Mean Value & Rolle’s Theorems3.2

Teddy Roosevelt National Park, North Dakota

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

Rolle’s Theorem

Let f be continuous on the closed interval [a, b] and differentiableon the open interval (a, b).

If f(a) = f(b)then there is at least one number c in (a, b) such that f ’(c)=0

So as long as the endpoints are equal, this guarantees an extreme value on the interior of the interval.

Ex. 1 Find the two x-intercepts of the function and show that f’(x) = 0 at some point between the two intercepts.

23)( 2 xxxf

Note that f(x) is differentiable and continuous over all real numbers.

0232 xx0)2)(1( xx

2,1 xx

0)2()1( ff

So since f(x) is continuous over [1, 2] and diff. over (1, 2) we can apply Rolle’s Thm. There must be a c in the interval

such that f’(c)=0.

How can we find c?

Ex. 2 Find all values of c in the interval (-2, 2) such that f’(c) = 024 2)( xxxf

First- Are the conditions for Rolle’s Thm. satisfied on the interval?Yes- continuous on closed interval differentiable on open interval

Are the endpoint values equal?Yes: f(-2) = f(2) = 8

044)(' 3 xxxf

0)1(4 2 xx

1,1,0 x

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 continuous over [a,b], and differentiable

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 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.

y

x0

A

B

a b

Slope of chord:

f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to chord.

c

Illustration of Mean Value Theorem

Ex. 3 Given the function find all values of c in the open interval (1, 4) such that

14

)1()4()('

ff

cf

)4(5)( xxf

Remember, this is the slope of the secant line through the points(1, f(1)) and (4, f(4)):

14

)1()4(

ff

114

14

Note that f satisfies the condition of MVT: it’s continuous on [1, 4]and differentiable on (1, 4).

So there is at least one point in the interval where f’(c) = 1.

14

)('2

xxf

Solve the equation:

2x2cSo since that is in the interval.

Ex. 5 Two stationary patrol cars equipped with radar are 5 milesapart on a highway. As a truck passes the first patrol car, its speedis clocked at 55 miles per hour. Four minutes later, when the truckpasses the second patrol car, its speed is clocked at 50 miles perhour. Prove that the truck must have exceeded the speed limit(of 55 miles per hour) at some time during the four minutes.

Let t = 0 be the time (in hours) when the truck passes thefirst patrol car.

When (in hours) does the truck pass the second patrol car?

hrt15

1

60

4

Find the average velocity for the truck:0)15/1(

05

v mph75

By the MVT, at some time on (0, 1/15) hr. the truck hit75 mph, and was therefore speeding.