Math 211, Section 02, Spring 2015
Solutions to Practice Problems 3
1. For the following integrals, sketch the region of integration, and then reverse the order of
integration.
1x2
1
f (x, y) dy dx is bounded by y = 1 x2 above and by y = 1 x
(a). The regio
Math 211, Section 01, Spring 2012
Final Exam, Friday, May 11, 2012
Instructions: Do all twelve numbered problems. If you wish, you may also attempt the three
optional bonus questions. Show all work, including scratch work. Little or no credit may
be award
Math 211, Section 02, Spring 2015
Final Exam, Wednesday, May 13, 2015
Instructions: Do all twelve numbered problems. If you wish, you may also attempt the three
optional bonus questions. Show all work, including scratch work. Little or no credit may
be aw
Math 211, Section 02, Spring 2015
Solutions to the Spring 2012 Final Exam
1. Find an equation for the line of intersection of the planes 2x y + 5z = 1 and x z = 3.
Answer. The planes have normals n1 = 2, 1, 5 and n2 = 1, 0, 1 , so the line must be paralle
Math 211, Section 02, Spring 2015
Solutions to Midterm Exam 1
1. Find an equation for the plane that contains the points P = (2, 0, 1), Q = (0, 2, 1), and R =
(3, 0, 2).
Answer. The vector from P to Q is v = Q P = 2, 2, 2 , and the vector from R to P is
w
Math 211, Section 02, Fall 2015
What you need to know for Exam 3
The exam will cover Chapter 15, excluding 15.6 and 15.10. (Of course, you should still know
Chapters 1214, but the exam will not directly test you on that material.) The following is a
list
Math 211, Section 02, Spring 2015
What you need to know for Exam 1
You should know everything from the beginning of the course up to (and including) Section 13.3.
The following is a list of most of the topics covered. THIS IS NOT A COMPREHENSIVE
LIST, BUT
Math 211, Section 02, Spring 2015
What you need to know for Exam 2
The exam will cover Chapter 14. (Of course, you should still know Chapters 12 and 13, but the
exam will not directly test you on that material.) The following is a list of most of the topi
Math 211, Section 02, Spring 2015
What you need to know for the Final Exam
The exam will cover the whole course, but with more weight on the second half of the semester.
The following is a list of most of the topics covered since Exam 3, i.e., Chapter 16.
Math 220, Section 01, Spring 2015
Practice Problems for Midterm Exam 2
(A little more dicult, and much longer, than the real exam)
For counting and probability problems, leave your answer as an expression involving short sums, dierences, products, and quo
Math 220, Section 01, Spring 2015
What you need to know for Exam 1
The exam (in class, Wednesday, Feb 25) will cover Chapter 1 and the rst two sections of
Chapter 2 of Richmond and Richmond. The following is a list of most of the topics covered.
THIS IS N
Math 220, Spring 2015
Solutions to Practice Problems 1
1. Let A = cfw_1, 2, 3, 4, 5, B = cfw_2, 5, 1, 7, C = cfw_3, 4, 4, 4, 7. Compute each of the following
sets:
Answers. Before starting, it will help to rewrite B = cfw_1, 2, 5, 7 and C = cfw_3, 4, 7. S
Math 220, Section 01, Spring 2015
Practice Problems for Midterm Exam 1
(A little more dicult, and much longer, than the real exam)
1. Let A = cfw_1, 2, 3, 4, 5, B = cfw_2, 5, 1, 7, C = cfw_3, 4, 4, 4, 7. Compute each of the following
sets:
a. A B
d. A B C
Math 220, Section 01, Spring 2015
Solutions to Practice Problems for the Final Exam
1. Prove that |(0, 1)| = |R|.
Proof. Dene f : (0, 1) R by f (x) = tan x
. Note that f is actually dened on (0, 1),
2
because if 0 < x < 1, then < x < , and so tan x
is d
Math 220, Section 01, Spring 2015
Graph Coloring [Based on notes by Professor Cox]
Throughout this handout, G will always be a simple graph, i.e., G has no loops and no
multiple edges.
1: Denitions and Basic Results
Denition. Let k be a positive integer.
