Calculus: Early Transcendentals 8th Edition

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

Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 302: 65

Answer

(a) Over time, the population increases at a faster and faster rate. The slope increases over time until the slope reaches a maximum at the inflection point. After the inflection point, the slope gradually decreases. This shows that the population is still increasing, but it is increasing at a slower and slower rate. (b) The rate of population increase is the highest after approximately 8 hours. (c) The population function is concave upward on the interval $(0,8)$ The population function is concave downward on the interval $(8, 18)$ (d) The inflection point is approximately at the coordinates $(8, 300)$

Work Step by Step

(a) Initially, the population increases slowly. Over time, the population increases at a faster and faster rate. We can see that the slope increases over time until the slope reaches a maximum at the inflection point. After the inflection point, the slope gradually decreases. This shows that the population is still increasing, but it is increasing at a slower and slower rate. (b) The rate of population increase is the highest when the slope of the graph is a maximum. This occurs after approximately 8 hours. (c) The population function is concave upward on the interval $(0,8)$ The population function is concave downward on the interval $(8, 18)$ (d) The inflection point is the point on the graph when it changes from concave upward to concave downward. The inflection point is approximately at the coordinates $(8, 300)$
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