Thomas' Calculus 13th Edition

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

Chapter 3: Derivatives - Section 3.1 - Tangents and the Derivative at a Point - Exercises 3.1 - Page 109: 44

Answer

a. See graph. b. No vertical tangent.
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Work Step by Step

a. See graph; it appears to have a vertical tangent at $x=0$ b. Evaluate the limit: $\lim_{h\to0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to0}\frac{h^{5/3}-5h^{2/3}}{h}=\lim_{h\to0}(h^{2/3}-5h^{-1/3})$. As $\lim_{h\to0^+}(h^{2/3}-5h^{-1/3})=-\infty$ and $\lim_{h\to0^-}(h^{2/3}-5h^{-1/3})=\infty\ne \lim_{h\to0^+}(h^{2/3}-5h^{-1/3})$ the function does not have a vertical tangent at $x=0$
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