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\centerline{Mathematics 41, section 1 -- Multivariable Calculus}
\centerline{Information on Exam 3}
\centerline{November 13, 1999}
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\noindent
{\it General Information}
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The third exam for the course will be given Friday, November 19, 
as announced in the course syllabus.  It will cover the material 
discussed in class since immediately before the second exam through
class on Monday, November 15.
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\noindent
{\it Topics to Be Covered}
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\item{a)} Critical points for functions $f(x,y)$, the ``First Derivative
Test'' for local maxima and minima -- see the Proposition on 
page 293 of the text (section 4.1)
\item{b)} The Second Derivative Test for local maxima and minima
of functions (section 4.2) 
\item{c)} The Lagrange Multiplier method for constrained max-min
problems (section 4.3)
\item{d)} Riemann sums and double integrals over rectangular
regions in ${\bf R}^2$ (section 5.1)
\item{e)} Double Integrals over non-rectangular regions via
iterated integrals with variable limits of integration (section 5.2)
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\noindent
There will again be 3 or 4 problems, possibly with several parts. 
Be prepared to:
\medskip 
\item{1)}  State the First and Second Derivative Tests (see class notes, 
and/or the Propositions on page 293, 318 of the text), 
and use them to classify critical 
points of a function.  
\item{2)} Set up and solve the Lagrange Multiplier equations for
constrained max-min problems
\item{3)} Use a Riemann sum to get an approximate value
for a double integral, and  
\item{4)} Use exact integration methods like substitution, integration
by parts, etc.  to evaluate double integrals
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\noindent
{\it Review Session}
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I hope to be able to schedule a problem session before
the exam, but I have some heavy schedule constraints this time.
Wednesday and Thursday in the early evening are not possible for me.
I have the Chamber Orchestra concert on Wednesday and another rehearsal
in Boston on Thursday evening.  Wednesday late afternoon (say 4 - 6 pm)
might work.  Wednesday at 9:30 pm or shortly thereafter would also be 
possible.  Thursday 3:30 - 5:30 pm is also possible, but that's getting 
pretty close to the exam!
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\noindent
{\it Suggested Practice Problems}
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From the text:  
\item{} Section 4.1/4,5,6,11,12
\item{} Section 4.2/3,7,10,11
\item{} Section 4.3/3,4,5,6
\item{} Section 5.1/3,4,5
\item{} Section 5.2/1,2,4,6,7,12,13,14
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\noindent
{\bf Practice Exam}
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\noindent
I.  All parts of this problem refer to $f(x,y) = x^3 + y^2 - 3x$.
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\item{A)}  Find all
critical points, and classify them using a plot of the gradient 
vector field.
\item{B)}  Use the Second Derivative test to verify your work 
from part A.
\item{C)}  Use the Lagrange Multiplier method to find the maxima 
and minima of $f$ on the constraint curve $x^2 + y^2 = 9$.  
\item{D)}   What are the biggest and smallest values of $f$ on 
the region $\{(x,y) : x^2 + y^2 \le 9\}$?
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\noindent
II.  
\item{A)} At the corners, the midpoints of the sides, and the center of
the rectangle $R = [1,3]\times [0,2]$ in the $xy$-plane, the
values of a function $f(x,y)$ are measured to be 
$$\matrix{&f(1,2) = 6 &f(2,2) = 8 &f(3,2) = 10\cr
          &f(1,1) = 3 &f(2,1) = 5 &f(3,1) = 7\cr
          &f(1,0) = 2 &f(2,0) = 4 &f(3,0) = 6\cr}$$
Estimate $\int\int_R f(x,y)\, dA$ using an appropriate Riemann sum.
\item{B)} The function $f(x,y)$ from part A was $f(x,y) = 2x + y^2$.
Compute $\int\int_R f(x,y)\, dA$ exactly.
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\noindent
III.  A region $S$ in the first and second quadrants of the $xy$-plane is
bounded by the lines $y = x, y = -x$, and by an arc of the circle 
$x^2 + y^2 = 9$.  
\item{A)}  Find the limits of integration to set up
$\int\int_S y\, dA$ as one or more iterated integrals of the form
$\int\int y\, dy dx$.  
DO NOT evaluate yet.
\item{B)}  Now find the limits of integration to set up
$\int\int_S y\, dA$ as one or more integrals of the form $\int\int y\, dx dy$. 
DO NOT evaluate this integral yet either.
\item{C)} Evaluate the integral of your choice from either A or B.
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