",${$self}{body})){
+        # remove \\ and anything following it
+        my $endPiece;
+        if($_ =~ m/(\\\\.*)/){
+            $_ =~ s/(\\\\.*)//;
+            $endPiece = $1;
+        }
+
+        my $numberOfAmpersands = () = $_ =~ /(?<!\\)&/g;
+        if($numberOfAmpersands == $maximumNumberOfAmpersands){
+            # remove any trailing comments
+            my $trailingComments;
+            if($_ =~ m/$trailingCommentRegExp/ ){
+                $_ =~ s/($trailingCommentRegExp)//;
+                $trailingComments = $1; 
+            }
+
+            # if the line finishes with an &, then add an empty space,
+            # otherwise the column count is off
+            $_ .= ($_ =~ m/(?<!\\)&$/ ? " ":q());
+
+            # loop through the columns
+            my $columnCount = 0;
+            my $strippedRow = '';
+            foreach my $column (split(/(?<!\\)&/,$_)){
+                # remove leading space
+                $column =~ s/^\h*//;
+
+                # remove trailing space
+                $column =~ s/\h*$//; 
+
+                # if a column has finished with a \ then we need to add a trailing space, 
+                # otherwise the \ can be put next to &. See test-cases/texexchange/112343-gonzalo for example
+                $column .= ($column =~ m/\\$/ ? " ": q());
+
+                # store the column size
+                $columnSizes[$columnCount] = length($column) if(length($column)>$columnSizes[$columnCount]);
+
+                # put the row back together, using " " if the column is empty
+                $strippedRow .= ($columnCount>0 ? "&" : q() ).(length($column)>0 ? $column: " ");
+
+                # move on to the next column
+                $columnCount++;
+            }
+
+            # store the information
+            push(@formattedBody,{
+                                row=>$strippedRow,
+                                format=>1,
+                                endPiece=>($endPiece ? $endPiece :q() ),
+                                trailingComment=>($trailingComments ? $trailingComments :q() )});
+        } else {
+            # otherwise simply store the row
+            push(@formattedBody,{
+                                row=>$_.($endPiece ? $endPiece : q() ),
+                                format=>0});
+        }
+    }
+
+    # output some of the info so far to the log file
+    $self->logger("Column sizes of horizontally stripped formatted block (${$self}{name}): @columnSizes") if $is_t_switch_active;
+
+    # README: printf( formatting, expression)
+    #
+    #   formatting has the form %-50s & %-20s & %-19s
+    #   (the numbers have been made up for example)
+    #       the - symbols mean that each column should be left-aligned
+    #       the numbers represent how wide each column is
+    #       the s represents string
+    #       the & needs to be inserted
+
+    # join up the maximum string lengths using "s %-"
+    my $fmtstring = join("s & %-",@columnSizes);
+
+    # add %- to the beginning and an s to the end
+    $fmtstring = "%-".$fmtstring."s ";
+
+    # log file info
+    $self->logger("Formatting string is: $fmtstring",'heading') if $is_t_switch_active;
+
+    # finally, reformat the body
+    foreach(@formattedBody){
+        if(${$_}{format} and ${$_}{row} !~ m/^\h*$/){
+            # format the row, and put the trailing \\ and trailing comments back into the row
+            ${$_}{row} = sprintf($fmtstring,split(/(?<!\\)&/,${$_}{row})).(${$_}{endPiece} ? ${$_}{endPiece} :q() ).(${$_}{trailingComment}? ${$_}{trailingComment} : q() );
+
+            # possibly remove space ahead of \\
+            ${$_}{row} =~ s/\h*\\\\/\\\\/ if(!${$self}{alignDoubleBackSlash});
+
+            # possibly insert spaces infront of \\
+            if(defined ${$self}{spacesBeforeDoubleBackSlash} and ${$self}{spacesBeforeDoubleBackSlash}>=0 and !${$self}{alignDoubleBackSlash}){
+                my $horizontalSpaceToInsert = " "x (${$self}{spacesBeforeDoubleBackSlash});
+                ${$_}{row} =~ s/\h*\\\\/$horizontalSpaceToInsert\\\\/;
+            }
+        }
+
+        # if we have an empty row, it's possible that it originally had an end piece (e.g \\) and/or trailing comments
+        if(${$_}{row} =~ m/^\h*$/){
+            ${$_}{row} .= (${$_}{endPiece} ? ${$_}{endPiece} :q() ).(${$_}{trailingComment}? ${$_}{trailingComment} : q() );
+        }
+    }
+
+    # to the log file
+    if($is_tt_switch_active){    
+        $self->logger(${$_}{row},'ttrace') for @formattedBody;
+    }
+
+    # delete the original body
+    ${$self}{body} = q();
+
+    # update the body
+    ${$self}{body} .= ${$_}{row}."\n" for @formattedBody;
+
+    # if the \end{} statement didn't originally have a line break before it, we need to remove the final 
+    # line break added by the above
+    ${$self}{body} =~ s/\h*\R$//s if !${$self}{linebreaksAtEnd}{body};
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Arguments.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Arguments.pm
new file mode 100644
index 00000000000..569cead9584
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Arguments.pm
@@ -0,0 +1,380 @@
+package LatexIndent::Arguments;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/get_arguments_regexp find_opt_mand_arguments get_numbered_arg_regexp construct_arguments_regexp $optAndMandRegExp/;
+our $ArgumentCounter;
+our $optAndMandRegExp; 
+our $optAndMandRegExpWithLineBreaks; 
+
+sub construct_arguments_regexp{
+    my $self = shift;
+
+    $optAndMandRegExp = $self->get_arguments_regexp;
+
+    $optAndMandRegExpWithLineBreaks = $self->get_arguments_regexp(mode=>"lineBreaksAtEnd");
+}
+
+sub indent{
+    my $self = shift;
+    $self->logger("Arguments object doesn't receive any direct indentation, but its children will...",'heading') if $is_t_switch_active;
+    return;
+}
+
+sub find_opt_mand_arguments{
+    my $self = shift;
+
+    $self->logger("Searching ${$self}{name} for optional and mandatory arguments",'heading') if $is_t_switch_active;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    # the command object allows () 
+    my $objectDependentOptAndMandRegExp = (defined ${$self}{optAndMandArgsRegExp} ? ${$self}{optAndMandArgsRegExp} : $optAndMandRegExpWithLineBreaks);
+
+    if(${$self}{body} =~ m/^$objectDependentOptAndMandRegExp\h*($trailingCommentRegExp)?/){
+        $self->logger("Optional/Mandatory arguments found in ${$self}{name}: $1",'heading') if $is_t_switch_active;
+
+        # create a new Arguments object
+        # The arguments object is a little different to most
+        # other objects, as it is created purely for its children,
+        # so some of the properties common to other objects, such 
+        # as environment, ifelsefi, etc do not exist for Arguments;
+        # they will, however, exist for its children: OptionalArgument, MandatoryArgument
+        my $arguments = LatexIndent::Arguments->new(begin=>"",
+                                                name=>${$self}{name}.":arguments",
+                                                parent=>${$self}{name},
+                                                body=>$1,
+                                                linebreaksAtEnd=>{
+                                                  end=>$2?1:0,
+                                                },
+                                                end=>"",
+                                                regexp=>$objectDependentOptAndMandRegExp,
+                                                endImmediatelyFollowedByComment=>$2?0:($3?1:0),
+                                              );
+
+        # give unique id
+        $arguments->create_unique_id;
+
+        # determine which comes first, optional or mandatory
+        if(${$arguments}{body} =~ m/.*?((?<!\\)\{|\[)/s){
+
+            if($1 eq "\["){
+                $self->logger("Searching for optional arguments, and then mandatory (optional found first)") if $is_t_switch_active;
+                # look for optional arguments
+                $arguments->find_optional_arguments;
+
+                # look for mandatory arguments
+                $arguments->find_mandatory_arguments;
+            } else {
+                $self->logger("Searching for mandatory arguments, and then optional (mandatory found first)") if $is_t_switch_active;
+                # look for mandatory arguments
+                $arguments->find_mandatory_arguments;
+
+                # look for optional arguments
+                $arguments->find_optional_arguments;
+            }
+
+        } else {
+                $self->logger("Searching for round brackets ONLY") if $is_t_switch_active;
+                # look for round brackets
+                $arguments->find_round_brackets;
+        }
+
+        # examine *first* child
+        #   situation: parent BodyStartsOnOwnLine >= 1, but first child has BeginStartsOnOwnLine == 0 || BeginStartsOnOwnLine == undef
+        #   problem: the *body* of parent actually starts after the arguments
+        #   solution: remove the linebreak at the end of the begin statement of the parent
+        if(defined ${$self}{BodyStartsOnOwnLine} and ${$self}{BodyStartsOnOwnLine}>=1){
+            if( !(defined ${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine} and ${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine}>=1) 
+                    and ${$self}{body} !~ m/^$blankLineToken/){
+                my $BodyStringLogFile = ${$self}{aliases}{BodyStartsOnOwnLine}||"BodyStartsOnOwnLine";
+                my $BeginStringLogFile = ${${${$arguments}{children}}[0]}{aliases}{BeginStartsOnOwnLine}||"BeginStartsOnOwnLine";
+                $self->logger("$BodyStringLogFile = 1 (in ${$self}{name}), but first argument should not begin on its own line (see $BeginStringLogFile)") if $is_t_switch_active;
+                $self->logger("Removing line breaks at the end of ${$self}{begin}") if $is_t_switch_active;
+                ${$self}{begin} =~ s/\R*$//s;
+                ${$self}{linebreaksAtEnd}{begin} = 0;
+            }
+        }
+
+        # situation: preserveBlankLines is active, so the body may well begin with a blank line token
+        #            which means that ${$self}{linebreaksAtEnd}{begin} *should be* 1
+        if(${${${$arguments}{children}}[0]}{body} =~ m/^($blankLineToken)/){
+            $self->logger("Updating {linebreaksAtEnd}{begin} for ${$self}{name} as $blankLineToken or blank line found at beginning of argument child") if $is_t_switch_active;
+            ${$self}{linebreaksAtEnd}{begin} = 1 
+          }
+
+        # examine *first* child
+        #   situation: parent BodyStartsOnOwnLine == -1, but first child has BeginStartsOnOwnLine == 1
+        #   problem: the *body* of parent actually starts after the arguments
+        #   solution: add a linebreak at the end of the begin statement of the parent so that
+        #              the child settings are obeyed.
+        #              BodyStartsOnOwnLine == 0 will actually be controlled by the last arguments' 
+        #              settings of EndFinishesWithLineBreak
+        if( ${$self}{linebreaksAtEnd}{begin} == 0
+           and ((defined ${$self}{BodyStartsOnOwnLine} and ${$self}{BodyStartsOnOwnLine}==-1) 
+                    or !(defined ${$self}{BodyStartsOnOwnLine})) 
+              ){
+            if(defined ${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine} and ${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine}>=1){
+                my $BodyStringLogFile = ${$self}{aliases}{BodyStartsOnOwnLine}||"BodyStartsOnOwnLine";
+                my $BeginStringLogFile = ${${${$arguments}{children}}[0]}{aliases}{BeginStartsOnOwnLine}||"BeginStartsOnOwnLine";
+                my $BodyValue = (defined ${$self}{BodyStartsOnOwnLine}) ? ${$self}{BodyStartsOnOwnLine} : "0";
+                $self->logger("$BodyStringLogFile = $BodyValue (in ${$self}{name}), but first argument *should* begin on its own line (see $BeginStringLogFile)") if $is_t_switch_active;
+
+                # possibly add a comment at the end of the begin statement
+                my $trailingCommentToken = q();
+                if(${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine}==1){
+                    $self->logger("Adding line breaks at the end of ${$self}{begin} (first argument, see $BeginStringLogFile == ${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine})") if $is_t_switch_active;
+                } elsif(${${${$arguments}{children}}[0]}{BeginStartsOnOwnLine}==2){
+                    $self->logger("Adding a % at the end of begin, ${$self}{begin} followed by a linebreak ($BeginStringLogFile == 2)") if $is_t_switch_active;
+                    $trailingCommentToken = "%".$self->add_comment_symbol;
+                    $self->logger("Removing trailing space on ${$self}{begin}") if $is_t_switch_active;
+                    ${$self}{begin} =~ s/\h*$//s;
+                }
+
+                # modification
+                ${$self}{begin} .= "$trailingCommentToken\n";
+                ${$self}{linebreaksAtEnd}{begin} = 1;
+            }
+        }
+
+        # the replacement text can be just the ID, but the ID might have a line break at the end of it
+        ${$arguments}{replacementText} = ${$arguments}{id};
+
+        # children need to receive ancestor information, see test-cases/commands/commands-triple-nested.tex
+        foreach (@{${$arguments}{children}}){
+            $self->logger("Updating argument children of ${$self}{name} to include ${$self}{id} in ancestors") if $is_t_switch_active;
+            push(@{${$_}{ancestors}},{ancestorID=>${$self}{id},ancestorIndentation=>${$self}{indentation},type=>"natural"});
+        }
+
+        # the argument object only needs a trailing line break if the *last* child
+        # did not add one at the end, and if BodyStartsOnOwnLine >= 1
+        if( (defined ${${${$arguments}{children}}[-1]}{EndFinishesWithLineBreak} and ${${${$arguments}{children}}[-1]}{EndFinishesWithLineBreak}<1)
+            and (defined ${$self}{BodyStartsOnOwnLine} and ${$self}{BodyStartsOnOwnLine}>=1) ){
+            $self->logger("Updating replacementtext to include a linebreak for arguments in ${$self}{name}") if $is_t_switch_active;
+            ${$arguments}{replacementText} .= "\n" if(${$arguments}{linebreaksAtEnd}{end});
+        }
+
+        # store children in special hash
+        push(@{${$self}{children}},$arguments);
+
+        # remove the environment block, and replace with unique ID
+        ${$self}{body} =~ s/${$arguments}{regexp}/${$arguments}{replacementText}/;
+
+        # delete the regexp, as there's no need for it
+        delete ${${${$self}{children}}[-1]}{regexp};
+
+        $self->logger(Dumper(\%{$arguments}),'ttrace') if($is_tt_switch_active);
+        $self->logger("replaced with ID: ${$arguments}{id}") if $is_t_switch_active;
+    } else {
+        $self->logger("... no arguments found") if $is_t_switch_active;
+    }
+
+}
+
+
+sub create_unique_id{
+    my $self = shift;
+
+    $ArgumentCounter++;
+    ${$self}{id} = "$tokens{arguments}$ArgumentCounter$tokens{endOfToken}";
+    return;
+}
+
+sub get_numbered_arg_regexp{
+
+    # for example #1 #2, etc
+    my $numberedArgRegExp = qr/(?:#\d\h*;?,?\/?)+/;
+    return $numberedArgRegExp;
+}
+
+sub get_arguments_regexp{
+
+    my $self = shift;
+    my %input = @_;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    # some calls to this routine need to account for the linebreaks at the end, some do not
+    my $lineBreaksAtEnd = (defined ${input}{mode} and ${input}{mode} eq 'lineBreaksAtEnd')?'\R*':q();
+
+    # for example #1 #2, etc
+    my $numberedArgRegExp = $self->get_numbered_arg_regexp;
+
+    # beamer special
+    my $beamerRegExp = qr/\<.*?\>/;
+
+    # commands are allowed strings between arguments, e.g node, decoration, etc, specified in stringsAllowedBetweenArguments
+    my $stringsBetweenArguments = q();
+
+    if(defined ${input}{stringBetweenArguments} and ${input}{stringBetweenArguments}==1){
+        # grab the strings allowed between arguments
+        my %stringsAllowedBetweenArguments = %{${$masterSettings{commandCodeBlocks}}{stringsAllowedBetweenArguments}};
+
+        while( my ($allowedStringName,$allowedStringValue)= each %stringsAllowedBetweenArguments){
+            # change + and - to escaped characters
+            $allowedStringName =~ s/\+/\\+/g;
+            $allowedStringName =~ s/\-/\\-/g;
+
+            # form the regexp
+            $stringsBetweenArguments .= ($stringsBetweenArguments eq '' ? q() : "|").$allowedStringName if $allowedStringValue; 
+        }
+
+        # report to log file
+        $self->logger("Strings allowed between arguments $stringsBetweenArguments (see stringsAllowedBetweenArguments)",'heading') if $is_t_switch_active;
+     }
+
+    if(defined ${input}{roundBrackets} and ${input}{roundBrackets}==1){
+            # arguments regexp
+            return qr/
+                                  (                          # capture into $1
+                                     (?:                  
+                                        (?:\h|\R|$blankLineToken|$trailingCommentRegExp|$numberedArgRegExp|$beamerRegExp|_|\^|$stringsBetweenArguments)* 
+                                        (?:
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \[
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\[|(?<!\\)\{) 
+                                                         ).
+                                                     )*?     # not including [, but \[ ok
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \]          # [optional arguments]
+                                             )
+                                             |               # OR
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \{
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\{|(?<!\\)\[) 
+                                                         ).
+                                                     )*?     # not including {, but \{ ok
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \}          # {mandatory arguments}
+                                             )
+                                             |               # OR
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \(\$
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\$) 
+                                                         ).
+                                                     )*?     # not including $
+                                                 \$
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \)          # {mandatory arguments}
+                                             )
+                                             |               # OR
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \(
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\() 
+                                                         ).
+                                                     )*?     # not including {, but \{ ok
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \)          # {mandatory arguments}
+                                             )
+                                        )
+                                     )
+                                     +                       # at least one of the above
+                                     # NOT followed by
+                                     (?!
+                                       (?:
+                                           (?:\h|\R|$blankLineToken|$trailingCommentRegExp|$numberedArgRegExp|$beamerRegExp|$stringsBetweenArguments)*  # 0 or more h-space, blanklines, trailing comments
+                                           (?:
+                                             (?:(?<!\\)\[)
+                                             |
+                                             (?:(?<!\\)\{)
+                                             |
+                                             (?:(?<!\\)\()
+                                           )
+                                       )
+                                     )
+                                     \h*
+                                     ($lineBreaksAtEnd)
+                                  )                  
+                                  /sx;
+        } else {
+            return qr/
+                                  (                          # capture into $1
+                                     (?:                  
+                                        (?:\h|\R|$blankLineToken|$trailingCommentRegExp|$numberedArgRegExp|$beamerRegExp|_|\^|$stringsBetweenArguments)* 
+                                        (?:
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \[
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\[|(?<!\\)\{) 
+                                                         ).
+                                                     )*?     # not including [, but \[ ok
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \]          # [optional arguments]
+                                             )
+                                             |               # OR
+                                             (?:
+                                                 \h*         # 0 or more spaces
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \{
+                                                     (?:
+                                                         (?!
+                                                             (?:(?<!\\)\{|(?<!\\)\[) 
+                                                         ).
+                                                     )*?     # not including {, but \{ ok
+                                                 (?<!\\)     # not immediately pre-ceeded by \
+                                                 \}          # {mandatory arguments}
+                                             )
+                                        )
+                                     )
+                                     +                       # at least one of the above
+                                     # NOT followed by
+                                     (?!
+                                       (?:
+                                           (?:\h|\R|$blankLineToken|$trailingCommentRegExp|$numberedArgRegExp|$beamerRegExp)*  # 0 or more h-space, blanklines, trailing comments
+                                           (?:
+                                             (?:(?<!\\)\[)
+                                             |
+                                             (?:(?<!\\)\{)
+                                           )
+                                       )
+                                     )
+                                     \h*
+                                     ($lineBreaksAtEnd)
+                                  )                  
+                                  /sx;
+        }
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/BackUpFileProcedure.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/BackUpFileProcedure.pm
new file mode 100644
index 00000000000..dcb3b5bb74e
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/BackUpFileProcedure.pm
@@ -0,0 +1,135 @@
+package LatexIndent::BackUpFileProcedure;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/%switches/;
+use File::Basename;             # to get the filename and directory path
+use File::Copy;                 # to copy the original file to backup (if overwrite option set)
+use Exporter qw/import/;
+our @EXPORT_OK = qw/create_back_up_file/;
+
+# copy main file to a back up in the case of the overwrite switch being active
+
+sub create_back_up_file{
+    my $self = shift;
+
+    return unless($switches{overwrite});
+
+    # if we want to over write the current file create a backup first
+    $self->logger("Backup procedure (-w flag active):",'heading');
+
+    my $fileName = ${$self}{fileName};
+
+    # grab the file extension preferences
+    my %fileExtensionPreference= %{$masterSettings{fileExtensionPreference}};
+
+    # sort the file extensions by preference 
+    my @fileExtensions = sort { $fileExtensionPreference{$a} <=> $fileExtensionPreference{$b} } keys(%fileExtensionPreference);
+
+    # backup file name is the base name
+    my $backupFile = basename(${$self}{fileName},@fileExtensions);
+    
+    # add the user's backup directory to the backup path
+    $backupFile = "${$self}{cruftDirectory}/$backupFile";
+
+    # local variables, determined from the YAML settings
+    my $onlyOneBackUp = $masterSettings{onlyOneBackUp};
+    my $maxNumberOfBackUps = $masterSettings{maxNumberOfBackUps};
+    my $cycleThroughBackUps= $masterSettings{cycleThroughBackUps};
+    my $backupExtension= $masterSettings{backupExtension};
+    
+    # if both ($onlyOneBackUp and $maxNumberOfBackUps) then we have
+    # a conflict- er on the side of caution and turn off onlyOneBackUp
+    if($onlyOneBackUp and $maxNumberOfBackUps>1) {
+        $self->logger("WARNING: onlyOneBackUp=$onlyOneBackUp and maxNumberOfBackUps: $maxNumberOfBackUps");
+        $self->logger("setting onlyOneBackUp=0 which will allow you to reach $maxNumberOfBackUps back ups");
+        $onlyOneBackUp = 0;
+    }
+    
+    # if the user has specified that $maxNumberOfBackUps = 1 then
+    # they only want one backup
+    if($maxNumberOfBackUps==1) {
+        $onlyOneBackUp=1 ;
+        $self->logger("FYI: you set maxNumberOfBackUps=1, so I'm setting onlyOneBackUp: 1 ");
+    } elsif($maxNumberOfBackUps<=0 and !$onlyOneBackUp) {
+        $onlyOneBackUp=0 ;
+        $maxNumberOfBackUps=-1;
+    }
+    
+    # if onlyOneBackUp is set, then the backup file will
+    # be overwritten each time
+    if($onlyOneBackUp) {
+        $backupFile .= $backupExtension;
+        $self->logger("copying $fileName to $backupFile");
+        $self->logger("$backupFile was overwritten (see onlyOneBackUp)") if (-e $backupFile);
+    } else {
+        # start with a backup file .bak0 (or whatever $backupExtension is present)
+        my $backupCounter = 0;
+        $backupFile .= $backupExtension.$backupCounter;
+    
+        # if it exists, then keep going: .bak0, .bak1, ...
+        while (-e $backupFile or $maxNumberOfBackUps>1) {
+            if($backupCounter==$maxNumberOfBackUps) {
+                $self->logger("maxNumberOfBackUps reached ($maxNumberOfBackUps, see maxNumberOfBackUps)");
+    
+                # some users may wish to cycle through back up files, e.g:
+                #    copy myfile.bak1 to myfile.bak0
+                #    copy myfile.bak2 to myfile.bak1
+                #    copy myfile.bak3 to myfile.bak2
+                #
+                #    current back up is stored in myfile.bak4
+                if($cycleThroughBackUps) {
+                    $self->logger("cycleThroughBackUps detected (see cycleThroughBackUps) ");
+                    for(my $i=1;$i<=$maxNumberOfBackUps;$i++) {
+                        # remove number from backUpFile
+                        my $oldBackupFile = $backupFile;
+                        $oldBackupFile =~ s/$backupExtension.*/$backupExtension/;
+                        my $newBackupFile = $oldBackupFile;
+    
+                        # add numbers back on
+                        $oldBackupFile .= $i;
+                        $newBackupFile .= $i-1;
+    
+                        # check that the oldBackupFile exists
+                        if(-e $oldBackupFile){
+                        $self->logger(" copying $oldBackupFile to $newBackupFile ");
+                            copy($oldBackupFile,$newBackupFile) or die "Could not write to backup file $backupFile. Please check permissions. Exiting.";
+                        }
+                    }
+                }
+    
+                # rest maxNumberOfBackUps
+                $maxNumberOfBackUps=1 ;
+                last; # break out of the loop
+            } elsif(!(-e $backupFile)) {
+                $maxNumberOfBackUps=1;
+                last; # break out of the loop
+            }
+            $self->logger(" $backupFile already exists, incrementing by 1... (see maxNumberOfBackUps and onlyOneBackUp)");
+            $backupCounter++;
+            $backupFile =~ s/$backupExtension.*/$backupExtension$backupCounter/;
+        }
+        $self->logger("copying $fileName to $backupFile");
+    }
+    
+    # output these lines to the log file
+    $self->logger("Backup file: ",$backupFile,"");
+    $self->logger("Overwriting file: ",$fileName,"");
+    copy($fileName,$backupFile) or die "Could not write to backup file $backupFile. Please check permissions. Exiting.";
+}
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/BlankLines.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/BlankLines.pm
new file mode 100644
index 00000000000..f4572dd6797
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/BlankLines.pm
@@ -0,0 +1,108 @@
+package LatexIndent::BlankLines;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/protect_blank_lines unprotect_blank_lines condense_blank_lines/;
+
+sub protect_blank_lines{
+    return unless $is_m_switch_active;
+    my $self = shift;
+
+    unless(${$masterSettings{modifyLineBreaks}}{preserveBlankLines}){
+        $self->logger("Blank lines will not be protected (preserveBlankLines=0)",'heading') if $is_t_switch_active;
+        return
+    }
+
+    $self->logger("Protecting blank lines (see preserveBlankLines)",'heading') if $is_t_switch_active;
+    ${$self}{body} =~ s/^(\h*)?\R/$tokens{blanklines}\n/mg;
+    return;
+}
+
+sub condense_blank_lines{
+    return unless $is_m_switch_active;
+
+    return unless ${$masterSettings{modifyLineBreaks}}{condenseMultipleBlankLinesInto}>0;
+
+    my $self = shift;
+
+    # if preserveBlankLines is set to 0, then the blank-line-token will not be present
+    # in the document -- we change that here
+    if(${$masterSettings{modifyLineBreaks}}{preserveBlankLines}==0){
+        # turn the switch on
+        ${$masterSettings{modifyLineBreaks}}{preserveBlankLines}=1;
+
+        # log file information
+        $self->logger("Updating body to inclued blank line token, this requires preserveBlankLines = 1",'ttrace') if($is_tt_switch_active);
+        $self->logger("(any blanklines that could have been removed, would have done so by this point)",'ttrace') if($is_tt_switch_active);
+
+        # make the call
+        $self->protect_blank_lines ;
+        $self->logger("body now looks like:\n${$self}{body}",'ttrace') if($is_tt_switch_active);
+     }
+
+    # grab the value from the settings
+    my $condenseMultipleBlankLinesInto = ${$masterSettings{modifyLineBreaks}}{condenseMultipleBlankLinesInto};
+
+    # grab the blank-line-token
+    my $blankLineToken = $tokens{blanklines};
+
+    # condense!
+    $self->logger("Condensing multiple blank lines into $condenseMultipleBlankLinesInto (see condenseMultipleBlankLinesInto)",'heading') if $is_t_switch_active;
+    my $replacementToken = $blankLineToken;
+    for (my $i=1; $i<$condenseMultipleBlankLinesInto; $i++ ){
+        $replacementToken .= "\n$blankLineToken";
+    }
+
+    $self->logger("blank line replacement token: $replacementToken",'ttrace') if($is_tt_switch_active);
+    ${$self}{body} =~ s/($blankLineToken\h*\R*\h*){1,}$blankLineToken/$replacementToken/mgs;
+    $self->logger("body now looks like:\n${$self}{body}",'ttrace') if($is_tt_switch_active);
+    return;
+}
+
+sub unprotect_blank_lines{
+    return unless $is_m_switch_active;
+
+    return unless ${$masterSettings{modifyLineBreaks}}{preserveBlankLines};
+    my $self = shift;
+
+    $self->logger("Unprotecting blank lines (see preserveBlankLines)",'heading') if $is_t_switch_active;
+    my $blankLineToken = $tokens{blanklines};
+
+    # loop through the body, looking for the blank line token
+    while(${$self}{body} =~ m/$blankLineToken/m){
+        # when the blank line token occupies the whole line
+        if(${$self}{body} =~ m/^\h*$blankLineToken$/m){
+            $self->logger("Replacing purely blank lines",'heading') if $is_tt_switch_active;
+            ${$self}{body} =~ s/^\h*$blankLineToken$//mg;
+            $self->logger("body now looks like:\n${$self}{body}",'ttrace') if($is_tt_switch_active);
+        }
+        # otherwise the blank line has been deleted, so we compensate with an extra
+        if(${$self}{body} =~ m/(^\h*)?$blankLineToken/m){
+            $self->logger("Replacing blank line token that doesn't take up whole line",'heading') if $is_tt_switch_active;
+            ${$self}{body} =~ s/(^\h*)?$blankLineToken/$1?"\n".$1:"\n"/me;
+            $self->logger("body now looks like:\n${$self}{body}",'ttrace') if($is_tt_switch_active);
+        }
+    }
+    $self->logger("Finished unprotecting lines (see preserveBlankLines)",'heading') if $is_t_switch_active;
+    $self->logger("body now looks like ${$self}{body}",'ttrace') if($is_tt_switch_active);
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Braces.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Braces.pm
new file mode 100644
index 00000000000..c5122fd1275
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Braces.pm
@@ -0,0 +1,188 @@
+package LatexIndent::Braces;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Command qw/$commandRegExp $commandRegExpTrailingComment $optAndMandAndRoundBracketsRegExpLineBreaks/;
+use LatexIndent::KeyEqualsValuesBraces qw/$key_equals_values_bracesRegExp $key_equals_values_bracesRegExpTrailingComment/;
+use LatexIndent::NamedGroupingBracesBrackets qw/$grouping_braces_regexp $grouping_braces_regexpTrailingComment/;
+use LatexIndent::UnNamedGroupingBracesBrackets qw/$un_named_grouping_braces_RegExp $un_named_grouping_braces_RegExp_trailing_comment/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_commands_or_key_equals_values_braces/;
+our $commandCounter;
+
+sub find_commands_or_key_equals_values_braces{
+
+    my $self = shift;
+
+    $self->logger("Searching for commands with optional and/or mandatory arguments AND key = {value}",'heading') if $is_t_switch_active ;
+
+    # match either a \\command or key={value}
+    while( ${$self}{body} =~ m/$commandRegExpTrailingComment/
+                            or  
+           ${$self}{body} =~ m/$key_equals_values_bracesRegExpTrailingComment/
+                            or
+           ${$self}{body} =~ m/$grouping_braces_regexpTrailingComment/
+                            or
+           ${$self}{body} =~ m/$un_named_grouping_braces_RegExp_trailing_comment/
+         ){
+      if(${$self}{body} =~ m/$commandRegExpTrailingComment/){ 
+
+        # global substitution
+        ${$self}{body} =~ s/
+                            $commandRegExpTrailingComment
+                          /
+                            # log file output
+                            $self->logger("command found: $2",'heading') if $is_t_switch_active ;
+
+                            # create a new command object
+                            my $command = LatexIndent::Command->new(begin=>$1.$2.($3?$3:q()).($4?$4:q()),
+                                                                    name=>$2,
+                                                                    body=>$5.($8?$8:($10?$10:q())),    # $8 is linebreak, $10 is trailing comment
+                                                                    end=>q(),
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$4?1:0,
+                                                                      end=>$8?1:0,            # $8 is linebreak before comment check, $10 is after
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"commands",
+                                                                    endImmediatelyFollowedByComment=>$8?0:($10?1:0),
+                                                                    aliases=>{
+                                                                      # begin statements
+                                                                      BeginStartsOnOwnLine=>"CommandStartsOnOwnLine",
+                                                                      # body statements
+                                                                      BodyStartsOnOwnLine=>"CommandNameFinishesWithLineBreak",
+                                                                    },
+                                                                    optAndMandArgsRegExp=>$optAndMandAndRoundBracketsRegExpLineBreaks,
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($command);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($8?($10?$10:q()):q());
+                         /xseg;
+
+      } elsif (${$self}{body} =~ m/$key_equals_values_bracesRegExpTrailingComment/){
+
+        # global substitution
+        ${$self}{body} =~ s/
+                              $key_equals_values_bracesRegExpTrailingComment
+                           /
+                           # log file output
+                           $self->logger("key_equals_values_braces found: $3",'heading') if $is_t_switch_active ;
+
+                           # create a new key_equals_values_braces object
+                           my $key_equals_values_braces = LatexIndent::KeyEqualsValuesBraces->new(
+                                                                   begin=>($2?$2:q()).$3.$4.($5?$5:q()),
+                                                                   name=>$3,
+                                                                   body=>$6.($9?$9:($10?$10:q()).($11?$11:q())),     # $9 is linebreak before comment check, $11 is trailing comment
+                                                                   end=>q(),
+                                                                   linebreaksAtEnd=>{
+                                                                     begin=>$5?1:0,
+                                                                     end=>$9?1:0,                # $9 is linebreak before comment check
+                                                                   },
+                                                                   modifyLineBreaksYamlName=>"keyEqualsValuesBracesBrackets",
+                                                                   beginningbit=>$1,
+                                                                   endImmediatelyFollowedByComment=>$9?0:($11?1:0),
+                                                                   aliases=>{
+                                                                     # begin statements
+                                                                     BeginStartsOnOwnLine=>"KeyStartsOnOwnLine",
+                                                                     # body statements
+                                                                     BodyStartsOnOwnLine=>"EqualsFinishesWithLineBreak",
+                                                                   },
+                                                                   additionalAssignments=>["EqualsStartsOnOwnLine"],
+                                                                 );
+                    
+                           # the settings and storage of most objects has a lot in common
+                           $self->get_settings_and_store_new_object($key_equals_values_braces);
+                           ${@{${$self}{children}}[-1]}{replacementText}.($9?($11?$11:q()):q());
+                           /xseg;
+                    
+      } elsif (${$self}{body} =~ m/$grouping_braces_regexpTrailingComment/){
+
+        # global substitution
+        ${$self}{body} =~ s/
+                            $grouping_braces_regexpTrailingComment
+                            /
+                            # log file output
+                            $self->logger("named grouping braces found: $2",'heading') if $is_t_switch_active ;
+
+                            # create a new key_equals_values_braces object
+                            my $grouping_braces = LatexIndent::NamedGroupingBracesBrackets->new(
+                                                                    begin=>$2.($3?$3:q()).($4?$4:q()),
+                                                                    name=>$2,
+                                                                    body=>$5.($8?$8:($9?$9:q())),    
+                                                                    end=>q(),
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$4?1:0,
+                                                                      end=>$8?1:0,
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"namedGroupingBracesBrackets",
+                                                                    beginningbit=>$1,
+                                                                    endImmediatelyFollowedByComment=>$8?0:($9?1:0),
+                                                                    aliases=>{
+                                                                      # begin statements
+                                                                      BeginStartsOnOwnLine=>"NameStartsOnOwnLine",
+                                                                      # body statements
+                                                                      BodyStartsOnOwnLine=>"NameFinishesWithLineBreak",
+                                                                    },
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($grouping_braces);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($8?($9?$9:q()):q());
+                           /xseg;
+
+    } elsif (${$self}{body} =~ m/$un_named_grouping_braces_RegExp_trailing_comment/) {
+        # global substitution
+        ${$self}{body} =~ s/
+                            $un_named_grouping_braces_RegExp_trailing_comment
+                          /
+                            # log file output
+                            $self->logger("UNnamed grouping braces found: (no name, by definition!)",'heading') if $is_t_switch_active ;
+
+                            # create a new Un-named-grouping-braces-brackets object
+                            my $un_named_grouping_braces = LatexIndent::UnNamedGroupingBracesBrackets->new(
+                                                                    begin=>q(),
+                                                                    name=>"always-un-named",
+                                                                    body=>$3.($6?$6:($8?$8:q())),    
+                                                                    end=>q(),
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$2?1:0,
+                                                                      end=>$6?1:0,
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"UnNamedGroupingBracesBrackets",
+                                                                    beginningbit=>$1.($2?$2:q()),
+                                                                    endImmediatelyFollowedByComment=>$6?0:($8?1:0),
+                                                                    # begin statements
+                                                                    BeginStartsOnOwnLine=>0,
+                                                                    # body statements
+                                                                    BodyStartsOnOwnLine=>0,
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($un_named_grouping_braces);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($6?($8?$8:q()):q());
+                         /xseg;
+
+    }
+  }
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Command.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Command.pm
new file mode 100755
index 00000000000..022064fd1d7
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Command.pm
@@ -0,0 +1,178 @@
+package LatexIndent::Command;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/construct_command_regexp $commandRegExp $commandRegExpTrailingComment $optAndMandAndRoundBracketsRegExpLineBreaks/;
+our $commandCounter;
+our $commandRegExp;
+our $commandRegExpTrailingComment; 
+our $optAndMandAndRoundBracketsRegExp; 
+our $optAndMandAndRoundBracketsRegExpLineBreaks; 
+
+# store the regular expresssion for matching and replacing 
+sub construct_command_regexp{
+    my $self = shift;
+
+    $optAndMandAndRoundBracketsRegExp = $self->get_arguments_regexp(
+                                                                    roundBrackets=>${$masterSettings{commandCodeBlocks}}{roundParenthesesAllowed},
+                                                                    stringBetweenArguments=>1);
+
+    $optAndMandAndRoundBracketsRegExpLineBreaks = $self->get_arguments_regexp(
+                                                                    roundBrackets=>${$masterSettings{commandCodeBlocks}}{roundParenthesesAllowed},
+                                                                    mode=>"lineBreaksAtEnd",
+                                                                    stringBetweenArguments=>1);
+
+    # construct the command regexp
+    $commandRegExp = qr/
+              (\\|@)   
+              (
+               [+a-zA-Z@\*0-9_\:]+? # lowercase|uppercase letters, @, *, numbers
+              )                
+              (\h*)
+              (\R*)?
+              ($optAndMandAndRoundBracketsRegExp)
+              (\R)?
+            /sx;
+
+    # command regexp with trailing comment
+    $commandRegExpTrailingComment = qr/$commandRegExp(\h*)((?:$trailingCommentRegExp\h*)*)/;
+
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # check for adding/removing linebreaks before =
+    $self->check_linebreaks_before_equals;
+
+    # search for arguments
+    $self->find_opt_mand_arguments;
+
+    # situation: ${${$self}{linebreaksAtEnd}}{end} == 1, and the argument container object
+    # still contains a linebreak at the end; in this case, we need to remove the linebreak from 
+    # the container object
+    if(${${$self}{linebreaksAtEnd}}{end} == 1 
+      and ${${${$self}{children}}[0]}{body} =~ m/\R$/s
+      and !${$self}{endImmediatelyFollowedByComment}){
