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: 2

Answer

When we think about the rate of change of a function on an interval, say $[a,x]$ we ask how much its value changes per unit length of that interval i.e. we divide the change of the function $f(x)-f(a)$ with the length of that interval which is $x-a$ which is exactly the slope of the secant.

Work Step by Step

The slope of the secant line through $(x,f(x))$ and $(a,f(a))$ is given by $$m_{sec} = \frac{f(x)-f(a)}{x-a}.$$ When we think about the rate of change of a function on an interval, say $[a,x]$ we ask how much its value changes per unit lenthg of that interval i.e. we divide the change of the function $f(x)-f(a)$ with the length of that interval which is $x-a$ which is exactly the slope of the secant.
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