TGM AL MAMUN
Teaching/Calculus
Course 03 · Undergraduate, with graduate extensions

Calculus

Single- and multivariable calculus with rigorous foundations, optimization and dynamics for economics and statistics.

LIMITS · SERIES · DIFFERENTIATION · INTEGRATION · MULTIVARIABLE CALCULUS

Lecture notesSyllabus
01 / OVERVIEW

About this course

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.

This course develops calculus from definitions and mathematical arguments through applications in economics, statistics, and dynamic modeling. Each lecture includes objectives, mathematical development, worked examples, problem sets, a computational exercise, and readings.

What you will learn

By the end of the course, students should be able to:

  1. Prove basic limit and continuity statements using epsilon-delta arguments.
  2. Analyze sequences and series using standard convergence criteria.
  3. Differentiate and integrate single-variable functions and interpret derivatives economically.
  4. Use Taylor approximations, convexity, and optimization conditions.
  5. Analyze multivariate functions using gradients, Jacobians, and Hessians.
  6. Solve equality- and inequality-constrained optimization problems.
  7. Apply implicit and inverse function theorems to comparative statics.
  8. Evaluate multiple integrals and use Jacobians in probability transformations.
  9. Analyze continuous- and discrete-time dynamic systems.
  10. Verify analytical results using symbolic and numerical computation.

How the course is organized

Each lecture session has a full set of written notes: learning objectives, formal development with derivations, worked examples, a problem set, a computational lab and readings. Problem sets with solution guidance and examination specifications are under Assignments and Exams.

Part I · Single-variable analysis Lectures 1–5

  1. Limits, Continuity, and the Logic of Analysis
  2. Sequences, Convergence, and Infinite Series
  3. Differentiation, Mean Value Theorems, and Comparative Rates
  4. Taylor Approximation, Convexity, and Single-Variable Optimization
  5. Integration, Riemann Sums, and the Fundamental Theorem

Part II · Multivariable calculus Lectures 6–8

  1. Multivariable Functions, Limits, and Continuity
  2. Partial Derivatives, Total Differentials, and the Chain Rule
  3. Hessians and Unconstrained Multivariable Optimization

Part III · Optimization and implicit functions Lectures 9–11

  1. Equality Constraints and Lagrange Multipliers
  2. Inequality Constraints and Karush–Kuhn–Tucker Conditions
  3. Implicit and Inverse Function Theorems

Part IV · Multiple integration Lecture 12

  1. Multiple Integrals, Change of Variables, and Probability

Part V · Dynamic systems Lectures 13–14

  1. Differential Equations and Continuous-Time Dynamics
  2. Difference Equations and Discrete-Time Dynamics

Assessment

Problem sets and proofs 30%; computational assignments 15%; midterm 20%; final 25%; applied mathematical modeling project 10%.

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).
02 / MATERIALS

Course materials

Everything needed to follow the course, organized as on an open courseware site.

01

Syllabus

Description, prerequisites, outcomes, assessment and policies.
02

Calendar

Session-by-session schedule with key activities.
03

Lecture Notes

Full written notes for every lecture session.
04

Readings

Required and further readings, organized by session.
05

Assignments

Problem sets with solution guidance.
06

Exams

Midterm and final examination specifications.