Franz W. Peren

Math for Business
and Economics
Compendium of Essential Formulas


Math for Business and Economics

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Franz W. Peren

Math for Business
and Economics
Compendium of Essential Formulas

www.pdfgrip.com


Franz W. Peren
Bonn-Rhein-Sieg University
Sankt Augustin, Germany

ISBN 978-3-662-63248-2
ISBN 978-3-662-63249-9 (eBook)
/>© Springer-Verlag GmbH Germany, part of Springer Nature 2021
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or
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The publisher, the authors, and the editors are safe to assume that the advice and information in
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For my father, Paul.

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Preface

The following book is based on the author’s expertise in the field of business mathematics. After completing his studies in business administration and mathematics, he started his career working for a global bank
and the German government. Later he became a professor of business
administration, specializing in quantitative methods. He has been a professor at the Bonn-Rhein-Sieg University in Sankt Augustin, Germany
since 1995, where he is mainly teaching business mathematics, business statistics, and operations research. He has also previously taught
and conducted research at the University of Victoria in Victoria, BC,
Canada and at Columbia University in New York City, New York, USA.

To the author’s best knowledge and beliefs, this formulary presents its
mathematical contents in a practical manner, as they are needed for
meaningful and relevant application in global business, as well as in
universities and economic practice.
The author would like to thank his academic colleagues who have contributed to this work and to many other projects with creativity, knowledge and dedication for more than 25 years. In particular, he would
like to thank Ms. Eva Siebertz and Mr. Nawid Schahab, who were instrumental in managing and creating this formulary. Special thanks are
given to Ms. Camilla Demuth, Ms. Linh Hoang, and Ms. Michelle Jarsen.
Should any mistakes remain, such errors shall be exclusively at the
expense of the author. The author is thankful in advance to all users of
this formulary for any constructive comments or suggestions.
Bonn, March 2021

Franz W. Peren

VII

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Contents

List of Abbreviations
1

XXI

Mathematical Signs and Symbols

1


1.1

Pragmatic Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

General Arithmetic Relations and Links . . . . . . . . . . . .

1

1.3

Sets of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.4

Special Numbers and Links . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4


1.6

Exponential Functions, Logarithm . . . . . . . . . . . . . . . .

4

1.7

Trigonometric Functions, Hyperbolic Functions . . . . .

4

1.8

Vectors, Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.9

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.10

Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7


1.11

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.12

Order Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.13

SI Multiplying and Dividing Prefixes . . . . . . . . . . . . . . .

8

1.14

Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

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X

2

3

Contents
Logic

11

2.1

Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Propositional Variable . . . . . . . . . . . . . . . . . . .
2.2.2
Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . .

11
11
12

Arithmetic

15


3.1

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2
Set Relations . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3
Set Operations . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4
Relations, Laws, Rules of Calculation for Sets
3.1.5
Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6
Numeral Systems . . . . . . . . . . . . . . . . . . . . . . .
3.1.6.1 Decimal System (Decadic System)
3.1.6.2 Dual System (Binary System) . . . .
3.1.6.3 Roman Numeral System . . . . . . . . .

15
15
16
17
19
21
22
23
23
24


3.2

Elementary Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1
Elementary Foundations . . . . . . . . . . . . . . . .
3.2.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . .
3.2.1.2 Factorisation . . . . . . . . . . . . . . . . . . .
3.2.1.3 Relations . . . . . . . . . . . . . . . . . . . . . .
3.2.1.4 Absolute Value, Signum . . . . . . . . .
3.2.1.5 Fractions . . . . . . . . . . . . . . . . . . . . . .
3.2.1.6 Polynomial Division . . . . . . . . . . . . .
3.2.2
Conversions of Terms . . . . . . . . . . . . . . . . . . .
3.2.2.1 Binomial Formulas . . . . . . . . . . . . . .
3.2.2.2 Binomial Theorem . . . . . . . . . . . . . .
3.2.2.3 General Binomial Theorem for Natural Exponents . . . . . . . . . . . . . . . . .
3.2.2.4 General Binomial Theorem for Real
Exponents . . . . . . . . . . . . . . . . . . . . .
3.2.2.5 Polynomial Terms . . . . . . . . . . . . . . .
3.2.3
Summation and Product Notation . . . . . . . . .
3.2.3.1 Summation Notation . . . . . . . . . . . .
3.2.3.2 Product Notation . . . . . . . . . . . . . . .

