Explanation Given: The function is f ( x ) = 10 x 1 + x 2 . Definition used: The interpolating polynomial : “If f ( x ) is the function defined by the following table of n + 1 points, then the interpolating polynomial for f ( x ) is the polynomial p ( x ) of degree less than or equal to n that satisfies p ( x 0 ) = y 0 = f ( x 0 ) , p ( x 1 ) = y 1 = f ( x 1 ) ⋯ ⋯ p ( x n ) = y n = f ( x n ) ”. Newton’s form: “The Newton’s form for the interpolating polynomial is p ( x ) = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) ( x − x 1 ) + ⋯ + a n ( x − x 0 ) ⋯ ( x − x n − 1 ) ”. Procedure used: The procedure of steps for finding the interpolating polynomial is as follows. “Step 1: Write Newton’s form for p ( x ) . Step 2: Use the conditions p ( x i ) = y i , i = 0 , 1 , … n to write n + 1 linear equations for the coefficients. Step 3: Starting with the first and proceeding down the list, solve each equation for the coefficient with the largest subscript and substitute all the previously determined coefficients. Calculation: Find the Interpolating Polynomial. Step 1: The newton’s form for interpolating polynomial p ( x ) is obtained as follows. From the given Table, x 0 = − 2 , x 1 = − 1 , x 2 = 0 and x 3 = 1 . Substitute the above values in the Newton’s form as, p ( x ) = a 0 + a 1 ( x + 2 ) + a 2 ( x + 2 ) ( x + 1 ) + a 3 x ( x + 2 ) ( x + 1 ) + a 4 x ( x + 2 ) ( x + 1 ) ( x − 1 ) . Step 2: The linear equations for the coefficients as follows, Substitute x = − 2 in p ( x ) as, p ( − 2 ) = { a 0 + a 1 ( − 2 + 2 ) + a 2 ( − 2 + 2 ) ( − 2 + 1 ) + a 3 ( − 2 ) ( − 2 + 2 ) ( − 2 + 1 ) + a 4 ( − 2 ) ( − 2 + 2 ) ( − 2 + 1 ) ( − 2 − 1 ) } = a 0 + 0 = a 0. That is, p ( − 2 ) = a 0 = f ( − 2 ) Substitute x = − 1 in p ( x ) , p ( − 1 ) = { a 0 + a 1 ( − 1 + 2 ) + a 2 ( − 1 + 2 ) ( − 1 + 1 ) + a 3 ( − 1 ) ( − 1 + 2 ) ( − 1 + 1 ) + a 4 ( − 1 ) ( − 1 + 2 ) ( − 1 + 1 ) ( − 1 − 1 ) } = a 0 + a 1 + 0 = a 0 + a 1 That is, p ( − 1 ) = a 0 + a 1 = f ( − 1 ) . Substitute x = 0 in p ( x ) , p ( 0 ) = { a 0 + a 1 ( 0 + 2 ) + a 2 ( 0 + 2 ) ( 0 + 1 ) + a 3 ( 0 ) ( 0 + 2 ) ( 0 + 1 ) + a 4 ( 0 ) ( 0 + 2 ) ( 0 + 1 ) ( 0 − 1 ) } = a 0 + 2 a 1 + 2 a 2 + 0 = a 0 + 2 a 1 + 2 a 2 That is, p ( 0 ) = a 0 + 2 a 1 + 2 a 2 = f ( 0 ) . Substitute x = 1 in p ( x ) , p ( 1 ) = { a 0 + a 1 ( 1 + 2 ) + a 2 ( 1 + 2 ) ( 1 + 1 ) + a 3 ( 1 ) ( 1 + 2 ) ( 1 + 1 ) + a 4 ( 1 ) ( 1 + 2 ) ( 1 + 1 ) ( 1 − 1 ) } = a 0 + 3 a 1 + ( 3 ) ( 2 ) a 2 + ( 3 ) ( 2 ) a 3 + 0 = a 0 + 3 a 1 + 6 a 2 + 6 a 3 That is, p ( 1 ) = a 0 + 3 a 1 + 6 a 2 + 6 a 3 = f ( 1 )

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter B.4, Problem 37E

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Chapter B.4, Problem 36E

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Chapter B.4 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. B.4 - Consider the points in Table 1. (A)Let...Ch. B.4 - Given the following polynomial and table,...Ch. B.4 - Prob. 1MP Ch. B.4 - Prob. 2MP Ch. B.4 - Prob. 3MP Ch. B.4 - Prob. 4MP Ch. B.4 - Prob. 1E Ch. B.4 - Prob. 2E Ch. B.4 - Prob. 3E Ch. B.4 - Prob. 4E

Ch. B.4 - Prob. 5E Ch. B.4 - Prob. 6E Ch. B.4 - Prob. 7E Ch. B.4 - Prob. 8E Ch. B.4 - Prob. 9E Ch. B.4 - Prob. 10E Ch. B.4 - Prob. 11E Ch. B.4 - Prob. 12E Ch. B.4 - Prob. 13E Ch. B.4 - In Problems 1320, use the interpolating polynomial...Ch. B.4 - Prob. 15E Ch. B.4 - Prob. 16E Ch. B.4 - Prob. 17E Ch. B.4 - Prob. 18E Ch. B.4 - Prob. 19E Ch. B.4 - Prob. 20E Ch. B.4 - Prob. 21E Ch. B.4 - Prob. 22E Ch. B.4 - In Problems 2130, find the interpolating...Ch. B.4 - Prob. 24E Ch. B.4 - Prob. 25E Ch. B.4 - Prob. 26E Ch. B.4 - Prob. 27E Ch. B.4 - Prob. 28E Ch. B.4 - Prob. 29E Ch. B.4 - Prob. 30E Ch. B.4 - Prob. 31E Ch. B.4 - Prob. 32E Ch. B.4 - Prob. 33E Ch. B.4 - Prob. 34E Ch. B.4 - The following table was obtained from the function...Ch. B.4 - Prob. 36E Ch. B.4 - Prob. 37E Ch. B.4 - The following table was obtained from the function...Ch. B.4 - Prob. 39E Ch. B.4 - Prob. 40E Ch. B.4 - Prob. 41E Ch. B.4 - Prob. 42E Ch. B.4 - Prob. 43E Ch. B.4 - Prob. 44E Ch. B.4 - Maximum revenue. The revenue R (in thousands of...Ch. B.4 - Minimum overage cost. The cost C (in thousands of...Ch. B.4 - Prob. 47E Ch. B.4 - Prob. 48E Ch. B.4 - Prob. 49E Ch. B.4 - Prob. 50E Ch. B.4 - Voter registration. The number N of registered...Ch. B.4 - Voter registration. The number N of registered...

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The interpolation polynomial for the given table; also sketch the graph of p ( x ) and f ( x ) .

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