Explanation Formula used: Newton’s Method : If the n th approximation is x n and f ′ ( x n ) ≠ 0 , then the next approximation is given by x n + 1 = x n − f ( x n ) f ′ ( x n ) . Calculation: The given function is x 2 ( 4 − x 2 ) = 4 x 2 + 1 . Rewrite the function as, x 2 ( 4 − x 2 ) − 4 x 2 + 1 = 0 4 x 2 − x 4 ( x 2 + 1 ) − 4 x 2 + 1 = 0 4 x 4 + 4 x 2 − x 6 − x 4 − 4 = 0 − x 6 + 3 x 4 + 4 x 2 − 4 = 0 Hence, take f ( x ) = − x 6 + 3 x 4 + 4 x 2 − 4 . This implies, f ′ ( x ) = − 6 x 5 + 12 x 3 + 8 x . Then, by the Newton’s method, x n + 1 = x n − − x n 6 + 3 x n 4 + 4 x n 2 − 4 − 6 x n 5 + 12 x n 3 + 8 x n . Draw the graph of f ( x ) = − x 6 + 3 x 4 + 4 x 2 − 4 and identify the initial points. From Figure 1 observe that the first zero is, ( − 1.993 , 0 ) . So, lets take the initial point as, ˗2. Let x 1 = − 2 and find the roots. x 1 + 1 = x 1 − − x 1 6 + 3 x 1 4 + 4 x 1 2 − 4 − 6 x 1 5 + 12 x 1 3 + 8 x 1 x 2 = − 2 − − ( − 2 ) 6 + 3 ( − 2 ) 4 + 4 ( − 2 ) 2 − 4 − 6 ( − 2 ) 5 + 12 ( − 2 ) 3 + 8 ( − 2 ) = − 2 − ( − 4 80 ) = − 1.95 Similarly, find the further values for n = 2, 3, 4… and approximate up to 8 decimal places. n x 2 − 1.94391018 3 − 1.94382539 4 − 1.94382538 5 − 1.94382538 Note that at n = 4 and 5 the number agrees to eight decimal places. Therefore, the first root of f ( x ) = − x 6 + 3 x 4 + 4 x 2 − 4 is − 1.94382538 _ . From Figure 1 observe that the second zero is, ( − 0.843 , 0 ) . So, let’s take the initial point as, − 1 . Let x 1 = − 1 and find the roots. x 1 + 1 = x 1 − − x 1 6 + 3 x 1 4 + 4 x 1 2 − 4 − 6 x 1 5 + 12 x 1 3 + 8 x 1 x 2 = − 1 − − ( − 1 ) 6 + 3 ( − 1 ) 4 + 4 ( − 1 ) 2 − 4 − 6 ( − 12 ) 5 + 12 ( − 1 ) 3 + 8 ( − 1 ) = − 1 − ( 2 − 14 ) = − 0

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 4.6, Problem 18E

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The roots of x2(4−x2)=4x2+1 correct to eight decimal places.

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Chapter 4.6, Problem 17E

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Essential Calculus: Early Transcendentals

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The roots of x 2 ( 4 − x 2 ) = 4 x 2 + 1 correct to eight decimal places.

Solution Summary: The author explains that the roots of x2(4- x ) are correct to eight decimal places, and the next approximation is given by

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The roots of x2(4−x2)=4x2+1 correct to eight decimal places.

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