MAT 1234 Calculus I

1. MAT 1234Calculus I Section 3.2 The Mean Value Theorem http://myhome.spu.edu/lauw

2. No Homework!!! • Take time to review problems from section 2.8 and/or • Start lab 04 -- be sure to read the info in the PPT. • Maple Lab tomorrow

3. Preview • Rolle’s Theorem • The Mean Value Theorem • Consequences of the Mean Value Theorem

4. Rolle’s Theorem Suppose f satisfies the following 3 conditions: 1. f is continuous on [a,b]. 2. f is differentiable on (a,b). 3. f(a) = f(b) Then there is a number c in (a,b) such that

5. Rolle’s Theorem a b

6. Rolle’s Theorem a b c

7. Example 1* Prove that has exactly one real root.

8. Example 1* (Q&A) Why do we need to show it when it is obvious from the graph?

9. We know… If f(x)=C on (a,b), then f’(x)=0 on (a,b)

10. T or F If f’(x)=0 on (a,b), then f(x)=C on (a,b)

11. The Mean Value Theorem Let f be a function satisfies the following conditions: 1. f is continuous on the closed interval [a,b]. 2. f is differentiable on the open interval (a,b). Then there is a number c in (a,b) such that or, equivalently,

12. The Mean Value Theorem a b c

13. The Mean Value Theorem a b c

14. Theorem (Consequence) If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b). Why?

15. Corollary (Important) a b

16. Corollary (Important) Why? a b

17. Possible Exam Questions • State the Rolle’s Theorem • State the Mean Value Theorem • Explain why the following is true