From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- .../contrib/probsoln/samples/prob-1stprncp.tex | 63 ++++++++++ .../latex/contrib/probsoln/samples/prob-args.tex | 49 ++++++++ .../latex/contrib/probsoln/samples/prob-easy.tex | 114 +++++++++++++++++ .../latex/contrib/probsoln/samples/prob-easy2.tex | 137 +++++++++++++++++++++ .../contrib/probsoln/samples/prob-implicit.tex | 52 ++++++++ .../contrib/probsoln/samples/prob-mchoice.tex | 37 ++++++ .../latex/contrib/probsoln/samples/prob-mixed.tex | 78 ++++++++++++ .../contrib/probsoln/samples/prob-newdata.tex | 27 ++++ .../latex/contrib/probsoln/samples/prob-nosoln.tex | 9 ++ .../contrib/probsoln/samples/prob-probspaces.tex | 33 +++++ .../contrib/probsoln/samples/prob-probspaces2.tex | 43 +++++++ .../contrib/probsoln/samples/prob-tabmchoice.tex | 18 +++ .../latex/contrib/probsoln/samples/prob-verb.tex | 28 +++++ 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b/macros/latex/contrib/probsoln/samples/prob-1stprncp.tex @@ -0,0 +1,63 @@ + % This file is public domain + % + % These problems all involve differentiating from 1st principles + +\newproblem{dfp:xcube}{% +Differentiate $f(x) = x^3$ with respect to $x$ by first principles.}{% +\begin{eqnarray*} +\frac{dy}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)^3-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)(x^2+2x\Delta x+(\Delta x)^2)-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x^3+3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3-x^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{3x^2\Delta x+3x(\Delta x)^2+(\Delta x)^3}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}3x^2+3x\Delta x + (\Delta x)^2\\ + & = & 3x^2 +\end{eqnarray*}} + +\newproblem{dfp:Ioverxsq}{% +Differentiate $\displaystyle f(x) = \frac{1}{x^2}$ with respect to $x$ by first principles.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{\frac{1}{(x+\Delta x)^2}-\frac{1}{x^2}}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{\frac{x^2-(x+\Delta x)^2}{x^2(x+\Delta x)^2}}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x^2-(x^2+2x\Delta x+(\Delta x)^2)}{x^2\Delta x(x+\Delta x)^2}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{-2x\Delta x-(\Delta x)^2}{x^2\Delta x(x+\Delta x)^2}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{-2x-\Delta x}{x^2(x+\Delta x)^2}\\ + & = & \frac{-2x}{x^2x^2}\\ + & = & -\frac{2}{x^3} +\end{eqnarray*}} + +\newproblem{dfp:sqrtx}{% +Differentiate from first principles $f(x) = \surd x$}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{\sqrt{x+\Delta x}-\surd x}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{(\sqrt{x+\Delta x}-\surd x)(\sqrt{x+\delta x}+\surd x)}{\Delta x(\sqrt{x+\Delta x}+\surd x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{x+\Delta x - x}{\Delta x(\sqrt{x+\Delta x}+\surd x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{\Delta x}{\Delta x(\sqrt{x+\Delta x}+\Delta x)}\\ + & = & \lim_{\Delta x\rightarrow 0}\frac{1}{\sqrt{x+\Delta x}+\surd x}\\ + & = & \frac{1}{2\surd x} +\end{eqnarray*}} + +\newproblem{dfp:cons}{% +Differentiate from first principles $f(x) = c$ where $c$ is a constant.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x\rightarrow 0}\frac{c-c}{\Delta x}\\ + & = & \lim_{\Delta x\rightarrow 0}0\\ + & = & 0 +\end{eqnarray*}} + +\newproblem{dfp:cosx}{% +Given +\begin{eqnarray*} +\lim_{x \rightarrow 0} \frac{\cos x - 1}{x} & = & 0\\ +\lim_{x \rightarrow 0} \frac{\sin x}{x} & = & 1 +\end{eqnarray*} +differentiate from first principles $f(x) = \cos x$.}{% +\begin{eqnarray*} +\frac{df}{dx} & = & \lim_{\Delta x \rightarrow 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos(x + \Delta x) - \cos(x)}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos x\cos\Delta x - \sin x\sin\Delta x - \cos x}{\Delta x}\\ + & = & \lim_{\Delta x \rightarrow 0} \frac{\cos x(\cos\Delta x - 1) - \sin x\sin\Delta x}{\Delta x}\\ + & = & \cos x\lim_{\Delta x \rightarrow 0}\frac{\cos\Delta x - 1}{\Delta x} + - \sin x\lim_{\Delta x \rightarrow 0}\frac{\sin\Delta x}{\Delta x}\\ + & = & -1 \qquad\mbox{(using given results)} +\end{eqnarray*}} diff --git a/macros/latex/contrib/probsoln/samples/prob-args.tex b/macros/latex/contrib/probsoln/samples/prob-args.tex new file mode 100644 index 0000000000..8276790feb --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-args.tex @@ -0,0 +1,49 @@ + % This file is public domain + % + % These problems require arguments +% + % Arguments: #1->a_2, #2->a_1 and #3->a_0 + % (Arguments