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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?

Chapter 3.3, Problem 58E, Coffee is being poured into the mug shown in the figure at a constant rate measured in volume per

To determine

To sketch:

(i) A rough graph of the depth of the coffee in the mug as a function of time

(ii) Give significance of the inflection point.

Explanation

1) Concept:

Point of inflection is a point on the graph where the concavity changes from upward to downward or downward to upward.

Draw the rough sketch for depth of the coffee in the mug and find the points where concavity changes, then give significance of the inflection point.

2) Calculation:

(i)

Initially, the depth increases slowly since the base of mug is wide

Then, the depth rises suddenly, as the cup is narrow, the coffee rises more quickly.

Thus, depth d increases at an increasing rate and its graph is concave upward.

The rate of increase of d has a maximum where the mug is narrowest, that is mug is half full.

It is the Inflection Point where the graph changes its concavity from concave upward to concave downward.

The rate of increase of d starts to decrease as the mug becomes wide and the graph is concave down

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