Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.7 - Derivatives and Rates of Change - 2.7 Exercises: 40

Answer

$f(x)=\frac{1}{x}$ and $a=\frac{1}{4}$

Work Step by Step

*Another way to write the derivative of a function $f$ at a number $a$ is $$f'(a)=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}\hspace{0.5cm}(1)$$ Here we have $$f'(a)=\lim\limits_{x\to 1/4}\frac{\frac{1}{x}-4}{x-\frac{1}{4}}$$ $$f'(a)=\lim\limits_{x\to 1/4}\frac{\frac{1}{x}-\frac{1}{1/4}}{x-1/4}$$ Now we match the formula found above with the formula of the derivative according to (1). We find that $a=\frac{1}{4}$, $f(a)=f(\frac{1}{4})=\frac{1}{1/4}$ and $f(x)=\frac{1}{x}$
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