Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 5.5, Problem 55E

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 55E

The solution of the equation x2−2x+1=sinx is about

x≈0.386237x≈1.961560

Explanation of Solution

Given information:

The given equation is x2−2x+1=sinx .

Formula used:

Newton’s method

xn+1=xn−f(xn)f′(xn)

Calculation :

Let f(x)=x2−2x+1−sinx , then f′(x)=2x−2−cosx and

The graph of f

Calculus: Graphical, Numerical, Algebraic, Chapter 5.5, Problem 55E

The graph of f(x) suggests that x=0.5 and x=2 are near solutions.

For x=0.5

xn+1=xn−f(xn)f′(xn)x1=0.5x2=0.5−f(0.5)f′(0.5)≈0.37780802x3=0.37780802−f(0.37780802)f′(0.37780802)≈0.38619812x4=0.38619812−f(0.38619812)f′(0.38619812)≈0.38623687x5=0.38623687−f(0.38623687)f′(0.38623687)≈0.38623687

For x=2

xn+1=xn−f(xn)f′(xn)x1=2x2=2−f(2)f′(2)≈1.96245983x3=1.96245983−f(1.96245983)f′(1.96245983)≈1.96156954x4=1.96156954−f(1.96156954)f′(1.96156954)≈1.96156903x≈1.961569

The xn for x≥5 appears to be equal to x4 , hence conclude that the solution of the equation x2−2x+1=sinx is about

x≈0.386237x≈1.961560

Chapter 5.5, Problem 54E

Chapter 5.5, Problem 56E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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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