Fall 2013
MATH 409-503: Advanced Calculus I
Time and venue: TR 11:10 a.m.-12:25 p.m., BLOC 161
Office hours (MILN 004):
Tuesday, 1:00-3:00 p.m.
Wednesday, 1:30-2:30 p.m.
by appointment
Help sessions (BLOC 111):
Monday - Thursday, 7:00-9:30 p.m.
Additional office hours (MILN 004):
Wednesday, December 4, 12:30-2:00 p.m.
Thursday, December 5, 12:00-2:00 p.m.
Friday, December 6, 10:00 a.m.-12:00 p.m.
Final exam: Friday, December 6, 3:00-5:00 p.m., BLOC 161
Rules for the test: no books, no lecture notes. Bring paper and a stapler.
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
Wade's book: Chapter 1, Appendix A
Lecture 1: Axioms of an ordered field.
Wade 1.1-1.2, Appendix A
Lecture 2: Properties of an ordered field. Absolute value. Supremum and infimum.
Wade 1.2-1.3, Appendix A
Lecture 3: Metric spaces. Completeness axiom. Existence of square roots.
Wade 1.3, Appendix A
Lecture 4: Intervals. Principle of mathematical induction. Inverse function.
Wade 1.4-1.5, Appendix A
Lecture 5: Binomial formula. Inverse function and inverse images. Countable and uncountable sets.
Wade 1.4-1.6
Part II: Limits and continuity
Limits of sequences
Bolzano-Weierstrass theorem
Cauchy sequences
Limits of functions
Continuity, uniform continuity
Wade's book: Chapters 2-3
Lecture 6: Limits of sequences. Limit theorems.
Wade 2.1-2.2
Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem.
Wade 2.3
Lecture 8: Monotone sequences (continued). Cauchy sequences. Limit points.
Wade 2.3-2.5
Lecture 9: Limit supremum and infimum. Limits of functions.
Wade 2.5, 3.1-3.2
Lecture 10: Continuity. Properties of continuous functions.
Wade 3.3
Lecture 11: More on continuous functions.
Wade 3.3
Lecture 12: Uniform continuity. Exponential functions.
Wade 3.4
Lecture 13: Review for Test 1.
Wade 1.1-1.6, 2.1-2.5, 3.1-3.4, Appendix A
Part III: 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
Wade's book: Chapters 4-5
Lecture 14: The derivative. Differentiability theorems.
Wade 4.1-4.2
Lecture 15: Derivatives of elementary functions. Derivative of the inverse function.
Wade 4.2, 4.5
Lecture 16: Mean value theorem. Taylor's formula.
Wade 4.3-4.4
Lecture 17: Applications of the mean value theorem. l'Hôpital's rule.
Wade 4.3-4.4
Lecture 18: Darboux sums. The Riemann integral.
Wade 5.1-5.2
Lecture 19: Riemann sums. Properties of integrals.
Wade 5.1-5.2
Lecture 20: The fundamental theorem of calculus. Change of the variable in an integral.
Wade 5.3
Lecture 21: Review for Test 2.
Wade 4.1-4.5, 5.1-5.3
Lecture 22: Improper Riemann integrals.
Wade 5.4
Part IV: Infinite series
Convergence of series
Absolute convergence
Alternating series
Tests of convergence
Wade's book: Chapter 6
Lecture 23: Convergence of infinite series.
Wade 6.1-6.2
Lecture 24: Alternating series. Absolute convergence of series.
Wade 6.3-6.4
Lecture 25: Review for the final exam.
Wade 1.1-1.6, Appendix A, 2.1-2.5, 3.1-3.4, 4.1-4.5, 5.1-5.4, 6.1-6.4