Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Review - Concept Check: 15

Answer

(a) The derivative exists at $x=a$. (b) The differentiability implies continuity but not vice versa. (c) The graph of $f(x)=|x-2|$ is on the figure below.
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Work Step by Step

(a) This is just another way to say that the derivative, i.e. the limit that defines the derivative exists at $x=a$. (b) All of the functions that are differentiable on some interval must be continuous on that interval. However, there are functions that are continuous on some interval but not differentiable on that same interval. See example at part (c) (c) The example would be $f(x)=|x-2|$. This function is continuous but at $x=a=2$ it has two different tangents and thus is not differentiable.
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