\bigskip
\noindent
I.  
\item{A)} Find the equation of the plane containing the point
$(0,2,-1)$ and perpendicular to the line passing through the
points $(0,1,-2)$ and $(3,2,2)$.
\item{B)} Give a parametrization of the line from part A.
\item{C)} At what point does the line from part A intersect 
your plane?
\bigskip
\noindent
II. All parts of this problem refer to the parametric curve
$$\alpha(t) = (t^2+t-2,t^3+t^2-2t)$$
(defined for all $t \in {\bf R}$).
\item{A)} Thinking of $\alpha(t)$ as the position of a moving
object as a function of time, at how many different times is 
the object at the location $(x,y) = (0,0)$?  What times are they?
\item{B)} Find all $t$ at which the tangent vector to the curve
is a horizontal vector.
\bigskip
\noindent
III.  Find the unit tangent vector and the curvature at a 
general point on the curve $\beta(t) = (t\cos t, t\sin t, 3t)$
in ${\bf R}^3$.  
\bigskip
\noindent
IV.  Consider the vector field on ${\bf R}^3$ given by 
$${\bf F}(x,y,z) = (-3x, y^2 - 1, z)$$
\item{A)} Find the derivative matrix $D({\bf F})$.
\item{B)} Now let $T : {\bf R}^3_{\rho,\phi,\theta} \to {\bf R}^3_{x,y,z}$
be the spherical coordinate mapping.  Compute $D(F\circ T)$ two ways -- once
by direct substitution and differentiation, once by the Chain Rule.  
\item{C)} Verify that $\alpha(t) = (e^{-3t},{e^{-t} - e^t \over e^{-t} + e^t},e^t)$
is a {\it flow line\/} of ${\bf F}$.
\item{D)} Compute the divergence and curl of ${\bf F}$ .
\item{E)} Find a function $f : {\bf R}^3 \to {\bf R}$ such that 
$\grad f = {\bf F}$.
\bigskip
\noindent
V.  Both parts of this question refer to $f(x,y) = x^4 - 3xy$.
\item{A)}  At the point $(1,2)$, what is the {\it direction of maximum
rate of increase\/} of $f$?  Express your result as a unit vector.
\item{B)}  Find the equation of the plane in ${\bf R}^3$ through
the point $(1,2,f(1,2))$, containing the vectors
$(1,0,{\partial f\over \partial x}(1,2))$ and 
$(0,1,{\partial f\over \partial y}(1,2))$.
\bigskip
\noindent
VI.  The vector field plotted above in question III is the gradient 
vector field of the function 
$$f(x,y) = y^3/3 + x^3/3 - y - x^2/2$$
\item{A)} What are the critical points of $f(x,y)$?
\item{B)} Apply the {\it Second Derivative Test\/} to each of 
the critical points and check your responses to part A.
\bigskip
\noindent
VII.  In the vicinity of Snowshoe Pass, which is located at coordinates
$(0,0)$, the elevation of the terrain
is given by the function $f(x,y) = xy + 0.9$ ($x,y,f$ in miles).
A group of hikers walks
along the path given by the equation $9x^2 + (y-1)^2 = 1$, making
one complete circuit of the path.
What are the maximum and minimum elevations that the hikers reach?  
\bigskip
\noindent
VIII.  A region $R$ in the plane is the set of all $(x,y)$ satisfying
$x^2 + y^2 \le 16$, $y \ge 0$, and $y \ge \sqrt{3} x$.
\item{A)}  Set up iterated integral(s) to compute $\int\!\int_R f(x,y)\ dA$
using {\it rectangular coordinates\/}.
(Use any convenient ordering of the variables: $dx\, dy$ or $dy\, dx$, then 
switch to the other for practice).
\item{B)}  Repeat part A, but using {\it polar coordinates\/}.
\item{C)}  A thin plate has the shape of  
$R$ and density $d(x,y) = y^2$ at the point $(x,y)$.  
Compute its {\it center of mass\/}.
\bigskip
\noindent
IX.   A hole is drilled through a solid sphere of radius
3 centered at the origin in ${\bf R}^3$, removing all the material that 
was at points satisfying $x^2 + y^2 \le 1$.  What is the volume of the 
remaining solid?
\bigskip
\noindent
X.  Let ${\bf F}$ be the vector field $(P(x,y),Q(x,y)) = (x^2 + 3y, y^2 - x)$.
\item{A)}  Let ${\bf x}(t) = (t,t^2)$, $0\le t \le 1$. Compute the line
integral $\int_{\bf x} {\bf F}\cdot d{\bf x}$.
\item{B)}  State Green's Theorem.
\item{C)}  Let $D$ be the closed disk $D = \{(x,y) : (x-1)^2 + y^2 \le 1\}$.
Verify that the conclusion of Green's Theorem holds for this region and this vector field.
