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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
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.
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 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
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
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