Essential Mathematics for Economic Analysis, 6th edition

Preface

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

6. Differentiation
   • 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

   Review Exercises

7. Derivatives in Use
   • 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'Hôpital's Rule Review Exercises

   Review Exercises

8. Concave and Convex Functions
   • Intuition
   • Definitions
   • General Properties
   • First Derivative Tests
   • Second Derivative Tests
   • Inflection Points

   Review Exercises

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

   Review Exercises

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

    Review Exercises

11. Topics in Finance and Dynamics
    • 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

    Review Exercises

III MULTI-VARIABLE ALGEBRA

12. Matrix Algebra
    • 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

    Review Exercises

13. Determinants, Inverses, and Quadratic Forms
    • 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

    Review Exercises

IV MULTI-VARIABLE CALCULUS

14. Multivariable Functions
    • 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

    Review Exercises

15. Partial Derivatives in Use
    • 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

    Review Exercises

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

    Review Exercises

V MULTI-VARIABLE OPTIMIZATION

17. Unconstrained 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

    Review Exercises

18. Equality Constraints
    • 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

    Review Exercises

19. Linear Programming
    • A Graphical Approach
    • Introduction to Duality Theory
    • The Duality Theorem
    • A General Economic Interpretation
    • Complementary Slackness

    Review Exercises

20. Nonlinear Programming
    • Two Variables and One Constraint
    • Many Variables and Inequality Constraints
    • Nonnegativity Constraints

    Review Exercises

Appendix

Solutions to the Exercises

Index

Publisher's Acknowledgments