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### Table of Contents

####

1 Introduction

1.1 Logic and Propositional Calculus

1.1.1 Propositions and their Composition

1.1.2 Universal and Existential Propositions

1.1.3 Types of Mathematical Proof

1.2 Sets and Operations on Sets

1.2.1 Basic Definitions

1.2.2 Operations on Sets

1.3 Combinatorics

1.4 Real and Complex Numbers

1.4.1 Real Numbers

1.4.2 Complex Numbers

2 Sequences, Series, Finance

2.1 Sequences

2.1.1 Basic Definitions

2.1.2 Limit of a Sequence

2.2 Series

2.2.1 Partial Sums

2.2.2 Series and Convergence of Series

2.3 Finance

2.3.1 Simple Interest and Compound Interest

2.3.2 Periodic Payments

2.3.3 Loan Repayments, Redemption Tables

2.3.4 Investment Projects

2.3.5 Depreciation

3 Relations, Mappings, Functions of a Real Variable

3.1 Relations

3.2 Mappings

3.3 Functions of a Real Variable

3.3.1 Basic Notions

3.3.2 Properties of Functions

3.3.3 Elementary Types of Functions

4 Differentiation

4.1 Limit and Continuity

4.1.1 Limit of a Function

4.1.2 Continuity of a Function

4.2 Difference Quotient and the Derivative

4.3 Derivatives of Elementary Functions, Differentiation Rules

4.4 Differential, Rate of Change and Elasticity

4.5 Graphing Functions

4.5.1 Monotonicity

4.5.2 Extreme Points

4.5.3 Convexity and Concavity

4.5.4 Limits

4.5.5 Further Examples

4.6 Mean-Value Theorem

4.7 Taylor Polynomials

4.8 Approximate Determination of Zeroes

5 Integration

5.1 Indefinite Integrals

5.2 Integration Formulas and Methods

5.2.1 Basic Integration Formulas

5.2.2 Integration by Substitution

5.2.3 Integration by Parts

5.3 The Definite Integral

5.4 Approximation of Definite Integrals

5.5 Improper Integrals

5.5.1 Infinite Limits of Integration

5.5.2 Unbounded Integrands

5.6 Some Applications of Integration

5.6.1 Present Value of a Continuous Future Income Flow

5.6.2 Lorenz Curves

5.6.3 Consumer and Producer Surplus

6 Vectors

6.1 Preliminaries

6.2 Operations on Vectors

6.3 Linear Dependence and Independence

6.4 Vector Spaces

7 Matrices and Determinants

7.1 Matrices

7.2 Matrix Operations

7.3 Determinants

7.4 Linear Mappings

7.5 The Inverse Matrix

7.6 An Economic Application: Input-Output Model

8 Linear Equations and Inequalities

8.1 Systems of Linear Equations

8.1.1 Preliminaries

8.1.2 Existence and Uniqueness of a Solution

8.1.3 Elementary Transformations, Solution Procedures

8.1.4 General Solution

8.1.5 Matrix Inversion

8.2 Systems of Linear Inequalities

8.2.1 Preliminaries

8.2.2 Properties of Feasible Solutions

8.2.3 A Solution Procedure

9 Linear Programming

9.1 Preliminaries

9.2 Graphical Solution

9.3 Properties of a Linear Programming Problem, Standard Form

9.4 Simplex Algorithm

9.5 Two-Phase Simplex Algorithm

9.6 Duality, Complementary Slackness

9.7 Dual Simplex Algorithm

10 Eigenvalue Problems and Quadratic Forms

10.1 Eigenvalues and Eigenvectors

10.2 Quadratic Forms and their Sign

11 Functions of Several Variables

11.1 Preliminaries

11.2 Partial Derivatives, Gradient

11.3 Total Differential

11.4 Generalized Chain Rule, Directional Derivatives

11.5 Partial Rate of Change and Elasticity, Homogeneous Functions

11.6 Implicit Functions

11.7 Unconstrained Optimization

11.7.1 Optimality Conditions

11.7.2 Method of Least Squares

11.7.3 Extreme Points of Implicit Functions

11.8 Constrained Optimization

11.8.1 Local Optimality Conditions

11.8.2 Global Optimality Conditions

11.9 Double Integrals

12 Differential Equations and Difference Equations

12.1 Differential Equations of the First Order

12.1.1 Graphical Solution

12.1.2 Separable Differential Equations

12.2 Linear Differential Equations of Order n

12.2.1 Properties of Solutions

12.2.2 Differential Equations with Constant Coefficients

12.3 Systems of Linear Differential Equations of the First Order

12.4 Linear Difference Equations

12.4.1 Definitions and Properties of Solutions

12.4.2 Linear Difference Equations of the First Order

12.4.3 Linear Difference Equations of the Second Order

List of identified misprints in the first edition (date: 10 March 2017)

<frank.werner@mathematik.uni-magdeburg.de>