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\begin{tabbing}
\hskip36pc \= Math.\ 301-02\\
\>Summer, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Final Examination \\
\it Saturday, July 30, 2005, 7:30AM--10:00AM
\end{center}

\noindent {\small {\bf Instructions:} This exam should be done on
your own paper. Your name should be on each sheet and on the back of
the last sheet; the answers should appear written carefully and in
order. If in doubt, show intermediate steps: Full credit may not be
given, even for correct answers, unless work is arranged clearly and
explained. This exam is closed book. You may leave after handing in
your exam paper, but be sure to check your answers carefully. You
may keep this exam sheet. Each entire problem is worth 16 points,
and 4 points are ``free."}

\begin{enumerate}
   \item \label{prob:compute-integrals}
      Compute the following integrals.  Show all work or explain.
      $$\begin{array}{rclrclrcl}
        \mbox{(a)} & \displaystyle
            \int_{x=0}^1 \frac{\arctan(x)}{x^2+1}\/dx & \quad
         \mbox{(b)} & \displaystyle
            \int_{x=0}^{\pi/2} x\sin(2x)\/dx & \quad
         \mbox{(c)} & \displaystyle
            \int_{x=-2}^{2} \frac{1}{\sqrt{4-x^2}}\/dx
      \end{array}$$
   \item
         Tell which of these integrals converge and which do not,
         and give reasons.  Of those integrals that converge,
         calculate the exact values for those that can be written in
         closed form.
         $$\begin{array}{rclrclrcl}
           \mbox{(a)} & \displaystyle
               \int_{x=0}^1 \frac{x+1}{\sqrt{x}}\/dx & \quad
           \mbox{(b)} & \displaystyle
               \int_{x=0}^1 \frac{\sqrt{x}}{x+1}\/dx & \quad
           \mbox{(c)} & \displaystyle
               \int_{x=\pi}^\infty \frac{\sin{(\sin{x})}}{x^2}\/dx & \\ \\
           \mbox{(d)} & \displaystyle
               \int_{x=0}^\infty xe^{-x}\/dx & \quad
           \mbox{(e)} & \displaystyle
               \int_{x=-\infty}^0 xe^{-x}\/dx & \quad
           \mbox{(f)} & \displaystyle
               \int_{x=-\infty}^\infty xe^{-x}\/dx
         \end{array}$$
   \item Compute the total force in pounds of a square plate of
         glass that forms a vertical wall of a freshwater aquarium.
         The length of one side of this plate is 6 feet.  (Note:
         The density of water is 62.4 pounds per cubic foot.)
   \item Compute the volume of the solid obtained by rotating the
         portion of the curve $y=e^x$ between $x=0$ and $x=1$ about
         the line $y=-1$.
   \item \begin{enumerate}
         \item Use the trapezoid rule with 1, 2, and 4 subintervals to
               compute approximations to the integral in
               Problem~\ref{prob:compute-integrals}(a).  Carry 4
               digits in the computations.
         \item Compare your approximations to the value obtained by
               computing a numerical value for your answer from
               Problem~\ref{prob:compute-integrals}(a).
         \end{enumerate}
   \item \begin{enumerate}
         \item Write down the Taylor polynomial $p_{8}(x)$ of degree 8 for
               $e^{-x^2}$, centered at $x=0$.
         \item Write down a bound for the error $e^{-x^2}-p_{8}(x)$
               for $-1\le x\le 1$.
         \item Evaluate $\int_{x=-1}^1 e^{-x^2}\/dx$ approximately
               by replacing $e^{-x^2}$ by $p_{8}(x)$ in the
               integrand.
         \item Do you think your approximation is good?  Why?
         \end{enumerate}
\end{enumerate}
\end{document}
