Explanation Given information: The antiderivative: ∫ e − x 2 d x Explanation: First to find the Taylor series expansion of e x at x = 0 by using the formula, f ( x ) = f ( 0 ) + f ' ( 0 ) 1 ! x + f ' ' ( 0 ) 2 ! x 2 + f ' ' ' ( 0 ) 3 ! x 3 + ... + f n ( 0 ) n ! x n + ... Let f ( x ) = e x Substitute x = 0 in f ( x ) = e x gives f ( 0 ) = e 0 = 1 Differentiate f ( x ) = e x with respect to x , f ' ( x ) = e x Substitute x = 0 in f ' ( x ) = e x gives f ' ( 0 ) = e 0 = 1 Differentiate f ' ( x ) = e x with respect to x , f ' ' ( x ) = e x Substitute x = 0 in f ' ' ( x ) = e x gives f ' ' ( 0 ) = e 0 = 1 Differentiate f ' ' ( x ) = e x with respect to x , f ' ' ' ( x ) = e x Substitute x = 0 in f ' ' ' ( x ) = e x gives f ' ' ' ( 0 ) = e 0 = 1 , Plug in the values of f ( 0 ) = 1 , f ' ( 0 ) = 1 , f ' ' ( 0 ) = 1 , f ' ' ' ( 0 ) = 1 , . . . in the Taylor series formula, ⇒ f ( x ) = 1 + 1 1 ! x + 1 2 ! x 2 + 1 3 ! x 3 + . . . ⇒ f ( x ) = 1 + x + 1 2 ! x 2 + 1 3 ! x 3 + . . . Thus, the Tylor series expansion of e x at x = 0 is 1 + x + 1 2 ! x 2 + 1 3 ! x 3 + . . . . ⇒ e x = 1 + x + 1 2 ! x 2 + 1 3 ! x 3 + . . . Replace x by − x 2 in above, ⇒ e − x 2 = 1 + ( − x 2 ) + 1 2 ! ( − x 2 ) 2 + 1 3 ! ( − x 2 ) 3 +

12th Edition

ISBN: 9780137590810

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

Publisher: PEARSON

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Chapter 11.5, Problem 35E

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The Taylor series expansion at x = 0 of the antiderivative ∫e− x2dx.

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Chapter 11.5, Problem 34E

Chapter 11.5, Problem 36E

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Calculus & Its Applications

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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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The Taylor series expansion at x = 0 of the antiderivative ∫ e − x 2 d x .

Solution Summary: The author explains how to determine the Taylor series expansion at x0 of the antiderivative.

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The Taylor series expansion at x = 0 of the antiderivative ∫e− x2dx.

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Chapter 11 Solutions