Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5.2, Problem 61E

To determine

To show : For and number a and b

|sinb−sina|≤|b−a| .

Expert Solution & Answer

Answer to Problem 61E

The possible explanation is shown below.

Explanation of Solution

Given information :

The given inequality is |sinb−sina|≤|b−a|

Calculation :

The Mean Value Theorem states that if f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b) , then there exists at least one c such that f'(c)=f(b)−f(a)b−a .

Let f(x)=sinx on the interval [a, b] .

Since, f(x)=sinx , so f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b) . Therefore, by Mean Value Theorem there must be at least c such that

f'(c)=f(b)−f(a)b−af'(c)=sinb−sinab−a ...... (1)

Differentiate f(x)=sinx with respect to x and replace x with c :

f(x)=sinxf'(x)=cosxf'(c)=cosc

Substitute f'(c) in equation (1) and solve for c :

cosc=sinb−sinab−a ...... (2)

Also,

−1≤cosc≤1 for all c|cosc|≤1

Take absolute Value on both the sides of equation (2):

|cosc|=|sinb−sinab−a|

Since, |cosc|≤1

Therefore,

|sinb−sinab−a|≤1|sinb−sina||b−a|≤1|sinb−sina|≤|b−a|

Hence,

|sinb−sina|≤|b−a|

Chapter 5.2, Problem 60E

Chapter 5.2, Problem 62E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

Ch. 5.1 - Prob. 11QR Ch. 5.1 - Prob. 12QR Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - 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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY

For and number a and b | sin b − sin a | ≤ | b − a | .

Concept explainers

Videos

To show : For and number a and b |sinb−sina|≤|b−a| .

Answer to Problem 61E

Explanation of Solution

Chapter 5 Solutions

Additional Math Textbook Solutions

To show : For and number a and b

|sinb−sina|≤|b−a| .