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\begin{document}

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\vspace{-44pt}

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\textbf{Math 107 \ Fall 2003 \hspace{42pt} FINAL EXAM \hspace{42pt} Name
\rule{170pt}{0.5pt}}

\vspace{3pt}
\textbf{Dec. 16, 2003 \hspace{125pt} Student ID Number:
\rule{170pt}{0.5pt}}

\vspace{15pt} Circle your instructor's name.

\qquad \textbf{Instructor:} \ \ Deng \ \ \ Haataja \ \ \ Rogge \ \ \ Thomas

Circle your TA's name, if applicable.

\qquad \textbf{TA:} \ \ Eubanks \ \ \ Evans \ \ \ Everson \ \ \ Harrington
\ \ \ Higgins \ \ \ Miller


\vspace{13pt}
\textbf{INSTRUCTIONS:}

\vspace{-8pt}
\begin{itemize}
  \item This exam should have 6 pages of questions plus this cover sheet;
please verify that it does.
  \item Work each problem completely and clearly in the space provided.
Show all your work for full credit. If you use the back of exam pages for
scratch work, make note of that in the space provided for the problem.
  \item Simplify as much as possible, except as noted. For example,
write $\sqrt{2}/2$ instead of $\cos(\pi/4)$.
  \item Give exact answers only, except as noted. For example, use $\pi$
instead of $3.1415$ if $\pi$ is the answer.
  \item Calculators are allowed (except TI-89's, TI-92's and similar models),
but \textbf{an answer will only be counted if it is supported by all the
work necessary to obtain that answer.}
  \item Do not spend too long on any one problem; note the point value of the
problem when deciding how much time to spend! You do not need to work the
problems in the order they appear.
\end{itemize}

\vspace{5pt}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Question & Points & Score \\ \hline
1 & 14 & \\ \hline
2 & 14 & \\ \hline
3 & 14 & \\ \hline
4 & 14 & \\ \hline
5 & 12 & \\ \hline
6 & 36 & \\ \hline
7 & 20 & \\ \hline
8 & 12 & \\ \hline
9 & 8 & \\ \hline
10 & 18 & \\ \hline
11 & 8 & \\ \hline
12 & 12 & \\ \hline
13 & 18 & \\ \hline
Total & 200 & \\ \hline
\end{tabular}
\end{center}

\newpage
\begin{enumerate}
\setlength{\itemsep}{130pt}
  \item (14 points) Construct a definite integral to determine the area
enclosed by the graphs of $x-y^2=1$ and $y=-x+3$.
\textbf{DO NOT EVALUATE THIS INTEGRAL.}

\vspace{10pt}
\includegraphics[height=2.0in]{graph_blank1.eps}

%\vspace{70pt}
%  \item (12 points) Set up an integral for computing the volume of the solid
%formed by rotating the region bounded by $y=x^2-1$ and $y=1-x^2$ about the
%line $x=2$. Use the method of \underline{cylindrical shells}.
%  \begin{enumerate}
%  \setlength{\itemsep}{160pt}
%    \item Sketch the region, carefully labeling all relevant quantities.
%    \item Set up an integral that yields the volume of this solid.
%\textbf{DO NOT EVALUATE THIS INTEGRAL.}
%  \end{enumerate}

\vspace{-80pt}
  \item (14 points) Set up integral(s) to compute the volume of the solid
formed by rotating the region bounded by $y=\frac{1}{2}x$, $y=x-2$, and the
$x$-axis about the line $x=-3$.
  \begin{enumerate}
%  \setlength{\itemsep}{20pt}
    \item Sketch the region, carefully labeling all relevant quantities.