must be integers) +\newcount\ctr +\newproblem[3][{1}{2}{3}]{diff:quad}{% +\(f(x) = +\ifnum#1=0 +\else +\ifnum#1=1\else#1\fi x^2 +\fi +\ifnum#2=0 +\else +\ifnum#2>0 \ifnum#1=0 \else + \fi \fi +\ifnum#2=1\else#2\fi x +\fi +\ifnum#3=0 +\else +\ifnum#3>0 \ifnum#2=0 \ifnum#1=0 \else + \fi \else + \fi\fi +#3 +\fi\) +}{% +\(f'(x) = +\ifnum#1=0 +\else +\ctr=2 +\multiply\ctr by #1 +\the\ctr x +\fi +\ifnum#2=0 +\else +\ifnum#2>0 \ifnum#1=0 \else + \fi \fi +#2 +\fi + % print 0 if both #1 and #2 are 0 +\ifnum#1=0 \ifnum#2=0 0 \fi\fi +\) +} + +\begin{defproblem}[1][{6}]{diff:sin} +\begin{onlyproblem} + \(f(x) = \sin(#1x)\) +\end{onlyproblem} +\begin{onlysolution} + \(f'(x) = #1\cos(#1x)\) +\end{onlysolution} +\end{defproblem} diff --git a/macros/latex/contrib/probsoln/samples/prob-easy.tex b/macros/latex/contrib/probsoln/samples/prob-easy.tex new file mode 100644 index 0000000000..0527e3e464 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-easy.tex @@ -0,0 +1,114 @@ + % This file is public domain + % + % These are all easy differentiation problems + +\newproblem{diffeasy:gpowh}{% +\(f(x) = g(x)^{h(x)}.\)}{% +\begin{eqnarray*} +f(x) & = & e^{\ln g(x)^{h(x)}}\\ + & = & e^{h(x)\ln g(x)}\\ +f'(x) & = & e^{h(x)\ln g(x)}(h'(x)\ln g(x) + h(x)\frac{g'(x)}{g(x)})\\ + & = & g(x)^{h(x)}(h'(x)\ln g(x) + \frac{h(x)g'(x)}{g(x)}) +\end{eqnarray*}} + +\newproblem{diffeasy:arcsin}{% +\(y = \arcsin(x)\)}{% +\[\sin(y) = x\] +diff. w.r.t. $x$: +\begin{eqnarray*} +\cos y \frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{1}{\cos y}\\ + & = & \frac{1}{\sqrt{1 - \sin^2y}}\\ + & = & \frac{1}{\sqrt{1-x^2}}. +\end{eqnarray*}} + +\newproblem{diffeasy:arccos}{% +$y = \arccos x$.}{% +\(\cos y = x\) +diff. w.r.t. $x$: +\begin{eqnarray*} +-\sin y \frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{-1}{\sin y}\\ + & = & \frac{-1}{\sqrt{1-\cos^2y}}\\ + & = & \frac{-1}{\sqrt{1-x^2}} +\end{eqnarray*}} + +\newproblem{diffeasy:tan}{% +\(y = \tan x\)}{% +\begin{eqnarray*} +y & = & \tan x\\ + & = & \frac{\sin x}{\cos x}\\ +\frac{dy}{dx} & = & \frac{\cos x}{\cos x} + \sin x\times\frac{-1}{\cos^2x}\times -\sin x\\ + & = & 1 + \tan^2x\\ + & = & \sec^2x. +\end{eqnarray*}} + +\newproblem{diffeasy:arctan}{% +\(y = \arctan x = \tan^{-1}x\)}{% +\[\tan y = x\] +diff w.r.t. $x$: +\begin{eqnarray*} +\sec^2y\frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{1}{\sec^2y}\\ + & = & \frac{1}{1+\tan^2y}\\ + & = & \frac{1}{1+x^2} +\end{eqnarray*}} + +\newproblem{diffeasy:cot}{% +\(y = (\tan x)^{-1} = \cot x\)}{% +\begin{eqnarray*} +\frac{dy}{dx} & = & -(\tan x)^{-2}\sec^2x\\ + & = & -\frac{\cos^2x}{\sin^2x}\cdot\frac{1}{\cos^2x}\\ + & = & \frac{-1}{\sin^2x}\\ + & = & -\csc^2x. +\end{eqnarray*}} + +\newproblem{diffeasy:cosxsqsinx}{% +$y = \cos(x^2)\sin x$.}{% +\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\]} + +\newproblem{diffeasy:xlnx}{% +$y = (x+1)\ln(x+1)$.}{% +\begin{eqnarray*} +\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\ + & = & 1 + \ln(x+1). +\end{eqnarray*}} + +\newproblem{diffeasy:glng}{% +$f(x) = g(x)\ln(g(x))$.}{% +\begin{eqnarray*} +f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\ + & = & g'(x)(1+\ln(g(x))). +\end{eqnarray*}} + +\newproblem{diffeasy:sinx/x}{% +$y = \frac{\sin x}{x}$.}{% +\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]} + +\newproblem{diffeasy:exp4x}{% + $y = \exp(4x)$ +}% +{% + \[\frac{dy}{dx} = 4\exp(4x)\] +} + +\newproblem{diffeasy:exp3x+2}{% + $y = \exp(3x+2)$ +}% +{% + \[\frac{dy}{dx} = 3\exp(3x+2)\] +} + +\newproblem{diffeasy:cubic}{% + $y=x^3 + 4x^2 - x + 3$ +}% +{% + \[\frac{dy}{dx} = 3x^2 + 8x - 1\] +} + +\newproblem{diffeasy:quad}{% + $y=2x^3 + 6x -1$ +}% +{% + \[\frac{dy}{dx} = 6x + 6 = 6(x+1)\] +} diff --git a/macros/latex/contrib/probsoln/samples/prob-easy2.tex b/macros/latex/contrib/probsoln/samples/prob-easy2.tex new file mode 100644 index 0000000000..d6edc7ee3c --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-easy2.tex @@ -0,0 +1,137 @@ + % This file is public domain + % + % These are all easy differentiation problems + +\begin{defproblem}{diffeasy:gpowh}% +\begin{onlyproblem}% +\(f(x) = g(x)^{h(x)}.\)% +\end{onlyproblem}% +\begin{onlysolution}% +\begin{eqnarray*} +f(x) & = & e^{\ln g(x)^{h(x)}}\\ + & = & e^{h(x)\ln g(x)}\\ +f'(x) & = & e^{h(x)\ln g(x)}(h'(x)\ln g(x) + h(x)\frac{g'(x)}{g(x)})\\ + & = & g(x)^{h(x)}(h'(x)\ln g(x) + \frac{h(x)g'(x)}{g(x)}) +\end{eqnarray*}% +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:arcsin}% +\begin{onlyproblem}% +\(y = \arcsin(x)\)% +\end{onlyproblem}% +\begin{onlysolution}% +\[\sin(y) = x\] +diff. w.r.t. $x$: +\begin{eqnarray*} +\cos y \frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{1}{\cos y}\\ + & = & \frac{1}{\sqrt{1 - \sin^2y}}\\ + & = & \frac{1}{\sqrt{1-x^2}}. +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:arccos}% +\begin{onlyproblem}% +$y = \arccos x$.