%% %% This is file `easy.tex', %% generated with the docstrip utility. %% %% The original source files were: %% %% probsoln.dtx (with options: `easy.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{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}{% $y = \frac{\sin x}{x}$.}{% \[\frac{dy}{dx} = \frac{\cos x}{x} - \frac{\sin x}{x^2}\]} \endinput %% %% End of file `easy.tex'.