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\begin{tabbing}
\hskip23pc \= Math.\ 270-03\\
\>Spring, 1997\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Final Examination \\
\it Thursday, May 8, 1997, 7:30--10:00AM
\end{center}

\noindent {\scriptsize
{\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
explanations are given. 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 your
exam sheet.  Each entire problem is worth 14 points, and 2 points
are ``free."}

\begin{enumerate}

\item Find:
$$\begin{array}{rlrl}
\mbox{(a)} & \frac{d}{dx}\Big(\ln\big(e^x\sin(x)\big)\Big) &
\qquad\mbox{(b)} & \frac{d}{dx}\left(\frac{x^2-1}{x^2+1}\right) \\
\mbox{\vphantom{\Huge I}}\qquad\mbox{(c)} & \frac{d}{dx}\sqrt{x^2+x+1}
\end{array}$$

\item One of the functions below is a quadratic, one is a
      cubic, and one is a trigonometric function.   Which is
      which?  State the reasons for your conclusions.

$$\begin{array}{ccc}
\mbox{$\begin{array}{r|r}
\hline
x & f(x) \\
\hline
0.2 & -0.42 \\
0.4 & -0.65 \\
0.6 &  0.96 \\
0.8 & -0.15 \\
1.2 &  0.84 \\
\hline
\end{array}$} & \qquad
\mbox{$\begin{array}{r|r}
\hline
x & f(x) \\
\hline
1.3 &  0.41 \\
1.7 &  0.81 \\
2.5 &  0.65 \\
3.0 & -0.10 \\
3.5 & -1.35 \\
\hline
\end{array}$} & \qquad
\mbox{$\begin{array}{r|r}
\hline
x & f(x) \\
\hline
0.5 & -1.13 \\
1.2 &  0.13 \\
1.8 &  0.03 \\
2.0 &  0.00 \\
2.2 &  0.05 \\
\hline
\end{array}$} \\
\mbox{(a)} & \qquad \mbox{(b)} & \qquad \mbox{(c)}
\end{array}$$

\begin{figure}
\input traffic.tcd
\caption{Traffic Along Speedway
         (for Problem~\protect\ref{prob:traffic})\label{fig:traffic}}
\end{figure}

\item \label{prob:traffic} To study traffic flow along a major road,
      the city installs a device at the edge of the road at 4:00AM.
      The device counts the cars driving past, and records the total
      periodically.  The resulting data is plotted on a graph, with
      time (in hours) on the horizontal axis and the number of cars
      on the vertical axis.  The graph is shown in
      Figure~\ref{fig:traffic};  it is the
      graph of the function
      $$C(t) = \mbox{Total number of cars that have passed by
                     after $t$ hours.}$$
      \begin{enumerate}
         \item When is the traffic flow the greatest?

         \item From the graph, estimate $C'(3)$.

         \item What is the meaning of $C'(3)$?  What are its
               units?  What does the value of $C'(3)$ you obtained
               mean in practical terms?
      \end{enumerate}

\item A function defined for all $x$ has the following properties.
      \begin{itemize}
         \item $f$ is increasing.
         \item $f$ is concave down.
         \item $f(5)=2$.
         \item $f'(5)=1/2$.
      \end{itemize}
      \begin{enumerate}
         \item Sketch a possible graph for $f(x)$
         \item How many zeros does $f(x)$ have and where are they
               located?  Justify your answer.
         \item Is it possible that $f'(1)=\frac{1}{4}$?  Justify
               your answer.
      \end{enumerate}

\item Find the
      \begin{enumerate}
         \item global maximum and
         \item global minimum
      \end{enumerate}
      of the function $f(x) = x^3-x$ over the interval
      $-1\le x\le 1$.

\item Eight hundred feet of fencing are available to enclose a
      rectangular area, one of whose sides is along a river bank
      (so that the fence comprises three sides of the
      river).  What should the dimensions of the rectangle be
      to maximize the enclosed area?

\item Compute
              $$\int_0^1 5 x^4 \/dx$$
      \begin{enumerate}
         \item by using the Fundamental Theorem of Calculus;
         \item approximately, by using a Riemann sum with
               four subdivisions.
      \end{enumerate}
      Use the monotonicity of the integrand to compute bounds
      on the integral, then compare to the
      exact answer.



\end{enumerate}

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