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 organ
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 organiz
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 th
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 pr
Carnegie Mellon University
Department of Mathematical Sciences
21-259 Calculus in 3 Dimensions
Practice Exam 3, Version 1, Fall, 2017
Warning: The following is a practice exam. Do not expect the actua
1. Find the limit, or show it does not exist. Should the limit exists, you must determine the limit
and provide the appropriate argument. If the limit does not exist, you must clearly explain why,
sho
I-Inlnewnrll. 3 Eullliul
Section. 14.1
18. 111(2 3:) is dened 01113:" when 2 :1: 3r 0, er 3: c: 2- In
addition, 9 is net dened if 1 3:2 y2 = U cfw_i
:32 | y: = 1. Thus the domain efg is
cfw_($133) I s
IPrrpdch EMA/- 3/ Salaam;
Note Title
1. (a) Find the area of the parallelogram with vertices P(l_. l_.2)_. Qi, 3,:l_. Rll, l, 13],
and 5(5,3,9).
(h) Find the angle at each vertex of this parallelogr
1. Let P1 and P2 be planes given by the equations 2x + y
z = 4 and 4x + y
2z = 0 respectively.
(a) Find the parametric
2 equations
3 2 for the
3 line
2 ` of 3intersection of the planes P1 and P2
2
4
1
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 area of the parallelogram with vertices P (1; 1; 2), Q(
Exam 3, Practice Exam 1, Solutions.
R4R2
3
1. Evaluate 0 px ey dy dx.
p
Solution: The iterated integral corresponds to a type 1 region D = f(x; y) : 0 x 4; x y
which as a type 2 region is given by D =
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 4
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Suppose f (x; y) is
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 2
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Reduce the equation
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. Ple
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
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
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 2 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Reduce th
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 5 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find the
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 3 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Determine
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 8
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Let C be the line s
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 5
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find the equation o
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 1 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find an e
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 1
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find an equation of
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 4 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Suppose f
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 3
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Determine the domai
Solutions: Please let me know of any mistakes you nd!.
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
Carnegie Mellon University
Department of Mathematical Sciences
21-259, Fall 2017
Homework 7
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find the area of th