Explanation Formula used: “ Newton’s Method: 1. Pick an initial estimate x 0 of the root c . 2. Find a new estimate using the iterative formula, x n + 1 = x n − f ( x n ) f ′ ( x n ) ( n = 0 , 1 , 2 , … ) 3. Compute | x n − x n + 1 | . If this number is less than a prescribed positive number, stop. The required approximation to the root x = c is x = x n + 1 . Obtain the expression for r as follows.” Calculation: Let a be a real number. If x is the nth root of a , then, x n = a x n − a = 0 Let the function, f ( x ) = x n − a , then f ( x i ) = x i n − a . Also, if the derivative of the function is, f ′ ( x ) = n x n − 1 , then f ′ ( x i ) = n x i n − 1 . Apply Newton method to the function f ( x ) as follows b. To determine The value of 42 4 using the result obtained in the previous part.

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Author: Tan

Publisher: INTER CENG

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Question

Chapter 11.7, Problem 41E

To determine

To show: The iterative formula for finding the nth root of a real number a is, xi+1=(n−1)nxi+anxin−1.

To determine

The value of 424 using the result obtained in the previous part.

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Chapter 11.7, Problem 40E

Chapter 11.7, Problem 1TE

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Chapter 11 Solutions

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Ch. 11.1 - Prob. 1CQ Ch. 11.1 - Prob. 2CQ Ch. 11.1 - Prob. 1E Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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The iterative formula for finding the nth root of a real number a is, x i + 1 = ( n − 1 ) n x i + a n x i n − 1 .

Solution Summary: The author explains Newton's method for finding the nth root of a real number.