Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.1 Limits, Rates of Change, and Tangent Lines - Exercises - Page 45: 16



Work Step by Step

We are given the function: $f(x)=e^x$ The average rate of change on $[x_0,x_1]$ is: $\dfrac{\Delta f}{\Delta x}=\dfrac{f(x_1)-f(x_0)}{x_1-x_0}$ The instantaneous rate of change is the limit of the average rate of change. In order to estimate the instantaneous rate of change at $x=e$, consider intervals $[x_1,x_0],[x_0,x_1]$ for $x_1$ close to $x_0$, where $x_0=e$: $[e-0.1,e]$: $\dfrac{\Delta f}{\Delta x}=\dfrac{f(e-0.1)-f(e)}{e-0.1-e}=\dfrac{e^{e-0.1}-e^e}{-0.1}\approx 14.421187$ See table From the table we find that the instantaneous rate of of change at $x=e$ is approximately $15.15$.
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