1. Calculus 1 Rolle’s Theroem And the Mean Value Theorem for Derivatives Mrs. Kessler 3.2

2. Let f (x) be continuous function over [ a, b ], and differentiable on (a, b). If f(a) = f(b), c then there is at least one number c in (a, b) such that f ′(c) = 0 Differentiable implies that the function is also continuous, but notice continuous is closed, differentiable interval is open Rolle’s Theorem for Derivatives (1691)

3. 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

4. If f (x) is continuous on [a, b] and a differentiable function over ( a, b ), then a point c between a and b : Mean Value Theorem for Derivatives This goes one step further than Rolle’s Theorem

5. If f (x) is a differentiable function over ( a, b ), then at some point between a and b : Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the interior of the closed interval, the slope equals the average slope.

6. Slope of chord: Slope of tangent: Tangent parallel to chord.

7. Alternate form of the Mean Value Theorem for Derivatives

8. Determine if the mean value theorem applies, and if so find the value of c. f is continuous on [ 1/2, 2 ], and differentiable on (1/2, 2). This should equal f ’(x) at the point c. Now find f ’(x).

9. Determine if the mean value theorem applies, and if so find the value of c.

11. 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 ?

13. 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 72 mph