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\begin{tabbing}
\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Fifth Examination \\
\it Thursday, December 1, 2005
\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. Each entire problem is worth 25 points. You may
keep this exam sheet.}

\begin{enumerate}
   \item \label{prob:balloon-position} A weather balloon is
          released, and its velocity in the eastward direction is
          measured.  (That is, positive velocities represent motion
          to the east, and negative velocities represent motion to
          the west.)  A graph of the balloon's velocity in miles per
          hour as a function of hours since the balloon's release is
          shown in Figure~\ref{fig:subspeed}.
          \begin{figure}[h]
            \centerline{\includegraphics[scale=0.5]{subspeed.eps}}
            \caption{\label{fig:subspeed} The eastward velocity of the
            weather balloon in
            Problem~\protect{\ref{prob:balloon-position}}}
          \end{figure}
          \begin{enumerate}
             \item Approximately how many hours after release is the
                   balloon moving fastest in the eastward direction,
                   and what is its approximate speed at that point?
             \item Does the balloon ever change direction?  If so,
                   approximately how many hours after its release
                   does it do so?
             \item Approximately how far east does the balloon go
                   from its starting position?  (That is, what is
                   the balloon's maximum distance east of its
                   starting position?)
             \item Approximately how many miles east of its starting
                   position is the balloon
                   \begin{enumerate}
                      \item 2 hours after release?
                      \item 4 hours after release?
                      \item 6 hours after release?
                   \end{enumerate}
          \end{enumerate}
   \item Compute exactly
         $$\begin{array}{rlrlrl}
            \textrm{(a)} & \ds\int_{-1}^1 x e^{-x^2}\/dx & \quad
            \textrm{(b)} & \ds\int_{-1}^1 x^2 \/dx & \quad
            \textrm{(c)} & \ds\int_{-1}^1 x^2+x\/dx
         \end{array}$$
   \item Suppose $f$ is odd, $g$ is even, and
         $$\begin{array}{rlrlrl}
             & \ds\int_1^2 f(x)\/dx = 1, & \quad
            \ & \ds\int_0^1 g(x)\/dx = 2, & \quad
            \textrm{and} & \ds\int_1^2 g(x)\/dx = 3.
         \end{array}$$
         Then compute $\ds\int_{-1}^2 2f(x) + g(x)\/dx$.
   \item Compute the average value of $x^2$ over the interval $0$ to
         $4$.
\end{enumerate}
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