a. Explanation Given information: The function f is given by; f ( x ) = { x 2 sin ( 1 x ) f o r x ≠ 0 0 f o r x = 0 Formula used: The definition of the derivative; f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x . Proof: Using the definition of derivative, f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x To find the function f differentiable at x = 0 , Consider, f ′ ( x ) = lim x → 0 + f ( x ) − f ( 0 ) ( x − 0 ) = lim Δ x → 0 f ( 0 + Δ x ) − 0 0 + Δ x = lim Δ x → 0 ( 0 + Δ x ) 2 sin ( 1 0 + Δ x ) − 0 0 + Δ x = lim Δ x → 0 Δ x b. To determine Calculate f ′ ( x ) for x ≠ 0 . c. To determine Prove that f ′ is not continuous at x = 0 .

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3, Problem 74SP

To determine

Show f is differentiable.

To determine

Calculate f′(x) for x≠0 .

To determine

Prove that f′ is not continuous at x=0 .

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Chapter 3, Problem 73SP

Chapter 3, Problem 75SP

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Chapter 3 Solutions

Calculus

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Show f is differentiable.

Solution Summary: The author explains how the function f is differentiable.

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Show f is differentiable.

Calculate f′(x) for x≠0 .

Prove that f′ is not continuous at x=0 .

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Chapter 3 Solutions