% +\end{onlyproblem}% +\begin{onlysolution}% +\(\cos y = x\) +diff. w.r.t. $x$: +\begin{eqnarray*} +-\sin y \frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{-1}{\sin y}\\ + & = & \frac{-1}{\sqrt{1-\cos^2y}}\\ + & = & \frac{-1}{\sqrt{1-x^2}} +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:tan}% +\begin{onlyproblem}% +\(y = \tan x\)% +\end{onlyproblem} +\begin{onlysolution}% +\begin{eqnarray*} +y & = & \tan x\\ + & = & \frac{\sin x}{\cos x}\\ +\frac{dy}{dx} & = & \frac{\cos x}{\cos x} + \sin x\times\frac{-1}{\cos^2x}\times -\sin x\\ + & = & 1 + \tan^2x\\ + & = & \sec^2x. +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:arctan}% +\begin{onlyproblem}% +\(y = \arctan x = \tan^{-1}x\)% +\end{onlyproblem}% +\begin{onlysolution}% +\[\tan y = x\] +diff w.r.t. $x$: +\begin{eqnarray*} +\sec^2y\frac{dy}{dx} & = & 1\\ +\frac{dy}{dx} & = & \frac{1}{\sec^2y}\\ + & = & \frac{1}{1+\tan^2y}\\ + & = & \frac{1}{1+x^2} +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:cot}% +\begin{onlyproblem}% +\(y = (\tan x)^{-1} = \cot x\)% +\end{onlyproblem} +\begin{onlysolution}% +\begin{eqnarray*} +\frac{dy}{dx} & = & -(\tan x)^{-2}\sec^2x\\ + & = & -\frac{\cos^2x}{\sin^2x}\cdot\frac{1}{\cos^2x}\\ + & = & \frac{-1}{\sin^2x}\\ + & = & -\csc^2x. +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:cosxsqsinx}% +\begin{onlyproblem}% +$y = \cos(x^2)\sin x$.% +\end{onlyproblem}% +\begin{onlysolution}% +\[\frac{dy}{dx} = -\sin(x^2)2x\sin x + \cos(x^2)\cos x\] +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:xlnx}% +\begin{onlyproblem}% +$y = (x+1)\ln(x+1)$. +\end{onlyproblem}% +\begin{onlysolution}% +\begin{eqnarray*} +\frac{dy}{dx} & = & \ln(x+1) + \frac{x+1}{x+1}\\ + & = & 1 + \ln(x+1). +\end{eqnarray*} +\end{onlysolution} +\end{defproblem} + +\begin{defproblem}{diffeasy:glng}% +\begin{onlyproblem}% +$f(x) = g(x)\ln(g(x))$. +\end{onlyproblem} +\begin{onlysolution}% +\begin{eqnarray*} +f'(x) & = & g'(x)\ln(g(x)) + \frac{g(x)}{g(x)}g'(x)\\ + & = & g'(x)(1+\ln(g(x))). +\end{eqnarray*} +\end{onlysolution}% +\end{defproblem} + +\begin{defproblem}{diffeasy:sinx/x} +\begin{onlyproblem}% +$y = \frac{\sin x}{x}$. +\end{onlyproblem} +\begin{onlysolution}% +\[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\] +\end{onlysolution}% +\end{defproblem} + diff --git a/macros/latex/contrib/probsoln/samples/prob-implicit.tex b/macros/latex/contrib/probsoln/samples/prob-implicit.tex new file mode 100644 index 0000000000..dd847d96e3 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-implicit.tex @@ -0,0 +1,52 @@ + % This file is public domain + % + % These are all implicit differentiation problems + +\newproblem{imd:circ}{% +Find the gradient of the unit circle ($x^2 + y^2 = 1$).}{% +Differentiating with respect to $x$ gives: +\begin{eqnarray*} +2x + 2y\frac{dy}{dx} & = & 0\\ +\frac{dy}{dx} & = & \frac{-2x}{2y}\\ + & = & \frac{-x}{\sqrt{1-x^2}}. +\end{eqnarray*}} + +\newproblem{imd:ellipse}{% +Find the gradient of the ellipse given by $4x^2 + 3y^2 = 25$.}{% +Differentiating with respect to $x$ gives: +\begin{eqnarray*} +8x + 6y\frac{dy}{dx} & = & 0\\ +\frac{dy}{dx} & = & \frac{-8x}{6y}\\ + & = & \frac{-4x}{\sqrt{25-4x^2}}. +\end{eqnarray*}} + +\newproblem{imd:ysq:xcuov2mx}{% +Find $\frac{dy}{dx}$, given +\begin{displaymath} +y^2 = \frac{x^3}{2-x} +\end{displaymath}}{% +Differentiating both sides w.r.t.\ $x$: +\begin{eqnarray*} +2y\frac{dy}{dx} & = & \frac{(2-x)3x^2 - x^3(-1)}{(2-x)^2}\\ + & = & \frac{3x^2(2-x) + x^3}{(2-x)^2}\\ + & = & \frac{6x^2 - 3x^3 + x^3}{(2-x)^2}\\ + & = & \frac{6x^2-2x^3}{(2-x)^2}\\ + & = & 2x^2\frac{3-x}{(2-x)^2} +\end{eqnarray*} +Therefore +\begin{displaymath} +y\frac{dy}{dx} = x^2\frac{3-x}{(2-x)^2} +\end{displaymath}} + +\newproblem{imd:exy:IIxay}{% +Differentiate w.r.t.\ $x$: +\begin{displaymath} +e^{xy} = 2x + y +\end{displaymath}}{% +Differentiating both sides w.r.t.