Calculus (3rd Edition)

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

Chapter 4 - Applications of the Derivative - 4.1 Linear Approximation and Applications - Exercises - Page 172: 7



Work Step by Step

Recall that for small $\Delta x $, $\Delta y\approx dy=f'(x)dx=f'(x)\Delta x $ Or $ f(a+\Delta x)-f(a)\approx f'(a)\Delta x $ Here, $ f(x)=\sqrt[3] x $, $ a=27$, $\Delta x=0.2$ and $ f'(x)=\frac{1}{3}\times x^{\frac{1}{3}-1}=\frac{1}{3}x^{-\frac{2}{3}}$ $\implies f(27+0.2)-f(27)\approx f'(27)\Delta x $ That is, $\sqrt[3] {27.2}-\sqrt[3] {27}\approx \frac{1}{3}(27)^{-\frac{2}{3}}\times0.2=0.0074074$
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