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: 41

Answer

$f(x)=\cos x$ and $a=\pi$

Work Step by Step

*By definition, the derivative of a function $f$ at a number $a$ is $$f'(a)=\lim\limits_{h\to0}\frac{f(a+h)-f(a)}{h}\hspace{0.5cm}(1)$$ Here we have $$f'(a)=\lim\limits_{h\to0}\frac{\cos{(\pi+h)}+1}{h}$$ $$f'(a)=\lim\limits_{h\to0}\frac{\cos{(\pi+h)}-(-1)}{h}$$ $$f'(a)=\lim\limits_{h\to0}\frac{\cos{(\pi+h)}-\cos\pi}{h}$$ Now we match the formula found above with the formula of the derivative by definition. We find that $a=\pi$, $f(a)=f(\pi)=\cos\pi$ and $f(x)=\cos x$
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