   Chapter 3.2, Problem 31E

Chapter
Section
Textbook Problem

# Use the Mean Value Theorem to prove the inequality | sin a − sin b | ≤ | a − b | for all a and b

To determine

To prove:

Explanation

1) Concept:

Using the Mean Value Theorem simplify

2) Theorem:

The Mean Value Theorem: Let f be the function that satisfies the following hypothesis:

f is continuous on the closed interval [a, b]

f is differentiable on the open interval (a, b)

Then, there is a number c in (a, b) such that

f'c=fb-f(a)b-a

3) Given:

sina-sinb|a-b|

4) Calculation:

Consider the function fx=sinx

Since sinx is differentiable everywhere, use Mean Value Theorem.

Take derivative to get f'(c)

f'x=cosx

Therefore f'c=cosc

By using Mean Value Theorem,

cosc=sinb-sin

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