Math 213 Calculus III
Final Exam
Spring 2002
Name:
You must show your work and justify your answers to receive full credit.
# Points
1
16
2
16
3
17
4
17
5
17
6
17
Total 100
Score
Spherical Coordinates
x = sin cos ,
y = sin sin ,
z = cos
1
1. Let
2x2 + y
Math 113 Calculus III
Final Exam Practice Problems
Spring 2003
1. Let
2x2 + y 2 + 4z 2 .
g (x, y, z ) =
(a) Describe the shapes of the level surfaces of g .
(b) In three dierent graphs, sketch the three cross sections to the level surface
g (x, y, z ) = 1
Math 113 Calculus III SOLUTIONS: Exam 2 Practice Problems Spring 2003
1. Suppose u is a unit vector, and v and w are two more vectors that are not necessarily
unit vectors. Simplify the following expression as much as possible:
(v u)u) (v w) (w v ) (v (u
Math 213 Calculus III
Practice Exam 2 Solutions
1. Let
Fall 2002
2
g (x, y, z ) = e(x+y) + z 2 (x + y ).
(a) What is the instantaneous rate of change of g at the point (2, 2, 1) in the direction of
the origin?
We want the directional derivative of g at (2
Math 113 Calculus III
Exam 1 Practice Problems
Spring 2004
Name:
1. Match each function to one of the given graphs, and to one of the given contour
diagrams. The plots are given in a separate handout.
f (x, y )
3y
2
2
x+
Graph
2
1
x2 y 2
2
x+
y
2
x+
y
2
s
Math 113 Calculus III
Practice Exam 1
Spring 2003
Name:
1. Match each function to one of the given graphs, and to one of the given contour
diagrams. The plots are given in a separate handout.
f (x, y )
3y
2
2
x+
Graph
2
1
x2 y 2
2
x+
y
2
x+
y
2
sin(x)
xy
Math 113 Calculus III
Exam 2 Practice Problems
Spring 2003
1. Suppose u is a unit vector, and v and w are two more vectors that are not necessarily
unit vectors. Simplify the following expression as much as possible:
(v u)u) (v w) (w v ) (v (u v )u).
2. L
Math 213 Calculus III
Practice Exam 2
1. Let
Fall 2002
2
g (x, y, z ) = e(x+y) + z 2 (x + y ).
(a) What is the instantaneous rate of change of g at the point (2, 2, 1) in the direction
of the origin?
(b) Suppose that a piece of fruit is sitting on a table
Math 113 Calculus III
Exam 4 Practice Problems
Spring 2004
1. The energy stored in an idealized stretched spring is proportional to the square of the
length of the string.
Suppose three springs lie in the xy plane. One end of the rst spring is xed at (1,
Math 113 Calculus III
Exam 3 Practice Problems
1. Let
Spring 2004
2
g (x, y, z ) = e(x+y) + z 2 (x + y ).
Suppose that a piece of fruit is sitting on a table in a room, and at each point (x, y, z )
in the space within the room, g (x, y, z ) gives the stre
Math 113 Calculus III
Exam 2 Practice Problems
Spring 2004
1. A train is traveling northwest at 10 miles per hour. A person in the train walks at 2
miles per hour from a window on the left side to a window directly across the train on
the right side. (Lef