I PRELIMINARIES

1. Essentials of Logic and Set Theory
   • Essentials of Set Theory
   • Essentials of Logic
   • Mathematical Proofs
   • Mathematical Induction
   Review Exercises
2. Algebra
   • The Real Numbers
   • Integer Powers
   • Rules of Algebra
   • Fractions
   • Fractional Powers
   • Inequalities
   • Intervals and Absolute Values
   • Sign Diagrams
   • Summation Notation
   • Rules for Sums
   • Newton's Binomial Formula
   • Double Sums
   Review Exercises
3. Solving Equations
   • Solving Equations
   • Equations and Their Parameters
   • Quadratic Equations
   • Some Nonlinear Equations
   • Using Implication Arrows
   • Two Linear Equations in Two Unknowns
   Review Exercises
4. Functions of One Variable
   • Introduction
   • Definitions
   • Graphs of Functions
   • Linear Functions
   • Linear Models
   • Quadratic Functions
   • Polynomials
   • Power Functions
   • Exponential Functions
   • Logarithmic Functions
   Review Exercises
5. Properties of Functions
   • Shifting Graphs
   • New Functions From Old
   • Inverse Functions
   • Graphs of Equations
   • Distance in The Plane
   • General Functions
   Review Exercises

II SINGLE-VARIABLE CALCULUS

• Slopes of Curves
• Tangents and Derivatives
• Increasing and Decreasing Functions
• Economic Applications
• A Brief Introduction to Limits
• Simple Rules for Differentiation
• Sums, Products, and Quotients
• The Chain Rule
• Higher-Order Derivatives
• Exponential Functions
• Logarithmic Functions

• Implicit Differentiation
• Economic Examples
• The Inverse Function Theorem
• Linear Approximations
• Polynomial Approximations
• Taylor's Formula
• Elasticities
• Continuity
• More on Limits
• The Intermediate Value Theorem
• Infinite Sequences
• L'H�pital's Rule Review Exercises

• Intuition
• Definitions
• General Properties
• First Derivative Tests
• Second Derivative Tests
• Inflection Points

• Extreme Points
• Simple Tests for Extreme Points
• Economic Examples
• The Extreme and Mean Value Theorems
• Further Economic Examples
• Local Extreme Points

• Indefinite Integrals
• Area and Definite Integrals
• Properties of Definite Integrals
• Economic Applications
• Integration by Parts
• Integration by Substitution
• Infinite Intervals of Integration

• Interest Periods and Effective Rates
• Continuous Compounding
• Present Value
• Geometric Series
• Total Present Value
• Mortgage Repayments
• Internal Rate of Return
• A Glimpse at Difference Equations
• Essentials of Differential Equations
• Separable and Linear Differential Equations

• Matrices and Vectors
• Systems of Linear Equations
• Matrix Addition
• Algebra of Vectors
• Matrix Multiplication
• Rules for Matrix Multiplication
• The Transpose
• Gaussian Elimination
• Geometric Interpretation of Vectors
• Lines and Planes

• Determinants of Order 2
• Determinants of Order 3
• Determinants in General
• Basic Rules for Determinants
• Expansion by Cofactors
• The Inverse of a Matrix
• A General Formula for The Inverse
• Cramer's Rule
• The Leontief Mode
• Eigenvalues and Eigenvectors
• Diagonalization
• Quadratic Forms

IV MULTI-VARIABLE CALCULUS

• Functions of Two Variables
• Partial Derivatives with Two Variables
• Geometric Representation
• Surfaces and Distance
• Functions of More Variables
• Partial Derivatives with More Variables
• Convex Sets
• Concave and Convex Functions
• Economic Applications
• Partial Elasticities

• A Simple Chain Rule
• Chain Rules for Many Variables
• Implicit Differentiation Along A Level Curve
• Level Surfaces
• Elasticity of Substitution
• Homogeneous Functions of Two Variables
• Homogeneous and Homothetic Functions
• Linear Approximations
• Differentials
• Systems of Equations
• Differentiating Systems of Equations

• Double Integrals Over Finite Rectangles
• Infinite Rectangles of Integration
• Discontinuous Integrands and Other Extensions
• Integration Over Many Variables

V MULTI-VARIABLE OPTIMIZATION

• Two Choice Variables: Necessary Conditions
• Two Choice Variables: Sufficient Conditions
• Local Extreme Points
• Linear Models with Quadratic Objectives
• The Extreme Value Theorem
• Functions of More Variables
• Comparative Statics and the Envelope Theorem

• The Lagrange Multiplier Method
• Interpreting the Lagrange Multiplier
• Multiple Solution Candidates
• Why Does the Lagrange Multiplier Method Work?
• Sufficient Conditions
• Additional Variables and Constraints
• Comparative Statics

• A Graphical Approach
• Introduction to Duality Theory
• The Duality Theorem
• A General Economic Interpretation
• Complementary Slackness

• Two Variables and One Constraint
• Many Variables and Inequality Constraints
• Nonnegativity Constraints