24
24
25
25
26

26
27
27
29
29
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30
30
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Contents

XI
3.2.4
3.2.5
3.2.6
3.2.7

4

Powers, Roots . . . . . . . . . . . . . . . . . . . . . . . . . .
Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Binomial Coefficient . . . . . . . . . . . . . . . . . . . . .

33
36
38
39

3.3

Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2
Limit of a Sequence . . . . . . . . . . . . . . . . . . . .
3.3.3
Arithmetic and Geometric Sequences . . . . .

41
41
44
46

3.4

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2
Arithmetic and Geometric Series . . . . . . . . . .


46
46
47

Algebra

51

4.1

Fundamental Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2

Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1
Linear Equations with One Variable . . . . . . .
4.2.2
Linear Inequalities with One Variable . . . . . .
4.2.3
Linear Equations with Multiple Variables . . .
4.2.4
Systems of Linear Equations . . . . . . . . . . . . .
4.2.5
Linear Inequalities with Multiple Variables . .

53
53

56
56
57
61

4.3

Non-linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1
Quadratic Equations with One Variable . . . .
4.3.2
Cubic Equations with One Variable . . . . . . .
4.3.3
Biquadratic Equations . . . . . . . . . . . . . . . . . . .
4.3.4
Equations of the nth Degree . . . . . . . . . . . . . .
4.3.5
Radical Equations . . . . . . . . . . . . . . . . . . . . . .

62
62
65
67
68
69

4.4

Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . .
4.4.1

Exponential Equations . . . . . . . . . . . . . . . . . .
4.4.2
Logarithmic Equations . . . . . . . . . . . . . . . . . .

71
71
73

4.5

Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1
Regula falsi (Secant Method) . . . . . . . . . . . .
4.5.2
Newton’s Method (Tangent Method) . . . . . . .

75
75
77

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Contents
4.5.3

5


General Approximation Method (Fixed-point
Iteration) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Linear Algebra

87

5.1

Fundamental Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2
Equality/Inequality of Matrices . . . . . . . . . . .
5.1.3
Transposed Matrix . . . . . . . . . . . . . . . . . . . . . .
5.1.4
Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5
Special Matrices and Vectors . . . . . . . . . . . .

87
87
88
89
89
92


5.2

Operations with Matrices . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1
Addition of Matrices . . . . . . . . . . . . . . . . . . . . .
5.2.2
Multiplication of Matrices . . . . . . . . . . . . . . . . .
5.2.2.1 Multiplication of a Matrix with a
Scalar . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.2 The Scalar Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.3 Multiplication of a Matrix by a Column Vector . . . . . . . . . . . . . . . . . . . .
5.2.2.4 Multiplication of a Row Vector by
a Matrix . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.5 Multiplication of Two Matrices . . . .

94
94
96

5.3

96
98
100
102
103

The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2
Determination of the Inverse with the Usage
of the Gaussian Elimination Method . . . . . .

107
107

5.4

The Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2
Determination of the Rank of a Matrix . . . . .

113
113
113

5.5

The Determinant of a Matrix . . . . . . . . . . . . . . . . . . . . .
5.5.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2
Calculation of Determinants . . . . . . . . . . . . . .
5.5.3
Characteristics of Determinants . . . . . . . . . . .

117

117
118
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Contents
5.6

6

7

XIII
The Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2
Determination of the Inverse with the Usage
of the Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . .

125
125
127

Combinatorics


129

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

6.2

Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

6.3

Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

6.4

Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136

Financial Mathematics

141


7.1

Calculation of Interest . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1
Fundamental Terms . . . . . . . . . . . . . . . . . . . . .
7.1.2
Annual Interest . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2.1 Simple Interest Calculation . . . . . . .
7.1.2.2 Compound Computation of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2.3 Composite Interest . . . . . . . . . . . . .
7.1.3
Interest During the Period . . . . . . . . . . . . . . .
7.1.3.1 Simple Interest Calculation (linear)
7.1.3.2 Simple Interest Using the Nominal Annual Interest Rate . . . . . . . . .
7.1.3.3 Compound Interest (exponential) .
7.1.3.4 Interest with Compound Interest
Using a Conforming Annual Interest Rate . . . . . . . . . . . . . . . . . . . . . .
7.1.3.5 Mixed Interest . . . . . . . . . . . . . . . . .
7.1.3.6 Steady Interest Rate . . . . . . . . . . . .

