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MATH 241:
CALCULUS III
Department of Mathematics, UMCP
Fall 2007
Practice Exam 2
Handed out
: Tuesday, 10/23/07
WORK ON ALL PROBLEMS. Justify your answers. Cross out what is not meant to be part
of your Fnal answer. This test should take approx. 90 min.
The actual exam will have 4 problems
with fewer questions.
1. (10 pts) A function
z
satisFes Laplace’s equation in the variables
x
and
y
if
∂
2
z
∂x
2
+
∂
2
z
∂y
2
= 0
.
Show
that
z
(
x, y
) = tan

1
(
y/x
) +
Ax
+
By
+
Cxy
satisFes Laplace’s equation where
A
,
B
and
C
are arbitrary constants.
Hint:
Recall that
d
dx
tan

1
x
= (1 +
x
2
)

1
.
2. (10 pts) Consider the function
f
(
x, y
) =
xy
n
x
4
+
y
4
,
for (
x, y
)
n
= (0
,
0)
0
,
for (
x, y
) = (0
,
0)
,
where
n
is a nonnegative integer (
n
= 0
,
1
,
2
, . . .
). Show
that
f
(
x, y
) is continuous
at the
point (0
,
0) if
n
≥
4
.
3. (10 pts) Let
w
=
f
(
x, y
) where
x
=
e

s
cos
t
and
y
=
e

s
sin
t
.
(a)(4 pts) By use of the Chain Rule, compute
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