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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?
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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 times at which the instantaneous velocity of 
the object is a horizontal vector.
\item{C)} A thin wire has the shape of the portion of the 
curve $\alpha$ for $t \in [0,2]$ and density $d(t) = t^2$. 
Set up an integral to compute its total mass.  (You do not need
to evaluate!)
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III.  The following plot shows the vector field
$$F(x,y) = (x^2 - x, y^2 - 1)$$
and some flow lines:
\item{A)}  Using the component functions of $F$, determine all the
{\it critical points\/} of $F$. 
\item{B)}  Using the plot, classify each as a either a sink, 
a source, a saddle point, or a center.
\item{C)} Verify that $\alpha(t) = (0,{e^{-t} - e^t \over e^{-t} + e^t})$
is a {\it flow line\/} of the vector field $F$.
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IV.  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 
$(0,1,{\partial f\over \partial y}(1,2))$.
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V.  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)} Using the plot of $\nabla f(x,y)$, and your answers to III B,
classify each of the critical points of $f$ as a maximum, minimum, or neither.
\item{B)}  Apply the {\it Second Derivative Test\/} to each of 
the critical points and check your responses to part A.
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VI.  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?  
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VII.  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 the total accumulation
of a function $f(x,y)$ over $R$ using {\it rectangular coordinates\/}.
(Use any convenient ordering of the variables: $dx\, dy$ or $dy\, dx$).
\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\/}.
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VIII.   A hole is drilled through a solid sphere of radius
3 centered at the origin in ${\bf R}^3$.  
\item{A)}  If all the material that 
was at points satisfying $x^2 + y^2 \le 1$ is removed, 
what is the volume of the remaining solid? (Hint:  use cylindrical coordinates.)
\item{B)}  If all the material that was at points with $z^2 > x^2 + y^2$
is removed, what is the volume of the remaining solid?  (Hint: use
spherical coordinates.)
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IX.  Let $\alpha$ be the simple, closed, positively oriented curve
in the plane given by the semicircle of radius 4 with
center at $(0,0)$ and $y \ge 0$ (traversed counterclockwise), followed by the portion
of the $x$-axis from $(-4,0)$ to $(4,0)$.  Let $F(x,y) = (x^2 - y^2,xy)$.
\item{A)}  Compute the line integral $\int_\alpha F\cdot T\ ds$ directly
from the parametrizations.
\item{B)}  Verify that the conclusion of Green's theorem holds 
for this $F$ and $\alpha$ by computing an appropriate double integral.

