Explanation Method 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 the number is less than a prescribed positive number, then stop the iteration. The required approximation to the root x = c is x = x n + 1 . Calculation: The pair of functions is given by f ( x ) = e − x 2 and g ( x ) = x 2 . Use online graphing calculator, graph the pair of functions f ( x ) = e − x 2 and g ( x ) = x 2 as shown below in Figure 1. From Figure 1, it is observed that the points of intersection of these graphs are ( 0.753 , 0.567 ) and ( − 0.753 , 0.567 ) To refine the approximation of the x -coordinate of the point of intersection, equate the two functions f ( x ) = e − x 2 and g ( x ) = x 2 . e − x 2 = x 2 e − x 2 − x 2 = 0 h ( x ) = e − x 2 − x 2 Find the derivative of h ( x ) = e − x 2 − x 2 as follows. h ′ ( x ) = − 2 e − x 2 x − 2 x The required iterative formula is, x n + 1 = x n − e − x n 2 − x n 2 − 2 e − x n 2 x n − 2 x n = − 2 e − x n 2 x n 2 − 2 x n 2 − e − x n 2 + x n 2 − 2 e − x n 2 x n − 2 x n = − 2 e − x n 2 x n 2 − x n 2 − e − x n 2 − 2 e − x n 2 x n − 2 x n By using the above mentioned Newton’s method, find x 1 at x 0 = 1 . x 1 = − 2 e − ( 1 ) 2 ( 1 ) 2 − ( 1 ) 2 − e − ( 1 ) 2 − 2 e − ( 1 ) 2 ( 1 ) − 2 ( 1 ) = 3 + e 2 + 2 e = 0.7689 Find x 2 by substituting x 1 = 0.7689 as follows. x 2 = − 2 e − ( 0

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 11.7, Problem 24E

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To make: A rough sketch of the pair of the graphs of f(x)=e−x2 and g(x)=x2 to approximate the points of intersection of the two graphs, and then apply Newton’s method to refine the approximation of the x-coordinate of the point of intersection.

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Chapter 11.7, Problem 23E

Chapter 11.7, Problem 25E

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Chapter 11 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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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To make: A rough sketch of the pair of the graphs of f ( x ) = e − x 2 and g ( x ) = x 2 to approximate the points of intersection of the two graphs, and then apply Newton’s method to refine the approximation of the x -coordinate of the point of intersection.

Solution Summary: The author illustrates Newton's method of calculating the x-coordinate of the points of intersection of two graphs using an online graphing calculator.

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To make: A rough sketch of the pair of the graphs of f(x)=e−x2 and g(x)=x2 to approximate the points of intersection of the two graphs, and then apply Newton’s method to refine the approximation of the x-coordinate of the point of intersection.

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Chapter 11 Solutions