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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Information on Exam 3}
\centerline{November 18, 2013}
\bigskip
\noindent
{\it General Information}
\bigskip
The third exam for the course will be given Tuesday, November 26 (the 
day before Thanksgiving break), 
as announced in the course syllabus.  It will cover the material 
from Problem Sets 7, 8, and 9.  As always, though, {\it there are
major dependencies of this material on things we discussed earlier
in the semester.  You should, in fact, be prepared to do any 
sort of computation we have covered, including vector operations,
finding equations of planes, finding parametrizations, determining critical points 
of vector fields, etc.)}
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\noindent
{\it Topics to Be Covered}
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\item{a)} The second degree Taylor polynomial and 
the Second Derivative Test for critical points
of functions (section 4.2) 
\item{b)} The Lagrange Multiplier method for constrained max-min
problems (sections 4.3 and 4.4), including max-min questions on
closed, bounded subsets of the plane or space.  
\item{c)} Riemann sums and double integrals over
regions in ${\bf R}^2$ (section 5.1)
\item{d)} Double Integrals over regions via
iterated integrals with variable limits of integration -- Fubini's Theorem
(section 5.2) with applications to volumes, average values, centers of mass.
\item{e)}  Polar coordinates in the plane and polar
double integrals, converting rectangular double integrals
to polar coordinates -- the Change of Variables Theorem (section 5.3)
\item{f)}  Triple integrals (section 5.4).  Note that finding a 
volume using polar coordinates in the $xy$-plane and the rectangular
$z$-coordinate is the same as using cylindrical coordinates.  So there
is a big overlap between things from sections 5.3 and 5.5 and a lot
of questions in section 5.5 are things I could ask on this exam too.  
\medskip
\noindent
There will again be 3 or 4 problems, possibly with several parts. 
Be prepared to:
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\item{1)} State the Second Derivative Test (see class notes, 
and/or the Proposition 4.6 on pages 242-243  of the text), 
and use it 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 the Second Derivative Test and some 
Lagrange Multiplier calculations to find the maximum 
and/or minimum values of a function on a closed set.  
\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.
\bigskip
\noindent
{\it Review Session}
\bigskip
I would be happy to schedule a problem session before
the exam.  We can discuss the date and time before you leave
for Thanksgiving break.
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\noindent
{\it Suggested Practice Problems}
\bigskip
From the text:  
\item{} End of Chapter 4 Exercises/1,2,3,8,12b
\item{} End of Chapter 5 Exercises/1, 5, 6ab, 7abcde
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\noindent
{\it Practice Exam Questions}
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\noindent
{\it Note:}  As usual, the actual exam questions may be 
organized differently and may include different types of 
examples.  The actual exam will also be somewhat shorter.
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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 of $f$.
\item{B)}  Give a precise statement of the Second Derivative
Test.  
\item{C)}  Use the Second Derivative test to classify each
critical point of $f$ as a local maximum, local minimum, or a saddle point.
\item{D)}  Use the Lagrange Multiplier method to find the maxima 
and minima of $f$ on the constraint curve $x^2 + y^2 = 9$.  
\item{E)}   What are the biggest and smallest values of $f$ on 
the region $\{(x,y) : x^2 + y^2 \le 9, x \ge 0, y\ge 0\}$?
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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.
\item{C)}  An solid object has ``shadow'' equal to $R$ from 
parts A and B in the $xy$-plane.  It is bounded above by the 
surface $z = 3x^2 + y^2 e^y$ and below by the surface $z = 2\cos(\pi(x)) + y^2$.
Find its volume.  
\item{D)}  What is the $x$-coordinate of the center of mass of the 
volume from part $C$ if the mass density at each point is a constant
$c$?   
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\noindent
III.  A region $S$ in the first and fourth 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
$\int\int y\, dy dx$.  Evaluate.
\item{B)}  Now find the limits of integration to set up
$\int\int_S y\, dA$ as one or more integrals of the form $\int\int y\, dx dy$. 
Evaluate
\item{C)} Find the limits of integration to express
$\int\int_S y\, dA$ in polar coordinates.  Evaluate.
\item{D)}  Your three answers to parts A,B,C should be equal.  
What theorems tell you that must be the case?
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\noindent 
IV. A region in ${\bf R}^3$ $R$ is bounded defined by 
$$R = \{(x,y,z) \mid x^2 + y^2 + z^2 \le 1, 4x^2 + 4y^2 \ge 1\}$$
Find the volume of $R$.  
\bye


 
