   Chapter 11.11, Problem 13E

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 ) = 1 / x ,     a = 1 ,     n = 2 ,     0.7 ≤ x ≤ 1.3

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=1x,   a=1,  n=2,  0.7x1.3

3) Calculation:

f0x=1/x and f01=11=1

f1x=-1/x2 and f11=-112=-1

f2x=2/x3 and f21=2

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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