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authorKarl Berry <karl@freefriends.org>2006-07-20 18:47:23 +0000
committerKarl Berry <karl@freefriends.org>2006-07-20 18:47:23 +0000
commit7a01b2f895cca6398088b5ec490c896dee03d2fd (patch)
tree51938b8ee0bdb7c1f29d8d114ac15adbe53a6dc1 /Master/texmf-dist/doc/latex/probsoln/1stprncp.tex
parentc1bc79c2f31e5ca9ba713e8cd5582a4b0a7f3fbc (diff)
probsoln update (3mar06)
git-svn-id: svn://tug.org/texlive/trunk@1885 c570f23f-e606-0410-a88d-b1316a301751
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+%%
+%% This is file `1stprncp.tex',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% probsoln.dtx (with options: `1stprncp.tex,package')
+%% Copyright (C) 2006 Nicola Talbot, all rights reserved.
+%% If you modify this file, you must change its name first.
+%% You are NOT ALLOWED to distribute this file alone. You are NOT
+%% ALLOWED to take money for the distribution or use of either this
+%% file or a changed version, except for a nominal charge for copying
+%% etc.
+%% \CharacterTable
+%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%% Digits \0\1\2\3\4\5\6\7\8\9
+%% Exclamation \! Double quote \" Hash (number) \#
+%% Dollar \$ Percent \% Ampersand \&
+%% Acute accent \' Left paren \( Right paren \)
+%% Asterisk \* Plus \+ Comma \,
+%% Minus \- Point \. Solidus \/
+%% Colon \: Semicolon \; Less than \<
+%% Equals \= Greater than \> Question mark \?
+%% Commercial at \@ Left bracket \[ Backslash \\
+%% Right bracket \] Circumflex \^ Underscore \_
+%% Grave accent \` Left brace \{ Vertical bar \|
+%% Right brace \} Tilde \~}
+
+\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*}}
+\endinput
+%%
+%% End of file `1stprncp.tex'.