+        $self->logger("Removing linebreak from argument container of ${$self}{name}") if $is_t_switch_active;
+        ${${${$self}{children}}[0]}{body} =~ s/\R$//s;
+        ${${${${$self}{children}}[0]}{linebreaksAtEnd}}{body} = 0;
+    }
+
+    # situation: ${${$self}{linebreaksAtEnd}}{end} == 1 and the last argument specifies
+    # EndFinishesWithLineBreaks = 0 (see test-cases/commands/just-one-command-mod10.tex)
+    if(${${$self}{linebreaksAtEnd}}{end} == 1 
+        and defined ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} 
+        and ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} == -1
+        ){
+          $self->logger("Switching linebreaksAtEnd{end} to be 0 in command ${$self}{name} as last argument specifies EndFinishesWithLineBreak == 0") if $is_t_switch_active;
+          ${${$self}{linebreaksAtEnd}}{end} = 0;
+          ${$self}{EndFinishesWithLineBreak} = -1;
+        }
+    
+    # if the last argument finishes with a linebreak, it won't get interpreted at 
+    # the right time (see test-cases/commands/commands-one-line-nested-simple-mod1.tex for example)
+    # so this little bit fixes it
+    if(${${${${${$self}{children}}[0]}{children}[-1]}{linebreaksAtEnd}}{end} and ${${$self}{linebreaksAtEnd}}{end} == 0 
+        and defined ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} 
+        and ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} >= 1 
+        and !${$self}{endImmediatelyFollowedByComment}){
+
+        # update the Command object
+        $self->logger("Adjusting linebreaksAtEnd in command ${$self}{name}") if $is_t_switch_active;
+        ${${$self}{linebreaksAtEnd}}{end} = ${${${${${$self}{children}}[0]}{children}[-1]}{linebreaksAtEnd}}{end};
+        ${$self}{replacementText} .= "\n";
+
+        # update the argument object
+        $self->logger("Adjusting argument object in command, ${$self}{name}") if $is_t_switch_active;
+        ${${${${$self}{children}}[0]}{linebreaksAtEnd}}{body} = 0;
+        ${${${$self}{children}}[0]}{body} =~ s/\R$//s;
+
+        # update the last mandatory/optional argument
+        $self->logger("Adjusting last argument in command, ${$self}{name}") if $is_t_switch_active;
+        ${${${${${$self}{children}}[0]}{children}[-1]}{linebreaksAtEnd}}{end} = 0;
+        ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} = -1;
+        ${${${${$self}{children}}[0]}{children}[-1]}{replacementText} =~ s/\R$//s;
+
+        # output to log file
+        $self->logger(Dumper(${${${$self}{children}}[0]}{children}[-1])) if $is_t_switch_active;
+    }
+
+    # situation: ${${$self}{linebreaksAtEnd}}{end} == 1 and the last argument has added 
+    # a line break, which can result in a bogus blank line (see test-cases/commands/just-one-command.tex with mand-args-mod1.yaml)
+    if(${${$self}{linebreaksAtEnd}}{end} == 1 
+        and defined ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} 
+        and ${${${${$self}{children}}[0]}{children}[-1]}{EndFinishesWithLineBreak} >= 1 
+        and ${${${${$self}{children}}[0]}{children}[-1]}{replacementText}=~m/\R$/s
+        and !${$self}{endImmediatelyFollowedByComment}){
+    
+        # last argument adjustment
+        $self->logger("Adjusting last argument in command, ${$self}{name} to avoid double line break") if $is_t_switch_active;
+        ${${${${$self}{children}}[0]}{children}[-1]}{replacementText}=~s/\R$//s;
+        ${${${${${$self}{children}}[0]}{children}[-1]}{linebreaksAtEnd}}{end} = 0;
+
+        # argument object adjustment
+        $self->logger("Adjusting argument object in command, ${$self}{name} to avoid double line break") if $is_t_switch_active;
+        ${${${${$self}{children}}[0]}{linebreaksAtEnd}}{body} = 0;
+        ${${${$self}{children}}[0]}{body}=~s/\R$//s;
+    } 
+
+    # the arguments body might finish with horizontal space, in which case, we need to transfer this 
+    # to the parent object replacement text.
+    #
+    # see ../test-cases/texexchange/5461.tex which was the first example to demonstrate the need for this
+    if(!${${${$self}{children}}[0]}{endImmediatelyFollowedByComment} and ${${${$self}{children}}[0]}{body} =~ m/\h*$/ and ${$self}{replacementText} !~ m/\R$/){
+        $self->logger("${$self}{name}: trailling horizontal space found in arguments -- removing it from arguments, adding to replacement text") if $is_t_switch_active;
+        ${${${$self}{children}}[0]}{body} =~ s/(\h*)$//s; 
+        ${$self}{replacementText} .= "$1";
+    }
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for special begin/end
+    $self->find_special;
+
+}
+
+sub check_linebreaks_before_equals{
+    # empty routine, which allows the above routine to function (this routine kicks in for KeyEqualsValuesBraces)
+    return;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $commandCounter++;
+    ${$self}{id} = "$tokens{command}$commandCounter";
+    return;
+}
+
+sub align_at_ampersand{
+    # need an empty routine here for commands; see
+    # test-cases/matrix1.tex for example
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Document.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Document.pm
new file mode 100644
index 00000000000..7c951ec7995
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Document.pm
@@ -0,0 +1,335 @@
+package LatexIndent::Document;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use Data::Dumper;
+
+# gain access to subroutines in the following modules
+use LatexIndent::Switches qw/storeSwitches %switches $is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::LogFile qw/logger output_logfile processSwitches/;
+use LatexIndent::GetYamlSettings qw/readSettings modify_line_breaks_settings get_indentation_settings_for_this_object get_every_or_custom_value get_indentation_information get_object_attribute_for_indentation_settings alignment_at_ampersand_settings/;
+use LatexIndent::FileExtension qw/file_extension_check/;
+use LatexIndent::BackUpFileProcedure qw/create_back_up_file/;
+use LatexIndent::BlankLines qw/protect_blank_lines unprotect_blank_lines condense_blank_lines/;
+use LatexIndent::ModifyLineBreaks qw/modify_line_breaks_body modify_line_breaks_end remove_line_breaks_begin adjust_line_breaks_end_parent/;
+use LatexIndent::TrailingComments qw/remove_trailing_comments put_trailing_comments_back_in add_comment_symbol construct_trailing_comment_regexp/;
+use LatexIndent::HorizontalWhiteSpace qw/remove_trailing_whitespace remove_leading_space/;
+use LatexIndent::Indent qw/indent wrap_up_statement determine_total_indentation indent_begin indent_body indent_end_statement final_indentation_check  get_surrounding_indentation indent_children_recursively check_for_blank_lines_at_beginning put_blank_lines_back_in_at_beginning add_surrounding_indentation_to_begin_statement/;
+use LatexIndent::Tokens qw/token_check %tokens/;
+use LatexIndent::HiddenChildren qw/find_surrounding_indentation_for_children update_family_tree get_family_tree check_for_hidden_children/;
+use LatexIndent::AlignmentAtAmpersand qw/align_at_ampersand find_aligned_block/;
+use LatexIndent::DoubleBackSlash qw/dodge_double_backslash un_dodge_double_backslash/;
+
+# code blocks
+use LatexIndent::Verbatim qw/put_verbatim_back_in find_verbatim_environments find_noindent_block find_verbatim_commands  put_verbatim_commands_back_in/;
+use LatexIndent::Environment qw/find_environments/;
+use LatexIndent::IfElseFi qw/find_ifelsefi/;
+use LatexIndent::Arguments qw/get_arguments_regexp find_opt_mand_arguments get_numbered_arg_regexp construct_arguments_regexp/;
+use LatexIndent::OptionalArgument qw/find_optional_arguments/;
+use LatexIndent::MandatoryArgument qw/find_mandatory_arguments get_mand_arg_reg_exp/;
+use LatexIndent::RoundBrackets qw/find_round_brackets/;
+use LatexIndent::Item qw/find_items construct_list_of_items/;
+use LatexIndent::Braces qw/find_commands_or_key_equals_values_braces/;
+use LatexIndent::Command qw/construct_command_regexp/;
+use LatexIndent::KeyEqualsValuesBraces qw/construct_key_equals_values_regexp/;
+use LatexIndent::NamedGroupingBracesBrackets qw/construct_grouping_braces_brackets_regexp/;
+use LatexIndent::UnNamedGroupingBracesBrackets qw/construct_unnamed_grouping_braces_brackets_regexp/;
+use LatexIndent::Special qw/find_special construct_special_begin/;
+use LatexIndent::Heading qw/find_heading construct_headings_levels/;
+use LatexIndent::FileContents qw/find_file_contents_environments_and_preamble/;
+use LatexIndent::Preamble;
+
+sub new{
+    # Create new objects, with optional key/value pairs
+    # passed as initializers.
+    #
+    # See Programming Perl, pg 319
+    my $invocant = shift;
+    my $class = ref($invocant) || $invocant;
+    my $self = {@_};
+    bless ($self,$class);
+    return $self;
+}
+
+sub latexindent{
+    my $self = shift;
+    $self->storeSwitches;
+    $self->processSwitches;
+    $self->readSettings;
+    $self->file_extension_check;
+    $self->operate_on_file;
+}
+
+sub operate_on_file{
+    my $self = shift;
+
+    $self->create_back_up_file;
+    $self->token_check;
+    $self->construct_regular_expressions;
+    $self->find_noindent_block;
+    $self->find_verbatim_commands;
+    $self->find_aligned_block;
+    $self->remove_trailing_comments;
+    $self->find_verbatim_environments;
+    $self->protect_blank_lines;
+    $self->remove_trailing_whitespace(when=>"before");
+    $self->find_file_contents_environments_and_preamble;
+    $self->dodge_double_backslash;
+    $self->remove_leading_space;
+    $self->process_body_of_text;
+    $self->remove_trailing_whitespace(when=>"after");
+    $self->condense_blank_lines;
+    $self->unprotect_blank_lines;
+    $self->un_dodge_double_backslash;
+    $self->put_verbatim_back_in;
+    $self->put_trailing_comments_back_in;
+    $self->put_verbatim_commands_back_in;
+    $self->output_indented_text;
+    $self->output_logfile;
+    return
+}
+
+sub construct_regular_expressions{
+    my $self = shift;
+    $self->construct_trailing_comment_regexp;
+    $self->construct_list_of_items;
+    $self->construct_special_begin;
+    $self->construct_headings_levels;
+    $self->construct_arguments_regexp;
+    $self->construct_command_regexp;
+    $self->construct_key_equals_values_regexp;
+    $self->construct_grouping_braces_brackets_regexp;
+    $self->construct_unnamed_grouping_braces_brackets_regexp;
+
+}
+
+sub output_indented_text{
+    my $self = shift;
+
+    # output to screen, unless silent mode
+    print ${$self}{body} unless $switches{silentMode};
+
+    $self->logger("Output routine",'heading');
+
+    # if -overwrite is active then output to original fileName
+    if($switches{overwrite}) {
+        $self->logger("Overwriting file ${$self}{fileName}");
+        open(OUTPUTFILE,">",${$self}{fileName});
+        print OUTPUTFILE ${$self}{body};
+        close(OUTPUTFILE);
+    } elsif($switches{outputToFile}) {
+        $self->logger("Outputting to file $switches{outputToFile}");
+        open(OUTPUTFILE,">",$switches{outputToFile});
+        print OUTPUTFILE ${$self}{body};
+        close(OUTPUTFILE);
+    } else {
+        $self->logger("Not outputting to file; see -w and -o switches for more options.");
+    }
+    return;
+}
+
+sub process_body_of_text{
+    my $self = shift;
+
+    # find objects recursively
+    $self->logger('Phase 1: searching for objects','heading');
+    $self->find_objects;
+
+    # find all hidden child
+    $self->logger('Phase 2: finding surrounding indentation','heading');
+    $self->find_surrounding_indentation_for_children;
+
+    # indentation recursively
+    $self->logger('Phase 3: indenting objects','heading');
+    $self->indent_children_recursively;
+
+    # final indentation check
+    $self->logger('Phase 4: final indentation check','heading');
+    $self->final_indentation_check;
+
+    return;
+}
+
+sub find_objects{
+    my $self = shift;
+
+    # search for environments
+    $self->logger('looking for ENVIRONMENTS');
+    $self->find_environments;
+
+    # search for ifElseFi blocks
+    $self->logger('looking for IFELSEFI');
+    $self->find_ifelsefi;
+
+    # search for headings (part, chapter, section, setc)
+    $self->logger('looking for HEADINGS (chapter, section, part, etc)');
+    $self->find_heading;
+    
+    # search for commands with arguments
+    $self->logger('looking for COMMANDS and key = {value}');
+    $self->find_commands_or_key_equals_values_braces;
+
+    # search for special begin/end
+    $self->logger('looking for SPECIAL begin/end');
+    $self->find_special;
+
+    # if there are no children, return
+    if(${$self}{children}){
+        $self->logger("Objects have been found.",'heading');
+    } else {
+        $self->logger("No objects found.");
+        return;
+    }
+
+    # logfile information
+    $self->logger(Dumper(\%{$self}),'ttrace') if($is_tt_switch_active);
+    $self->logger("Operating on: ${$self}{name}",'heading');
+    $self->logger("Number of children:",'heading');
+    $self->logger(scalar (@{${$self}{children}}));
+
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+    $self->logger("There are no tasks particular to ${$self}{name}") if $is_t_switch_active;
+}
+
+sub get_settings_and_store_new_object{
+    my $self = shift;
+
+    # grab the object to be operated upon
+    my ($latexIndentObject) = @_;
+
+    # there are a number of tasks common to each object
+    $latexIndentObject->tasks_common_to_each_object(%{$self});
+      
+    # tasks particular to each object
+    $latexIndentObject->tasks_particular_to_each_object;
+
+    # store children in special hash
+    push(@{${$self}{children}},$latexIndentObject);
+
+}
+
+sub tasks_common_to_each_object{
+    my $self = shift;
+
+    # grab the parent information
+    my %parent = @_;
+
+    # update/create the ancestor information
+    if($parent{ancestors}){
+      $self->logger("Ancestors *have* been found for ${$self}{name}") if($is_t_switch_active);
+      push(@{${$self}{ancestors}},@{$parent{ancestors}});
+    } else {
+      $self->logger("No ancestors found for ${$self}{name}") if($is_t_switch_active);
+      if(defined $parent{id} and $parent{id} ne ''){
+        $self->logger("Creating ancestors with $parent{id} as the first one") if($is_t_switch_active);
+        push(@{${$self}{ancestors}},{ancestorID=>$parent{id},ancestorIndentation=>\$parent{indentation},type=>"natural",name=>${$self}{name}});
+      }
+    }
+
+    # natural ancestors
+    ${$self}{naturalAncestors} = q();
+    if(${$self}{ancestors}){
+      ${$self}{naturalAncestors} .= "---".${$_}{ancestorID}."\n" for @{${$self}{ancestors}};
+    }
+
+    # in what follows, $self can be an environment, ifElseFi, etc
+
+    # count linebreaks in body
+    my $bodyLineBreaks = 0;
+    $bodyLineBreaks++ while(${$self}{body} =~ m/\R/sxg);
+    ${$self}{bodyLineBreaks} = $bodyLineBreaks;
+
+    # get settings for this object
+    $self->get_indentation_settings_for_this_object;
+
+    # give unique id
+    $self->create_unique_id;
+
+    # add trailing text to the id to stop, e.g LATEX-INDENT-ENVIRONMENT1 matching LATEX-INDENT-ENVIRONMENT10
+    ${$self}{id} .= $tokens{endOfToken};
+
+    # the replacement text can be just the ID, but the ID might have a line break at the end of it
+    $self->get_replacement_text;
+
+    # the above regexp, when used below, will remove the trailing linebreak in ${$self}{linebreaksAtEnd}{end}
+    # so we compensate for it here
+    $self->adjust_replacement_text_line_breaks_at_end;
+
+    # modify line breaks on body and end statements
+    $self->modify_line_breaks_body;
+
+    # modify line breaks end statements
+    $self->modify_line_breaks_end;
+
+    # check the body for current children
+    $self->check_for_hidden_children;
+
+    return;
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text can be just the ID, but the ID might have a line break at the end of it
+    ${$self}{replacementText} = ${$self}{id};
+    return;
+}
+
+sub adjust_replacement_text_line_breaks_at_end{
+    my $self = shift;
+
+    # the above regexp, when used below, will remove the trailing linebreak in ${$self}{linebreaksAtEnd}{end}
+    # so we compensate for it here
+    $self->logger("Putting linebreak after replacementText for ${$self}{name}") if($is_t_switch_active);
+    if(defined ${$self}{horizontalTrailingSpace}){
+        ${$self}{replacementText} .= ${$self}{horizontalTrailingSpace} unless(!${$self}{endImmediatelyFollowedByComment} and defined ${$self}{EndFinishesWithLineBreak} and ${$self}{EndFinishesWithLineBreak}==2);
+    }
+    ${$self}{replacementText} .= "\n" if(${$self}{linebreaksAtEnd}{end});
+
+}
+
+sub count_body_line_breaks{
+    my $self = shift;
+
+    my $oldBodyLineBreaks = (defined ${$self}{bodyLineBreaks})? ${$self}{bodyLineBreaks} : 0;
+
+    # count linebreaks in body
+    my $bodyLineBreaks = 0;
+    $bodyLineBreaks++ while(${$self}{body} =~ m/\R/sxg);
+    ${$self}{bodyLineBreaks} = $bodyLineBreaks;
+    $self->logger("bodyLineBreaks ${$self}{bodyLineBreaks}")  if((${$self}{bodyLineBreaks} != $oldBodyLineBreaks) and  $is_t_switch_active);
+}
+
+sub wrap_up_tasks{
+    my $self = shift;
+
+    # most recent child object
+    my $child = @{${$self}{children}}[-1];
+
+    # check if the last object was the last thing in the body, and if it has adjusted linebreaks
+    $self->adjust_line_breaks_end_parent;
+
+    $self->logger(Dumper(\%{$child})) if($is_tt_switch_active);
+    $self->logger("replaced with ID: ${$child}{id}") if $is_t_switch_active;
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/DoubleBackSlash.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/DoubleBackSlash.pm
new file mode 100644
index 00000000000..8b8be3e25ec
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/DoubleBackSlash.pm
@@ -0,0 +1,45 @@
+package LatexIndent::DoubleBackSlash;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::Tokens qw/%tokens/;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/dodge_double_backslash un_dodge_double_backslash/;
+
+# some code can contain, e.g
+#       cycle list={blue,mark=none\\},
+# see test-cases/texexchange/29293-christian-feuersanger.tex
+#
+# This is problematic, as the argument regexp won't count the right } because it has a 
+# backslash immediately infront of it!
+sub dodge_double_backslash{
+    my $self = shift;
+
+    ${$self}{body} =~ s/(?:\\\\(\{|\}|\]))/$tokens{doubleBackSlash}$1/sg;
+    return;
+}
+
+# this routine replaces the token with the \\\\
+sub un_dodge_double_backslash{
+    my $self = shift;
+
+    ${$self}{body} =~ s/$tokens{doubleBackSlash}/\\\\/sg;
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Environment.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Environment.pm
new file mode 100755
index 00000000000..1b0ad27d550
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Environment.pm
@@ -0,0 +1,128 @@
+package LatexIndent::Environment;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_environments/;
+our $environmentCounter;
+
+# store the regular expresssion for matching and replacing the \begin{}...\end{} statements
+our $environmentRegExp = qr/
+            (
+                \\begin\{
+                        (
+                         [a-zA-Z@\*0-9_\\]+ # lowercase|uppercase letters, @, *, numbers
+                        )                 # environment name captured into $2
+                       \}                 # \begin{<something>} statement
+                       \h*                # horizontal space
+                       (\R*)?             # possible line breaks (into $3)
+            )                             # begin statement captured into $1
+            (
+                (?:                       # cluster-only (), don't capture 
+                    (?!                   # don't include \begin in the body
+                        (?:\\begin)       # cluster-only (), don't capture
+                    ).                    # any character, but not \\begin
+                )*?                       # non-greedy
+                        (\R*)?            # possible line breaks (into $5)
+            )                             # environment body captured into $4
+            (
+                \\end\{\2\}               # \end{<something>} statement
+            )                             # captured into $6
+            (\h*)?                        # possibly followed by horizontal space
+            (\R)?                         # possibly followed by a line break 
+            /sx;
+
+sub find_environments{
+    my $self = shift;
+
+
+    while( ${$self}{body} =~ m/$environmentRegExp\h*($trailingCommentRegExp)?/){
+
+      # global substitution
+      ${$self}{body} =~ s/
+                $environmentRegExp(\h*)($trailingCommentRegExp)?
+             /
+                # log file output
+                $self->logger("environment found: $2",'heading') if $is_t_switch_active;
+
+                # create a new Environment object
+                my $env = LatexIndent::Environment->new(begin=>$1,
+                                                        name=>$2,
+                                                        body=>$4,
+                                                        end=>$6,
+                                                        linebreaksAtEnd=>{
+                                                          begin=>$3?1:0,
+                                                          body=>$5?1:0,
+                                                          end=>$8?1:0,
+                                                        },
+                                                        modifyLineBreaksYamlName=>"environments",
+                                                        endImmediatelyFollowedByComment=>$8?0:($10?1:0),
+                                                        horizontalTrailingSpace=>$7?$7:q(),
+                                                      );
+
+                # the settings and storage of most objects has a lot in common
+                $self->get_settings_and_store_new_object($env);
+                ${@{${$self}{children}}[-1]}{replacementText}.($9?$9:q()).($10?$10:q());
+                /xseg;
+    $self->adjust_line_breaks_end_parent;
+    } 
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # if the environment is empty, we may need to update linebreaksAtEnd{body}
+    if(${$self}{body} =~ m/^\h*$/s and ${${$self}{linebreaksAtEnd}}{begin}){
+          $self->logger("empty environment body (${$self}{name}), updating linebreaksAtEnd{body} to be 1") if($is_t_switch_active);
+          ${${$self}{linebreaksAtEnd}}{body} = 1;
+    }
+
+    # search for items as the first order of business
+    $self->find_items;
+
+    # search for headings (important to do this before looking for commands!)
+    $self->find_heading;
+
+    # search for commands, keys, named grouping braces
+    $self->find_commands_or_key_equals_values_braces;
+
+    # search for arguments
+    $self->find_opt_mand_arguments;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for special begin/end
+    $self->find_special;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $environmentCounter++;
+    ${$self}{id} = "$tokens{environment}$environmentCounter";
+    return;
+}
+
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/FileContents.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/FileContents.pm
new file mode 100644
index 00000000000..dc8aa53f206
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/FileContents.pm
@@ -0,0 +1,230 @@
+package LatexIndent::FileContents;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/find_file_contents_environments_and_preamble/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our $fileContentsCounter;
+
+sub find_file_contents_environments_and_preamble{
+    my $self = shift;
+
+    # store the file contents blocks in an array which, depending on the value 
+    # of indentPreamble, will be put into the verbatim hash, or otherwise 
+    # stored as children to be operated upon
+    my @fileContentsStorageArray; 
+
+    # fileContents environments
+    $self->logger('looking for FILE CONTENTS environments (see fileContentsEnvironments)','heading');
+    $self->logger(Dumper(\%{$masterSettings{fileContentsEnvironments}})) if($is_t_switch_active);
+    while( my ($fileContentsEnv,$yesno)= each %{$masterSettings{fileContentsEnvironments}}){
+        if($yesno){
+            $self->logger("looking for $fileContentsEnv:$yesno environments");
+
+            # the trailing * needs some care
+            if($fileContentsEnv =~ m/\*$/){
+                $fileContentsEnv =~ s/\*$//;
+                $fileContentsEnv .= '\*';
+            }
+
+            my $fileContentsRegExp = qr/
+                            (
+                            \\begin\{
+                                    $fileContentsEnv       
+                                   \}                     
+                            )
+                            (
+                                .*?
+                            )                            
+                            (
+                                \\end\{$fileContentsEnv\}  
+                                \h*
+                            )                    
+                            (\R)?  
+                        /sx;
+
+            while( ${$self}{body} =~ m/$fileContentsRegExp/sx){
+
+              # create a new Environment object
+              my $fileContentsBlock = LatexIndent::FileContents->new( begin=>$1,
+                                                    body=>$2,
+                                                    end=>$3,
+                                                    name=>$fileContentsEnv,
+                                                    linebreaksAtEnd=>{
+                                                      begin=>0,
+                                                      body=>0,
+                                                      end=>$4?1:0,
+                                                    },
+                                                    modifyLineBreaksYamlName=>"filecontents",
+                                                    );
+              # give unique id
+              $fileContentsBlock->create_unique_id;
+
+              # the replacement text can be just the ID, but the ID might have a line break at the end of it
+              $fileContentsBlock->get_replacement_text;
+
+              # count body line breaks
+              $fileContentsBlock->count_body_line_breaks;
+
+              # the above regexp, when used below, will remove the trailing linebreak in ${$self}{linebreaksAtEnd}{end}
+              # so we compensate for it here
+              $fileContentsBlock->adjust_replacement_text_line_breaks_at_end;
+
+              # store the fileContentsBlock, and determine location afterwards
+              push(@fileContentsStorageArray,$fileContentsBlock);
+
+              # log file output
+              $self->logger("FILECONTENTS environment found: $fileContentsEnv");
+
+              # remove the environment block, and replace with unique ID
+              ${$self}{body} =~ s/$fileContentsRegExp/${$fileContentsBlock}{replacementText}/sx;
+
+              $self->logger("replaced with ID: ${$fileContentsBlock}{id}");
+            } 
+      } else {
+            $self->logger("*not* looking for $fileContentsEnv as $fileContentsEnv:$yesno");
+      }
+    }
+
+    # determine if body of document contains \begin{document} -- if it does, then assume
+    # that the body has a preamble
+    my $preambleRegExp = qr/
+                        (.*?)
+                        (\R*)?            # linebreaks at end of body into $2
+                        \\begin\{document\}
+                /sx;
+    my $preamble = q();
+
+    my $needToStorePreamble = 0;
+
+    # try and find the preamble
+    if( ${$self}{body} =~ m/$preambleRegExp/sx and ${$masterSettings{lookForPreamble}}{${$self}{fileExtension}}){
+
+        $self->logger("\\begin{document} found in body (after searching for filecontents)-- assuming that a preamble exists");
+
+        # create a preamble object
+        $preamble = LatexIndent::Preamble->new( begin=>q(),
+                                              body=>$1,
+                                              end=>q(),
+                                              name=>"preamble",
+                                              linebreaksAtEnd=>{
+                                                begin=>0,
+                                                body=>$2?1:0,
+                                                end=>0,
+                                              },
+                                              afterbit=>($2?$2:q())."\\begin{document}",
+                                              modifyLineBreaksYamlName=>"preamble",
+                                              );
+
+        # give unique id
+        $preamble->create_unique_id;
+
+        # get the replacement_text
+        $preamble->get_replacement_text;
+
+        # log file output
+        $self->logger("preamble found: $preamble");
+
+        # remove the environment block, and replace with unique ID
+        ${$self}{body} =~ s/$preambleRegExp/${$preamble}{replacementText}/sx;
+
+        $self->logger("replaced with ID: ${$preamble}{replacementText}");
+        # indentPreamble set to 1
+        if($masterSettings{indentPreamble}){
+            $self->logger("storing ${$preamble}{id} for indentation (see indentPreamble)");
+            $needToStorePreamble = 1;
+        } else {
+            # indentPreamble set to 0
+            $self->logger("NOT storing ${$preamble}{id} for indentation -- will store as VERBATIM object (see indentPreamble)");
+            $preamble->unprotect_blank_lines;
+            ${$self}{verbatim}{${$preamble}{id}} = $preamble;
+        }
+    }
+
+    # loop through the fileContents array, check if it's in the preamble
+    foreach(@fileContentsStorageArray){
+              my $indentThisChild = 0;
+              # verbatim children go in special hash
+              if($preamble ne '' and ${$preamble}{body} =~ m/${$_}{id}/){
+                $self->logger("filecontents (${$_}{id}) is within preamble");
+                # indentPreamble set to 1
+                if($masterSettings{indentPreamble}){
+                    $self->logger("storing ${$_}{id} for indentation (indentPreamble is 1)");
+                    $indentThisChild = 1;
+                } else {
+                    # indentPreamble set to 0
+                    $self->logger("Storing ${$_}{id} as a VERBATIM object (indentPreamble is 0)");
+                    ${$self}{verbatim}{${$_}{id}}=$_;
+                }
+              } else {
+                    $self->logger("storing ${$_}{id} for indentation (${$_}{name} found outside of preamble)");
+                    $indentThisChild = 1;
+              }
+              # store the child, if necessary
+              if($indentThisChild){
+                    $_->remove_leading_space;
+                    $_->get_indentation_settings_for_this_object;
+                    $_->tasks_particular_to_each_object;
+                    push(@{${$self}{children}},$_);
+              }
+    }
+
+    if($needToStorePreamble){
+        $preamble->dodge_double_backslash;
+        $preamble->remove_leading_space;
+        $preamble->find_commands_or_key_equals_values_braces if($masterSettings{preambleCommandsBeforeEnvironments});
+        $preamble->tasks_particular_to_each_object;
+        push(@{${$self}{children}},$preamble);
+    }
+    return;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $fileContentsCounter++;
+    ${$self}{id} = "$tokens{filecontents}$fileContentsCounter$tokens{endOfToken}";
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # search for environments
+    $self->find_environments;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for headings (part, chapter, section, setc)
+    $self->find_heading;
+    
+    # search for commands with arguments
+    $self->find_commands_or_key_equals_values_braces;
+
+    # search for special begin/end
+    $self->find_special;
+
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/FileExtension.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/FileExtension.pm
new file mode 100644
index 00000000000..8dcb66dd80c
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/FileExtension.pm
@@ -0,0 +1,99 @@
+package LatexIndent::FileExtension;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use File::Basename; # to get the filename and directory path
+use Exporter qw/import/;
+our @EXPORT_OK = qw/file_extension_check/;
+
+sub file_extension_check{
+    my $self = shift;
+
+    # grab the filename
+    my $fileName = ${$self}{fileName};
+
+    # grab the file extension preferences
+    my %fileExtensionPreference= %{$masterSettings{fileExtensionPreference}};
+
+    # sort the file extensions by preference 
+    my @fileExtensions = sort { $fileExtensionPreference{$a} <=> $fileExtensionPreference{$b} } keys(%fileExtensionPreference);
+    
+    # get the base file name, allowing for different extensions (possibly no extension)
+    my ($dir, $name, $ext) = fileparse($fileName, @fileExtensions);
+
+    # check to make sure given file type is supported
+    if( -e $fileName  and !$ext ){
+        my $message = "The file $fileName exists , but the extension does not correspond to any given in fileExtensionPreference; consinder updating fileExtensionPreference.";
+        $self->logger($message,'heading');
+        $self->output_logfile;
+        die($message);
+    }
+
+    # if no extension, search according to fileExtensionPreference
+    if (!$ext) {
+        $self->logger("File extension work:",'heading');
+        $self->logger("latexindent called to act upon $fileName with an, as yet, unrecognised file extension;");
+        $self->logger("searching for file with an extension in the following order (see fileExtensionPreference):");
+        $self->logger(join("\n",@fileExtensions));
+
+        my $fileFound = 0;
+        # loop through the known file extensions (see @fileExtensions)
+        foreach (@fileExtensions ){
+            if ( -e $fileName.$_ ) {
+               $self->logger("$fileName$_ found!");
+               $fileName .= $_;
+               $self->logger("Updated fileName to $fileName");
+               ${$self}{fileName} = $fileName ;
+               $fileFound = 1;
+               $ext = $_;
+               last;
+            }
+        }
+        unless($fileFound){
+          $self->logger("I couldn't find a match for $fileName in fileExtensionPreference (see defaultSettings.yaml)");
+          foreach (@fileExtensions ){
+            $self->logger("I searched for $fileName$_");
+          }
+          $self->logger("but couldn't find any of them.");
+          $self->logger("Consider updating fileExtensionPreference. Error: Exiting, no indendation done.");
+          $self->output_logfile;
+          die "I couldn't find a match for $fileName in fileExtensionPreference.\nExiting, no indendation done."; 
+        }
+      } else {
+        # if the file has a recognised extension, check that the file exists
+        unless( -e $fileName ){
+          my $message = "Error: I couldn't find $fileName, are you sure it exists?. No indentation done. Exiting.";
+          $self->logger($message);
+          $self->output_logfile;
+          die $message;
+        }
+      }
+
+    # store the file extension
+    ${$self}{fileExtension} = $ext;
+
+    # read the file into the Document body
+    my @lines;
+    open(MAINFILE, $fileName) or die "Could not open input file, $fileName";
+    push(@lines,$_) while(<MAINFILE>);
+    close(MAINFILE);
+
+    # the all-important step: update the body
+    ${$self}{body} = join("",@lines);
+}
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/GetYamlSettings.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/GetYamlSettings.pm
new file mode 100644
index 00000000000..4c93bc80426
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/GetYamlSettings.pm
@@ -0,0 +1,471 @@
+package LatexIndent::GetYamlSettings;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use strict;
+use warnings;
+use LatexIndent::Switches qw/%switches $is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use YAML::Tiny;                # interpret defaultSettings.yaml and other potential settings files
+use File::Basename;            # to get the filename and directory path
+use File::HomeDir;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/readSettings modify_line_breaks_settings get_indentation_settings_for_this_object get_every_or_custom_value get_indentation_information get_object_attribute_for_indentation_settings alignment_at_ampersand_settings %masterSettings/;
+
+# Read in defaultSettings.YAML file
+our $defaultSettings;
+
+# master yaml settings is a hash, global to this module
+our %masterSettings;
+
+# previously found settings is a hash, global to this module
+our %previouslyFoundSettings;
+
+sub readSettings{
+  my $self = shift;
+  
+  $defaultSettings = YAML::Tiny->new;
+  $defaultSettings = YAML::Tiny->read( "$FindBin::RealBin/defaultSettings.yaml" );
+  $self->logger("YAML settings read",'heading');
+  $self->logger("Reading defaultSettings.yaml from $FindBin::RealBin/defaultSettings.yaml");
+  
+  # if latexindent.exe is invoked from TeXLive, then defaultSettings.yaml won't be in 
+  # the same directory as it; we need to navigate to it
+  if(!$defaultSettings) {
+    $defaultSettings = YAML::Tiny->read( "$FindBin::RealBin/../../texmf-dist/scripts/latexindent/defaultSettings.yaml");
+    $self->logger("Reading defaultSettings.yaml (2nd attempt, TeXLive, Windows) from $FindBin::RealBin/../../texmf-dist/scripts/latexindent/defaultSettings.yaml");
+  }
+
+  # need to exit if we can't get defaultSettings.yaml
+  die "Could not open defaultSettings.yaml" if(!$defaultSettings);
+   
+  # master yaml settings is a hash, global to this module
+  our %masterSettings = %{$defaultSettings->[0]};
+
+  # scalar to read user settings
+  my $userSettings;
+
+  # array to store the paths to user settings
+  my @absPaths;
+  
+  # we'll need the home directory a lot in what follows
+  my $homeDir = File::HomeDir->my_home; 
+  
+  # get information about user settings- first check if indentconfig.yaml exists
+  my $indentconfig = "$homeDir/indentconfig.yaml";
+
+  # if indentconfig.yaml doesn't exist, check for the hidden file, .indentconfig.yaml
+  $indentconfig = "$homeDir/.indentconfig.yaml" if(! -e $indentconfig);
+
+  # messages for indentconfig.yaml and/or .indentconfig.yaml
+  if ( -e $indentconfig and !$switches{onlyDefault}) {
+        $self->logger("Reading path information from $indentconfig");
+        # if both indentconfig.yaml and .indentconfig.yaml exist
+        if ( -e File::HomeDir->my_home . "/indentconfig.yaml" and  -e File::HomeDir->my_home . "/.indentconfig.yaml") {
+              $self->logger("$homeDir/.indentconfig.yaml has been found, but $indentconfig takes priority");
+        } elsif ( -e File::HomeDir->my_home . "/indentconfig.yaml" ) {
+              $self->logger("(Alternatively $homeDir/.indentconfig.yaml can be used)");
+        } elsif ( -e File::HomeDir->my_home . "/.indentconfig.yaml" ) {
+              $self->logger("(Alternatively $homeDir/indentconfig.yaml can be used)");
+        }
+
+        # read the absolute paths from indentconfig.yaml
+        $userSettings = YAML::Tiny->read( "$indentconfig" );
+
+        # output the contents of indentconfig to the log file
+        $self->logger(Dump \%{$userSettings->[0]});
+
+        # update the absolute paths
+        @absPaths = @{$userSettings->[0]->{paths}};
+  } else {
+     if($switches{onlyDefault}) {
+        $self->logger("Only default settings requested, not reading USER settings from $indentconfig");
+        $self->logger("Ignoring $switches{readLocalSettings} (you used the -d switch)") if($switches{readLocalSettings});
+        $switches{readLocalSettings}=0;
+     } else {
+       # give the user instructions on where to put indentconfig.yaml or .indentconfig.yaml
+       $self->logger("Home directory is $homeDir (didn't find either indentconfig.yaml or .indentconfig.yaml)");
+       $self->logger("To specify user settings you would put indentconfig.yaml here: $homeDir/indentconfig.yaml");
+       $self->logger("Alternatively, you can use the hidden file .indentconfig.yaml as: $homeDir/.indentconfig.yaml");
+     }
+  }
+
+  # get information about LOCAL settings, assuming that $readLocalSettings exists
+  my $directoryName = dirname (${$self}{fileName});
+  
+  # local settings can be separated by ,
+  # e.g  
+  #     -l = myyaml1.yaml,myyaml2.yaml
+  # and in which case, we need to read them all
+  my @localSettings;
+  if($switches{readLocalSettings} =~ m/,/){
+        $self->logger("Multiple localSettings found, separated by commas:",'heading');
+        @localSettings = split(/,/,$switches{readLocalSettings});
+  } else {
+    push(@localSettings,$switches{readLocalSettings}) if($switches{readLocalSettings});
+  }
+
+  # add local settings to the paths, if appropriate
+  foreach (@localSettings) {
+    if ( (-e "$directoryName/$_") and !(-z "$directoryName/$_")) {
+        $self->logger("Adding $directoryName/$_ to YAML read paths");
+        push(@absPaths,"$directoryName/$_");
+    } elsif ( !(-e "$directoryName/$_") ) {
+          $self->logger("WARNING yaml file not found: $directoryName/$_ not found");
+          $self->logger("Proceeding without it.");
+    }
+  }
+
+  # read in the settings from each file
+  foreach my $settings (@absPaths) {
+    # check that the settings file exists and that it isn't empty
+    if (-e $settings and !(-z $settings)) {
+        $self->logger("Reading USER settings from $settings");
+        $userSettings = YAML::Tiny->read( "$settings" );
+  
+        # if we can read userSettings
+        if($userSettings) {
+              # update the MASTER setttings to include updates from the userSettings
+              while(my($firstLevelKey, $firstLevelValue) = each %{$userSettings->[0]}) {
+                      # the update approach is slightly different for hashes vs scalars/arrays
+                      if (ref($firstLevelValue) eq "HASH") {
+                          while(my ($secondLevelKey,$secondLevelValue) = each %{$userSettings->[0]{$firstLevelKey}}) {
+                            if (ref $secondLevelValue eq "HASH"){
+                                # if masterSettings already contains a *scalar* value in secondLevelKey
+                                # then we need to delete it (test-cases/headings-first.tex with indentRules1.yaml first demonstrated this)
+                                if(ref $masterSettings{$firstLevelKey}{$secondLevelKey} ne "HASH"){
+                                    $self->logger("masterSettings{$firstLevelKey}{$secondLevelKey} currently contains a *scalar* value, but it needs to be updated with a hash (see $settings); deleting the scalar") if($is_t_switch_active);
+                                    delete $masterSettings{$firstLevelKey}{$secondLevelKey} ;
+                                }
+                                while(my ($thirdLevelKey,$thirdLevelValue) = each %{$secondLevelValue}) {
+                                    if (ref $thirdLevelValue eq "HASH"){
+                                        # similarly for third level
+                                        if (ref $masterSettings{$firstLevelKey}{$secondLevelKey}{$thirdLevelKey} ne "HASH"){
+                                            $self->logger("masterSettings{$firstLevelKey}{$secondLevelKey}{$thirdLevelKey} currently contains a *scalar* value, but it needs to be updated with a hash (see $settings); deleting the scalar") if($is_t_switch_active);
+                                            delete $masterSettings{$firstLevelKey}{$secondLevelKey}{$thirdLevelKey} ;
+                                        }
+                                        while(my ($fourthLevelKey,$fourthLevelValue) = each %{$thirdLevelValue}) {
+                                            $masterSettings{$firstLevelKey}{$secondLevelKey}{$thirdLevelKey}{$fourthLevelKey} = $fourthLevelValue;
+                                        }
+                                    } else {
+                                        $masterSettings{$firstLevelKey}{$secondLevelKey}{$thirdLevelKey} = $thirdLevelValue;
+                                    }
+                                }
+                            } else {
+                                $masterSettings{$firstLevelKey}{$secondLevelKey} = $secondLevelValue;
+                            }
+                          }
+                      } else {
+                            $masterSettings{$firstLevelKey} = $firstLevelValue;
+                      }
+              }
+
+              # output settings to $logfile
+              if($masterSettings{logFilePreferences}{showEveryYamlRead}){
+                  $self->logger(Dump \%{$userSettings->[0]});
+              } else {
+                  $self->logger("Not showing settings in the log file (see showEveryYamlRead and showAmalgamatedSettings).");
+              }
+         } else {
+               # otherwise print a warning that we can not read userSettings.yaml
+               $self->logger("WARNING $settings contains invalid yaml format- not reading from it");
+         }
+    } else {
+        # otherwise keep going, but put a warning in the log file
+        $self->logger("WARNING: $homeDir/indentconfig.yaml");
+        if (-z $settings) {
+            $self->logger("specifies $settings but this file is EMPTY -- not reading from it");
+        } else {
+            $self->logger("specifies $settings but this file does not exist - unable to read settings from this file");
+        }
+    }
+  }
+
+  # some users may wish to see showAmalgamatedSettings
+  # which details the overall state of the settings modified
+  # from the default in various user files
+  if($masterSettings{logFilePreferences}{showAmalgamatedSettings}){
+      $self->logger("Amalgamated/overall settings to be used:",'heading');
