﻿ Wei-Chi Yang  |  Radford University ### Remember:

Be sure to get all your homework done as soon as possible after class as this will lead to better grades.

- Flash Movies: Chile 07, Chile 05, Korea 05.
- eJMT
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Whitt 208; Phone: (540) 831-5232

• Implicit Differentiation (A flash)
• Derivative of a polar equation (A flash)
1. Course Contract
2. We will cover some topics from chapters 3 and 4 for review. The main content for 430 starts from chapter 5.
3. Proving 1=2 (what went wrong?)
4. Proving All People in Canada are the Same Age (what went wrong? (Need principle of induction)
5. Interactive Real Analysis.
6. Maple command to mean value theorem.
7. Page 46-47 (Maple file)
8. Homework: page 25, page 28, page 31.
9. A problem from page 65 (Maple file).
10. Mathematical Induction.
11. Countable and uncountable sets (1)
12. Page68#17(b).mws
14. Definition of field/ring.
15. Solving inequalities graphically. (page 83).
16. Explore the set of rational number is dense in R. (a Maple file).
17. limit points and closed set.
18. Hints for problems on Final.
19. Hints to a homework.
Math 431
1. Hint on an old Test 1.
2. Using Maple to learn sequences.
4. Using Fixed Point or Newton's method? When will 2^x>x^10? Corresponding Maple file
5. Another Recursive Sequence with Maple.
6. Newton's Method
7. Homework page 150
8. Homework page 155
9. Cauchy Sequence
10. The speed of convergence of two series. (Maple file)
11. A link to an online Real Analysis course.
12. Using Maple to explore the limit of a function at  point. (Maple file).
13. Epsilon-delta concept.
14. A ruler function
15. Taylor polynomial, Fourier Series and Bernstein Polynomial.
16. Another look at exploring the limit of a function at  point. (Maple file).
17. A proof to the squeezing principle.
18. Homework set 1 (Exercises on Cantor Theorem)
19. Recall the relationship between a closed set and its limit points.
20. Solution to page 175.
21. Solution to page 195
22. Understand the proofs of the followings:
• A continuous function sends a closed and bounded set to a closed and bounded set.
• If  f  is continuous on a closed and bounded set, then  f  assumes its maximum and minimum.
• If  f  is continuous on a closed and bounded set, then  f  assumes all its intermediate value.
23. Solution to (continuous functions on closed and bounded set).
24. Solution to page 216.
25. Some exercises on uniform continuous functions.
26. About continuity and uniform continuity of a function.
28. Continuity and Differentiability
29. *A nowhere differentiable function
30. Mean Value and Cauchy Mean Value Theorems (Dr. Yang's eJMT paper. A video clip for the proof of MVT.
31. Converse of Mean Value Theorem (Dr. Yang's).
32. Java applet on Mean Value Theorem.
33. Cauchy Mean Value Theorem and L'Hopital's Rule
34. Solution to page 237
35. Taylor's Theorem.
• Dr. Yang's eJMT paper regarding Mean-Value and Cauchy Mean Value Theorems.
• A Maple template for take home test 2, problem 5.
• Power Series and Taylor's Series-local but not global. (Maple file)
• Ratio test and interval of convergence (html).
36. Second Partial Derivative Test (html).
37. Fourier Series approximation is global but not local (Maple).
38. Homework on Taylor Polynomial and its Remainder. (PDF)
• Maple solution.
39. Motivation for Riemann Integration
40. Riemann Integration Theory
41. Uneven partition and numerical integrations with singularities
42. My own adaptive quadratures, good for functions that are monotone with singularities or highly oscillatory.
• 1 dimensional closed quadrature (Maple, Matlab)
• f(x)=1/sqrt(1-x^2) in [-1,0] Maple
• 1 dimensional open quadrature (Maple, Matlab)
• 2 dimensional closed quadrature (Maple, Matlab)
• 2 dimensional open quadrature (Maple)
43. Fubini's Theorem
44. *Animations on sequence of functions (Maple)
45. Romberg Integration