9.3

Taylor’s Theorem

Quick Review

2

Find the smallest number that bounds from above on the

interval (that is, find the smallest such that ( ) for

all in ).

1. ( ) 2cos(3 ), -2 , 2

2. ( ) 3 1, 2

3. ( ) 2 -3,0

4.

x

M f

I M f x M

x I

f x x I

f x x I

f x I

2

( ) -2, 21

xf x I

x

2

71

2

1

Quick Review

2

-

3

2

Tell whether the function has derivatives of all orders at the given values of .

5. , 01

6. 4 , 2

7. sin cos ,

8. , 0

9. , 0

x

a

xa

xx a

x x a

e a

x a

Tell whether the function has derivatives of all orders at the given values of a.

Yes

No

Yes

Yes

No

What you’ll learn about Taylor Polynomials The Remainder Remainder Estimation Theorem Euler’s Formula

Essential QuestionsHow do we determine the error in the approximationof a function represented by a power series by itsTaylor polynomials?

Example Approximating a Function to Specifications

1. Find a Taylor polynomial that will serve as an adequate substitute for sin x on the interval [– , ].

Choose Pn(x) so that |Pn(x) – sin x| < 0.0001 for every x in the interval [– , ].

We need to make |Pn() – sin | < 0.0001, because then Pn then will be adequate throughout the interval

0001.0sin nP

0001.0nP

Evaluate partial sums at x = , adding one term at a time.

! 3

3

! 5

5

! 7

7

! 9

9

! 11

11

! 31

13 510114256749.2

Taylor’s Theorem with RemainderLet f has a derivative of all orders in an open interval I containing a, then for each positive integer n and for each x in I

. and between somefor ! 1

where 11

xacaxn

cfxR n

n

n

,

! ! 2 2 xRax

n

afax

afaxafafxf n

nn

Example Proving Convergence of a Maclaurin Series

. real allfor sin toconverges ! 12

1 series that theProve 2.0

12

xxk

x

k

kk

Consider Rn(x) as n → ∞. By Taylor’s Theorem,

where f (n+1)(c) is the (n + 1)st derivative of sin x evaluated at some c between x and 0.

11

! 1

nn

n axn

cfxR

11 Since, 1 cf n

11

0 ! 1

nn

n xn

cfxR

1

! 1

1

nx

n ! 1

1

n

xn

As n → ∞, the factorial growth is larger in the bottom than the exp. growth in the top.

. allfor 0! 1

as Therefore,1

xn

xn

n

This means that Rn(x) → 0 for all x.

Remainder Estimation TheoremIf there are positive constants M and r such that

. ! 1

11

n

axrMxR

nn

n

11 nn Mrtf

for all t between a and x, then the remainder Rn(x) in Taylor’s Theorem

satisfies the inequality

If these conditions hold for every n and all the other conditions of

Taylor’s Theorem are satisfied by f , then the series converges to f (x).

Example Proving Convergence

3. Use the Remainder Estimation Theorem to prove the following for all real x.

0 ! k

kx

k

xe

We have already shown this to be the Taylor series generated by e x at x = 0.

We must verify Rn(x) → 0 for all x.

To do this we must find M and r such that .arbitrary and 0between for by bounded is 11 xtMretf ntn

Let M be the maximum value for e t and let r = 0.

If the interval is [0, x ], let M = e x .

If the interval is [x, 0 ], let M = e 0 = 1.

In either case, e x < M throughout the interval, and the Remainder Estimation

Theorem guarantees convergence.

Euler’s Formula

xixeix sincos

Quick Quiz Sections 9.1-9.3

20

2

2

2

1. Which of the following is the sum of the series ?

(A) -

(B) -

(C) -e

(D) -

(E) The series diverges

n

nn ee

e

e

e

e

Quick Quiz Sections 9.1-9.3

20

2

2

2

1. Which of the following is the sum of the series ?

(

(

A) -

(B) -

(C) -e

(E) The series diverges

D) -

n

nn e

e

e

e

e

e

Quick Quiz Sections 9.1-9.3

2

2. Assume that has derivatives of all orders for all real numbers ,

(0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following

is the third order Taylor polynomial for at 0?

(A) 2 3

f x

f f f f

f x

x x

3

2 3

2 3

2 3

2

2

(B) 2 6 12

1(C) 2 3 2

2(D) 2 3 2

(E) 2 6

x

x x x

x x x

x x x

x x

Quick Quiz Sections 9.1-9.3

2

2. Assume that has derivatives of all orders for all real numbers ,

(0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following

is the third order Taylor polynomial for at 0?

( ) 2A 3

f x

f f f f

f

x

x

x

2 3

2 3

2

3

3

2

(B) 2 6 12

1(C) 2 3 2

2(D) 2 3 2

(E) 2

2

6

x x x

x x x

x x x

x

x

x

Quick Quiz Sections 9.1-9.3

0

0

0

0

0

3. Which of the following is the Taylor series generated by

( ) 1/ at 1?

(A) 1

(B) 1

(C) 1 1

1(D) 1

!

(E) 1 1

n

n

n n

n

n n

n

n

n

n

n n

n

f x x x

x

x

x

x

n

x

Quick Quiz Sections 9.1-9.3

0

0

0

0

0

3. Which of the following is the Taylor series generated by

( ) 1/ at 1?

(A) 1

(B) 1

(C) 1 1

1(D) 1

!

(E) 1 1

n

n

n n

n

n n

n

n

n

n

n n

n

f x x x

x

x

x

x

n

x

Pg. 386, 7.1 #1-25 odd