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MAT 1234 Calculus I

MAT 1234 Calculus I. Section 3.2 The Mean Value Theorem. http://myhome.spu.edu/lauw. No Homework!!! Take time to review problems from section 2.8. Make sure you can do all the problems without any help. Start lab 04 -- be sure to read the info in the PPT. Maple Lab tomorrow. 11/8.

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MAT 1234 Calculus I

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  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. Make sure you can do all the problems without any help. • Start lab 04 -- be sure to read the info in the PPT. • Maple Lab tomorrow

  3. 11/8 • Last day to withdraw from a class. • Talk to me (or your advisor) if you have any questions. • I and your advisor are here to help you make a informed decision

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

  5. Preview • Rolle’s Theorem • The Mean Value Theorem • Consequences of the Mean Value Theorem • We are going to convince ourselves that these results are correct • Sometimes, a diagram is sufficient • Other times, we need more details

  6. 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

  7. Rolle’s Theorem a b

  8. Rolle’s Theorem a b c

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

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

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

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

  13. 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,

  14. The Mean Value Theorem a b c

  15. The Mean Value Theorem a b c

  16. Theorem (Consequence) If for all in an interval , then is constant on . Why?

  17. Why? (1) Let , be two numbers in . Assuming , draw a diagram to illustrate the positions of the 4 points.

  18. Why? (2) is differentiable on implies that (i) is continuous on (why?) (ii) is differentiable on (why?)

  19. Why? (3) By the MVT, there is no. in such that () What is the value of ?

  20. Why? (4) By the MVT, there is no. in such that () What is the is the relationship between and ?

  21. Corollary (Important) a b

  22. Corollary (Important) Why? a b

  23. Wai? (1) Let Find .

  24. Wai? (2) for some constant . (Wai?)

  25. Wai? (3) for some constant . Then what?

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

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