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CalculusCalculus10th Edition

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Chapter 9.7, Problem 43E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Problem 59E
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Problem 66E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Approximating a Function Value In Exercises 39-44, approximate the function at the given value of x , using the polynomial found in the indicated exercise.

f ( x ) = ln x ,       f ( 2.1 ) , , Exercise 31

To determine

To calculate: The approximation of the function f(x)=lnx at the value of x=2.1 using the 4th Taylor polynomial as obtained in Exercise 31.

Explanation

Given:

The function is f(x)=lnx and the value of x at which the function is to be approximated is 2.1.

Formula used:

The standard Taylor series is:

P4(x)=f(c)+(x−c)1!f′(c)+(x−c)22!f″(c)+(x−c)33!f‴(c)+(x−c)44!f4(c)+........

Calculation:

Consider the following function:

f(x)=lnx

Centered at c=2, substitute x=2,

f(2)=ln2

First derivative of the function is,

f′(x)=1x

Centered at c=2, substitute x=2,

f(2)=12

Second derivative of the function is,

f″(x)=−1x2

Centered at c=2, substitute x=2,

f″(2)=−14

Third derivative of the function is,

f‴(x)=2x3

Centered at c=2, substitute x=2,

f″(2)=14

Fourth derivative of the function is,

f4(x)=−6x4

Centered at c=2, substitute x=2,

f″(−2)=−38

Substitute these derivative values into Taylor series as follows:

P4(x)=f(c)+(x−c)1!f′(c)+(x−c)22!f″(c)+(x−c)33!f‴(c)+(x−c)44!f4(c)=ln2+12(x−2)−12!(x−2)2

