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\title{ESP Workshop, Worksheet \#18}
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{\bf \Large ESP Workshop, Worksheet \#18}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
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As promised, here are some more old exam problems...
\begin{enumerate}
\item Use the Chain Rule to find $\partial u/ \partial s$ if
\begin{displaymath}
u = x^2 -y^2,\ x = s\cos t,\ \textrm{and}\ y = t\cos s
\end{displaymath}
Write the answer in terms of $s$ and $t$.

\item The length $x$ and width $y$ of a rectangular parallelopiped are
  increasing at a rate of 3 cm per second, and the height $z$ is
  increasing at a rate of 2 centimeters per second. At what rate is
  the volume changing when the length, width, and height are 20, 15,
  and 10 cm respectively?

\item Find and simplify $\partial z/\partial y$ if $x e^z -z e^y =0$.

\item The radius $r$ and height $h$ of a right circular cylinder are
  changing in such a way that the volume $V$ is increasing at a rate
  of 2 cubic centimeters per second when the height is 12 centimeters
  and the radius is 3 centimeters. Find the rate of change of the
  height if the radius is increasing at a rate of 2 centimeters per
  second. ($V = \pi r^2 h$)

\item Find the directional derivative of $f(x,y) =xy^2$ at (2,-1) in
  the direction $\mathbf{v} = \mathbf{i} - \mathbf{j}$.

\item Find the equation of the tangent plane to the surface $x^2 -y^2
  -z^2 = 4$ at the point (3,1,-2).

\item Find the path of a heat-seeking particle placed at the point
  (-2,3) on a metal plate with the temperature field $T(x,y) = 20 -x^2
  -4y^2$.

\item Assume $f(x,y) = xe^y$.
\begin{enumerate}
\item Find the gradient of $f$.
\item Find the directional derivative of $f$ at (-2,1) in the
  direction $3\mathbf{i} +4\mathbf{j}$.

\item What is the maximum value of the directional derivative of $f$
  at (-2,1)?

\end{enumerate}

\item (Challenge!) Among all planes that are tangent to the surface
  $x y^2 z^2 = 1$, find the ones that are farthest from the origin.

\item (Challenge!) Suppose that $f$ is a differentiable function of
  one variable. Show that all tangent planes to the surface $z = x
  f(y/x)$ intersect in a common point.

\end{enumerate}



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