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,
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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
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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 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
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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
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
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
Carnegie Mellon University
Department of Mathematical Sciences
21-259 Calculus in 3 Dimensions
Review Exam 1, Fall, 2016
Your exam shall consist of problems similar to the homework problems. Please prepare by practicing these
problems. Calculators, notes,
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
:
(
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
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