(a) Explanation Given information: The function y = cosh x , where cosh x = 1 2 ( e x + e − x ) Formula used: Tylor series 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 ) = y = cosh x = 1 2 ( e x + e − x ) ⇒ f ( x ) = 1 2 ( e x + e − x ) and a = 0 Substitute x = 0 in f ( x ) = 1 2 ( e x + e − x ) , it gives f ( 0 ) = 1 2 ( e 0 + e − ( 0 ) ) = 1 2 ( 1 + 1 ) = 1 2 ( 2 ) = 1 ⇒ f ( 0 ) = 1 . By differentiating f ( x ) with respect to x , f ' ( x ) = 1 2 ( e x − e − x ) Substitute x = 0 in f ' ( x ) = 1 2 ( e x − e − x ) , it gives f ' ( 0 ) = 1 2 ( e 0 − e − ( 0 ) ) = 1 2 ( 1 − 1 ) = 1 2 ( 0 ) = 0 ⇒ f ' ( 0 ) = 0 . By differentiating f ' ( x ) with respect to x , f ' ' ( x ) = 1 2 ( e x + e − x ) Substitute x = 0 in f ' ' ( x ) = 1 2 ( e x + e − x ) , it gives f ' ' ( 0 ) = 1 2 ( e 0 + e 0 ) = 1 2 ( 1 + 1 ) = 1 2 ( 2 ) = 1 ⇒ f ' ' ( 0 ) = 1 By differentiating f ' ' ( x ) with respect to x , f ' ' ' ( x ) = 1 2 ( e x − e − x ) Substitute x = 0 in f ' ' ' ( x ) = 1 2 ( e x − e − x ) , it gives f ' ' ' ( 0 ) = 1 2 ( e 0 − e 0 ) = 1 2 ( 1 − 1 ) = 1 2 ( 0 ) = 0 ⇒ f ' ' ' ( 0 ) = 0 By differentiating f ' ' ' ( x ) with respect to x , f ( 4 ) ( x ) = 1 2 ( e x + e − x ) Substitute x = 0 in f ( 4 ) ( x ) = 1 2 ( e x + e − x ) , it gives f ( 4 ) (b) To determine To calculate: The Tylor series for cosh x at x = 0 , by using known Tylor series for e x , Where cosh x = 1 2 ( e x + e − x ) .

Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137400096

Author: Larry Goldstein, David Lay

Publisher: PEARSON+

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Chapter 11.5, Problem 23E

(a)

To determine

To calculate: First four non zero terms in the Tylor series of y=coshx at x=0, by using differentiation and the definition of Tylor series, where coshx= 12(ex + e− x).

(b)

To determine

To calculate: The Tylor series for coshx at x=0, by using known Tylor series for ex, Where coshx= 12(ex + e− x).

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Chapter 11.5, Problem 22E

Chapter 11.5, Problem 24E

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First four non zero terms in the Tylor series of y = cosh x at x = 0 , by using differentiation and the definition of Tylor series, where cosh x = 1 2 ( e x + e − x ) .

Solution Summary: The author explains how to calculate the first four non-zero terms in the Tylor series of y=mathrmcoshx.

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To calculate: First four non zero terms in the Tylor series of y=coshx at x=0, by using differentiation and the definition of Tylor series, where coshx= 12(ex + e− x).

To calculate: The Tylor series for coshx at x=0, by using known Tylor series for ex, Where coshx= 12(ex + e− x).

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Chapter 11 Solutions