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\def\pt#1{\hbox{#1) }}
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\input amssym.tex
\input epsf
\def\R{{\Bbb R}}
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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 36\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 36} 
\bigskip

\item{1.}  True or False.  If false, make the statement true.

\medskip
\itemitem{a)}  If $f$ is a differentiable function of $x$ and $y$,
then the directional derivative of $f$ in the direction of the unit
vector ${\bf u}$ is $D_{\bf u}f(x,y)=\nabla f(x,y)\cdot{\bf u}$.

\medskip
\itemitem{b)} ${\bf a}\cdot{\bf b}=||{\bf a}||\,||{\bf b}||\sin\theta$.

\medskip
\itemitem{c)}  The minimum value of ${\bf a}\cdot{\bf b}$ is $||{\bf
a}||$.

\medskip
\itemitem{d)}  The maximum value of $D_{\bf u}f(x,y)$ is $\pi$ for any
$f(x,y)$. 

\medskip
\itemitem{e)}  $D_{\bf u}f(x,y)=0$ for ${\bf u}={\nabla
f(x,y)\over||\nabla f(x,y)||}$.

\medskip
\itemitem{f)}  The direction of {\bf maximum} increase of $f$ is given
by $\nabla f(x,y)$.
  

\bigskip 
\item{2.}  The temperature distribution around a certain point (set
here to be the origin $(0,0,0)$ in 3-space) is given by
$T(x,y,z)=\sqrt{x^2+y^2+z^2}$.  That is, the temperature at any point
is equal to the distance from that point to the origin.

\medskip
\itemitem{a)}  Find $\nabla T$.

\medskip
\itemitem{b)}  At the point $(1,2,-2)$, what direction would you travel
to experience the greatest increase in temperature?  the least?

\medskip
\itemitem{c)}  From the point $(1,2,-2)$, indicate two directions in
which the rate of change in temperature would be zero.  (Make these
two directions which are NOT simply $180^\circ$ apart.)

\medskip
\itemitem{d)}  Describe the level set consisting of all points
$(x,y,z)$ where the temperature is 3 units, that is, where
$T(x,y,z)=3$.

\medskip
\itemitem{e)}  Find an equation for the plane which is tangent to the
surface you describe in part d) at the point $(1,2,-2)$.

\medskip
\itemitem{f)}  A particle traces the path ${\bf x}(t)=(\cos t,\sin
t,t)$ for $0\leq t\leq2\pi$.  At what point of the path is the lowest
temperature experienced? 

\bigskip
\item{3.}  Let $f(x,y,z)=x^2e^{-yz}$.  Compute the rate of change of
$f$ in the direction of the unit vector
$$
{\bf v}=\left({1\over\sqrt3},{1\over\sqrt3},{1\over\sqrt3}\right)
\quad\hbox{at}\quad(1,0,0).
$$


\eject
\item{4.}  Assume that $\nabla f({\bf x})\neq{\bf 0}$.  Prove that
$\nabla f({\bf x})$ points in the direction along which $f$ is
increasing the fastest.
\item{} (Hint:  Think directional derivative.)


\bigskip
\item{5.}  Below is the graph of a function $f\colon\R^2\to\R$.
Sketch $\nabla f$ as a vector field on $\R^2$.

\centerline{
\epsfxsize=3.5in\epsfbox{/home/oehrtman/m210s96/3dgraph.ps}}


\bigskip
\item{6.}  Sketch the level set $f(x,y,z)=1$ for the function $f$
given in Problem 3.  Find two nonparallel vectors at the point
$(-1,0,0)$ that are tangent to this level set.

\medskip
\item{}  Hint:  to sketch the level surface, first consider what the
intersections of this surface with the horizontal planes
$z=$ constant look like.



\bye