Explanation Given Data: Write the given function as follows. f ″ ( x ) = − 2 + 12 x − 12 x 2 , f ( 0 ) = 4 , f ′ ( 0 ) = 12 Formula used: The antiderivative function for the function x n is x n + 1 n + 1 + C . Here, C is the constant. Calculation of f ′ ( x ) : Rewrite the function f ″ ( x ) = − 2 + 12 x − 12 x 2 as follows. f ″ ( x ) = − 2 x 0 + 12 x 1 − 12 x 2 (1) From the antiderivative function formula, the antiderivative for the function in equation (1) is written as follows. f ′ ( x ) = − 2 ( x 0 + 1 0 + 1 ) + 12 ( x 1 + 1 1 + 1 ) − 12 ( x 2 + 1 2 + 1 ) + C = − 2 x + 12 ( x 2 2 ) − 12 ( x 3 3 ) + C f ′ ( x ) = − 2 x + 6 x 2 − 4 x 3 + C (2) As f ′ ( 0 ) = 12 , substitute 0 for x in equation (2), f ′ ( 0 ) = − 2 ( 0 ) + 6 ( 0 ) 2 − 4 ( 0 ) 3 + C = 0 + 0 − 0 + C = C Rewrite the expression as follows. C = f ′ ( 0 ) Substitute 12 for f ′ ( 0 ) , C = 12 Substitute 12 for C in equation (2), f ′ ( x ) = − 2 x + 6 x 2 − 4 x 3 + 12 Calculation of f ( x ) : Rewrite the function f ′ ( x ) = − 2 x + 6 x 2 − 4 x 3 + 12 as follows

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 4.7, Problem 29E

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To find: The general antiderivative for the function f″(x)=−2+12x−12x2, f(0)=4, f′(0)=12 .

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Chapter 4.7, Problem 28E

Chapter 4.7, Problem 30E

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The general antiderivative for the function f ″ ( x ) = − 2 + 12 x − 12 x 2 , f ( 0 ) = 4 , f ′ ( 0 ) = 12 .

Solution Summary: The author explains the general antiderivative for the function xn.

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To find: The general antiderivative for the function f″(x)=−2+12x−12x2, f(0)=4, f′(0)=12 .

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Chapter 4 Solutions