\ $x$: +\begin{eqnarray*} +e^{xy}(1y + x\frac{dy}{dx}) & = & 2 + \frac{dy}{dx}\\ +xe^{xy}\frac{dy}{dx} - \frac{dy}{dx} & = & 2 - ye^{xy} \\ +\frac{dy}{dx}(xe^{xy}-1) & = & 2 - ye^{xy}\\ +\frac{dy}{dx} & = & \frac{2-ye^{xy}}{xe^{xy}-1} +\end{eqnarray*}} diff --git a/macros/latex/contrib/probsoln/samples/prob-mchoice.tex b/macros/latex/contrib/probsoln/samples/prob-mchoice.tex new file mode 100644 index 0000000000..22886ce9b2 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-mchoice.tex @@ -0,0 +1,37 @@ + % This file is public domain + % + % These problems are for multiple choice questions +\newproblem{mc:prod}{% +Which of the following is the derivative of $x\sin(x)$? +(Circle the correct answer.) +\ifthenelse{\boolean{showanswers}}{}% +{\begin{description} +\item[A] $\sin(x)$ +\item[B] $x\cos(x)$ +\item[C] $\sin(x) + x\cos(x)$ +\end{description} +}}{% +\begin{description} +\item[A] $\sin(x)$ +\item[B] $x\cos(x)$ +\item[\textcircled{C}] $\sin(x) + x\cos(x)$ (product rule). +\end{description} +} + +\newproblem{mc:quot}{% +Which of the following is the derivative of $\frac{\sin(x)}{x}$? +(Circle the correct answer.) +\ifthenelse{\boolean{showanswers}}{}% +{\begin{description} +\item[A] $\sin(x)$ +\item[B] $\cos(x)$ +\item[C] $\frac{\cos(x)x-\sin(x)}{x^2}$ +\end{description} +}}{% +\begin{description} +\item[A] $\sin(x)$ +\item[B] $\cos(x)$ +\item[\textcircled{C}] $\frac{\cos(x)x-\sin(x)}{x^2}$ +(quotient rule) +\end{description} +} diff --git a/macros/latex/contrib/probsoln/samples/prob-mixed.tex b/macros/latex/contrib/probsoln/samples/prob-mixed.tex new file mode 100644 index 0000000000..1f43968b8a --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-mixed.tex @@ -0,0 +1,78 @@ + % This file is public domain + % + % These problems are a mixture of essay-style and questions with + % answers. One of these problems requires the tikz package + +\newproblem*{oop}{Describe what is meant by object-oriented +programming.} + +\begin{defproblem}{inheritance} + Describe what is meant by the term \emph{inheritance} in + object-oriented programming. Use examples. +\end{defproblem} + +\begin{defproblem}{weightedcoin}% + \begin{onlyproblem} + A coin is weighted so that heads is four times as likely + as tails. Find the probability that: + \begin{textenum} + \item tails appears, + \item heads appears + \end{textenum}% + \end{onlyproblem}% + \begin{onlysolution} + Let $p=P(T)$, then $P(H)=4p$. We require $P(H)+P(T)=1$, + so $4p+p=1$, hence $p=\frac{1}{5}$. Therefore: + \begin{textenum} + \item $P(T)=\frac{1}{5}$, + \item $P(H)=\frac{4}{5}$ + \end{textenum} + \end{onlysolution} +\end{defproblem} + +\begin{defproblem}{validprobspaces} +\begin{onlyproblem}% +Under which of the following functions does +$S=\{a_1,a_2\}$ become a probability space? +\par +\begin{textenum} +\begin{tabular}{ll} +\item $P(a_1)=\frac{1}{3}$, $P(a_2)=\frac{1}{2}$ +& +\item\label{validprobspacescorrect1} $P(a_1)=\frac{3}{4}$, +$P(a_2)=\frac{1}{4}$ +\\ +\item\label{validprobspacescorrect2} $P(a_1)=1$, $P(a_2)=0$ +& +\item $P(a_1)=\frac{5}{4}$, $P(a_2)=-\frac{1}{4}$ +\end{tabular} +\end{textenum} +\end{onlyproblem}% +\begin{onlysolution}% +\ref{validprobspacescorrect1} and \ref{validprobspacescorrect2}% +\end{onlysolution} +\end{defproblem} + +\begin{defproblem}{digraph} + \begin{onlyproblem}\label{ex:digraph} + Identify, if any, the sinks and sources of the digraph shown in Figure~\ref{fig:digraph}. + + \begin{figure}[tbh] + \centering + \begin{tikzpicture}[every node/.style={draw,circle}] + \path (0,0) node (A) {$A$} + (1,0) node (B) {$B$} + (0,1) node (C) {$C$}; + \draw[->] (A) -- (B); + \draw[->] (B) -- (C); + \draw[->] (A) -- (C); + \end{tikzpicture} + \par + \caption{Digraph for Question~\ref{ex:digraph}} + \label{fig:digraph} + \end{figure} + \end{onlyproblem} + \begin{onlysolution} + $A$ is a souce and $C$ is a sink. + \end{onlysolution} +\end{defproblem} diff --git a/macros/latex/contrib/probsoln/samples/prob-newdata.tex b/macros/latex/contrib/probsoln/samples/prob-newdata.tex new file mode 100644 index 0000000000..946f0e1124 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-newdata.tex @@ -0,0 +1,27 @@ + % This file is public domain +\begin{defproblem}{sample} +\begin{onlyproblem} +Differentiate $y=\sin x$ +\end{onlyproblem} +\begin{onlysolution} +$y'=\cos x$ +\end{onlysolution} +\end{defproblem} + +\begin{defproblem}[1]{sample2} +\begin{onlyproblem} +Differentiate $y = \sin(#1x)$ +\end{onlyproblem} +\begin{onlysolution} +$y'=#1\cos #1x$ +\end{onlysolution} +\end{defproblem} + +\begin{defproblem}{sample3} +\begin{onlyproblem} +Differentiate $y = x^2$. +\end{onlyproblem} +\begin{onlysolution} +$y' = 2x$ +\end{onlysolution} +\end{defproblem} diff --git a/macros/latex/contrib/probsoln/samples/prob-nosoln.tex b/macros/latex/contrib/probsoln/samples/prob-nosoln.tex new file mode 100644 index 0000000000..9909651b86 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-nosoln.tex @@ -0,0 +1,9 @@ + % This file is public domain + % + % these problems don't have solutions + +\newproblem*{oop}{Describe what is meant by object-oriented +programming.