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 174: 62


$$ \frac{65}{32},\ \ \ 0.0348\%$$

Work Step by Step

Given $$(17)^{1 / 4}$$ Consider $f(x)=x^{1 / 4}, a=16,$ and $\Delta x=1$, since \begin{align*} f^{\prime}(x)&=\frac{1}{4}x^{-3/4}\\ f^{\prime}(16)&=\frac{1}{32} \end{align*} Then the linearization to $f (x)$ is given by \begin{align*} L(x)&=f^{\prime}(a)(x-a)+f(a)\\ &=\frac{1}{32}(x-16)+2\\ &=\frac{3}{2}+\frac{x}{32} \end{align*} Since \begin{align*} L(17) &=\frac{3}{2}+\frac{17}{32}\\ &\approx \frac{65}{32} \end{align*} Hence the error is given by $$ | (17)^{1/4}-\frac{65}{32}|= 0.00070$$ and the percentage is $$\frac{0.00070}{(17)^{1/4}}\times 100 \% \approx 0.0348\%$$
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