Answer Guide 1
Math 408C: Unique Numbers 56975, 56980, and 56985
Tuesday, September 1, 2009
Homework problems
Section 1.1
22.
A spherical balloon with radius
r
+ 1
has volume
V
(
r
+ 1) =
4
3
(
r
+ 1)
3
=
4
3
(
r
3
+ 3
r
2
+ 3
r
+ 1)
:
r
to radius
r
+ 1
is
V
(
r
+ 1)
V
(
r
) =
4
3
(3
r
2
+ 3
r
+ 1)
:
(
F
)
24.
Because
f
(
x
) =
x
3
, we have
f
(
a
+
h
) =
a
3
+ 3
a
2
h
+ 3
ah
2
+
h
3
:
Therefore,
f
(
a
+
h
)
f
(
a
)
h
=
3
a
2
h
+ 3
ah
2
+
h
3
h
= 3
a
2
+ 3
ah
+
h
2
:
(
F
)
26.
We compute that
f
(
x
)
f
(1)
x
1
=
x
+3
x
+1
4
2
x
1
=
(
x
+ 3)
2(
x
+ 1)
(
x
1)(
x
+ 1)
=
x
+ 1
(
x
1)(
x
+ 1)
=
1
x
+ 1
:
(
F
)
36.
Notice that
H
(
t
) =
4
t
2
2
t
=
(2 +
t
)(2
t
)
2
t
:
The domain of
H
is
f
t
:
t
6
= 2
g
. As long as
t
6
= 2
, we can cancel, obtaining
H
(
t
) = 2 +
t
.
The graph
y
=
H
(
t
)
is the same as the graph
y
= 2 +
t
with the point
2
;
4)
removed.
54.
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 Fall '06
 McAdam
 Math, Calculus

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