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SECTION
15.3
Partial Derivatives
(ET Section 14.3)
647
Hence,
V
∂
P
∂
T
=
nR
⇒
∂
P
∂
T
=
nR
V
.
47.
The volume of a right-circular cone of radius
r
and height
h
is
V
=
π
3
r
2
h
. Calculate
∂
V
∂
r
and
∂
V
∂
h
.
SOLUTION
We obtain the following derivatives:
∂
V
∂
r
=
∂
∂
r
³
3
r
2
h
´
=
h
3
∂
∂
r
r
2
=
h
3
·
2
r
=
2
hr
3
∂
V
∂
h
=
∂
∂
h
³
3
r
2
h
´
=
3
r
2
48.
A right-circular cone has
r
=
h
=
12 cm. What leads to a greater increase in
V
, a 1-cm increase in
r
or 1-cm
increase in
h
? Argue using partial derivatives.
SOLUTION
The partial derivatives
∂
V
∂
r
and
∂
V
∂
h
can be estimated for small values of
1
r
and
1
h
by
∂
V
∂
r
≈
V
(
r
+
1
r
,
h
)
−
V
(
r
,
h
)
1
r
and
∂
V
∂
h
≈
V
(
r
,
h
+
1
h
)
−
V
(
r
,
h
)
1
h
Hence,
V
(
r
+
1
r
,
h
)
−
V
(
r
,
h
)
≈
∂
V
∂
r
1
r
V
(
r
,
h
+
1
h
)
−
V
(
r
,
h
)
≈
∂
V
∂
h
1
h
Therefore an increase
1
r
=
1cmin
r
leads to an increase of
∂
V
∂
r

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