Leibniz’s Rule In applications, we sometimes | Ch 5 - 38AAE

Thomas' Calculus: Early Transcendentals | 13th Edition | ISBN: 9780321884077 | Authors: George B. Thomas Jr., Maurice D. Weir, Joel R. Hass

Problem 38AAE Chapter 5

Thomas' Calculus: Early Transcendentals | 13th Edition

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Thomas' Calculus: Early Transcendentals | 13th Edition | ISBN: 9780321884077 | Authors: George B. Thomas Jr., Maurice D. Weir, Joel R. Hass

Thomas' Calculus: Early Transcendentals | 13th Edition

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Problem 38AAE

Leibniz’s Rule In applications, we sometimes encounter functions like  defined by integrals that have variable upper limits of integration and variable lower limits of integration at the same time. The first integral can be evaluated directly, but the second cannot. We may find the derivative of either integral, however, by a formula called Leibniz’s Rule.

Leibniz’s Rule

If ƒ is continuous on [a, b] and if u(x) and (x) are differentiable functions of x whose values lie in [a, b], then

Figure 5.33 gives a geometric interpretation of Leibniz’s Rule. It shows a carpet of variable width ƒ(t) that is being rolled up at the left at the same time x as it is being unrolled at the right. (In this interpretation, time is x, not t.) At time x, the floor is covered from u(x) to (x). The rate  at which the carpet is being rolled up need not be the same as the rate at which the carpet is being laid down. At any given time x, the area covered by carpet is

FIGURE 5.33 Rolling and unrolling a carpet: a geometric interpretation of Leibniz’s Rule:

At what rate is the covered area changing? At the instant x, A(x) is increasing by the width ƒ( (x)) of the unrolling carpet times the rate at which the carpet is being unrolled. That is, A(x) is being increased at the rate

At the same time, A is being decreased at the rate

the width at the end that is being rolled up times the rate . The net rate of change in A is

which is precisely Leibniz’s Rule.

To prove the rule, let F be an antiderivative of ƒ on [a, b]. Then.

Differentiating both sides of this equation with respect to x gives the equation we want:

Use Leibniz’s Rule to find the derivatives of the functions.

Step-by-Step Solution:

Step 1 of 3:

In this problem we have to find the derivative of the given function by using Leibniz's rule

Step 2 of 2

Chapter 5, Problem 38AAE is Solved
Textbook: Thomas' Calculus: Early Transcendentals
Edition: 13th
Author: George B. Thomas Jr., Maurice D. Weir, Joel R. Hass
ISBN: 9780321884077

This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13th. Since the solution to 38AAE from 5 chapter was answered, more than 222 students have viewed the full step-by-step answer. This full solution covers the following key subjects: Being, rate, rule, carpet, leibniz. This expansive textbook survival guide covers 138 chapters, and 9198 solutions. The answer to “Leibniz’s Rule In applications, we sometimes encounter functions like defined by integrals that have variable upper limits of integration and variable lower limits of integration at the same time. The first integral can be evaluated directly, but the second cannot. We may find the derivative of either integral, however, by a formula called Leibniz’s Rule.Leibniz’s RuleIf ƒ is continuous on [a, b] and if u(x) and (x) are differentiable functions of x whose values lie in [a, b], then Figure 5.33 gives a geometric interpretation of Leibniz’s Rule. It shows a carpet of variable width ƒ(t) that is being rolled up at the left at the same time x as it is being unrolled at the right. (In this interpretation, time is x, not t.) At time x, the floor is covered from u(x) to (x). The rate at which the carpet is being rolled up need not be the same as the rate at which the carpet is being laid down. At any given time x, the area covered by carpet is FIGURE 5.33 Rolling and unrolling a carpet: a geometric interpretation of Leibniz’s Rule: At what rate is the covered area changing? At the instant x, A(x) is increasing by the width ƒ( (x)) of the unrolling carpet times the rate at which the carpet is being unrolled. That is, A(x) is being increased at the rate At the same time, A is being decreased at the rate the width at the end that is being rolled up times the rate . The net rate of change in A is which is precisely Leibniz’s Rule.To prove the rule, let F be an antiderivative of ƒ on [a, b]. Then. Differentiating both sides of this equation with respect to x gives the equation we want: Use Leibniz’s Rule to find the derivatives of the functions.” is broken down into a number of easy to follow steps, and 306 words. The full step-by-step solution to problem: 38AAE from chapter: 5 was answered by Sieva Kozinsky, our top Calculus solution expert on 08/01/17, 02:37PM. Thomas' Calculus: Early Transcendentals was written by Sieva Kozinsky and is associated to the ISBN: 9780321884077.

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