Analysis and Linear Algebra: An Introduction for Economists

Preface to the Fifth Edition
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Elementary Basics
1.1 Elementary From Propositional Logic
1.2 Set Theory
1.3 Basic Arithmetic Operations
1.4 Equations
1.5 Trigonometry
1.6 Test
References
2 Functions
2.1 Definition and Representation of Functions
2.2 Some Elementary Functions
2.2.1 Linear Function
2.2.2 Quadratic Function
2.2.3 Integer Rational Functions or Polynomials
2.2.4 Power Function
2.2.5 Fractional Rational Functions
2.2.6 Hyperbolic Function
2.2.7 Root Function
2.2.8 Exponential Function
2.2.9 Logarithm Function
2.2.10 Trigonometric Functions
2.2.11 Sectionally Defined Functions
2.2.12 Some Economic Functions
2.3 The Inverse Function
2.4 Chained Functions
2.5 Properties of Functions
2.5.1 Limitation
2.5.2 Monotonicity
2.5.3 Symmetry
2.5.4 Injectivity, Surjectivity and Bijectivity
2.6 Limits
2.6.1 Convergence and Limits of Sequences and Series
2.6.2 The Limit Concept for Functions
2.6.3 Cauchy’s Definition of the Limit of Functions
2.6.4 Limit Considerations of Some Elementary Functions
2.6.5 Calculation Rules for Limits
2.6.6 Examples of Limit Considerations
2.7 Continuity of Functions
2.8 Exercises
References
3 Differential Calculus
3.1 The Concept of the Derivative
3.2 Derivatives of Elementary Functions
3.3 Derivative Rules
3.4 Differentiability
3.5 Higher Derivatives, Extreme Values and Turning Points
3.6 Applications of Differential Calculus
3.6.1 L’Hospital’s Rule
3.6.2 Determination of Zeros with the Newton Method
3.6.3 Taylor Series
3.6.4 Curve Discussion
3.6.5 Limit Functions
3.6.6 Elasticity of Functions
3.7 Exercises
References
4 Integral Calculus
4.1 The Indefinite Integral
4.1.1 Primitives of Elementary Functions
4.1.2 Linearity of the indefinite integral
4.2 The Definite Integral
4.2.1 Properties of the Definite Integral
4.2.2 Value of an Integral
4.2.3 Area Between Two Curves
4.2.4 Improper Integrals
4.2.5 Partial Integration
4.2.6 Integration by Substitution
4.3 Application of Integral Calculus
4.3.1 Determination of the Economic Function from the Marginal Function
4.3.2 Consumer Rent
4.3.3 Producer Surplus
4.3.4 Numerical Integration
4.4 Exercises
5 Linear Algebra
5.1 Vectors
5.1.1 Definition of Vectors
5.1.2 The Linear Combination of Vectors
5.1.3 Scalar Product of Two Vectors
5.2 Matrices
5.2.1 Definition of a Matrix
5.2.2 Addition of Matrices
5.2.3 Multiplication by a Scalar
5.2.4 Matrix Multiplication
5.2.5 Calculation Rules of the Matrix Product
5.2.6 Inverse Matrix
5.3 Systems of Linear Equations
5.3.1 Basic Considerations
5.3.2 Solution Methods for Systems of Linear Equations
5.3.3 Standardized form of Systems of Linear Equations
5.3.4 Matrix Inversion
5.3.5 Business Applications
5.3.6 Eigenvalues of a Matrix
5.4 Exercises
References
6 Functions with Several Variables
6.1 Introduction and Representation
6.2 Differential Calculus for Functions with Several Variables
6.2.1 Partial Derivative
6.2.2 The Total Differential
6.3 Extremum Values of Functions with Several Variables
6.3.1 Extremum without Boundary Conditions
6.3.2 Extremum Values with Boundary Conditions
6.4 Exercises
References
7 Financial Mathematics
7.1 Interest Calculation
7.1.1 Simple Interest
7.1.2 Compound Interest
7.1.3 Annuity Calculation
7.1.4 Yearly Interest
7.2 Repayment Calculation
7.3 Exercises
Appendix A
References