\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1
\def\r{{\bf r}}
\def\a{{\bf a}}
\def\v{{\bf v}}
\def\A{{\bf A}}
\def\B{{\bf B}}
\def\C{{\bf C}}


\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 34\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 34} 
\centerline{\bf a.k.a. Practice Exam 2}
\bigskip

$$
\hbox{1.  Show that }{\partial^2 z\over\partial x\partial
y}={\partial^2 z\over\partial y\partial x} \hbox{ for } z=2^{x^3y}.
\hskip2.7in
$$

\bigskip
\item{2.}  Find the plane perpendicular to the path $(t^2,t,\cos2\pi
t)$ at time $t=1/4$.

\bigskip
\item{3.}  An object is traveling along the path $(a\sin t,b\cos
t,t)$ for $0\leq t<2\pi$ where $a$ and $b$ are constants with $a>b$.
At what location(s) is the object moving the fastest?

\bigskip
\item{4.}    Consider the paths 
$$\eqalign{
\gamma(t)&=(\cos t,t,t^2) \cr
\hbox{and}\qquad
\sigma(t)&=(t^2+1,-t,\sin t). \cr}
$$
Find the equation of a plane which is tangent to both of these curves
at a point of their intersection.

\bigskip
\itemitem{5.\hskip12pt a)}  Sketch a graph in polar coordinates of
$r=1-2\sin\theta$.  

\medskip
\itemitem{b)}  Find the area of the smallest loop.  

\medskip
\itemitem{c)}  Set up the integral which gives the arclength around
this loop.

\bigskip
\item{6.}  Let $A$, $B$, and $C$ be the vertices of a triangle.  Let
$\A=\vec{OA}$, $\B=\vec{OB}$, and $\C=\vec{OC}$.  Let $P$ be the point
that is on the line segment joining $A$ to the midpoint of the edge
$BC$ and twice as far from $A$ as from the midpoint.  Show that
$\vec{OP}=\A+\B+\C)/3$. 

\bigskip
\item{7.}  Given that the orbit of a planet about a sun (located at
the origin) is 
$$
r={k\over 1+e\cos\theta}
$$
for some constants $k>1$ and $0