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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

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BuyFindarrow_forward

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
Chapter 10.4, Problem 32E
Textbook Problem
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Finding Taylor and Maclaurin Series In Exercises 25–32, apply Taylor’s Theorem to find the power series for the function centered at c. Then find the radius of convergence. See Examples 4 and 5.

F u n c t i o n C e n t e r f ( x ) = x c = 4

To determine

To calculate: The power series for the function f(x)=x centered at 4 with the help of Taylor’s theorem and then estimate the radius of convergence.

Explanation of Solution

Given Information:

The provided function f(x)=x with center c=4.

Formula used:

The simple power rule of derivative ddx(xn)=nxn1.

The power series of a function f centered at c from Taylor’s theorem is

f(x)=n=0f(n)(c)(xc)nn!=f(c)+f'(c)(xc)1!+f''(c)(xc)22!+f'''(c)(xc)33!+

Ratio test:

For an infinite series n=1an with nonzero terms,

a) If limn|an+1an|<1 then the series converges.

b) If limn|an+1an|>1 or limn|an+1an|= then the series diverges.

c) If limn|an+1an|=1 then test is inconclusive.

Convergence of a Power Series:

If a power series is centered at a constant c, then either of the following condition is true.

a) The radius of convergence is R=0 if the series converges only at c.

b) The radius of convergence R exist and is a positive real number such that series converges for |xc|<R and diverges for |xc|>R.

c) The radius of convergence is R= if the series converges for all x.

The chain rule for derivative of composite function is,

ddxf(g(x))=f'(x)ddxg(x)=f'(x)g'(x)

Calculation:

Consider the function f(x)=x with center c=4.

Substitute 4 for x in the function f(x)=x.

f(4)=4=2

Therefore, the value of f(4) is 2.

Use the simple power rule to differentiate the provided function.

ddxf(x)=ddx(x)f(x)=ddx(x12)=(12)x121=12x12

Substitute 4 for x in the above first derivative function.

f(4)=12(4)12=12(1(4)12)=122=14

Therefore, the value of f(4) is 14.

Use the simple power rule to differentiate the first derivative f(x)=12x.

ddx(f(x))=ddx(12x)f(x)=12(12)x121=(14)x32=14x32

Substitute 4 for x in the above second derivative function.

f(4)=14432=132

Use the simple power rule to differentiate the second derivative f(x)=14x32.

ddx(f(x))=ddx(14x32)f(x)=(14)(32)x321=(14)(32)x52=38x52

Substitute 4 for x in the above third derivative function.

f(4)=38452=3256

Use the simple power rule to differentiate the third derivative f(x)=38x52.

ddx(f(x))=ddx(38x52)f(4)(x)=(38)(52)x521=(38)(52)x72=1516x72

Substitute 4 for x in the above fourth derivative function.

f(4)(4)=1516(472)=151627=152048

From the above derivative pattern, the nth derivative of the provided function in general for (x2) can be written as,

f(n)(x)=(1)n(135(2n3))2nx2n12

Substitute 4 for x in the above nth derivative function

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Chapter 10 Solutions

Calculus: An Applied Approach (MindTap Course List)
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Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Think About It Consider the geometric series...Ch. 10.4 - HOW DO YOU SEE IT? Match the graph of the first 10...Ch. 10.5 - Find the 12th-degree Taylor polynomial for...Ch. 10.5 - Use the fourth-degree Taylor polynomial from...Ch. 10.5 - Checkpoint 3 Worked-out solution available at...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - Using Taylor Polynomials In Exercises 1 and 2, use...Ch. 10.5 - Using Taylor Polynomials In Exercises 1 and 2, use...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 15 and 16,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 15 and 16,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 17 and 18,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 17 and 18,...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Finding the Maximum Error In Exercises 27 and 28,...Ch. 10.5 - Finding the Maximum Error In Exercises 27 and 28,...Ch. 10.5 - Using the Maximum Error In Exercises 29 and 30,...Ch. 10.5 - Using the Maximum Error In Exercises 29 and 30,...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Comparing Taylor Polynomials Consider the...Ch. 10.5 - HOW DO YOU SEE IT? The graphs show first-,...Ch. 10.6 - Calculate three iterations of Newton's Method to...Ch. 10.6 - Repeat Example 2 for f(x)=x3+2x+1. Use Newtons...Ch. 10.6 - Repeat Example 3 for y=ex2andy=x. 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For what value(s) will Newtons...Ch. 10.6 - True or False? In Exercises 29 and 30, determine...Ch. 10.6 - True or False? 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Multiplier Effect In Exercises 51 54, use the...Ch. 10 - Multiplier Effect In Exercises 51 54, use the...Ch. 10 - Multiplier Effect In Exercises 51 54, use the...Ch. 10 - Multiplier Effect In Exercises 51 54, use the...Ch. 10 - Depreciation A company buys a machine for $...Ch. 10 - Salary You accept a job that pays a salary of...Ch. 10 - Determining Convergence or Divergence In Exercises...Ch. 10 - Determining Convergence or Divergence In Exercises...Ch. 10 - Determining Convergence or Divergence In Exercises...Ch. 10 - Determining Convergence or Divergence In Exercises...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Using the Ratio Test In Exercises 61-66, use the...Ch. 10 - Matching In Exercises 67-70, match the series with...Ch. 10 - 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