Ch01: Essentials of Logic and Set Theory

1.1 Essentials of set theory

1.2 Some aspects of logic

1.3 Mathematical proofs

1.4 Mathematical induction

Ch02: 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 Summation

2.9 Rules for sums

2. 10 Newton’s binomial formula

2. 11 Double sums

Ch03: Solving Equations

3.1 Solving equations

3.2 Equations and their parameters

3.3 Quadratic equations

3.4 Nonlinear equations

3.5 Using implication arrows

3.6 Two linear equations in two unknowns

Ch04: Functions of One Variable

4.1 Introduction

4.2 Basic 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

Ch05: 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

Ch06: Differentiation

6.1 Slopes of curves

6.2 Tangents and derivatives

6.3 Increasing and decreasing functions

6.4 Rates of change

6.5 A dash of 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

Ch07: Derivatives in Use

7.1 Implicit differentiation

7.2 Economic examples

7.3 Differentiating the inverse

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 and Newton’s method

7. 11 Infinite sequences

7. 12 L'Hôpital's Rule

Ch08: Single-Variable Optimization

8.1 Extreme points

8.2 Simple tests for extreme points

8.3 Economic examples

8.4 The Extreme Value Theorem

8.5 Further economic examples

8.6 Local extreme points

8.7 Inflection points

Ch09: Integration

9.1 Indefinite integrals

9.2 Area and definite integrals

9.3 Properties of definite integrals

9.4 Economic applications

9.5 Integration by parts

9.6 Integration by substitution

9.7 Infinite intervals of integration

9.8 A glimpse at differential equations

9.9 Separable and linear differential equations

Ch10: Topics in Financial Mathematics

10.1   Interest periods and effective rates

10.2   Continuous compounding

10.3   Present value

10.4   Geometric series

10.5   Total present value

10.6   Mortgage repayments

10.7   Internal rate of return

10.8   A glimpse at difference equations

Ch11: Functions of Many Variables

11.1   Functions of two variables

11.2   Partial derivatives with two variables

11.3   Geometric representation

11.4   Surfaces and distance

11.5   Functions of more variables

11.6   Partial derivatives with more variables

11.7   Economic applications

11.8   Partial elasticities

Ch12: Tools for Comparative Statics

12.1   A simple chain rule

12.2   Chain rules for many variables

12.3   Implicit differentiation along a level curve

12.4   More general cases

12.5   Elasticity of substitution

12.6   Homogeneous functions of two variables

12.7   Homogeneous and homothetic functions

12.8   Linear approximations

12.9   Differentials

12. 10 Systems of equations

12. 11 Differentiating systems of equations

Ch13: Multivariable Optimization

13.1   Two variables: necessary conditions

13.2   Two variables: sufficient conditions

13.3   Local extreme points

13.4   Linear models with quadratic objectives

13.5   The Extreme Value Theorem

13.6   The general case

13.7   Comparative statics and the envelope theorem

Ch14: Constrained Optimization

14.1   The Lagrange Multiplier Method

14.2   Interpreting the Lagrange multiplier

14.3   Multiple solution candidates

14.4   Why the Lagrange method works

14.5   Sufficient conditions

14.6   Additional variables and constraints

14.7   Comparative statics

14.8   Nonlinear programming: a simple case

14.9   Multiple inequality constraints

14. 10 Nonnegativity constraints

Ch15: Matrix and Vector Algebra

15.1   Systems of linear equations

15.2   Matrices and matrix operations

15.3   Matrix multiplication

15.4   Rules for matrix multiplication

15.5   The transpose

15.6   Gaussian elimination

15.7   Vectors

15.8   Geometric interpretation of vectors

15.9   Lines and planes

Ch16: Determinants and Inverse Matrices

16.1   Determinants of order 2

16.2   Determinants of order 3

16.3   Determinants in general

16.4   Basic rules for determinants

16.5   Expansion by cofactors

16.6   The inverse of a matrix

16.7   A general formula for the inverse

16.8   Cramer's Rule

16.9   The Leontief Model

Ch17: Linear Programming

17.1   A graphical approach

17.2   Introduction to Duality Theory

17.3   The Duality Theorem

17.4   A general economic interpretation

17.5   Complementary slackness

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