Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5.5, Problem 54E

To determine

To estimate the all real solutions of the equation using Newton’s method, accurate to 6 decimal places.

Expert Solution & Answer

Answer to Problem 54E

The solution of the equation x4+x−3=0 is about

x≈−1.152627x≈1.164035

Explanation of Solution

Given information:

The given equation is x4+x−3=0 .

Formula used:

Newton’s method

xn+1=xn−f(xn)f′(xn)

Calculation :

Let f(x)=x4+x−3 , then f′(x)=4x3+1 and

The graph of f

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.5, Problem 54E

The graph of f(x) suggests that x=−1.4 and x=1.2 are near solutions.

For x=−1.4

xn+1=xn−f(xn)f′(xn)x2=−1.4−f(−1.4)f′(−1.4)≈−1.455974x3=−1.455974−f(−1.455974)f′(−1.455974)≈−1.452639x4=−1.452639−f(−1.452639)f′(−1.452639)≈−1.452627x5=−1.452627−f(−1.452627)f′(−1.452627)≈−1.452627

For x=1.2

xn+1=xn−f(xn)f′(xn)x2=1.2−f(1.2)f′(1.2)≈1.165420x3=1.165420−f(1.165420)f′(1.165420)≈1.164037x4=1.164037−f(1.164037)f′(1.164037)≈1.164035x5=1.164035−f(1.164035)f′(1.164035)≈1.164035

The xn for x≥5 appears to be equal to x4 , hence conclude that the solution of the equation x4+x−3=0 is about

x≈−1.152627x≈1.164035

Chapter 5.5, Problem 53E

Chapter 5.5, Problem 55E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

Ch. 5.1 - Prob. 11QR Ch. 5.1 - Prob. 12QR Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - 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