} + +\newproblem*{inheritance}{Describe what is meant by the term +\emph{inheritance} in object-oriented programming. Use examples.} diff --git a/macros/latex/contrib/probsoln/samples/prob-probspaces.tex b/macros/latex/contrib/probsoln/samples/prob-probspaces.tex new file mode 100644 index 0000000000..4862e5caef --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-probspaces.tex @@ -0,0 +1,33 @@ + % This file is public domain + % + % Finite probability spaces +\newproblem{weightedcoin}{% +A coin is weighted so that heads is four times as likely +as tails. Find the probability that: +\begin{textenum} +\item tails appears, +\item heads appears +\end{textenum}}{% +Let $p=P(T)$, then $P(H)=4p$. We require $P(H)+P(T)=1$, +so $4p+p=1$, hence $p=\frac{1}{5}$. Therefore: +\begin{textenum} +\item $P(T)=\frac{1}{5}$, +\item $P(H)=\frac{4}{5}$ +\end{textenum}} + +\newproblem*{validprobspaces}{% +Under which of the following functions does +$S=\{a_1,a_2\}$ become a probability space? +\par +\begin{textenum} +\begin{tabular}{ll} +\incorrectitem $P(a_1)=\frac{1}{3}$, $P(a_2)=\frac{1}{2}$ +& +\correctitem $P(a_1)=\frac{3}{4}$, $P(a_2)=\frac{1}{4}$ +\\ +\correctitem $P(a_1)=1$, $P(a_2)=0$ +& +\incorrectitem $P(a_1)=\frac{5}{4}$, $P(a_2)=-\frac{1}{4}$ +\end{tabular} +\end{textenum} +} diff --git a/macros/latex/contrib/probsoln/samples/prob-probspaces2.tex b/macros/latex/contrib/probsoln/samples/prob-probspaces2.tex new file mode 100644 index 0000000000..dacddb510c --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-probspaces2.tex @@ -0,0 +1,43 @@ + % This file is public domain + % + % Finite probability spaces +\begin{defproblem}{weightedcoin} +\begin{onlyproblem}% +A coin is weighted so that heads is four times as likely +as tails. Find the probability that: +\begin{textenum} +\item tails appears, +\item heads appears +\end{textenum} +\end{onlyproblem} +\begin{onlysolution}% +Let $p=P(T)$, then $P(H)=4p$. We require $P(H)+P(T)=1$, +so $4p+p=1$, hence $p=\frac{1}{5}$. Therefore: +\begin{textenum} +\item $P(T)=\frac{1}{5}$, +\item $P(H)=\frac{4}{5}$ +\end{textenum} +\end{onlysolution} +\end{defproblem} + +\begin{defproblem}{validprobspaces} +\begin{onlyproblem}% +Under which of the following functions does +$S=\{a_1,a_2\}$ become a probability space? +\par +\begin{textenum} +\begin{tabular}{ll} +\item $P(a_1)=\frac{1}{3}$, $P(a_2)=\frac{1}{2}$ +& +\item\label{validprobspacescorrect1} $P(a_1)=\frac{3}{4}$, $P(a_2)=\frac{1}{4}$ +\\ +\item\label{validprobspacescorrect2} $P(a_1)=1$, $P(a_2)=0$ +& +\item $P(a_1)=\frac{5}{4}$, $P(a_2)=-\frac{1}{4}$ +\end{tabular} +\end{textenum} +\end{onlyproblem}% +\begin{onlysolution}% +\ref{validprobspacescorrect1} and \ref{validprobspacescorrect2}% +\end{onlysolution} +\end{defproblem} diff --git a/macros/latex/contrib/probsoln/samples/prob-tabmchoice.tex b/macros/latex/contrib/probsoln/samples/prob-tabmchoice.tex new file mode 100644 index 0000000000..9555b97dd3 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-tabmchoice.tex @@ -0,0 +1,18 @@ + % This file is public domain + % + % These problems are designed to be placed in a + % tabular environment + % +\newproblem{tab:1}{% +What is $(3+2)\times5$? & +25 \ifshowanswers\selected\else\notselected\fi & +13 \notselected & +10 \notselected & +}{Brackets come first}% + +\newproblem{tab:2}{% +What is $-1+2\times3$? & +3 \notselected & +-7 \notselected & +5 \ifshowanswers\selected\else\notselected\fi & +}{Multiplication comes first}% diff --git a/macros/latex/contrib/probsoln/samples/prob-verb.tex b/macros/latex/contrib/probsoln/samples/prob-verb.tex new file mode 100644 index 0000000000..07e0f83513 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/prob-verb.tex @@ -0,0 +1,28 @@ +\begin{defproblem}{code-helloworld} +This problem has a code fragment. +\begin{onlyproblem} +\lstset{language=Java} +\begin{lstlisting} +public class HelloWorld +{ + public static void main(String[] args) + { + System.out.println("Hello World!"); + } +} +\end{lstlisting} +\end{onlyproblem} +\begin{onlysolution} +\lstset{language=Java} +\begin{lstlisting} +public class HelloWorld +{ + public static void main(String[] args) + { + System.out.println("Hello " + + (args.length==0 ? "anon" : args[0])+"!"); + } +} +\end{lstlisting} +\end{onlysolution} +\end{defproblem} diff --git a/macros/latex/contrib/probsoln/samples/sample-exclude.pdf b/macros/latex/contrib/probsoln/samples/sample-exclude.pdf new file mode 100644 index 0000000000..a0e6571cac Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample-exclude.