Section 3.2 – Rolle’s Theorem and the Mean Value Theorem

 

 

Mean Value TheoremThe Mean Value Theorem can be interpreted geometrically as follows:

f b f a

b a

y f x

,a f a

,b f b

Is the slope of the line segment joining the points where x=a

and x=b.

 

,c f c 'f cIs the slope of the line tangent to the graph at the point x=c. This is equal to the slope of

the line segment.MVT says at least one point like x=c

must exist. It does NOT find where it is.

This is a continuous and differentiable function.

Mean Value TheoremLet f be a function that satisfies the following hypotheses:

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

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

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

Example: If a car travels smoothly down a straight level road with average velocity 60 mph, we would expect the speedometer reading to be exactly 60 mph at least once during the trip.

' f b f a

b af c

The derivative must equal the slope between the endpoints for at least one value of x.

There is at least one point where the instantaneous rate of change equals the average rate of change.

White Board Challenge

Sketch a graph of the function with the following characteristic:

The function is only defined on [2,8].

The function is differentiable on (2,8).

f(2) = f(8)

Does anyone have a graph that does not have a horizontal tangent line?

Rolle’s TheoremLet f be a function that satisfies the following three hypotheses:

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

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

3. f(a) = f(b)

Then there is a number c in (a,b) such that f '(c) = 0.

Example: If a car begins and ends at the same place, then somewhere during its journey, it must reverse direction.

f(x)

xa bc

There must be at least one

critical number between

the endpoints.

Start and end at the same y-value

At least one critical point.

Example 1Show that the function satisfies the

hypotheses of the MVT on the closed interval [9,25], and explain what conclusions you can draw from it.

25 9

25 9' f ff c

f x x

The function f is not differentiable at x≤0. BUT x≤0 is not in the interval [9,25]. Thus, the function is continuous on

[9,25] and differentiable on (9,25). The hypotheses of the MVT are satisfied.

Use the Theorem:

25 916 2

16 18

Therefore the Mean Value Theorem guarantees at least one x–value in (9,25) at which the instantaneous rate of change

of f is equal to 1/8.

Example 2Show that the function satisfies the

hypotheses of the MVT on the closed interval [1,2], and find a number c between 1 and 2 so that:

2 1

2 1' f ff c

3 2f x x x

Because f is a polynomial function, it is differentiable and also continuous on the entire interval [1,2]. Thus, the hypotheses of the MVT are satisfied.

3 2' ddxf x x x

Find the derivative.

2' 3 2f x x x

2 1

2 1' f ff c

Find the value of c.

3 2 3 22 2 1 1213 2c c

23 2 10c c 23 2 10 0c c

22 2 4 3 10

2 3c

2 1246

1.523

2.189

Example 3Discuss why Rolle’s Theorem can not be applied to the

functions below.

2 0,4f x x on tan 0,2f x x on

Not differentiable at x=2

Not continuous

The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by h(x) = f(g(x)) – 6.

Explain why there must be a value of c for 1< c < 3 such that h'(c) = -5.

3 1

3 1

h h 3 1

2

f g f g 4 2

2

f f 1 92

5

Since h is continuous and differentiable, by the MVT, there exists a value c, 1 < c < 3, such

that h' (c) = -5.

Notice how every part of the MVT is discussed

(Continuity AND Differentiability).

We already learned that this implies continuity.

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