Math 220, Section 01, Spring 2015
Practice Problems for the Final Exam
These problems only cover material from Section 6.3 and on. This list, in combination with
the lists of practice problems for the two midterm exams, will give a good set of practice
pr
Math 220, Section 01, Spring 2015
Solutions to Extra Practice Problems on Functions
1. Dene f : R R by f (x) = 2x2 7. Prove that:
3
3
(a) f 1 [2, 5) = 6, , 6 , (b) f 1 (10, 5] = [1, 1],
2
2
(c) f (3, 2) = [7, 11)
Proofs. (a): (): Given x f 1 [2, 5) , we
Math 211, Section 02, Spring 2015
Solutions to the Final Exam
1. Calculate the equation of the tangent plane to the surface x2 y + y 2 z + z 2 x = 7 at the point (1, 1, 2).
Answer. We have fx = 2xy + z 2 , fy = x2 + 2yz, and fz = y 2 + 2xz. Thus, f (1, 1,
Math 211, Section 02, Spring 2015
Midterm Exam 2, Tuesday, March 24, 2015
Instructions: Do all six numbered problems. If you wish, you may also attempt the two
optional bonus questions. Show all work, including scratch work. Little or no credit may
be awa
Math 211, Section 02, Spring 2015
Solutions to Practice Problems for the Final Exam
x+z
and C is the path from (1, 1, 1) to (4, 4, 8) along the
y+z
C
curve given by the equations y = x and z = x3/2 .
Answer. Parametrize C by r(t) = t, t, t3/2 , for 1 t 4.
Math 211, Section 02, Fall 2015
Practice Problems for Midterm Exam 3
(A little harder than, and about three times as long as, the real exam)
1. For the following integrals, sketch the region of integration, and then reverse the order of
integration.
1x2
1
Math 211, Section 02, Spring 2015
Determinants
For our purposes, determinants are certain specic recipes for adding, subtracting, and
multiplying quantities arranged in a square grid. Although any n n square grid of numbers
has a determinant formula, we a
Math 211, Section 02, Spring 2015
Practice Problems for Midterm Exam 2
(A little harder, and substantially longer, than the real exam)
1. Each of the following limits converges. Compute them.
x4 3y 2
x4 + 2x2 + x2 y 2 + 2y 2
ey + x sin y
lim
lim
lim
(x,y)
Math 211, Section 02, Spring 2015
Solutions to Midterm Exam 2
x2 y(x y)
diverges.
(x,y)(0,0) x4 + y 4
x2 y(x y)
Answer. Write f (x, y) =
.
x4 + y 4
0
On the x-axis, lim f (x, 0) = lim 4 = 0 = lim 0 = 0.
x0
x0 x
x0
x2 (x)(2x)
2x4
On the line y = x, lim f (
Math 211, Section 02, Spring 2015
Practice Problems for the Final Exam
(Warning: these problems only cover the material since Midterm 3. The actual nal exam
will be comprehensive.)
x+z
and C is the path from (1, 1, 1) to (4, 4, 8)
y+z
C
along the curve gi
Math 211, Section 02, Spring 2015
Conservative Vector Fields
This handout is about four nice properties that a vector eld F (x, y) = P (x, y), Q(x, y) on
D R2 or F (x, y, z) = P (x, y, z), Q(x, y, z), R(x, y, z) on D R3 may or may not have.
Property 1. F
Math 211, Section 02, Spring 2015
Solutions to Practice Problems 1
1. By its equation, the line (and hence the plane) is parallel to the vector v = 2, 1, 3 . Plugging
in t = 0, the line also contains the point (0, 3, 1). The plane therefore also parallel
Math 211, Section 02, Spring 2015
Practice Problems for Midterm Exam 1
(A little harder than, and two to three times as long as, the real exam)
1. Find an equation for the plane that passes throught the point (1, 2, 2) and contains the
line r(t) = 2ti + (
Math 211, Section 02, Spring 2015
Solutions to the Fall 2014 Final Exam
1. A particle is travelling in such a way that its velocity vector at time t is given by r (t) = t2 , 2t, 1 .
(1a) How far does the particle travel from time t = 1 to time t = 2?
(1b)