Math 151A HW1 Solutions
Will Feldman
1. We want to find the third order Taylor polynomial for the function
f
(
x
) =
√
1 +
x
about
x
0
= 0. Recall the formula for the Taylor polynomial of order
n
centered at
x
0
is given by
P
n
(
x
) =
n
X
k
=0
f
(
k
)
(
x
0
)
k
!
(
x

x
0
)
k
.
Calculating some derivatives of
f
you will find that
P
3
(
x
) = 1 +
x
2

x
2
8
+
x
3
16
.
Now we want to use this to approximate some values of the square root. Note that
√
y
=
f
(
y

1), so that (for example)
√
0
.
5 =
f
(

0
.
5)
≈
P
3
(

0
.
5)
≈
0
.
711. The same
method applies to the other values of square root.
2. We can calculate the Taylor series for
f
(
x
) = (1

x
)

1
by calculating that
f
(
n
)
(
x
) =
n
!(1

x
)

(
n
+1)
. Then we have
P
n
(
x
) =
n
X
k
=0
x
k
.
In order to calculate the error of using
P
n
(
x
) to approximate
f
on
x
∈
[0
, .
5] note that
we can calculate
P
n
(
x
) exactly since
xP
n
(
x
) =
x
n
+1
+
P
n
(
x
)

1
P
n
(
x
) =
x
n
+1

1
x

1
.
So that we have,

f
(
x
)

P
n
(
x
)

=
x
n
+1
1

x
≤
1
2
n
.
Then just choose
n >
6 log 10
/
log 2 so that (1
/
2)
n
<
10

6
.
3. Solution can be found in Burden and Faires.
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 Spring '08
 hitrik
 Math, Taylor Series, pn, Feldman, 3 digit

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