Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.1 Introducing the Derivative - 3.1 Execises: 6

Answer

At the given point of the graph the instantaneous rate of change, the slope of the tangent and the value of the derivative are equal to each other.

Work Step by Step

The instantaneous rate of change at $a$ is $$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The slope of the tangent at the given point $(a,f(a))$ is $$m_{tan}=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ The value of the derivative of the function $f$ at the point $a$ is given by deffinition as $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ We see that at the given point of the graph the instantaneous rate of change, the slope of the tangent and the value of the derivative are equal to each other.
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