Last updated: 12/10/2008.
Topics already covered:
1. Monday, 08/25:
Introduction.
Section 1.1: Sets and functions.
2. Wednesday, 08/27:
Section 1.1: Sets and functions (continued).
Section 1.2: Mathematical induction.
3. Friday, 08/29:
Section 1.2: Strong induction.
Section 1.3: Sets and cardinality. Equipollent sets.
See also the notes on the subject.
The notes were last updated
Wednesday, Sep. 3.
4. Wednesday, 09/03:
Section 1.3: Countable and uncountable sets.
See also the notes on the subject.
The notes were last updated
Wednesday, Sep. 3.
Discussion on Homework 1.
5. Friday, 09/05:
Section 1.3: Countable sets (conclusion).
Sections 2.1-2: Axioms of real numbers: algebraic operations. Rational and irrational numbers.
6. Monday, 09/08:
Sections 2.1-2: Axioms of real numbers: order. Arithmetic and geometric mean.
Absolute value.
Section 2.3: Completeness properties. Supremum and infimum.
7. Wednesday, 09/10:
Section 2.3: Completeness properties. Supremum and infimum.
Section 2.4: Consequences of completeness. Archimedean Property.
Discussion on Homework 2.
8. Friday, 09/12:
Section 2.4: Consequences of completeness. Square roots. Density theorems.
Section 2.5: Intervals.
9. Monday, 09/15:
Section 2.5: Intervals.
Section 3.1: Sequences and their limits.
10. Wednesday, 09/17:
Section 3.1: Sequences and their limits.
Section 3.2: Limit theorems.
Discussion on Homework 3.
11. Friday, 09/19:
Section 3.2: Limit theorems (continued).
Section 3.3: Monotone sequences.
12. Monday, 09/22:
Section 3.3: Monotone sequences. The number e.
Section 3.4: Subsequences.
13. Wednesday, 09/24:
Section 3.4: Bolzano-Weierstrass Theorem.
Section 3.5: The Cauchy criterion for convergence.
Discussion on Homework 4.
14. Friday, 09/26:
Section 3.5: The Cauchy criterion for convergence. Contractive sequences.
Section 3.7: Infinite series.
15. Monday, 09/29:
Section 3.7: Infinite series.
Section 2.5: Binary and decimal expansions.
Section 4.1: Limits of functions.
16. Wednesday, 10/01:
Section 4.1: Limits of functions.
Discussion on Homework 5.
17. Friday, 10/03:
Section 4.1: Limits of functions.
Section 4.2: Limit theorems.
18. Monday, 10/06:
Discussion on the midterm.
19. Wednesday, 10/08:
The midterm.
20. Friday, 10/10:
Brief review of the midterm.
Section 5.1: Continuity of functions. Definition and examples.
21. Monday, 10/13:
Section 5.1: Continuity of functions. Definition and examples.
Section 5.2: Properties of continuous functions.
Section 5.3: Functions, continuous on intervals.
22. Wednesday, 10/15:
Section 5.3: Functions, continuous on intervals.
Discussion on Homework 6.
23. Friday, 10/17:
Section 5.3: Functions, continuous on intervals. Preservation of intervals.
Section 5.4: Uniform continuity.
24. Monday, 10/20:
Section 5.4: Uniform continuity. Extensions of functions.
25. Wednesday, 10/22:
Section 4.3: One-sided limits (a quick review).
Section 5.6: Monotone functions.
Discussion on Homework 7.
26. Friday, 10/24:
Section 5.6: Monotone functions and their discontinuities. Inverse functions.
27. Monday, 10/27:
Section 6.1: The derivative. Differentiation versus continuity. Rules of differentiation.
28. Wednesday, 10/29:
Section 6.1: The Chain Rule of differentiation. Differentiation of inverse functions.
Discussion on Homework 8.
29. Friday, 10/31:
Section 6.1: Differentiation of inverse functions.
Section 6.2: Mean Value Theorem.
30. Monday, 11/03:
Section 6.2: Mean Value Theorem. Applications.
31. Wednesday, 11/05:
Section 6.2: More on Mean Value Theorem. Intermediate Value Theorem for Derivatives.
Discussion on Homework 9.
32. Friday, 11/07:
Section 7.1: Riemann integral. Definition and examples.
33. Monday, 11/10:
Section 7.1: Riemann integral. Basic properties
Section 7.2: More on Riemann integrals.
34. Wednesday, 11/12:
Section 7.2: More on Riemann integrals. Step functions. Continuous functions are Riemann integrable.
Discussion on Homework 10.
35. Friday, 11/14:
Section 7.2: More on Riemann integrals. Monotone functions are Riemann integrable.
Additivity of Riemann integrals.
36. Monday, 11/17:
Discussion on Midterm 2.
37. Wednesday, 11/19:
Midterm 2.
38. Friday, 11/21:
Brief review of the midterm.
Section 7.2: More on Riemann integrals. Additivity of Riemann integrals (continued).
Exercise 11.
39. Monday, 12/01:
Section 7.2: Exercise 11 (completed).
Section 7.3: Fundamental Theorem of Calculus.
Discussion on Homework 11.
40. Wednesday, 12/03:
Section 7.3: Fundamental Theorem of Calculus.
41. Friday, 12/05:
Section 7.3: Null sets. Lebesgue criterion of integrability.
42. Monday, 12/08:
Section 6.4: Taylor series.
Discussion on Homework 12.
43. Wednesday, 12/10:
Review for the final.
Coming up:
44. Thursday, 12/11:
Review for the final (12-12:50, AH 145).