Math 251, Quiz #7 Solutions, December 9, 2014
1. Let F(x, y, z) = (xyz, x2 , y2 + z).
a) Show that H = F = (2y, xy, 2x xz).
i
j
k
= (2y 0, (0 xy), 2x xz) = (2y, xy, 2x xz).
Solution: F = x
y
z
xyz x2 y 2 + z
ZZ
H dS where S is the hemisphere x2 +
Math 251, Quiz #0 Solutions, September 9, 2014
1. Parametrize all vectors perpendicular to v = [1, 2, 3]. Describe this set of points. Find a
vector in this set that is also perpendicular to w = [3, 1, 4].
Solution: If the vector [r, s, t] is perpendicula
Math 251, Quiz #1 Solutions, September 23, 2014
1. A particle travels along the curve r(t) = (cos(t), sin(t), t) as t ranges from 0 to 8. Describe this path in words. How far did the particle travel? Compare this to the particles
net displacement over thi
Math 251, Quiz #5 Solutions, November 4, 2014
1. Evaluate
ZZZ
(x2 + y 2 + z 2 )dV
E
where
pE is the region bounded by the xy-plane and the hemispheres z =
z = 9 x2 y 2 .
p
1 x2 y 2 and
Solution: We will parameterize E in spherical coordinates and then com
Math 251, Quiz #2 Solutions, October 7, 2014
1. At which of the following three points on the surface z = x2 + 3y2 is the terrain steepest?
(1, 1, 4), (0, 2, 12), (2, 1, 7)
Solution: For the surface z = f (x, y), the steepness can be found by computing f
640:251:0103
REVIEW PROBLEMS FOR FINAL EXAMPage 1
FALL 2014
This set of problems concentrates primarily on material from Chapters 16 and 17 of Rogawski, with
some questions material from earlier chapters. To review for the final you should study also the
Math 251, Quiz #6 Solutions, November 11, 2014
1. Let D be the parallelogram spanned by vectors h7, 2i and h4, 4i anchored at the origin.
Let G(u, v) = h7u + 4v, 2u + 4vi.
a) Show that G maps the square [0, 1] [0, 1] to D.
Solution: We know linear maps se
Math 251, Quiz #4, October 28, 2014
1. Compute the average value of f (x, y) = xy over the square [0, 1] [0, 1]. Recall that the mean
value theorem for integrals says that (since f is continues and the region is closed, bounded and connected) that this av
Math 251, Quiz #4 Solutions, October 21, 2014
1. Use Lagrange multipliers to find the maximum value of f (x, y, z) = xyz subject to the
condition x + 2y + 3z = 9.
Solution: The question is misworded. There is no maximum value. To see this, we will
interse
Math 251, Midterm Two Review Questions
1. Find the max and min values of f (x, y, z) = x+y+2z on the ellipsoid x2 +4y 2 +9z 2 = 1.
2. Consider the cone frustum show below
Figure 1: Dont get FRUSTrated!
Compute the volume of this cone by parameterizing the