CHAPTER 3 SECTION 3.2 ROLLE’S THEOREM AND THE MEAN VALUE THEOREM

Theorem 3.3 Rolle's Theorem and Figure 3.8

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

Apply Rolle's Theorem Apply Rolle's Theorem to the following function f and compute the location c.

Theorem 3.4 The Mean Value Theorem and

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

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.

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. 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: 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.

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

Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b) then there exists a value, c, in (a,b) such that

Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b) then there exists a value, c, in (a,b) such that c can’t be an endpoint Slope of a tangent line Slope of the line through the endpoints Instantaneous rate of change Average rate of change

1. Apply the MVT to on [-1,4].

1. Apply the MVT to on [-1,4]. f(x) is continuous on [-1,4]. MVT applies! f(x) is differentiable on [-1,4].

2. Apply the MVT to on [-1,2].

2. Apply the MVT to on [-1,2]. MVT does not apply! f(x) is continuous on [-1,2]. f(x) is not differentiable at x = 0. MVT does not apply!

the Mean Value Theorem for Derivatives 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.

the Mean Value Theorem for Derivatives 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

AP QUESTION

AP QUESTION