Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.1 - Areas and Distances - 5.1 Exercises - Page 376: 13

Answer

A lower estimate for the distance that the runner traveled is $34.7~feet$ during these three seconds. An upper estimate for the distance that the runner traveled is $44.8~feet$ during these three seconds.

Work Step by Step

The interval $[0,3.0]$ is divided into 6 subintervals. $\Delta t = \frac{b-a}{n} = \frac{3.0-0}{6} = 0.5$ To find a lower estimate for the distance, we can use the left endpoint of each subinterval: $t_1 = 0$ $t_2 = 0.5$ $t_3 = 1.0$ $t_4 = 1.5$ $t_5 = 2.0$ $t_6 = 2.5$ We can find a lower estimate for the distance: $\sum_{i=1}^{6} v(t_i)~\Delta t$ $= (0+6.2+10.8+14.9+18.1+19.4)~(0.5)$ $= 34.7$ A lower estimate for the distance that the runner traveled is $34.7~feet$ during these three seconds. To find an upper estimate for the distance, we can use the right endpoint of each subinterval: $t_1 = 0.5$ $t_2 = 1.0$ $t_3 = 1.5$ $t_4 = 2.0$ $t_5 = 2.5$ $t_6 = 3.0$ We can find an upper estimate for the distance: $\sum_{i=1}^{6} v(t_i)~\Delta t$ $= (6.2+10.8+14.9+18.1+19.4+20.2)~(0.5)$ $= 44.8$ An upper estimate for the distance that the runner traveled is $44.8~feet$ during these three seconds.
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