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Sect-9.1 P-1ESect-9.1 P-2ESect-9.1 P-3ESect-9.1 P-4ESect-9.1 P-5ESect-9.1 P-6ESect-9.1 P-7ESect-9.1 P-8ESect-9.1 P-9ESect-9.1 P-10E
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P-36ESect-9.9 P-37ESect-9.9 P-38ESect-9.9 P-39ESect-9.9 P-40ESect-9.9 P-41ESect-9.9 P-42ESect-9.9 P-43ESect-9.9 P-44ESect-9.9 P-45ESect-9.9 P-46ESect-9.9 P-47ESect-9.9 P-48ESect-9.9 P-49ESect-9.9 P-50ESect-9.9 P-51ESect-9.9 P-52ESect-9.9 P-53ESect-9.9 P-54ESect-9.9 P-55ESect-9.9 P-56ESect-9.9 P-57ESect-9.9 P-58ESect-9.9 P-59ESect-9.9 P-60ESect-9.9 P-61ESect-9.9 P-62ESect-9.10 P-81ESect-9.10 P-1ESect-9.10 P-2ESect-9.10 P-3ESect-9.10 P-4ESect-9.10 P-5ESect-9.10 P-6ESect-9.10 P-7ESect-9.10 P-8ESect-9.10 P-9ESect-9.10 P-10ESect-9.10 P-11ESect-9.10 P-12ESect-9.10 P-13ESect-9.10 P-14ESect-9.10 P-15ESect-9.10 P-16ESect-9.10 P-19ESect-9.10 P-17ESect-9.10 P-18ESect-9.10 P-20ESect-9.10 P-21ESect-9.10 P-22ESect-9.10 P-23ESect-9.10 P-24ESect-9.10 P-25ESect-9.10 P-26ESect-9.10 P-27ESect-9.10 P-28ESect-9.10 P-29ESect-9.10 P-30ESect-9.10 P-31ESect-9.10 P-32ESect-9.10 P-33ESect-9.10 P-34ESect-9.10 P-35ESect-9.10 P-36ESect-9.10 P-37ESect-9.10 P-38ESect-9.10 P-39ESect-9.10 P-40ESect-9.10 P-45ESect-9.10 P-46ESect-9.10 P-41ESect-9.10 P-42ESect-9.10 P-43ESect-9.10 P-44ESect-9.10 P-47ESect-9.10 P-48ESect-9.10 P-49ESect-9.10 P-50ESect-9.10 P-51ESect-9.10 P-52ESect-9.10 P-53ESect-9.10 P-54ESect-9.10 P-55ESect-9.10 P-56ESect-9.10 P-57ESect-9.10 P-58ESect-9.10 P-59ESect-9.10 P-60ESect-9.10 P-61ESect-9.10 P-62ESect-9.10 P-63ESect-9.10 P-64ESect-9.10 P-65ESect-9.10 P-66ESect-9.10 P-67ESect-9.10 P-68ESect-9.10 P-69ESect-9.10 P-70ESect-9.10 P-71ESect-9.10 P-72ESect-9.10 P-73ESect-9.10 P-74ESect-9.10 P-75ESect-9.10 P-76ESect-9.10 P-77ESect-9.10 P-78ESect-9.10 P-79ESect-9.10 P-80ESect-9.10 P-82ESect-9.10 P-83ESect-9.10 P-84ESect-9.10 P-85ESect-9.10 P-86ESect-9.10 P-87ESect-9.10 P-88ESect-9.10 P-89ESect-9.10 P-90ESect-9.10 P-91ESect-9.10 P-92ESect-9.10 P-93ESect-9.10 P-94ESect-9.10 P-95ESect-9.10 P-96ECh-9 P-1RECh-9 P-2RECh-9 P-3RECh-9 P-4RECh-9 P-5RECh-9 P-6RECh-9 P-7RECh-9 P-8RECh-9 P-9RECh-9 P-10RECh-9 P-14RECh-9 P-11RECh-9 P-15RECh-9 P-13RECh-9 P-12RECh-9 P-16RECh-9 P-17RECh-9 P-18RECh-9 P-19RECh-9 P-20RECh-9 P-21RECh-9 P-22RECh-9 P-23RECh-9 P-24RECh-9 P-25RECh-9 P-26RECh-9 P-27RECh-9 P-28RECh-9 P-29RECh-9 P-30RECh-9 P-31RECh-9 P-32RECh-9 P-33RECh-9 P-34RECh-9 P-35RECh-9 P-36RECh-9 P-37RECh-9 P-38RECh-9 P-40RECh-9 P-39RECh-9 P-41RECh-9 P-42RECh-9 P-43RECh-9 P-44RECh-9 P-45RECh-9 P-46RECh-9 P-47RECh-9 P-48RECh-9 P-49RECh-9 P-50RECh-9 P-51RECh-9 P-52RECh-9 P-53RECh-9 P-54RECh-9 P-55RECh-9 P-56RECh-9 P-57RECh-9 P-58RECh-9 P-59RECh-9 P-60RECh-9 P-61RECh-9 P-62RECh-9 P-65RECh-9 P-63RECh-9 P-64RECh-9 P-66RECh-9 P-67RECh-9 P-68RECh-9 P-69RECh-9 P-70RECh-9 P-72RECh-9 P-71RECh-9 P-73RECh-9 P-74RECh-9 P-75RECh-9 P-76RECh-9 P-77RECh-9 P-78RECh-9 P-79RECh-9 P-80RECh-9 P-81RECh-9 P-82RECh-9 P-83RECh-9 P-84RECh-9 P-85RECh-9 P-86RECh-9 P-87RECh-9 P-88RECh-9 P-89RECh-9 P-90RECh-9 P-91RECh-9 P-92RECh-9 P-93RECh-9 P-94RECh-9 P-95RECh-9 P-96RECh-9 P-97RECh-9 P-98RECh-9 P-99RECh-9 P-100RECh-9 P-101RECh-9 P-102RECh-9 P-103RECh-9 P-104RECh-9 P-105RECh-9 P-106RECh-9 P-107RECh-9 P-108RECh-9 P-109RECh-9 P-110RECh-9 P-1PSCh-9 P-2PSCh-9 P-3PSCh-9 P-4PSCh-9 P-5PSCh-9 P-6PSCh-9 P-7PSCh-9 P-8PSCh-9 P-9PSCh-9 P-10PSCh-9 P-11PSCh-9 P-12PSCh-9 P-13PSCh-9 P-14PSCh-9 P-15PSCh-9 P-16PS

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