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Math 8: Problem Solving in Calculus

Course Information:

• Instructor: Padraic Bartlett.
• Class Time/Location: MWF 2-3, Sloan 151.
• Office Hours: 3-4 MWF, Sloan 151; other, more individual sessions are also possible! Just email me.
• Email: padraic at caltech dot edu.

Course Description:

Math 8 is a support course for Math 1a; where Math 1a is (by design!) a highly abstract course, Math 8 is a much more hands-on and example-oriented series of lectures. Specifically, Math 8's lectures are designed to serve as a "how-to" guide on how to craft clear, well-written mathematical proofs; students who are new to the concept of formal proofs or who are otherwise concerned with meeting the levels of rigor in Math 1a are encouraged to attend.

Course Policies:

• Grades: Math 8 is a pass-fail 3-unit course, graded entirely on attendance. Registered students are required to attend at least 3/4 of the lectures to pass the course -- effectively, this means you're allowed to miss at most 7 classes over the quarter. Absences are excused only if they are accompanied by a note from the Deans or from the Student Health Center; make-up work for missed classes is not available.
• Unregistered Students: Unregistered students are always welcome to attend class and office hours as they desire! I am always willing to teach more people mathematics.
• Textbooks: This course has none. If you're looking for interesting mathematical texts on proofs and calculus, however, the following three texts are remarkably enjoyable reads:
• Calculus, by Caltech's own Tom Apostol; one of the few well-written rigorous calculus texts in existence. Your Ma1a textbook and best friend for the next two quarters.
• Mathematical Thinking: Problem Solving and Proofs, by John P. D'Angelo and Doug West. A fantastic introduction to the world of proof-based mathematics; well-written and full of interesting problems.
• Calculus, by Michael Spivak. One of the few other calculus texts worth its salt; contains some of the best exposition in mathematics that I've read. Also, a ton of excellent problems.

Course Notes:

• Week 1:
• Monday: Propositional logic, deductive proofs, and contradictive proofs.
• Wednesday: More on proofs by contradiction; proofs by induction.
• Friday: Proofs by contrapositive; constructing finite fields.
• Week 2:
• Monday: The size of infinity.
• Wednesday: Finding two infinite sets of different sizes.
• Friday: A proper introduction to sequences.
• Week 3:
• Monday: More sequence tools; series.
• Wednesday: Various theorems for manipulating series.
• Friday: Absolute and conditional convergence: also, the constant e.
• Week 4:
• Monday: Limits and continuity.
• Wednesday: Various discontinuous functions: (censored), (uncensored).
• Friday: One-sided limits, limits at infinity, the IVT, open sets, and closed sets.
• Week 5:
• Monday: An introduction to the derivative.
• Wednesday: Extrema and derivatives; C¹ functions.
• Friday: An everywhere-continuous, nowhere-differentiable function.
• Midterm review notes:
• Midterm review notes!
• Week 6:
• Monday: An introduction to the integral.
• Wednesday: The integral of x^p.
• Friday: How to integrate discontinuous functions.
• Week 7:
• Monday: The two fundamental theorems of calculus.
• Wednesday: Integration by parts; integration by substitution.
• Friday: Polar functions, parametric functions, and their integrals.
• Week 8:
• Monday: A curious substitution.
• Wednesday: Logarithmic differentiation, L'Hôpital's rule, and the exp function.
• Friday: The integral test.
• Week 9:
• Monday: Taylor series, e, and the binomial formula.
• Wednesday: A Taylor series that never converges.
• Friday: (thanksgiving! => no lecture. Instead: a cat!)
• Week 10:
• Monday: Roots of unity.
• Wednesday: (illness => no lecture. Perhaps some cute ducks instead?)
• Friday: Partial fractions; complex power series.
• Final review notes:
• Definitions!
• Examples!

email: padraic at math dot ucsb dot edu | mailing address: south hall, room 6516, ucsb