Final Exam, Math. 20C
Dr. Cristian D. Popescu
March 14, 2005
Name:
Student ID:
Section Number:
Note:
There are 5 problems on this exam. Each of them is worth 40 points. You
will not receive credit unless you show all your work. No books, calculators, notes
or tables are permitted.
I. (40 points)
Let
f
(
x, y
) be the function given by
f
(
x, y
) =
x
4
+
y
4

4
xy
+ 1
.
(1) Determine the critical points of
f
(
x, y
).
(2) Classify the critical points of
f
(
x, y
) into local maxima, local minima, and
saddle points, respectively.
(3) Determine the global maximum and minimum points of
f
(
x, y
) on the
(closed and bounded) domain
D
=
{
(
x, y
)

0
≤
x
≤
3
,
0
≤
y
≤
2
}
.
(4) Use the method of Lagrange multipliers to find minimum value of
f
(
x, y
)
along the curve given by the equation 4
xy
= 1
.
Is there a maximum value
for
f
along the given curve ?
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II. (40 points)
(1) Compute the integral
ZZ
D
f
(
x, y
)
dA ,
where
f
(
x, y
) =
y
cos(
x
2
) and
D
is the plane region situated above the
x
–axis and bounded by the curves
y
= 0,
x
=
y
2
, and
x
= 9.
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 Fall '07
 BoonYeap
 Osculating circle, Dr. Cristian D. Popescu, Cristian D. Popescu

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