   Chapter 11.R, Problem 57E

Chapter
Section
Textbook Problem

(a) Approximate f by a Taylor polynomial with degree n at the number a.(b) Graph f and T n on a common screen.(c) Use Taylor’s Inequality to estimate the accuracy of the approximation f ( x ) ≈ T ( x ) when x lies in the given interval.(d) Check your result in part (c) by graphing | R n ( x ) | . f ( x ) = 1 ,    a = 1 ,    n = 3 ,    0.9 ≤ x ≤ 1.1

To determine

(a)

To approximate:

f  by a Taylor polynomial with degree 3

Explanation

1) Concept:

The Taylor series of function  f at  a,

fx=n=0fnan!x-an

=fa+f'a1!x-a+f''a2!x-a2+f'''a3!x-a3+

2) Given:

fx=x,  a=1,  n=3,  0.9x1.1

3) Calculation:

It is given that,

fx=x,  a=1,  n=3,  0.9x1.1

f1=1=1

Differentiate fx with respect to x.

f'x=12x-12

f'1=121-12=12

Differentiate again.

f''x=-14x-32

f''1=-141-32=-14

f'''x<

To determine

(b)

To draw:

The graph of fx & T3x

To determine

(c)

To estimate:

The accuracy of fxT3x by using the Taylor’s inequality

To determine

(d)

To check:

R3x  by using the graph.

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