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

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Presentation transcript:

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

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)

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

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

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.

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

Alternate form of the Mean Value Theorem for Derivatives

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).

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

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 ?

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