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 - PowerPoint PPT PresentationTRANSCRIPT
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 cb 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
( ) cot2xf 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: MVTTwo 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