+      $self->logger(Dump \%masterSettings);
+  }
+
+  return;
+}
+
+sub get_indentation_settings_for_this_object{
+    my $self = shift;
+
+    # create a name for previously found settings
+    my $storageName = ${$self}{name}.${$self}{modifyLineBreaksYamlName};
+
+    # check for storage of repeated objects
+    if ($previouslyFoundSettings{$storageName}){
+        $self->logger("Using stored settings for $storageName") if($is_t_switch_active);
+    } else {
+        my $name = ${$self}{name};
+        $self->logger("Storing settings for $storageName") if($is_t_switch_active);
+
+        # check for noAdditionalIndent and indentRules
+        # otherwise use defaultIndent
+        my $indentation = $self->get_indentation_information;
+
+        # check for alignment at ampersand settings
+        $self->alignment_at_ampersand_settings;
+
+        # check for line break settings
+        $self->modify_line_breaks_settings;
+
+        # store the settings
+        %{${previouslyFoundSettings}{$storageName}} = (
+                        indentation=>$indentation,
+                        BeginStartsOnOwnLine=>${$self}{BeginStartsOnOwnLine},
+                        BodyStartsOnOwnLine=>${$self}{BodyStartsOnOwnLine},
+                        EndStartsOnOwnLine=>${$self}{EndStartsOnOwnLine},
+                        EndFinishesWithLineBreak=>${$self}{EndFinishesWithLineBreak},
+                      );
+
+        # don't forget alignment settings!
+        ${${previouslyFoundSettings}{$storageName}}{lookForAlignDelims} = ${$self}{lookForAlignDelims} if(defined ${$self}{lookForAlignDelims});
+        ${${previouslyFoundSettings}{$storageName}}{alignDoubleBackSlash} = ${$self}{alignDoubleBackSlash} if(defined ${$self}{alignDoubleBackSlash});
+        ${${previouslyFoundSettings}{$storageName}}{spacesBeforeDoubleBackSlash} = ${$self}{spacesBeforeDoubleBackSlash} if(defined ${$self}{spacesBeforeDoubleBackSlash});
+
+        # some objects, e.g ifElseFi, can have extra assignments, e.g ElseStartsOnOwnLine
+        # these need to be stored as well!
+        foreach (@{${$self}{additionalAssignments}}){
+            ${${previouslyFoundSettings}{$storageName}}{$_} = ${$self}{$_};
+        }
+
+    }
+
+    # append indentation settings to the current object
+    while( my ($key,$value)= each %{${previouslyFoundSettings}{$storageName}}){
+            ${$self}{$key} = $value;
+    }
+
+    return;
+}
+
+sub alignment_at_ampersand_settings{
+    my $self = shift;
+
+    # if the YamlName is, for example, optionalArguments, mandatoryArguments, heading, then we'll be looking for information about the *parent*
+    my $name = (defined ${$self}{nameForIndentationSettings}) ? ${$self}{nameForIndentationSettings} : ${$self}{name};
+
+    # check, for example,
+    #   lookForAlignDelims:
+    #      tabular: 1
+    # or
+    #
+    #   lookForAlignDelims:
+    #      tabular: 
+    #         delims: 1
+    #         alignDoubleBackSlash: 1
+    #         spacesBeforeDoubleBackSlash: 2
+    return unless ${$masterSettings{lookForAlignDelims}}{$name}; 
+
+    ## check, for example,
+    ##   lookForAlignDelims:
+    ##      tabular:
+    ##         body: 1
+    #return unless $self->get_object_attribute_for_indentation_settings;
+    
+    if(ref ${$masterSettings{lookForAlignDelims}}{$name} eq "HASH"){
+      ${$self}{lookForAlignDelims} = (defined ${${$masterSettings{lookForAlignDelims}}{$name}}{delims} ) ? ${${$masterSettings{lookForAlignDelims}}{$name}}{delims} : 1;
+      ${$self}{alignDoubleBackSlash} = (defined ${${$masterSettings{lookForAlignDelims}}{$name}}{alignDoubleBackSlash} ) ? ${${$masterSettings{lookForAlignDelims}}{$name}}{alignDoubleBackSlash} : 1;
+      ${$self}{spacesBeforeDoubleBackSlash} = (defined ${${$masterSettings{lookForAlignDelims}}{$name}}{spacesBeforeDoubleBackSlash} ) ? ${${$masterSettings{lookForAlignDelims}}{$name}}{spacesBeforeDoubleBackSlash} : -1;
+    } else {
+      ${$self}{lookForAlignDelims} = 1;
+      ${$self}{alignDoubleBackSlash} = 1;
+      ${$self}{spacesBeforeDoubleBackSlash} = -1;
+    }
+    return;
+}
+
+sub modify_line_breaks_settings{
+    # return with undefined values unless the -m switch is active
+    return unless $is_m_switch_active;
+
+    my $self = shift;
+
+    # details to the log file
+    $self->logger("-m modifylinebreaks switch active, looking for settings for ${$self}{name} ",'heading') if $is_t_switch_active;
+
+    # some objects, e.g ifElseFi, can have extra assignments, e.g ElseStartsOnOwnLine
+    my @toBeAssignedTo = ${$self}{additionalAssignments} ? @{${$self}{additionalAssignments}} : ();
+
+    # the following will *definitley* be in the array, so let's add them
+    push(@toBeAssignedTo,("BeginStartsOnOwnLine","BodyStartsOnOwnLine","EndStartsOnOwnLine","EndFinishesWithLineBreak"));
+
+    # we can effeciently loop through the following
+    foreach (@toBeAssignedTo){
+                    $self->get_every_or_custom_value(
+                                    toBeAssignedTo=>$_,
+                                    toBeAssignedToAlias=> ${$self}{aliases}{$_} ?  ${$self}{aliases}{$_} : $_,
+                                  );
+      }
+    return;
+}
+
+sub get_every_or_custom_value{
+  my $self = shift;
+  my %input = @_;
+
+  my $toBeAssignedTo = $input{toBeAssignedTo};
+  my $toBeAssignedToAlias = $input{toBeAssignedToAlias};
+
+  # alias
+  if(${$self}{aliases}{$toBeAssignedTo}){
+        $self->logger("aliased $toBeAssignedTo using ${$self}{aliases}{$toBeAssignedTo}") if($is_t_switch_active);
+  }
+
+  # name of the object in the modifyLineBreaks yaml (e.g environments, ifElseFi, etc)
+  my $YamlName = ${$self}{modifyLineBreaksYamlName};
+
+  # if the YamlName is either optionalArguments or mandatoryArguments, then we'll be looking for information about the *parent*
+  my $name = ($YamlName =~ m/Arguments/) ? ${$self}{parent} : ${$self}{name};
+
+  # these variables just ease the notation what follows
+  my $everyValue = ${${$masterSettings{modifyLineBreaks}}{$YamlName}}{$toBeAssignedToAlias};
+  my $customValue = ${${${$masterSettings{modifyLineBreaks}}{$YamlName}}{$name}}{$toBeAssignedToAlias};
+
+  # check for the *custom* value
+  if (defined $customValue){
+      $self->logger("$name: $toBeAssignedToAlias=$customValue, (*custom* value) adjusting $toBeAssignedTo") if($is_t_switch_active);
+      ${$self}{$toBeAssignedTo} = $customValue !=0 ? $customValue : undef;
+   } else {
+      # check for the *every* value
+      if (defined $everyValue and $everyValue != 0){
+          $self->logger("$name: $toBeAssignedToAlias=$everyValue, (*every* value) adjusting $toBeAssignedTo") if($is_t_switch_active);
+          ${$self}{$toBeAssignedTo} = $everyValue;
+      }
+   }
+  return;
+}
+
+sub get_indentation_information{
+    my $self = shift;
+
+    #**************************************
+    # SEARCHING ORDER:
+    #   noAdditionalIndent *per-name* basis
+    #   indentRules *per-name* basis
+    #   noAdditionalIndentGlobal
+    #   indentRulesGlobal
+    #**************************************
+
+    # noAdditionalIndent can be a scalar or a hash, e.g
+    #
+    #   noAdditionalIndent:
+    #       myexample: 1
+    #
+    # OR
+    #
+    #   noAdditionalIndent:
+    #       myexample: 
+    #           body: 1
+    #           optionalArguments: 1
+    #           mandatoryArguments: 1
+    # 
+    # specifying as a scalar with no field (e.g myexample: 1)
+    # will be interpreted as noAdditionalIndent for *every*
+    # field, so the body, optional arguments and mandatory arguments
+    # will *all* receive noAdditionalIndent 
+    #
+    # indentRules can also be a scalar or a hash, e.g
+    #   indentRules:
+    #       myexample: "\t"
+    #
+    # OR
+    #
+    #   indentRules:
+    #       myexample:
+    #           body: "  "
+    #           optionalArguments: "\t \t"
+    #           mandatoryArguments: ""
+    #
+    # specifying as a scalar with no field will
+    # mean that *every* field will receive the same treatment
+
+    # if the YamlName is, for example, optionalArguments, mandatoryArguments, heading, then we'll be looking for information about the *parent*
+    my $name = (defined ${$self}{nameForIndentationSettings}) ? ${$self}{nameForIndentationSettings} : ${$self}{name};
+
+    # if the YamlName is not optionalArguments, mandatoryArguments, heading (possibly others) then assume we're looking for 'body'
+    my $YamlName = $self->get_object_attribute_for_indentation_settings;
+
+    my $indentationInformation;
+    foreach my $indentationAbout ("noAdditionalIndent","indentRules"){
+        # check that the 'thing' is defined
+        if(defined ${$masterSettings{$indentationAbout}}{$name}){
+            if(ref ${$masterSettings{$indentationAbout}}{$name} eq "HASH"){
+                $self->logger("$indentationAbout indentation specified with multiple fields for $name, searching for $name: $YamlName (see $indentationAbout)") if $is_t_switch_active ;
+                $indentationInformation = ${${$masterSettings{$indentationAbout}}{$name}}{$YamlName};
+            } else {
+                $indentationInformation = ${$masterSettings{$indentationAbout}}{$name};
+                $self->logger("$indentationAbout indentation specified for $name (for *all* fields, body, optionalArguments, mandatoryArguments, afterHeading), using '$indentationInformation' (see $indentationAbout)") if $is_t_switch_active ;
+            }
+            # return, after performing an integrity check
+            if(defined $indentationInformation){
+                if($indentationAbout eq "noAdditionalIndent" and $indentationInformation == 1){
+                        $self->logger("Found! Using '' (see $indentationAbout)") if $is_t_switch_active;
+                        return q();
+                } elsif($indentationAbout eq "indentRules" and $indentationInformation=~m/^\h*$/){
+                        $self->logger("Found! Using '$indentationInformation' (see $indentationAbout)") if $is_t_switch_active;
+                        return $indentationInformation ;
+                }
+            }
+        }
+    }
+
+    # gather information
+    $YamlName = ${$self}{modifyLineBreaksYamlName};
+
+    foreach my $indentationAbout ("noAdditionalIndent","indentRules"){
+        # global assignments in noAdditionalIndentGlobal and/or indentRulesGlobal
+        my $globalInformation = $indentationAbout."Global";
+        next if(!(defined ${$masterSettings{$globalInformation}}{$YamlName})); 
+        if( ($globalInformation eq "noAdditionalIndentGlobal") and ${$masterSettings{$globalInformation}}{$YamlName}==1){
+            $self->logger("$globalInformation specified for $YamlName (see $globalInformation)") if $is_t_switch_active;
+            return q();
+        } elsif($globalInformation eq "indentRulesGlobal") {
+            if(${$masterSettings{$globalInformation}}{$YamlName}=~m/^\h*$/){
+                $self->logger("$globalInformation specified for $YamlName (see $globalInformation)") if $is_t_switch_active;
+                return ${$masterSettings{$globalInformation}}{$YamlName};
+            } else {
+                $self->logger("$globalInformation specified (${$masterSettings{$globalInformation}}{$YamlName}) for $YamlName, but it needs to only contain horizontal space -- I'm ignoring this one") if $is_t_switch_active;
+          }
+        }
+    }
+
+    # return defaultIndent, by default
+    $self->logger("Using defaultIndent for $name") if $is_t_switch_active;
+    return $masterSettings{defaultIndent};
+}
+
+sub get_object_attribute_for_indentation_settings{
+    # when looking for noAdditionalIndent or indentRules, we may need to determine
+    # which thing we're looking for, e.g
+    #
+    #   chapter:
+    #       body: 0
+    #       optionalArguments: 1
+    #       mandatoryArguments: 1
+    #       afterHeading: 0
+    #
+    # this method returns 'body' by default, but the other objects (optionalArgument, mandatoryArgument, afterHeading)
+    # return their appropriate identifier.
+    return "body";
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Heading.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Heading.pm
new file mode 100644
index 00000000000..be7ea36661d
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Heading.pm
@@ -0,0 +1,240 @@
+package LatexIndent::Heading;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::Switches qw/$is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_heading construct_headings_levels/;
+our $headingCounter;
+our @headingsRegexpArray;
+our $allHeadingsRegexp = q();
+
+sub construct_headings_levels{
+    my $self = shift;
+
+    # grab the heading levels
+    my %headingsLevels = %{$masterSettings{indentAfterHeadings}};
+
+    # delete the values that have indentAfterThisHeading set to 0
+    while( my ($headingName,$headingInfo)= each %headingsLevels){
+        if(!${$headingsLevels{$headingName}}{indentAfterThisHeading}){
+            $self->logger("Not indenting after $headingName (see indentAfterThisHeading)",'heading') if $is_t_switch_active;
+            delete $headingsLevels{$headingName};
+        } else {
+            # *all heading* regexp, remembering put starred headings at the front of the regexp
+            if($headingName =~ m/\*/){
+                 $self->logger("Putting $headingName at the beginning of the allHeadings regexp, as it contains a *");
+                 $allHeadingsRegexp = $headingName.($allHeadingsRegexp eq '' ?q():"|$allHeadingsRegexp");
+            } else {
+                 $self->logger("Putting $headingName at the END of the allHeadings regexp, as it contains a *");
+                 $allHeadingsRegexp .= ($allHeadingsRegexp eq '' ?q():"|").$headingName ;
+            }
+        }
+    }
+
+    # check for a * in the name
+    $allHeadingsRegexp =~ s/\*/\\\*/g;
+
+    # sort the file extensions by preference 
+    my @sortedByLevels = sort { ${$headingsLevels{$a}}{level} <=> $headingsLevels{$b}{level} } keys(%headingsLevels);
+
+    # it could be that @sortedByLevels is empty;
+    return if !@sortedByLevels;
+
+    $self->logger("All headings regexp: $allHeadingsRegexp",'heading'); 
+    $self->logger("Now to construct headings regexp for each level:",'heading'); 
+
+    # loop through the levels, and create a regexp for each (min and max values are the first and last values respectively from sortedByLevels)
+    for(my $i = ${$headingsLevels{$sortedByLevels[0]}}{level}; $i <= ${$headingsLevels{$sortedByLevels[-1]}}{level}; $i++ ){
+        # level regexp
+        my @tmp = grep { ${$headingsLevels{$_}}{level} == $i } keys %headingsLevels;
+        if(@tmp){
+            my $headingsAtThisLevel = q();
+            foreach(@tmp){
+               # put starred headings at the front of the regexp
+               if($_ =~ m/\*/){
+                    $self->logger("Putting $_ at the beginning of this regexp, as it contains a *");
+                    $headingsAtThisLevel = $_.($headingsAtThisLevel eq '' ?q():"|$headingsAtThisLevel");
+               } else {
+                    $self->logger("Putting $_ at the END of this regexp, as it contains a *");
+                    $headingsAtThisLevel .= ($headingsAtThisLevel eq '' ?q():"|").$_ ;
+               }
+            }
+
+            # make the stars escaped correctly
+            $headingsAtThisLevel =~ s/\*/\\\*/g;
+            push(@headingsRegexpArray,$headingsAtThisLevel);
+            $self->logger("Heading level regexp for level $i will contain: $headingsAtThisLevel");
+        }
+    }
+  }
+
+sub find_heading{
+
+    # if there are no headings regexps, there's no point going any further
+    return if!@headingsRegexpArray;
+
+    my $self = shift;
+
+    # otherwise loop through the headings regexp
+    $self->logger("Searching for special begin/end (see specialBeginEnd)");
+
+    # loop through each headings match; note that we need to 
+    # do it in *reverse* so as to ensure that the lower level headings get matched first of all
+    foreach(reverse(@headingsRegexpArray)){
+        
+        # the regexp
+        my $headingRegExp = qr/
+                              (
+                                  \\($_)        # name stored into $2
+                              )                 # beginning bit into $1
+                              (
+                                  .*?                 
+                              )                 # body into $3      
+                              (\R*)?            # linebreaks at end of body into $4
+                              ((?:\\(?:$allHeadingsRegexp))|$)  # up to another heading, or else the end of the file
+                           /sx;
+        
+        while(${$self}{body} =~ m/$headingRegExp/){
+
+            # log file output
+            $self->logger("heading found: $2",'heading');
+
+            ${$self}{body} =~ s/
+                                $headingRegExp
+                               /
+                                # create a new heading object
+                                my $headingObject = LatexIndent::Heading->new(begin=>q(),
+                                                                        body=>$1.$3,
+                                                                        end=>q(),
+                                                                        afterbit=>($4?$4:q()).($5?$5:q()),
+                                                                        name=>$2.":heading",
+                                                                        parent=>$2,
+                                                                        nameForIndentationSettings=>$2,
+                                                                        linebreaksAtEnd=>{
+                                                                          begin=>0,
+                                                                          body=>0,
+                                                                          end=>0,
+                                                                        },
+                                                                        modifyLineBreaksYamlName=>"afterHeading",
+                                                                        endImmediatelyFollowedByComment=>0,
+                                                                      );
+
+                                # the settings and storage of most objects has a lot in common
+                                $self->get_settings_and_store_new_object($headingObject);
+                                ${@{${$self}{children}}[-1]}{replacementText};
+                              /xse;
+        }
+     }
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text for a heading (chapter, section, etc) needs to put the trailing part back in
+    $self->logger("Custom replacement text routine for heading ${$self}{name}");
+    ${$self}{replacementText} = ${$self}{id}.${$self}{afterbit};
+    delete ${$self}{afterbit};
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $headingCounter++;
+
+    ${$self}{id} = "$tokens{heading}$headingCounter";
+    return;
+}
+
+sub adjust_replacement_text_line_breaks_at_end{
+    return;
+}
+
+sub get_object_attribute_for_indentation_settings{
+    # when looking for noAdditionalIndent or indentRules, we may need to determine
+    # which thing we're looking for, e.g
+    #
+    #   chapter:
+    #       body: 0
+    #       optionalArguments: 1
+    #       mandatoryArguments: 1
+    #       afterHeading: 0
+    #
+    # this method returns 'body' by default, but the other objects (optionalArgument, mandatoryArgument, afterHeading)
+    # return their appropriate identifier.
+    my $self = shift;
+    
+    return ${$self}{modifyLineBreaksYamlName};
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # search for commands, keys, named grouping braces
+    $self->find_commands_or_key_equals_values_braces;
+
+    # we need to transfer the details from the modifyLineBreaks of the command
+    # child object to the heading object.
+    #
+    # for example, if we have
+    #
+    #   \chapter{some heading here}
+    #
+    # and we want to modify the linebreak before the \chapter command using, for example,
+    #
+    # commands:
+    #     CommandStartsOnOwnLine: 1
+    #
+    # then we need to transfer this information to the heading object
+    if($is_m_switch_active){
+        $self->logger("Searching for linebreak preferences immediately infront of ${$self}{parent}",'heading');
+        foreach(@{${$self}{children}}){
+            if(${$_}{name} eq ${$self}{parent}){
+                $self->logger("Named child found: ${$_}{name}");
+                if(defined ${$_}{BeginStartsOnOwnLine}){
+                    $self->logger("Transferring information from ${$_}{id} (${$_}{name}) to ${$self}{id} (${$self}{name}) for BeginStartsOnOwnLine");
+                    ${$self}{BeginStartsOnOwnLine} = ${$_}{BeginStartsOnOwnLine};
+                } else {
+                    $self->logger("No information found in ${$_}{name} for BeginStartsOnOwnLine");
+                }
+                last;
+            }
+        }
+    }
+
+    # search for special begin/end
+    $self->find_special;
+
+    return;
+}
+
+sub add_surrounding_indentation_to_begin_statement{
+    # almost all of the objects add surrounding indentation to the 'begin' statements, 
+    # but some (e.g HEADING) have their own method
+    my $self = shift;
+    
+    $self->logger("Adding surrounding indentation after (empty, by design!) begin statement of ${$self}{name} (${$self}{id})");
+    ${$self}{begin} .= ${$self}{surroundingIndentation};  # add indentation
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/HiddenChildren.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/HiddenChildren.pm
new file mode 100644
index 00000000000..594e7dbd14e
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/HiddenChildren.pm
@@ -0,0 +1,190 @@
+package LatexIndent::HiddenChildren;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::Tokens qw/%tokens/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/find_surrounding_indentation_for_children update_family_tree get_family_tree check_for_hidden_children %familyTree/;
+
+# hiddenChildren can be stored in a global array, it doesn't matter what level they're at
+our %familyTree;
+our %allChildren;
+
+
+#----------------------------------------------------------
+#   Discussion surrounding hidden children
+#
+#   Consider the following latex code
+#
+#   \begin{one}
+#       body of one 
+#       body of one 
+#       body of one 
+#       \begin{two}
+#          body of two
+#          body of two
+#          body of two
+#          body of two
+#       \end{two}
+#   \end{one}
+#
+#   From the visual perspective, we might say that <one> and <two> are *nested* children;
+#   from the persepective of latexindent.pl, however, they actually have *the same level*.
+#
+#   Graphically, you might represent it as follows
+#
+#                     *
+#                   /  \
+#                  /    \
+#                 /      \ 
+#                O        O
+#
+#   where * represents the 'root' document object, and each 'O' is an environment object; the
+#   first one, on the left, represents <two> and the second one, on the right, represents <one>. 
+#   (Remember that the environment regexp does not allow \begin within its body.)
+#
+#   When processing the document, <one> will be processed *before* <two>. Furthermore, because
+#   <one> and <two> are at the same level, they are not *natural* ancestors of each other; as such, 
+#   we say that <two> is a *hidden* child, and that its 'adopted' ancestor is <one>. 
+#
+#   We need to go to a lot of effort to make sure that <two> knows about its ancestors and its 
+#   surrounding indentation (<one> in this case). The subroutines in this file do that effort.
+#----------------------------------------------------------
+
+sub find_surrounding_indentation_for_children{
+    my $self = shift;
+
+    # output to logfile
+    $self->logger("FamilyTree before update:",'heading') if $is_t_switch_active;
+    $self->logger(Dumper(\%familyTree)) if($is_t_switch_active);
+
+    # update the family tree with ancestors
+    $self->update_family_tree;
+
+    # output information to the logfile
+    $self->logger("FamilyTree after update:",'heading') if $is_t_switch_active;
+    $self->logger(Dumper(\%familyTree)) if($is_t_switch_active);
+
+    while( my ($idToSearch,$ancestorToSearch) = each %familyTree){
+          $self->logger("Hidden child ID: ,$idToSearch, here are its ancestors:",'heading') if $is_t_switch_active;
+          foreach(@{${$ancestorToSearch}{ancestors}}){
+              $self->logger("ID: ${$_}{ancestorID}") if($is_t_switch_active);
+              my $tmpIndentation = ref(${$_}{ancestorIndentation}) eq 'SCALAR'?${${$_}{ancestorIndentation}}:${$_}{ancestorIndentation};
+              $tmpIndentation = $tmpIndentation ? $tmpIndentation : q(); 
+              $self->logger("indentation: '$tmpIndentation'") if($is_t_switch_active);
+              }
+          }
+
+    return;
+}
+
+sub update_family_tree{
+    my $self = shift;
+
+    # loop through the hash
+    $self->logger("Updating FamilyTree...",'heading') if $is_t_switch_active;
+    while( my ($idToSearch,$ancestorToSearch)= each %familyTree){
+          foreach(@{${$ancestorToSearch}{ancestors}}){
+              my $ancestorID = ${$_}{ancestorID};
+              $self->logger("current ID: $idToSearch, ancestor: $ancestorID") if($is_t_switch_active);
+              if($familyTree{$ancestorID}){
+                  $self->logger("$ancestorID is a key within familyTree, grabbing its ancestors") if($is_t_switch_active);
+                  my $naturalAncestors = q();
+                  foreach(@{${$familyTree{$idToSearch}}{ancestors}}){
+                      $naturalAncestors .= "---".${$_}{ancestorID} if(${$_}{type} eq "natural");
+                  }
+                  foreach(@{${$familyTree{$ancestorID}}{ancestors}}){
+                      $self->logger("ancestor of *hidden* child: ${$_}{ancestorID}") if($is_t_switch_active);
+                      my $newAncestorId = ${$_}{ancestorID};
+                      my $type;
+                      if($naturalAncestors =~ m/$ancestorID/){
+                            $type = "natural";
+                      } else {
+                            $type = "adopted";
+                      }
+                      my $matched = grep { $_->{ancestorID} eq $newAncestorId } @{${$familyTree{$idToSearch}}{ancestors}};
+                      push(@{${$familyTree{$idToSearch}}{ancestors}},{ancestorID=>${$_}{ancestorID},ancestorIndentation=>${$_}{ancestorIndentation},type=>$type}) unless($matched);
+                  }
+              } else {
+                    my $naturalAncestors = q();
+                    foreach(@{${$familyTree{$idToSearch}}{ancestors}}){
+                        $naturalAncestors .= "---".${$_}{ancestorID} if(${$_}{type} eq "natural");
+                    }
+                    $self->logger("natural ancestors of $ancestorID: $naturalAncestors") if($is_t_switch_active);
+                    foreach(@{${$allChildren{$ancestorID}}{ancestors}}){
+                        my $newAncestorId = ${$_}{ancestorID};
+                        my $type;
+                        if($naturalAncestors =~ m/$newAncestorId/){
+                            $type = "natural";
+                        } else {
+                            $type = "adopted";
+                        }
+                        my $matched = grep { $_->{ancestorID} eq $newAncestorId } @{${$familyTree{$idToSearch}}{ancestors}};
+                        unless($matched){
+                            $self->logger("ancestor of UNHIDDEN child: ${$_}{ancestorID}") if($is_t_switch_active);
+                            push(@{${$familyTree{$idToSearch}}{ancestors}},{ancestorID=>${$_}{ancestorID},ancestorIndentation=>${$_}{ancestorIndentation},type=>$type});
+                        }
+                    }
+              } 
+          }
+    }
+
+}
+
+sub check_for_hidden_children{
+
+    my $self = shift;
+
+    # if there are no hidden children, then exit
+    return if ${$self}{body} !~ m/$tokens{beginOfToken}/;
+
+    # grab the matches
+    my @matched = (${$self}{body} =~ /((?:$tokens{ifelsefiSpecial})?$tokens{beginOfToken}.[-a-z0-9]+?$tokens{endOfToken})/ig);
+
+    # log file
+    $self->logger("Hidden children check") if $is_t_switch_active;
+    $self->logger(join("|",@matched)) if $is_t_switch_active;
+
+    my $naturalAncestors = ${$self}{naturalAncestors}; 
+
+    # loop through the hidden children
+    foreach my $match (@matched){
+        # update the family tree with ancestors of self
+        if(${$self}{ancestors}){
+            foreach(@{${$self}{ancestors}}){
+                my $newAncestorId = ${$_}{ancestorID};
+                unless (grep { $_->{ancestorID} eq $newAncestorId } @{${$familyTree{$match}}{ancestors}}){
+                    my $type = ($naturalAncestors =~ m/${$_}{ancestorID}/ ) ? "natural" : "adopted";
+                    $self->logger("Adding ${$_}{ancestorID} to the $type family tree of $match") if($is_t_switch_active);
+                    push(@{$familyTree{$match}{ancestors}},{ancestorID=>${$_}{ancestorID},ancestorIndentation=>${$_}{ancestorIndentation},type=>$type});
+                }
+            }
+        }
+
+        # update the family tree with self
+        unless (grep { $_->{ancestorID} eq ${$self}{id}} @{${$familyTree{$match}}{ancestors}}){
+                    my $type = ($naturalAncestors =~ m/${$self}{id}/ ) ? "natural" : "adopted";
+                    $self->logger("Adding ${$self}{id} to the $type family tree of hiddenChild $match") if($is_t_switch_active);
+                    push(@{$familyTree{$match}{ancestors}},{ancestorID=>${$self}{id},ancestorIndentation=>${$self}{indentation},type=>$type});
+        }
+    }
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/HorizontalWhiteSpace.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/HorizontalWhiteSpace.pm
new file mode 100644
index 00000000000..360a8077022
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/HorizontalWhiteSpace.pm
@@ -0,0 +1,62 @@
+package LatexIndent::HorizontalWhiteSpace;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/remove_trailing_whitespace remove_leading_space/;
+
+sub remove_trailing_whitespace{
+    my $self = shift;
+    my %input = @_;
+
+    # this method can be called before the indendation, and after, depending upon the input
+    if($input{when} eq "before"){
+        return unless(${$masterSettings{removeTrailingWhitespace}}{beforeProcessing});
+        $self->logger("Removing trailing white space *before* the document is processed (see removeTrailingWhitespace: beforeProcessing)",'heading') if $is_t_switch_active;
+    } elsif($input{when} eq "after"){
+        return unless(${$masterSettings{removeTrailingWhitespace}}{afterProcessing});
+        $self->logger("Removing trailing white space *after* the document is processed (see removeTrailingWhitespace: afterProcessing)",'heading') if $is_t_switch_active;
+    } else {
+        return;
+    }
+
+    ${$self}{body} =~ s/
+                       \h+  # followed by possible horizontal space
+                       $    # up to the end of a line
+                       //xsmg;
+
+    $self->logger("Processed body, *$input{when}* indentation (${$self}{name}):") if($is_t_switch_active);
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+
+}
+
+sub remove_leading_space{
+    my $self = shift;
+    $self->logger("Removing leading space from ${$self}{name} (verbatim/noindentblock already accounted for)",'heading') if $is_t_switch_active;
+    ${$self}{body} =~ s/
+                        (   
+                            ^           # beginning of the line
+                            \h*         # with 0 or more horizontal spaces
+                        )?              # possibly
+                        //mxg;
+    return;
+}
+
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/IfElseFi.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/IfElseFi.pm
new file mode 100755
index 00000000000..e1ac17b3613
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/IfElseFi.pm
@@ -0,0 +1,276 @@
+package LatexIndent::IfElseFi;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_ifelsefi/;
+our %previouslyFoundSettings;
+our $ifElseFiCounter;
+
+# store the regular expresssion for matching and replacing the \if...\else...\fi statements
+# note: we search for \else separately in an attempt to keep this regexp a little more managable
+our $ifElseFiRegExp = qr/
+                (
+                    \\
+                        (@?if[a-zA-Z@]*?)
+                    \h*
+                    (\R*)
+                )                           # begin statement, e.g \ifnum, \ifodd
+                (
+                    \\(?!if)|\R|\h|\#|!-!   # up until a \\, linebreak # or !-!, which is 
+                )                           # part of the tokens used for latexindent
+                (
+                    (?: 
+                        (?!\\if).
+                    )*?                     # body, which can't include another \if
+                )
+                (\R*)                       # linebreaks after body
+                (
+                    \\fi(?![a-zA-Z])                    # \fi statement 
+                )
+                (\h*)                       # 0 or more horizontal spaces
+                (\R)?                       # linebreaks after \fi
+/sx;
+
+
+sub indent{
+    my $self = shift;
+
+    # determine the surrounding and current indentation
+    $self->determine_total_indentation;
+
+    # line break checks after \if statement, can get messy if we 
+    # have, for example
+    #       \ifnum
+    #               something
+    # which might be changed into
+    #       \ifnumsomething
+    # which is undeserible
+    if (defined ${$self}{BodyStartsOnOwnLine}
+        and ${$self}{BodyStartsOnOwnLine}==-1 
+        and ${$self}{body} !~ m/^(\h|\\|(?:!-!))/s
+    ){
+        ${$self}{begin} .= " ";
+    }
+
+    # indent the body
+    $self->indent_body;
+
+    # calculate and grab the surrounding indentation for the \else statement adjustment
+    $self->get_surrounding_indentation;
+    my $surroundingIndentation = ${$self}{surroundingIndentation}?${$self}{surroundingIndentation}:q();
+
+    if(${$self}{elsePresent} and ${$self}{linebreaksAtEnd}{ifbody}){
+            $self->logger("Adding surrounding indentation to \\else statement ('$surroundingIndentation')")if $is_t_switch_active;
+            ${$self}{body} =~ s/\h*\\else/$surroundingIndentation\\else/; 
+            $self->logger("Body (${$self}{name}) after \\else adjustment:\n${$self}{body}") if $is_t_switch_active;
+    }
+
+    # indent the end statement
+    $self->indent_end_statement;
+
+    # wrap-up statement
+    $self->wrap_up_statement;
+
+    # line break checks *after* \end{statement}
+    if (defined ${$self}{EndFinishesWithLineBreak}
+        and ${$self}{EndFinishesWithLineBreak}==-1 
+        ) {
+        # add a single horizontal space after the child id, otherwise we can end up 
+        # with things like
+        #       before: 
+        #               \fi
+        #                   text
+        #       after:
+        #               \fitext
+        $self->logger("Adding a single space after \\fi statement (otherwise \\fi can be comined with next line of text in an unwanted way)",'heading') if $is_t_switch_active;
+        ${$self}{end} =${$self}{end}." ";
+    }
+
+    return $self;
+}
+
+sub find_ifelsefi{
+    my $self = shift;
+
+    while( ${$self}{body} =~ m/$ifElseFiRegExp\h*($trailingCommentRegExp)?/){
+
+      ${$self}{body} =~ s/
+                $ifElseFiRegExp(\h*)($trailingCommentRegExp)?
+                    /
+                        # log file output
+                        $self->logger("IfElseFi found: $2",'heading')if $is_t_switch_active;
+           
+                        # create a new IfElseFi object
+                        my $ifElseFi = LatexIndent::IfElseFi->new(begin=>$1.(($4 eq "\n" and !$3)?"\n":q()),
+                                                                name=>$2,
+                                                                # if $4 is a line break, don't count it twice (it will already be in 'begin')
+                                                                body=>($4 eq "\n") ? $5.$6 : $4.$5.$6,
+                                                                end=>$7,
+                                                                linebreaksAtEnd=>{
+                                                                  begin=>(($4 eq "\n")||$3)?1:0,
+                                                                  body=>$6?1:0,
+                                                                  end=>$9?1:0,
+                                                                },
+                                                                aliases=>{
+                                                                  # begin statements
+                                                                  BeginStartsOnOwnLine=>"IfStartsOnOwnLine",
+                                                                  # end statements
+                                                                  EndStartsOnOwnLine=>"FiStartsOnOwnLine",
+                                                                  # after end statements
+                                                                  EndFinishesWithLineBreak=>"FiFinishesWithLineBreak",
+                                                                },
+                                                                elsePresent=>0,
+                                                                modifyLineBreaksYamlName=>"ifElseFi",
+                                                                additionalAssignments=>["ElseStartsOnOwnLine","ElseFinishesWithLineBreak"],
+                                                                endImmediatelyFollowedByComment=>$9?0:($11?1:0),
+                                                                horizontalTrailingSpace=>$8?$8:q(),
+                                                              );
+                        # the settings and storage of most objects has a lot in common
+                        $self->get_settings_and_store_new_object($ifElseFi);
+                        ${@{${$self}{children}}[-1]}{replacementText}.($10?$10:q()).($11?$11:q());
+                        /xse;
+
+    } 
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # check for existence of \else statement, and associated line break information
+    $self->check_for_else_statement;
+
+    # search for headings (important to do this before looking for commands!)
+    $self->find_heading;
+
+    # search for commands, keys, named grouping braces
+    $self->find_commands_or_key_equals_values_braces;
+
+    # search for arguments
+    $self->find_opt_mand_arguments;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for special begin/end
+    $self->find_special;
+
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $ifElseFiCounter++;
+
+    ${$self}{id} = "$tokens{ifelsefi}$ifElseFiCounter";
+    return;
+}
+
+sub check_for_else_statement{
+    my $self = shift;
+    $self->logger("Looking for \\else statement (${$self}{name})",'heading') if $is_t_switch_active;
+    if(${$self}{body} =~ m/
+                            (\R*)   # possible line breaks before \else statement
+                            \\else  
+                            \h*     # possible horizontal space
+                            (\R*)   # possible line breaks after \else statement
+                            /x){
+      $self->logger("found \\else statement, storing line break information:") if($is_t_switch_active);
+
+      # linebreaks *before* \else statement
+      ${$self}{linebreaksAtEnd}{ifbody} = $1?1:0;
+      $self->logger("linebreaksAtEnd of ifbody: ${$self}{linebreaksAtEnd}{ifbody}") if($is_t_switch_active);
+
+      # linebreaks *after* \else statement
+      ${$self}{linebreaksAtEnd}{else} = $2?1:0;
+      $self->logger("linebreaksAtEnd of else: ${$self}{linebreaksAtEnd}{else}") if($is_t_switch_active);
+      ${$self}{elsePresent}=1;
+
+      # check that \else isn't the first thing in body
+      if(${$self}{body} =~ m/^\\else/s and ${$self}{linebreaksAtEnd}{begin}){
+        ${$self}{linebreaksAtEnd}{ifbody} = 1;
+        $self->logger("\\else *begins* the ifbody, linebreaksAtEnd of ifbody: ${$self}{linebreaksAtEnd}{ifbody}") if($is_t_switch_active);
+      }
+
+      # check if -m switch is active
+      return unless $is_m_switch_active;
+
+      # possibly modify line break *before* \else statement
+      if(defined ${$self}{ElseStartsOnOwnLine}){
+          if(${$self}{ElseStartsOnOwnLine}>=1 and !${$self}{linebreaksAtEnd}{ifbody}){
+              # by default, assume that no trailing comment token is needed
+              my $trailingCommentToken = q();
+              if(${$self}{ElseStartsOnOwnLine}==2){
+                $self->logger("Adding a % immediately before else statement of ${$self}{name} (ElseStartsOnOwnLine==2)") if $is_t_switch_active;
+                $trailingCommentToken = "%".$self->add_comment_symbol;
+              }
+
+              # add a line break after ifbody, if appropriate
+              $self->logger("Adding a linebreak before the \\else statement (see ElseStartsOnOwnLine)");
+              ${$self}{body} =~ s/\\else/$trailingCommentToken\n\\else/s;
+              ${$self}{linebreaksAtEnd}{ifbody} = 1;
+          } elsif (${$self}{ElseStartsOnOwnLine}==-1 and ${$self}{linebreaksAtEnd}{ifbody}){
+              # remove line break *after* ifbody, if appropriate
+              $self->logger("Removing linebreak before \\else statement (see ElseStartsOnOwnLine)");
+              ${$self}{body} =~ s/\R*(\h*)\\else/$1\\else/sx;
+              ${$self}{linebreaksAtEnd}{ifbody} = 0;
+          }
+      }
+
+      # possibly modify line break *before* \else statement
+      if(defined ${$self}{ElseFinishesWithLineBreak}){
+          if(${$self}{ElseFinishesWithLineBreak}>=1 and !${$self}{linebreaksAtEnd}{else}){
+              # by default, assume that no trailing comment token is needed
+              my $trailingCommentToken = q();
+              if(${$self}{ElseFinishesWithLineBreak}==2){
+                return if(${$self}{body} =~ m/\\else\h*$trailingCommentRegExp/s);
+                $self->logger("Adding a % immediately after else statement of ${$self}{name} (ElseFinishesWithLineBreak==2)") if $is_t_switch_active;
+                $trailingCommentToken = "%".$self->add_comment_symbol;
+              }
+
+              # add a line break after else, if appropriate
+              $self->logger("Adding a linebreak after the \\else statement (see ElseFinishesWithLineBreak)")if $is_t_switch_active;
+              ${$self}{body} =~ s/\\else\h*/\\else$trailingCommentToken\n/s;
+              ${$self}{linebreaksAtEnd}{else} = 1;
+          } elsif (${$self}{ElseFinishesWithLineBreak}==-1 and ${$self}{linebreaksAtEnd}{else}){
+              # remove line break *after* else, if appropriate, 
+              # note the space so that, for example,
+              #     \else
+              #             some text
+              # becomes
+              #     \else some text
+              # and not
+              #     \elsesome text
+              $self->logger("Removing linebreak after \\else statement (see ElseFinishesWithLineBreak)")if $is_t_switch_active;
+              ${$self}{body} =~ s/\\else\h*\R*/\\else /sx;
+              ${$self}{linebreaksAtEnd}{else} = 0;
+          }
+      }
+
+      return;
+    } else {
+      $self->logger("\\else statement not found") if($is_t_switch_active);
+    }
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Indent.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Indent.pm
new file mode 100644
index 00000000000..600aed5040f
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Indent.pm
@@ -0,0 +1,371 @@
+package LatexIndent::Indent;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::Switches qw/$is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+use LatexIndent::HiddenChildren qw/%familyTree/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/indent wrap_up_statement determine_total_indentation indent_begin indent_body indent_end_statement final_indentation_check push_family_tree_to_indent get_surrounding_indentation indent_children_recursively check_for_blank_lines_at_beginning put_blank_lines_back_in_at_beginning add_surrounding_indentation_to_begin_statement/;
+our %familyTree;
+
+sub indent{
+    my $self = shift;
+
+    # determine the surrounding and current indentation
+    $self->determine_total_indentation;
+
+    # indent the begin statement
+    $self->indent_begin;
+
+    # indent the body
+    $self->indent_body;
+
+    # indent the end statement
+    $self->indent_end_statement;
+
+    # output the completed object to the log file
+    $self->logger("Complete indented object (${$self}{name}) after indentation:\n${$self}{begin}${$self}{body}${$self}{end}") if $is_t_switch_active;
+
+    # wrap-up statement
+    $self->wrap_up_statement;
+    return $self;
+}
+
+sub wrap_up_statement{
+    my $self = shift;
+    $self->logger("Finished indenting ${$self}{name}",'heading') if $is_t_switch_active;
+    return $self;
+  }
+
+sub determine_total_indentation{
+    my $self = shift;
+
+    # calculate and grab the surrounding indentation
+    $self->get_surrounding_indentation;
+
+    # logfile information
+    my $surroundingIndentation = ${$self}{surroundingIndentation};
+    $self->logger("indenting object ${$self}{name}") if($is_t_switch_active);
+    (my $during = $surroundingIndentation) =~ s/\t/TAB/g;
+    $self->logger("indentation *surrounding* object: '$during'") if($is_t_switch_active);
+    ($during = ${$self}{indentation}) =~ s/\t/TAB/g;
