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Mathematics 23 Final Exam
December 14, 2002
Solutions
This document was created on December 11, 2007 at 1:36pm.
These solutions are presented without warranty. If you ﬁnd any typos or errors, please
inform Ken at
ken.marks@lehigh.edu
.
1. The cross product of any two vectors on the plane gives us a vector which is perpen
dicular to the plane.
h
1
,
2
,
1
i × h
0
,

2
,
0
i
=
h
2
,
0
,

2
i
.
2. (a) Maximum rate of change at (1
,
0) is
∇
f
(1
,
0)

=
√
1 + 16 =
√
17.
(b) Direction of maximum rate of change is
∇
f
(1
,
0) =
h
1
,
4
i
.
3. In spherical coordinates the cone is
φ
=
π/
4 and the sphere is
ρ
= cos
φ
. Thus the
volume of the solid is
V
=
Z
2
π
0
Z
π/
4
0
Z
cos
φ
0
ρ
2
sin
φ dρ dφ dθ.
4. Use Lagrange multipliers to maximize
f
(
x, y, z
) =
xyz
subject to the constraint
g
(
x, y, z
) =
x
+ 3
y
+ 2
z
= 6. Note that
x, y, z
are all nonzero. We solve the sys
tem of equations
yz
=
λ,
xz
= 3
λ,
xy
= 2
λ,
x
+ 3
y
+ 2
z
= 6
.
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 Spring '06
 YUKICH
 Math

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