Spring 2020
- MATH 409-501: Advanced Calculus I
Time and venue: MWF 11:30 a.m.-12:20 p.m., ZOOM meeting
Office hours (ZOOM meeting):
- MWF 10:15-11:15 a.m.
- MWF 1:00-2:00 p.m.
- by appointment
Help sessions (ZOOM meeting):
- Tuesday 7:30-9:00 p.m.
- Wednesday 7:30-9:00 p.m.
Final exam: Tuesday, May 5, 10:00 a.m-1:00 p.m.
Course outline:
Part I: Axiomatic model of the real numbers
- Axioms of an ordered field
- Completeness axiom
- Principle of mathematical induction
- Countable and uncountable sets
Thomson/Bruckner/Bruckner: Chapter 1, Appendix A, Section 2.3
Lecture 1: Axioms of a field.
Lecture 2: Properties of ordered fields.
Lecture 3: Supremum and infimum. Completeness axiom.
Lecture 4: Archimedean principle. Mathematical induction. Binomial formula.
Lecture 5: Intervals. Density of the rational numbers. Existence of square roots.
Lecture 6: Functions. Countable and uncountable sets.
Lecture 7a: Absolute value. Metric spaces.
Part II: Sequences and infinite sums
- Limits of sequences
- Bolzano-Weierstrass theorem
- Cauchy sequences
- Convergence of series
- Tests for convergence
- Absolute convergence
Thomson/Bruckner/Bruckner: Chapters 2-3
Lecture 7b: Limit of a sequence.
Lecture 8: Properties of limits. Divergent sequences.
Lecture 9: Algebra of limits.
Lecture 10: Monotonic sequences.
Lecture 11: More examples of limits.
Lecture 12: Bolzano-Weierstrass theorem. Cauchy sequences.
Lecture 13: Limit points. Upper and lower limits.
Lecture 14: Convergence of infinite series.
- T/B/B 3.1-3.2, 3.4, 3.5.1-3.5.2, 3.6.1-3.6.2
Lecture 15: Tests for convergence.
Lecture 16: Tests for convergence (continued). Alternating series.
Lecture 17: Summation by parts. Absolute convergence of series.
- T/B/B 3.2, 3.5.3, 3.6.13-3.6.14, 3.7
Lecture 18: Review for Test 1.
- T/B/B 1.1-1.10, 2.1-2.13, 3.1-3.2, 3.4-3.7
Lecture 19a: Review of Test 1.
- T/B/B 1.1-1.10, 2.1-2.13, 3.1-3.2, 3.4-3.7
Part III: Continuity
- Topology of the real line
- Limits of functions
- Continuouos functions
- Uniform continuity
Thomson/Bruckner/Bruckner: Chapters 4-5
Lecture 19b: Topology of the real line: classification of points.
Lecture 20: Topology of the real line: open and closed sets.
Lecture 21: Open and closed sets (continued). Compact sets.
Lecture 22: Limits of functions.
Lecture 23: Limits of functions (continued).
Lecture 24: Continuity. Properties of continuous functions.
Lecture 25: More on continuous functions. Points of discontinuity.
Lecture 26: Monotonic functions. Exponential function. Uniform continuity.
Part IV: Differential and integral calculus
- Differentiability, properties of the derivative
- The mean value theorem
- Taylor's theorem
- Riemann sums, the Riemann integral
- The fundamental theorem of calculus
Thomson/Bruckner/Bruckner: Chapter 7-8
Lecture 27: The derivative. Differentiability theorems.
Lecture 28: Differentiability theorems (continued). Derivatives of elementary functions.
Lecture 29: Mean value theorem.
Lecture 30: L'Hôpital's rule. Taylor's formula.
Lecture 31: Review for Test 2.
- T/B/B 4.1-4.7, 5.1-5.2, 5.4-5.10, 7.1-7.7, 7.9, 7.11-7.13
Lecture 32: Riemann integral. Riemann sums and Darboux sums.
Lecture 33: Properties of the integral.
Lecture 34: Fundamental theorem of calculus. Indefinite integral.
Lecture 35: Integration by parts. Integration by substitution.
Lecture 36: Improper Riemann integrals.
Lecture 37: Review for the final exam.
- T/B/B 1.1-1.10, 2.1-2.13, 3.1-3.2, 3.4-3.7, 4.1-4.7, 5.1-5.2, 5.4-5.9, 7.1-7.7, 7.9, 7.11-7.12, 8.1-8.8