Newton’s Method In calculus you will learn that if P ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0 is a polynomial function, then the derivative of is P ' ( x ) = n a n x n − 1 + ( n − 1 ) a n − 1 x n − 2 + ... + 2 a 2 x + a 1 Newton’s Method is an efficient method for approximating the x -intercepts (or real zeros) of a function, such as p ( x ) . The following steps outline Newton’s Method. STEP 1: Select an initial value x 0 that is somewhat close to the x -intercept being sought. STEP 2: Find values for x using the relation x n + 1 = x n − P ( x n ) P ' ( x n ) n = 1, 2, ... until you get two consecutive values x n and x n + 1 that agree to whatever decimal place accuracy you desire. STEP 3: The approximate zero will be x n + 1 . Consider the polynomial P ( x ) = x 3 − 7 x − 40 . a. Evaluate p ( 5 ) and p ( − 3 ) . b. What might we conclude about a zero of p ? Explain. c. Use Newton’s Method to approximate an x -intercept , r , − 3 < r < 5 , of p ( x ) to four decimal places. d. Use a graphing utility to graph p ( x ) and verify your answer in part . e. Using a graphing utility, evaluate p ( r ) to verify your result.
Newton’s Method In calculus you will learn that if P ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0 is a polynomial function, then the derivative of is P ' ( x ) = n a n x n − 1 + ( n − 1 ) a n − 1 x n − 2 + ... + 2 a 2 x + a 1 Newton’s Method is an efficient method for approximating the x -intercepts (or real zeros) of a function, such as p ( x ) . The following steps outline Newton’s Method. STEP 1: Select an initial value x 0 that is somewhat close to the x -intercept being sought. STEP 2: Find values for x using the relation x n + 1 = x n − P ( x n ) P ' ( x n ) n = 1, 2, ... until you get two consecutive values x n and x n + 1 that agree to whatever decimal place accuracy you desire. STEP 3: The approximate zero will be x n + 1 . Consider the polynomial P ( x ) = x 3 − 7 x − 40 . a. Evaluate p ( 5 ) and p ( − 3 ) . b. What might we conclude about a zero of p ? Explain. c. Use Newton’s Method to approximate an x -intercept , r , − 3 < r < 5 , of p ( x ) to four decimal places. d. Use a graphing utility to graph p ( x ) and verify your answer in part . e. Using a graphing utility, evaluate p ( r ) to verify your result.
Solution Summary: The author explains how Newton's method approximates an x-intercept, p = 3 r to four decimal places.
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