Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 64

Answer

$$0.04$$

Work Step by Step

Since $$ f(x) =x^{1 / 3} e^{x-1} ,\ \ \ \ \Delta x= 0.03,\ \ a=1 $$ and \begin{align*} f'(x)& = \frac{d}{dx}\left(x^{\frac{1}{3}}\right)e^{x-1}+\frac{d}{dx}\left(e^{x-1}\right)x^{\frac{1}{3}}\\ &= \frac{e^{x-1}+3xe^{x-1}}{3x^{\frac{2}{3}}}\\ f'(1)&=\frac{4}{3} \end{align*} Then \begin{align*} \Delta f&=f(a+\Delta x)-f(a)\\ &\approx f'(a) \Delta x\\ &\approx \frac{4(0.03)}{3}\\ &\approx 0.04 \end{align*}
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