Explanation Given information: The given function is f ( x ) = 2 x − x 2 + 1 . The given value of x 0 is 0. The given value of x 0 is 2. Calculation: Express the given function as follows: f ( x ) = 2 x − x 2 + 1 (1) Express the formula to calculate the approximation as follows: x n + 1 = x n − f ( x ) f ′ ( x ) Differentiate Equation (1) with respect to x and it is written as follows: f ′ ( x ) = 2 − 2 x Determine the value of x 1 if the value of x 0 is equal to 0 using the relation; x n + 1 = x n − f ( x ) f ′ ( x ) Substitute 2 x n − x n 2 + 1 for f ( x ) and 2 − 2 x n for f ′ ( x ) . x n + 1 = x n − 2 x n − x n 2 + 1 2 − 2 x n Substitute 0 for x n . x 0 + 1 = x 0 − 2 x 0 − x 0 2 + 1 2 − 2 x 0 x 1 = x 0 − 2 x 0 − x 0 2 + 1 2 − 2 x 0 Substitute 0 for x 0 . x 1 = 0 − 2 ( 0 ) + ( 0 ) 2 + 1 2 − 2 ( 0 ) = 0 − 1 2 = − 1 2 Determine the value of x 2 if the value of x 1 is equal to − 1 2 using the relation; x n + 1 = x n − f ( x ) f ′ ( x ) Substitute 1 for n . x 1 + 1 = x 1 − f ( x ) f ′ ( x ) x 2 = x 1 − f ( x ) f ′ ( x ) Substitute − 1 2 for x 1 , 2 x n − x n 2 + 1 for f ( x ) , and 2 − 2 x n for f ′ ( x )

3rd Edition

ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

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Chapter 4.7, Problem 4E

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Find the solution of the given Equation using Newton’s method.

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Chapter 4 Solutions

University Calculus

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