   Chapter 11.11, Problem 16E

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 ) = sin x ,     a = π / 6 ,     n = 4 ,     0 ≤ x ≤ π / 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=sinx,   a=π6,  n=4,  0xπ3

3) Calculation:

f0x=sinx and f0π6=12

f1x=cosx and f1π6=32

f2x=-sinx and f2π6=-12

f3x=-cosx and f3π6=-32

f4x=sinx and f4π6=12

f5x=cosx

Now for n=4

The Taylor polynomial for the function fx=sinxT4(x)

T4x=i=04fiπ6

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)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 