MAT157Y Analysis I

1. Preliminaries.
   • Proofs by contradiction, proofs by induction
   • General comments on sets and functions
   • General comments on logic, mathematical notation
   • Axioms of real numbers

2. Foundations
   • Functions of a real variable
   • Graphing functions
   • Limits
   • Continuity
   • Rolle's theorem, intermediate value theorem
   • Least upper bounds

3. Differentiation
   • Definition of derivative
   • Differentiation of functions
   • Rules of computation of derivatives
   • Derivatives of inverse functions

4. Integration
   • Riemannn sums
   • Definition of integral
   • Fundamental theorem of calculus
   • Defining triginomoetric functions
   • Defining exponential and logarithmic functions
   • Techniques of integration

5. Infinite sequences and infinite series
   • Infinite sequences of numbers
   • Infinite series of functions
   • Taylor approximations
   • Uniform convergence
   • Power series