Section 11.6 – Taylor’s Formula with Remainder

The Lagrange Remainder of a Taylor Polynomial

n 1n 1

n

f zR x x

n 1 !

where z is some number between x and c

The Error of a Taylor Polynomial

n 1

nR x b cn 1 !

M

where M is the maximum value of n 1f x

on the interval [b, c] or [c, b]

Let f be a function that has derivatives of all orders on the Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and 4f x 6 for all x in the interval (0, 1).

a. Find the third-degree Taylor polynomial about x = 0 for f.

2 33

1 1/ 4 3 / 8p x 1 x x x

2 2! 3!

2 33

1 1 1p x 1 x x x

2 8 16

b. Use your answer to part a to estimate the value of f(0.5)

2 3

3

1 10.5 0.5 0.5

1p 1 0.5

2 8 16

3

157p 0.5 1.227

128

Let f be a function that has derivatives of all orders on the Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and 4f x 6 for all x in the interval (0, 1).

c. What is the maximum possible error for the approximation made in part b?

3 1

3R x b c3 1 !

M

4

3 1

3R x b c3 1 !

f x

4

3R x 01

40.016

6

6

!0

4.5

3

157p 0.5 1.227

128

Estimate the error that results when arctan x is replaced by

34x

x if x 0.2 and3

f x 4

3 1

3R x b c3 1 !

M

3 1

3R x 0.2 03 1 !

4

4

3

1R x 0.2 0.000267

6

Estimate the error that results when ln(x + 1) is replaced by 21

x x if x 0.12

2

3

f x ln x 1

1f ' x

x 11

f " x

2f "' x

x 1

x 1

F ‘’’ (x) has a maximum value at x = -0.1

2 1

2R x 0 0.12 1 !

f ''' 0.1

2R x 0.000457

Find an approximation of ln 1.1 that is accurate to three decimalplaces.

We just determined that the error using the second degreeexpansion is 0.000457.

22

1p x x x

2

2

2

1p 0.1 0.1 0.1

2

2p 0.1 0.095

Use a Taylor Polynomial to estimate cos(0.2) to 3 decimal places2 4 6x x x

cosx 1 ...2! 4! 6!

If x = 0.2, Alternating Series Test works for convergence

2n0.2

0.0052n !

20.21 0.98

2os

!c x

Use a Taylor Polynomial to estimate 1

0

sinxdx

x with three decimal

place accuracy.1 3 5

0

1 x xx ... dx

x 3! 5!

1 2 4

0

x x1 ... dx

3! 5!

3 5 71o

x x xx ... |

3 3! 5 5! 7 7!

1 1 11 ...

3 3! 5 5! 7 7!

Satisfies Alternating Series Test

1

0.0052n 1 2n 1 !

1 11 0.946

3 3! 5 5!

Suppose the function f is defined so that

2

32

2 3x 11 1f 1 , f ' 1 , f " x

2 2 x 1

a. Write a second degree Taylor polynomial for f about x = 1

2

2

1 1 1/ 2p x x 1

2!x 1

2 2

2

2

1 1 1p x x 1 x 1

2 2 4

b. Use the result from (a) to approximate f(1.5)

2

2 0.311 1 1

p 1.5 1.5 1 1.5 12 2 4

25

Suppose the function f is defined so that

2

32

2 3x 11 1f 1 , f ' 1 , f " x

2 2 x 1

c. 1If f " x

2 for all x in [1, 1.5], find an upper bound for the

approximation error in part b if

n 1

nR x b cn 1 !

M

2

42

24x 1 xf "' x

x 1

12

2R x1 !

M

21.5 1

3

2

0

6MR

.5x

3

2

0.R x

60.42 009 77

5.008

The first four derivatives of 1f x are

1 x

3 / 2

5 / 2

7 / 2

9 / 2

1f ' x

2 1 x

3f " x

4 1 x

15f "' x

8 1 x

105f "" x

16 1 x

a. Find the third-degree Taylor approximation to f at x = 0

b. Use your answer in (a) to find an approximation of f(0.5)

c. Estimate the error involved in the approximation in (b). Show your reasoning.

The first four derivatives of 1f x are

1 x

3 / 2

5 / 2

7 / 2

9 / 2

1f ' x

2 1 x

3f " x

4 1 x

15f "' x

8 1 x

105f "" x

16 1 x

a. Find the third-degree Taylor approximation to f at x = 0

1 3 15f 0 1 f ' 0 f " 0 f "' 0

2 4 8

2 33

1 3 / 4 15 / 8p x 1 x x x

2 2! 3!

2 33

1 3 5p x 1 x x x

2 8 16

b. Use your answer in (a) to find an approximation of f(0.5)

2 3

3

1 3 5p 0.5 1 0.5 0.5 0.5

2 8 16

3

103p 0.5

128

The first four derivatives of 1f x are

1 x

3 / 2

5 / 2

/ 2

9

7

/ 2

1f ' x

2 1 x

3f " x

4 1 x

15f "' x

105f "" x

16 1

x

x

8 1

c. Estimate the error involved in the approximation in (b). Show your reasoning.

13

3R x1 !

M

30.5 0

13

3 3R x

1 !0

105 /16.5 0

4

3

105 350.5 0.017

384 2048R x