\vspace{10pt}
\includegraphics[height=2.0in]{graph_blank1.eps}
    \item Set up integral(s) to compute the volume of this solid.
\textbf{DO NOT EVALUATE THIS INTEGRAL.}
  \end{enumerate}

\vspace{60pt}
  \item (14 points) A water tank in the shape of a right circular cylinder
has a base radius of 5 feet and a height of 10 feet. Suppose that the tank
contains 7 feet of water. Set up an integral to determine the work done in
pumping all of the water out of the top of the tank.
\textbf{DO NOT EVALUATE THIS INTEGRAL.}

%\vspace{90pt}
%  \item \begin{enumerate}
%  \setlength{\itemsep}{90pt}
%    \item (6 points) Find the derivative of the function $y=f(x)=x^5-x-1$ and
%use it to show $f$ is invertible over the interval $[1,\infty)$.
%    \item (6 points) Show that the point $(1,-1)$ is on the graph of the
%function $f$, and find an equation of the tangent line at the point $(-1,1)$
%for the inverse function $y=f^{-1}(x)$.
%  \end{enumerate}

\vspace{10pt}
  \item \begin{enumerate}
  \setlength{\itemsep}{70pt}
    \item (7 points) Find the exact value of $\sin^{-1}(\sin(3\pi/4))$
    \item (7 points) Find $\frac{d}{dx}\sin^{-1}(\tan x)$
  \end{enumerate}

\vspace{-30pt}
  \item (12 points) Determine whether the improper integral converges or
diverges. If it converges, find the limit.

$\displaystyle{\int_0^1\frac{x}{x^2-1}dx }$

\vspace{70pt}
  \item (12 points each) Evaluate the integrals.
  \begin{enumerate}
  \setlength{\itemsep}{170pt}
    \item $\int\cos^3 x\sin^2 x dx $
    \item $\int_1^e (\ln x)^2dx $
\vspace{15pt}
    \item $\int\frac{x-4}{x^2+2x}dx $
  \end{enumerate}

\vspace{90pt}
  \item (10 points each) Given the Taylor series centered at $0$ (the
Maclaurin series) for

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$, for all $x$.

  \begin{enumerate}
  \setlength{\itemsep}{160pt}
    \item Use this series to estimate $e$ accurate to three decimal places.
How many terms are required? You must show your work.

    \item Use the first three terms of the Maclaurin series for $e^x$ to
estimate $\int_0^{1}e^{-x^2}\,dx$. You must show your work.
  \end{enumerate}

\vspace{30pt}
  \item (12 points) A rubber ball, when dropped on concrete, rebounds 90
percent of the distance it falls. If it is dropped from a height of 6 feet,
how far does it travel, both up and down, before coming to rest?

\vspace{40pt}
  \item (8 points) Find the radius of convergence of this power series:
$\sum_{k=0}^{\infty} \frac{k^2}{3^k}x^k$

  \item (9 points each) Determine if each of the following series converges
absolutely, converges conditionally, or diverges. Show enough work to justify
your answer.
  \begin{enumerate}
  \setlength{\itemsep}{130pt}
    \item $\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{\sqrt{k^2+3}}$
    \item $\sum_{k=1}^{\infty}\frac{5k}{(k+2)2^k}$
  \end{enumerate}

%\vspace{10pt}
  \item (8 points) Find an $x$-$y$ equation for the curve given by $x=4\cos t$,
$y=\sin t$.

  \item Consider the curve given by the parametric equations $x=t^2+3$,
$y=\cos t$, where $0\leq t\leq\pi$.
  \begin{enumerate}
  \setlength{\itemsep}{90pt}
    \item (6 points) Compute the slope of the tangent line of this curve at
$t=\pi /2$.
    \item (6 points) Set up an integral to compute the arc length of this
curve.
\newline\textbf{DO NOT EVALUATE THIS INTEGRAL.}
  \end{enumerate}

\vspace{-10pt}
  \item \begin{enumerate}
%  \setlength{\itemsep}{150pt}
    \item (6 points) Sketch the graph of $r=1-2\cos(2\theta)$.

\vspace{10pt}
\includegraphics[height=2.0in]{graph_blank1.eps}
    \item (12 points) Set up an integral to find the area of \underline{one
large loop} of this graph. \textbf{DO NOT EVALUATE THIS INTEGRAL.}
  \end{enumerate}


\end{enumerate}
\end{document}
