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\large
\begin{tabbing}
\hskip23pc \= Math.\ 270-03\\
\>Spring, 1997\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Second Examination \\
\it Friday, March 7, 1997
\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. 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.}

\begin{enumerate}
\begin{table}[h] \caption{Average temperature by Month\label{tbl:ave-temp}}
\bigskip
\centerline{\parbox{2.8in}{$\begin{array}{|r|r|r|r|r|r|r|} \hline
  & \mbox{Jan} & \mbox{Feb} & \mbox{Mar} & \mbox{Apr} & \mbox{May} & \mbox{Jun} \\ \hline
\phantom{.}^{\circ}\mbox{F} & 51.4 & 54.1 & 60.8 & 67.5 & 73.9 & 79.7 \\ \hline
 \hline
 & \mbox{Jul} & \mbox{Aug} & \mbox{Sep} & \mbox{Oct} & \mbox{Nov} & \mbox{Dec} \\ \hline
\phantom{.}^{\circ}\mbox{F} & 81.7 & 81.5 & 77.4 & 67.8 & 58.8 & 53.1 \\\hline
\end{array}$}}
\end{table}

\begin{figure}
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\caption{Plot of the data from
         Table~\protect{\ref{tbl:ave-temp}}\label{fig:temp-data}}
\end{figure}

\item \label{prob:lafayette-temp}
      Table~\ref{tbl:ave-temp} gives average temperature by month for
      a certain city.  Use the data to answer the following
      questions. (Also refer to the plot in Figure~\ref{fig:temp-data},
      where the temperature $T$ is plotted as function of the number
      of months $t$ from the beginning of the year.)

      \begin{enumerate}

      \item  At what rate in degrees per month, approximately, is the
              average temperature
             increasing during March?

      \item  In what month does the maximum temperature occur?
             (Note:  The maximum and minimum temperatures occur in a months
              in which the rate of change of temperature is
              equal to 0.)

      \item In what month does the minimum temperature occur?

      \item In what month is the temperature increasing most rapidly?
            What is, approximately, the maximum rate of increase?

      \item In what month is the temperature decreasing most rapidly?
            What is, approximately, the maximum rate of decrease?
      \end{enumerate}

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\end{picture}
\caption{The graph for
Problem~\protect{\ref{prob:sinc}\label{pic:sinc}}}
\end{figure}

\item \label{prob:sinc}
      The graph in Figure~\ref{pic:sinc} shows
$$f(x)=\left\{\begin{array}{rl} \sin(x)/x & \mbox{if $x\ne0$;} \\
                                       1  & \mbox{if $x=0.$}
              \end{array}\right.$$
      Refer to that graph to answer the following questions.
      \begin{enumerate}
      \item Give, approximately, the values of $x$ between $-10$ and
            $10$ at which $f'(x)=0$.
      \item Give, approximately, a value of $x$ between $-10$ and
            $10$ for which $f'(x)$ is maximum.  What can you
            say about $f''$ there? What is this  maximum $f'$?
      \item Give, approximately, a value of $x$ between $-10$ and
            $10$ for which $f'(x)$ is minimum.  What is this
            minimum $f'$?
      \item Give approximate ranges of $x$ between $-10$ and $10$
            for which $f''(x)>0$.
      \item Give approximate ranges of $x$ between $-10$ and $10$
            for which $f''(x)<0$.
      \end{enumerate}

\item Use the definition of the derivative as a limit of a
      difference quotient, that is, use
      $$f'(x) = \lim_{h\rightarrow0} \frac{f(x+h)-f(x)}{h}$$
      to compute the derivative function $f'(x)$ to $f(x) = x^3$.

\item A person with a certain liver disease first exhibits larger and
      larger concentrations of certain enzymes (called SGOT and SGPT)
      in the blood.  As the
      disease progressses, the concentration of these enzymes drops,
      first to the predisease level and eventually to zero
      (when almost all of the liver cells have died).  Monitoring the
      levels of these enzymes allows doctors to track the progress
      of a patient with this disease.  If $C=f(t)$ is the concentration
      of the enzymes in the blood as a function of time:
      \begin{enumerate}
      \item Sketch a possible graph of $C = f(t)$.
      \item Mark on the graph the intervals where $f'>0$ and where
            $f'<0$.
      \item What does $f'(t)$ represent, in practical terms?
      \end{enumerate}



\end{enumerate}

\end{document}
