1. AP Calculus BC
September 23, 2016

2. Entry Task
Sketch a rectangular coordinate plane on a piece of paper. Label the points (1, 3) and (5, 3). Use your pencil to draw the graph of a differentiable function f that starts at (1, 3) and ends at (5, 3).
Is there at least one point where 𝑓 ′ =0?
Is it possible to draw the graph so that there isn’t a point for which 𝑓 ′ =0? Explain your reasoning.

3. Learning Targets
I understand Rolle’s Theorem and the Mean Value Theorem and how they relate to derivatives of functions.
I solved a variety of problems involving applications of derivatives with Rolle’s Theorem and the MVT.

4. Section 3.1 – Extrema on and INterval
What are extrema?
The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum.

5. Rolle’s Theorem
Revisit exit Ticket

6. Rolle’s Theorem 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.

7. MVT 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:
𝑓 ′ 𝑐 = 𝑓 𝑏 −𝑓 𝑎 𝑏−𝑎

8. 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 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the four minutes.

9. Assignment #10
Page : 15, 17, 21, 41, 44, 59, 64, 77-80