1. 3.1/3.2 Extrema on an interval & Mean value Theorem
Ms. Clark
11/2/2016

2. Warm up:
Use the definition of the derivative to find the slope of the tangent line to the graph 𝑓 𝑥 =3 𝑥 2 −2𝑥+6 at the point (2,14) 2.) What is the derivative of 𝑔 𝑥 = tan cos 𝑥 3 +6𝑥 ? 3.) Simplify csc 𝑥− cos 𝑥 cot 𝑥

3. Extrema on an interval
Section 3.1 You are the proud owner of a business that orders widgets and sells them to the public. You have determined that the cost of ordering and storing x bundles of widgets is: 𝐶 𝑥 =2𝑥 𝑥 per bundle of widgets. The delivery truck can bring at most 450 bundles per order. Find the order size that will minimize the cost per bundle.

4. Relative vs absolute extrema
Definition of Extrema:
Minimum:
Maximum:
Relative Extrema:
Absolute Extrema:

5. Does the derivative exist at each of these extrema?

8. Extreme value theorem:

14. Critical numbers: Relative Extrema occur only at critical numbers!!!
(Endpoints are not technically critical numbers)

15. Using the extreme value theorem
Guidelines for finding Extrema on a closed interval:

16. Example 1
Find the absolute extrema of 𝑓 𝑥 =3 𝑥 4 −4 𝑥 3 on the interval [-1, 2]

18. Example 2
Find the absolute extrema of 𝑓 𝑥 =2𝑥−3 𝑥 2/3 on the interval [-1, 3]

19. Example 3
Find the absolute extrema of 𝑓 𝑥 =2 sin 𝑥− cos 2𝑥 on the interval [0, 2𝜋]

20. The mean value theorem Rolle’s Theorem See pictures on Pg 172

21. Example 4
For the graph of 𝑓 𝑥 =5− 4 𝑥 , find all values of c on the interval [1,4] where the Mean Value Theorem applies.