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: 66

Answer

See graph and explanations.
1575974388

Work Step by Step

a. See graph for the piecewise function. b. To see if the function is continuous at $x=0$, we need to evaluate the left and right limits at this point and compare with the function value. We have $\lim_{x\to0^-}x=0$, $\lim_{x\to0^+}tanx=0$, and $f(0)=0$. Since these values are equal, we conclude that the function is continuous at $x=0$. c. To see if the function is differentiable at $x=0$, we need to evaluate the left and right derivatives at this point. We have $\lim_{x\to0^-}f'(x)=\lim_{x\to0^-}1=1$ and $\lim_{x\to0^+}f'(x)=\lim_{x\to0^+}sec^2)(x)=1$. Since these values are equal, we conclude that the function is differentiable at $x=0$.
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