% LaTex2e \documentclass[12pt]{article} \usepackage{amssymb,latexsym} \def\easy{4} \def\medium{15} \def\hard{8} \oddsidemargin 5pt \evensidemargin 5pt \marginparwidth 68pt \marginparsep 10pt \topmargin 10pt \headheight 10pt \headsep 15pt \footskip 35pt \textheight = 625pt \textwidth 440pt \columnsep 10pt \columnseprule 5pt \pagestyle{myheadings} \def\la{\langle} \def\ra{\rangle} \def\header{{\hfill Exam 1 \hfill A. Miller \hfill Fall 99 \hfill Math 211\hfill}} \markboth\header\header \newcount\probno \newcount\total \newwrite\examTBLFile \probno=0 \def\prob#1{\advance\probno by 1 \vfill\newpage % \par \noindent\the\probno . (#1 pts)\advance\total by #1 \immediate\write\examTBLFile{\the\probno \string& #1 \string& \string\qquad \string\\ \string\hline}} \def\probx#1{\advance\probno by 1 \vfill % \par \noindent\the\probno . (#1 pts)\advance\total by #1 \immediate\write\examTBLFile{\the\probno \string& #1 \string& \string\qquad \string\\ \string\hline}} \def\finishtbl{ \immediate\write\examTBLFile{Total \string& \the\total \string& \qquad \string\\ \string\hline} } \newwrite\ans \immediate\openout\ans=answers \outer\def\answer{\par\medbreak \immediate\write\ans{} \immediate\write\ans{\string\bigskip \the\probno . } \copytoblankline} \outer\def\putanswer{\par\medbreak \immediate\write\ans{} \copytoblankline} \def\copytoblankline{\begingroup\setupcopy\copyans} \def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials \catcode`\|=12\obeylines} {\obeylines \gdef\copyans#1 {\def\next{#1}% \ifx\next\empty\let\next=\endgroup% \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}} \def\printanswers{\newpage\begin{center} Answers \end{center} \immediate\closeout\ans\input answers} \begin{document} \bigskip \begin{center} {%\Large No Calculators. \\ Show all your work. \\ Write clearly and carefully. \\ Hand in your exam to your TA.\\ } \end{center} \par\vskip .25in \begin{center} Name\makebox[3in]{\hrulefill} \\ \end{center} \begin{center} Circle your TA section: \\ \end{center} \begin{verbatim} 211 002 321 Lang, Mike 7:45 TR B333 Van Vleck 211 002 322 Lang, Mike 8:50 TR 202 Bascom 211 002 324 Rhee, Kyoungwon 9:55 TR 301 Educ Sci 211 002 326 Rhee, Kyoungwon 11:00 TR 3331 Sterling 211 002 327 Hartwig, Brian 12:05 TR 346 Birge 211 002 328 Lee, Byungyoon 12:05 TR 341 Van Vleck 211 002 329 Lee, Byungyoon 1:20 TR 341 Van Vleck 211 002 331 Hartwig, Brian 2:25 TR 114 Ingraham 211 002 332 Kethireddypally 2:25 TR 223 Ingraham 211 002 334 Kethireddypally 3:30 TR B325 Van Vleck \end{verbatim} % end file with \finishtbl %\bigskip \begin{center} \Large \begin{tabular}{||c|c|c||} \hline\hline Problem & Points & Score \\ \hline\hline \input{exam.tbl} \hline \hline \end{tabular} \end{center} \immediate\openout\examTBLFile=exam.tbl \setcounter{page}{0} %\prob{4} % 1-50 %Let $f(x)=x^3+2$, $g(x)= \sqrt{x+3}$, and $u(x)=4x$. Find %equations for the functions, $h(x)$ and $k(x)$: %\par\bigskip %(a) $h=u\circ f\circ g$ %\par\vfill %(b) $k=(f-g)\over u$ %\par\vfill \prob{4} % 1-66 Hay production in the United States increased from 12 million bales in 1991 to 19 million bales in 1994. Assuming production of hay to follow a linear equation, find the production level predicted by these data for the year 1999. \answer $30{2\over 3}$ million bales. \prob{4} % 2-10,13 / 3-38 Find the indicated limits: \par\bigskip (a) $\lim_{x\to 0}{{(2+x)^2-4}\over x}$ \par\vfill (b) $\lim_{x\to 3^+}{{x^2-4x-3}\over{x-3}}$ \par\vfill \answer (a) 4 \par (b) $-\infty$ \prob{4} % 2-26,38,44 Find the derivative of the given function: \par\bigskip (a) $f(x)=\sqrt{9+x^2}$ \par\vfill (b) $f(x)=(1-x^2)^3(x^2+9)^4$ \par\vfill \answer (a) $x\over{\sqrt{9+x^2}}$ \par (b) $3(1-x^2)(-2x)(x^2+9)^4\;\;+\;\;(1-x^2)^3(x^2+9)^3(8x)$ \prob{4} % 2-54 \par\bigskip The revenue function $R(x)$ is defined by $$R(x)=\left\{ \begin{array}{ll} 4x & 0< x < 10 \\ 140-(x-20)^2 & 10 \leq x < 20 \\ 160-x & 20 \leq x < 160 \\ \end{array}\right.$$ \par At which points is $R(x)$ continuous? differentiable? What is the marginal revenue function, $MR(x)$? \answer $R(x)$ is continuous in the interval where it is defined, $(0,160)$ and differentiable in the same interval except at $x=10$ and $x=20$. $$MR(x)=\left\{ \begin{array}{ll} 4 & 0< x < 10 \\ -2(x-20) & 10 < x < 20 \\ -1 & 20 < x < 160 \\ \mbox{undefined} & x=10,20 \\ \end{array}\right.$$ \prob{4} % 3.8-6 The cost of operating a small aircraft is calculated to be \$400 per hour plus the cost of fuel. Fuel costs are ${s\over 100}$ dollars per kilometer, where $s$ represents speed in kilometers per hour. Find the speed that minimizes the cost per kilometer. \answer $200$ kph. \prob{4} % 3-6 $$f(x)={x\over{x^2+1}}$$ \par\bigskip Find the intervals on which $f$ is increasing or decreasing. \answer increasing on $(-1,1)$, decreasing on $(-\infty,-1)$ and $(1,\infty)$. %\prob{4} % 3-21 %$$f(x)=x^4-2x^2$$ %Find the intervals on which $f$ is concave up or concave down. \prob{4} % 3.11-8 Find a value of $c$ that satisfies the Mean Value Theorem for the given function and interval. $$f(x)=x^3-2x+4,\;\;\;\mbox{ for } x \mbox{ in } [0,2]$$ \answer $2\over{\sqrt{3}}$ \prob{4} % 3-64 A publisher currently sells 2000 subscriptions annually for a professional journal at a rate of \$40 per year. Its marketing department predicts that for each \$1 reduction in subscription price an additional 50 subscriptions can be sold. Under this assumption what subscription rate maximizes revenue? \answer \$40 \prob{4} % 3.9-20 Find the value of the derivative ${dy\over dx}$ at the indicated point. $$(y-x)^2=x \mbox{ at } (9,12)$$ \answer $1{1\over6}$ \prob{4} A policeman 300 yards away from a straight highway has a radar gun pointed at your car on the highway. He finds that when your car is 500 yards away from him that the distance between him and your car is increasing at the rate of 60 miles per hour. How fast is your car traveling? \answer 75 mph. \finishtbl \printanswers \end{document}