%% %% 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'.