1. Rolle’s Theorem & the Mean Value Theorem (3.2)
October 20th, 2016

2. I. Rolle’s Theorem
Thm. 3.3: 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.

3. II. The Mean Value Theorem
Thm. 3.4: The Mean Value Theorem: If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that .

4. Ex. 1: Use the following table of values that models the distance x(t) from Sacramento in miles and velocity v(t) in miles per hour of a train traveling from Sacramento to San Francisco to answer the following questions. Assume that the velocity of the train is a differentiable function on the interval (0,2]. Be sure to justify your reasoning. t
0.3
0.45
0.75 1
1.65 2
x(t) 10 19 42 51 74 90
v(t) 35 39 60 58 85
During the interval (0,2] hours, is there any given time at which the train is neither speeding up nor slowing down?
During the interval (0,2] hours, is there any given time at which the velocity of the train is 40 mph?
During the interval (0,2] hours, is there any given time at which the acceleration of the train is 70 miles per hour per hour?