In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. Σ(-1)"x" 87. f (x) 88. f (x) = = 1+x = ∞ n=0 ∞ Σ n=0 x²n ∞ In the following exercises, given that=x", use term-by-term differentiation or integration to find power n=0 series for each function centered at the given point. 95. f (x) = In x centered at x = 1 (Hint: x = 1 -(1-x)) 96. In (1-x) at x = 0 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered a the given point. 116. f (x) = 1 + x + x² at a = 1 117. f (x) = 1 + x + x² at a = -1
In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. Σ(-1)"x" 87. f (x) 88. f (x) = = 1+x = ∞ n=0 ∞ Σ n=0 x²n ∞ In the following exercises, given that=x", use term-by-term differentiation or integration to find power n=0 series for each function centered at the given point. 95. f (x) = In x centered at x = 1 (Hint: x = 1 -(1-x)) 96. In (1-x) at x = 0 In the following exercises, find the Taylor polynomials of degree two approximating the given function centered a the given point. 116. f (x) = 1 + x + x² at a = 1 117. f (x) = 1 + x + x² at a = -1
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Subject: calculus
Transcribed Image Text:In the following exercises, differentiate the given series expansion of f term-by-term to obtain the corresponding
series expansion for the derivative of f.
87. f (x) = x = Σ (-1)"x"
1+x
n=0
∞
- - Σ
=
n=0
88. ƒ (x) = 1⁄²
2n
Σx²n
In the following exercises, given that=x", use term-by-term differentiation or integration to find power
n=0
series for each function centered at the given point.
95. f (x) = ln x centered at x = 1 (Hint: x = 1 − (1 − x))
96. In (1-x) at x = 0
In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at
the given point.
116. f (x) = 1 + x + x² at a = 1
117. f (x) = 1 + x + x² at a = -1
118. f (x) = cos (2x) at a = π
119. f (x) = sin (2x) at a = /2
120. f (x)=√x at a = 4
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