141
141
142
142

7.2

Annual Percentage Rate . . . . . . . . . . . . . . . . . . . . . . . .

164


7.3

Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1
Time Depreciation . . . . . . . . . . . . . . . . . . . . . .
7.3.1.1 Linear Depreciation . . . . . . . . . . . . .

169
169
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146
155
155
156
157
157
159
160


XIV

Contents
7.3.1.2
7.3.1.3

7.3.2
7.3.3

7.4

7.5

7.6

Arithmetic-Degressive Depreciation
Geometric-Degressive Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units of Production Depreciation . . . . . . . . .
Extraordinary Depreciation . . . . . . . . . . . . . .

Annuity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1
Fundamental Terms . . . . . . . . . . . . . . . . . . . . .
7.4.2
Finite, Regular Annuity . . . . . . . . . . . . . . . . . .
7.4.2.1 Annual Annuity with Annual Interest
7.4.2.2 Annual Annuity with Sub-Annual
Interest . . . . . . . . . . . . . . . . . . . . . . .
7.4.2.3 Sub-Annual Annuity with Annual
Interest . . . . . . . . . . . . . . . . . . . . . . .
7.4.2.4 Sub-Annual Annuity with Sub-Annual
Interest . . . . . . . . . . . . . . . . . . . . . . .
7.4.3
Finite, Variable Annuity . . . . . . . . . . . . . . . . . .
7.4.3.1 Irregular Annuity . . . . . . . . . . . . . . .
7.4.3.2 Arithmetic Progressive Annuity . . .

7.4.3.3 Geometric Progressive Annuity . . .
7.4.4
Perpetuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sinking Fund Calculation . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1
Fundamental Terms . . . . . . . . . . . . . . . . . . . . .
7.5.2
Annuity Repayment . . . . . . . . . . . . . . . . . . . . .
7.5.3
Repayment by Instalments . . . . . . . . . . . . . .
7.5.4
Repayment with Premium . . . . . . . . . . . . . . .
7.5.4.1 Annuity Repayment with Premium
7.5.4.2 Repayment of an Instalment Debt
with Premium . . . . . . . . . . . . . . . . . .
7.5.5
Grace Periods . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.6
Rounded Annuities . . . . . . . . . . . . . . . . . . . . .
7.5.6.1 Percentage Annuity . . . . . . . . . . . . .
7.5.6.2 Repayment of Bonds . . . . . . . . . . .
7.5.7
Repayment During the Year . . . . . . . . . . . . . .
7.5.7.1 Annuity Repayment During the Year
7.5.7.2 Repayment by Instalments During
the Year . . . . . . . . . . . . . . . . . . . . . . .
Investment Calculation . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1
Fundamental Terms . . . . . . . . . . . . . . . . . . . . .


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172
174
175
176
176
179
179
183
186
190
192
192
193
195
198
199
200
202
204
206
206
210
212
214
214
217
222

222
229
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Contents

XV
7.6.2
7.6.3
7.6.4

8

9

Fundamentals of Financial Mathematics . . .
Methods of Static Investment Calculation . .
Methods of Dynamic Investment Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.4.1 Net Present Value Method
(Net Present Value, Amount of
Capital, Final Asset Value) . . . . . . .
7.6.4.2 Annuity Method . . . . . . . . . . . . . . . .
7.6.4.3 Internal Rate of Return Method . .

238
240
241
241

244
246

Optimisation of Linear Models

249

8.1

Lagrange Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2
Formation of the Lagrange Function . . . . . . .
8.1.3
Determination of the Solution . . . . . . . . . . . . .
8.1.4
Interpretation of l . . . . . . . . . . . . . . . . . . . . . . .

