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INSTRUCTOR: JOS
´
E MANUEL G
´
OMEZ
Solutions Homework 8.
Please show all your work.
(1) (5 Points) Consider the function
f
(
x
) =
√
2
x
+ 2. Use a suitable the linear approxi
mation on the function
f
(
x
) to estimate
√
4
.
2.
Solution:
Note that
√
4
.
2 =
f
(1
.
1). Thus we can use the linear approximation of
f
(
x
) at
x
= 1 to approximate
f
(1
.
1). By deFnition the linear approximation of
f
(
x
)
at
x
= 1 is
L
(
x
) =
f
(1) +
f
′
(1)(
x
−
1)
.
In this case
f
(1) =
√
4 = 2. Also
f
′
(
x
) =
1
2
(2
x
+ 2)
−
1
/
2
(2) =
1
√
2
x
+ 2
Therefore
f
(1) =
1
√
4
=
1
2
. It follows that the linear approximation of
f
(
x
) at
x
= 1
is
L
(
x
) = 2 +
1
2
(
x
−
1) =
3
2
+
x
2
.
We can estimate
f
(1
.
1) with
L
(1
.
1) to obtain
√
4
.
2 =
f
(1
.
1)
∼
=
L
(1
.
1) =
3
2
+
1
.
1
2
= 2
.
05
(2) (10 Points) Suppose that
w
(
x
) =
x
2
x
2
+ 1
.
±ind the intervals where
w
(
x
) is increasing, decreasing, concave up and concave down.
Solution:
We have
w
′
(
x
) =
(
x
2
+ 1)(2
x
)
−
x
2
(2
x
)
(
x
2
+ 1)
2
=
2
x
(
x
2
+ 1)
2
.
Note that
w
′
(
x
) exists for every real number
x
. Also
w
′
(
x
) = 0 if and only if
2
x
(
x
2
+ 1)
2
= 0
and this occurs if and only if
x
= 0. We see that
f
′
(
x
)
>
0 for
x
in the interval
(0
,
∞
) and
f
′
(
x
)
<
0 for
x
on the interval (
−∞
,
0). Therefore
w
is increasing on the
1
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INSTRUCTOR: JOS
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 Fall '11
 JoseGomez
 Calculus, Approximation, Linear Approximation

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