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: 70

Answer

Initially, the graph is concave up, because the depth increases at a faster and faster rate. The slope of the graph is gradually increasing. Above the halfway point, the depth increases at a slower rate because the mug gets wider and wider. Thus the slope of the graph gradually decreases. Above the halfway point, the graph is concave downward. The inflection point represents the moment when the depth of the coffee has reached the halfway point in the mug, which is the narrowest part of the mug.
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Work Step by Step

Initially, when we start pouring the coffee, the depth increases slowly because the bottom of the mug is wider. As we keep pouring coffee, the depth increases at a faster rate because the mug gets narrower. Thus the slope of the graph increases until the depth reaches the halfway point, which is the narrowest part of the mug. Initially, the graph is concave up, because the depth increases at a faster and faster rate. The slope of the graph is gradually increasing. Above the halfway point, the depth increases at a slower rate because the mug gets wider and wider. Thus the slope of the graph gradually decreases. Above the halfway point, the graph is concave downward. The inflection point represents the moment when the depth of the coffee has reached the halfway point in the mug, which is the narrowest part of the mug.
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