Mathematics 241, section 1 -- Multivariable Calculus
Review Sheet -- Exam 2
October 25, 2013
General Information
The second exam for the course will be given next Friday, November 1,
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 25:
- a) Vector fields and flow lines, applications
- b) Critical points of vector fields, and the "source, center, sink,
saddle point" classification
- c) Graphs and contour plots for
f(x,y)
- d) Directional derivatives (know the definition
of the directional derivative
Du f(x0,y0) and how to
compute it with the gradient formula),
- e) first and second order
partial derivatives
- f) the tangent plane and the idea of differentiability for f(x,y)
(know in particular that it is not enough just to have both partial derivatives
exist in order for f to be differentiable)
- f) The gradient vector field and the classification
of critical points of f(x,y) via critical points of the gradient
vector field (''first derivative test'')
This includes material from sections 2.4, 2.5, 3.1, 3.2, 3.4, 3.5, 4.1.
In the cases where a section includes material we have not discussed
in class, you are not responsible for that portion, of course.
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,
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 the contours of f for the values
c = -4, -3, -2.
- B) Compute the directional derivative of f in the direction
of a general unit vector u = (u1, u2)
at (x0, y0) = (3,2).
- C) 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.
- A) Sketch or describe each of the following sets.
- 1) S = {(x,y) | x < 1, y < 3}
- 2) T = {(x,y) | x2 + y2 < 4, x <= 1 }
- 3) U = {(x,y) | x and y are both rational numbers }
- 4) V = {(x,y) | (x - 1)2 + (y - 1)2 >=1, |y| > 4}
- B) A subset S of R2 is said to be open
if for each point in the set, there is an open circular disc of positive radius centered at
that point that is completely contained in S. Which of the sets in part A are open?
Explain.
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) Find the partial derivatives of f with respect to
x and y at (0,0) by using the definition.
V. Let f(x,y) = xy e-x2 - y2.
- A) Find all the critical points of f
- B) Given a plot of the gradient vector field, determine what type of critical point is
each one (local maximum, local minimum, saddle).