+    $self->logger("indentation *of* object: '$during'") if($is_t_switch_active);
+    ($during = $surroundingIndentation.${$self}{indentation}) =~ s/\t/TAB/g;
+    $self->logger("*total* indentation to be added: '$during'") if($is_t_switch_active);
+
+    # form the total indentation of the object
+    ${$self}{indentation} = $surroundingIndentation.${$self}{indentation};
+
+}
+
+sub get_surrounding_indentation{
+    my $self = shift;
+
+    my $surroundingIndentation = q();
+
+    if($familyTree{${$self}{id}}){
+        $self->logger("Adopted ancestors found!") if($is_t_switch_active);
+        foreach(@{${$familyTree{${$self}{id}}}{ancestors}}){
+            if(${$_}{type} eq "adopted"){
+                my $newAncestorId = ${$_}{ancestorID};
+                $self->logger("ancestor ID: $newAncestorId, adding indentation of $newAncestorId to surroundingIndentation of ${$self}{id}") if($is_t_switch_active);
+                $surroundingIndentation .= ref(${$_}{ancestorIndentation}) eq 'SCALAR'
+                                                    ?
+                                            (${${$_}{ancestorIndentation}}?${${$_}{ancestorIndentation}}:q())
+                                                    :
+                                            (${$_}{ancestorIndentation}?${$_}{ancestorIndentation}:q());
+            }
+        }
+    }
+    ${$self}{surroundingIndentation} = $surroundingIndentation;
+
+}
+
+sub indent_begin{
+    # for most objects, the begin statement is just one line, but there are exceptions, e.g KeyEqualsValuesBraces
+    return;
+}
+
+sub indent_body{
+    my $self = shift;
+
+    # grab the indentation of the object
+    my $indentation = ${$self}{indentation};
+
+    # output to the logfile
+    $self->logger("Body (${$self}{name}) before indentation:\n${$self}{body}") if $is_t_switch_active;
+
+    # last minute check for modified bodyLineBreaks
+    $self->count_body_line_breaks if $is_m_switch_active;
+
+    # some objects need to check for blank line tokens at the beginning
+    $self->check_for_blank_lines_at_beginning if $is_m_switch_active; 
+
+    # some objects can format their body to align at the & character
+    $self->align_at_ampersand if ${$self}{lookForAlignDelims};
+
+    # body indendation
+    if(${$self}{linebreaksAtEnd}{begin}==1){
+        if(${$self}{body} =~ m/^\h*$/s){
+            $self->logger("Body of ${$self}{name} is empty, not applying indentation") if $is_t_switch_active;
+        } else {
+            # put any existing horizontal space after the current indentation
+            $self->logger("Entire body of ${$self}{name} receives indendentation") if $is_t_switch_active;
+            ${$self}{body} =~ s/^(\h*)/$indentation$1/mg;  # add indentation
+        }
+    } elsif(${$self}{linebreaksAtEnd}{begin}==0 and ${$self}{bodyLineBreaks}>0) {
+        if(${$self}{body} =~ m/
+                            (.*?)      # content of first line
+                            \R         # first line break
+                            (.*$)      # rest of body
+                            /sx){
+            my $bodyFirstLine = $1;
+            my $remainingBody = $2;
+            $self->logger("first line of body: $bodyFirstLine",'heading') if $is_t_switch_active;
+            $self->logger("remaining body (before indentation):\n'$remainingBody'") if($is_t_switch_active);
+    
+            # add the indentation to all the body except first line
+            $remainingBody =~ s/^/$indentation/mg unless($remainingBody eq '');  # add indentation
+            $self->logger("remaining body (after indentation):\n$remainingBody'") if($is_t_switch_active);
+    
+            # put the body back together
+            ${$self}{body} = $bodyFirstLine."\n".$remainingBody; 
+        }
+    }
+
+    # if the routine check_for_blank_lines_at_beginning has been called, then the following routine
+    # puts blank line tokens back in 
+    $self->put_blank_lines_back_in_at_beginning if $is_m_switch_active; 
+
+    # the final linebreak can be modified by a child object; see test-cases/commands/figureValign-mod5.tex, for example
+    if($is_m_switch_active and defined ${$self}{linebreaksAtEnd}{body} and ${$self}{linebreaksAtEnd}{body}==1 and ${$self}{body} !~ m/\R$/){
+        $self->logger("Updating body for ${$self}{name} to contain a linebreak at the end (linebreaksAtEnd is 1, but there isn't currently a linebreak)") if($is_t_switch_active);
+        ${$self}{body} .= "\n";
+    }
+
+    # output to the logfile
+    $self->logger("Body (${$self}{name}) after indentation:\n${$self}{body}") if $is_t_switch_active;
+    return $self;
+}
+
+sub check_for_blank_lines_at_beginning{
+    # some objects need this routine
+    return;
+}
+
+sub put_blank_lines_back_in_at_beginning{
+    # some objects need this routine
+    return;
+}
+
+sub indent_end_statement{
+    my $self = shift;
+    my $surroundingIndentation = (${$self}{surroundingIndentation} and $familyTree{${$self}{id}})
+                                            ?
+                                 (ref(${$self}{surroundingIndentation}) eq 'SCALAR'?${${$self}{surroundingIndentation}}:${$self}{surroundingIndentation})
+                                            :q();
+
+    # end{statement} indentation, e.g \end{environment}, \fi, }, etc
+    if(${$self}{linebreaksAtEnd}{body}){
+        ${$self}{end} =~ s/^\h*/$surroundingIndentation/mg;  # add indentation
+        $self->logger("Adding surrounding indentation to ${$self}{end} (${$self}{name}: '$surroundingIndentation')") if($is_t_switch_active);
+     }
+    return $self;
+}
+
+sub final_indentation_check{
+    # problem:
+    #       if a tab is appended to spaces, it will look different 
+    #       from spaces appended to tabs (see test-cases/items/spaces-and-tabs.tex)
+    # solution:
+    #       move all of the tabs to the beginning of ${$self}{indentation}
+    # notes;
+    #       this came to light when studying test-cases/items/items1.tex
+
+    my $self = shift;
+
+    my $indentation;
+    my $numberOfTABS; 
+    my $after;
+    ${$self}{body} =~ s/
+                        ^((\h*|\t*)((\h+)(\t+))+)
+                        /   
+                        # fix the indentation
+                        $indentation = $1;
+
+                        # count the number of tabs
+                        $numberOfTABS = () = $indentation=~ \/\t\/g;
+                        $self->logger("Number of tabs: $numberOfTABS") if($is_t_switch_active);
+
+                        # log the after
+                        ($after = $indentation) =~ s|\t||g;
+                        $after = "TAB"x$numberOfTABS.$after;
+                        $self->logger("Indentation after: '$after'") if($is_t_switch_active);
+                        ($indentation = $after) =~s|TAB|\t|g;
+                        $indentation;
+                       /xsmeg;
+
+}
+
+sub indent_children_recursively{
+    my $self = shift;
+
+    unless(defined ${$self}{children}) {
+        $self->logger("No child objects (${$self}{name})") if $is_t_switch_active;
+        return;
+    }
+
+    $self->logger('Pre-processed body:','heading') if $is_t_switch_active;
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+
+    # send the children through this indentation routine recursively
+    if(defined ${$self}{children}){
+        foreach my $child (@{${$self}{children}}){
+            $self->logger("Indenting child objects on ${$child}{name}") if $is_t_switch_active;
+            $child->indent_children_recursively;
+        }
+    } 
+
+    $self->logger("Replacing ids with begin, body, and end statements:",'heading') if $is_t_switch_active;
+
+    # loop through document children hash
+    while( scalar (@{${$self}{children}}) > 0 ){
+          my $index = 0;
+          # we work through the array *in order*
+          foreach my $child (@{${$self}{children}}){
+            $self->logger("Searching ${$self}{name} for ${$child}{id}...",'heading') if $is_t_switch_active;
+            if(${$self}{body} =~ m/${$child}{id}/s){
+                # we only care if id is first non-white space character 
+                # and if followed by line break 
+                # if m switch is active 
+                my $IDFirstNonWhiteSpaceCharacter = 0;
+                my $IDFollowedImmediatelyByLineBreak = 0;
+
+                # update the above two, if necessary
+                if ($is_m_switch_active){
+                    $IDFirstNonWhiteSpaceCharacter = (${$self}{body} =~ m/^${$child}{id}/m 
+                                                            or 
+                                                         ${$self}{body} =~ m/^\h\h*${$child}{id}/m
+                                                        ) ?1:0;
+                    $IDFollowedImmediatelyByLineBreak = (${$self}{body} =~ m/${$child}{id}\h*\R*/m) ?1:0;
+               }
+
+                # log file info
+                $self->logger("${$child}{id} found!") if($is_t_switch_active);
+                $self->logger("Indenting  ${$child}{name} (id: ${$child}{id})",'heading') if $is_t_switch_active;
+                $self->logger("looking up indentation scheme for ${$child}{name}") if($is_t_switch_active);
+
+                # line break checks *after* <end statement>
+                if (defined ${$child}{EndFinishesWithLineBreak}
+                    and ${$child}{EndFinishesWithLineBreak}==-1 
+                    and $IDFollowedImmediatelyByLineBreak) {
+                    # remove line break *after* <end statement>, if appropriate
+                    my $EndStringLogFile = ${$child}{aliases}{EndFinishesWithLineBreak}||"EndFinishesWithLineBreak";
+                    $self->logger("Removing linebreak after ${$child}{end} (see $EndStringLogFile)") if $is_t_switch_active;
+                    ${$self}{body} =~ s/${$child}{id}(\h*)?(\R|\h)*/${$child}{id}$1/s;
+                    ${$child}{linebreaksAtEnd}{end} = 0;
+                }
+
+                # perform indentation
+                $child->indent;
+
+                # surrounding indentation is now up to date
+                my $surroundingIndentation = (${$child}{surroundingIndentation} and ${$child}{hiddenChildYesNo})
+                                                        ?
+                                             (ref(${$child}{surroundingIndentation}) eq 'SCALAR'?${${$child}{surroundingIndentation}}:${$child}{surroundingIndentation})
+                                                        :q();
+
+                # line break checks before <begin statement>
+                if(defined ${$child}{BeginStartsOnOwnLine}){
+                    my $BeginStringLogFile = ${$child}{aliases}{BeginStartsOnOwnLine}||"BeginStartsOnOwnLine";
+                    if(${$child}{BeginStartsOnOwnLine}>=1 and !$IDFirstNonWhiteSpaceCharacter){
+                        # by default, assume that no trailing comment token is needed
+                        my $trailingCommentToken = q();
+                        if(${$child}{BeginStartsOnOwnLine}==2){
+                            $self->logger("Removing space immediately before ${$child}{id}, in preparation for adding % ($BeginStringLogFile == 2)") if $is_t_switch_active;
+                            ${$self}{body} =~ s/\h*${$child}{id}/${$child}{id}/s;
+                            $self->logger("Adding a % at the end of the line that ${$child}{begin} is on, then a linebreak ($BeginStringLogFile == 2)") if $is_t_switch_active;
+                            $trailingCommentToken = "%".$self->add_comment_symbol;
+                        } else {
+                            $self->logger("Adding a linebreak at the beginning of ${$child}{begin} (see $BeginStringLogFile)") if $is_t_switch_active;
+                        }
+
+                        # the trailing comment/linebreak magic
+                        ${$child}{begin} = "$trailingCommentToken\n".${$child}{begin};
+                        $child->add_surrounding_indentation_to_begin_statement;
+
+                        # remove surrounding indentation ahead of %
+                        ${$child}{begin} =~ s/^(\h*)%/%/ if(${$child}{BeginStartsOnOwnLine}==2);
+                    } elsif (${$child}{BeginStartsOnOwnLine}==-1 and $IDFirstNonWhiteSpaceCharacter){
+                        # important to check we don't move the begin statement next to a blank-line-token
+                        my $blankLineToken = $tokens{blanklines};
+                        if(${$self}{body} !~ m/$blankLineToken\R*\h*${$child}{id}/s){
+                            $self->logger("Removing linebreak before ${$child}{begin} (see $BeginStringLogFile in ${$child}{modifyLineBreaksYamlName} YAML)") if $is_t_switch_active;
+                            ${$self}{body} =~ s/(\h*)(?:\R*|\h*)+${$child}{id}/$1${$child}{id}/s;
+                        } else {
+                            $self->logger("Not removing linebreak ahead of ${$child}{begin}, as blank-line-token present (see preserveBlankLines)") if $is_t_switch_active;
+                        }
+                    }
+                }
+
+                $self->logger(Dumper(\%{$child}),'ttrace') if($is_tt_switch_active);
+
+                # replace ids with body
+                ${$self}{body} =~ s/${$child}{id}/${$child}{begin}${$child}{body}${$child}{end}/;
+
+                # log file info
+                $self->logger("Body (${$self}{name}) now looks like:",'heading') if $is_t_switch_active;
+                $self->logger(${$self}{body}) if($is_t_switch_active);
+
+                # remove element from array: http://stackoverflow.com/questions/174292/what-is-the-best-way-to-delete-a-value-from-an-array-in-perl
+                splice(@{${$self}{children}}, $index, 1);
+
+                # output to the log file
+                $self->logger("deleted child key ${$child}{name} (parent is: ${$self}{name})") if $is_t_switch_active;
+
+                # restart the loop, as the size of the array has changed
+                last;
+              } else {
+                $self->logger("${$child}{id} not found") if($is_t_switch_active);
+              }
+
+              # increment the loop counter
+              $index++;
+            }
+    }
+
+    # logfile info
+    $self->logger("${$self}{name} has this many children:",'heading') if $is_t_switch_active;
+    $self->logger(scalar @{${$self}{children}}) if $is_t_switch_active;
+    $self->logger("Post-processed body (${$self}{name}):") if($is_t_switch_active);
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+
+}
+
+sub add_surrounding_indentation_to_begin_statement{
+    # almost all of the objects add surrounding indentation to the 'begin' statements, 
+    # but some (e.g HEADING) have their own method
+    my $self = shift;
+
+    my $surroundingIndentation = ${$self}{surroundingIndentation};
+    ${$self}{begin} =~ s/^(\h*)?/$surroundingIndentation/mg;  # add indentation
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Item.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Item.pm
new file mode 100644
index 00000000000..d85f4e24c31
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Item.pm
@@ -0,0 +1,149 @@
+package LatexIndent::Item;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_items construct_list_of_items/;
+our $itemCounter;
+our $listOfItems = q();
+our $itemRegExp; 
+
+sub construct_list_of_items{
+    my $self = shift;
+
+    # put together a list of the items
+    while( my ($item,$lookForThisItem)= each %{$masterSettings{itemNames}}){
+        $listOfItems .= ($listOfItems eq "")?"$item":"|$item" if($lookForThisItem);
+    }
+
+    # detail items in the log
+    $self->logger("List of items: $listOfItems (see itemNames)",'heading');
+
+    $itemRegExp = qr/
+                          (
+                              \\($listOfItems)
+                              \h*
+                              (\R*)?
+                          )
+                          (
+                              (?:                 # cluster-only (), don't capture 
+                                  (?!             
+                                      (?:\\(?:$listOfItems)) # cluster-only (), don't capture
+                                  ).              # any character, but not \\$item
+                              )*                 
+                          )                       
+                          (\R)?
+                       /sx;
+    
+
+    return;
+}
+
+sub find_items{
+    # no point carrying on if the list of items is empty
+    return if($listOfItems eq "");
+
+    my $self = shift;
+
+    return unless ${$masterSettings{indentAfterItems}}{${$self}{name}};
+
+    # otherwise loop through the item names
+    $self->logger("Searching for items (see itemNames) in ${$self}{name} (see indentAfterItems)") if $is_t_switch_active;
+    $self->logger(Dumper(\%{$masterSettings{itemNames}})) if $is_t_switch_active;
+
+    while(${$self}{body} =~ m/$itemRegExp\h*($trailingCommentRegExp)?/){
+
+        # log file output
+        $self->logger("Item found: $2",'heading') if $is_t_switch_active;
+
+        ${$self}{body} =~ s/
+                            $itemRegExp(\h*)($trailingCommentRegExp)?
+                           /
+                            # create a new Item object
+                            my $itemObject = LatexIndent::Item->new(begin=>$1,
+                                                                    body=>$4,
+                                                                    end=>q(),
+                                                                    name=>$2,
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$3?1:0,
+                                                                      body=>$5?1:0,
+                                                                    },
+                                                                    aliases=>{
+                                                                      # begin statements
+                                                                      BeginStartsOnOwnLine=>"ItemStartsOnOwnLine",
+                                                                      # body statements
+                                                                      BodyStartsOnOwnLine=>"ItemFinishesWithLineBreak",
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"items",
+                                                                    endImmediatelyFollowedByComment=>$5?0:($7?1:0),
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($itemObject);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($6?$6:q()).($7?$7:q());
+                           /xseg;
+    }
+}
+
+
+sub create_unique_id{
+    my $self = shift;
+
+    $itemCounter++;
+
+    ${$self}{id} = "$tokens{item}$itemCounter";
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # the item body could hoover up line breaks; we do an additional check
+    ${${$self}{linebreaksAtEnd}}{body}=1 if(${$self}{body} =~ m/\R+$/s );
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for headings (part, chapter, section, setc)
+    $self->find_heading;
+    
+    # search for commands with arguments
+    $self->find_commands_or_key_equals_values_braces;
+
+    # search for special begin/end
+    $self->find_special;
+
+}
+
+sub remove_line_breaks_begin{
+    # the \item command can need a trailing white space if the line breaks have been removed after it and
+    # there is no white space
+    my $self = shift;
+    my $BodyStringLogFile = ${$self}{aliases}{BodyStartsOnOwnLine}||"BodyStartsOnOwnLine";
+    $self->logger("Removing linebreak at the end of begin (see $BodyStringLogFile)");
+    ${$self}{begin} =~ s/\R*$//sx;
+    ${$self}{begin} .= " " unless(${$self}{begin} =~ m/\h$/s or ${$self}{body} =~ m/^\h/s or ${$self}{body} =~ m/^\R/s );
+    ${$self}{linebreaksAtEnd}{begin} = 0;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/KeyEqualsValuesBraces.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/KeyEqualsValuesBraces.pm
new file mode 100755
index 00000000000..41fcb981a15
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/KeyEqualsValuesBraces.pm
@@ -0,0 +1,122 @@
+package LatexIndent::KeyEqualsValuesBraces;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_m_switch_active/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Command"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/construct_key_equals_values_regexp $key_equals_values_bracesRegExp $key_equals_values_bracesRegExpTrailingComment/;
+our $key_equals_values_braces_Counter;
+our $key_equals_values_bracesRegExp; 
+our $key_equals_values_bracesRegExpTrailingComment; 
+
+sub construct_key_equals_values_regexp{
+    my $self = shift;
+
+    # grab the arguments regexp
+    my $optAndMandRegExp = $self->get_arguments_regexp;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    # store the regular expresssion for matching and replacing 
+    $key_equals_values_bracesRegExp = qr/
+                  (
+                    (?:
+                       (?:(?<!\\)\{)
+                           |
+                           ,
+                           |
+                       (?:(?<!\\)\[)
+                     )
+                     (?:\h|\R|$blankLineToken|$trailingCommentRegExp)*
+                  )                                                     # $1 pre-key bit: could be { OR , OR [                                 
+                  (\\)?                                                 # $2 possible backslash
+                  (
+                   [a-zA-Z@\*0-9_\/.\h\{\}:\#-]+?                               # lowercase|uppercase letters, @, *, numbers, forward slash, dots
+                  )                                                     # $3 name
+                  (
+                    (?:\h|\R|$blankLineToken|$trailingCommentRegExp)*
+                    =\h*
+                    (?:\d*\:?)
+                  )                                                     # $4 = symbol
+                  (\R*)?                                                # $5 linebreak after =
+                  ($optAndMandRegExp)                                   # $6 opt|mand arguments
+                  (\R)?                                                 # $9 linebreak at end
+                /sx;
+
+    $key_equals_values_bracesRegExpTrailingComment = qr/$key_equals_values_bracesRegExp(\h*)((?:$trailingCommentRegExp\h*)*)?/;
+}
+
+sub indent_begin{
+    my $self = shift;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    if(${$self}{begin} =~ /\R=/s or ${$self}{begin} =~ /$blankLineToken\h*=/s ){
+        $self->logger("= found on own line in ${$self}{name}, adding indentation") if $is_t_switch_active;
+        ${$self}{begin} =~ s/=/${$self}{indentation}=/s;
+    }
+}
+
+sub check_linebreaks_before_equals{
+    # check if -m switch is active
+    return unless $is_m_switch_active;
+    
+    my $self = shift;
+
+    # linebreaks *infront* of = symbol
+    if(${$self}{begin} =~ /\R\h*=/s){
+          if(defined ${$self}{EqualsStartsOnOwnLine} and ${$self}{EqualsStartsOnOwnLine}==-1){
+            $self->logger("Removing linebreak before = symbol in ${$self}{name} (see EqualsStartsOnOwnLine)") if $is_t_switch_active;
+            ${$self}{begin} =~ s/(\R|\h)*=/=/s;
+          }
+    } else {
+      if(defined ${$self}{EqualsStartsOnOwnLine} and ${$self}{EqualsStartsOnOwnLine}==1){
+            $self->logger("Adding a linebreak before = symbol for ${$self}{name} (see EqualsStartsOnOwnLine)") if $is_t_switch_active;
+            ${$self}{begin} =~ s/=/\n=/s;
+      } elsif(defined ${$self}{EqualsStartsOnOwnLine} and ${$self}{EqualsStartsOnOwnLine}==2){
+            $self->logger("Adding a % linebreak immediately before = symbol for ${$self}{name} (see EqualsStartsOnOwnLine)") if $is_t_switch_active;
+            ${$self}{begin} =~ s/\h*=/%\n=/s;
+      }
+    }
+    return;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $key_equals_values_braces_Counter++;
+    ${$self}{id} = "$tokens{key_equals_values_braces}$key_equals_values_braces_Counter";
+    return;
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text for a key = {value} needes to accomodate the leading [ OR { OR % OR , OR any combination thereof
+    $self->logger("Custom replacement text routine for ${$self}{name}") if $is_t_switch_active;
+    ${$self}{replacementText} = ${$self}{beginningbit}.${$self}{id};
+    delete ${$self}{beginningbit};
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/LogFile.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/LogFile.pm
new file mode 100644
index 00000000000..89cffa58a81
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/LogFile.pm
@@ -0,0 +1,166 @@
+package LatexIndent::LogFile;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/%switches/;
+use FindBin; 
+use File::Basename; # to get the filename and directory path
+use Exporter qw/import/;
+our @EXPORT_OK = qw/logger output_logfile processSwitches/;
+our @logFileNotes;
+
+# log file methods
+# log file methods
+# log file methods
+#   reference: http://stackoverflow.com/questions/6736998/help-calling-a-sub-routine-from-a-perl-module-and-printing-to-logfile
+
+sub logger{
+    shift;
+    my $line = shift;
+    my $infoLevel = shift;
+    push(@logFileNotes,{line=>$line,level=>$infoLevel?$infoLevel:'default'});
+    return
+}
+
+sub processSwitches{
+    my $self = shift;
+
+    # details of the script to log file
+    $self->logger("$FindBin::Script version 3.0, a script to indent .tex files",'heading');
+    $self->logger("$FindBin::Script lives here: $FindBin::RealBin/");
+
+    # time the script is used
+    my $time = localtime();
+    $self->logger("$time");
+
+    if(scalar(@ARGV) < 1 or $switches{showhelp}) {
+    print <<ENDQUOTE
+latexindent.pl version 3.0
+usage: latexindent.pl [options] [file][.tex|.sty|.cls|.bib|...]
+      -h, --help
+          help (see the documentation for detailed instructions and examples)
+      -o, --outputfile
+          output to another file; sample usage:
+                latexindent.pl -o outputfile.tex myfile.tex 
+                latexindent.pl -o=outputfile.tex myfile.tex 
+      -w, --overwrite
+          overwrite the current file; a backup will be made, but still be careful
+      -s, --silent
+          silent mode: no output will be given to the terminal
+      -t, --trace
+          tracing mode: verbose information given to the log file
+      -l, --local[=myyaml.yaml]
+          use localSettings.yaml (assuming it exists in the directory of your file);
+          alternatively, use myyaml.yaml, if it exists; sample usage:
+                latexindent.pl -l some.yaml myfile.tex 
+                latexindent.pl -l=another.yaml myfile.tex 
+                latexindent.pl -l=some.yaml,another.yaml myfile.tex 
+      -d, --onlydefault
+          ONLY use defaultSettings.yaml, ignore ALL (yaml) user files
+      -g, --logfile
+          used to specify the name of logfile (default is indent.log)
+      -c, --cruft=<cruft directory> 
+          used to specify the location of backup files and indent.log
+      -m, --modifylinebreaks
+          modify linebreaks before, during, and at the end of code blocks; 
+          trailing comments can also be added using this feature
+ENDQUOTE
+    ;
+    exit(2);
+}
+
+    # log the switches from the user
+    $self->logger('Processing switches','heading');
+
+    # check on the trace mode switch (should be turned on if ttrace mode active)
+    $switches{trace} = $switches{ttrace} ?  1 : $switches{trace};
+
+    # output details of switches
+    $self->logger('-t|--trace: Trace mode active (you have used either -t or --trace)') if($switches{trace} and !$switches{ttrace});
+    $self->logger('-tt|--ttrace: TTrace mode active (you have used either -tt or --ttrace)') if($switches{tracingModeVeryDetailed});
+    $self->logger('-s|--silent: Silent mode active (you have used either -s or --silent)') if($switches{silentMode});
+    $self->logger('-d|--onlydefault: Only defaultSettings.yaml will be used (you have used either -d or --onlydefault)') if($switches{onlyDefault});
+    $self->logger("-w|--overwrite: Overwrite mode active, will make a back up of ${$self}{fileName} first") if($switches{overwrite});
+    $self->logger("-l|--localSettings: Read localSettings YAML file") if($switches{readLocalSettings});
+    $self->logger("-o|--outputfile: output to file") if($switches{outputToFile});
+    $self->logger("-m|--modifylinebreaks: modify line breaks") if($switches{modifyLineBreaks});
+    $self->logger("-g|--logfile: logfile name") if($switches{logFileName});
+    $self->logger("-c|--cruft: cruft directory") if($switches{cruftDirectory});
+
+    # check if overwrite and outputfile are active similtaneously
+    if($switches{overwrite} and $switches{outputToFile}){
+        $self->logger("Options check",'heading');
+        $self->logger("You have called latexindent.pl with both -o and -w");
+        $self->logger("-o (output to file) will take priority, and -w (over write) will be ignored");
+        $switches{overwrite}=0;
+    }
+
+    # cruft directory
+    ${$self}{cruftDirectory} = $switches{cruftDirectory}||(dirname ${$self}{fileName});
+    die "Could not find directory ${$self}{cruftDirectory}\nExiting, no indentation done." if(!(-d ${$self}{cruftDirectory}));
+    my $logfileName = $switches{logFileName}||"indent.log";
+    $self->logger("Directory for backup files and $logfileName: ${$self}{cruftDirectory}",'heading');
+
+    # output location of modules
+    if($FindBin::Script eq 'latexindent.pl' or ($FindBin::Script eq 'latexindent.exe' and $switches{trace} )) {
+        my @listOfModules = ('FindBin', 'YAML::Tiny', 'File::Copy', 'File::Basename', 'Getopt::Long','File::HomeDir');
+        $self->logger("Perl modules are being loaded from the following directories:",'heading');
+        foreach my $moduleName (@listOfModules) {
+                (my $file = $moduleName) =~ s|::|/|g;
+                require $file . '.pm';
+                $self->logger($INC{$file .'.pm'});
+              }
+        $self->logger("Latex Indent perl modules are being loaded from, for example:",'heading');
+                (my $file = 'LatexIndent::Document') =~ s|::|/|g;
+                require $file . '.pm';
+                $self->logger($INC{$file .'.pm'});
+    }
+    return;
+}
+
+sub output_logfile{
+  my $self = shift;
+  my $logfile;
+  my $logfileName = $switches{logFileName}||"indent.log";
+
+  open($logfile,">","${$self}{cruftDirectory}/$logfileName") or die "Can't open $logfileName";
+
+  # put the final line in the logfile
+  $self->logger("${$masterSettings{logFilePreferences}}{endLogFileWith}",'heading');
+
+  # github info line
+  $self->logger("Please direct all communication/issues to: ",'heading') if ${$masterSettings{logFilePreferences}}{showGitHubInfoFooter};
+  $self->logger("https://github.com/cmhughes/latexindent.pl") if ${$masterSettings{logFilePreferences}}{showGitHubInfoFooter};
+
+  # output the logfile
+  foreach my $line (@logFileNotes){
+        if(${$line}{level} eq 'heading'){
+            print $logfile ${$line}{line},"\n";
+          } elsif(${$line}{level} eq 'default') {
+            # add tabs to the beginning of lines 
+            # for default logfile lines
+            ${$line}{line} =~ s/^/\t/mg;
+            print $logfile ${$line}{line},"\n";
+          } 
+  }
+
+  close($logfile);
+}
+
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/MandatoryArgument.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/MandatoryArgument.pm
new file mode 100644
index 00000000000..172e002eda4
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/MandatoryArgument.pm
@@ -0,0 +1,130 @@
+package LatexIndent::MandatoryArgument;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_mandatory_arguments get_mand_arg_reg_exp/;
+our $mandatoryArgumentCounter;
+
+sub find_mandatory_arguments{
+    my $self = shift;
+
+    my $mandArgRegExp = $self->get_mand_arg_reg_exp;
+
+    # pick out the mandatory arguments
+    while(${$self}{body} =~ m/$mandArgRegExp\h*($trailingCommentRegExp)*(.*)/s){
+        # log file output
+        $self->logger("Mandatory argument found, body in ${$self}{name}",'heading') if $is_t_switch_active;
+        $self->logger("(last argument)") if($9 eq '' and $is_t_switch_active);
+
+        ${$self}{body} =~ s/
+                            $mandArgRegExp(\h*)($trailingCommentRegExp)*(.*)
+                           /
+                            # create a new Mandatory Argument object
+                            my $mandatoryArg = LatexIndent::MandatoryArgument->new(begin=>$1,
+                                                                    name=>${$self}{name}.":mandatoryArgument",
+                                                                    nameForIndentationSettings=>${$self}{parent},
+                                                                    parent=>${$self}{parent},
+                                                                    body=>$3.($4?$4:q()),
+                                                                    end=>$5,
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$2?1:0,
+                                                                      body=>$4?1:0,
+                                                                      end=>$7?1:0,
+                                                                    },
+                                                                    aliases=>{
+                                                                      # begin statements
+                                                                      BeginStartsOnOwnLine=>"LCuBStartsOnOwnLine",
+                                                                      # body statements
+                                                                      BodyStartsOnOwnLine=>"MandArgBodyStartsOnOwnLine",
+                                                                      # end statements
+                                                                      EndStartsOnOwnLine=>"RCuBStartsOnOwnLine",
+                                                                      # after end statements
+                                                                      EndFinishesWithLineBreak=>"RCuBFinishesWithLineBreak",
+                                                                    },
+                                                                    horizontalTrailingSpace=>$6?$6:q(),
+                                                                    modifyLineBreaksYamlName=>"mandatoryArguments",
+                                                                    # the last argument (determined by $10 eq '') needs information from the argument container object
+                                                                    endImmediatelyFollowedByComment=>($10 eq '')?${$self}{endImmediatelyFollowedByComment}:($9?1:0),
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($mandatoryArg);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($8?$8:q()).($9?$9:q()).($10?$10:q());
+                            /xseg;
+    $self->wrap_up_tasks;
+        }
+  }
+
+sub create_unique_id{
+    my $self = shift;
+
+    $mandatoryArgumentCounter++;
+    ${$self}{id} = "$tokens{mandatoryArgument}$mandatoryArgumentCounter";
+    return;
+}
+
+sub get_mand_arg_reg_exp{
+
+    my $mandArgRegExp = qr/      
+                                   (?<!\\)     # not immediately pre-ceeded by \
+                                   (
+                                    \{
+                                       \h*
+                                       (\R*)   # linebreaks after { into $2
+                                   )           # { captured into $1
+                                   (
+                                       (?:
+                                           (?!
+                                               (?:(?<!\\)\{) 
+                                           ).
+                                       )*?     # not including {, but \{ ok
+                                   )            # body into $3
+                                   (\R*)       # linebreaks after body into $4
+                                   (?<!\\)     # not immediately pre-ceeded by \
+                                   (
+                                    \}         # {mandatory arguments}
+                                   )           # } into $5
+                                   (\h*)
+                                   (\R)?       # linebreaks after } into $6
+                               /sx;
+
+    return $mandArgRegExp;
+}
+
+sub get_object_attribute_for_indentation_settings{
+    my $self = shift;
+    
+    return ${$self}{modifyLineBreaksYamlName};
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for special begin/end
+    $self->find_special;
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/ModifyLineBreaks.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/ModifyLineBreaks.pm
new file mode 100644
index 00000000000..bcfb2336264
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/ModifyLineBreaks.pm
@@ -0,0 +1,174 @@
+package LatexIndent::ModifyLineBreaks;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use Data::Dumper;
+use Exporter qw/import/;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+our @EXPORT_OK = qw/modify_line_breaks_body modify_line_breaks_end adjust_line_breaks_end_parent remove_line_breaks_begin/;
+our @allObjects;
+
+sub modify_line_breaks_body{
+    my $self = shift;
+
+    # add a line break after \begin{statement} if appropriate
+    if(defined ${$self}{BodyStartsOnOwnLine}){
+      my $BodyStringLogFile = ${$self}{aliases}{BodyStartsOnOwnLine}||"BodyStartsOnOwnLine";
+      if(${$self}{BodyStartsOnOwnLine}>=1 and !${$self}{linebreaksAtEnd}{begin}){
+          if(${$self}{BodyStartsOnOwnLine}==1){
+            # modify the begin statement
+            $self->logger("Adding a linebreak at the end of begin, ${$self}{begin} (see $BodyStringLogFile)") if $is_t_switch_active;
+            ${$self}{begin} .= "\n";       
+            ${$self}{linebreaksAtEnd}{begin} = 1;
+            $self->logger("Removing leading space from body of ${$self}{name} (see $BodyStringLogFile)") if $is_t_switch_active;
+            ${$self}{body} =~ s/^\h*//;       
+          } elsif(${$self}{BodyStartsOnOwnLine}==2){
+            # by default, assume that no trailing comment token is needed
+            my $trailingCommentToken = q();
+            if(${$self}{body} !~ m/^\h*$trailingCommentRegExp/s){
+                # modify the begin statement
+                $self->logger("Adding a % at the end of begin, ${$self}{begin}, followed by a linebreak ($BodyStringLogFile == 2)") if $is_t_switch_active;
+                $trailingCommentToken = "%".$self->add_comment_symbol;
+                ${$self}{begin} =~ s/\h*$//;       
+                ${$self}{begin} .= "$trailingCommentToken\n";       
+                ${$self}{linebreaksAtEnd}{begin} = 1;
+                $self->logger("Removing leading space from body of ${$self}{name} (see $BodyStringLogFile)") if $is_t_switch_active;
+                ${$self}{body} =~ s/^\h*//;       
+            } else {
+                $self->logger("Even though $BodyStringLogFile == 2, ${$self}{begin} already finishes with a %, so not adding another.") if $is_t_switch_active;
+            }
+          } 
+       } elsif (${$self}{BodyStartsOnOwnLine}==-1 and ${$self}{linebreaksAtEnd}{begin}){
+          # remove line break *after* begin, if appropriate
+          $self->remove_line_breaks_begin;
+       }
+    }
+  }
+
+sub remove_line_breaks_begin{
+    my $self = shift;
+    my $BodyStringLogFile = ${$self}{aliases}{BodyStartsOnOwnLine}||"BodyStartsOnOwnLine";
+    $self->logger("Removing linebreak at the end of begin (see $BodyStringLogFile)") if $is_t_switch_active;
+    ${$self}{begin} =~ s/\R*$//sx;
+    ${$self}{linebreaksAtEnd}{begin} = 0;
+}
+
+sub modify_line_breaks_end{
+    my $self = shift;
+
+    # possibly modify line break *before* \end{statement}
+    if(defined ${$self}{EndStartsOnOwnLine}){
+          my $EndStringLogFile = ${$self}{aliases}{EndStartsOnOwnLine}||"EndStartsOnOwnLine";
+          if(${$self}{EndStartsOnOwnLine}>=1 and !${$self}{linebreaksAtEnd}{body}){
+              # add a line break after body, if appropriate
+              $self->logger("Adding a linebreak at the end of body (see $EndStringLogFile)") if $is_t_switch_active;
+
+              # by default, assume that no trailing comment token is needed
+              my $trailingCommentToken = q();
+              if(${$self}{EndStartsOnOwnLine}==2){
+                $self->logger("Adding a % immediately after body of ${$self}{name} ($EndStringLogFile==2)") if $is_t_switch_active;
+                $trailingCommentToken = "%".$self->add_comment_symbol;
+                ${$self}{body} =~ s/\h*$//s;
+              }
+              
+              # modified end statement
+              if(${$self}{body} =~ m/^\h*$/s and ${$self}{BodyStartsOnOwnLine} >=1 ){
+                ${$self}{linebreaksAtEnd}{body} = 0;
+              } else {
+                ${$self}{body} .= "$trailingCommentToken\n";
+                ${$self}{linebreaksAtEnd}{body} = 1;
+              }
+          } elsif (${$self}{EndStartsOnOwnLine}==-1 and ${$self}{linebreaksAtEnd}{body}){
+              # remove line break *after* body, if appropriate
+
+              # check to see that body does *not* finish with blank-line-token, 
+              # if so, then don't remove that final line break
+              if(${$self}{body} !~ m/$tokens{blanklines}$/s){
+                $self->logger("Removing linebreak at the end of body (see $EndStringLogFile)") if $is_t_switch_active;
+                ${$self}{body} =~ s/\R*$//sx;
+                ${$self}{linebreaksAtEnd}{body} = 0;
+              } else {
+                $self->logger("Blank line token found at end of body (${$self}{name}), see preserveBlankLines, not removing line break before ${$self}{end}") if $is_t_switch_active;
+              }
+          }
+    }
+
+    # possibly modify line break *after* \end{statement}
+    if(defined ${$self}{EndFinishesWithLineBreak}
+       and ${$self}{EndFinishesWithLineBreak}>=1 
+       and !${$self}{linebreaksAtEnd}{end}){
+              my $EndStringLogFile = ${$self}{aliases}{EndFinishesWithLineBreak}||"EndFinishesWithLineBreak";
+              if(${$self}{EndFinishesWithLineBreak}==1){
+                $self->logger("Adding a linebreak at the end of ${$self}{end} (see $EndStringLogFile)") if $is_t_switch_active;
+                ${$self}{linebreaksAtEnd}{end} = 1;
+
+                # modified end statement
+                ${$self}{replacementText} .= "\n";
+              } elsif(${$self}{EndFinishesWithLineBreak}==2){
+                if(${$self}{endImmediatelyFollowedByComment}){
+                  # no need to add a % if one already exists
+                  $self->logger("Even though $EndStringLogFile == 2, ${$self}{end} is immediately followed by a %, so not adding another; not adding line break.") if $is_t_switch_active;
+                } else {
+                  # otherwise, create a trailing comment, and tack it on 
+                  $self->logger("Adding a % immediately after, ${$self}{end} ($EndStringLogFile==2)") if $is_t_switch_active;
+                  my $trailingCommentToken = "%".$self->add_comment_symbol;
+                  ${$self}{end} =~ s/\h*$//s;
+                  ${$self}{replacementText} .= "$trailingCommentToken\n";
+                  ${$self}{linebreaksAtEnd}{end} = 1;
+                }
+              }
+    }
+
+}
+
+sub adjust_line_breaks_end_parent{
+    # when a parent object contains a child object, the line break
+    # at the end of the parent object can become messy
+    return unless $is_m_switch_active;
+
+    my $self = shift;
+
+    # most recent child object
+    my $child = @{${$self}{children}}[-1];
+
+    # adjust parent linebreaks information
+    if(${$child}{linebreaksAtEnd}{end} and ${$self}{body} =~ m/${$child}{replacementText}\h*\R*$/s and !${$self}{linebreaksAtEnd}{body}){
+        $self->logger("ID: ${$child}{id}") if($is_t_switch_active);
+        $self->logger("${$child}{begin}...${$child}{end} is found at the END of body of parent, ${$self}{name}, avoiding a double line break:") if($is_t_switch_active);
+        $self->logger("adjusting ${$self}{name} linebreaksAtEnd{body} to be 1") if($is_t_switch_active);
+        ${$self}{linebreaksAtEnd}{body}=1;
+      }
+
+    # the modify line switch can adjust line breaks, so we need another check, 
+    # see for example, test-cases/environments/environments-remove-line-breaks-trailing-comments.tex
+    if(defined ${$child}{linebreaksAtEnd}{body} 
+        and !${$child}{linebreaksAtEnd}{body} 
+        and ${$child}{body} =~ m/\R(?:$trailingCommentRegExp\h*)?$/s ){
+        # log file information
+        $self->logger("Undisclosed line break at the end of body of ${$child}{name}: '${$child}{end}'") if($is_t_switch_active);
+        $self->logger("Adding a linebreak at the end of body for ${$child}{id}") if($is_t_switch_active);
+        
+        # make the adjustments
+        ${$child}{body} .= "\n";
+        ${$child}{linebreaksAtEnd}{body}=1;
+    }
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/NamedGroupingBracesBrackets.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/NamedGroupingBracesBrackets.pm
new file mode 100755
index 00000000000..ea2c5323cf3
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/NamedGroupingBracesBrackets.pm
@@ -0,0 +1,71 @@
+package LatexIndent::NamedGroupingBracesBrackets;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Command"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/construct_grouping_braces_brackets_regexp $grouping_braces_regexp $grouping_braces_regexpTrailingComment/;
+our $groupingBracesCounter;
+our $grouping_braces_regexp; 
+our $grouping_braces_regexpTrailingComment; 
+
+sub construct_grouping_braces_brackets_regexp{
+    my $self = shift;
+
+    # grab the arguments regexp
