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\begin{tabbing}
\hskip36pc \= Math.\ 302-01\\
\>Spring, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Final Examination \\
\it Monday, May 2, 2005, 10:15AM to 12:45PM
\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. You may keep this exam sheet.  Each problem is
worth 16 points, and 4 points are ``free."}

\begin{enumerate}
   \item  A person swimming in the ocean, one mile south of the
          beach, encounters a cross-current of 2 miles per hour
          flowing west, parallel to the beach. How fast and in
          what direction would the person need to swim to
          follow a line back to the beach that is perpendicular
          to the beach, such that the person gets back to the
          beach in an hour? Give the direction the person must
          swim as an angle with respect to east. {\em Hint: It
          may help to draw a picture.}
   \item  An object is traveling along a trajectory such that,
          at time $t$ seconds after starting, it is at $x(t) =
          t^2$, $y(t) = t^2$, $z(t)=t^{3/2}$ meters from its
          starting position. On the other hand, the temperature
          at point $(x,y)$ from the object's starting position
          is somehow known to be $T(x,y,z) = 25 - .0001x - .0002y
          -.005z$ degrees centigrade.
          \begin{enumerate}
             \item  Find the temperature at the object's
                    position one minute after the object has
                    started.
             \item  Find the rate of change of temperature at
                    the object's position one minute after the
                    object has started,
                    \begin{enumerate}
                       \item  by directly plugging in, and
                       \item  by using the chain rule for
                              functions of several variables.
                    \end{enumerate}
          \end{enumerate}
   \item An industrial container in the shape of a box (with
         rectangular sides, intersecting at right angles) is to be
         constructed.  The volume of the box must be 10 cubic
         meters.  The cost of the bottom of the box is \$10 per
         square meters, the cost of the front and back is \$4 per square
         meter, the cost of the right and left sides is \$2 per
         square meter, and the cost of the top is \$1 per square
         meter.  What should the dimensions of the box be to
         minimize the cost?
   \item Use Green's Theorem to evaluate the work needed to traverse
         the unit circle centered at the origin in a
         counterclockwise direction, through a vector field given by
         $\vec F(x,y) = (0, x+y)$.
   \item Compute the flux of the vector field $\vec F(x,y,z) =
         (x,y,z)$ through that portion of the paraboloid of revolution
         $z = x^2 + y^2$ for $-1\le x\le 1$, $-1\le y\le 1$;
         assume that the
         paraboloid surface is oriented outward from the $z$-axis.
   \item Use the divergence theorem to compute the flux through
         the sphere $x^2 + y^2 + z^2 = 1$ of the vector field
         $\vec F(x,y,z) = (y^2/2, -xy + y, xz - yz).$
\end{enumerate}
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