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\begin{tabbing}
\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Third Examination \\
\it Tuesday, October 25, 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.}

\large
\begin{enumerate}
   \item  For the following functions $f$, compute $df/dx$
          by hand.  You will not be
          given full credit unless you show all steps in your
          computations.
          $$\begin{array}{rlrlrl}
          \textrm{(a)} &
             \ds{}f(x) = \sin(x^2) \qquad &
          \textrm{(b)} &
             \ds{} f(x) = \frac{x^2+1}{x^3+1} \qquad &
          \textrm{(c)} & f(x) = \log(x^2\sin(x))
          \end{array}$$
   \item Suppose a balloon is rising from sea level, and that
         the height of the balloon at time $t$ minutes from its
         launching is approximately
         $h(t) = 10000\left(1 - e^{-t/60}\right)$ feet.  Suppose
         further that the temperature at height $h$ feet above sea
         level at the site of the balloon launching is
         approximately $T(h) = 90 - 0.7\sqrt{h}$ degrees
         Fahrenheit.
         \begin{enumerate}
            \item What is the altitude one hour after launching?
            \item At what rate, in degrees per minute, is the
                  temperature in the balloon decreasing one hour
                  after launching?
         \end{enumerate}
   \item If $f(x) = (\sin(x))^3$ for
         $x\in[-\left({\pi}/{2}\right)^3,
                     \left({\pi}/{2}\right)^3]$,
         then compute the instantaneous
         rate of change of $f^{-1}(y)$ with respect to $y$ when
         $y=(\sqrt{2}/2)^3$.
   \item Consider the function
         $$f(x) = \frac{1}{3} x^3 - x.$$
         Use $f'(x)$ to answer the following questions. For full
         credit, be sure to explain fully.
         \begin{enumerate}
            \item Is $f$ increasing or decreasing for
                  $x\in[-1,1]$?
            \item If $\ds{}g(x)=(1/3)x^4-x$, then $g(1)=f(1)=0$.
                  Explain why $g(x) \ge f(x)$ for $x\ge1$.
         \end{enumerate}


\end{enumerate}
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