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MAT 3749 Introduction to Analysis

MAT 3749 Introduction to Analysis. Section 2.2 Part II Intermediate Value Theorem. http://myhome.spu.edu/lauw. Preview. Prove Intermediate Value Theorem (statement 2). The first proof with substantial length As usual, we will go through the analysis carefully

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MAT 3749 Introduction to Analysis

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  1. MAT 3749Introduction to Analysis Section 2.2 Part II Intermediate Value Theorem http://myhome.spu.edu/lauw

  2. Preview • Prove Intermediate Value Theorem (statement 2). • The first proof with substantial length • As usual, we will go through the analysis carefully • I will have questions to ask you. Please try to answer some questions.

  3. References • Section 2.2

  4. IVT Statement 1 • Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b) • Then there is a no. c in (a,b) such that f(c)=N

  5. IVT Statement 2 • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

  6. Intermediate Value Theorem • There are usually two type of proofs. • Use sequences • Use contradictions to argue that and

  7. Recall Continuity (e-d)

  8. Continuous from the Right

  9. Recall Supremum

  10. The Completeness Axiom

  11. Analysis • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

  12. Analysis: Step 1 Without

  13. Analysis: Step 2 Without

  14. Analysis: Step 2

  15. Analysis: Step 3

  16. Analysis: Step 4 Why does ? Why does ?

  17. Analysis: Step 5

  18. Analysis: Step 5

  19. Analysis: Step 5 What’s wrong?

  20. Analysis: Step 5

  21. Analysis: Step 5 Why is not an upper bound of ?

  22. Analysis: Step 5 What is the consequence if is not an upper bound of ?

  23. Analysis: Step 5 Why

  24. Analysis: Step 5 Why

  25. Analysis: Step 5 If so, what is the contradiction?

  26. Analysis: Step 6 Can ? Why?

  27. Lemma 1

  28. Lemma 2

  29. Proof • Suppose f is continuous on [a,b] and that f(a), f(b) are with different signs • Then there is a no. c in (a,b) such that f(c)=0

  30. Proof Without

  31. Proof

  32. Proof

  33. Proof

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