Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Chapter 5.9, Problem 2E
To determine

To match the rules of estimates of the integral.

Expert Solution & Answer
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Answer to Problem 2E

The value of L2, R2 and M2 is 6, 12 and 9.6

Explanation of Solution

Given information: I=∫02f(x) dx

  1. Since f(x) is decreasing function and is concave up.

    2. The large part of the upper right corner of the rectangles are outside the region, so larger estimate is Ln=0.9540

    All rectangles are inside the region with extra space left between the tops of the rectangles, so smaller estimate is Rn=0.7811

    The trapezoidal rule will have the top sides of the trapezoids connecting two points on the curve, the estimate slightly overestimate with the midpoint as: Tn=0.8675

    Midpoint number is slightly less than trapezoidal and the estimate is Mn=0.8632

  2. Here, Ln=0.9540; Rn=0.7811; Tn= 0.8675; Mn=0.8632

The value of the integral lie between trapezoidal and the midpoint.

Thus the order of the integral is Ln> Tn>I>Mn>Rn

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Chapter 5.9, Problem 1E
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Chapter 5.9, Problem 3E

Chapter 5 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 5.1 - Prob. 1ECh. 5.1 - (a) Use six rectangles to find estimates of each...Ch. 5.1 - (a) Estimate the area under the graph of f(x) =...Ch. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - The speed of a runner increased steadily during...Ch. 5.1 - Prob. 12E
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