Mean Value & Rolle’s Theorems

1. 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

2. 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.

3. 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?

4. 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

5. If f (x) is a differentiable function over [a,b], then at some point between a and b: Mean Value Theorem for Derivatives

6. 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.

7. 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.

8. 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

9. Illustration of Mean Value Theorem Tangent parallel to chord. Slope of tangent: Slope of chord:

10. 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.

11. Solve the equation: So since that is in the interval.

12. 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.