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 303: 69

Answer

Over time, we gain more knowledge at a slower rate as we approach the maximum level. Since the slope of the graph is decreasing, then $K''(t) \lt 0$. Therefore, the graph is concave downward. In the graph of the learning curve, we can see that the graph is concave downward. Thus, $K(3) - K(2)$ would be larger than $K(8) - K(7)$.

Work Step by Step

Initially, when we first start studying, we can gain new knowledge quite quickly. Over time, we gain more knowledge at a slower rate as we approach the maximum level. Therefore, the slope of the $K(t)$ graph has a steep positive slope initially and then gradually becomes less positive. Since the slope of the graph is decreasing, that is $K'(t)$ is decreasing, then $K''(t) \lt 0$. Therefore, the graph is concave downward. In the graph of the learning curve, we can see that the graph is concave downward. Thus, $K(3) - K(2)$ would be larger than $K(8) - K(7)$.
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