**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