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\centerline{Mathematics 41, section 1 -- Analysis 3}
\centerline{Final Examination}
\centerline{May 6, 1998}
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{\bf Directions}
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You have until 11:30 am to work on this exam.
Do all your work in the blue exam booklet.  Do not place anything you 
want considered for credit on these sheets.  The exam consists of problems 
I - VII, and {\it either problem VIII or VIII'\/}.  If you submit solutions 
for both, one will be counted as Extra Credit.  There are 200 possible
points.  Good hunting!
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I.  
\item{A)} (10) 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)} (5) Give a parametrization of the line from part A.
\item{C)} (5) 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)} (10) 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)} (10) Find all times at which the instantaneous velocity of 
the object is a horizontal vector.
\item{C)}  (10)  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:
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\item{A)} (10)  Using the component functions of $F$, determine all the
{\it critical points\/} of $F$. 
\item{B)} (10)  Using the plot, classify each as a either a sink, 
a source, a saddle point, or a center.
\item{C)} (10) 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)}  (10) 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)}  (10)  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)} (10) 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)} (15)  Apply the {\it Second Derivative Test\/} to each of 
the critical points and check your responses to part A.
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\noindent
VI.  (20) 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)} (10) 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)}  (10) Repeat part A, but using {\it polar coordinates\/}.
\item{C)}  (15) 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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{\it Do either problem VIII, or problem VIII' below.}
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VIII.  (25)  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?
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VIII'.  
\item{A)}  (10) Compute the line integral of the vector field
$F(x,y) = (x+y^2, xy - 3)$ around the ellipse 
$\alpha(t) = (2\cos(t), 3\sin(t))$, $t \in [0,2\pi]$.  
\item{B)}  (15) State Green's Theorem and verify that its conclusion is
valid for this $F$ and $\alpha$.
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\centerline{\it Have a safe, productive, and enjoyable summer!}
\bye