+    my $optAndMandRegExp = $self->get_arguments_regexp;
+
+    # store the regular expresssion for matching and replacing 
+    $grouping_braces_regexp = qr/
+                  (
+                     \h|\R|\{|\[|\$|\)
+                  )
+                  (
+                   [0-9a-zA-Z@\*><]+?  # lowercase|uppercase letters, @, *, numbers, forward slash, dots
+                  )                    # $2 name
+                  (\h*)                # $3 h-space
+                  (\R*)                # $4 linebreaks
+                  ($optAndMandRegExp)  # $5 mand|opt arguments (at least one)
+                  (\R)?                # $8 linebreak 
+                /sx;
+
+    # something {value} grouping braces with trailing comment
+    $grouping_braces_regexpTrailingComment = qr/$grouping_braces_regexp\h*((?:$trailingCommentRegExp\h*)*)?/;
+
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $groupingBracesCounter++;
+    ${$self}{id} = "$tokens{groupingBraces}$groupingBracesCounter";
+    return;
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text for a key = {value} needes to accomodate the leading [ OR { OR % OR , OR any combination thereof
+    $self->logger("Custom replacement text routine for ${$self}{name}") if $is_t_switch_active;
+    ${$self}{replacementText} = ${$self}{beginningbit}.${$self}{id};
+    delete ${$self}{beginningbit};
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/OptionalArgument.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/OptionalArgument.pm
new file mode 100644
index 00000000000..b949e9b7e92
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/OptionalArgument.pm
@@ -0,0 +1,115 @@
+package LatexIndent::OptionalArgument;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_optional_arguments/;
+our $optionalArgumentCounter;
+our $optArgRegExp = qr/      
+                               (?<!\\)     # not immediately pre-ceeded by \
+                               (
+                                \[
+                                   \h*
+                                   (\R*)
+                               )
+                               (.*?)
+                               (\R*)
+                               (?<!\\)     # not immediately pre-ceeded by \
+                               (
+                                \]          # [optional arguments]
+                               )
+                               (\h*)
+                               (\R)?
+                           /sx;
+
+sub find_optional_arguments{
+    my $self = shift;
+
+    # pick out the optional arguments
+    while(${$self}{body} =~ m/$optArgRegExp\h*($trailingCommentRegExp)*(.*)/s){
+        # log file output
+        $self->logger("Optional argument found, body in ${$self}{name}",'heading') if $is_t_switch_active;
+        $self->logger("(last argument)") if($9 eq '' and $is_t_switch_active);
+
+        ${$self}{body} =~ s/
+                            $optArgRegExp(\h*)($trailingCommentRegExp)*(.*)
+                           /
+                            # create a new Optional Argument object
+                            my $optionalArg = LatexIndent::OptionalArgument->new(begin=>$1,
+                                                                    name=>${$self}{name}.":optionalArgument",
+                                                                    nameForIndentationSettings=>${$self}{parent},
+                                                                    parent=>${$self}{parent},
+                                                                    body=>$3.($4?$4:q()),
+                                                                    end=>$5,
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$2?1:0,
+                                                                      body=>$4?1:0,
+                                                                      end=>$7?1:0,
+                                                                    },
+                                                                    aliases=>{
+                                                                      # begin statements
+                                                                      BeginStartsOnOwnLine=>"LSqBStartsOnOwnLine",
+                                                                      # body statements
+                                                                      BodyStartsOnOwnLine=>"OptArgBodyStartsOnOwnLine",
+                                                                      # end statements
+                                                                      EndStartsOnOwnLine=>"RSqBStartsOnOwnLine",
+                                                                      # after end statements
+                                                                      EndFinishesWithLineBreak=>"RSqBFinishesWithLineBreak",
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"optionalArguments",
+                                                                    # the last argument (determined by $10 eq '') needs information from the argument container object
+                                                                    endImmediatelyFollowedByComment=>($10 eq '')?${$self}{endImmediatelyFollowedByComment}:($9?1:0),
+                                                                    horizontalTrailingSpace=>$6?$6:q(),
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($optionalArg);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($8?$8:q()).($9?$9:q()).($10?$10:q());
+                            /xseg;
+        }
+  }
+
+sub get_object_attribute_for_indentation_settings{
+    my $self = shift;
+    
+    return ${$self}{modifyLineBreaksYamlName};
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $optionalArgumentCounter++;
+    ${$self}{id} = "$tokens{optionalArgument}$optionalArgumentCounter";
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for special begin/end
+    $self->find_special;
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Preamble.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Preamble.pm
new file mode 100644
index 00000000000..a3a62dd3f56
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Preamble.pm
@@ -0,0 +1,64 @@
+package LatexIndent::Preamble;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our $preambleCounter;
+
+sub create_unique_id{
+    my $self = shift;
+
+    $preambleCounter++;
+    ${$self}{id} = "$tokens{preamble}$preambleCounter$tokens{endOfToken}";
+    return;
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text for preamble needs to put the \\begin{document} back in
+    $self->logger("Custom replacement text routine for preamble ${$self}{name}");
+    ${$self}{replacementText} = ${$self}{id}.${$self}{afterbit};
+    delete ${$self}{afterbit};
+}
+
+sub indent{
+    # preamble doesn't receive any additional indentation
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    my $self = shift;
+
+    # search for environments
+    $self->find_environments;
+
+    # search for ifElseFi blocks
+    $self->find_ifelsefi;
+
+    # search for commands with arguments
+    $self->find_commands_or_key_equals_values_braces if(!$masterSettings{preambleCommandsBeforeEnvironments});
+
+    # search for special begin/end
+    $self->find_special;
+
+
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/RoundBrackets.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/RoundBrackets.pm
new file mode 100644
index 00000000000..f320d007fa2
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/RoundBrackets.pm
@@ -0,0 +1,100 @@
+package LatexIndent::RoundBrackets;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_round_brackets/;
+our $roundBracketCounter;
+our $roundBracketRegExp = qr/      
+                           (?<!\\)
+                           (
+                               \(
+                              \h*
+                              (\R*)
+                           )
+                            (.*?)
+                            (\R*)
+                            (?<!\\)     # not immediately pre-ceeded by \
+                            (
+                             \)          # [optional arguments]
+                            )
+                            (\h*)
+                            (\R)?
+                           /sx;
+
+sub indent{
+    return;
+}
+
+sub find_round_brackets{
+    my $self = shift;
+
+    # pick out the optional arguments
+    while(${$self}{body} =~ m/$roundBracketRegExp\h*($trailingCommentRegExp)*(.*)/s){
+        # log file output
+
+        ${$self}{body} =~ s/
+                            $roundBracketRegExp(\h*)($trailingCommentRegExp)*(.*)
+                           /
+                            # create a new Optional Argument object
+                            my $roundBracket = LatexIndent::RoundBrackets->new(begin=>$1,
+                                                                    name=>${$self}{name}.":roundBracket",
+                                                                    nameForIndentationSettings=>${$self}{parent},
+                                                                    parent=>${$self}{parent},
+                                                                    body=>$3.($4?$4:q()),
+                                                                    end=>$5,
+                                                                    linebreaksAtEnd=>{
+                                                                      begin=>$2?1:0,
+                                                                      body=>$4?1:0,
+                                                                      end=>$7?1:0,
+                                                                    },
+                                                                    modifyLineBreaksYamlName=>"roundBracket",
+                                                                    # the last argument (determined by $10 eq '') needs information from the argument container object
+                                                                    endImmediatelyFollowedByComment=>($10 eq '')?${$self}{endImmediatelyFollowedByComment}:($9?1:0),
+                                                                    horizontalTrailingSpace=>$6?$6:q(),
+                                                                  );
+
+                            # the settings and storage of most objects has a lot in common
+                            $self->get_settings_and_store_new_object($roundBracket);
+                            ${@{${$self}{children}}[-1]}{replacementText}.($8?$8:q()).($9?$9:q()).($10?$10:q());
+                            /xseg;
+        }
+  }
+
+sub get_object_attribute_for_indentation_settings{
+    my $self = shift;
+    
+    return ${$self}{modifyLineBreaksYamlName};
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $roundBracketCounter++;
+    ${$self}{id} = "$tokens{roundBracket}$roundBracketCounter";
+    return;
+}
+
+sub tasks_particular_to_each_object{
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Special.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Special.pm
new file mode 100644
index 00000000000..ba7b905b492
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Special.pm
@@ -0,0 +1,184 @@
+package LatexIndent::Special;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/find_special construct_special_begin/;
+our $specialCounter;
+our $specialBegins = q();
+our $specialAllMatchesRegExp = q();
+our %individualSpecialRegExps;
+
+sub construct_special_begin{
+    my $self = shift;
+
+    # put together a list of the begin terms in special
+    while( my ($specialName,$BeginEnd)= each %{$masterSettings{specialBeginEnd}}){
+      # only append the regexps if lookForThis is 1
+      $specialBegins .= ($specialBegins eq ""?q():"|").${$BeginEnd}{begin} if(${$BeginEnd}{lookForThis});
+    }
+
+    # put together a list of the begin terms in special
+    while( my ($specialName,$BeginEnd)= each %{$masterSettings{specialBeginEnd}}){
+
+      # only append the regexps if lookForThis is 1
+      if(${$BeginEnd}{lookForThis}){
+        # the beginning parts
+        $specialBegins .= ($specialBegins eq ""?q():"|").${$BeginEnd}{begin};
+
+        # the overall regexp
+        $specialAllMatchesRegExp .= ($specialAllMatchesRegExp eq ""?q():"|")
+                                    .qr/
+                                    ${$BeginEnd}{begin}
+                                    (?:                        # cluster-only (), don't capture 
+                                        (?!             
+                                            (?:$specialBegins) # cluster-only (), don't capture
+                                        ).                     # any character, but not anything in $specialBegins
+                                    )*?                 
+                                    ${$BeginEnd}{end}
+                             /sx;
+
+        # store the individual special regexp
+        $individualSpecialRegExps{$specialName} = qr/
+                                  (
+                                      ${$BeginEnd}{begin}
+                                      \h*
+                                      (\R*)?
+                                  )
+                                  (
+                                      (?:                        # cluster-only (), don't capture 
+                                          (?!             
+                                              (?:$specialBegins) # cluster-only (), don't capture
+                                          ).                     # any character, but not anything in $specialBegins
+                                      )*?                 
+                                     (\R*)?
+                                  )                       
+                                  (
+                                    ${$BeginEnd}{end}
+                                  )
+                                  (\h*)
+                                  (\R)?
+                               /sx
+
+        } else {
+            $self->logger("The special regexps won't include anything from $specialName (see lookForThis)",'heading') if $is_t_switch_active ;
+        }
+    }
+
+    # move $$ to the beginning
+    if($specialBegins =~ m/\|\\\$\\\$/){
+      $specialBegins =~ s/\|(\\\$\\\$)//;
+      $specialBegins = $1."|".$specialBegins; 
+    }
+
+    # info to the log file
+    $self->logger("The special beginnings regexp is: $specialBegins (see specialBeginEnd)",'heading') if $is_t_switch_active;
+
+    # overall special regexp
+    $self->logger("The overall special regexp is: $specialAllMatchesRegExp(see specialBeginEnd)",'heading') if $is_t_switch_active;
+
+  }
+
+sub find_special{
+    my $self = shift;
+
+    # no point carrying on if the list of specials is empty
+    return if($specialBegins eq "");
+
+    # otherwise loop through the special begin/end
+    $self->logger("Searching for special begin/end (see specialBeginEnd)") if $is_t_switch_active ;
+    $self->logger(Dumper(\%{$masterSettings{specialBeginEnd}})) if $is_tt_switch_active;
+
+    # keep looping as long as there is a special match of some kind
+    while(${$self}{body} =~ m/$specialAllMatchesRegExp/sx){
+
+        # loop through each special match
+        while( my ($specialName,$BeginEnd)= each %{$masterSettings{specialBeginEnd}}){
+
+            # log file
+            if(${$BeginEnd}{lookForThis}){
+                $self->logger("Looking for $specialName",'heading') if $is_t_switch_active ;
+            } else {
+                $self->logger("Not looking for $specialName (see lookForThis)",'heading') if $is_t_switch_active ;
+                next;
+            }
+
+            # the regexp
+            my $specialRegExp = $individualSpecialRegExps{$specialName};
+            
+            while(${$self}{body} =~ m/$specialRegExp(\h*)($trailingCommentRegExp)?/){
+
+                # global substitution
+                ${$self}{body} =~ s/
+                                    $specialRegExp(\h*)($trailingCommentRegExp)?
+                                   /
+                                    # log file output
+                                    $self->logger("special found: $specialName",'heading') if $is_t_switch_active;
+
+                                    # create a new special object
+                                    my $specialObject = LatexIndent::Special->new(begin=>$1,
+                                                                            body=>$3,
+                                                                            end=>$5,
+                                                                            name=>$specialName,
+                                                                            linebreaksAtEnd=>{
+                                                                              begin=>$2?1:0,
+                                                                              body=>$4?1:0,
+                                                                              end=>$7?1:0,
+                                                                            },
+                                                                            aliases=>{
+                                                                              # begin statements
+                                                                              BeginStartsOnOwnLine=>"SpecialBeginStartsOnOwnLine",
+                                                                              # body statements
+                                                                              BodyStartsOnOwnLine=>"SpecialBodyStartsOnOwnLine",
+                                                                              # end statements
+                                                                              EndStartsOnOwnLine=>"SpecialEndStartsOnOwnLine",
+                                                                              # after end statements
+                                                                              EndFinishesWithLineBreak=>"SpecialEndFinishesWithLineBreak",
+                                                                            },
+                                                                            modifyLineBreaksYamlName=>"specialBeginEnd",
+                                                                            endImmediatelyFollowedByComment=>$7?0:($9?1:0),
+                                                                            horizontalTrailingSpace=>$6?$6:q(),
+                                                                          );
+
+                                    # the settings and storage of most objects has a lot in common
+                                    $self->get_settings_and_store_new_object($specialObject);
+                                    ${@{${$self}{children}}[-1]}{replacementText}.($8?$8:q()).($9?$9:q());
+                                    /xseg;
+
+    $self->wrap_up_tasks;
+            }
+         }
+     }
+}
+
+
+sub create_unique_id{
+    my $self = shift;
+
+    $specialCounter++;
+
+    ${$self}{id} = "$tokens{special}$specialCounter";
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Switches.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Switches.pm
new file mode 100644
index 00000000000..a5b07936a6d
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Switches.pm
@@ -0,0 +1,37 @@
+package LatexIndent::Switches;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/%switches storeSwitches $is_m_switch_active $is_t_switch_active $is_tt_switch_active/;
+our %switches;
+our $is_m_switch_active;
+our $is_t_switch_active;
+our $is_tt_switch_active;
+
+sub storeSwitches{
+    my $self = shift;
+
+    # copy document switches into hash local to this module
+	%switches = %{${$self}{switches}};
+    $is_m_switch_active = defined $switches{modifyLineBreaks}?$switches{modifyLineBreaks}: 0;
+    $is_t_switch_active = defined $switches{trace}?$switches{trace}: 0;
+    $is_tt_switch_active = defined $switches{ttrace}?$switches{ttrace}: 0;
+    $is_t_switch_active = $is_tt_switch_active ? $is_tt_switch_active : $is_t_switch_active;
+    delete ${$self}{switches};
+  }
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Tokens.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Tokens.pm
new file mode 100644
index 00000000000..8a9a100dc26
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Tokens.pm
@@ -0,0 +1,68 @@
+package LatexIndent::Tokens;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use Exporter qw/import/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+our @EXPORT_OK = qw/token_check %tokens/;
+
+# each of the tokens begins the same way -- this is exploited during the hidden Children routine
+my $beginningToken = "LTXIN-TK-";
+my $ifelsefiSpecial = "!-!";
+
+# the %tokens hash is passed around many modules
+our %tokens = (
+                environment=>$beginningToken."ENVIRONMENT",
+                ifelsefiSpecial=>$ifelsefiSpecial,
+                ifelsefi=>$ifelsefiSpecial.$beginningToken."IFELSEFI", 
+                item=>$beginningToken."ITEMS",
+                trailingComment=>"latexindenttrailingcomment", 
+                blanklines=>$beginningToken."blank-line",
+                arguments=>$beginningToken."ARGUMENTS",
+                optionalArgument=>$beginningToken."OPTIONAL-ARGUMENT",
+                mandatoryArgument=>$beginningToken."MANDATORY-ARGUMENT",
+                roundBracket=>$beginningToken."ROUND-BRACKET",
+                verbatim=>$beginningToken."VERBATIM",
+                command=>$beginningToken."COMMAND",
+                key_equals_values_braces=>$beginningToken."KEY-VALUE-BRACES",
+                groupingBraces=>$beginningToken."GROUPING-BRACES",
+                unNamedgroupingBraces=>$beginningToken."UN-NAMED-GROUPING-BRACES",
+                special=>$beginningToken."SPECIAL",
+                heading=>$beginningToken."HEADING",
+                filecontents=>$beginningToken."FILECONTENTS",
+                preamble=>$beginningToken."preamble",
+                beginOfToken=>$beginningToken,
+                doubleBackSlash=>$beginningToken."DOUBLEBACKSLASH",
+                alignmentBlock=>$beginningToken."ALIGNMENTBLOCK",
+                endOfToken=>"-END",
+              );
+
+sub token_check{
+    my $self = shift;
+
+    $self->logger("Token check",'heading') if $is_t_switch_active;
+    # we use tokens for trailing comments, environments, commands, etc, so check that they're not in the body
+    foreach( keys %tokens){
+        while(${$self}{body} =~ m/$tokens{$_}/si){
+            $self->logger("Found $tokens{$_} within body, updating replacement token to $tokens{$_}-LIN") if($is_t_switch_active);
+            $tokens{$_} .= "-LIN";
+        }
+    }
+}
+
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/TrailingComments.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/TrailingComments.pm
new file mode 100644
index 00000000000..f8987d271f1
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/TrailingComments.pm
@@ -0,0 +1,112 @@
+package LatexIndent::TrailingComments;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/remove_trailing_comments put_trailing_comments_back_in $trailingCommentRegExp add_comment_symbol construct_trailing_comment_regexp/;
+our @trailingComments;
+our $commentCounter = 0;
+our $trailingCommentRegExp;
+
+sub construct_trailing_comment_regexp{
+    $trailingCommentRegExp = qr/(?<!\\)%$tokens{trailingComment}\d+$tokens{endOfToken}/;
+}
+
+sub add_comment_symbol{
+    # add a trailing comment token after, for example, a square brace [
+    # or a curly brace { when, for example, BeginStartsOnOwnLine == 2
+    my $self = shift;
+
+    # increment the comment counter
+    $commentCounter++;
+
+    # store the comment -- without this, it won't get processed correctly at the end
+    push(@trailingComments,{id=>$tokens{trailingComment}.$commentCounter.$tokens{endOfToken},value=>q()});
+
+    # log file info
+    $self->logger("Updating trailing comment array",'heading')if $is_t_switch_active;
+    $self->logger(Dumper(\@trailingComments),'ttrace') if($is_tt_switch_active);
+
+    # the returned value
+    return $tokens{trailingComment}.$commentCounter.$tokens{endOfToken};
+}
+
+sub remove_trailing_comments{
+    my $self = shift;
+    $self->logger("Storing trailing comments",'heading')if $is_t_switch_active;
+
+    # perform the substitution
+    ${$self}{body} =~ s/
+                            (?<!\\)  # not preceeded by a \
+                            %        # % 
+                            (
+                                \h*? # followed by possible horizontal space
+                                .*?  # and anything else
+                            )
+                            $        # up to the end of a line
+                        /   
+                            # increment comment counter and store comment
+                            $commentCounter++;
+                            push(@trailingComments,{id=>$tokens{trailingComment}.$commentCounter.$tokens{endOfToken},value=>$1});
+
+                            # replace comment with dummy text
+                            "%".$tokens{trailingComment}.$commentCounter.$tokens{endOfToken};
+                       /xsmeg;
+    if(@trailingComments){
+        $self->logger("Trailing comments stored in:") if($is_t_switch_active);
+        $self->logger(Dumper(\@trailingComments)) if($is_t_switch_active);
+    } else {
+        $self->logger("No trailing comments found") if($is_t_switch_active);
+    }
+    return;
+}
+
+sub put_trailing_comments_back_in{
+    my $self = shift;
+    return unless( @trailingComments > 0 );
+
+    $self->logger("Returning trailing comments to body",'heading')if $is_t_switch_active;
+
+    # loop through trailing comments in reverse so that, for example, 
+    # latexindenttrailingcomment1 doesn't match the first 
+    # part of latexindenttrailingcomment18, which would result in an 8 left over (bad)
+    while( my $comment = pop @trailingComments){
+      my $trailingcommentID = ${$comment}{id};
+      my $trailingcommentValue = ${$comment}{value};
+      if(${$self}{body} =~ m/%$trailingcommentID
+                              (
+                                  (?!          # not immediately preceeded by 
+                                      (?<!\\)  # \
+                                      %        # %
+                                  ).*?
+                              )                # captured into $1
+                              (\h*)?$                
+                          /mx and $1 ne ''){
+          $self->logger("Comment not at end of line $trailingcommentID, moving it to end of line")if $is_t_switch_active;
+          ${$self}{body} =~ s/%$trailingcommentID(.*)$/$1%$trailingcommentValue/m;
+      } else {
+          ${$self}{body} =~ s/%$trailingcommentID/%$trailingcommentValue/;
+      }
+      $self->logger("replace %$trailingcommentID with %$trailingcommentValue") if($is_t_switch_active);
+    }
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/UnNamedGroupingBracesBrackets.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/UnNamedGroupingBracesBrackets.pm
new file mode 100644
index 00000000000..268f5b5adfa
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/UnNamedGroupingBracesBrackets.pm
@@ -0,0 +1,135 @@
+package LatexIndent::UnNamedGroupingBracesBrackets;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::TrailingComments qw/$trailingCommentRegExp/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Exporter qw/import/;
+our @ISA = "LatexIndent::Command"; # class inheritance, Programming Perl, pg 321
+our @EXPORT_OK = qw/construct_unnamed_grouping_braces_brackets_regexp $un_named_grouping_braces_RegExp $un_named_grouping_braces_RegExp_trailing_comment/;
+our $unNamedGroupingBracesCounter;
+our $un_named_grouping_braces_RegExp;
+our $un_named_grouping_braces_RegExp_trailing_comment; 
+
+sub construct_unnamed_grouping_braces_brackets_regexp{
+    my $self = shift;
+
+    # grab the arguments regexp
+    my $optAndMandRegExp = $self->get_arguments_regexp;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    # for example #1 #2, etc
+    my $numberedArgRegExp = $self->get_numbered_arg_regexp;
+
+    # store the regular expresssion for matching and replacing 
+    $un_named_grouping_braces_RegExp = qr/
+                  # NOT
+                  (?!
+                      (?:
+                        (?:(?<!\\)\]) 
+                        |
+                        (?:(?<!\\)\}) 
+                      )
+                      (?:\h|\R|$blankLineToken|$trailingCommentRegExp|$numberedArgRegExp)*  # 0 or more h-space, blanklines, trailing comments
+                  )
+                  # END of NOT
+                  (
+                     (?:
+                        \{|\[|,|&|\)|\(|\$ # starting with { OR [ OR , OR & OR ) OR ( OR $
+                     )
+                     \h*
+                  )                     # $1 into beginning bit
+                  (\R*)                 # $2 linebreaksAtEnd of begin
+                  ($optAndMandRegExp)   # $3 mand|opt arguments (at least one) stored into body
+                  (\R)?                 # $6 linebreak 
+                /sx;
+
+    $un_named_grouping_braces_RegExp_trailing_comment = qr/$un_named_grouping_braces_RegExp(\h*)((?:$trailingCommentRegExp\h*)*)?/; 
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $unNamedGroupingBracesCounter++;
+    ${$self}{id} = "$tokens{unNamedgroupingBraces}$unNamedGroupingBracesCounter";
+    return;
+}
+
+sub get_replacement_text{
+    my $self = shift;
+
+    # the replacement text for a key = {value} needes to accomodate the leading [ OR { OR % OR , OR any combination thereof
+    $self->logger("Custom replacement text routine for ${$self}{name}") if $is_t_switch_active;
+
+    # the un-named object is a little special, as it doesn't have a name; as such, if there are blank lines before
+    # the braces/brackets, we have to insert them
+    #
+    # also, the argument reg-exp can pick up a leading comment (with line break), which needs to be put
+    # into the replacement text (see documentation/demonstrations/pstricks.tex and test-cases/unnamed-braces/unnamed.tex for example)
+    ${$self}{body} =~ s/(.*?)(\{|\[)/$2/s;
+    ${$self}{replacementText} = ${$self}{beginningbit}.($1 ne ''?$1:q()).${$self}{id};
+
+    # but now turn off the switch for linebreaksAtEnd{begin}, otherwise the first brace gets too much indentation
+    # (see, for example, test-cases/namedGroupingBracesBrackets/special-characters-minimal.tex)
+    ${${$self}{linebreaksAtEnd}}{begin} = 0;
+    $self->logger("Beginning bit is: ${$self}{beginningbit}") if($is_t_switch_active);
+    delete ${$self}{beginningbit};
+}
+
+sub check_for_blank_lines_at_beginning{
+    # some examples can have blank line tokens at the beginning of the body, 
+    # which can confuse the routine below 
+    # See, for example, 
+    #       test-cases/namedGroupingBracesBrackets/special-characters-minimal-blank-lines-m-switch.tex
+    #   compared to
+    #       test-cases/namedGroupingBracesBrackets/special-characters-minimal-blank-lines-default.tex
+    my $self = shift;
+
+    # blank line token
+    my $blankLineToken = $tokens{blanklines};
+
+    # if the body begins with 2 or more blank line tokens
+    if(${$self}{body} =~ m/^((?:$blankLineToken\R){2,})/s){
+
+        # remove them
+        ${$self}{body} =~ s/^((?:$blankLineToken\R)+)//s;
+
+        # store
+        my $blank_line_tokens_at_beginning_of_body = $1;
+
+        # and count them, for use after the indentation routine
+        ${$self}{blankLinesAtBeginning} = () = $blank_line_tokens_at_beginning_of_body =~ /$blankLineToken\R/sg
+    }
+    return;
+}
+
+sub put_blank_lines_back_in_at_beginning{
+    my $self = shift;
+
+    # some bodies have blank lines at the beginning
+    if(${$self}{blankLinesAtBeginning}){
+        for(my $i=0; $i<${$self}{blankLinesAtBeginning}; $i++){
+            ${$self}{body} = $tokens{blanklines}.${$self}{body};
+        }
+    }
+    return
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/LatexIndent/Verbatim.pm b/Master/texmf-dist/scripts/latexindent/LatexIndent/Verbatim.pm
new file mode 100644
index 00000000000..f3804aaf94d
--- /dev/null
+++ b/Master/texmf-dist/scripts/latexindent/LatexIndent/Verbatim.pm
@@ -0,0 +1,296 @@
+package LatexIndent::Verbatim;
+#	This program is free software: you can redistribute it and/or modify
+#	it under the terms of the GNU General Public License as published by
+#	the Free Software Foundation, either version 3 of the License, or
+#	(at your option) any later version.
+#
+#	This program is distributed in the hope that it will be useful,
+#	but WITHOUT ANY WARRANTY; without even the implied warranty of
+#	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+#	GNU General Public License for more details.
+#
+#	See http://www.gnu.org/licenses/.
+#
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
+use strict;
+use warnings;
+use LatexIndent::Tokens qw/%tokens/;
+use LatexIndent::GetYamlSettings qw/%masterSettings/;
+use LatexIndent::Switches qw/$is_t_switch_active $is_tt_switch_active/;
+use Data::Dumper;
+use Exporter qw/import/;
+our @EXPORT_OK = qw/put_verbatim_back_in find_verbatim_environments find_noindent_block find_verbatim_commands put_verbatim_commands_back_in/;
+our @ISA = "LatexIndent::Document"; # class inheritance, Programming Perl, pg 321
+our $verbatimCounter;
+
+sub find_noindent_block{
+    my $self = shift;
+
+    # noindent block
+    $self->logger('looking for NOINDENTBLOCk environments (see noIndentBlock)','heading') if $is_t_switch_active;
+    $self->logger(Dumper(\%{$masterSettings{noIndentBlock}})) if($is_t_switch_active);
+    while( my ($noIndentBlock,$yesno)= each %{$masterSettings{noIndentBlock}}){
+        if($yesno){
+            $self->logger("looking for $noIndentBlock:$yesno environments") if $is_t_switch_active;
+
+            my $noIndentRegExp = qr/
+                            (
+                                (?!<\\)
+                                %
+                                \h*                     # possible horizontal spaces
+                                \\begin\{
+                                        $noIndentBlock  # environment name captured into $2
+                                       \}               # %* \begin{noindentblock} statement
+                            )
+                            (
+                                .*?
+                            )                           # non-greedy match (body)
+                            (
+                                (?!<\\)
+                                %                       # %
+                                \h*                     # possible horizontal spaces
+                                \\end\{$noIndentBlock\} # \end{noindentblock}
+                            )                           # %* \end{<something>} statement
+                        /sx;
+
+            while( ${$self}{body} =~ m/$noIndentRegExp/sx){
+
+              # create a new Environment object
+              my $noIndentBlock = LatexIndent::Verbatim->new( begin=>$1,
+                                                    body=>$2,
+                                                    end=>$3,
+                                                    name=>$noIndentBlock,
+                                                    );
+            
+              # give unique id
+              $noIndentBlock->create_unique_id;
+
+              # verbatim children go in special hash
+              ${$self}{verbatim}{${$noIndentBlock}{id}}=$noIndentBlock;
+
+              # log file output
+              $self->logger("NOINDENTBLOCK environment found: $noIndentBlock") if $is_t_switch_active;
+
+              # remove the environment block, and replace with unique ID
+              ${$self}{body} =~ s/$noIndentRegExp/${$noIndentBlock}{id}/sx;
+
+              $self->logger("replaced with ID: ${$noIndentBlock}{id}") if $is_t_switch_active;
+            } 
+      } else {
+            $self->logger("*not* looking for $noIndentBlock as $noIndentBlock:$yesno") if $is_t_switch_active;
+      }
+    }
+    return;
+}
+
+sub find_verbatim_environments{
+    my $self = shift;
+
+    # verbatim environments
+    $self->logger('looking for VERBATIM environments (see verbatimEnvironments)','heading') if $is_t_switch_active;
+    $self->logger(Dumper(\%{$masterSettings{verbatimEnvironments}})) if($is_t_switch_active);
+    while( my ($verbEnv,$yesno)= each %{$masterSettings{verbatimEnvironments}}){
+        if($yesno){
+            $self->logger("looking for $verbEnv:$yesno environments") if $is_t_switch_active;
+
+            my $verbatimRegExp = qr/
+                            (
+                            \\begin\{
+                                    $verbEnv     # environment name captured into $1
+                                   \}            # \begin{<something>} statement
+                            )
+                            (
+                                .*?
+                            )                    # any character, but not \\begin
+                            (
+                                \\end\{$verbEnv\}# \end{<something>} statement
+                            )                    
+                        /sx;
+
+            while( ${$self}{body} =~ m/$verbatimRegExp/sx){
+
+              # create a new Environment object
+              my $verbatimBlock = LatexIndent::Verbatim->new( begin=>$1,
+                                                    body=>$2,
+                                                    end=>$3,
+                                                    name=>$verbEnv,
+                                                    );
+              # give unique id
+              $verbatimBlock->create_unique_id;
+
+              # verbatim children go in special hash
+              ${$self}{verbatim}{${$verbatimBlock}{id}}=$verbatimBlock;
+
+              # log file output
+              $self->logger("VERBATIM environment found: $verbEnv") if $is_t_switch_active;
+
+              # remove the environment block, and replace with unique ID
+              ${$self}{body} =~ s/$verbatimRegExp/${$verbatimBlock}{id}/sx;
+
+              $self->logger("replaced with ID: ${$verbatimBlock}{id}") if $is_t_switch_active;
+            } 
+      } else {
+            $self->logger("*not* looking for $verbEnv as $verbEnv:$yesno") if $is_t_switch_active;
+      }
+    }
+    return;
+}
+
+sub find_verbatim_commands{
+    my $self = shift;
+
+    # verbatim commands need to be treated separately to verbatim environments;
+    # note that, for example, we could quite reasonably have \lstinline!%!, which 
+    # would need to be found *before* trailing comments have been removed. Similarly, 
+    # verbatim commands need to be put back in *after* trailing comments have been put 
+    # back in
+    $self->logger('looking for VERBATIM commands (see verbatimCommands)','heading');
+    $self->logger(Dumper(\%{$masterSettings{verbatimCommands}})) if($is_tt_switch_active);
+    while( my ($verbCommand,$yesno)= each %{$masterSettings{verbatimCommands}}){
+        if($yesno){
+            $self->logger("looking for $verbCommand:$yesno Commands") if $is_t_switch_active;
+
+            my $verbatimCommandRegExp = qr/
+                            (
+                                \\$verbCommand     
+                                \h*                                             
+                            )                                                   # name of command into $1
+                            (
+                                \[
+                                    (?:
+                                        (?!
+                                            (?:(?<!\\)\[) 
+                                        ).
+                                    )*?     # not including [, but \[ ok
+                                (?<!\\)     # not immediately pre-ceeded by \
+                                \]          # [optional arguments]
+                                \h*
+                            )?                                                  # opt arg into $2
+                            (
+                              .
+                            )                                                   # delimiter into $3
+                            (
+                              .*?
+                            )                                                   # body into $4
+                            \3
+                        /mx;
+
+            while( ${$self}{body} =~ m/$verbatimCommandRegExp/){
+
+              # create a new Environment object
+              my $verbatimCommand = LatexIndent::Verbatim->new( begin=>$1.($2?$2:q()).$3,
+                                                    body=>$4,
+                                                    end=>$3,
+                                                    name=>$verbCommand,
+                                                    optArg=>$2?$2:q(),
+                                                    );
+              # give unique id
+              $verbatimCommand->create_unique_id;
+
+              # output, if desired
+              $self->logger(Dumper($verbatimCommand),'ttrace') if($is_tt_switch_active);
+
+              # verbatim children go in special hash
+              ${$self}{verbatimCommands}{${$verbatimCommand}{id}}=$verbatimCommand;
+
+              # log file output
+              $self->logger("VERBATIM command found: $verbCommand") if $is_t_switch_active;
+
+              # remove the environment block, and replace with unique ID
+              ${$self}{body} =~ s/$verbatimCommandRegExp/${$verbatimCommand}{id}/sx;
+
+              $self->logger("replaced with ID: ${$verbatimCommand}{id}") if $is_t_switch_active;
+            } 
+      } else {
+            $self->logger("*not* looking for $verbCommand as $verbCommand:$yesno") if $is_t_switch_active;
+      }
+    }
+    return;
+
+}
+
+sub  put_verbatim_back_in {
+    my $self = shift;
+
+    # if there are no verbatim children, return
+    return unless(${$self}{verbatim});
+
+    # search for environments/commands
+    $self->logger('Putting verbatim back in, here is the pre-processed body:','heading') if $is_t_switch_active;
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+
+    # loop through document children hash
+    while( (scalar keys %{${$self}{verbatim}})>0 ){
+          while( my ($key,$child)= each %{${$self}{verbatim}}){
+            if(${$self}{body} =~ m/${$child}{id}/mx){
+
+                # replace ids with body
+                ${$self}{body} =~ s/${$child}{id}/${$child}{begin}${$child}{body}${$child}{end}/;
+
+                # log file info
+                $self->logger('Body now looks like:','heading') if $is_tt_switch_active;
+                $self->logger(${$self}{body},'ttrace') if($is_tt_switch_active);
+
+                # delete the hash so it won't be operated upon again
+                delete ${$self}{verbatim}{${$child}{id}};
+                $self->logger("deleted key") if $is_t_switch_active;
+              }
+            }
+    }
+
+    # logfile info
+    $self->logger("Number of children:",'heading') if $is_t_switch_active;
+    $self->logger(scalar keys %{${$self}{verbatim}}) if $is_t_switch_active;
+    $self->logger('Post-processed body:','heading') if $is_t_switch_active;
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+    return;
+}
+
+sub  put_verbatim_commands_back_in {
+    my $self = shift;
+
+    # if there are no verbatim children, return
+    return unless(${$self}{verbatimCommands});
+
+    # search for environments/commands
+    $self->logger('Putting verbatim commands back in, here is the pre-processed body:','heading') if $is_t_switch_active;
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+
+    # loop through document children hash
+    while( (scalar keys %{${$self}{verbatimCommands}})>0 ){
+          while( my ($key,$child)= each %{${$self}{verbatimCommands}}){
+            if(${$self}{body} =~ m/${$child}{id}/mx){
+
+                # replace ids with body
+                ${$self}{body} =~ s/${$child}{id}/${$child}{begin}${$child}{body}${$child}{end}/;
+
+                # log file info
+                $self->logger('Body now looks like:','heading') if $is_tt_switch_active;
+                $self->logger(${$self}{body},'ttrace') if($is_tt_switch_active);
+
+                # delete the hash so it won't be operated upon again
+                delete ${$self}{verbatimCommands}{${$child}{id}};
+                $self->logger("deleted key") if $is_t_switch_active;
+              }
+            }
+    }
+
+    # logfile info
+    $self->logger("Number of children:",'heading');
+    $self->logger(scalar keys %{${$self}{verbatimCommands}});
+    $self->logger('Post-processed body:','heading') if $is_t_switch_active;
+    $self->logger(${$self}{body}) if($is_t_switch_active);
+    return;
+}
+
+sub create_unique_id{
+    my $self = shift;
+
+    $verbatimCounter++;
+    ${$self}{id} = "$tokens{verbatim}$verbatimCounter$tokens{endOfToken}";
+    return;
+}
+
+1;
diff --git a/Master/texmf-dist/scripts/latexindent/defaultSettings.yaml b/Master/texmf-dist/scripts/latexindent/defaultSettings.yaml
index ef620846f4f..20040817b12 100755
--- a/Master/texmf-dist/scripts/latexindent/defaultSettings.yaml
+++ b/Master/texmf-dist/scripts/latexindent/defaultSettings.yaml
@@ -1,5 +1,5 @@
 # 
-# defaultSettings.yaml for latexindent.pl,
+# defaultSettings.yaml for latexindent.pl, version 3.0
 #                      a script that aims to
 #                      beautify .tex, .sty, .cls files
 #
@@ -13,8 +13,17 @@
 #
 # Please read the manual first to understand what each switch does :)
 
