Explanation Concave up and increasing function over [ a , b ] is shown below in Figure 1. From the Figure 1, it is noticed that for the function that is concave up and increasing, the curve lies below the all rectangle. Therefore, the exact area overestimates for the function which is concave up and increasing. Concave up and decreasing function over [ a , b ] is shown below in Figure 2. From Figure 2, it is noticed that for the function that is concave up and decreasing, each rectangle of the right Riemann sum will completely lie below the curve. Therefore, the exact area for concave up and decreasing function is underestimates. Concave down and increasing function over [ a , b ] is shown below in Figure 3. From the Figure 3, it is noticed that for the function that is concave down and increasing, the curve lies below the all rectangle. Therefore, the exact area overestimates for the function which is concave up and increasing

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 5.1, Problem 79E

To determine

To fill: The given table with the resulting approximation underestimates or overestimates the exact area for the given four cases where the area is calculated using the right Riemann sum; explain the reason with the appropriate graph.

Increasing on [a,b] Decreasing on [a,b]
Concave up on [a,b] _____
Concave down on [a,b] _ ____

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Chapter 5.1, Problem 78E

Chapter 5.2, Problem 1E

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