Rolle’s and The Mean Value Theorem

BC Calculus

Mean Value and Rolle’s Theorems

The Mean-Value Theorem ( and its special case ) Rolle’s Theorem are

Existence Theorems - - - The basis of many other concepts

Existence Theorems insure the existence of one of more numbers having a specific property . They DO NOT identify the point . . . . . .

[instead - - - It leads to attempts to find the value guaranteed by

the theorems]

Existence Theorems:

Completeness Postulate and Exponents

Zero Locator Theorem - Intermediate Value Theorem

Mean Value(Very Important)

IF f (x) is: 1.  Continuous on [a,b] , and

2.  Differentiable on (a,b)

THEN There exists a point c in (a,b) such that

*LAYMAN: The slope of the tangent at c

equals the slope of the secant

through f (a), and f (b)

*[The instantaneous rate of change

equals the average rate of change] 

( ) ( )'( )

f b f af c

b a

Example 1: Mean Value Theorem

Determine whether satisfies the conditions of the

Mean Value Theorem on [ 0, 2]

Determine whether satisfies the conditions of the

Mean Value Theorem on

( ) 1f x x

( ) cot2

xf x

,3

Example 2: M V T

Find the “ c ” guaranteed by the Mean Value Theorem.

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

Example 3: M V T

Find the “ c ” guaranteed by the Mean Value Theorem.

<< calculator dependent.>> 3 2( ) 2 [1,3]f x x x x on

Example 4: MVT

Two police patrol a highway with a 70 mph speed limit. The cars have radar and are in radio contact. They are stationed 5 miles apart. As a truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car its speed is clocked at 50 mph. The second patrolman pulls the truck over and issues a citation for excessive speed.

WHY?

Rolle’sROLLE’S THEOREM:

IF f (x) is 1. Continuous on closed interval [a,b],

2. Differentiable on (a,b), and

3. f (a) = f (b)

THEN: There exists at least one pt. “c”in (a,b)

Such that f / (c) = 0

Example 1: Rolle’s Theorem

Show that satisfies the conditions

of Rolle’s Theorem on [ 1, 2]

3 2( ) 2 2f x x x x

Example 2: Rolle’s Theorem

Find the “ c” guaranteed by Rolle’s Therorem.

4 2( ) 2 3 [ 2,2]f x x x on

Example 3: Rolle’s Theorem

Find the “ c” guaranteed by Rolle’s Theorem.

( ) sin(2 ) [0, ]2

f x x on

Last Update:

12/10/07