9.2

Taylor Series

Quick Review

Find a formula for the nth derivative of the function.xe3 .1

x3 .2

nx .3

xln .4

xn e33

nx 3ln3

! n

nn xn ! 11 1

Quick Review

! .5

n

xy

n

dx

dy Find

! 121 .6

12

n

xy

nn

!

1 .7

n

xy

n

! 1

1

n

x

dx

dy n

! 2

1 2

n

x

dx

dy nn

! 1

1 1

n

x

dx

dy n

What you’ll learn about Constructing a Series Series for sin x and cos x Beauty Bare Maclaurin and Taylor Series Combining Taylor Series Table of Maclaurin Series

Essential QuestionHow can we use partial sums of a Taylor series toapproximate the function represented by the series?

Taylor Series Generated by f at x=0 (Maclaurin Series)

Let f be a function with derivatives of all orders throughout some open interval containing 0. Then the Taylor series generated by f at x = 0 is

n

n

xn

fx

fxff

!

0

! 2

0 00 2

.

!

0

0

k

k

k

xk

f

This series is also called the Maclaurin series generated by f.

k

n

k

k

n xk

fxP

!

0 sum partial The

0

is the Taylor polynomial of order n for f at x = 0.

Example Constructing a Power Series for sin x1. Construct the seventh order Taylor polynomial and the Taylor series

for sin x at x = 0. 00sin

0nsi 0cos 1 0nsi 0sin 0 0nsi 0cos 1

0sin 4 0sin 0 10sin 5 00sin 6 10sin 7

07 xP x1 20x 3

! 3

1x 40x 5

! 5

1x 60x 7

! 7

1x

753

! 7

1

! 5

1

! 3

1xxxx

9753

! 9

1

! 7

1

! 5

1

! 3

1xxxxx

1

12

! 121

n

nn

n

xThe Taylor series:

Example Approximating a Function near 02. Find the fourth order Taylor polynomial that approximates y = cos 2x

near x = 0. 10cos 0sco 0sin 0

0sco 0cos 1

0sco 0sin 0 0cos 4 0cos 1

42

! 4

1

! 2

11cos xxx

42 2! 4

12

! 2

112cos xxx

24

16

2

41

42 xx

14 xP 22x 4

3

2x

Taylor Series Generated by f at x=aLet f be a function with derivatives of all orders throughout some open interval containing a. Then the Taylor series generated by f at x = a is

n

n

axn

afax

afaxafaf

! ! 2 2

. ! 0

k

k

k

axk

af

k

n

k

k

n axk

afxP

!

sum partial The0

is the Taylor polynomial of order n for f at x = a.

Example A Taylor Series at x = 1

3. Find the Taylor series generated by f (x) = e x at x = 1.

eef 11 eef 11

een 1 xe 21! 2

xe nx

n

e1

!

0 ! k k

e .1 kx

Example A Taylor Polynomial for a Polynomial

4. Find the third order Taylor polynomial for f (x) = x 3 + 3x

2 – x + 1 at x = 0.

Since the polynomial is already written in powers of x and is of degree three,

then it is its own third order Taylor polynomial at x = 0.

5. Find the third order Taylor polynomial for f (x) = x 3 + 3x

2 – x + 1 at x = 1.

1

23 131

x

xxxf 4

1

2 163

x

xx 8

1f

1f

166

xx 12 1f 6

43 xP 18 x 21! 2

12 x 31

! 3

6 x

31 x 216 x 18 x 4

Maclaurin Series

x1

1 nxxx 21

0n

nx 1x

x1

1 nn xxx 11 2

0

1n

nn x 1x

ne ! ! 2

12

n

xxx

n

0 ! n

n

n

x x real all

xsin

! 121

! 5! 3

1253

n

xxxx

nn

0

12

! 121

n

nn

n

x

xcos ! 2

1! 4! 2

1242

n

xxx nn

0

2

! 21

n

nn

n

x x real all

x real all

Maclaurin Series

x1ln

n

xxxx

nn 1

32

132

0

11n

nn

n

x

11 x

x1tan

121

53

1253

n

xxxx

nn

0

12

121

n

nn

n

x

1x

Pg. 386, 9.2 #1-25 odd