Mathematics 241, section 1 -- Multivariable Calculus
Review Sheet -- Exam 2
October 26, 2010
General Information
The second exam for the course will be given next Friday, November 5,
as announced in the course syllabus. It will cover the material
discussed in class since immediately before the first exam, through
the material from class on Friday October 29:
- a) Vector fields and flow lines, applications
- b) Critical points of vector fields, and the "source, center, sink,
saddle point" classification
- c) Graphs, contour plots, and density plots for
f(x,y)
- d) Directional derivatives (know the definition
of the directional derivative
Du f(x0,y0) and how to
compute it using the definition), first and second order
partial derivatives
- e) Open and closed sets, limits, continuity, and differentiability
for f(x,y)
(the ``total derivative'' and the tangent plane)
- f) The gradient vector field and the classification
of critical points of f(x,y) via critical points of the gradient
vector field.
This includes material from sections 2.4, 2.5, 3.1, 3.2, 3.3, 3.4, 3.5, 4.1.
In the cases where a section includes material we have not discussed
in class, though, you are not responsible for that.
There will again be 3 or 4 problems, each with several parts. Some
may ask you to analyze a graph (e.g. for the result of a calculation;
others may ask for a short description or explanation of some phenomenon
(similar to the questions from Lab assignments).
I will be happy to schedule an problem session before
the exam. (Is Wednesday afternoon or evening good?)
Suggested Practice Problems
From the text:
- Section 2.6 (End of Chapter Exercises)/11, 12, 13
- Section 3.6 (End of Chapter Exercises)/1, 3, 5, 6, 7, 9, 11, 14,
18, 19, 20, 21,
- Section 4.1/ 1,2,3,4,5
You should also review the problems from the labs and problem set on
this material.
Practice Exam Questions
Note: The actual exam will be shorter than this, of course. The
following questions show the range of different topics that might
be included.
I. All parts of this problem refer to the vector field
F(x,y) = (x,y2-1).
- A) Find all the critical points of F.
- B) Verify that α(t) = (0,(1-3e2t)/(1+3e2t))
and β(t) = (et/10,(1-3e2t)/(1+3e2t)) are
both flow lines for the vector field F, and
find α(0), β(0).
- C) What does part B) tell you about the type(s) of the critical
point(s) of F?
- D) Show that F(x,y) is the gradient field of the
function f(x,y) = x2/2 + y3/3 - y + c
for any constant c.
Note: In other problems dealing with flow lines, I might give you
a Maple plot of a vector field and flow lines, and ask you
to determine the types of critical points.
II. All parts of this question refer to the function f(x,y)
defined by f(x,y) = x2 - 3 y2.
- A) Sketch a density plot for f on the rectangle
[-2,2] x [-2,2] using a 4 x 4 grid of pixels, and 6 different
gray scale values. Give
a table showing how your gray scale corresponds to values of f.
- B) Sketch the contours of f for the values
c = -4, -3, -2.
- C) Compute the directional derivative of f in the direction
of a general unit vector u = (u1, u2)
at (x0, y0) = (3,2).
- D) In which directions is the directional derivative from C zero?
In which directions is the directional derivative positive?
Shade the corresponding half-plane in your graph from part B.
III. Sketch or describe each of the following sets and determine which of them are open, closed, or neither.
If the set is open, show that that is true. If not, say why not.
- A) S = {(x,y) | x < 1, y < 3}
- B) T = {(x,y) | x2 + y2 < 4, x <= 1 }
- C) U = {(x,y) | x and y are both rational numbers }
- D) V = {(x,y) | (x - 1)2 + (y - 1)2 >=1, |y| > 4}
IV. All parts of this question refer to the function defined by
f(x,y) = (2x5+y4)/(x4 + y4)
if (x,y) <> (0,0) and f(0,0) = 0
- A) Compute the tangent plane to the graph of f at
(x,y) = (1,1).
- B) Find the second order partial derivatives of f(x,y) at
(x,y) = (1,1).
- C) Compute the limits lim x -> 0 f(x,0) and
lim y -> 0 f(0,y). Does
lim(x,y) -> (0,0) f(x,y) exist? Explain.
- D) Is f(x,y) continuous at (0,0)? Explain.
- E) Find the partial derivatives of f with respect to
x and y at (0,0) by using the definition.
V.
- A) State the definition of differentiability of f(x,y)
at a point.
- B) Using the definition, show that f(x,y) = |x y|
is differentiable at (0,0).
- C) Examine the graph of some function
z = f(x,y). From that evidence, does
f appear to be
differentiable at (0,0)? Explain.
(On the actual exam, I would give you the Maple plot, of course!)