Rolle’s and The Mean Value Theorem

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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

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## Rolle’s and The Mean Value Theorem

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### 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]

We have irrational values

Existence Theorems:

Completeness Postulate and Exponents

Zero Locator Theorem - Intermediate Value Theorem

If f has the following values:

And f is continuous

(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]

Mean Value

Example 1: Mean Value Theorem

Determine whether satisfies the conditions of the

Mean Value Theorem on [ 0, 2]

1) Is continuous

2) Not differential --

we have a cusp

Determine whether satisfies the conditions of the

Mean Value Theorem on

discontinuous at which is an internal point MVT does not apply

Example 2: M V T

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

1) cont.?

2) diff?

Polynomial – continuous and smoothmeets MVT

Part 1: a c in[-1,3] such that

Part 2: (find it)

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

<< calculator dependent.>>

Example 3: M V T

Step 2: a c in [1,3] s.t.

Step 1:

cont.?

diff.?

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?

d=rt

Step 1:

Cont?

Diff?

The average speed is 75mph a point c where he was going 75mph

ROLLE’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

Rolle’s
Example 1: Rolle’s Theorem

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

Step 1:

MVT

Cont?

Diff?

Example 2: Rolle’s Theorem

Show that satisfies the conditions

of Rolle’s Theorem on [ 1, 2]

Step 1:

Cont?

Diff?

Example 3: Rolle’s Theorem

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

Step 1:

Cont?

Rolle’s

Diff?

MVT b/c 0 Rolle’s

or

Example 3: Rolle’s Theorem

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

on [0,

Step 1:

Cont?

Diff?