249
249
249
250
251

8.2

Linear Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2.2
The Linear Programming Approach . . . . . . .
8.2.3
Graphical Solution . . . . . . . . . . . . . . . . . . . . . .
8.2.4
Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . .
8.2.5
Simplex Tableau (Basic Structure) . . . . . . . . .

254
254
254
255
258
259

Functions

265

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1
Composition of Functions . . . . . . . . . . . . . . .
9.1.2
Inverse Function . . . . . . . . . . . . . . . . . . . . . . .

265
269

271

9.2

Classification of Functions . . . . . . . . . . . . . . . . . . . . . .
9.2.1
Rational Functions . . . . . . . . . . . . . . . . . . . . .
9.2.1.1 Polynomial Functions . . . . . . . . . . .
9.2.1.2 Broken Rational Functions . . . . . . .
9.2.2
Non-rational Functions . . . . . . . . . . . . . . . . . .
9.2.2.1 Power Functions . . . . . . . . . . . . . . . .
9.2.2.2 Root Function . . . . . . . . . . . . . . . . .

273
274
274
274
278
278
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Contents
9.2.2.3
9.2.2.4


Transcendental Functions . . . . . . . .
9.2.2.3.1 Exponential Functions .
9.2.2.3.2 Logarithmic Functions . .
Trigonometric Functions (Angle Functions/Circular Functions) . . . . . . . . .

282
282
288
294

9.3

Characteristics of Real Functions . . . . . . . . . . . . . . . .
9.3.1
Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2.1 Axial Symmetry . . . . . . . . . . . . . . . .
9.3.2.2 Point Symmetry . . . . . . . . . . . . . . . .
9.3.3
Transformations . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3.1 Vertex Form . . . . . . . . . . . . . . . . . . .
9.3.4
Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.5
Infinite Discontinuities . . . . . . . . . . . . . . . . . . .
9.3.6
Removable Discontinuities . . . . . . . . . . . . . . .
9.3.7

Jump Discontinuities . . . . . . . . . . . . . . . . . . . .
9.3.8
Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.9
Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.10 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.11 Local Extremes . . . . . . . . . . . . . . . . . . . . . . . .
9.3.12 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.13 Concavity and Convexity | Inflection Points .
9.3.14 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.14.1 Horizontal Asymptotes . . . . . . . . . .
9.3.14.2 Vertical Asymptote . . . . . . . . . . . . .
9.3.14.3 Oblique Asymptote . . . . . . . . . . . . .
9.3.14.4 Asymptotic Curve . . . . . . . . . . . . . . .
9.3.15 Tangent Lines to a Curve . . . . . . . . . . . . . . . .
9.3.16 Normal Lines to a Curve . . . . . . . . . . . . . . . .

322
322
324
324
326
329
331
334
334
336
337
338
339

339
340
341
342
344
345
347
348
349
350
351

9.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

352

10 Differential Calculus
10.1

Differentiation of Functions with One Independent
Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 First Derivative of Elementary Functions . . .
10.1.3 Derivation Rules . . . . . . . . . . . . . . . . . . . . . . .

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357
357
360
362


Contents

XVII
10.1.4
10.1.5
10.1.6

10.2

Differentiation of Functions with More Than One Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1
10.2.2
10.2.3

10.2.4
10.3

Higher Derivations . . . . . . . . . . . . . . . . . . . . . .
Differentiation of Functions with Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Curve Sketching . . . . . . . . . . . . . . . . . . . . . . .

Partial Derivatives (1st Order) . . . . . . . . . . . . .
Partial Derivatives (2nd Order) . . . . . . . . . . . .
Local Extrema of the Function f = f (x, y) . .

10.2.3.1 Relative Extrema without Constraint
of the Function f = f (x, y) . . . . . . .
10.2.3.2 Relative Extrema with m Constraints
of the Function f = f (x1 , . . . , xn )
with m < n . . . . . . . . . . . . . . . . . . . . .
Differentials of the Function f = f (x1 , ..., xn )

Theorems of Differentiable Functions . . . . . . . . . . . . .
10.3.1 Mean Value Theorem for Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Generalized Mean Value Theorem for Differential Calculus . . . . . . . . . . . . . . . . . . . . . . .
10.3.3 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . .
10.3.5 Bounds Theorem for Differential Calculus . .

