Explanation Given information: The function f ( x ) = x 3 cos 7 x . Formula used: Taylor series of function f ( x ) at x = a : f ( x ) = f ( a ) + f ' ( a ) 1 ! ( x − a ) + f ' ' ( a ) 2 ! ( x − a ) 2 + f ' ' ' ( a ) 3 ! ( x − a ) 3 + ... + f n ( a ) n ! ( x − a ) n + ... Calculation: Here, f ( x ) = cos x and a = 0 Substitute x = 0 in f ( x ) = cos x , it gives f ( 0 ) = cos ( 0 ) = 1 ⇒ f ( 0 ) = 1 . By differentiating f ( x ) with respect to x , f ' ( x ) = − sin x Substitute x = 0 in f ' ( x ) = − sin x , it gives f ' ( 0 ) = − sin ( 0 ) = 0 ⇒ f ' ( 0 ) = 0 . By differentiating f ' ( x ) with respect to x , f ' ' ( x ) = − cos x Substitute x = 0 in f ' ' ( x ) = − cos x , it gives f ' ' ( 0 ) = − cos ( 0 ) = − 1 ⇒ f ' ' ( 0 ) = − 1 By differentiating f ' ' ( x ) with respect to x , f ' ' ' ( x ) = sin x Substitute x = 0 in f ' ' ' ( x ) = sin x , it gives f ' ' ' ( 0 ) = sin ( 0 ) = 0 ⇒ f ' ' ' ( 0 ) = 0 By differentiating f ' ' ' ( x ) with respect to x , f ( 4 ) ( x ) = cos x Substitute x = 0 in f ( 4 ) ( x ) = cos x , it gives f ( 4 ) ( 0 ) = cos ( 0 ) = 1 ⇒ f ( 4 ) ( 0 ) = 1 By differentiating f ( 4 ) ( x ) with respect to x , f ( 5 ) ( x ) = − sin x Substitute x = 0 in f ( 5 ) ( x ) = − sin x , it gives f ( 5 ) ( 0 ) = − sin ( 0 ) = 0 ⇒ f ( 5 ) ( 0 ) = 0 Now substitute the values of f ( x ) , f ( 0 ) , f ' ( 0 ) , f ' ' ( 0 ) , f ' ' ' ( 0 ) , f ( 4 ) ( 0 ) , f ( 5 ) ( 0 ) and a = 0 in f ( x ) = f ( a ) + f ' ( a ) 1 ! ( x − a ) + f ' ' ( a ) 2 ! ( x − a ) 2 + f ' ' ' ( a ) 3 ! ( x − a ) 3 +

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

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Chapter 11.5, Problem 4CYU

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To calculate: f(5)(0), where f(x)=x3cos7x.

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Chapter 11.5, Problem 3CYU

Chapter 11.5, Problem 1E

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Chapter 11 Solutions

Calculus & Its Applications

Ch. 11.1 - Determine the third Taylor polynomial of f(x)=cosx...Ch. 11.1 - Prob. 2CYU Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E 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 - Prob. 9E 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 - Prob. 15E Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Determine the third and fourthTaylor polynomial...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 - Graph the function Y1=11x and its fourth Taylor...Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.2 - Prob. 1CYU Ch. 11.2 - Prob. 2CYU Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 3E Ch. 11.2 - Prob. 4E Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 6E Ch. 11.2 - Prob. 7E Ch. 11.2 - Prob. 8E Ch. 11.2 - Sketch the graph of y=x3+2x+2, and use the...Ch. 11.2 - 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f ( 5 ) ( 0 ) , where f ( x ) = x 3 cos 7 x .

Solution Summary: The author calculates f(x)=x3mathrmcos7x with the following formula.

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To calculate: f(5)(0), where f(x)=x3cos7x.

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Chapter 11 Solutions