\documentclass[11pt]{article} \usepackage{multicol} \renewcommand{\thesection}{\Alph{section}.} \renewcommand{\labelenumi}{\bf{\theenumi.}} \renewcommand{\labelenumii}{\bf{\theenumii)}} \newcommand{\ins}[1]{{\sc#1}} \newcommand{\unit}[1]{{\,\small\sf#1}} \pagestyle{empty} \pagenumbering{roman} %\setcounter{page}{-1} %\setlength{\marginparwidth}{0pt} %\setlength{\marginparsep}{0pt} %\setlength{\topmargin}{0pt} %\setlength{\headsep}{0pt} %\setlength{\headheight}{0pt} %\setlength{\footskip}{0pt} %\setlength{\textheight}{530pt} %\setlength{\textwidth}{350pt} \oddsidemargin=0in % 1in margins at left and right \evensidemargin=0in \textwidth=6.7in % US paper is 8.5in wide \headheight=-.3in % 1in margins at top and bottom \headsep=0pt \topmargin=0in \textheight=9.6in % Us paper is 11.0in high %\marginparwidth=0.5in %\setlength{\voffset}{-30pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \newcommand{\ds}{\displaystyle} \newcommand{\be}{\begin{enumerate}} \newcommand{\ee}{\end{enumerate}} \begin{center} {\bf APPM 1360 --- Final Exam --- May 9, 1997} \end{center} On the front of your bluebook, print your name, student number, the name of your instructor (Dougherty, Halburd, Lomeli), the time of your class and a grading table. There are 10 questions plus one extra credit problem. It is very important that you show all your work in the bluebook. \fbox{Box} in your answers, if possible. A correct answer with no relevant work may receive no credit, while an incorrect answer accompanied by some correct work may receive partial credit. Text books and class notes are NOT permitted. A calculator and a one-page crib sheet are allowed. {\bf Please start each new problem on a new page.} % You should not find integrals using your calculator. \begin{enumerate} %********************************************************************* \item (15 points) Find $dy/dx$ for each of the following: \begin{multicols}{2} \begin{enumerate} \item $\ds y= \ln(2^x + 3^x)$ \item $\ds y = \frac{\ln(10^{(x^2)})}{\ln(3^x)}$ \item $\ds x(t) = 4 \sin(t) + \sin(5)$, \newline $\ds y(t) = \sqrt{3}\cos(t)$ \end{enumerate} \end{multicols} \item (15 points) Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage $V$ across its terminals and that, if $t$ is measured in seconds, $\ds \frac{dV}{dt} = - \frac{1}{40}V$. Solve this equation for $V$, using $V_0$ to denote the value of $V$ when $t=0$. How long will it take the voltage to drop to 10\% of its original value? \item (24 points) Evaluate each of the following integrals. \begin{multicols}{2} \begin{enumerate} \item $\ds \int \frac{1}{x^2+6x+25} \; dx$ \item $\ds \int \frac{1}{x^2 + 6x + 8} \; dx$ \item $\ds \int_0^2 \sin(\tan^{-1} x) \; dx$ \newline (Hint: reference triangles) \end{enumerate} \end{multicols} \item (16 points) Explain whether each of the following integrals converge or diverge. \begin{multicols}{2} \begin{enumerate} \item $\displaystyle \int_1^\infty\frac{x^2+1}{x^5+x+1}\, dx $ \item $\ds \int_0^2 \frac{1}{1-x} \; dx$ \end{enumerate} \end{multicols} \item (30 points) True or False. If the statement is true, write the word TRUE and provide a short explanation of why it is true. If it is false, write the word FALSE and give an example showing that it is false. \begin{enumerate} \item If $\displaystyle \lim_{n \to\infty} a_n = 0$ then $\displaystyle \sum_{n=1}^{\infty} a_n $ converges. \item If $\displaystyle \lim_{n \to\infty} \frac{a_{n+1}}{a_n} = 1$ then $\displaystyle \sum_{n=1}^{\infty} a_n $ diverges. \item If $f(x)$ is a differentiable function such that $f(0)=1$, $f(5)=2$ and $\int_0^5f(x)dx=3$ then $ \ds \int_0^5xf'(x) \;dx = 7 $ \item The series $\ds \sum_{n=1}^\infty n(1+n^2)^p $ converges if $p < -1$. \item The Taylor series for $e^x$ centered at $a=1$ is $\ds \sum_{k=0}^{\infty} \frac{(x-1)^k}{k!}$. \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \item (12 points) Do the following series converge absolutely, converge conditionally or diverge? Give reasons for your answers. \begin{multicols}{2} \be \item $\displaystyle \sum_{n=1}^\infty (-1)^n\frac{\ln n}{n+1}$ \item $\ds \sum_{n=1}^{\infty} \frac{\tan^{-1}(1+n^2)}{n^3}$ \ee \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item (24 points) For what values of $x$ do the following series (i) converge absolutely, (ii) converge conditionally? \begin{multicols}{3} \begin{enumerate} \item $\displaystyle \sum_{n=1}^\infty (\ln x)^n$ \item $\displaystyle \sum_{n=1}^\infty \frac{n^5(x-3)^{2n}}{4^n} $ \item $\ds \sum_{n=1}^{\infty} \frac{2^n x^n}{(3n)! n!}$ \end{enumerate} \end{multicols} \item (18 points) \be \item Use the Binomial Theorem to calculate the first four non-zero terms in the Maclaurin series for $\ds \frac{1}{\sqrt{1-x^2}}$. \item Using integration and your answer from part (a), find the first four terms in the Maclaurin series for $\ds \sin^{-1} x$. \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item (16 points) Let $ \ds x(t)=\frac{1}{3}(2t+3)^{3/2}$ and $\ds y(t) = t + \frac{t^2}{2}$ for $0 \leq t \leq 3$. \be \item Use your calculator to sketch this curve. \item Find the length of the curve. \ee %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item (30 points) Let $\ds r_1 = \cos \theta$ and $r_2= 1-\cos\theta$ be two equations given in polar coordinates. \be \item Calculate the length of the perimeter of $\ds r_1 = \cos \theta$. \item Find all the intersection points between $r_1$ and $r_2$. \item Using your calculator, graph the curves described by $r_1$ and $r_2$. Carefully label the intersection points on your graph. \item Find the area inside the graph of $r_1$ and outside the graph of $r_2$. \ee \item (EXTRA CREDIT, 10 points) Let $\ds f(n) = \int_0^{\infty} t^{n-1}e^{-t} \; dt$ for all positive integers $n$. \be \item Show that $f(1)=1$. \item Use integration by parts to show that $f(n+1)= nf(n)$. \item Use part (b) to give a simple formula for $f(n)$, for any integer $n \geq 1$. \ee \end{enumerate} \end{document}