-# Default value of indentation
-defaultIndent: "\t"
+# latexindent can be called without a file extension, 
+# e.g, simply
+#       latexindent myfile
+# in which case the choice of file extension is chosen
+# according to the choices made in fileExtensionPreference
+# Other file extensions can be added.
+fileExtensionPreference:
+    .tex: 1
+    .sty: 2
+    .cls: 3
+    .bib: 4
 
 # default file extension of backup file (if original is overwritten with -w switch)
 # for example, if your .tex file is called
@@ -50,19 +59,62 @@ maxNumberOfBackUps: 0
 # the back up will be written to myfile.bak4
 cycleThroughBackUps: 0
 
+# preferences for information displayed in the log file
+logFilePreferences:
+    showEveryYamlRead: 1
+    showAmalgamatedSettings: 0
+    endLogFileWith: '--------------' 
+    showGitHubInfoFooter: 1
+
+#  verbatim environments- environments specified 
+#  in this hash table will not be changed at all!
+verbatimEnvironments:
+    verbatim: 1
+    lstlisting: 1
+
+#  verbatim commands such as \verb! body !, \lstinline$something else$
+verbatimCommands:
+    verb: 1
+    lstinline: 1
+
+#  no indent blocks (not necessarily verbatim 
+#  environments) which are marked as %\begin{noindent}
+#  or anything else that the user puts in this hash
+#  table
+noIndentBlock:
+    noindent: 1
+    cmhtest: 1
+
+# remove trailing whitespace from all lines 
+removeTrailingWhitespace:
+    beforeProcessing: 0
+    afterProcessing: 1
+
+# \begin{document} and \end{document} are treated differently
+# by latexindent within filecontents environments
+fileContentsEnvironments:
+    filecontents: 1
+    filecontents*: 1
+
 # indent preamble
 indentPreamble: 0
 
