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Study Guide for Stewart's Multivar...

8th Edition
Stewart + 1 other
Publisher: Brooks Cole
ISBN: 9781305271845

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Section
BuyFindarrow_forward

Study Guide for Stewart's Multivar...

8th Edition
Stewart + 1 other
Publisher: Brooks Cole
ISBN: 9781305271845
Chapter 11.11, Problem 3PT
Textbook Problem
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What degree Taylor polynomial about a = 1 is needed to approximate e1.05 accurate to within 0.0001?

  1. a) n = 2
  2. b) n = 3
  3. c) n = 4
  4. d) n = 5

To determine

For what value of n, |Rn(1.05)|<0.0001 if f(x)=ex about a=1.

Explanation of Solution

Result used:

Taylor’s inequality: If |f(n+1)(x)|M for |xa|d, then the nth remainder Rn(x) of the Taylor series satisfies the inequality |Rn(x)|M(n+1)!|xa|n+1for |xa|d.

Calculation:

For what value of n, |Rn(1.05)|<0.0001 if f(x)=ex about a=1.

That is, find the degree of the Taylor’s polynomial for f(x)=ex about a=1 is needed to approximate the value of e1.05.

The nth term of the Taylor’s polynomial for f(x)=ex about a = 1 is, fn+1(x)=e.

Therefore, M=e.

By Taylor’s inequality, |Rn(1.05)|e(n+1)!|1.051|n+1.

Since e2.72, |Rn(1.05)|<2.72(n+1)!|0.05|n+1        (1)

Given that |Rn(1.05)|<0.0001        (2)

From inequalities (1) and (2),

2.72(n+1)!|0.05|n+10.00012

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Chapter 11 Solutions

Study Guide for Stewart's Multivariable Calculus, 8th
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