CHAPTER 3SECTION 3.2

ROLLE’S THEOREM AND

THE MEAN VALUE THEOREM

Theorem 3.3 Rolle's Theorem and Figure 3.8

Rolle’s Theorem for Derivatives

Example: Determine whether Rolle’s Theorem can be applied to f(x) = (x - 3)(x + 1)2 on [-1,3]. Find all values of c such that f ′(c )= 0.

f(-1)= f(3) = 0 AND f is continuous on [-1,3] and diff on (1,3) therefore Rolle’s Theorem applies.

f ′(x )= (x-3)(2)(x+1)+ (x+1)2 FOIL and Factor

f ′(x )= (x+1)(3x-5) , set = 0 c = -1 ( not interior on the interval) or 5/3

c = 5/3

Apply Rolle's TheoremApply Rolle's Theorem to the following function f and compute the location c.

3

2

2

2

2

2 13

1 13 3

( ) [0, 1]

( ) 3 1

(0) (1) 0

' [0, 1]

( ) 3 1 0

3 1 0

3 1

, [ ]

f x x x on

f x x

f f

By Rolle s Theorem there is a c in such that

f c c

c

c

c

c

Theorem 3.4 The Mean Value Theorem and

If f (x) is a differentiable function over [a,b], then

at some point between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

If f (x) is a differentiable function over [a,b], then

at some point between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

Differentiable implies that the function is also continuous.

If f (x) is a differentiable function over [a,b], then

at some point between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

Differentiable implies that the function is also continuous.

The Mean Value Theorem only applies over a closed interval.

If f (x) is a differentiable function over [a,b], then

at some point between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

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

Mean Value Theorem

4

2

-2

-4

-5 5

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

a b

'

fc

b

b af

f a

Mean Value Theorem

4

2

-2

-4

-5 5

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

a b

'

fc

b

b af

f a

Slope of the line through the endpoints

Slope of a tangent line

c can’t be an endpoint

Average rate of changeInstantaneous rate of change

1c 2c 3c 4c

1. Apply the MVT to on [-1,4]. 2 4f x x

1. Apply the MVT to on [-1,4]. 2 4f x x f(x) is continuous on [-1,4].

' 2f x x f(x) is differentiable on [-1,4].

12

4

4 1

f fc

12

5

5c

32c 3

2c

MVT applies!

2. Apply the MVT to on [-1,2]. 23f x x

2. Apply the MVT to on [-1,2]. 23f x x

f(x) is continuous on [-1,2].

132

3'f x x

f(x) is not differentiable at x = 0.

MVT does not apply!

13

2

3x

( ) ( ) ( ) '( )f b f a b a f c

Alternate form of the Mean Value Theorem for Derivatives

Determine if the mean value theorem applies, and if Determine if the mean value theorem applies, and if so find the value of so find the value of cc..

1 1( ) , 2

2

xf x on

x

f is continuous on [ 1/2, 2 ], and differentiable on (1/2, 2).

1 3(2) 32 2 1

1 32

2 2

f f

This should equal f ’(x) at the point c. Now find f ’(x).

2 2

(1) ( 1)(1) 1'( )

x xf x

x x

Determine if the mean value theorem applies, and if so find the value of c. 1 1

( ) , 22

xf x on

x

1 3(2) 32 2 1

1 32

2 2

f f

2 2

(1) ( 1)(1) 1'( )

x xf x

x x

2

11

x

2 1

1

1

x

x

c

Application of the Mean Value Theorem for Derivatives

You are driving on I 595 at 55 mph when you pass a police car with radar. Five minutes later, 6 miles down the road, you pass another police car with radar and you are still going 55 mph. She pulls you over and gives you a ticket for speeding citing the mean value theorem as proof.

WHY ?

Application of the Mean Value Theorem for Derivatives

You are driving on I 595 at 55 mph when you pass a police car with radar. Five minutes later, 6 miles down the road you pass another police car with radar and you are still going 55mph.

He pulls you over and gives you a ticket for speeding citing the mean value theorem as proof.

Let t = 0 be the time you pass PC1. Let s = distance traveled. Five minutes later is 5/60 hour = 1/12 hr. and 6 mi later, you pass PC2. There is some point in time c where your average velocity is defined by

(1/12) (0) 6Average Vel. =

(1/12 0) 1/12

s s mi

hr

72 mph

f b f a

b a

AP QUESTION

AP QUESTION