Math 2110, Discussion, Week 14
1. Compute curl F if F =
2. T
F
(xyz).
For any twice dierentiable function f (x, y, z), if F =
f then curl F = 0.
3. Compute div F if F = curl y 2 , z 2 , x2 .
4. T F For any twice dierentiable vector function G, if F = curl
Name:
Discussion section:
(circle one)
9:35 Tuesday | 11:00 Tuesday | 1:20 Tuesday | 9:35 Thursday | 11:00 Thursday | 1:20 Thursday
Math 2210 Calculus III Exam IV
Spring 2014 Thursday, May 8
Instructions:
Except for problem 1, answers that show little or
Calculus III Practice Exam 1
1. V 4I 7J W I 5J are two vectors in the plane. a) Find the angle between V and W. b) Find the angle between V and W . c) Find the area of the parallelogram spanned by W and W . 2. Find the distance of the point (2,-2) from th
Name:
Discussion section:
(circle one)
9:35 Tuesday | 11:00 Tuesday | 1:20 Tuesday | 9:35 Thursday | 11:00 Thursday | 1:20 Thursday
Math 2210 Calculus III Mock Exam IIIc
Instructions:
Except for problems 1 and 5, answers that show little or no work will
Math 2110, Discussion, Week 15
A paraboloid has the parametrization r(x, y) = x, y, 4 x2 y 2 .
1. Compute dS for the paraboloid. Simplify your answer as much as possible.
ANSWER: because rx ry = 2x, 2y, 1 ,
dS = | 2x, 2y, 1 dx dy | =
1 + 4x2 + 4y 2 dx dy.
Math 2110, Discussion, Week 15
A paraboloid has the parametrization r(x, y) = x, y, 4 x2 y 2 .
1. Compute dS for the paraboloid. Simplify your answer as much as possible.
2. Compute the surface area of the part of the paraboloid where z 0. Hint: change to
Math 2110, Discussion, Week 13
1. Compute the line integral
8x ds where C is the curve parametrized by
C
r(t) = t, 1 t2 , 2 + t2 from P (0, 1, 2) to Q(2, 3, 6).
ANSWER:
The parametric equations for the curve are x = t, y = 1 t2 , and z = 2 + t2 where
0 t
Calculus III Practice Exam 2, Answers
1. A conic in the plane is given by the equation x2 a) What conic is it? Answer. Since B2 square to obtain: x 4AC 2 y 2 2x y
2
2xy
4 1 1 0, this is a parabola. We can verify that by completing the 0, which is the para
Calculus III Practice Exam 3, Answers
1. Find the volume under the plane z
x
2y 1 over the triangle bounded by the lines y
Answer. We view this as the region lying over the triangle T and under the plane z type 1 domain 0 x 1 0 y 2x. Thus Volume The inner
Calculus III Practice Exam 1, Answers
1. V
4I 7J W
I5J are two vectors in the plane.
a) Find the angle between V and W. Answer. Let be the angle. Then cos so arccos 7508 V W VW 41 75 65 26
31
1690
7508
7165 radians.
b) Find the angle between V and W . An