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Photo by Vickie Kelly, 2002. Greg Kelly, Hanford High School, Richland, Washington. 3.2. Mean Value & Rolle’s Theorems. Teddy Roosevelt National Park, North Dakota. Rolle’s Theorem. Let f be continuous on the closed interval [ a , b ] and differentiable on the open interval ( a , b ).

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slide1

Photo by Vickie Kelly, 2002

Greg Kelly, Hanford High School, Richland, Washington

3.2

Mean Value & Rolle’s Theorems

Teddy Roosevelt National Park, North Dakota

slide2

Rolle’s Theorem

Let f be continuous on the closed interval [a, b] and differentiable

on 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.

slide3

Ex. 1 Find the two x-intercepts of the function and show that

f’(x) = 0 at some point between the two intercepts.

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

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?

slide4

Ex. 2 Find all values of c in the interval (-2, 2) such that f’(c) = 0

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

slide5

If f (x) is a differentiable function over [a,b], then at some point between a and b:

Mean Value Theorem for Derivatives

slide6

If f (x) is a differentiable function over [a,b], then at some point between a and b:

Mean Value Theorem for Derivatives

Differentiable implies that the function is also continuous.

slide7

If f (x) is continuous over [a,b], and differentiable over (a, b) then at some point between a and b:

Mean Value Theorem for Derivatives

The Mean Value Theorem only applies over a closed interval.

slide8

If f (x) is a differentiable function over [a,b], then at some point between a and b:

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

Mean Value Theorem for Derivatives

slide9

Illustration of Mean Value Theorem

Tangent parallel to chord.

Slope of tangent:

Slope of chord:

slide10

Ex. 3 Given the function

find all values of c in the open interval (1, 4) such that

Remember, this is the slope of the secant line through the points

(1, f(1)) and (4, f(4)):

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.

slide11

Solve the equation:

So

since that is in the interval.

slide12

Ex. 5 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 miles per hour. Four minutes later, when the truck

passes the second patrol car, its speed is clocked at 50 miles per

hour. 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 the

first patrol car.

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

Find the average velocity for the truck:

By the MVT, at some time on (0, 1/15) hr. the truck hit

75 mph, and was therefore speeding.