Math 324A
Sample Quiz 5 SOLUTIONS
1
Let S be the surface y = x2 + z 2 with 0 y 1.
(a) Give a parametrization of S, including stating the domain.
(b) Orient S in the direction of the positive y axis, and let F = x, y, 0 . Set up (but do
not evaluate) the s
Math 324A
Quiz 3 Sample Answers
Autumn 2011
1
1. Let S be the portion of the surface y = x2 where 0 z x 2.
(a) Give a parametrization of S, including its domain.
Because the surface is a graph, y = f (x, z) = x2 , We should let x = u and z = v, and
then y
Math 324A
Solutions for Q4 Sample questions
Autumn 2011
1
1) Du f (2, 2) = f (2, 2) u = | f (2, 2)| cos() where is the angle between the two vectors.
The scale of on this graph can be determined by the fact that u is a unit vector, or by
the location of t
Math 324A
Quiz 4 Solutions
November, 2011
1
IMPORTANT GENERAL COMMENT: Be careful about which quantities
are vectors and which are scalars, especially if I wrote a comment about this
on your paper.
1. a) Compute
f at (1, 3, 10).
f (1, 3, 10) = 2xy, x2 + 3
Math 324A
Answers for Quiz 1 Samples
3
3
Answers for Last years quiz about the integral
Winter 2010
0
9y 2
1
x dx dy.
SKETCH THE DOMAIN OF INTEGRATION to see the limits for the other order and
for polar! Answers:
0
Question 1)
3
9x2
0
3/2
9x2
x dy dx;
r
Math 324
Final Exam
Page 1 of 8
1. Let S be the part of the plane 3x + 2y + z = 6 which lies above the square 1 x, y 1 in the
xy-plane, oriented in the positive x direction. Compute the surface integral
x, y, z dS.
S
Solution. Viewing S as part of the gra
Final Exam Review
This covers Chapters 15 and 16. For the sections from Chapter 14 we covered,
see the midterm 2 review. This does overlap with the midterm 2 review, but its
organized in a dierent way: having nished Chapter 16, hindsight allows us to
thin
Extra Credit 1
Consider the double integral
region
1 x1
0 0
1 dx2 dx1 . This is the area of the
S = cfw_(x1 , x2 ) : 0 x1 1, x2 x1 1
= cfw_(x1 , x2 ) : 0 x2 x1 1 :
(0, 1)
(1, 1)
(0, 0)
(1, 0)
The picture makes it obvious that the value of the integral is
Extra Credit 2
1. We know that the gradient vector of a function, at each point, points in the
direction of greatest change for the function. This leads to a simple algorithm
for nding local maxima (or minima) of functions numerically. Pick a point,
and c
Extra Credit 2 Solution
Problem 1: Write a computer program which uses gradient ascent, starting with
p0 = (0, 0) and using a step size of h = .2 with 100 steps, to estimate the maximum
of
f (x, y) = 2xy 2 2x2 y x4 x2 + 2x y 4 3y 2 .
Your program should o
Extra Credit 1 Solution
1 x1
0 0
Consider the double integral
1 dx2 dx1 . This is the area of the region
S = cfw_(x1 , x2 ) : 0 x1 1, x2 x1 1
= cfw_(x1 , x2 ) : 0 x2 x1 1 :
(0, 1)
(1, 1)
(0, 0)
(1, 0)
The picture makes it obvious that the value of the int
Midterm 2 Review
Derivatives (14.5, 14.6)
Gradients
f points in the direction of greatest change for f , and is 0 at extrema
f is perpendicular to the level curves/surfaces of f
Directional derivatives
Du f = u f gives the rate of change of f in the di