rolle’s theorem and the mean value theorem3.2 teddy roosevelt national park, north dakota greg...
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Rolle’s Theorem and the Mean Value Theorem 3.2
Teddy Roosevelt National Park, North Dakota
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
Let f (x) be continuous on [a,b] and differentiable on (a,b). If f (a)= f (b) then there is at least one number c in (a,b) such that f '(c)=0.
Rolle’s Theorem for Derivatives
Conditions:
1. f is continuous on [a,b]2. f is differentiable on (a,b)3. f(a) = f(b)
The slope has to be zero somewhere between a and b.
Continuous
Not differentiable
ContinuousDifferentiablef(a)=f(b)
a b
Does Rolle’s Theorem Apply?
1. Continuous?
2. Differentiable?
3. f(1)=f(2)
2( ) 3 2 [1,2]f x x x
32x
Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.
yes
yes (=0)
yes
( ) 2 3 0f x x
32c
1. Continuous?
2. Differentiable?
3. f(-2)=f(2)
4 2( ) 2 [ 2,2]f x x x
0,1, 1x
Determine whether Rolle’s Theorem can be applied. If so, find all values such that f '(c)=0.
yes
yes
yes 3( ) 4 4 0f x x x
1,0,1c
2( ) 4 ( 1) 0f x x x
( ) 4 ( 1)( 1) 0f x x x x
If f (x) is continuous on [a,b] and differentiable on
(a,b), then there exists a number c in (a,b) such that: f b f a
f cb a
Mean Value Theorem
If f (x) is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that:
f b f af c
b a
Mean Value Theorem
The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.Or the instantaneous rate of change equals the average rate of change.
Example:
Find all the values of c in (1,4) such
that
4( ) 5 f x x
24( ) f x
x
Average rate of change
(4) (1)( ) .
4 1
f ff c
(4) (1) 4 11
4 1 4 1
f f
Instantaneous rate of change
24 1
x
24 x
2x
So, c=2
(-2 is not in (1,4))
5 miles
5 miles1/15 hour0 miles
0 minutes
1 0 515avg. vel. 75 mph1 1015 15
f f
Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit of 55 mph at some time during the four minutes.
At time 0, the truck passes the 1st patrol car.
At time 4 minutes (1/15 hr), the truck passes the 2nd patrol car.
So, the truck must have been traveling at a rate of 75 mph sometime during the four minutes!