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(12.12) Taylor Polynomials
Partial sums of Taylor series are called Taylor polynomials. Here, we study how closely
they approximate a function. Recall if
f
(
x
)=
a
0
+
a
1
(
x

c
)+
a
2
(
x

c
)
2
+
a
3
(
x

3)
3
+
a
4
(
x

c
)
4
+
···
then
a
n
=
f
(
n
)
(
c
)
n
!
.
We deFne the
n
th
Taylor polynomial
T
n
(
x
) to be the
n
th partial sum of the Taylor
series, so
T
n
(
x
∑
n
k
=0
a
k
(
x

c
)
k
(where
a
k
=
f
(
k
)
(
c
)
k
!
). We deFne the
n
th
remainder
to be
R
n
(
x
f
(
x
)

T
n
(
x
).
We will study how closely
T
n
(
x
) approximates
f
(
x
) using
Taylor’s inequality
:
If

f
(
n
+1)
(
x
)
≤
M
for all
x
in the interval (
c

d, c
+
d
), then for all
x
in that interval we
have

R
n
(
x
)
M
(
n
+ 1)!

x

c

n
+1
.
Example
Let
f
(
x
) = sin
x
. Expand
f
(
x
) as a Taylor series around
c
= 0. Recall
sin
x
=
x

1
3!
x
3
+
1
5!
x
5

1
7!
x
7
+
.
Sketch the graph of
T
5
(
x
x

1
6
x
3
+
1
120
x
5
with the graph of sin
x
. Notice how close the
graphs are near
x
=0.
Now we ask the question, how close does
T
5
(
x
) approximate sin
x
for
x
in the interval
[

0
.
5
,
0
.
5]?
To use Taylor’s inequality, we need to Fnd out how big
f
(6)
(
x
) can be on that interval. We
have
f
(6)
(
x

sin
x
.On[

0
.
5
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 Fall '08
 VALDIMARSSON
 Polynomials, Taylor Series

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