18.02 FINAL EXAM
BJORN POONEN
December 14, 2010, 9:00am12:00 (3 hours)
Please turn cell phones off completely and put them away.
No books, notes, or electronic devices are permitted during this exam.
Generally, you must show your work to receive credit.
Name:
Student ID number:
Recitation leader’s last name:
Recitation time (e.g., 10am):
1
out of 25
2
out of 15
3
out of 15
4
out of 15
5
out of 15
6
out of 10
7
out of 25
8
out of 20
9
out of 20
10
out of 25
11
out of 25
12
out of 20
13
out of 20
Total
out of 250
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1) For each of (a)(e) below: If the statement is true, write TRUE. If the statement is
false, write FALSE. (Please do not use the abbreviations T and F.) No explanations are
required in this problem.
(a) (5 pts.) If
f
(
x, y
) is a continuously differentiable function on
R
2
, and
∂
2
f
∂x
2
= 0 at every
point of
R
2
, then there exist constants
a
and
b
such that
f
(
x, y
) =
ax
+
b
for all
x
and
y
.
(b) (5 pts.) The annulus in
R
2
defined by 9
≤
x
2
+
y
2
≤
16 is simply connected.
(c) (5 pts.)
If
f
(
x, y
) is a function whose second derivatives exist and are continuous
everywhere on
R
2
, and
f
(0
,
0) =
f
x
(0
,
0) =
f
y
(0
,
0) =
f
xx
(0
,
0) =
f
yy
(0
,
0) = 0
and
f
xy
(0
,
0) = 0, then
f
has a saddle point at (0
,
0).
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 Fall '08
 Auroux
 Derivative, Multivariable Calculus, Force, pts, scratch work, BJORN POONEN

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