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\begin{tabbing}
\hskip23pc \= Math.\ 270-03\\
\>Spring, 1997\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Third Examination \\
\it Wednesday, April 2, 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 20 points.}

\begin{enumerate}

\item
Suppose that the mileage that an automobile gets decreases gradually as the
automobile engine ages, according to the following table:
$$\begin{array}{|r|c||r|c|}
\hline
\mbox{Mileage} & \mbox{Miles per gallon} & \mbox{Mileage} &
\mbox{Miles per gallon} \\ \hline
0       & 35      &  60,000 & 30 \\ \hline
20,000  & 33      &  80,000 & 27 \\ \hline
40,000  & 32      & 100,000 & 24 \\ \hline \hline \end{array}$$
Based on this table, give lower and upper bounds on the total amount of
gasoline the automobile would use if it were driven 100,000 miles.
{\em Hint: You should first convert miles per gallon to gallons per mile.}

\item \label{prob:area-integral}
If $f$ is as in Figure~\ref{fig:integral-area}, then give an
approximation to
$$\int_0^8 f(x) \/dx.$$

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\caption{\label{fig:integral-area}The function $f$ for Problem~\protect{\ref{prob:area-integral}}}
\end{figure}

\item Find the average value of $f(t) = 1 + 2t$ over the interval
$t=0$ to $t=2$.

\item \label{prob:fundamental-theorem}
If $F(x) = x^4/4$, then it is known that $F'(x)=x^3$.  Use
this fact to compute the following integrals.
$$
\mbox{(a)} \quad\int_{-1}^1 x^3\/dx  \qquad\qquad\mbox{(b)} \quad\int_0^2 x^3 \/dx
$$

\item Compute left and right Riemann sums to $$\int_{-1}^1 x^3\/dx$$ with 4 subintervals of length $\frac{1}{2}$. Compare the result to your result in Problem~\ref{prob:fundamental-theorem}.

\end{enumerate}

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