Name:
MAC 2313
Exam 4 Version A
10/17/2011
SHOW ALL YOUR WORK. Answers without procedure will be graded zero.
[5] 1.The radius
r
and height
h
of a cylinder change with time. At a certain instant the dimensions are
r
= 1
and
h
= 6, and
r
is
increasing
at a rate of 2 m/s while
h
is
decreasing
at a rate of 1 m/s. At that instant ﬁnd
the rate at which the volume of the cylinder is changing.
[5] 2. Find
∂z
∂u
, if
z
= cos
x
sin
y
,
x
=
u

v
and
y
=
u
2
+
v
2
. Express your answer in terms of
u
and
v
.
[5] 3. Suppose that
z
=
z
(
x,y
) (
z
is implicitly deﬁned as a function of
x
and
y
), and also
y
=
y
(
x,z
) through
some equation
F
(
x,y,z
) = 0. Suppose that at some point
P
,
F
x
= 2,
F
y
=

4 and
F
z
= 3, which of the following
is the smallest at
P
?
∂z
∂y
,
∂z
∂x
or
∂y
∂x
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View Full Document[5] 4. Find the directional derivative of
f
(
x,yz
) =
z
2
e
xy
at the point (

1
,
0
,
3), in the direction of
v
=
h
3
,
1
,

5
i
[5] 5. What is the maximum value of the directional derivative of
f
(
x,y,z
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 Spring '08
 Keeran
 Calculus, Geometry, Differential Calculus, Extra Credit Problem, lagrange multipliers

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