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Problem

If $ f'(x) = c $ ($ c $ a constant) for all $ x $…

01:56

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Problem 31 Hard Difficulty

Use the Mean Value Theorem to prove the inequality
$ | \sin a - \sin b | \leqslant | a - b | $ for all $ a $ and $ b $


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Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 2

The Mean Value Theorem

Related Topics

Derivatives

Differentiation

Volume

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Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

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06:14

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Video Transcript

So we're being asked to use the mean value item To prove the inequality, you have to sign a very minus sign of peace lesson equally after value A minus B for A and B since F. Beck, we can let f of x equal sign of X and we know that a derivative of effort Beck is close on X and we can actually, since signing close eye necks are for domestic, I mean, our continuous indefensible. We can actually apply them in various healing right away so we can write some C in the prime of eggs, which is co sign ex co sign of C. Their existence. See, just at the average slope is equal to the function was a sign of B minus sign of a over B minus A s O. We also know that the co sign just statement right here. So this statement, I hear co sign that that's him. But negative is bounded, which were negative one and a positive one and this statement canasta be rewritten in terms of ah, absolute value signed because the absolute value symbol actually represent distance. You can think of absolute values of distant and so this statement right here or the exact are the same. So we're actually going to rewrite the statement here, And since we applied the absent very function to this inside, we apply it to, um to the right side as well. So we have to We write this as, um, absolutely frozen, See is equal to the absolute value of sign of B miners. Sign of a all over baby might say. And now we're gonna go to the next page. Recall that the absolute coast I'm c'est les an equal one. So we're actually gonna substitute that one into our equation. So we're going to sign a minus. I mean, I'm so sorry, Sinan. Kind of b finest kind of a all over. Okay, B minus is less than equal to one. And then we're going to multiply, be minus a on both sides. We're going to sign a oh, my anti sign of B for that minus the sign of a And this is also still in absolute value. And then the B minus days also an absolute value. So I banned him. And then we can rewrite this as sign a minus a sign of the A minus B. And that is our proof. We have now proven our statement

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Calculus: Early Transcendentals

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Related Topics

Derivatives

Differentiation

Volume

Top Calculus 2 / BC Educators
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

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