rolle s and the mean value theorem n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Rolle’s and The Mean Value Theorem PowerPoint Presentation
Download Presentation
Rolle’s and The Mean Value Theorem

Loading in 2 Seconds...

play fullscreen
1 / 14

Rolle’s and The Mean Value Theorem - PowerPoint PPT Presentation


  • 199 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

Rolle’s and The Mean Value Theorem


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
mean value and rolle s theorems
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

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

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] 

Mean Value
slide6

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
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)

example 3 m v t

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
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

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
Example 1: Rolle’s Theorem

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

Step 1:

MVT

Cont?

Diff?

example 2 rolle s theorem
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
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 theorem1
Example 3: Rolle’s Theorem

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

on [0,

Step 1:

Cont?

Diff?