Course level: Undergraduate mathematical foundations with graduate extensions for economics and statistics
Duration: 14 weeks
Prerequisites: Algebra, functions, and elementary trigonometry
Course description
This course develops single- and multivariable calculus with rigorous foundations and applications to economics, statistics, and dynamic modeling. It emphasizes definitions, derivations, mathematical reasoning, optimization, and computational verification.
Learning outcomes
By the end of the course, students should be able to:
- Prove basic limit and continuity statements using epsilon-delta arguments.
- Analyze sequences and series using standard convergence criteria.
- Differentiate and integrate single-variable functions and interpret derivatives economically.
- Use Taylor approximations, convexity, and optimization conditions.
- Analyze multivariate functions using gradients, Jacobians, and Hessians.
- Solve equality- and inequality-constrained optimization problems.
- Apply implicit and inverse function theorems to comparative statics.
- Evaluate multiple integrals and use Jacobians in probability transformations.
- Analyze continuous- and discrete-time dynamic systems.
- Verify analytical results using symbolic and numerical computation.
Weekly schedule
| Week | Topic | Main output |
|---|---|---|
| 1 | Limits and continuity | Epsilon-delta proof |
| 2 | Sequences and series | Convergence analysis |
| 3 | Differentiation and mean value theorems | Derivative derivations |
| 4 | Taylor approximation, convexity, optimization | Single-variable optimization |
| 5 | Integration and fundamental theorem | Integration assignment |
| 6 | Multivariable limits and continuity | Surface and level-set analysis |
| 7 | Partial derivatives, differentials, chain rule | Gradient/Jacobian exercises |
| 8 | Hessians and unconstrained optimization | Midterm examination |
| 9 | Lagrange multipliers | Equality-constrained application |
| 10 | KKT conditions | Inequality-constrained application |
| 11 | Implicit and inverse function theorems | Comparative statics |
| 12 | Multiple integrals and Jacobians | Probability integration |
| 13 | Differential equations | Continuous-time dynamics |
| 14 | Difference equations | Discrete-time stability project |
Assessment
- Problem sets and mathematical proofs: 30%
- Computational assignments: 15%
- Midterm examination: 20%
- Final examination: 25%
- Applied mathematical modeling project: 10%
Expectations
Solutions must show mathematical reasoning, not only final answers. State domains, regularity assumptions, and constraint qualifications. Computational results must be reproducible and checked against analytical arguments where possible.
Core references
- Apostol, Calculus, Volumes 1 and 2.
- Spivak, Calculus.
- Simon and Blume, Mathematics for Economists.
- Chiang and Wainwright, Fundamental Methods of Mathematical Economics.
- Rudin, Principles of Mathematical Analysis (selected advanced topics).