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\begin{tabbing}
\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Fourth Examination \\
\it Friday, November 11, 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 20 points. You may
keep this exam sheet.}

\Large
\begin{enumerate}
   \item \label{prob:crit-points}
         Find and classify all critical points of
         $$f(x) = \frac{1}{3}x^3 - \frac{3}{2}x^2 + 2x.$$
   \item Find the global maximum and global minimum of the
         function in problem~\ref{prob:crit-points}, over the
         interval $x\in[-3,3]$.
   \item A farmer needs to fence in a pasture for his cattle, and
         decides to make a rectangular area.  The cattle require
         100,000 square feet of space.  The cost for the fencing
         on the back of the pasture is $\$5$ per foot, the cost
         for the front is $\$10$ per foot, while the cost for each
         side is $\$3$ per foot.  What should the dimensions of
         the pasture be to minimize the farmer's total cost?
   \item A voltage $V$ across a resistance $R$ generates a current
         $I=V/R$.  If a constant voltage of 12 volts is put across
         a resistance that is increasing at a rate of 0.5 ohms per
         second when the resistance is 2 ohms, at what rate is the
         current changing then?
   \item Use L'Hopital's rule to compute the following limits.
         $$\begin{array}{rlrlrl}
         \mbox{(a)} &
         \ds\lim_{x\rightarrow0} \frac{\sin^2(x)}{x^2} &
         \quad\mbox{(b)} &
         \ds\lim_{x\rightarrow\infty} \frac{\ln(x)}{x} &
         \quad\mbox{(c)} &
         \ds\lim_{x\rightarrow\infty}\frac{x^2 + 3x+2}{x^2\ln(x)}
         \end{array}$$
\end{enumerate}
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