1 Essentials of Logic and Set Theory
1.1 Essentials of Set Theory
1.2 Essentials of Logic
1.3 Mathematical Proofs
1.4 Mathematical Induction
Review Exercises
2 Algebra
2.1 The Real Numbers
2.2 Integer Powers
2.3 Rules of Algebra
2.4 Fractions
2.5 Fractional Powers
2.6 Inequalities
2.7 Intervals and Absolute Values
2.8 Sign Diagrams
2.9 Summation Notation
2.10 Rules for Sums
2.11 Newton’s Binomial Formula
2.12 Double Sums
Review Exercises
3 Solving Equations
3.1 Solving Equations
3.2 Equations and Their Parameters
3.3 Quadratic Equations
3.4 Some Nonlinear Equations
3.5 Using Implication Arrows
3.6 Two Linear Equations in Two Unknowns
Review Exercises
4 Functions of One Variable
4.1 Introduction
4.2 Definitions
4.3 Graphs of Functions
4.4 Linear Functions
4.5 Linear Models
4.6 Quadratic Functions
4.7 Polynomials
4.8 Power Functions
4.9 Exponential Functions
4.10 Logarithmic Functions
Review Exercises
5 Properties of Functions
5.1 Shifting Graphs
5.2 New Functions From Old
5.3 Inverse Functions
5.4 Graphs of Equations
5.5 Distance in The Plane
5.6 General Functions
Review Exercises
II SINGLE-VARIABLE CALCULUS
6 Differentiation
6.1 Slopes of Curves
6.2 Tangents and Derivatives
6.3 Increasing and Decreasing Functions
6.4 Economic Applications
6.5 A Brief Introduction to Limits
6.6 Simple Rules for Differentiation
6.7 Sums, Products, and Quotients
6.8 The Chain Rule
6.9 Higher-Order Derivatives
6.10 Exponential Functions
6.11 Logarithmic Functions
Review Exercises
7 Derivatives in Use
7.1 Implicit Differentiation
7.2 Economic Examples
7.3 The Inverse Function Theorem
7.4 Linear Approximations
7.5 Polynomial Approximations
7.6 Taylor’s Formula
7.7 Elasticities
7.8 Continuity
7.9 More on Limits
7.10 The Intermediate Value Theorem
7.11 Infinite Sequences
7.12 L’Hôpital’s Rule
Review Exercises
8 Concave and Convex Functions
8.1 Intuition
8.2 Definitions
8.3 General Properties
8.4 First Derivative Tests
8.5 Second Derivative Tests
8.6 Inflection Points
Review Exercises
9 Optimization
9.1 Extreme Points
9.2 Simple Tests for Extreme Points
9.3 Economic Examples
9.4 The Extreme and Mean Value Theorems
9.5 Further Economic Examples
9.6 Local Extreme Points
Review Exercises
10 Integration
10.1 Indefinite Integrals
10.2 Area and Definite Integrals
10.3 Properties of Definite Integrals
10.4 Economic Applications
10.5 Integration by Parts
10.6 Integration by Substitution
10.7 Infinite Intervals of Integration
Review Exercises
11 Topics in Finance and Dynamics
11.1 Interest Periods and Effective Rates
11.2 Continuous Compounding
11.3 Present Value
11.4 Geometric Series
11.5 Total Present Value
11.6 Mortgage Repayments
11.7 Internal Rate of Return
11.8 A Glimpse at Difference Equations
11.9 Essentials of Differential Equations
11.10 Separable and Linear Differential Equations
Review Exercises
III MULTI-VARIABLE ALGEBRA
12 Matrix Algebra
12.1 Matrices and Vectors
12.2 Systems of Linear Equations
12.3 Matrix Addition
12.4 Algebra of Vectors
12.5 Matrix Multiplication
12.6 Rules for Matrix Multiplication
12.7 The Transpose
12.8 Gaussian Elimination
12.9 Geometric Interpretation of Vectors
12.10 Lines and Planes
Review Exercises
13 Determinants, Inverses, and Quadratic Forms
13.1 Determinants of Order 2
13.2 Determinants of Order 3
13.3 Determinants in General
13.4 Basic Rules for Determinants
13.5 Expansion by Cofactors
13.6 The Inverse of a Matrix
13.7 A General Formula for The Inverse
13.8 Cramer’s Rule
13.9 The Leontief Model
13.10 Eigenvalues and Eigenvectors
13.11 Diagonalization
13.12 Quadratic Forms
Review Exercises
IV MULTI-VARIABLE CALCULUS
14 Multivariable Functions
14.1 Functions of Two Variables
14.2 Partial Derivatives with Two Variables
14.3 Geometric Representation
14.4 Surfaces and Distance
14.5 Functions of More Variables
14.6 Partial Derivatives with More Variables
14.7 Convex Sets
14.8 Concave and Convex Functions
14.9 Economic Applications
14.10 Partial Elasticities
Review Exercises
15 Partial Derivatives in Use
15.1 A Simple Chain Rule
15.2 Chain Rules for Many Variables
15.3 Implicit Differentiation Along A Level Curve
15.4 Level Surfaces
15.5 Elasticity of Substitution
15.6 Homogeneous Functions of Two Variables
15.7 Homogeneous and Homothetic Functions
15.8 Linear Approximations
15.9 Differentials
15.10 Systems of Equations
15.11 Differentiating Systems of Equations
Review Exercises
16 Multiple Integrals
16.1 Double Integrals Over Finite Rectangles
16.2 Infinite Rectangles of Integration
16.3 Discontinuous Integrands and Other Extensions
16.4 Integration Over Many Variables
V MULTI-VARIABLE OPTIMIZATION
17 Unconstrained Optimization
17.1 Two Choice Variables: Necessary Conditions
17.2 Two Choice Variables: Sufficient Conditions
17.3 Local Extreme Points
17.4 Linear Models with Quadratic Objectives
17.5 The Extreme Value Theorem
17.6 Functions of More Variables
17.7 Comparative Statics and the Envelope Theorem
Review Exercises
18 Equality Constraints
18.1 The Lagrange Multiplier Method
18.2 Interpreting the Lagrange Multiplier
18.3 Multiple Solution Candidates
18.4 Why Does the Lagrange Multiplier Method Work?
18.5 Sufficient Conditions
18.6 Additional Variables and Constraints
18.7 Comparative Statics
Review Exercises
19 Linear Programming
19.1 A Graphical Approach
19.2 Introduction to Duality Theory
19.3 The Duality Theorem
19.4 A General Economic Interpretation
19.5 Complementary Slackness
Review Exercises
20 Nonlinear Programming
20.1 Two Variables and One Constraint
20.2 Many Variables and Inequality Constraints
20.3 Nonnegativity Constraints
Review Exercises
Appendix: Geometry
Solutions to the Exercises
Index