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample-exclude.tex b/macros/latex/contrib/probsoln/samples/sample-exclude.tex new file mode 100644 index 0000000000..0120d310ed --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample-exclude.tex @@ -0,0 +1,79 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{probsoln} + %uncomment the following line to display the answers + %\showanswers + + % Set the academic year to be 2011/12 (if omitted assume current + % academic year): + \SetStartYear{2011} + + % Set the seed for the random number generator: + \PSNrandseed{\GetStartYear} + + % Exclude problems that have been used this year or the previous + % two academic years. (Creates a file called 'excluded.tex' to + % store labels of used problems. Also creates a file called + % \jobname.prb that stores labels of problems used in this + % document, so they don't get excluded on subsequent runs.) + \ExcludePreviousFile[3]{excluded} + + % To clear the used problems file (\jobname.prb) of the labels + % generated in the previous run, comment the above and uncomment + % below: + % \ClearUsedFile{excluded} + + % Load 1 problem randomly selected from the given file + % and store in database 'firstprinciples': +\loadrandomproblems[firstprinciples]{1}{prob-1stprncp} + + % Load 5 problems randomly selected from the given file + % and store in database 'easy': +\loadrandomproblems[easy]{5}{prob-easy} + + % Load 2 problems randomly selected from the given file + % and store in database 'implicit': +\loadrandomproblems[implicit]{2}{prob-implicit} + + % Load 2 problems randomly selected from the given file + % and store in database 'probspaces': +\loadrandomproblems[probspaces]{2}{prob-probspaces} + + % Load 1 problem randomly selected from the given file + % and store in database 'mchoice': +\loadrandomproblems[mchoice]{1}{prob-mchoice} + + % Load 1 problem randomly selected from the given file + % and store in database 'nosoln': +\loadrandomproblems[nosoln]{1}{prob-nosoln} + + +\begin{document} +\title{Sample Problem Sheet} +\author{Nicola Talbot} +\maketitle + +\begin{enumerate} + % Differentiation from first principles + \foreachproblem[firstprinciples]{\item \thisproblem} + +\item Differentiate the following functions: +\begin{enumerate} + \foreachproblem[easy]{\item \thisproblem} +\end{enumerate} + + % Implicit differentiation +\foreachproblem[implicit]{\item \thisproblem} + + % Finite probability spaces +\foreachproblem[probspaces]{\item \thisproblem} + + % Multiple choice question +\foreachproblem[mchoice]{\item \thisproblem} + + % select one essay style question that doesn't come with a solution + +\foreachproblem[nosoln]{\item \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample.pdf b/macros/latex/contrib/probsoln/samples/sample.pdf new file mode 100644 index 0000000000..3a44bed21a Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample.tex b/macros/latex/contrib/probsoln/samples/sample.tex new file mode 100644 index 0000000000..b6bd665825 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample.tex @@ -0,0 +1,64 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{probsoln} + % uncomment the following line to display the answers + % \showanswers + + % The following will generate a different set of + % problems each year + %\PSNrandseed{\year} + + % Load 1 problem randomly selected from the given file + % and store in database 'firstprinciples': +\loadrandomproblems[firstprinciples]{1}{prob-1stprncp} + + % Load 5 problems randomly selected from the given file + % and store in database 'easy': +\loadrandomproblems[easy]{5}{prob-easy} + + % Load 2 problems randomly selected from the given file + % and store in database 'implicit': +\loadrandomproblems[implicit]{2}{prob-implicit} + + % Load 2 problems randomly selected from the given file + % and store in database 'probspaces': +\loadrandomproblems[probspaces]{2}{prob-probspaces} + + % Load 1 problem randomly selected from the given file + % and store in database 'mchoice': +\loadrandomproblems[mchoice]{1}{prob-mchoice} + + % Load 1 problem