   Chapter 11.11, Problem 15E

Chapter
Section
Textbook Problem

(a) Approximate f by a Taylor polynomial with degree n at the number a.(b) Use Taylor’s Inequality to estimate the accuracy of the approximation f ( x ) ≈ T n ( x ) when x lies in the given interval.(c) Check your result in part (b) by graphing | R n ( x ) | . f ( x ) = x 2 / 3 ,     a = 1 ,     n = 3 ,     0.8 ≤ x ≤ 1.2

To determine

(a)

To approximate:

The function f by a Taylor polynomial with degree n at the number a

Explanation

1) Concept:

The Taylor polynomial centred at a is

Tnx=i=0nfiai!x-ai

=f0a0!x-a0+fa1!x-a+f2a2!x-a2+f3a3!x-a3+

2) Given:

fx=x2/3,   a=1,  n=3,  0.8x1.2

3) Calculation:

f0x=x2/3 and f01=1

f1x=23x-1/3 and f11=23

f2x=-29 x-4/3 and f21=-29

f3x=827x-7/3 and f31=827

f4x=-5681

To determine

(b)

To estimate:

The accuracy of the approximation fxTn(x) by using Taylor’s inequality.

To determine

(c)

To sketch:

The graph of Rnx to check it with part (b)

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