Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 3: Derivatives - Practice Exercises - Page 178: 67

Answer

See graph and explanations.
1575974896

Work Step by Step

a. See graph for the piecewise function. b. To see if the function is continuous at $x=1$, we need to evaluate the left and right limits at this point and compare with the function value. We have $\lim_{x\to1^-}x=1$, $\lim_{x\to1^+}(2-x)=1$, and $f(1)=1$. Since these values are equal, we conclude that the function is continuous at $x=1$. c. To see if the function is differentiable at $x=1$, we need to evaluate the left and right derivatives at this point. We have $\lim_{x\to1^-}f'(x)=\lim_{x\to1^-}1=1$ and $\lim_{x\to1^+}f'(x)=\lim_{x\to1^+}(-1)=-1$. Since these values are not equal, we conclude that the function is not differentiable at $x=1$.
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