-# always look for split { }, which means that the user doesn't
-# have to complete checkunmatched, checkunmatchedELSE
-alwaysLookforSplitBraces: 1
+# assume no preamble in cls, sty, by default
+lookForPreamble:
+    .tex: 1
+    .sty: 0
+    .cls: 0
+    .bib: 0
 
-# always look for split [ ], which means that the user doesn't
-# have to complete checkunmatchedbracket
-alwaysLookforSplitBrackets: 1
+# some preambles can contain \begin and \end statements
+# that are not in their 'standard environment block', for example,
+# consider the following key = values:
+#    preheadhook={\begin{mdframed}[style=myframedstyle]},
+#    postfoothook=\end{mdframed},
+preambleCommandsBeforeEnvironments: 0
 
-# remove trailing whitespace from all lines 
-removeTrailingWhitespace: 0
+# Default value of indentation
+defaultIndent: "\t"
 
 # environments that have tab delimiters, add more 
 # as needed
@@ -91,48 +143,43 @@ lookForAlignDelims:
    dcases: 1
    listabla: 1
 
-# if you have indent rules for particular environments
-# or commands, put them in here; for example, you might just want 
-# to use a space " " or maybe a double tab "\t\t"
-indentRules:
-   myenvironment: "\t\t"
-   anotherenvironment: "\t\t\t\t"
-   chapter: " "
-   section: " "
-   item: "      "
-
-#  verbatim environments- environments specified 
-#  in this hash table will not be changed at all!
-verbatimEnvironments:
-    verbatim: 1
-    lstlisting: 1
+# if you want the script to look for \item commands 
+# and format it, as follows (for example),
+#       \begin{itemize}
+#           \item content here
+#                 next line is indented
+#                 next line is indented
+#           \item another item
+#       \end{itemize}
+# then populate indentAfterItems. See also itemNames
+indentAfterItems:
+    itemize: 1
+    enumerate: 1
+    list: 1
 
-#  no indent blocks (not necessarily verbatim 
-#  environments) which are marked as %\begin{noindent}
-#  or anything else that the user puts in this hash
-#  table
-noIndentBlock:
-    noindent: 1
-    cmhtest: 1
+# if you want to use other names for your items (such as, for example, part)
+# then populate them here- note that you can trick latexindent.pl
+# into indenting all kinds of commands (within environments specified in 
+# indentAfterItems) using this technique.
+itemNames:
+    item: 1
+    myitem: 1
 
-# if you don't want to have additional indentation 
-# in an environment put it in this hash table; note that
-# environments in this hash table will inherit 
-# the *current* level of indentation they just won't
-# get any *additional*.
-noAdditionalIndent:
-    myexample: 1
-    mydefinition: 1
-    problem: 1
-    exercises: 1
-    mysolution: 1
-    foreach: 0
-    widepage: 1
-    comment: 1
-    \[: 0
-    \]: 0
-    document: 1
-    frame: 0
+# specialBeginEnd is mainly mathmode focus, although
+# there's no restrictions
+specialBeginEnd:
+    displayMath:
+        begin: '\\\['
+        end: '\\\]'
+        lookForThis: 1
+    inlineMath:
+        begin: '(?<!\$)(?<!\\)\$(?!\$)'
+        end: '(?<!\\)\$(?!\$)'
+        lookForThis: 1
+    displayMathTeX:
+        begin: '\$\$'
+        end: '\$\$'
+        lookForThis: 1
 
 # if you want to add indentation after
 # a heading, such as \part, \chapter, etc
@@ -144,139 +191,220 @@ noAdditionalIndent:
 # or add your own title command
 indentAfterHeadings:
     part:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 1
     chapter: 
-       indent: 0
+       indentAfterThisHeading: 0
        level: 2
     section:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 3
     subsection:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 4
     subsection*:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 4
     subsubsection:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 5
     paragraph:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 6
     subparagraph:
-       indent: 0
+       indentAfterThisHeading: 0
        level: 7
 
-# if you want the script to look for \item commands 
-# and format it, as follows (for example),
-#       \begin{itemize}
-#           \item content here
-#                 next line is indented
-#                 next line is indented
-#           \item another item
-#       \end{itemize}
-# then populate indentAfterItems. See also itemNames
-indentAfterItems:
-    itemize: 1
-    enumerate: 1
-    list: 1
-
-# if you want to use other names for your items (such as, for example, part)
-# then populate them here- note that you can trick latexindent.pl
-# into indenting all kinds of commands (within environments specified in 
-# indentAfterItems) using this technique.
-itemNames:
-    item: 1
-    myitem: 1
+# if you don't want to have additional indentation 
+# in an environment put it in this hash table; note that
+# environments in this hash table will inherit 
+# the *current* level of indentation they just won't
+# get any *additional*.
+noAdditionalIndent:
+    myexample: 1
+    mydefinition: 1
+    problem: 1
+    exercises: 1
+    mysolution: 1
+    foreach: 0
+    widepage: 1
+    comment: 1
+    document: 1
+    frame: 0
 
-# if you want to indent if, else, fi constructs such as, for example,
-#
-#       \ifnum#1=2
-#               something
-#       \else
-#               something else
-#       \fi
-#
-# then populate them in constructIfElseFi
-constructIfElseFi:
-    ifnum: 1
-    ifdim: 1
-    ifodd: 1
-    ifvmode: 1
-    ifhmode: 1
-    ifmmode: 1
-    ifinner: 1
-    if: 1
-    ifcat: 1
-    ifx: 1
-    ifvoid: 1
-    ifeof: 1
-    iftrue: 1
-    ifcase: 1
+# if you have indent rules for particular environments
+# or commands, put them in here; for example, you might just want 
+# to use a space " " or maybe a double tab "\t\t"
+indentRules:
+   myenvironment: "\t\t"
+   anotherenvironment: "\t\t\t\t"
+   chapter: " "
+   section: " "
+   item: "      "
+   myitem: "        "
 
-# latexindent can be called without a file extension, 
-# e.g, simply
-#       latexindent myfile
-# in which case the choice of file extension is chosen
-# according to the choices made in fileExtensionPreference
-# Other file extensions can be added.
-fileExtensionPreference:
-    .tex: 1
-    .sty: 2
-    .cls: 3
-    .bib: 4
+# set noAdditionalIndent globally for codeblocks
+noAdditionalIndentGlobal:
+    environments: 0
+    commands: 1
+    optionalArguments: 0
+    mandatoryArguments: 0
+    ifElseFi: 0
+    items: 0
+    keyEqualsValuesBracesBrackets: 0
+    namedGroupingBracesBrackets: 0
+    UnNamedGroupingBracesBrackets: 0
+    specialBeginEnd: 0
+    afterHeading: 0
+    filecontents: 0
 
-# preferences for information displayed in the log file
-logFilePreferences:
-    showEveryYamlRead: 1
-    showAlmagamatedSettings: 0
-    endLogFileWith: '--------------' 
-    traceModeIncreaseIndent: '>>'
-    traceModeAddCurrentIndent: '||'
-    traceModeDecreaseIndent: '<<'
-    traceModeBetweenLines: "\n"
+# set indentRules globally for codeblocks; these need 
+# to be horizontal spaces, if they are to be used
+indentRulesGlobal:
+    environments: 0
+    commands: 0
+    optionalArguments: 0
+    mandatoryArguments: 0
+    ifElseFi: 0
+    items: 0
+    keyEqualsValuesBracesBrackets: 0
+    namedGroupingBracesBrackets: 0
+    UnNamedGroupingBracesBrackets: 0
+    specialBeginEnd: 0
+    afterHeading: 0
+    filecontents: 0
 
-# \begin{document} and \end{document} are treated differently
-# by latexindent within filecontents environments
-fileContentsEnvironments:
-    filecontents: 1
-    filecontents*: 1
+# command code block details
+commandCodeBlocks:
+    roundParenthesesAllowed: 1
+    stringsAllowedBetweenArguments:
+        node: 1
+        at: 1
+        to: 1
+        decoration: 1
+        ++: 1
+        --: 1
 
-    
-# *** NOTE ***
-# If you have specified alwaysLookforSplitBraces: 1
-# and alwaysLookforSplitBrackets: 1 then you don't need
-# to worry about completing
+# modifyLineBreaks will only be searched if the -m 
+# switch is active
+#    BeginStartsOnOwnLine: 1
+#       if a begin statement is not already on its own line,
+#       then, with -m active, it will be moved to its own line.
 #
-#       checkunmatched
-#       checkunmatchedELSE
-#       checkunmatchedbracket
+#       when set to 1, e.g 
+#               some text some text \begin{myenvironment}
+#           will be changed to
+#               some text some text 
+#               \begin{myenvironment}
+#       when set to -1, e.g 
+#               some text some text 
+#               \begin{myenvironment}
+#           will be changed to
+#               some text some text \begin{myenvironment}
+#       when set to 0, the switch is ignored
 #
-# in other words, you don't really need to edit anything 
-# below this line- it used to be necessary for older 
-# versions of the script, but not anymore :)
-#***      ***
-
-# commands that might split {} across lines
-# such as \parbox, \marginpar, etc
-checkunmatched:
-    parbox: 1
-    vbox: 1
-
-# very similar to %checkunmatched except these 
-# commands might have an else construct
-checkunmatchedELSE:
-    pgfkeysifdefined: 1
-    DTLforeach: 1
-    ifthenelse: 1
-
-# commands that might split []  across lines
-# such as \pgfplotstablecreatecol, etc
-checkunmatchedbracket:
-    pgfplotstablecreatecol: 1
-    pgfplotstablesave: 1
-    pgfplotstabletypeset: 1
-    mycommand: 1
-    psSolid: 1
-
+#    BodyStartsOnOwnLine: 1
+#       if body does not already start on its own line, then with 
+#       -m active, it will be moved to its own line.
+#
+#       when set to 1, e.g 
+#               \begin{myenv}body text body text
+#           will be changed to
+#               \begin{myenv}
+#                   body text body text
+#       when set to -1, e.g 
+#               \begin{myenv}
+#                   body text body text
+#           will be changed to
+#               \begin{myenv}body text body text
+#       when set to 0, the switch is ignored
+#
+#    EndStartsOnOwnLine: 1
+#       if an end statement is not already on its own line,
+#       then, with -m active, it will be moved to its own line.
+#
+#       when set to 1, e.g 
+#               some text some text \end{myenvironment}
+#           will be changed to
+#               some text some text 
+#               \end{myenvironment}
+#       when set to -1, e.g 
+#               some text some text 
+#               \end{myenvironment}
+#           will be changed to
+#               some text some text \end{myenvironment}
+#       when set to 0, the switch is ignored
+#
+#    EndFinishesWithLineBreak: 1
+#       this switch adjusts line breaks immmediately 
+#       after an end statement (with -m active)
+#       
+#       when set to 1, e.g 
+#               \end{myenvironment}some text some text 
+#           will be changed to
+#                   \end{myenvironment}
+#                   some text some text 
+#       when set to -1, e.g 
+#               \end{myenvironment}
+#               some text some text 
+#           will be changed to
+#                   \end{myenvironment}some text some text 
+#       when set to 0, the switch is ignored
+#
+# Naturally, you can specify settings for individual environments,
+# commands, headings, etc, populate as you wish.
+modifyLineBreaks:
+    preserveBlankLines: 1
+    condenseMultipleBlankLinesInto: 1
+    environments:
+        BeginStartsOnOwnLine: 0
+        BodyStartsOnOwnLine: 0
+        EndStartsOnOwnLine: 0
+        EndFinishesWithLineBreak: 0
+        equation*:
+            BeginStartsOnOwnLine: 0
+            BodyStartsOnOwnLine: 0
+            EndStartsOnOwnLine: 0
+            EndFinishesWithLineBreak: 0
+    ifElseFi:
+        IfStartsOnOwnLine: 0
+        BodyStartsOnOwnLine: 0
+        ElseStartsOnOwnLine: 0
+        ElseFinishesWithLineBreak: 0
+        FiStartsOnOwnLine: 0
+        FiFinishesWithLineBreak: 0
+        ifnum:
+            IfStartsOnOwnLine: 0
+            BodyStartsOnOwnLine: 0
+            ElseStartsOnOwnLine: 0
+            ElseFinishesWithLineBreak: 0
+            FiStartsOnOwnLine: 0
+            FiFinishesWithLineBreak: 0
+    commands:
+        CommandStartsOnOwnLine: 0
+        CommandNameFinishesWithLineBreak: 0
+    optionalArguments:
+        LSqBStartsOnOwnLine: 0
+        OptArgBodyStartsOnOwnLine: 0
+        RSqBStartsOnOwnLine: 0
+        RSqBFinishesWithLineBreak: 0
+    mandatoryArguments:
+        LCuBStartsOnOwnLine: 0
+        MandArgBodyStartsOnOwnLine: 0
+        RCuBStartsOnOwnLine: 0
+        RCuBFinishesWithLineBreak: 0
+    keyEqualsValuesBracesBrackets:
+        KeyStartsOnOwnLine: 0
+        EqualsStartsOnOwnLine: 0
+        EqualsFinishesWithLineBreak: 0
+    items:
+        ItemStartsOnOwnLine: 0
+        ItemFinishesWithLineBreak: 0
+    namedGroupingBracesBrackets:
+        NameStartsOnOwnLine: 0
+        NameFinishesWithLineBreak: 0
+    specialBeginEnd:
+        SpecialBeginStartsOnOwnLine: 0
+        SpecialBodyStartsOnOwnLine: 0
+        SpecialEndStartsOnOwnLine: 0
+        SpecialEndFinishesWithLineBreak: 0
diff --git a/Master/texmf-dist/scripts/latexindent/latexindent.pl b/Master/texmf-dist/scripts/latexindent/latexindent.pl
index cdd979a8ccf..06adc58a1b8 100755
--- a/Master/texmf-dist/scripts/latexindent/latexindent.pl
+++ b/Master/texmf-dist/scripts/latexindent/latexindent.pl
@@ -11,57 +11,34 @@
 #
 #	See http://www.gnu.org/licenses/.
 #
-#	For details of how to use this file, please see readme.txt
+#	Chris Hughes, 2017
+#
+#	For all communication, please visit: https://github.com/cmhughes/latexindent.pl
 
-# load packages/modules: assume strict and warnings are part of every perl distribution
 use strict;
 use warnings;
+use FindBin;                   # help find defaultSettings.yaml
+use Getopt::Long;              # to get the switches/options/flags
 
-# list of modules
-my @listOfModules = ('FindBin','YAML::Tiny','File::Copy','File::Basename','Getopt::Long','File::HomeDir');
-
-# check the other modules are available
-foreach my $moduleName (@listOfModules) {
-    # references:
-    #       http://stackoverflow.com/questions/251694/how-can-i-check-if-i-have-a-perl-module-before-using-it
-    #       http://stackoverflow.com/questions/1917261/how-can-i-dynamically-include-perl-modules-without-using-eval
-    eval {
-        (my $file = $moduleName) =~ s|::|/|g;
-        require $file . '.pm';
-        $moduleName->import();
-        1;
-    } or die "$moduleName Perl Module not currently installed; please install the module, and then try running latexindent.pl again; exiting";
-}
-
-# now that we have confirmed the modules are available, load them
-use FindBin;        # help find defaultSettings.yaml
-use YAML::Tiny;     # interpret defaultSettings.yaml and other potential settings files
-use File::Copy;     # to copy the original file to backup (if overwrite option set)
-use File::Basename; # to get the filename and directory path
-use Getopt::Long;   # to get the switches/options/flags
-use File::HomeDir;  # to get users home directory, regardless of OS
+# use lib to make sure that @INC contains the latexindent directory
+use lib $FindBin::RealBin;
+use LatexIndent::Document;
 
 # get the options
-my $overwrite;
-my $outputToFile;
-my $silentMode;
-my $tracingMode;
-my $tracingModeVeryDetailed;
-my $readLocalSettings=0;
-my $onlyDefault;
-my $showhelp;
-my $cruftDirectory;
+my %switches = (readLocalSettings=>0);
 
 GetOptions (
- "overwrite|w"=>\$overwrite,
-"outputfile|o"=>\$outputToFile,
-"silent|s"=>\$silentMode,
-"trace|t"=>\$tracingMode,
-"ttrace|tt"=>\$tracingModeVeryDetailed,
-"local|l:s"=>\$readLocalSettings,
-"onlydefault|d"=>\$onlyDefault,
-"help|h"=>\$showhelp,
-"cruft|c=s"=>\$cruftDirectory,
+    "silent|s"=>\$switches{silentMode},
+    "trace|t"=>\$switches{trace},
+    "ttrace|tt"=>\$switches{ttrace},
+    "local|l:s"=>\$switches{readLocalSettings},
+    "onlydefault|d"=>\$switches{onlyDefault},
+    "overwrite|w"=>\$switches{overwrite},
+    "outputfile|o=s"=>\$switches{outputToFile},
+    "modifylinebreaks|m"=>\$switches{modifyLineBreaks},
+    "logfile|g=s"=>\$switches{logFileName},
+    "help|h"=>\$switches{showhelp},
+    "cruft|c=s"=>\$switches{cruftDirectory},
 );
 
 # check local settings doesn't interfer with reading the file;
@@ -74,20 +51,9 @@ GetOptions (
 #
 # In such circumstances, we correct the mistake by assuming that 
 # the only argument is the file to be indented, and place it in @ARGV
-if($readLocalSettings and scalar(@ARGV) < 1) {
-    push(@ARGV,$readLocalSettings);
-    $readLocalSettings = '';
-}
-
-# this can also happen if the script is called as
-#
-#       latexindent.pl -o -l myfile.tex outputfile.tex
-#
-# in which case, the GetOptions routine mistakes myfile.tex
-# as the optional parameter to the l flag.
-if($readLocalSettings and scalar(@ARGV) < 2 and $outputToFile) {
-    unshift(@ARGV,$readLocalSettings);
-    $readLocalSettings = '';
+if($switches{readLocalSettings} and scalar(@ARGV) < 1) {
+    push(@ARGV,$switches{readLocalSettings});
+    $switches{readLocalSettings} = '';
 }
 
 # default value of readLocalSettings
@@ -95,2054 +61,10 @@ if($readLocalSettings and scalar(@ARGV) < 2 and $outputToFile) {
 #       latexindent -l myfile.tex
 #
 # means that we wish to use localSettings.yaml
-if(defined($readLocalSettings) and ($readLocalSettings eq '')){
-    $readLocalSettings = 'localSettings.yaml';
-}
-
-# detailed tracing mode also implies regular tracing mode
-$tracingMode = $tracingModeVeryDetailed ? 1 : $tracingMode;
-
-# version number
-my $versionNumber = "2.2";
-
-# Check the number of input arguments- if it is 0 then simply
-# display the list of options (like a manual)
-if(scalar(@ARGV) < 1 or $showhelp) {
-    print <<ENDQUOTE
-latexindent.pl version $versionNumber
-usage: latexindent.pl [options] [file][.tex]
-      -h, --help
-          help (see the documentation for detailed instructions and examples)
-      -o, --outputfile
-          output to another file; sample usage
-                latexindent.pl -o myfile.tex outputfile.tex
-      -w, --overwrite
-          overwrite the current file- a backup will be made, but still be careful
-      -s, --silent
-          silent mode- no output will be given to the terminal
-      -t, --trace
-          tracing mode- verbose information given to the log file
-      -l, --local[=myyaml.yaml]
-          use localSettings.yaml (assuming it exists in the directory of your file);
-          alternatively, use myyaml.yaml, if it exists
-      -d, --onlydefault
-          ONLY use defaultSettings.yaml, ignore ALL user files
-      -c, --cruft=<cruft directory> 
-          used to specify the location of backup files and indent.log
-ENDQUOTE
-    ;
-    exit(2);
-}
-
-# set up default for cruftDirectory using the one from the input file,
-# unless it has been specified using -c="/some/directory"
-$cruftDirectory=dirname $ARGV[0] unless(defined($cruftDirectory));
-
-die "Could not find directory $cruftDirectory\nExiting, no indentation done." if(!(-d $cruftDirectory));
-
-# we'll be outputting to the logfile and to standard output
-my $logfile;
-my $out = *STDOUT;
-
-# open the log file
-open($logfile,">","$cruftDirectory/indent.log") or die "Can't open indent.log";
-
-# output time to log file
-my $time = localtime();
-print $logfile $time;
-
-# output version to log file
-print $logfile <<ENDQUOTE
-
-$FindBin::Script version $versionNumber, a script to indent .tex files
-$FindBin::Script lives here: $FindBin::RealBin/
-
-ENDQUOTE
-;
-
-# latexindent.exe is a standalone executable, and caches 
-# the required perl modules onto the users system; they will
-# only be displayed if the user specifies the trace option
-if($FindBin::Script eq 'latexindent.exe' and !$tracingMode ) {
-print $logfile <<ENDQUOTE
-$FindBin::Script is a standalone script and caches the required perl modules
-onto your system. If you'd like to see their location in your log file, indent.log, 
-call the script with the tracing option, e.g latexindent.exe -t myfile.tex
-
-ENDQUOTE
-;
-}
-
-# output location of modules
-if($FindBin::Script eq 'latexindent.pl' or ($FindBin::Script eq 'latexindent.exe' and $tracingMode )) {
-    print $logfile "Modules are being loaded from the following directories:\n ";
-    foreach my $moduleName (@listOfModules) {
-            (my $file = $moduleName) =~ s|::|/|g;
-            require $file . '.pm';
-            print $logfile "\t",$INC{$file .'.pm'},"\n";
-          }
-}
-
-# a quick options check
-if($outputToFile and $overwrite) {
-    print $logfile <<ENDQUOTE
-
-WARNING:
-\t You have called latexindent.pl with both -o and -w
-\t -o (output to file) will take priority, and -w (over write) will be ignored
-
-ENDQUOTE
-;
-    $overwrite = 0;
-}
-
-# can't call the script with MORE THAN 2 files
-if(scalar(@ARGV)>2) {
-    for my $fh ($out,$logfile) {print $fh <<ENDQUOTE
-
-ERROR:
-\t You're calling latexindent.pl with more than two file names
-\t The script can take at MOST two file names, but you
-\t need to call it with the -o switch; for example
-
-\t latexindent.pl -o originalfile.tex outputfile.tex
-
-No indentation done :(
-Exiting...
-ENDQUOTE
-    };
-    exit(2);
-}
-
-# don't call the script with 2 files unless the -o flag is active
-if(!$outputToFile and scalar(@ARGV)==2)
-{
-for my $fh ($out,$logfile) {
-print $fh <<ENDQUOTE
-
-ERROR:
-\t You're calling latexindent.pl with two file names, but not the -o flag.
-\t Did you mean to use the -o flag ?
-
-No indentation done :(
-Exiting...
-ENDQUOTE
-};
-    exit(2);
-}
-
-# if the script is called with the -o switch, then check that
-# a second file is present in the call, e.g
-#           latexindent.pl -o myfile.tex output.tex
-if($outputToFile and scalar(@ARGV)==1) {
-    for my $fh ($out,$logfile) {print $fh <<ENDQUOTE
-ERROR: When using the -o flag you need to call latexindent.pl with 2 arguments
-
-latexindent.pl -o "$ARGV[0]" [needs another name here]
-
-No indentation done :(
-Exiting...
-ENDQUOTE
-};
-    exit(2);
+if(defined($switches{readLocalSettings}) and ($switches{readLocalSettings} eq '')){
+    $switches{readLocalSettings} = 'localSettings.yaml';
 }
 
