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\hskip36pc \= Math.\ 302-01\\
\>Spring, 2005\\
\>R. B. Kearfott
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\bf Third Examination \\
\it Tuesday, February 22, 2005
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\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. You
may keep this exam sheet.  Each entire problem is worth 30 points,
while 10 points are free.}

\Large
\begin{enumerate}
   \item \label{ex:compute-partials}
         Write down all five first and second-order partial
         derivatives of $f(x,y)=x^2\sin(x+y)$.
   \item Compute the directional derivative of the function in
         problem~\ref{ex:compute-partials} at $(x,y) =
         (\pi/2,\pi/2)$ in the direction $\vec u =
         (1/\sqrt{2},-1/\sqrt{2})$.
   \item Suppose the relative humidity $H$ at a point $x$ units to the right,
         $y$ units forward, and $z$ units up from a corner of a
         room, $x\in[0,20]$, $y\in[0,20]$, $z\in[0,12]$
         is modelled by the equation
         $$H(x,y,z) = 100
         \left(\frac{(z-6)^2}{72}+.5\right)e^{-\frac{x^2+(y-10)^2}{500}}.$$
         A biologist has determined that a mosquito will fly in the
         direction of maximum increase in relative humidity.  If a
         mosquito is at $(5,10,6)$, then:
         \begin{enumerate}
            \item What is the relative humidity at the point where the
                  mosquito is?
            \item In what direction will the mosquito fly? Give the
                  direction as a vector.
         \end{enumerate}
\end{enumerate}

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