11 Integral Calculus

364
365
365
375
375
378
380
380
384
388
390
390
391
391

392
393
395

11.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

11.2

The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Definition/Determining the Antiderivative . . .
11.2.2 Elementary Calculation Rules for the Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . .

396
396

The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Relationship between the Definite and the
Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . .

400
400

11.3

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399

404


XVIII

Contents
11.3.3

Special Techniques of Integration . . . . . . . . .
11.3.3.1 Partial Integration . . . . . . . . . . . . . .
11.3.3.2 Integration by Substitution . . . . . . .

409
409
411

11.4

Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

412

11.5

Integral Calculus and Economic Problems . . . . . . . . .
11.5.1 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.2 Revenue Function (= Sales Function) . . . . . .

11.5.3 Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . .

413
413
415
416

12 Elasticities

421

12.1

Definition of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . .

421

12.2

Arc Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

422

12.3

Point Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

427

12.4


Price Elasticity of Demand exp . . . . . . . . . . . . . . . . . . .

430

12.5

Cross Elasticity of Demand exA pB . . . . . . . . . . . . . . . . .

435

12.6

Income Elasticity of Demand exy . . . . . . . . . . . . . . . . . .

436

13 Economic Functions

439

13.1

Supply Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

439

13.2

Demand Function / Inverse Demand Function . . . . . .


441

13.3

Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

443

13.4

Buyer’s Market and Seller’s Market . . . . . . . . . . . . . . .

444

13.5

Supply Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

13.6

Demand Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

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Contents

XIX

13.7

Revenue Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

446

13.8

Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

452

13.9

Neoclassical Cost Function . . . . . . . . . . . . . . . . . . . . . .

460

13.10 Cost Function According to the Law of Diminishing
Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467

13.11 Direct Costs versus Indirect Costs . . . . . . . . . . . . . . . .
13.11.1 One-Dimensional Cost Allocation Principles
13.11.2 Multi-Dimensional Cost Allocation Principles


478
481
483

13.12 Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

486

14 The Peren Theorem
The Mathematical Frame in Which We Live

495

A Financial Mathematical Factors

503

B Bibliography

549

Index

553

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List of Abbreviations

ACT

Advance Corporation Tax

a.m.

ante meridiem

APR

Annual Percentage Rate

AU

area unit(s)

BCD

Binary Coded Decimal

BEP

break-even point

BGB

German Civil Code

bit


binary digit

calcul.

calculate, calculation

cf.

confer

CFPB

Consumer Financial Protection Bureau

cm

centimetre(s)

CM

Contribution Margin

const.

constant

c.p.

ceteris paribus


det

determinant

e

Euler’s number

EC

European Commission

ED

edition

e.g.

exempli gratia
XXI

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XXII

List of Abbreviations

etc.


et cetera

EU

European Union

gal.

gallon

gen.

general

h

hour(s)

ICMA

International Capital Market Association

i.e.

id est

incl.

includes, including


inf

infimum

int

integer function

IP

inflection point(s)

IRR

internal rate of return

ISDA

International Swaps and Derivatives Association

j

relative periodic interest rate

kbyte

kilobyte

kg


kilogram(s)

l

litre(s)

lb

pound(s)

lim

limit

ln

natural logarithm

log

logarithm

ltd.

limited

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List of Abbreviations


XXIII

LU

length unit(s)

max

maximum, maximise

mbyte

megabyte

min

minimise, minimum

min(s)

minute(s)

mm

millimetre(s)

norm

normal


NPV

net present value

opt

optimise, optimisation

oz

ounce(s)

p.a.

per annum

p.m.

post meridiem

QU

quantity unit(s)

rad

radius

regen


regeneration

rep.

repetition

resp.

respectively

sgn

signum

SI

Système International d’Unités

SS

solution set

SU

universal set

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XXIV

List of Abbreviations

sup

supremum

TU

time unit(s)

USP

unique selling proposition

U.S.