-# yaml work
-print $logfile "YAML files:\n";
-
-# Read in defaultSettings.YAML file
-my $defaultSettings = YAML::Tiny->new;
-
-# Open defaultSettings.yaml
-$defaultSettings = YAML::Tiny->read( "$FindBin::RealBin/defaultSettings.yaml" );
-print $logfile "\tReading defaultSettings.yaml from $FindBin::RealBin/defaultSettings.yaml\n\n" if($defaultSettings);
-
-# if latexindent.exe is invoked from TeXLive, then defaultSettings.yaml won't be in 
-# the same directory as it; we need to navigate to it
-if(!$defaultSettings) {
-    $defaultSettings = YAML::Tiny->read( "$FindBin::RealBin/../../texmf-dist/scripts/latexindent/defaultSettings.yaml");
-    print $logfile "\tReading defaultSettings.yaml (2nd attempt, TeXLive, Windows) from $FindBin::RealBin/../../texmf-dist/scripts/latexindent/defaultSettings.yaml\n\n" if($defaultSettings);
-}
-
-# if both of the above attempts have failed, we need to exit
-if(!$defaultSettings) {
-  for my $fh ($out,$logfile) {
- print $fh <<ENDQUOTE
- ERROR  There seems to be a yaml formatting error in defaultSettings.yaml
-        Please check it for mistakes- you can find a working version at https://github.com/cmhughes/latexindent.pl
-        if you would like to overwrite your current version
-
-        Exiting, no indendation done.
-ENDQUOTE
-};
- exit(2);
-}
-
-# the MASTER settings will initially be from defaultSettings.yaml
-# and we update them with USER settings (if any) below
-my %masterSettings = %{$defaultSettings->[0]};
-
-# empty array to store the paths
-my @absPaths;
-
-# scalar to read user settings
-my $userSettings;
-
-# get information about user settings- first check if indentconfig.yaml exists
-my $indentconfig = File::HomeDir->my_home . "/indentconfig.yaml";
-# if indentconfig.yaml doesn't exist, check for the hidden file, .indentconfig.yaml
-$indentconfig = File::HomeDir->my_home . "/.indentconfig.yaml" if(! -e $indentconfig);
-
-if ( -e $indentconfig and !$onlyDefault ) {
-      print $logfile "\tReading path information from $indentconfig\n";
-      # if both indentconfig.yaml and .indentconfig.yaml exist
-      if ( -e File::HomeDir->my_home . "/indentconfig.yaml" and  -e File::HomeDir->my_home . "/.indentconfig.yaml") {
-            print $logfile File::HomeDir->my_home,"/.indentconfig.yaml has been found, but $indentconfig takes priority\n";
-      } elsif ( -e File::HomeDir->my_home . "/indentconfig.yaml" ) {
-            print $logfile "\tAlternatively, ",File::HomeDir->my_home,"/.indentconfig.yaml can be used\n";
-
-      } elsif ( -e File::HomeDir->my_home . "/.indentconfig.yaml" ) {
-            print $logfile "\tAlternatively, ",File::HomeDir->my_home,"/indentconfig.yaml can be used\n";
-      }
-
-      # read the absolute paths from indentconfig.yaml
-      $userSettings = YAML::Tiny->read( "$indentconfig" );
-
-      # integrity check
-      if($userSettings) {
-        print $logfile "\t",Dump \%{$userSettings->[0]};
-        print $logfile "\n";
-        @absPaths = @{$userSettings->[0]->{paths}};
-      } else {
-        print $logfile <<ENDQUOTE
-WARNING:  $indentconfig
-          contains some invalid .yaml formatting- unable to read from it.
-          No user settings loaded.
-ENDQUOTE
-;
-      }
-} else {
-      if($onlyDefault) {
-        print $logfile "\tOnly default settings requested, not reading USER settings from $indentconfig\n";
-        print $logfile "\tIgnoring $readLocalSettings\n" if($readLocalSettings);
-        $readLocalSettings = 0;
-      } else {
-        # give the user instructions on where to put indentconfig.yaml or .indentconfig.yaml
-        print $logfile "\tHome directory is ",File::HomeDir->my_home,"\n";
-        print $logfile "\tTo specify user settings you would put indentconfig.yaml here: \n\t",File::HomeDir->my_home,"/indentconfig.yaml\n\n";
-        print $logfile "\tAlternatively, you can use the hidden file .indentconfig.yaml as: \n\t",File::HomeDir->my_home,"/.indentconfig.yaml\n\n";
-      }
-}
-
-# get information about LOCAL settings, assuming that $readLocalSettings exists
-my $directoryName = dirname $ARGV[0];
-
-# add local settings to the paths, if appropriate
-if ( (-e "$directoryName/$readLocalSettings") and $readLocalSettings and !(-z "$directoryName/$readLocalSettings")) {
-    print $logfile "\tAdding $directoryName/$readLocalSettings to paths\n\n";
-    push(@absPaths,"$directoryName/$readLocalSettings");
-} elsif ( !(-e "$directoryName/$readLocalSettings") and $readLocalSettings) {
-      print $logfile "\tWARNING yaml file not found: \n\t$directoryName/$readLocalSettings not found\n";
-      print $logfile "\t\tcarrying on without it.\n";
-}
-
-# read in the settings from each file
-foreach my $settings (@absPaths) {
-  # check that the settings file exists and that it isn't empty
-  if (-e $settings and !(-z $settings)) {
-      print $logfile "\tReading USER settings from $settings\n";
-      $userSettings = YAML::Tiny->read( "$settings" );
-
-      # if we can read userSettings
-      if($userSettings) {
-            # update the MASTER setttings to include updates from the userSettings
-            while(my($userKey, $userValue) = each %{$userSettings->[0]}) {
-                    # the update approach is slightly different for hashes vs scalars/arrays
-                    if (ref($userValue) eq "HASH") {
-                        while(my ($userKeyFromHash,$userValueFromHash) = each %{$userSettings->[0]{$userKey}}) {
-                          $masterSettings{$userKey}{$userKeyFromHash} = $userValueFromHash;
-                        }
-                    } else {
-                          $masterSettings{$userKey} = $userValue;
-                    }
-            }
-            # output settings to $logfile
-            if($masterSettings{logFilePreferences}{showEveryYamlRead}){
-                print $logfile Dump \%{$userSettings->[0]};
-                print $logfile "\n";
-            } else {
-                print $logfile "\t\tNot showing settings in the log file, see showEveryYamlRead.\n";
-            }
-       } else {
-             # otherwise print a warning that we can not read userSettings.yaml
-             print $logfile "WARNING\n\t$settings \n\t contains invalid yaml format- not reading from it\n";
-       }
-  } else {
-      # otherwise keep going, but put a warning in the log file
-      print $logfile "\nWARNING\n\t",File::HomeDir->my_home,"/indentconfig.yaml\n";
-      if (-z $settings) {
-          print $logfile "\tspecifies $settings \n\tbut this file is EMPTY- not reading from it\n\n"
-      } else {
-          print $logfile "\tspecifies $settings \n\tbut this file does not exist- unable to read settings from this file\n\n"
-      }
-  }
-}
-
-# some people may wish to see showAlmagamatedSettings
-# which details the overall state of the settings modified
-# from the default in various user files
-if($masterSettings{logFilePreferences}{showAlmagamatedSettings}){
-    print $logfile "Almagamated/overall settings to be used:\n";
-    print $logfile Dump \%masterSettings ;
-}
-
-# scalar variables
-my $defaultIndent = $masterSettings{defaultIndent};
-my $alwaysLookforSplitBraces = $masterSettings{alwaysLookforSplitBraces};
-my $alwaysLookforSplitBrackets = $masterSettings{alwaysLookforSplitBrackets};
-my $backupExtension = $masterSettings{backupExtension};
-my $indentPreamble = $masterSettings{indentPreamble};
-my $onlyOneBackUp = $masterSettings{onlyOneBackUp};
-my $maxNumberOfBackUps = $masterSettings{maxNumberOfBackUps};
-my $removeTrailingWhitespace = $masterSettings{removeTrailingWhitespace};
-my $cycleThroughBackUps = $masterSettings{cycleThroughBackUps};
-
-# hash variables
-my %lookForAlignDelims= %{$masterSettings{lookForAlignDelims}};
-my %indentRules= %{$masterSettings{indentRules}};
-my %verbatimEnvironments= %{$masterSettings{verbatimEnvironments}};
-my %noIndentBlock= %{$masterSettings{noIndentBlock}};
-my %checkunmatched= %{$masterSettings{checkunmatched}};
-my %checkunmatchedELSE= %{$masterSettings{checkunmatchedELSE}};
-my %checkunmatchedbracket= %{$masterSettings{checkunmatchedbracket}};
-my %noAdditionalIndent= %{$masterSettings{noAdditionalIndent}};
-my %indentAfterHeadings= %{$masterSettings{indentAfterHeadings}};
-my %indentAfterItems= %{$masterSettings{indentAfterItems}};
-my %itemNames= %{$masterSettings{itemNames}};
-my %constructIfElseFi= %{$masterSettings{constructIfElseFi}};
-my %fileExtensionPreference= %{$masterSettings{fileExtensionPreference}};
-my %fileContentsEnvironments= %{$masterSettings{fileContentsEnvironments}};
-
-# original name of file
-my $fileName = $ARGV[0];
-
-# sort the file extensions by preference 
-my @fileExtensions = sort { $fileExtensionPreference{$a} <=> $fileExtensionPreference{$b} } keys(%fileExtensionPreference);
-
-# get the base file name, allowing for different extensions (possibly no extension)
-my ($dir, $name, $ext) = fileparse($fileName, @fileExtensions);
-
-# quick check to make sure given file type is supported
-if( -e $ARGV[0] and !$ext ){
-for my $fh ($out,$logfile) {print $fh <<ENDQUOTE
-The file $ARGV[0] exists , but the extension does not correspond to any given in fileExtensionPreference;
-consinder updating fileExtensionPreference.
-
-Exiting, no indentation done.
-ENDQUOTE
-};
-exit(2);
-}
-
-# if no extension, search according to fileExtensionPreference
-if (!$ext) {
-    print $logfile "File extension work:\n";
-    print $logfile "\tlatexindent called to act upon $fileName with an, as yet, unrecognised file extension;\n";
-    print $logfile "\tsearching for file with an extension in the following order (see fileExtensionPreference):\n\t\t";
-    print $logfile join("\n\t\t",@fileExtensions),"\n";
-    my $fileFound = 0;
-    # loop through the known file extensions (see @fileExtensions)
-    foreach my $fileExt (@fileExtensions ){
-        if ( -e $fileName.$fileExt ) {
-           print $logfile "\t",$fileName,$fileExt," found!\n";
-           $fileName .= $fileExt;
-           print $logfile "\tUpdated $ARGV[0] to ",$fileName,"\n";
-           $fileFound = 1;
-           last;
-        }
-    }
-    unless($fileFound){
-      print $logfile "\tI couldn't find a match for $ARGV[0] in fileExtensionPreference (see defaultSettings.yaml)\n";
-      foreach my $fileExt (@fileExtensions ){
-        print $logfile "\t\tI searched for $ARGV[0]$fileExt\n";
-      }
-      print $logfile "\tbut couldn't find any of them.\n";
-      print $logfile "\tConsider updating fileExtensionPreference. \nError: Exiting, no indendation done.";
-      die "I couldn't find a match for $ARGV[0] in fileExtensionPreference.\nExiting, no indendation done.\n" 
-    }
-  } else {
-    # if the file has a recognised extension, check that the file exists
-    unless( -e $ARGV[0] ){
-      print $logfile "Error: I couldn't find $ARGV[0], are you sure it exists?. No indentation done. \nExiting.\n";
-      die "Error: I couldn't find $ARGV[0], are you sure it exists?. Exiting.\n" ;
-    }
-  }
-
-# if we want to over write the current file create a backup first
-if ($overwrite) {
-    print $logfile "\nBackup procedure:\n";
-    # cruft directory
-    print $logfile "\tDirectory for backup files and indent.log: $cruftDirectory\n\n";
-
-    my $backupFile; 
-
-    # backup file name is the base name
-    $backupFile = basename($fileName,@fileExtensions);
-
-    # add the user's backup directory to the backup path
-    $backupFile = "$cruftDirectory/$backupFile";
-
-    # if both ($onlyOneBackUp and $maxNumberOfBackUps) then we have
-    # a conflict- er on the side of caution and turn off onlyOneBackUp
-    if($onlyOneBackUp and $maxNumberOfBackUps>1) {
-        print $logfile "\t WARNING: onlyOneBackUp=$onlyOneBackUp and maxNumberOfBackUps: $maxNumberOfBackUps\n";
-        print $logfile "\t\t setting onlyOneBackUp=0 which will allow you to reach $maxNumberOfBackUps back ups\n";
-        $onlyOneBackUp = 0;
-    }
-
-    # if the user has specified that $maxNumberOfBackUps = 1 then
-    # they only want one backup
-    if($maxNumberOfBackUps==1) {
-        $onlyOneBackUp=1 ;
-        print $logfile "\t FYI: you set maxNumberOfBackUps=1, so I'm setting onlyOneBackUp: 1 \n";
-    } elsif($maxNumberOfBackUps<=0 and !$onlyOneBackUp) {
-        $onlyOneBackUp=0 ;
-        $maxNumberOfBackUps=-1;
-    }
-
-    # if onlyOneBackUp is set, then the backup file will
-    # be overwritten each time
-    if($onlyOneBackUp) {
-        $backupFile .= $backupExtension;
-        print $logfile "\tcopying $fileName to $backupFile\n";
-        print $logfile "\t$backupFile was overwritten\n\n" if (-e $backupFile);
-    } else {
-        # start with a backup file .bak0 (or whatever $backupExtension is present)
-        my $backupCounter = 0;
-        $backupFile .= $backupExtension.$backupCounter;
-
-        # if it exists, then keep going: .bak0, .bak1, ...
-        while (-e $backupFile or $maxNumberOfBackUps>1) {
-            if($backupCounter==$maxNumberOfBackUps) {
-                print $logfile "\t maxNumberOfBackUps reached ($maxNumberOfBackUps)\n";
-
-                # some users may wish to cycle through back up files, e.g:
-                #    copy myfile.bak1 to myfile.bak0
-                #    copy myfile.bak2 to myfile.bak1
-                #    copy myfile.bak3 to myfile.bak2
-                #
-                #    current back up is stored in myfile.bak4
-                if($cycleThroughBackUps) {
-                    print $logfile "\t cycleThroughBackUps detected (see cycleThroughBackUps) \n";
-                    for(my $i=1;$i<=$maxNumberOfBackUps;$i++) {
-                        # remove number from backUpFile
-                        my $oldBackupFile = $backupFile;
-                        $oldBackupFile =~ s/$backupExtension.*/$backupExtension/;
-                        my $newBackupFile = $oldBackupFile;
-
-                        # add numbers back on
-                        $oldBackupFile .= $i;
-                        $newBackupFile .= $i-1;
-
-                        # check that the oldBackupFile exists
-                        if(-e $oldBackupFile){
-                        print $logfile "\t\t copying $oldBackupFile to $newBackupFile \n";
-                            copy($oldBackupFile,$newBackupFile) or die "Could not write to backup file $backupFile. Please check permissions. Exiting.\n";
-                        }
-                    }
-                }
-
-                # rest maxNumberOfBackUps
-                $maxNumberOfBackUps=1 ;
-                last; # break out of the loop
-            } elsif(!(-e $backupFile)) {
-                $maxNumberOfBackUps=1 ;
-                last; # break out of the loop
-            }
-            print $logfile "\t $backupFile already exists, incrementing by 1...\n";
-            $backupCounter++;
-            $backupFile =~ s/$backupExtension.*/$backupExtension$backupCounter/;
-        }
-        print $logfile "\n\t copying $fileName to $backupFile\n\n";
-    }
-
-    # output these lines to the log file
-    print $logfile "\tBackup file: ",$backupFile,"\n";
-    print $logfile "\tOverwriting file: ",$fileName,"\n\n";
-    copy($fileName,$backupFile) or die "Could not write to backup file $backupFile. Please check permissions. Exiting.\n";
-}
-
-if(!($outputToFile or $overwrite)) {
-    print $logfile "Just out put to the terminal :)\n\n" if !$silentMode  ;
-}
-
-
-# scalar variables
-my $line;                   # $line: takes the $line of the file
-my $inpreamble=!$indentPreamble;
-                            # $inpreamble: switch to determine if in
-                            #               preamble or not
-my $inverbatim=0;           # $inverbatim: switch to determine if in
-                            #               a verbatim environment or not
-my $delimiters=0;           # $delimiters: switch that governs if
-                            #              we need to check for & or not
-my $trailingcomments;       # $trailingcomments stores the comments at the end of
-                            #           a line
-my $lineCounter=0;          # $lineCounter keeps track of the line number
-my $inIndentBlock=0;        # $inindentblock: switch to determine if in
-                            #               a inindentblock or not
-my $inFileContents=0;       # $inFileContents: switch to determine if we're in a filecontents environment
-
-# array variables
-my @lines;                  # @lines: stores the newly indented lines
-my @mainfile;               # @mainfile: stores input file; used to
-                            #            grep for \documentclass
-
-# array of hashes, containing details of commands & environments
-my @masterIndentationArrayOfHashes;
-
-# check to see if the current file has \documentclass, if so, then
-# it's the main file, if not, then it doesn't have preamble
-open(MAINFILE, $fileName) or die "Could not open input file, $fileName";
-    @mainfile=<MAINFILE>;
-close(MAINFILE);
-
-# if the MAINFILE doesn't have a \documentclass statement, then
-# it shouldn't have preamble
-if(scalar(@{[grep(m/^\s*\\documentclass/, @mainfile)]})==0) {
-    $inpreamble=0;
-
-    print $logfile "Trace:\tNo documentclass detected, assuming no preamble\n" if($tracingMode);
-} else {
-    print $logfile "Trace:\t documentclass detected, assuming preamble\n" if($tracingMode);
-}
-
-# the previous OPEN command puts us at the END of the file
-open(MAINFILE, $fileName) or die "Could not open input file, $fileName";
-
-# loop through the lines in the INPUT file
-while(<MAINFILE>) {
-    # increment the line counter
-    $lineCounter++;
-
-    # very detailed output to logfile
-    if($tracingModeVeryDetailed){
-        if( @masterIndentationArrayOfHashes){
-            print $logfile "\nLine $lineCounter\t (detailed trace) indentation hash: \n" if($tracingMode);
-            for my $href ( @masterIndentationArrayOfHashes) {
-                   print $logfile Dump \%{$href};
-            }
-        }
-    }
-
-    # tracing mode
-    print $logfile $masterSettings{logFilePreferences}{traceModeBetweenLines} if($tracingMode and !($inpreamble or $inverbatim or $inIndentBlock));
-
-    # check to see if we're still in the preamble
-    # or in a verbatim environment or in IndentBlock
-    if(!($inpreamble or $inverbatim or $inIndentBlock)) {
-        # if not, remove all leading spaces and tabs
-        # from the current line, assuming it isn't empty
-        s/^\t*// if($_ !~ /^((\s*)|(\t*))*$/);
-        s/^\s*// if($_ !~ /^((\s*)|(\t*))*$/);
-
-        # tracing mode
-        print $logfile "Line $lineCounter\t removing leading spaces\n" if($tracingMode);
-    } else {
-        # otherwise check to see if we've reached the main
-        # part of the document
-        if(m/^\s*\\begin\{document\}/ and !$inFileContents and !$inverbatim) {
-            $inpreamble = 0;
-
-            # tracing mode
-            print $logfile "Line $lineCounter\t \\begin{document} found, switching indentation searches on. \n" if($tracingMode);
-        } else {
-            # tracing mode
-            if($inpreamble) {
-                print $logfile "Line $lineCounter\t still in PREAMBLE, leaving exisiting leading space (see indentPreamble)\n" if($tracingMode);
-            } elsif($inverbatim) {
-                print $logfile "Line $lineCounter\t in VERBATIM-LIKE environment, leaving exisiting leading space\n" if($tracingMode);
-            } elsif($inIndentBlock) {
-                print $logfile "Line $lineCounter\t in NO INDENT BLOCK, leaving exisiting leading space\n" if($tracingMode);
-            }
-        }
-    }
-
-    # \END{ENVIRONMENTS}, or CLOSING } or CLOSING ]
-    # \END{ENVIRONMENTS}, or CLOSING } or CLOSING ]
-    # \END{ENVIRONMENTS}, or CLOSING } or CLOSING ]
-
-    # check to see if we're ending a filecontents environment
-    if( $_ =~ m/^\s*\\end\{(.*?)\}/ and  $fileContentsEnvironments{$1} and $inFileContents){
-        print $logfile "Line $lineCounter\t Found END of filecontents environment (see fileContentsEnvironments)\n" if($tracingMode);
-        $inFileContents = 0;
-    }
-
-    # set the delimiters switch
-    $delimiters = @masterIndentationArrayOfHashes?$masterIndentationArrayOfHashes[-1]{alignmentDelimiters}:0;
-
-    if($inverbatim){
-        print $logfile "Line $lineCounter\t $masterSettings{logFilePreferences}{traceModeDecreaseIndent} PHASE 1: in VERBATIM-LIKE environment, looking for $masterIndentationArrayOfHashes[-1]{end} \n" if($tracingMode);
-    } elsif($inIndentBlock) {
-        print $logfile "Line $lineCounter\t in NO INDENT BLOCK, doing nothing\n" if($tracingMode);
-    } elsif($delimiters) {
-        print $logfile "Line $lineCounter\t $masterSettings{logFilePreferences}{traceModeDecreaseIndent} PHASE 1: in ALIGNMENT BLOCK environment, looking for $masterIndentationArrayOfHashes[-1]{end}\n" if($tracingMode);
-    } elsif($inpreamble and !$inFileContents) {
-        print $logfile "Line $lineCounter\t In preamble, looking for \\begin{document}\n" if($tracingMode);
-    } elsif($inpreamble and $inFileContents) {
-        print $logfile "Line $lineCounter\t In preamble, in filecontents environment\n" if($tracingMode);
-    } else {
-        print $logfile "Line $lineCounter\t $masterSettings{logFilePreferences}{traceModeDecreaseIndent} PHASE 1: looking for reasons to DECREASE indentation of CURRENT line \n" if($tracingMode);
-    }
-
-    # check to see if we have \end{something} or \]
-    &at_end_of_env_or_eq() unless ($inpreamble or $inIndentBlock);
-
-    # check to see if we have %* \end{something} for alignment blocks
-    # outside of environments
-    &end_command_with_alignment();
-
-    # check to see if we're at the end of a noindent
-    # block %\end{noindent}
-    &at_end_noindent();
-
-    # only check for unmatched braces if we're not in
-    # a verbatim-like environment or in the preamble or in a
-    # noIndentBlock or in a delimiter block
-    if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters)) {
-        # The check for closing } and ] relies on counting, so
-        # we have to remove trailing comments so that any {, }, [, ]
-        # that are found after % are not counted
-        #
-        # note that these lines are NOT in @lines, so we
-        # have to store the $trailingcomments to put
-        # back on after the counting
-        #
-        # note the use of (?<!\\)% so that we don't match \%
-        if ( $_=~ m/(?<!\\)%.*/) {
-            s/((?<!\\)%.*)//;
-            $trailingcomments=$1;
-
-            # tracing mode
-            print $logfile "Line $lineCounter\t Removed trailing comments to count braces and brackets: $1\n" if($tracingMode);
-        }
-
-        # check to see if we're at the end of a \parbox, \marginpar
-        # or other split-across-lines command and check that
-        # we're not starting another command that has split braces (nesting)
-        &end_command_or_key_unmatched_braces();
-
-        # check to see if we're at the end of a command that splits
-        # [ ] across lines
-        &end_command_or_key_unmatched_brackets();
-
-        # check for a heading such as \chapter, \section, etc
-        &indent_heading();
-
-        # check for \item
-        &indent_item();
-
-        # check for \else or \fi
-        &indent_if_else_fi();
-
-        # add the trailing comments back to the end of the line
-        if(scalar($trailingcomments)) {
-            # some line break magic, http://stackoverflow.com/questions/881779/neatest-way-to-remove-linebreaks-in-perl
-            s/\R//;
-            $_ = $_ . $trailingcomments."\n" ;
-
-            # tracing mode
-            print $logfile "Line $lineCounter\t counting braces/brackets complete\n" if($tracingMode);
-            print $logfile "Line $lineCounter\t Adding trailing comments back on: $trailingcomments\n" if($tracingMode);
-
-            # empty the trailingcomments
-            $trailingcomments='';
-
-        }
-        # remove trailing whitespace
-        if ($removeTrailingWhitespace) {
-            print $logfile "Line $lineCounter\t removing trailing whitespace (see removeTrailingWhitespace)\n" if ($tracingMode);
-            s/\s+$/\n/;
-        }
-    }
-
-    # ADD CURRENT LEVEL OF INDENTATION
-    # ADD CURRENT LEVEL OF INDENTATION
-    # ADD CURRENT LEVEL OF INDENTATION
-    # (unless we're in a delimiter-aligned block)
-    if(!$delimiters) {
-        # make sure we're not in a verbatim block or in the preamble
-        if($inverbatim or $inpreamble or $inIndentBlock) {
-           # just push the current line as is
-           push(@lines,$_);
-        } else {
-            # add current value of indentation to the current line
-            # and output it
-            # unless this would only create trailing whitespace and the
-            # corresponding option is set
-            unless ($_ =~ m/^$/ and $removeTrailingWhitespace){
-                $_ = &current_indentation().$_;
-            }
-            push(@lines,$_);
-            # tracing mode
-            print $logfile "Line $lineCounter\t $masterSettings{logFilePreferences}{traceModeAddCurrentIndent} PHASE 2: Adding current level of indentation: ",&current_indentation_names(),"\n" if($tracingMode);
-        }
-    } else {
-        # output to @block (within masterIndentationArrayOfHashes) if we're in a delimiter block
-        push(@{$masterIndentationArrayOfHashes[-1]{block}},$_);
-
-        # tracing mode
-        print $logfile "Line $lineCounter\t In delimeter block ($masterIndentationArrayOfHashes[-1]{name}), waiting for block formatting\n" if($tracingMode);
-    }
-
-    # \BEGIN{ENVIRONMENT} or OPEN { or OPEN [
-    # \BEGIN{ENVIRONMENT} or OPEN { or OPEN [
-    # \BEGIN{ENVIRONMENT} or OPEN { or OPEN [
-
-    # check to see if we're beginning a filecontents environment
-    if( ($_ =~ m/^\s*\\begin\{(.*?)\}/ and  $fileContentsEnvironments{$1} and !$inverbatim)){
-        print $logfile "Line $lineCounter\t Found filecontents environment (see fileContentsEnvironments)\n" if($tracingMode);
-        $inFileContents = 1;
-    }
-
-    # only check for new environments or commands if we're
-    # not in a verbatim-like environment or in the preamble
-    # or in a noIndentBlock, or delimiter block
-    if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters)) {
-
-        print $logfile "Line $lineCounter\t $masterSettings{logFilePreferences}{traceModeIncreaseIndent} PHASE 3: looking for reasons to INCREASE indentation of SUBSEQUENT lines \n" if($tracingMode);
-
-        # check if we are in a
-        #   % \begin{noindent}
-        # block; this is similar to a verbatim block, the user
-        # may not want some blocks of code to be touched
-        #
-        # IMPORTANT: this needs to go before the trailing comments
-        # are removed!
-        &at_beg_noindent();
-
-        # check for
-        #   %* \begin{tabular}
-        # which might be used to align blocks that contain delimeters that
-        # are NOT contained in an alignment block in the usual way, e.g
-        #   \matrix{
-        #       %* \begin{tabular}
-        #           1 & 2 \\
-        #           3 & 4 \\
-        #       %* \end{tabular}
-        #           }
-        &begin_command_with_alignment();
-        if(@masterIndentationArrayOfHashes){
-               $delimiters = $masterIndentationArrayOfHashes[-1]{alignmentDelimiters}||0;
-             }
-
-        # remove trailing comments so that any {, }, [, ]
-        # that are found after % are not counted
-        #
-        # note that these lines are already in @lines, so we
-        # can remove the trailing comments WITHOUT having
-        # to put them back in
-        #
-        # Note that this won't match \%
-        s/(?<!\\)%.*// if( $_=~ m/(?<!\\)%.*/);
-
-        # tracing mode
-        print $logfile "Line $lineCounter\t Removing trailing comments for brace count (line is already stored)\n" if($tracingMode);
-
-        # check to see if we have \begin{something} or \[
-        &at_beg_of_env_or_eq() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-        if(@masterIndentationArrayOfHashes){
-               $delimiters = $masterIndentationArrayOfHashes[-1]{alignmentDelimiters}||0;
-             }
-
-        # check to see if we have \parbox, \marginpar, or
-        # something similar that might split braces {} across lines,
-        # specified in %checkunmatched hash table
-        &start_command_or_key_unmatched_braces() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-
-        # check for an else statement (braces, not \else)
-        &check_for_else() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-
-        # check for a command that splits [] across lines
-        &start_command_or_key_unmatched_brackets();
-
-        # check for a heading
-        &indent_after_heading() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-
-        # check for \item
-        &indent_after_item() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-
-        # check for \if or \else command
-        &indent_after_if_else_fi() if(!($inverbatim or $inpreamble or $inIndentBlock or $delimiters));
-
-        # tracing mode
-        if($tracingMode){
-            if(scalar(@masterIndentationArrayOfHashes)){
-                print $logfile "Line $lineCounter\t Indentation array: ",&current_indentation_names(),"\n";
-              } else {
-                print $logfile "Line $lineCounter\t Indentation array empty\n";
-              }
-        }
-    }
-}
-
-# close the main file
-close(MAINFILE);
-
-# put line count information in the log file
-print $logfile "Line Count of $fileName: ",scalar(@mainfile),"\n";
-print $logfile "Line Count of indented $fileName: ",scalar(@lines);
-if(scalar(@mainfile) != scalar(@lines))
-{
-  print $logfile <<ENDQUOTE
-WARNING: \t line count of original file and indented file does
-\t not match- consider reverting to a back up, see $backupExtension;
-ENDQUOTE
-;
-} else {
-    print $logfile "\n\nLine counts of original file and indented file match.\n";
-}
-
-# output the formatted lines to the terminal
-print @lines if(!$silentMode);
-
-# if -w is active then output to $ARGV[0]
-if($overwrite) {
-    open(OUTPUTFILE,">",$fileName);
-    print OUTPUTFILE @lines;
-    close(OUTPUTFILE);
-}
-
-# if -o is active then output to $ARGV[1]
-if($outputToFile) {
-    open(OUTPUTFILE,">",$ARGV[1]);
-    print OUTPUTFILE @lines;
-    close(OUTPUTFILE);
-    print $logfile "Output from indentation written to $ARGV[1].\n";
-}
-
-# final line of the logfil
-print $logfile "\n",$masterSettings{logFilePreferences}{endLogFileWith};
-
-# close the log file
-close($logfile);
-
+my $document = LatexIndent::Document->new(name=>"masterdocument",fileName=>$ARGV[0],switches=>\%switches);
+$document->latexindent;
 exit(0);
-
-sub indent_if_else_fi{
-    # PURPOSE: set indentation of line that contains \else, \fi  command
-    #
-    #
-
-    # @masterIndentationArrayOfHashes could be empty -- if so, exit
-    return unless @masterIndentationArrayOfHashes;
-    return unless $constructIfElseFi{$masterIndentationArrayOfHashes[-1]{name}};
-
-    # look for \fi
-    if( $_ =~ m/^\s*\\fi/) {
-        # tracing mode
-        print $logfile "Line $lineCounter\t \\fi command found, matching: \\",$masterIndentationArrayOfHashes[-1]{name}, "\n" if($tracingMode);
-        &decrease_indent($masterIndentationArrayOfHashes[-1]{name});
-    } 
-    # look for \else or \or
-    elsif( $_ =~ m/^\s*\\else/ or $_ =~ m/^\s*\\or/ ) {
-        # tracing mode
-        print $logfile "Line $lineCounter\t \\else command found, matching: \\",$masterIndentationArrayOfHashes[-1]{name}, "\n" if($tracingMode);
-        print $logfile "Line $lineCounter\t decreasing indent, still looking for \\fi to match \\",&current_indentation_names(), "\n" if($tracingMode);
-
-        # finding an \else or \or command removes the *indentation*, but not the entry from the master hash
-        $masterIndentationArrayOfHashes[-1]{indent}="";
-    }
-}
-
-sub indent_after_if_else_fi{
-    # PURPOSE: set indentation *after* \if construct such as
-    #
-    #               \ifnum\x=2
-    #                   <stuff>
-    #                   <stuff>
-    #               \else
-    #                   <stuff>
-    #                   <stuff>
-    #               \fi
-    #
-    #   How to read /^\s*\\(if.*?)(\s|\\|\#)
-    #
-    #       ^\s*        begins with multiple spaces (possibly none)
-    #       \\(if.*?)(\s|\\|\#)   matches \if... up to either a
-    #                             space, a \, or a #
-    #   Note: this won't match \if.*\fi
-    if( $_ =~ m/^\s*\\(if.*?)(\s|\\|\#)/ and $_ !~ m/^\s*\\(if.*?\\fi)/ and $constructIfElseFi{$1}) {
-        # tracing mode
-        print $logfile "Line $lineCounter\t ifelsefi construct found: $1 \n" if($tracingMode);
-        &increase_indent({name=>$1,type=>"ifelseif"});
-    } elsif(@masterIndentationArrayOfHashes) {
-        if( ($_ =~ m/^\s*\\else/ or $_ =~ m/^\s*\\or/ ) and $constructIfElseFi{$masterIndentationArrayOfHashes[-1]{name}}) {
-            # tracing mode
-            print $logfile "Line $lineCounter\t setting indent *after* \\else or \\or command found for $masterIndentationArrayOfHashes[-1]{name} \n" if($tracingMode);
-
-            # recover the indentation to be implemented *after* the \else or \or
-            $masterIndentationArrayOfHashes[-1]{indent}=$indentRules{$masterIndentationArrayOfHashes[-1]{name}}||$defaultIndent unless ($noAdditionalIndent{$masterIndentationArrayOfHashes[-1]{name}});
-        }
-  }
-}
-
-sub indent_item{
-    # PURPOSE: when considering environments that can contain items, such 
-    #          as enumerate, itemize, etc, this subroutine sets the indentation for the item *itself*
-
-    return unless(scalar(@masterIndentationArrayOfHashes)>1);
-    return unless $indentAfterItems{$masterIndentationArrayOfHashes[-2]{name}};
-
-    if( $_ =~ m/^\s*\\(.*?)(\[|\s)/ and $itemNames{$1}){
-        # tracing mode
-        print $logfile "Line $lineCounter\t $1 found within ",$masterIndentationArrayOfHashes[-1]{name}," environment (see indentAfterItems and itemNames)\n" if($tracingMode);
-        if($itemNames{$masterIndentationArrayOfHashes[-1]{name}}) {
-            print $logfile "Line $lineCounter\t $1 found - neutralizing indentation from previous ",$masterIndentationArrayOfHashes[-1]{name},"\n" if($tracingMode);
-            &decrease_indent($1);
-        }
-    }
-
-}
-
-sub indent_after_item{
-    # PURPOSE: Set the indentation *after* the item
-    #          This matches a line that begins with
-    #
-    #               \item
-    #               \item[
-    #               \myitem
-    #               \myitem[
-    #
-    #           or anything else specified in itemNames
-    #
-    return unless @masterIndentationArrayOfHashes;
-    return unless $indentAfterItems{$masterIndentationArrayOfHashes[-1]{name}};
-
-    if( $_ =~ m/^\s*\\(.*?)(\[|\s)/
-            and $itemNames{$1}) {
-        # tracing mode
-        print $logfile "Line $lineCounter\t $1 found within ",$masterIndentationArrayOfHashes[-1]{name}," environment (see indentAfterItems and itemNames)\n" if($tracingMode);
-        &increase_indent({name=>$1,type=>"item"});
-    }
-}
-
-sub begin_command_with_alignment{
-    # PURPOSE: This matches
-    #           %* \begin{tabular}
-    #          with any number of spaces (possibly none) between
-    #          the * and \begin{noindent}.
-    #
-    #          the comment symbol IS indended!
-    #
-    #          This is to align blocks that contain delimeters that
-    #          are NOT contained in an alignment block in the usual way, e.g
-    #             \matrix{
-    #                 %* \begin{tabular}
-    #                     1 & 2 \\
-    #                     3 & 4 \\
-    #                 %* \end{tabular}
-    #                     }
-
-    if( $_ =~ m/^\s*%\*\s*\\begin\{(.*?)\}/ and $lookForAlignDelims{$1}) {
-           # increase the indentation
-           &increase_indent({name=>$1,
-                             alignmentDelimiters=>1,
-                             type=>"environment",
-                             begin=>"\\begin{$1}",
-                             end=>"\\end{$1}"});
-           # tracing mode
-           print $logfile "Line $lineCounter\t Delimiter environment started: $1 (see lookForAlignDelims)\n" if($tracingMode);
-    }
-}
-
-sub end_command_with_alignment{
-    # PURPOSE: This matches
-    #           %* \end{tabular}
-    #          with any number of spaces (possibly none) between
-    #          the * and \end{tabular} (or any other name used from
-    #          lookFroAlignDelims)
-    #
-    #          Note: the comment symbol IS indended!
-    #
-    #          This is to align blocks that contain delimeters that
-    #          are NOT contained in an alignment block in the usual way, e.g
-    #             \matrix{
-    #                 %* \begin{tabular}
-    #                     1 & 2 \\
-    #                     3 & 4 \\
-    #                 %* \end{tabular}
-    #                     }
-    return unless @masterIndentationArrayOfHashes;
-    return unless $masterIndentationArrayOfHashes[-1]{alignmentDelimiters};
-
-    if( $_ =~ m/^\s*%\*\s*\\end\{(.*?)\}/ and $lookForAlignDelims{$1}) {
-        # same subroutine used at the end of regular tabular, align, etc
-        # environments
-        if($delimiters) {
-            &print_aligned_block();
-            &decrease_indent($1);
-        } else {
-            # tracing mode
-            print $logfile "Line $lineCounter\t FYI: did you mean to start a delimiter block on a previous line? \n" if($tracingMode);
-            print $logfile "Line $lineCounter\t      perhaps using %* \\begin{$1}\n" if($tracingMode);
-        }
-    }
-}
-
-sub indent_heading{
-    # PURPOSE: This matches
-    #           \part
-    #           \chapter
-    #           \section
-    #           \subsection
-    #           \subsubsection
-    #           \paragraph
-    #           \subparagraph
-    #
-    #           and anything else listed in indentAfterHeadings
-    #
-    #           This subroutine specifies the indentation for the
-    #           heading itself, i.e the line that has \chapter, \section etc
-    if( $_ =~ m/^\s*\\(.*?)(\[|{)/ and $indentAfterHeadings{$1}){
-       # tracing mode
-       print $logfile "Line $lineCounter\t Heading found: $1 \n" if($tracingMode);
-
-       # get the heading settings, it's a hash within a hash
-       my %currentHeading = %{$indentAfterHeadings{$1}};
-
-       # $previousHeadingLevel: scalar that stores which heading
-       # we are under: \part, \chapter, etc
-       my $previousHeadingLevel=0;         
-
-       # form an array of the headings available
-       my @headingStore=();
-       foreach my $env (@masterIndentationArrayOfHashes){
-           if($env->{type} eq 'heading'){
-               push(@headingStore,$env->{name});
-               # update heading level
-               $previousHeadingLevel= $env->{headinglevel};
-             }
-         }
-
-       # if current heading level < old heading level,
-       if($currentHeading{level}<$previousHeadingLevel) {
-            # decrease indentation, but only if
-            # specified in indentHeadings. Note that this check
-            # needs to be done here- decrease_indent won't
-            # check a nested hash
-
-            if(scalar(@headingStore)) {
-               while($currentHeading{level}<$previousHeadingLevel and scalar(@headingStore)) {
-                    my $higherHeadingName = pop(@headingStore);
-                    my %higherLevelHeading = %{$indentAfterHeadings{$higherHeadingName}};
-
-                    # tracing mode
-                    print $logfile "Line $lineCounter\t stepping UP heading level from $higherHeadingName \n" if($tracingMode);
-
-                    &decrease_indent($higherHeadingName) if($higherLevelHeading{indent});
-                    $previousHeadingLevel=$higherLevelHeading{level};
-               }
-               # put the heading name back in to storage
-               push(@headingStore,$1);
-            }
-       } elsif($currentHeading{level}==$previousHeadingLevel) {
-            if(scalar(@headingStore)) {
-                 my $higherHeadingName = pop(@headingStore);
-                 my %higherLevelHeading = %{$indentAfterHeadings{$higherHeadingName}};
-                 &decrease_indent($higherHeadingName) if($higherLevelHeading{indent});
-            }
-       } 
-    }
-}
-
-sub indent_after_heading{
-    # PURPOSE: This matches
-    #           \part
-    #           \chapter
-    #           \section
-    #           \subsection
-    #           \subsubsection
-    #           \paragraph
-    #           \subparagraph
-    #
-    #           and anything else listed in indentAfterHeadings
-    #
-    #           This subroutine is specifies the indentation for
-    #           the text AFTER the heading, i.e the body of conent
-    #           in each \chapter, \section, etc
-    if( $_ =~ m/^\s*\\(.*?)(\[|{)/ and $indentAfterHeadings{$1}) {
-       # get the heading settings- it's a hash within a hash
-       my %currentHeading = %{$indentAfterHeadings{$1}};
-
-       &increase_indent({name=>$1,type=>"heading",headinglevel=>$currentHeading{level}}) if($currentHeading{indent});
-    }
-}
-
-sub at_end_noindent{
-    # PURPOSE: This matches
-    #           % \end{noindent}
-    #          with any number of spaces (possibly none) between
-    #          the comment and \end{noindent}.
-    #
-    #          the comment symbol IS indended!
-    #
-    #          This is for blocks of code that the user wants
-    #          to leave untouched- similar to verbatim blocks
-
-    if( $_ =~ m/^%\s*\\end\{(.*?)\}/ and $noIndentBlock{$1}) {
-            $inIndentBlock=0;
-            # tracing mode
-            print $logfile "Line $lineCounter\t % \\end{no indent block} found, switching inIndentBlock OFF \n" if($tracingMode);
-    }
-}
-
-sub at_beg_noindent{
-    # PURPOSE: This matches
-    #           % \begin{noindent}
-    #          with any number of spaces (possibly none) between
-    #          the comment and \begin{noindent}.
-    #
-    #          the comment symbol IS indended!
-    #
-    #          This is for blocks of code that the user wants
-    #          to leave untouched- similar to verbatim blocks
-
-    if( $_ =~ m/^%\s*\\begin\{(.*?)\}/ and $noIndentBlock{$1}) {
-           $inIndentBlock = 1;
-           # tracing mode
-           print $logfile "Line $lineCounter\t % \\begin{no indent block} found, switching inIndentBlock ON \n" if($tracingMode);
-    }
-}
-
-sub start_command_or_key_unmatched_brackets{
-    # PURPOSE: This matches
-    #              \pgfplotstablecreatecol[...
-    #
-    #              or any other command/key that has brackets [ ]
-    #              split across lines specified in the
-    #              hash tables, %checkunmatchedbracket
-    #
-    # How to read: ^\s*(\\)?(.*?)(\[\s*)
-    #
-    #       ^       line begins with
-    #       \s*     any (or no)spaces
-    #       (\\)?   matches a \ backslash but not necessarily
-    #       (.*?)   non-greedy character match and store the result
-    #       ((?<!\\)\[\s*) match [ possibly leading with spaces
-    #                      but it WON'T match \[
-
-    if ($_ =~ m/^\s*(\\)?(.*?)(\s*(?<!\\)\[)/
-        and ($checkunmatchedbracket{$2} or $alwaysLookforSplitBrackets)) {
-            # store the command name, because $2
-            # will not exist after the next match
-            my $commandname = $2;
-            my $matchedBRACKETS=0;
-
-            # match [ but don't match \[
-            $matchedBRACKETS++ while ($_ =~ /(?<!\\)\[/g);
-            # match ] but don't match \]
-            $matchedBRACKETS-- while ($_ =~ /(?<!\\)\]/g);
-
-            # set the indentation
-            if($matchedBRACKETS != 0 ) {
-                  # tracing mode
-                  print $logfile "Line $lineCounter\t Found opening BRACKET [ $commandname\n" if($tracingMode);
-
-                  &increase_indent({name=>$commandname,matchedBRACKETS=>$matchedBRACKETS,type=>'splitBrackets'});
-            }
-        }
-}
-
-sub end_command_or_key_unmatched_brackets{
-    # PURPOSE:  Check for the closing BRACKET of a command that
-    #           splits its BRACKETS across lines, such as
-    #
-    #               \pgfplotstablecreatecol[ ...
-    #
-    #           It works by checking if we have any entries
-    #           in the array @masterIndentationArrayOfHashes, and making
-    #           sure that we're not starting another command/key
-    #           that has split BRACKETS (nesting).
-    #
-    #           It also checks that the line is not commented.
-    #
-    #           We count the number of [ and ADD to the counter
-    #                                  ] and SUBTRACT to the counter
-    return unless @masterIndentationArrayOfHashes;
-    return unless ($masterIndentationArrayOfHashes[-1]{type} eq 'splitBrackets');
-    print $logfile "Line $lineCounter\t Searching for closing BRACKET ] $masterIndentationArrayOfHashes[-1]{name}\n" if($tracingMode);
-
-    if(!($_ =~ m/^\s*(\\)?(.*?)(\s*\[)/
-        and ($checkunmatchedbracket{$2} or $alwaysLookforSplitBrackets))
-        and $_ !~ m/^\s*%/) {
-
-       # get the details of the most recent command name
-       my $commandname =  $masterIndentationArrayOfHashes[-1]{name};
-       my $matchedBRACKETS = $masterIndentationArrayOfHashes[-1]{matchedBRACKETS};
-
-       # match [ but don't match \[
-       $matchedBRACKETS++ while ($_ =~ m/(?<!\\)\[/g);
-
-       # match ] but don't match \]
-       $matchedBRACKETS-- while ($_ =~ m/(?<!\\)\]/g);
-
-       # if we've matched up the BRACKETS then
-       # we can decrease the indent by 1 level
-       if($matchedBRACKETS == 0){
-            # tracing mode
-            print $logfile "Line $lineCounter\t Found closing BRACKET ] $commandname\n" if($tracingMode);
-
-            # decrease the indentation (if appropriate)
-            &decrease_indent($commandname);
-       } else {
-           # otherwise we need to enter the new value
-           # of $matchedBRACKETS and the value of $command
-           # back into storage
-           $masterIndentationArrayOfHashes[-1]{matchedBRACKETS} = $matchedBRACKETS;
-
-           # tracing mode
-           print $logfile "Line $lineCounter\t Searching for closing BRACKET ] $commandname\n" if($tracingMode);
-       }
-     }
-}
-
-sub start_command_or_key_unmatched_braces{
-    # PURPOSE: This matches
-    #              \parbox{...
-    #              \parbox[..]..{
-    #              empty header/.style={
-    #              \foreach \something
-    #              etc
-    #
-    #              or any other command/key that has BRACES
-    #              split across lines specified in the
-    #              hash tables, %checkunmatched, %checkunmatchedELSE
-    #
-    # How to read: ^\s*(\\)?(.*?)(\[|{|\s)
-    #
-    #       ^                  line begins with
-    #       \s*                any (or no) spaces
-    #       (\\)?              matches a \ backslash but not necessarily
-    #       (.*?)              non-greedy character match and store the result
-    #       (\[|}|=|(\s*\\))   either [ or { or = or space \
-
-    if ($_ =~ m/^\s*(\\)?(.*?)(\[|{|=|(\s*\\))/
-            and ($checkunmatched{$2} or $checkunmatchedELSE{$2}
-                 or $alwaysLookforSplitBraces)
-        ) {
-            # store the command name, because $2
-            # will not exist after the next match
-            my $commandname = $2;
-            my $matchedbraces=0;
-
-            # by default, don't look for an else construct
-            my $lookforelse=$checkunmatchedELSE{$2}||0;
-
-            # match { but don't match \{
-            $matchedbraces++ while ($_ =~ /(?<!\\){/g);
-
-            # match } but don't match \}
-            $matchedbraces-- while ($_ =~ /(?<!\\)}/g);
-
-            # tracing mode
-            print $logfile "Line $lineCounter\t matchedbraces = $matchedbraces\n" if($tracingMode);
-
-            # set the indentation
-            if($matchedbraces > 0 ) {
-                  # tracing mode
-                  print $logfile "Line $lineCounter\t Found opening BRACE { $commandname\n" if($tracingMode);
-
-                  &increase_indent({name=>$commandname,
-                                      matchedbraces=>$matchedbraces,
-                                      lookforelse=>$lookforelse,
-                                      countzeros=>0,
-                                      type=>"splitbraces"});
-            } elsif($matchedbraces<0) {
-                # if $matchedbraces < 0 then we must be matching
-                # braces from a previous split-braces command
-
-                # keep matching { OR }, and don't match \{ or \}
-                while ($_ =~ m/(((?<!\\){)|((?<!\\)}))/g) {
-
-                     # store the match, either { or }
-                     my $braceType = $1;
-
-                     # exit the loop if @masterIndentationArrayOfHashes[-1] is empty
-                     last if(!@masterIndentationArrayOfHashes);
-
-                     # exit the loop if we're not looking for split braces
-                     last if($masterIndentationArrayOfHashes[-1]{type} ne 'splitbraces');
-
-                     # get the details of the most recent command name
-                     $commandname =  $masterIndentationArrayOfHashes[-1]{name};
-                     $matchedbraces = $masterIndentationArrayOfHashes[-1]{'matchedbraces'};
-                     my $countzeros = $masterIndentationArrayOfHashes[-1]{'countzeros'};
-                     $lookforelse= $masterIndentationArrayOfHashes[-1]{'lookforelse'};
-
-                     $matchedbraces++ if($1 eq "{");
-                     $matchedbraces-- if($1 eq "}");
-
-                     # update the matched braces count
-                     $masterIndentationArrayOfHashes[-1]{matchedbraces} = $matchedbraces;
-
-                     # if we've matched up the braces then
-                     # we can decrease the indent by 1 level
-                     if($matchedbraces == 0) {
-                          $countzeros++ if $lookforelse;
-
-                          # tracing mode
-                          print $logfile "Line $lineCounter\t Found closing BRACE } $1\n" if($tracingMode);
-
-                          # decrease the indentation (if appropriate)
-                          &decrease_indent($commandname);
-
-                         if($countzeros==1) {
-                              $masterIndentationArrayOfHashes[-1]{'matchedbraces'} = $matchedbraces;
-                              $masterIndentationArrayOfHashes[-1]{'countzeros'} = $countzeros;
-                              $masterIndentationArrayOfHashes[-1]{'lookforelse'} = $lookforelse;
-                         }
-                     } 
-                }
-            }
-        }
-}
-
-sub end_command_or_key_unmatched_braces{
-    # PURPOSE:  Check for the closing BRACE of a command that
-    #           splits its BRACES across lines, such as
-    #
-    #               \parbox{ ...
-    #
-    #           or one of the tikz keys, such as
-    #
-    #              empty header/.style={
-    #
-    #           It works by checking if we have any entries
-    #           in the array @masterIndentationArrayOfHashes, and making
-    #           sure that we're not starting another command/key
-    #           that has split BRACES (nesting).
-    #
-    #           It also checks that the line is not commented.
-    #
-    #           We count the number of { and ADD to the counter
-    #                                  } and SUBTRACT to the counter
-    return unless @masterIndentationArrayOfHashes;
-    return unless ($masterIndentationArrayOfHashes[-1]{type} eq 'splitbraces');
-    print $logfile "Line $lineCounter\t Searching for closing BRACE } $masterIndentationArrayOfHashes[-1]{name}\n" if($tracingMode);
-
-    if(!($_ =~ m/^\s*(\\)?(.*?)(\[|{|=|(\s*\\))/
-        and ($checkunmatched{$2} or $checkunmatchedELSE{$2} or $alwaysLookforSplitBraces))
-        and $_ !~ m/^\s*%/
-       ) {
-       # keep matching { OR }, and don't match \{ or \}
-       while ($_ =~ m/(((?<!\\){)|((?<!\\)}))/g) {
-            # store the match, either { or }
-            my $braceType = $1;
-
-            # exit the loop if @masterIndentationArrayOfHashes[-1] is empty
-            last if(!@masterIndentationArrayOfHashes);
-
-            # exit the loop if we're not looking for split braces
-            last if($masterIndentationArrayOfHashes[-1]{type} ne 'splitbraces');
-
-            # get the details of the most recent command name
-            my $commandname =  $masterIndentationArrayOfHashes[-1]{name};
-            my $matchedbraces = $masterIndentationArrayOfHashes[-1]{matchedbraces};
-            my $countzeros = $masterIndentationArrayOfHashes[-1]{countzeros};
-            my $lookforelse= $masterIndentationArrayOfHashes[-1]{lookforelse};
-
-            $matchedbraces++ if($1 eq "{");
-            $matchedbraces-- if($1 eq "}");
-
-            # update the matched braces count
-            $masterIndentationArrayOfHashes[-1]{matchedbraces} = $matchedbraces;
-
-            # if we've matched up the braces then
-            # we can decrease the indent by 1 level
-            if($matchedbraces == 0) {
-                 $countzeros++ if $lookforelse;
-
-                 # tracing mode
-                 print $logfile "Line $lineCounter\t Found closing BRACE } $commandname\n" if($tracingMode);
-
-                 # decrease the indentation (if appropriate)
-                 &decrease_indent($commandname);
-
-                if($countzeros==1){
-                    $masterIndentationArrayOfHashes[-1]{'matchedbraces'} = $matchedbraces;
-                    $masterIndentationArrayOfHashes[-1]{'countzeros'} = $countzeros;
-                    $masterIndentationArrayOfHashes[-1]{'lookforelse'} = $lookforelse;
-                }
-            } 
-            
-            if(@masterIndentationArrayOfHashes){
-                if($masterIndentationArrayOfHashes[-1]{'type'} eq 'splitbraces'){
-                   # tracing mode
-                   print $logfile "Line $lineCounter\t Searching for closing BRACE } $masterIndentationArrayOfHashes[-1]{name}\n" if($tracingMode);
-                }
-             }
-        }
-     }
-}
-
-sub check_for_else{
-    # PURPOSE: Check for an else clause
-    #
-    #          Some commands have the form
-    #
-    #               \mycommand{
-    #                   if this
-    #               }
-    #               {
-    #                   else this
-    #               }
-    #
-    #          so we need to look for the else bit, and set
-    #          the indentation appropriately.
-    #
-    #          We only perform this check if there's something
-    #          in the array @masterIndentationArrayOfHashes, and if
-    #          the line itself is not a command, or comment,
-    #          and if it begins with {
-
-    if(scalar(@masterIndentationArrayOfHashes)
-        and  !($_ =~ m/^\s*(\\)?(.*?)(\[|{|=)/
-                    and ($checkunmatched{$2} or $checkunmatchedELSE{$2}
-                         or $alwaysLookforSplitBraces))
-        and $_ =~ m/^\s*{/
-        and $_ !~ m/^\s*%/
-       ) {
-       # get the details of the most recent command name
-       my $matchedbraces = $masterIndentationArrayOfHashes[-1]{'matchedbraces'};
-       my $countzeros = $masterIndentationArrayOfHashes[-1]{'countzeros'};
-       my $lookforelse= $masterIndentationArrayOfHashes[-1]{'lookforelse'};
-
-       # increase indentation
-       if($lookforelse and $countzeros==1) {
-         #&increase_indent($commandname);
-       }
-
-       # put the array back together
-       $masterIndentationArrayOfHashes[-1]{'matchedbraces'} = $matchedbraces;
-       $masterIndentationArrayOfHashes[-1]{'countzeros'} = $countzeros;
-       $masterIndentationArrayOfHashes[-1]{'lookforelse'} = $lookforelse;
-    }
-}
-
-sub at_beg_of_env_or_eq{
-    # PURPOSE: Check if we're at the BEGINning of an environment
-    #          or at the BEGINning of a displayed equation \[
-    #
-    #          This subroutine checks for matches of the form
-    #
-    #               \begin{environmentname}
-    #          or
-    #               \[
-    #
-    #          It also checks to see if the current environment
-    #          should have alignment delimiters; if so, we need to turn
-    #          ON the $delimiter switch
-
-    # How to read
-    #  m/^\s*(\$)?\\begin{(.*?)}/
-    #
-    #   ^               beginning of a line
-    #   \s*             any white spaces (possibly none)
-    #   (\$)?           possibly a $ symbol, but not required
-    #   \\begin{(.*)?}  \begin{environmentname}
-    #
-    # How to read
-    #  m/^\s*()(\\\[)/
-    #
-    #  ^        beginning of a line
-    #  \s*      any white spaces (possibly none)
-    #  ()       empty just so that $1 and $2 are defined
-    #  (\\\[)   \[  there are lots of \ because both \ and [ need escaping
-    #  \\begin{\\?(.*?)}  \begin{something} where something could start
-    #                     with a backslash, e.g \my@env@ which can happen
-    #                     in a style or class file, for example
-
-    if( (   ( $_ =~ m/^\s*(\$)?\\begin\{\\?(.*?)\}/ and $_ !~ m/\\end\{$2\}/)
-         or ($_=~ m/^\s*()(\\\[)/ and $_ !~ m/\\\]/) )
-        and $_ !~ m/^\s*%/ ) {
-       # tracing mode
-       print $logfile "Line $lineCounter\t \\begin{environment} found: $2 \n" if($tracingMode);
-
-       # increase the indentation
-       &increase_indent({name=>$2,
-                         type=>"environment",
-                         begin=>"\\begin{$2}",
-                         end=>"\\end{$2}"});
-
-       # check for verbatim-like environments
-       if($verbatimEnvironments{$2}){
-           $inverbatim = 1;
-           # tracing mode
-           print $logfile "Line $lineCounter\t \\begin{verbatim-like} found, $2, switching ON verbatim \n" if($tracingMode);
-
-           # remove the key and value from %lookForAlignDelims hash
-           # to avoid any further confusion
-           if($lookForAlignDelims{$2}) {
-                print $logfile "WARNING\n\t Line $lineCounter\t $2 is in *both* lookForAlignDelims and verbatimEnvironments\n";
-                print $logfile "\t\t\t ignoring lookForAlignDelims and prioritizing verbatimEnvironments\n";
-                print $logfile "\t\t\t Note that you only get this message once per environment\n";
-                delete $lookForAlignDelims{$2};
-           }
-       }
-    }
-}
-
-sub at_end_of_env_or_eq{
-    # PURPOSE: Check if we're at the END of an environment
-    #          or at the END of a displayed equation \]
-    #
-    #          This subroutine checks for matches of the form
-    #
-    #               \end{environmentname}
-    #          or
-    #               \]
-    #
-    #          Note: environmentname can begin with a backslash
-    #                which might happen in a sty or cls file.
-    #
-    #          It also checks to see if the current environment
-    #          had alignment delimiters; if so, we need to turn
-    #          OFF the $delimiter switch
-
-    return unless @masterIndentationArrayOfHashes;
-    print $logfile "Line $lineCounter\t looking for \\end{$masterIndentationArrayOfHashes[-1]{name}} \n" if($tracingMode);
-
-    if( ($_ =~ m/^\s*\\end\{\\?(.*?)\}/ or $_=~ m/^(\\\])/) and $_ !~ m/\s*^%/) {
-       # check if we're at the end of a verbatim-like environment
-       if($verbatimEnvironments{$1}) {
-           $inverbatim = 0;
-            # tracing mode
-
-            print $logfile "Line $lineCounter\t \\end{verbatim-like} found: $1, switching off verbatim \n" if($tracingMode);
-            print $logfile "Line $lineCounter\t removing leading spaces \n" if($tracingMode);
-            #s/^\ *//;
-            s/^\t+// if($_ ne "");
-            s/^\s+// if($_ ne "");
-       }
-
-       # check if we're in an environment that is looking
-       # to indent after each \item
-       if(scalar(@masterIndentationArrayOfHashes) and $itemNames{$masterIndentationArrayOfHashes[-1]{name}}) {
-            &decrease_indent($masterIndentationArrayOfHashes[-1]{name});
-       }
-
-       # if we're at the end of an environment that receives no additional indent, log it, and move on
-       if($noAdditionalIndent{$1}){
-            print $logfile "Line $lineCounter\t \\end{$1} finished a no-additional-indent environment (see noAdditionalIndent)\n" if($tracingMode);
-       }
-
-       # some commands contain \end{environmentname}, which
-       # can cause a problem if \begin{environmentname} was not
-       # started previously; if @masterIndentationArrayOfHashes is empty,
-       # then we don't need to check for \end{environmentname}
-       if(@masterIndentationArrayOfHashes) {
-          # check to see if \end{environment} fits with most recent \begin{...}
-          my %previousEnvironment = %{$masterIndentationArrayOfHashes[-1]};
-
-          # check to see if we need to turn off alignment
-          # delimiters and output the current block
-          if($masterIndentationArrayOfHashes[-1]{alignmentDelimiters} and ($previousEnvironment{name} eq $1)) {
-               &print_aligned_block();
-          }
-
-          # tracing mode
-          print $logfile "Line $lineCounter\t \\end{environment} found: $1 \n" if($tracingMode and !$verbatimEnvironments{$1});
-
-          # check to see if \end{environment} fits with most recent \begin{...}
-          if($previousEnvironment{name} eq $1) {
-               # decrease the indentation (if appropriate)
-               print $logfile "Line $lineCounter\t removed $1 from Indentation array\n" if($tracingMode); 
-               &decrease_indent($1);
-          } else {
-              # otherwise put the environment name back on the stack
-              print $logfile "Line $lineCounter\t WARNING: \\end{$1} found on its own line, not matched to \\begin{$previousEnvironment{name}}\n" unless ($delimiters or $inverbatim or $inIndentBlock or $1 eq "\\\]");
-          }
-
-          # need a special check for \[ and \]
-          if($1 eq "\\\]") {
-               &decrease_indent($1);
-          }
-       }
-
-       # if we're at the end of the document, we remove all current
-       # indentation- this is especially prominent in examples that
-       # have headings, and the user has chosen to indentAfterHeadings
-       if($1 eq "document" and !$inFileContents and !$inpreamble and !$delimiters and !$inverbatim and !$inIndentBlock and @masterIndentationArrayOfHashes) {
-            @masterIndentationArrayOfHashes=();
-
-            # tracing mode
-            if($tracingMode) {
-                print $logfile "Line $lineCounter\t \\end{$1} found, emptying indentation array \n" unless ($delimiters or $inverbatim or $inIndentBlock or $1 eq "\\\]");
-            }
-       }
-    }
-}
-
-sub print_aligned_block{
-    # PURPOSE: this subroutine does a few things related
-    #          to printing blocks of code that contain
-    #          delimiters, such as align, tabular, etc
-    #
-    #          It does the following
-    #           - turns off delimiters switch
-    #           - processes the block
-    #           - deletes the block
-    $delimiters=0;
-
-    # tracing mode
-    print $logfile "Line $lineCounter\t Delimiter body FINISHED: $masterIndentationArrayOfHashes[-1]{name}\n" if($tracingMode);
-
-    # print the current FORMATTED block
-    my @block = &format_block(@{$masterIndentationArrayOfHashes[-1]{block}});
-    foreach $line (@block) {
-         # add the indentation and add the
-         # each line of the formatted block
-         # to the output
-         # unless this would only create trailing whitespace and the
-         # corresponding option is set
-         unless ($line =~ m/^$/ and $removeTrailingWhitespace) {
-             $line =&current_indentation().$line;
-         }
-         push(@lines,$line);
-    }
-}
-
-sub format_block{
-    #   PURPOSE: Format a delimited environment such as the
-    #            tabular or align environment that contains &
-    #
-    #   INPUT: @block               array containing unformatted block
-    #                               from, for example, align, or tabular
-    #   OUTPUT: @formattedblock     array containing FORMATTED block
-
-    # @block is the input
-    my @block=@_;
-
-    # tracing mode
-    print $logfile "\t\tFormatting alignment block: $masterIndentationArrayOfHashes[-1]{name}\n" if($tracingMode);
-
-    # step the line counter back to the beginning of the block-
-    # it will be increased back to the end of the block in the
-    # loop later on:  foreach $row (@tmpblock)
-    $lineCounter -= scalar(@block);
-
-    # local array variables
-    my @formattedblock;
-    my @tmprow=();
-    my @tmpblock=();
-    my @maxmstringsize=();
-    my @ampersandCount=();
-
-    # local scalar variables
-    my $alignrowcounter=-1;
-    my $aligncolcounter=-1;
-    my $tmpstring;
-    my $row;
-    my $column;
-    my $maxmcolstrlength;
-    my $i;
-    my $j;
-    my $fmtstring;
-    my $linebreak;
-    my $maxNumberAmpersands = 0;
-    my $currentNumberAmpersands;
-    my $trailingcomments;
-
-    # local hash table
-    my %stringsize=();
-
-    # loop through the block and count & per line- store the biggest
-    # NOTE: this needs to be done in its own block so that
-    # we can know what the maximum number of & in the block is
-    foreach $row (@block) {
-       # delete trailing comments
-       $trailingcomments='';
-       if($row =~ m/((?<!\\)%.*$)/) {
-            $row =~ s/((?<!\\)%.*)/%TC/;
-            $trailingcomments=$1;
-       }
-
-       # reset temporary counter
-       $currentNumberAmpersands=0;
-
-       # count & in current row (exclude \&)
-       $currentNumberAmpersands++ while ($row =~ /(?<!\\)&/g);
-
-       # store the ampersand count for future
-       push(@ampersandCount,$currentNumberAmpersands);
-
-       # overwrite maximum count if the temp count is higher
-       $maxNumberAmpersands = $currentNumberAmpersands if($currentNumberAmpersands > $maxNumberAmpersands );
-
-       # put trailing comments back on
-       if($trailingcomments){
-            $row =~ s/%TC/$trailingcomments/;
-       }
-    }
-
-    # tracing mode
-    print $logfile "\t\tmaximum number of & in any row: $maxNumberAmpersands\n" if($tracingMode);
-
-    # loop through the lines in the @block
-    foreach $row (@block){
-        # get the ampersand count
-        $currentNumberAmpersands = shift(@ampersandCount);
-
-        # increment row counter
-        $alignrowcounter++;
-
-        # clear the $linebreak variable
-        $linebreak='';
-
-        # check for line break \\
-        # and don't mess with a line that doesn't have the maximum
-        # number of &
-        if($row =~ m/\\\\/ and $currentNumberAmpersands==$maxNumberAmpersands ) {
-          # remove \\ and all characters that follow
-          # and put it back in later, once the measurement
-          # has been done
-          $row =~ s/(\\\\.*)//;
-          $linebreak = $1;
-        }
-
-        if($currentNumberAmpersands==$maxNumberAmpersands) {
-
-            # remove trailing comments
-            $trailingcomments='';
-            if($row =~ m/((?<!\\)%.*$)/) {
-                 $row =~ s/((?<!\\)%.*)/%TC/;
-                 $trailingcomments=$1;
-            }
-
-            # separate the row at each &, but not at \&
-            @tmprow = split(/(?<!\\)&/,$row);
-
-            # reset column counter
-            $aligncolcounter=-1;
-
-            # loop through each column element
-            # removing leading and trailing space
-            foreach $column (@tmprow) {
-               # increment column counter
-               $aligncolcounter++;
-
-               # remove leading and trailing space from element
-    	       $column =~ s/^\s+//;
-               $column =~ s/\s+$//;
-
-               # assign string size to the array
-               $stringsize{$alignrowcounter.$aligncolcounter}=length($column);
-               if(length($column)==0){
-                 $column=" ";
-               }
-
-               # put the row back together
-               if ($aligncolcounter ==0){
-                 $tmpstring = $column;
-               } else {
-                 $tmpstring .= "&".$column;
-               }
-            }
-
-
-            # put $linebreak back on the string, now that
-            # the measurement has been done
-            $tmpstring .= $linebreak;
-
-            # put trailing comments back on
-            if($trailingcomments) {
-                 $tmpstring =~ s/%TC/$trailingcomments/;
-            }
-
-            push(@tmpblock,$tmpstring);
-        } else {
-               # if there are no & then use the
-               # NOFORMATTING token
-               # remove leading space
-    	       s/^\s+//;
-               push(@tmpblock,$row."NOFORMATTING");
-        }
-    }
-
-    # calculate the maximum string size of each column
-    for($j=0;$j<=$aligncolcounter;$j++) {
-        $maxmcolstrlength=0;
-        for($i=0; $i<=$alignrowcounter;$i++) {
-            # make sure the stringsize is defined
-            if(defined $stringsize{$i.$j}) {
-                if ($stringsize{$i.$j}>$maxmcolstrlength) {
-                    $maxmcolstrlength = $stringsize{$i.$j};
-                }
-            }
-        }
-        push(@maxmstringsize,$maxmcolstrlength);
-    }
-
-    # README: printf( formatting, expression)
-    #
-    #   formatting has the form %-50s & %-20s & %-19s
-    #   (the numbers have been made up for example)
-    #       the - symbols mean that each column should be left-aligned
-    #       the numbers represent how wide each column is
-    #       the s represents string
-    #       the & needs to be inserted
-
-    # join up the maximum string lengths using "s %-"
-    $fmtstring = join("s & %-",@maxmstringsize);
-
-    # add an s to the end, and a newline
-    $fmtstring .= "s ";
-
-    # add %- to the beginning
-    $fmtstring = "%-".$fmtstring;
-
-    # process the @tmpblock of aligned material
-    foreach $row (@tmpblock) {
-        $linebreak='';
-        # check for line break \\
-        if($row =~ m/\\\\/) {
-          # remove \\ and all characters that follow
-          # and put it back in later
-          $row =~ s/(\\\\.*$)//;
-          $linebreak = $1;
-        }
-
-        if($row =~ m/NOFORMATTING/) {
-            $row =~ s/NOFORMATTING//;
-            $tmpstring=$row;
-
-            # tracing mode
-            print $logfile "\t\tLine $lineCounter\t maximum number of & NOT found- not aligning delimiters \n" if($tracingMode);
-        } else {
-          # remove trailing comments
-          $trailingcomments='';
-          if($row =~ m/((?<!\\)%.*$)/) {
-               $row =~ s/((?<!\\)%.*)/%TC/;
-               $trailingcomments=$1;
-          }
-
-          $tmpstring = sprintf($fmtstring,split(/(?<!\\)&/,$row)).$linebreak."\n";
-
-          # remove space before \\ if specified in alignDoubleBackSlash
-          if($masterIndentationArrayOfHashes[-1]{alignDoubleBackSlash}==0){
-                print $logfile "\t\tLine $lineCounter\t removing space before \\\\ (see $masterIndentationArrayOfHashes[-1]{name} alignDoubleBackSlash)\n" if($tracingMode);
-                $tmpstring =~ s/\s*\\\\/\\\\/;
-                # some users may like to put a number of spaces before \\
-                if($masterIndentationArrayOfHashes[-1]{spacesBeforeDoubleBackSlash}){
-                    my $spaceString;
-                    for($j=1;$j<=$masterIndentationArrayOfHashes[-1]{spacesBeforeDoubleBackSlash};$j++) {
-                        $spaceString .= ' ';
-                    }
-                    print $logfile "\t\tLine $lineCounter\t adding $masterIndentationArrayOfHashes[-1]{spacesBeforeDoubleBackSlash} ",$masterIndentationArrayOfHashes[-1]{spacesBeforeDoubleBackSlash}>1?"spaces":"space"," before \\\\ (see $masterIndentationArrayOfHashes[-1]{name} spacesBeforeDoubleBackSlash)\n" if($tracingMode);
-                    $tmpstring =~ s/\\\\/$spaceString\\\\/;
-                }
-          }
-
-          # put trailing comments back on
-          if($trailingcomments) {
-               $tmpstring =~ s/%TC/$trailingcomments/;
-          }
-
-          # tracing mode
-          print $logfile "\t\tLine $lineCounter\t Found maximum number of & so aligning delimiters\n" if($tracingMode);
-        }
-
-        # remove trailing whitespace
-        if ($removeTrailingWhitespace) {
-            print $logfile "\t\tLine $lineCounter\t removing trailing whitespace from delimiter aligned line\n" if ($tracingMode);
-            $tmpstring =~ s/\s+$/\n/;
-        }
-
-        push(@formattedblock,$tmpstring);
-
-        # increase the line counter
-        $lineCounter++;
-    }
-
-    # return the formatted block
-	@formattedblock;
-}
-
-sub increase_indent{
-       # PURPOSE: Adjust the indentation
-       #          of the current environment, command, etc;
-       #          check that it's not an environment
-       #          that doesn't want indentation.
-
-       my %infoHash = %{pop(@_)};
-       my $command = $infoHash{name};
-
-       # check for conflicting hash keys
-       &check_conflicting_keys($command);
-
-       # quick check for verbatim Environment
-       if($inverbatim){
-            print $logfile "Line $lineCounter\t currently inverbatim environment, not increasing indentation\n" if($tracingMode);
-            return;
-       }
-
-       if($indentRules{$command}) {
-          # tracing mode
-          print $logfile "Line $lineCounter\t increasing indent using rule for $command (see indentRules)\n" if($tracingMode);
-       } else {
-          # default indentation
-          if(!($noAdditionalIndent{$command} or $verbatimEnvironments{$command})) {
-            # tracing mode
-            print $logfile "Line $lineCounter\t increasing indent using defaultIndent\n" if($tracingMode);
-          } elsif($noAdditionalIndent{$command})  {
-            # tracing mode
-            print $logfile "Line $lineCounter\t no additional indent added for $command (see noAdditionalIndent)\n" if($tracingMode);
-          }
-       }
-
-       # add to the master array of hashes
-       push(@masterIndentationArrayOfHashes,\%infoHash);
-
-       # handle the keys slightly different when dealing with environments or commands
-       if($infoHash{type} eq 'environment'){
-            # environments
-            if(!$noAdditionalIndent{$command}){
-                 $masterIndentationArrayOfHashes[-1]{indent} = $indentRules{$command}||$defaultIndent;
-               } 
-            # check to see if we need to look for alignment delimiters
-            if($lookForAlignDelims{$command}){ 
-                # there are two ways to complete the lookForAlignDelims field, either as a scalar
-                # or as a hash, so that we can check for alignDoubleBackSlash. 
-                #
-                # tabular: 
-                #    delims: 1
-                #    alignDoubleBackSlash: 1
-                #
-                # or, simply,
-                #
-                # tabular: 1
-                #
-                # We need to perform a check to see which has been done.
-                if(ref($lookForAlignDelims{$command}) eq 'HASH'){
-                      # tabular: 
-                      #    delims: 1
-                      #    alignDoubleBackSlash: 1
-                      $masterIndentationArrayOfHashes[-1]{alignmentDelimiters}=defined $lookForAlignDelims{$command}{delims}?$lookForAlignDelims{$command}{delims}:1;
-                      $masterIndentationArrayOfHashes[-1]{alignDoubleBackSlash}=defined $lookForAlignDelims{$command}{alignDoubleBackSlash}?$lookForAlignDelims{$command}{alignDoubleBackSlash}:1;
-                      $masterIndentationArrayOfHashes[-1]{spacesBeforeDoubleBackSlash}=$lookForAlignDelims{$command}{spacesBeforeDoubleBackSlash}||0;
-                } else {
-                    # tabular: 1
-                    $masterIndentationArrayOfHashes[-1]{alignmentDelimiters}=1;
-                    $masterIndentationArrayOfHashes[-1]{alignDoubleBackSlash}=1;
-                }
-                if($masterIndentationArrayOfHashes[-1]{alignmentDelimiters}==1){
-                    # tracing mode
-                    print $logfile "Line $lineCounter\t Delimiter environment started: $command (see lookForAlignDelims)\n" if($tracingMode);
-                }
-            }
-       } else {
-            # commands, headings, etc
-            if(!$noAdditionalIndent{$command}){
-                $masterIndentationArrayOfHashes[-1]{indent} = $indentRules{$command}||$defaultIndent;
-             } 
-       }
-
-       # details of noAdditionalIndent to the main hash
-       if($noAdditionalIndent{$command}){
-             $masterIndentationArrayOfHashes[-1]{noAdditionalIndent} = 'yes';
-       }
-}
-
-sub decrease_indent{
-       # PURPOSE: Adjust the indentation
-       #          of the current environment;
-       #          check that it's not an environment
-       #          that doesn't want indentation.
-
-       # if there is no evidence of indentation, then return
-       return unless(scalar(@masterIndentationArrayOfHashes));
-
-       # otherwise get details of the most recent command, environment, item, if, heading, etc
-       my $command = pop(@_);
-
-       if(!$inverbatim) {
-            print $logfile "Line $lineCounter\t removing ", $masterIndentationArrayOfHashes[-1]{name}, " from masterIndentationArrayOfHashes\n" if($tracingMode);
-            pop(@masterIndentationArrayOfHashes);
-            # tracing mode
-            if($tracingMode) {
-                if(@masterIndentationArrayOfHashes) {
-                    print $logfile "Line $lineCounter\t decreasing masterIndentationArrayOfHashes to: ",&current_indentation_names(),"\n";
-                } else {
-                    print $logfile "Line $lineCounter\t masterIndentationArrayOfHashes now empty \n";
-              }
-            }
-       }
-}
-
-sub current_indentation{
-    # PURPOSE: loop through masterIndentationArrayOfHashes and 
-    #          pull out the indentation, and join it together
-
-    # if the masterIndentationArrayOfHashes is empty, return an empty string
-    return "" unless(@masterIndentationArrayOfHashes);
-
-    my $indent;
-    foreach my $env (@masterIndentationArrayOfHashes){
-        $indent .= defined($env->{indent})?$env->{indent}:'';
-      }
-    return $indent;
-}
-
-sub current_indentation_names{
-    # PURPOSE: loop through masterIndentationArrayOfHashes and 
-    #          pull out the list of environment/command names
-    return "masterIndentationArrayOfHashes empty" unless(@masterIndentationArrayOfHashes);
-
-    my $listOfNames;
-    foreach my $env (@masterIndentationArrayOfHashes){
-        $listOfNames .= $env->{name};
-        $listOfNames .= "," unless $env == $masterIndentationArrayOfHashes[-1];
-      }
-    return $listOfNames;
-}
-
-sub check_conflicting_keys{
-  # PURPOSE: users may sometimes put an environment in two
-  #          hash keys; for example, they might put lstlistings
-  #          in both indentRules and in noAdditionalIndent;
-  #          in which case, we need a hierachy.
-  #
-  #          This subroutine implements such a hierachy, 
-  #          and deletes the redundant key.
-
-  # if the user has specified $indentRules{$command} and
-  # $noAdditionalIndent{$command} then they are a bit confused-
-  # we remove the $indentRules{$command} and assume that they
-  # want $noAdditionalIndent{$command}
-
-  my $command = pop(@_);
-
-  if(scalar($indentRules{$command}) and $noAdditionalIndent{$command}) {
-       print $logfile "WARNING\n\t Line $lineCounter\t $command is in *both* indentRules and noAdditionalIndent\n";
-       print $logfile "\t\t\t ignoring indentRules and prioritizing noAdditionalIndent\n";
-       print $logfile "\t\t\t Note that you only get this message once per command/environment\n";
-
-       # remove the key and value from %indentRules hash
-       # to avoid any further confusion
-       delete $indentRules{$command};
-  }
-
-  # if the command is in verbatimEnvironments and in indentRules then
-  # remove it from %indentRules hash
-  # to avoid any further confusion
-  if($indentRules{$command} and $verbatimEnvironments{$command}) {
-       # remove the key and value from %indentRules hash
-       # to avoid any further confusion
-       print $logfile "WARNING\n\t Line $lineCounter\t $command is in *both* indentRules and verbatimEnvironments\n";
-       print $logfile "\t\t\t ignoring indentRules and prioritizing verbatimEnvironments\n";
-       print $logfile "\t\t\t Note that you only get this message once per environment\n";
-       delete $indentRules{$command};
-  }
-
-}