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MATH 51 FINAL EXAM
December 12, 2005
Name:
Numeric Student ID:
TA and Section No.:
I agree to abide by the terms of the honor code:
Signature:
Instructions:
Print your name, and student ID number in the space provided. You
may not use notes, or textbooks, or calculators. Read each question carefully. Some
questions continue to more than one page. Correct answers without justiﬁcation will
receive little or no credit. There are 12 questions. There is a 3 hour time limit on this
exam. Good luck.
Question
Score
Maximum
1
10
2
10
3
10
4
10
5
10
6
15
7
10
8
10
9
10
10
10
11
10
12
5
Total
120
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View Full Document Question 1 of 12, Page 2 of 17
Name:
1. Find all critical points of the function
f
(
x, y
) = 3
xy

x
2
y

2
xy
2
. Determine the
nature of the critical points (i.e. local min/local max/neither max nor min).
Question 2 of 12, Page 3 of 17
Name:
2. Compute the determinant
±
±
±
±
±
±
±
±

2
1
1

1
1

2

1
1
1

1

2
1

1
1
1

2
±
±
±
±
±
±
±
±
.
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View Full Document Question 3 of 12, Page 4 of 17
Name:
3. Consider the function
f
(
x, y
) =
y
x
2
.
(a) Draw three diﬀerent level sets of
f
. Brieﬂy describe the shape of the resulting
curves you ﬁnd.
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