Mathematical Analysis
- Derivative, Differentiation Rules, Algebraic Properties
- Mean Value Theorems and Their Applications
- Taylor's Theorem, Remainders, and Local Approximation
- The Riemann Integral, Upper and Lower Sums
- Sets of Measure Zero, Riemann-Lebesgue Integrability Criterion
- Fundamental Theorems of Calculus
- Improper Integrals and Their Convergence
- Midterm Exam & Review of Chapter 3
- Pointwise vs. Uniform Convergence
- Weierstrass M-test, Continuity and Uniform Limits
- Interchange of Limits with Integrals and Derivatives
- and the Sup-norm Metric
- The Weierstrass Approximation Theorem
- Equicontinuity and the Arzel-Ascoli Theorem
- The Contraction Mapping Principle & Fixed Point Theorems
- Applications to ODEs (Picard's Theorem) & Final Review