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: 68


a. any value of $m$. b. $m=2$.

Work Step by Step

a. For the function to be continuous at $x=0$, we need to evaluate the left and right limits and compare with the function value at this point. We have $\lim_{x\to0^-}sin2x=0$, $\lim_{x\to0^+}mx=0$, and $f(0)=0$. Since these values are equal, we conclude that the function is continuous at $x=0$ for any value of $m$. b. For the function to be differentiable at $x=0$, we need to evaluate the left and right derivatives at this point and let them equal. We have $\lim_{x\to0^-}f'(x)=\lim_{x\to0^-}2cos2x=2$ and $\lim_{x\to0^+}f'(x)=\lim_{x\to0^+}m=m$. Thus the function is differentiable at $x=0$ only if $m=2$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.