a. Explanation Given: The function f ( x ) = { e 1 x 2 if x ≠ 0 0 if x = 0 . Result used: The definition of derivative is f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h . Calculation: Obtain the derivative of f ( x ) . Consider the right side quadrant 0 + . By the Result stated above the right side limit computed as follows, f ′ ( x ) = lim x → 0 + e − 1 x 2 − 0 x = lim x → 0 + e − 1 x 2 x Substitute x = 1 y in above. f ′ ( x ) = lim y → 0 + e − 1 ( 1 y ) 2 ( 1 y ) = lim y → 0 + e − 1 ( 1 y ) ⋅ y = lim y → 0 + e − y ⋅ y = lim y → 0 + y e y Simplify further, f ′ ( x ) = 0 e 0 = 0 e 0 ( ∵ e − 0 = 1 ) = 0 1 = 0 Hence f ′ ( x ) = 0 for the right hand side limit b. To determine To write: The Taylor series for f centered at 0. c. To determine To explain: Why the Taylor series for f does not converge to f for x ≠ 0 .

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 9.3, Problem 90E

To determine

To show: The first derivative of the function f(x) is equal to 0.

To determine

To write: The Taylor series for f centered at 0.

To determine

To explain: Why the Taylor series for f does not converge to f for x≠0.

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Chapter 9.3, Problem 89E

Chapter 9.4, Problem 1E

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

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The first derivative of the function f ( x ) is equal to 0.

Solution Summary: The author explains that the first derivative of the function f(x) is equal to 0.

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To show: The first derivative of the function f(x) is equal to 0.

To write: The Taylor series for f centered at 0.

To explain: Why the Taylor series for f does not converge to f for x≠0.

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