1. 3.2: Rolle’s Theorem and the Mean Value Theorem
Homework: p.174 9, 11, 15, 19, 23, 29, 31, 35, 39, 41, 53, 55
Standards
EK2.4A1 – If a function f is continuous over the interval [a,b] and differentiable over the interval (a,b), the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.
Learning Objectives:
Understand and use Rolle’s Theorem
Understand and use the Mean Value Theorem

2. 1 2 3 4 5
Con TE
TEx2
SLE
SLEx2

3. Concept 1: Rolle’s Theorem
The Three Conditions of Rolle’s Theorem
𝑓(𝑥) must be continuous on [a,b]
𝑓(𝑥) must be differentiable on (a,b)
𝑓 𝑎 =𝑓(𝑏)

4. Example 1: Rolle’s Theorem with INTECEPTS
Find the two x-intercepts of ℎ 𝑥 = 𝑥 2 −2𝑥−35 and show that ℎ ′ 𝑥 =0 at some point between the two intercepts.

5. Student Led Example 1: Rolle’s Theorem with INTERCEPTS
Find the two x-intercepts of 𝑘 𝑥 = 𝑥 2 −3𝑥+2 and show that 𝑘 ′ 𝑥 =0 at some point between the two intercepts.

6. Example 2A: Rolle’s Theorem on a CLOSED INTERVAL
Find a value 𝑐 such that ℎ ′ 𝑐 =0 if ℎ 1 𝑥 = 1 3 𝑥 3 − 𝑥 2 −3𝑥+4 on the interval [−3, 3]

7. Student Led Example 2A: Rolle’s Theorem on a CLOSED INTERVAL
Find a value 𝑐 such that 𝑘 ′ 𝑐 =0 if 𝑘 𝑐 = 𝑥 2 +8𝑥+5 on the interval [2, 6]

8. Example 2B: Rolle’s Theorem on a CLOSED INTERVAL
Can Rolle’s Theorem be applied to ℎ 2 𝑥 = 𝑥 2 +2𝑥+3 𝑥+2 on the interval −1, 3 ?

9. Student Led Example 2B: Rolle’s Theorem on a CLOSED INTERVAL
Can Rolle’s Theorem be applied to
𝑘 2 𝑥 = 𝑥 2 +𝑥−6 𝑥−3 on the interval 0, 4 ?

10. Example 3: Rolle’s Theorem on a closed interval (TRIG)
Find a value for 𝑐 such that 𝑓 ′ 𝑐 =0 If ℎ 𝑥 = sec 𝑥 On the interval −𝜋, 𝜋
Warning!

11. Technology Test
You decide…
Is this the graph of − 𝑥 2 +2𝑥?
Warning!

12. Pictures are NOT proofs
1− 𝑥−1 2 − 𝑥−

13. Concept 4: Mean Value Theorem (but really, it will be concept 5)
The Three Conditions of the MVT
𝑓(𝑥) must be continuous on [a,b]
𝑓(𝑥) must be differentiable on (a,b)
There exists a 𝑐 such that
𝑓 ′ 𝑐 = 𝑓 𝑏 −𝑓(𝑎) 𝑏−𝑎

14. Example 4: The Mean Value Theorem
It’s basically the slope formula…moving on

15. Example 5: Instantaneous Rate of Change
A plane begins its take-off at 2:00pm on a 2500 mile flight. After 5 ½ hours, the plane arrives at its destination. Explain why there are least two times during the flight the speed of the plane is 400 miles per hour.

16. Student Led Example 5: Instantaneous Rate of Change
Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a semi-truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 miles per hours. If the speed limit is 55 miles per hour, should the driver of the truck get a speeding ticket?

17. Can you do it?
lim 𝑥→0 7𝑥− sin 𝑥 𝑥 2 + sin 3𝑥