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Applications of Differentiation

Rolle’s Theorem and the Mean Value Theorem

Copyright © Cengage Learning. All rights reserved.

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Understand and use Rolle’s Theorem.

Understand and use the Mean Value Theorem.

Objectives

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Rolle’s Theorem

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Rolle’s Theorem, named after the French mathematician Michel Rolle, gives conditions that guarantee the existence of an extreme value in the interior of a closed interval.

Rolle’s Theorem

Michel Rolle (1652–1719)

brain waves?

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Rolle’s Theorem

Summary: 1.If f(x) is both continuous on interval [a,b] and differentiable on (a,b) AND has the same value at the start and finish,2.then there is at least one relative extrema on (a,b) (i.e. slope = 0).

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From Rolle’s Theorem, you can see that if a function f is continuous on [a, b] and differentiable on (a, b), and if f(a) = f(b), there must be at least one x-value between a and b at which the graph of f has a horizontal tangent, as shown in Figure 3.8(a).

Figure 3.8(a)

Rolle’s Theorem

continuous?

differentiable?

f(a) = f(b)?

extrema exist!

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If the differentiability requirement is dropped from Rolle’s Theorem, f will still have a critical number in (a, b), but it may not yield a horizontal tangent. Such a case is shown in Figure 3.8(b).

Figure 3.8(b)

Rolle’s Theorem

continuous?

differentiable?

f(a) = f(b)?

extrema existsBUT no tangent line

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Example 1 – Illustrating Rolle’s Theorem

Find the two x-intercepts of

f(x) = x2 – 3x + 2

and show that f’(x) = 0 at some point between the twox-intercepts.

Solution:

Note that f is differentiable on the entire real line. Setting

f(x) equal to 0 produces

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Example 1 – Solution

So, f(1) = f(2) = 0, and from Rolle’s Theorem you know that there exists at least one c in the interval (1, 2) such that f'(c) = 0.

and determine that f'(x) = 0 when x =

To find such a c, you can solve the equation

cont’d

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Example 1 – Solution

Note that this x-value lies in the open interval (1, 2), as shown in Figure 3.9.

Figure 3.9

cont’d

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Assignment 19 Read pp 172-175 Do: p 176-178/1-9odd, 11-23eoo 27-53 odd, 59, 67, 73,

76 (26 problems) Due Tuesday at the start of class

Use worksheet for #31

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The Mean Value Theorem

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The Mean Value Theorem

Rolle’s Theorem was used to prove another theoremby Lagrange—the Mean Value Theorem.

Summary: 1.If f(x) is both continuous on interval [a,b] and differentiable on interval (a,b),2.then there is at least one point c in the interval that has a derivative (i.e. slope) equal to the mean value (avg) of the function on the interval.

Joseph-Louis Lagrange(1736-1813)

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What’s the theory on all the theorems?

“The intuition (just looking at the Mean Value Theorem) is pretty obvious, but they stick it in the math books. People are just trying to learn calculus and get what matters, and they put the Mean Value Theorem there. They have function notation and all these words...it just confuses people.”

‒ Sal Khan, in his video on MVThttps://www.khanacademy.org/math/differential-calculus/derivative_applications/mean_value_theorem/v/mean-value-theorem

Who said this?

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What’s the theory on all the theorems?

Calculus teachers are just like you. We ask why too.

A few reasons often given for why we (teachers) give you (students) theorems:

1.They’re historic. They were developed and used by great minds to move calculus along. In fact one thing leads to another. As I mentioned, Lagrande use Rolle’s Theorem to develop the MVT.

2.They’re fundamental. Wading through the notation and the thinking is one of the basic activities of doing math. If we can’t read, write and think in a foreign language, do we really know the language? Maybe the same applies to math?

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What’s the theory on all the theorems?

3. We’ll need ‘em later. Theorems that make it into a first-year calculus book are usually one’s we’ll need again....and quite often again and again. That’s true for the Mean Value Theorem, for instance. The MVT is the basis for another even more fundamental theorem that we’ll see soon (next chapter) called.....shockingly...The Fundamental Theorem of Calculus. Now THAT sounds fundamental!

4. They’re on the AP test (they are, sometimes in a disguised form, but they’re ALWAYS there)

...and...the ever popular....

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What’s the theory on all the theorems?

So....while we may not all be able to look at

theorems like Pierre de Fermat...

“I have found a very great number of exceeding beautiful theorems”

‒ Pierre de Fermat

...we should, at least, wade through a few of the most important ones. Let’s look at an example:

Pierre de Fermat(1601-1665)

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Example 3 – Finding a Tangent Line

Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) such that

Solution:The slope of the secant line through (1, f(1)) and (4, f(4)) is

Note that the function satisfies the conditions of the Mean Value Theorem. continuous?

differentiable?

Can you see this as an application of the MVT?

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Example 3 – Solution

That is, f is continuous on the interval [1, 4] and differentiable on the interval (1, 4).

So, there exists at least one number c in (1, 4) such that f'(c) = 1.

Solving the equation f'(x) = 1 yields

which implies that x = ±2.

cont’d

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So, in the interval (1, 4), you can conclude that c = 2, as

shown in Figure 3.13.

Figure 3.13

cont’dExample 3 – Solution

slope of the tangent line [f’(c)] equals the slope of the secant line [f(b)-f(a)]/[(b-a)

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Assignment 19 Read pp 172-175 Do: p 176-178/1-9odd, 11-23eoo 27-53 odd, 59, 67, 73,

76 (26 problems) Due Tuesday at the start of class

Use worksheet for #31