Second term

• Lecture 01 slides
  • Sigma notation.
  • Infima and suprema of sets and functions.
  • The definition of the definite integral.
• Lecture 02 slides
  • More on definite integrals.
  • Riemann sums.
  • Elementary properties of definite integrals.
  • Antiderivatives.
  • Introduction to the FTC.
• Lecture 03 slides
  • Proof of the FTC.
  • Areas.
  • Substitution (not on slides).
• Lecture 04 slides
  • More on substitution.
  • Volumes.
  • Integration by parts.
• Lecture 05 slides
  • More on integration by parts.
  • Integrating certain combinations of trig functions.
  • Simple partial fractions decompositions (not on slides).
  • Trigonometric substitutions (not on slides).
• Lecture 06 slides
  • A little more on trig substitutions.
  • Definition of sequences and examples of ways of defining them.
  • Increasing, non-increasing, monotone, bounded, unbounded...
  • Definition of sequence convergence.
  • Two big theorems about sequences.
• Lecture 07 slides
  • More on the Monotone Sequence Theorem.
  • The "Big Theorem".
  • Improper integrals.
  • The Basic Comparison Test.
  • The Limit Comparison Test (not on slides yet).
• Lecture 08 slides
  • Typos fixed!
  • Reminder of results about improper integrals.
  • Definition of series, and convergence of series.
  • Basic theorems about series.
  • Necessary condition test, geometric series.
  • The Basic Comparison Test and Limit Comparison Test for series.
• Lecture 09 slides
  • Reminder of results about series from last class.
  • Ratio test.
  • Absolutely convergent and conditionally convergent series.
  • The Alternating Series test.
  • Estimating using the Alternating Series test.
• Lecture 10 slides
  • Reminder of the discussion at the end of last class.
  • Definition of power series.
  • Radius and interval of convergence.
  • Absolutely convergent power series act like polynomials.
  • Taylor polynomials.
• Lecture 11 slides
  • Reminder of material from last class.
  • Definition of the Taylor series of a function.
  • When does a function equal its Taylor series?
  • Definition of C^n and analytic functions.
  • The Lagrange Remainder Theorem, and using it.
• Lecture 12 slides
  • Reminder of the six major Taylor series we know so far.
  • Using Taylor series to help compute limits.
  • Using Taylor series to help sum series.
  • Taylor series can help with integration too.