Practice Exam 1 Solutions
Most of the multiple choice answers are (A); the following are typos:
The answer to 2 is 8 , which is (C).
The lamina in 3 should be = cfw_(x, y ) : x2 + y 2 2 , y 0 (NOT x2 + y 2 = ).
The answer to this problem is (A).
In 7,

Answer Key 4
MATH 20550: Calculus III
Exam II
Name:
October 26, 2010
Instructor:
As a member of the Notre Dame community, I will not participate in or tolerate academic dishonesty.
Please sign
Record your answers to the multiple choice problems by placing

2
Practice Exam 2
The multichoice answers are all A.
Problem (#11). Let f (x, y ) = x2 y + xy 2 + 3xy . Find the critical points of
f and tell what type each one is.
Solution. A point (x0 , y0 ) is a critical point of f just in case either f (x0 , y0 )
is

Math 251
Multivariable Calculus
Fall 2016, Sections 4, 6, and 7
Instructor: Matthew C. Russell. I invite you to call me Matthew.
Email: russell2@math.rutgers.edu
Meeting times and place: Monday and Thursday, 10:20 a.m. 11:40 a.m. in the T.
Alexander Pond

Case Analysis: The Oil Curse
The political risk is high and the Nigeria government seems to be
incapable to deliver protection for the foreign companies. The
hazards are moderately well known and companies are mostly
accountable for offering their own saf

Wednesday, July 1, 2015
Math 251 Practice for Exam 2
From the book (mostly for the later material):
Section 15.6 problems 35, 38, 39, and if youve taken linear algebra: 49, 50, 51 (these
prove Equation 14 which you use in 39)
Chapter 15 Chapter Review E

Practice problems for the final Exam, Math 251, Fall 2014
2
2
1. Evaluate R (x+y)ex y dA, Where R, is the rectangle enclosed by the lines x-y=0,x-y=2,x+y=0,x+y=3.
(ANS:(e6 7)/4. Hint: use the change of variable formula)
R 2 R 4y2 R 4x2 y2 2
2. Evaluate u

Sunday, July 12, 2015
Major concepts since the last exam (not exhaustive)
parametrizing surfaces: basic idea, some common parametrizations (cylinders, spheres, cones, graph
of a function z = f (x, y), etc.), grid curves as level curves with respect to ea

Math 251 Practice for Exam 1
Sunday, June 14, 2015
Topics/concepts
Chapter 12: vector; 2-vector; 3-vector; n-vector; standard basis vectors i, j, k; definitions,
algebraic properties and geometric interpretations of vector addition and scalar multiplicati

Wednesday, June 17, 2015
Answers (not full solutions) to exam 1 practice problems
3.
a. u = hu1 , u2 , u3 i, v = hv1 , v2 , v3 i, w = hw1 , w2 , w3 i.
b. Think about vector subtraction.
c. The relevant property is that the cross product of two vectors is

640:251:0103
REVIEW PROBLEMS FOR FINAL EXAMPage 1
FALL 2014
This set of problems concentrates primarily on material from Chapters 16 and 17 of Rogawski, with
some questions material from earlier chapters. To review for the final you should study also the

Practice Exam 2 Solutions
The answer to evey multiple choice question is a.
9. Under the change of variables x = s2 t2 , y = 2st the quarter circular region in the
st-plane given by s2 + t2 1, s 0, t 0 is mapped onto a certain region D of the xy -plane.
E

Problem 1: If the vectors a and b have lengths 2 and 9, and the angle between them is
a b.
() 9
() 2
() 0
() 18
, nd
3
() 5.5
Problem 2: Find the maximum value of the function f (x, y ) = 2x +3y on the surface x2 + xy + y 2 =
21.
() 14
() 5
() 19
() 24
()

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rYiqngYWv%)C!%BY4Gm7QC94CE94C4 H m9YWiq7b
P w U u F
v 8 x 35 y xb A 8 3 3 b v db 3 3 1 o x P V U T DF 3 1 A F D 3 q j y A 8b x A d 8 D y x 8 D w U u A x j
4B7)7CaC9nd427%C94f4CqCBl4C&Gla2a7

s
Yk s
jd
j d
6 dqYf
cpc
gk
a2
6
Q
Y
bx g c p g c v p V b b ex c h g
odBSg QdGYSIX
j k g d c f e hx q h `x p c p g p ` c v g X b hx c v p V b g b c h c i j k g d
Yh5Y22dwwqc BSxutSg QdwHYIvYGYh5It n&r2
R 1 E
jj d d jj d d
vd5W od5W on 7 B

FINAL EXAM Math 251
Name:
PROBLEM 2: The two space curves r1 (t) =
PROBLEM 1 (5 pts): Find the equation of the plane containing the points (1, 0, 1), (1, 1, 2) and (1, 2, 9). r2 (s) = 2s, s3 2s + 1, 5s 4 intersect in exactly one point. a (3 pts): Find the

Formula Sheet (Final Exam)
The length of r = x, y, z is r = x2 + y 2 + z 2
The distance from r0 = (x0 , y0 , z0 ) to r1 = (x1 , y1 , z1 ) is
(x1 x0 )2 + (y1 y0 )2 + (z1 z0 )2
The line through the point r0 = (x0 , y0 , z0 ) with direction vector v = (a, b,

Math 233
Exam 3
Page 1
Name:
ID:
This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientic
calculator and a 3 5 inch note card.
1. Find
C
F , where F (x, y ) = y , x2 and C goes along the parabola y = x2 from (0, 0

Sunday July 6, 2015
Answers (not full solutions) to my exam 2 practice problems
2. The region is below; the integral evaluates to 1/3.
3. The region is below. Use Fubinis Theorem to switch the order! The integral evaluates to
Z
4. The setup is
2
2
Z
q
2
1