Rolle’s Theorem and the Mean Value Theorem 3.2

Teddy Roosevelt National Park, North Dakota

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

Objectives

• Understand and use Rolle's Theorem

• Understand and use the Mean Value Theorem

Let f (x) be continuous on [a,b] and differentiable on (a,b). If f (a)= f (b) then there is at least one number c in (a,b) such that f '(c)=0.

Rolle’s Theorem for Derivatives

Conditions:

1. f is continuous on [a,b]2. f is differentiable on (a,b)3. f(a) = f(b)

The slope has to be zero somewhere between a and b.

Continuous

Not differentiable

ContinuousDifferentiablef(a)=f(b)

a b

Does Rolle’s Theorem Apply?

1. Continuous?

2. Differentiable?

3. f(1)=f(2)

2( ) 3 2 [1,2]f x x x

32x

Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.

yes

yes (=0)

yes

( ) 2 3 0f x x

32c

1. Continuous?

2. Differentiable?

3. f(-2)=f(2)

4 2( ) 2 [ 2,2]f x x x

0,1, 1x

Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.

yes

yes

yes 3( ) 4 4 0f x x x

1,0,1c

2( ) 4 ( 1) 0f x x x

( ) 4 ( 1)( 1) 0f x x x x

If f (x) is continuous on [a,b] and differentiable on

(a,b), then there exists a number c in (a,b) such that: f b f a

f cb a

Mean Value Theorem

If f (x) is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that:

f b f af c

b a

Mean Value Theorem

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.Or the instantaneous rate of change equals the average rate of change.

y

x0

A

B

a b

Slope of chord:

f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to chord.

c

Example:

Find all the values of c in (1,4) such

that

4( ) 5 f x x

24( ) f x

x

Average rate of change

(4) (1)( ) .

4 1

f ff c

(4) (1) 4 11

4 1 4 1

f f

Instantaneous rate of change

24 1

x

24 x

2x

So, c=2

(-2 is not in (1,4))

5 miles

5 miles1/15 hour0 miles

0 minutes

1 0 515avg. vel. 75 mph1 1015 15

f f

Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit of 55 mph at some time during the four minutes.

At time 0, the truck passes the 1st patrol car.

At time 4 minutes (1/15 hr), the truck passes the 2nd patrol car.

So, the truck must have been traveling at a rate of 75 mph sometime during the four minutes!

Homework

3.2 (page 176)

#1-5 odd

#9, 11, 15, 19

#23, 25, 29, 33, 35

#39-47 odd,

#53, 59