Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.1 - Rates of Change and Tangents to Curves - Exercises 2.1 - Page 47: 20


a. (i) $-\frac{1}{6}$, (ii)$-\frac{1}{2T}$ b. See Table. c. $-0.25$ d. $-\frac{1}{4}$

Work Step by Step

a. Given $f(t)=1/t$ for $t\ne0$, (i) the average rate of change from $t=2$ to $t=3$ is $R_1=\frac{f(3)-f(2)}{3-2}=\frac{1}{3}-\frac{1}{2}=-\frac{1}{6}$, and (ii) from $t=2$ to $t=T$ is $R_T=\frac{f(T)-f(2)}{T-2}=\frac{1/T-1/2}{T-2}=\frac{2-T}{2T(T-2)}=-\frac{1}{2T}$ b. See Table. c. The table indicates that the rate of change of $f$ with respect to $t$ at $t=2$ is $-0.25$ d. From the results in part a, we have $\lim_{T\to 2}R_T=\lim_{T\to 2}(-\frac{1}{2T})=-\frac{1}{4}$
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