Final sample exam
If you have any questions, or think you’ve spotted an error / typo, please get in
touch with me. Solutions will appear by the end of Tuesday (hopefully before,
but that’s probably excessively optimistic).
1. Find an equation describing the collection of points that is equidistant
between the sphere center the origin and radius 2, and the origin itself. (6
points)
2. Find a scalar equation for the plane that is perpendicular to the line
segment from
P
= (1
,
2
,
3) to
Q
= (3
,
2
,
1) and contains
Q
. (6 points)
3. Find the cosine of the angle between the plane containing the points
(1
,
2
,
3), (3
,
2
,
1) and (0
,
0
,
0), and the
xy
plane. (6 points)
4. Are the lines
~
r
1
(
t
) =
h
1 +
t,
2
,
1

t
i
and
~
r
2
(
t
) =
h
3
t, t,

2
t
i
parallel, skew,
or do they intersect at a unique point?
If the last, find the point.
(8
points)
5. Let
P
denote the parallelepiped formed from the vectors
~a
=
h
1
,
0
,
0
i
,
~
b
=
h
0
,
1
,
0
i
and
~
c
=
h
0
,
1
,
1
i
.
(a) Use a triple integral to write down an expression for the volume of
P
. (10 points)
(b) Using your answer to part (a), or otherwise, compute the volume of
P
. (5 points)
6. The force due to gravity coming from a planet at the origin is given by
~
F
=
∇
f
, where
f
(
x, y, z
) =
1
p
x
2
+
y
2
+
z
2
.
A spaceship flies along a (clockwise) circular path in the
yz
plane at dis
tance 2 from the planet, starting at the point (0
,
2
,
0).
(a) Show that the direction of motion of the spaceship is always perpen
dicular to the gravitational forcefield. (7 points)
(b) What can you conclude about the integral
R
C
~
F
·
d~
r
, where
C
is the
path followed by the spaceship? (2 points)
(c) Say the spaceship flies instead around the (complete) circle
C
0
, center
(0
,
0
,
4), radius 3 and parallel to the
xy
plane.
What can you say
about
R
C
0
~
F
·
d~
r
? Justify your answer (in no more than 30 words).
(5 points)
7. Say
~
r
(
t
) =
h
3+
t,
2
,
1

t
i
, 1
≤
t
≤
4 represents the path of a sloth through
the jungle (...
a particularly clear bit of jungle where he can go in a
straight line, so not terribly realistic).
1
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(a) Find the arclength function
s
(
t
).
What does this represent physi
cally? (5 points)
(b) Find
~
r
00
(
t
). What does this tell you about the Sloth’s motion? (3
points)
8. Let
f
(
x, y
) =
xy
x
2
+
y
2
.
Is
f
continuous at the origin? Justify your answer. (5 points)
9. Let
f
(
x, y
) be a smooth (i.e. infinitely differentiable) function of two vari
ables, which is defined on all of
R
2
and has at least two critical points.
For each of the following, label T (true for all such f), F (false for all such
f), or M (true for some such
f
and not for others). (2 points each  partial
credit for giving a sensible reason for a wrong answer).
(a)
f
xyx
=
f
yxy
.
(b) The graph of
f
is unbounded in the
z
direction.
(c) The graph of
f
is unbounded in the
x
direction.
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 Spring '08
 ALDROUBI
 Coordinate system, Spherical coordinate system, Multiple integral, Polar coordinate system

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