Math 208-002
Exam #2
November 4, 2013
UNL Student ID Number:
Name:
Turn off your cell phones.
Page:
1
2
3
4
5
6
7
Total
Points:
20
10
10
15
15
15
15
100
Score:
Instructions:
1. No books or notes may be used on the exam.
2. Credit or partial credit will be given only when the appropriate explanation and/or work is
shown.
3. Read and follow directions carefully.
4. This exam has 8 questions, for a total of 100 points. There are 7 pages (in addition to this
cover sheet).
5. You will have 50 minutes to complete the exam.
6. ALL CELL PHONES MUST BE TURNED OFF DURING THE EXAM. Phones
that ring, vibrate, or otherwise cause a distraction will be confiscated and may result in the
student receiving no credit for the exam.
DO NOT OPEN YOUR EXAM UNTIL INSTRUCTED TO DO SO!!
(10 points) 1. A particle passes through the point P = (5, 4, −2) at time t = 4, moving with constant velocity
~v = 2~i − 3~j + ~k. Give a parametric equation (in vector form) for its motion.
(10 points) 2. Evaluate the following iterated integral:
Z
0
1Z 1
√
y
3 cos(x3 ) dx dy.
(10 points) 3. Set up, but do notpevaluate, an iterated integral to give the mass of the solid that lies
outside the cone z = x2 + y 2 and inside the sphere of radius 16 centered at the origin, where
2
2
2
the density at a point (x, y, z) in the solid is given by δ(x, y, z) = ex +y +z .
(Note: This solid has portions above and below the xy-plane.)
Page 2
(10 points) 4. The velocity of a flow at the point (x, y) is F~ (x, y) = 2y~i + 4~j. If an object is at (1, 3) in the
flow at time t = 1, then where is the object at time t = 2?
(Hint: First find the object’s path of motion.)
Page 3
(15 points) 5. Use Lagrange multipliers to find the maximum value of the function f (x, y) = x2 + 2y 2 subject
to the constraint x2 + y 2 ≤ 4.
Page 4
(15 points) 6. Find the point on the surface y = x + 2z that is closest to the point (1, 0, 2). You do not need
to verify that the point you find is the closest.
(Hint: Minimize the square of the distance between a point (x, y, z) on the surface and
the point (1, 0, 2).)
Page 5
(15 points) 7. Evaluate the following triple integral:
Z
1
−1
√
Z
−
1−x2
√
1−x2
Z
0
1
1
dz dy dx.
(x2 + y 2 )1/2
Page 6
(15 points) 8. Let R be the region in 2-space bounded by the curves x2 + y 2 = 1 and x2 + y 2 = 9. Find the
average distance from the origin of the points of R.
Page 7