Mathematics 241 -- Multivariable Calculus
Review Sheet for Exam 3
November 16, 2007
General Information
The third exam for the course will be given Friday, November 30,
as announced in the course syllabus. It will cover the material
we have discussed in class (including the lab on the geometry of Lagrange multipliers)
through and including the material on the change of variables formula
for double and triple integrals from class on November 19 and 20.
The topics are covered in sections 3.4, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 5.4, 5.5.
of the text. (But note that there is quite a bit more in the text than we
have discussed in class. You are only responsible for things we have done
in class.) Here's a breakdown by topic:
- Gradient, divergence, curl, and the del operator (a ``left-over''
topic from Exam 2).
- The first and second order Taylor polynomials of f(x,y)
at a point (a,b) and how they can be used to understand
the shape of a graph z = f(x,y) for (x,y) near (a,b).
- Extrema of functions -- finding critical points, using the Hessian
Criterion (Second Derivative Test) to classify them as local maxima, local minima, or neither.
(We only did these calculations for functions of 2 and 3 variables. Know
what how to apply the second derivative test in these cases.)
- Constrained optimization and Lagrange multipliers. (We only discussed
the case of a single constraint equation; the text also has an extensive
discussion of such problems involving several constraints -- see general comment above.)
Also know how to combine the previous item and Lagrange multipliers to determine
the absolute maximum or minimum of a function on a compact region in R2
(e.g. the region on and inside a simple closed curve).
- Double integrals over rectangles and more general regions.
- Interchanging order of integration.
- Triple integrals over rectangular boxes and more general regions.
- The change of variables formula in double and triple integrals.
Concentrate here on how to set up and evaluate integrals over regions
that are complicated to describe in rectangular coordinates, but are
simplified by going
to polar coordinates in the plane, or
cylindrical or spherical coordinates in space.
There will be 3 or 4 problems, each with several parts. Some may
ask for a graph or how to set up an integral (if the problem
does not say ``evaluate'' the integral, it is not necessary to do
that). Other questions may ask for
a description or explanation of some phenomenon (similar to some
questions from labs).
Review Session
Scheduling a review session before the exam this time is going
to be tricky again: I have off-campus commitments Tuesday and Friday
evenings, and an exam for another class starting at 6:00pm on
Wednesday. Earlier in the afternoon on Wednesday, or Thursday evening
are possible.
Suggested Review Problems
For practice, it would be a very good idea to
try a few problems from all sections we
have covered.
Here are some good ones.
- Section 3.4/1, 3, 7, 9, 14, 15.
- Section 4.1/8 and 17, 15 and 18 (this should say p2(x,y,z)).
- Section 4.2/3, 7, 11, 15, 33.
- Section 4.3/ 2, 4, 18, 21, 31.
- Section 5.1/3, 5, 9.
- Section 5.2/7, 9, 21, 25.
- Section 5.3/4, 6, 8, 10, 17.
- Section 5.4/3, 7, 9, 12, 15.
- Section 5.5/16, 17, 26, 29.
Sample Exam Questions
Note: the actual exam will not be this long. Also, the actual
exam questions may ask things in different ways and may involve
examples with different properties.
I. Let F(x,y,z) = (x + yz , y + xz , z + xy) be a vector field on R3.
- A) Compute the divergence of F.
- B) Compute the curl of F.
- C) Is there a scalar-valued function f(x,y,z) such
that F = ∇ f? Why or why not?
II. All parts of this question refer to the function
f(x,y) = xe-x2cos(y).
- A) Compute the second degree Taylor polynomial of f
at (a,b) = (0,0). Give both the form with the
derivative and Hessian matrices and the expanded (``messy, practical'') form.
- B) f(x,y) has exactly three critical points with
0 ≤ y ≤ 2 π /3. Find them.
- C) Classify the critical points from part B as
local maxima, local minima, or saddle points using the Second Derivative
Test (Hessian Criterion).
III. Let
f(x,y) = (x - 1)2 + y2
and g(x,y) = x2/4 + y2.
- A) Find the largest and smallest values of f on the constraint
curve defined by g(x,y) = 1
using the method of Lagrange Multipliers.
- B) Sketch the level curves of f for the values c = 11/16, 4, 9
together with the curve
g(x,y) = 1 and
explain how your sketch relates to the result of part A.
- C) What are the absolute maximum and minimum values of
h(x,y) = 10 - f(x,y)
on the region defined by g(x,y) ≤ 1?
IV. All parts of this question refer to the region R in the
plane inside the circle x2 + y2 = 5,
below the
parabola y = x2 + 1, and above the x-axis.
- A) Set up iterated integral(s) to integrate a
general function f(x,y) over R,
integrating first with respect to x.
- B) Same question as A, but interchanging the order of
integration (integrating first with respect to
y).
V.
- A) State the change of variables formula for double integrals in general.
- B) Where does the ``extra r'' in the polar
coordinate integral ∫ ∫ f(r cos θ, r sinθ) r dr dθ
come from?
- C) Change variables to polar coordinates to determine the integral
of f(x,y) = xy
over the quarter circle inside x2 + y2 = 4
in the second quadrant.
- D) Change variables to spherical coordinates in R3 to
evaluate the triple integral of the function x over the portion of the solid ball
of radius 1 with center at (0,0,0) in the first octant (i.e. the part of the
region inside the sphere of radius 1, with x, y, z all ≥ 0).
VI. Find the volume of the solid bounded above by
the cone z = (x2 + y2)1/2, below by z = 0,
and by 0 ≤ x2 + y2 ≤ 4.
You may use any convenient coordinate system to carry out the necessary
computations.