MTH 122 — Calculus II
Essex County College — Division of Mathematics and Physics
1
Lecture Notes #12 — Sakai Web Project Material
1
Taylor Polynomials
In the last class we actually generated several Taylor
2
polynomials and everyone should be clear
that our example followed a pattern, as follows:
Degree 1:
The Taylor Polynomial of degree 1 approximating
f
(
x
) for
x
near zero is:
f
(
x
)
≈
f
(0) +
f
0
(0)
x.
Degree 2:
The Taylor Polynomial of degree 2 approximating
f
(
x
) for
x
near zero is:
f
(
x
)
≈
f
(0) +
f
0
(0)
x
+
f
00
(0)
2!
x
2
.
Certainly if we continue this process an easy pattern emerges. For example if we have an
n
th
degree polynomial of the form,
f
(
x
)
≈
C
0
+
C
1
x
+
C
2
x
2
+
C
3
x
3
+
C
4
x
4
+
· · ·
+
C
n

1
x
n

1
+
C
9
x
n
,
and we follow the same process outlined in the prior worksheet, we’ll get:
C
0
=
f
(0)
C
1
=
f
0
(0)
1!
C
2
=
f
00
(0)
2!
C
3
=
f
000
(0)
3!
C
4
=
f
(4)
(0)
4!
.
.
.
=
.
.
.
C
n
=
f
(
n
)
(0)
n
!
Finally we have a Taylor polynomial of degree
n
approximating
f
(
x
) for
x
near 0 is,
f
(
x
)
≈
f
(0) +
f
0
(0)
x
+
f
00
(0)
2!
x
2
+
f
000
(0)
3!
x
3
+
f
(4)
(0)
4!
x
4
+
· · ·
+
f
(
n
)
(0)
n
!
x
n
.
1
This document was prepared by Ron Bannon (
[email protected]
) using L
A
T
E
X 2
ε
. Last revised
January 10, 2009.
2
Brooke Taylor was an English Mathematician (1685–1731), but these approximating polynomials were known
prior to Taylor’s exposition on the subject.
1
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1.1
Examples
1. Find the Taylor polynomial of degree 9 about
x
= 0 for the function
f
(
x
) =
e
x
.
Work:
This one is pretty easy, mainly because the derivative of
e
x
never changes. So we
have:
e
x
= 1 +
x
+
x
2
2!
+
x
3
3!
+
x
4
4!
+
x
5
5!
+
x
6
6!
+
x
7
7!
+
x
8
8!
+
x
9
9!
.
Here’s a graph of both
f
(
x
) =
e
x
(in black) and the ninth degree polynomial (in red). You
should notice that the fit is not perfect, but it looks damn good for
x >

3. Although not
obvious, the higher degree for this polynomial the better the fit becomes.
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 Spring '10
 Ban
 Calculus, Polynomials, Division, Power Series, Taylor Series, Taylor's theorem, Taylor Polynomial, Brooke Taylor

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