MATH 126 D & E
Exam II
Spring 2013
Name
Student ID #
Section
HONOR STATEMENT
“I affirm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized assistance
on this exam.”
SIGNATURE:
1
8
2
8
3
8
4
8
5
10
6
8
Total 50
• Your exam should consist of this cover sheet, followed by 6 problems on 5 pages. Check that
you have a complete exam.
• Pace yourself. You have 50 minutes to complete the exam and there are 5 pages. Try not
to spend more than about 10 minutes on each page.
• Unless otherwise indicated, show all your work and justify your answers.
• Unless otherwise indicated, your answers should be exact values rather than decimal approximations. (For example, π4 is an exact answer and is preferable to its decimal approximation
0.7854.)
• You may use a scientific calculator and one 8.5×11-inch sheet of handwritten notes. All
other electronic devices (including graphing calculators) are forbidden.
• The use of headphones or earbuds during the exam is not permitted.
• There are multiple versions of the exam, you have signed an honor statement, and cheating
is a hassle for everyone involved. DO NOT CHEAT.
• Turn your cell phone OFF and put it AWAY for the duration of the exam.
GOOD LUCK!
Math 126 — Spring 2013
1
1. (8 points) A moving object has acceleration a(t) = 8t i − et j + sin(4t) k, initial velocity
v(0) = 2i + 4j + 12 k, and initial position r(0) = i − k. Find its position at time t.
Math 126 — Spring 2013
2. (8 points) Find and classify all critical points of f (x, y) = 64x3 − 3xy + y 3 .
2
Math 126 — Spring 2013
3
3. (8 points) Suppose that f (x, y) is a continuous function and that
��
f (x, y) dA =
D
�
√
0
2
�
4−x2
f (x, y) dy dx.
x2
Sketch the region D and reverse the order of integration.
4. (8 points) Use linear approximation to estimate the value of
�
(2.1)3 (−0.8)3 + 10.
Math 126 — Spring 2013
4
5. (10 points) At right is a portion of the surface
�
z = 4 + x2 + y 2 .
�� �
(a) Let D = {(x, y) ∈ R : x + y ≤ 5}. Compute
4 + x2 + y 2 dA.
2
2
2
D
(b) Find the volume of the solid enclosed by z =
�
4 + x2 + y 2 and the plane z = 3.
Math 126 — Spring 2013
5
6. (8 points) A lamina in the xy-plane is in the shape of the region that lies inside the circle
r = sin(θ) and outside the cardiod r = 1 + cos(θ) (shown below). The density of the
lamina at a point (x, y) is the reciprocal of its distance from the origin. Compute the mass
of the lamina.
r = sin(!)
r = 1 + cos(!)