Fall 2013
  • MATH 409-503: Advanced Calculus I
    •

    Time and venue:  TR 11:10 a.m.-12:25 p.m., BLOC 161

    First day hand-out

    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.

    Sample problems for the final exam (Solutions)

    Sample problems for Test 2 (Solutions)

    Sample problems for Test 1 (Solutions)


    Homework assignments ##1-11


    Challenges ##1-17

    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