randomly selected from the given file + % and store in database 'nosoln': +\loadrandomproblems[nosoln]{1}{prob-nosoln} + + +\begin{document} +\title{Sample Problem Sheet} +\author{Nicola Talbot} +\maketitle + +\begin{enumerate} + % Differentiation from first principles + \foreachproblem[firstprinciples]{\item \thisproblem} + +\item Differentiate the following functions: +\begin{enumerate} + \foreachproblem[easy]{\item \thisproblem} +\end{enumerate} + + % Implicit differentiation +\foreachproblem[implicit]{\item \thisproblem} + + % Finite probability spaces +\foreachproblem[probspaces]{\item \thisproblem} + + % Multiple choice question +\foreachproblem[mchoice]{\item \thisproblem} + + % select one essay style question that doesn't come with a solution + +\foreachproblem[nosoln]{\item \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample2.pdf b/macros/latex/contrib/probsoln/samples/sample2.pdf new file mode 100644 index 0000000000..007f6d906b Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample2.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample2.tex b/macros/latex/contrib/probsoln/samples/sample2.tex new file mode 100644 index 0000000000..cd13c89f30 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample2.tex @@ -0,0 +1,48 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{probsoln} + %uncomment the following line to display the answers +\showanswers + + % The following will generate a different set of + % problems in subsequent years + %\PSNrandseed{\year} + + % The following will generate a different set of + % problems on every run (where each run is at + % least a minute apart from the previous one) +\PSNrandseed{\time} + +\begin{document} +\title{Sample Problem Sheet} +\author{Nicola Talbot} +\maketitle + +\begin{enumerate} + +\newcounter{numproblems} + % store a random number from 1 to 3 in the counter numproblems +\random{numproblems}{1}{3} + + % selects 1,2 or 3 problems from 1 of the given files: +\doforrandN{1}{\file}{prob-1stprncp,prob-implicit}{\selectrandomly{\file}{\value{numproblems}}} + +\item Differentiate the following functions: +\input{prob-args}% load database +\begin{enumerate} +\newcounter{A} +\newcounter{B} +\newcounter{C} +\random{A}{-4}{5} +\random{B}{-3}{3} +\random{C}{-1}{10} +\item \useproblem{diff:quad}{\arabic{A}}{\arabic{B}}{\arabic{C}} + +\random{A}{-5}{5} + % make sure its not 0 +\whiledo{\value{A}=0}{\random{A}{-5}{5}} +\item \useproblem{diff:sin}{\arabic{A}} +\end{enumerate} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample3.pdf b/macros/latex/contrib/probsoln/samples/sample3.pdf new file mode 100644 index 0000000000..d6f6ff4909 Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample3.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample3.tex b/macros/latex/contrib/probsoln/samples/sample3.tex new file mode 100644 index 0000000000..6922f9538e --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample3.tex @@ -0,0 +1,25 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{longtable} +\usepackage{probsoln} + + %\showanswers + +\newcounter{problem} +\renewcommand{\PSNitem}{\refstepcounter{problem}% +\theproblem. } +\renewcommand{\endPSNitem}{\\} + +\renewenvironment{solution}{}{} + +\newcommand{\selected}{\fbox{$\times$}} +\newcommand{\notselected}{\fbox{\phantom{$\times$}}} + +\begin{document} +\begin{longtable}{lrrrl} +\bfseries Question & \bfseries A & \bfseries B & +\bfseries C & \ifshowanswers \bfseries Reason\fi\\ +\selectrandomly{prob-tabmchoice}{2} +\end{longtable} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample4.pdf b/macros/latex/contrib/probsoln/samples/sample4.pdf new file mode 100644 index 0000000000..7aadf56ca1 Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample4.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample4.tex b/macros/latex/contrib/probsoln/samples/sample4.tex new file mode 100644 index 0000000000..b432182d01 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample4.tex @@ -0,0 +1,21 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{probsoln} + + % Sample file illustrating use of \loadallproblems + % All problems defined in specified file will be used + % in order of definition + + \loadallproblems{prob-easy} + %\loadselectedproblems{diffeasy:gpowh,diffeasy:arccos}{prob-easy} + %\loadexceptproblems{diffeasy:gpowh,diffeasy:arccos}{prob-easy} + + \showanswers + +\begin{document} +Differentiate the following functions: +\begin{enumerate} + \foreachproblem{\item \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample5.pdf b/macros/latex/contrib/probsoln/samples/sample5.pdf new file mode 100644 index 0000000000..f4ac8c7fdc Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample5.