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: 42

Answer

$f(\theta)=\sin\theta$ and $a=\frac{\pi}{6}$

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 $$f'(a)=\lim\limits_{\theta\to\pi/6}\frac{\sin\theta-\frac{1}{2}}{\theta-\pi/6}$$ $$f'(a)=\lim\limits_{\theta\to\pi/6}\frac{\sin\theta-\sin{\pi/6}}{\theta-\pi/6}$$ Now we match the formula found above with the formula of the derivative according to (1). We find that $a=\frac{\pi}{6}$, $f(\frac{\pi}{6})=f(a)=\sin{\frac{\pi}{6}}$ and $f(\theta)=\sin{\theta}$
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