Full Name:
MATH 259: Exam 2B
Section:
Wednesday March 7, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various resu
Full Name:
MATH 259: Exam 3A
Section:
Monday, April 9, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various result
Exam 2 Review
1. Sketch the region R, in the xy-plane, that corresponds to the domain of the given function.
x2 + y 2 1
(a) f (x, y) =
(b) f (x, y) = ln(4 xy)
2. For the following functions, sketch the level curves at the given heights
25 x2 y 2 , k = 0,
Full Name:
MATH 259: Exam 3B
Section:
Monday, April 9, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various result
Exam 1 Review
1. Let P = (2, 5, 7) and Q = (4, 3, 8)
(a) Find P Q
(b) Find P Q
(c) For the line l through P and Q nd
i. a vector equation of l
ii. parametric equations of l
iii. symmetric equations of l
2. Let P = (1, 2, 3) and let a = 2i 2j + k. Find a p
Full Name:
MATH 259: Exam 1B
Section:
Wednesday February 8, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various r
Full Name:
MATH 259: Exam 1A
Section:
Wednesday February 8, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various r
Full Name:
MATH 259: Exam 2A
Section:
Wednesday March 7, 2012
You are expected to justify your answers in a manner that an average 21-259 student could
understand.
Your work should be neat and organized and legible. It should NOT consist of various resu
Warning: The following is a practice exam. Do not expect the actual exam to necessarily consist
of the same or similar problems.
1
(a) Find the limit, or show it does not exist:
(b) Find the limit, or show it does not exist:
2. Let f (x; y) =
x2 y 2
p
:
(
Solutions: Please let me know of any mistakes you nd!.
1
x2 y 2
p
:
(x;y)!(0;0)
x4 + y 4
Ans: The limit exists, by the squeeze theorem:
p
p
4
4
x2 y 2
4
x
2
p
px +y y 2 = y 2
p
0
=
y
4 +y 4
4 +y 4
4
4
x
x
x +y
(a) Find the limit, or show it does not exist
Carnegie Mellon University
Department of Mathematical Sciences
21-259 Calculus in 3 Dimensions
Review Exam 2, Fall, 2016
Your exam shall consist problems similar to the homework and quiz problems. Please prepare by practicing these
problems. Monday, Octob
Exam 3 Review
1. Evaluate the following iterated integrals. You may nd it useful to reverse the order of
integration or transform to polar coordinates.
(e)
1 1 x+y
dy dx
0 0 e
1
1x2 x2 y 2
e
dy dx
0 0
1 1
2
2
0 x x y x dy dx
3 x
2
1 0 x2 +y 2 dy dx
3
4 2