18.02 PRACTICE FINAL EXAM A
BJORN POONEN
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1
out of 25
2
out of 10
3
out of 10
4
out of 15
5
out of 20
6
out of 10
7
out of 10
8
out of 25
9
out of 20
10
out of 20
11
out of 30
12
out of 20
13
out of 15
14
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) If
a
and
b
are vectors in
R
3
such that
a
×
b
=
0
, then either
a
=
0
or
b
=
0
.
(b) If the first row of a 3
×
3 matrix is the same as the second row, then the determinant
of the matrix is 0.
(c) If
T
is a unit tangent vector at
P
to a level curve of a differentiable function
f
(
x, y
),
then the directional derivative of
f
at
P
in the direction
T
is 0.
(d) Let
P
=
P
(
x, y
) and
Q
=
Q
(
x, y
) be continuously differentiable functions on the region
R
in
R
2
where 1
< x
2
+
y
2
<
4. If
P dx
+
Q dy
is an exact differential (i.e., the differential
of some function on
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 Fall '08
 Auroux
 Multivariable Calculus, Vector Calculus, South Pole, problem number, scratch work, BJORN POONEN

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