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample5.tex b/macros/latex/contrib/probsoln/samples/sample5.tex new file mode 100644 index 0000000000..ac294c11ca --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample5.tex @@ -0,0 +1,34 @@ + % This file is public domain +\documentclass[a4paper]{report} + +\usepackage{probsoln} + +\begin{document} +\hideanswers + +\chapter{Easy Problems} +\loadrandomproblems[easyproblems]{5}{prob-easy2} + +\renewcommand{\theenumi}{\thechapter.\arabic{enumi}} +\begin{enumerate} +\foreachproblem[easyproblems]{\item\label{prob:\thisproblemlabel}\thisproblem} +\end{enumerate} + +\chapter{Probability Spaces} +\loadrandomproblems[probspaces]{2}{prob-probspaces2} + +\renewcommand{\theenumi}{\thechapter.\arabic{enumi}} +\begin{enumerate} +\foreachproblem[probspaces]{\item\label{prob:\thisproblemlabel}\thisproblem} +\end{enumerate} + +\appendix +\chapter{Solutions} +\showanswers +\begin{enumerate} +\foreachdataset{\thisdataset}{% +\foreachproblem[\thisdataset]{\item[\ref{prob:\thisproblemlabel}]\thisproblem} +} +\end{enumerate} + +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample6.pdf b/macros/latex/contrib/probsoln/samples/sample6.pdf new file mode 100644 index 0000000000..2a8be97bbc Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample6.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample6.tex b/macros/latex/contrib/probsoln/samples/sample6.tex new file mode 100644 index 0000000000..65769d3c97 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample6.tex @@ -0,0 +1,11 @@ +\documentclass{article} + +\usepackage[usedefaultargs]{probsoln} + +\loadallproblems{prob-args} + +\begin{document} +\begin{enumerate} +\foreachproblem{\item \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample7.pdf b/macros/latex/contrib/probsoln/samples/sample7.pdf new file mode 100644 index 0000000000..12b0da7dad Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample7.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample7.tex b/macros/latex/contrib/probsoln/samples/sample7.tex new file mode 100644 index 0000000000..00ccf637d4 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample7.tex @@ -0,0 +1,20 @@ +\documentclass{article} + +\usepackage{tikz} +\usepackage{probsoln} + +\loadallproblems{prob-mixed} + +\begin{document} +\section{Exercises} +\begin{enumerate} + \foreachproblem{\item\label{q-\thisproblemlabel} \thisproblem} +\end{enumerate} + +\section{Solutions} +\showanswers + +\begin{enumerate} + \foreachsolution{\item[\ref{q-\thisproblemlabel}] \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample8.pdf b/macros/latex/contrib/probsoln/samples/sample8.pdf new file mode 100644 index 0000000000..f931368efa Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample8.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample8.tex b/macros/latex/contrib/probsoln/samples/sample8.tex new file mode 100644 index 0000000000..e68954daf2 --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample8.tex @@ -0,0 +1,15 @@ +\documentclass{article} + +\usepackage{listings} +\usepackage{probsoln} + +\setkeys{probsoln}{fragile} + +\loadallproblems{prob-verb} + +\showanswers +\begin{document} +\begin{enumerate} + \foreachproblem{\item \thisproblem} +\end{enumerate} +\end{document} diff --git a/macros/latex/contrib/probsoln/samples/sample9.pdf b/macros/latex/contrib/probsoln/samples/sample9.pdf new file mode 100644 index 0000000000..9632dc35df Binary files /dev/null and b/macros/latex/contrib/probsoln/samples/sample9.pdf differ diff --git a/macros/latex/contrib/probsoln/samples/sample9.tex b/macros/latex/contrib/probsoln/samples/sample9.tex new file mode 100644 index 0000000000..79ac7c5f5e --- /dev/null +++ b/macros/latex/contrib/probsoln/samples/sample9.tex @@ -0,0 +1,19 @@ + % This file is public domain +\documentclass[a4paper]{article} + +\usepackage{probsoln} + % uncomment the following line to display the answers + % \showanswers + + % Load 10 problems randomly selected from the given files +\loadrandomproblems{10}{prob-1stprncp,prob-easy,prob-implicit,prob-probspaces} + +\begin{document} +\title{Sample Problem Sheet} +\author{Nicola Talbot} +\maketitle + +\begin{enumerate} + \foreachproblem{\item \thisproblem} +\end{enumerate} +\end{document} -- cgit v1.2.3