Use newton's method with the specified initial approximation x_1 to find x_3, the third appromizmation to the root of the given equation x^5−x−1=0 , x_1=1

Bellenik3

Bellenik3

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2022-08-19

Use newton's method with the specified initial approximation x 1 to find x 3, the third appromizmation to the root of the given equation
x 5 x 1 = 0, x 1 = 1

Answer & Explanation

blerbintiy0

blerbintiy0

Beginner2022-08-20Added 8 answers

Choose f ( x ) = x 5 x 1, which makes f ( x ) = 5 x 4 1. The Newton's iteration take the form
x n + 1 = x n f ( x n ) f ( x n ) = x n x n 5 x n 1 5 x n 4 1
Starting with x 1 = 1 we get x 2 = 1.25, x 3 = 1.1785, x 4 = 1.1675, x 5 = 1.16730 which is the approximate root. As for your other question, the values x 1 = 0.5 or 0.25 will not converge to right answer. Notice that f ( 1 ) = 1 and f ( 2 ) = 29 So the root lies in ( 1 , 2 ). If anything you should increase the values of x 1

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