United States of America

w/

with

w/o

without

yd


yard(s)

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Chapter 1

Mathematical Signs and Symbols
Remark:

The signs and symbols are partly shown in applications. For definitions see dedicated passage.

1.1 Pragmatic Signs
a ⇡ b

a approximately similar to b

a

a large towards b

a⌧b

a small towards b, a can be neglected compared to b

a =
b b

a equivalent to b, e.g. 1 cm =
b 10 mm; 1 inch =

b 25.4 mm

b

a ^ b

a _ b
...

a and b
a or b

and so forth (until), omission

1.2 General Arithmetic Relations and Links
(a, b are figures, elements, objects)
a = b

a equals b, arithmetic fundamental term, identity

a 6= b

a unequal to b, no identity

a := b

a equals b by definition

a < b


a less than b, fundamental term, e.g.

a > b

a greater than b, e.g. 3 > 8

© Springer-Verlag GmbH Germany, part of Springer Nature 2021
F. W. Peren, Math for Business and Economics,
/>
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6< 2

1


2

1 Mathematical Signs and Symbols

a  b
a

b

a less than or (at most) equal to b, a  8 is equivalent to
•, 8]

]


a greater than or (at least) equal to b, is equivalent to b  a

a + b

a plus b, sum of a and b, arithmetic fundamental term

a

a minus b, difference between a and b, single-digit

b

linking sign
a · b
a
b
n

 ai

i=1

a times b, product of a and b, arithmetic fundamental term
a divided by b, quotient of a and b, e.g.

16
4

= 16 ÷ 4 = 4


Sum over ai of i equals 1 up to n,
n

 ai = a1 + a2 + a3 + ... + an

i=1
n

’ ai

i=1

n

Product over ai of i equals 1 up to n, ’ ai = a1 · a2 · ... · an
i=1

1.3 Sets of Numbers
N
N⇤
Z
Q
Q⇤
Q+

set of natural numbers, N = {0, 1, 2, ...}

set of positive natural numbers, N⇤ = N \{0} = {1, 2, 3, ...}
set of integers, Z = {...


2, 1, 0, 1, 2, ...}

set of rational numbers, Q = { ab | a, b 2 Z, b 6= 0}

set of rational numbers which vary from zero, Q⇤ = Q \{0}
set of positive rational numbers

Q+
0

set of positive rational numbers plus zero

R

set of real numbers

R⇤

set of real numbers which vary from zero

R+

set of positive real numbers

R+
0

set of positive real numbers plus zero

C


set of complex numbers

]a, b[

open interval from a to b {x | a < x < b}

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1.4 Special Numbers and Links
]a, •[

3

open, unbounded interval starting at a, {x | a < x}
closed interval from a to b, {x | a  x  b}

[a, b]
[a, •[

closed, unbounded interval starting at a, {x | a  x}

left-closed, right-open interval from a to b, {x | a  x < b}

[a, b[

1.4 Special Numbers and Links
(a, b 2 R; n, m 2 Z; s 2 N)
an

p
1
a = a2 = b
p
n

1
n

a=a =b

a to the power of n, n th power of a for n
root (square root) of a, equivalent to

b2

b

0, a

n th

root of a, equivalent to bn = a for b

0
= a for

0
’ni=1 ai


0, a

n!

n factorial, n! =

sgn a

signum of a (algebraic sign), e.g. sgn( 3) =

|a|

absolute value of a, e.g. |
i th

= 1 · 2 · 3 · ... · n
8| = 8

a[i]

a in the

position; e.g. 5; 6; 7; a[2] = 6

•

infinity, note: • is not a number

p


3.1415926...

e

Euler’s Number, e = 2.718281

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0
1


4

1 Mathematical Signs and Symbols

1.5 Limit
lim f (x) = a

x!0

a is the limit of the function f (x) for x towards 0,
i.e. x x!0 gradually approaches the value 0,
the function’s value f (x) converges (limits)
towards a

lim f (x) = b

b is the limit of the function f (x) for x towards •


lim f (x) = c

c is the limit of the function f (x) for x towards 5

x!•

x!5

1.6 Exponential Functions, Logarithm
ex

exponential function of x, e to the power of x

ln x

natural logarithm of x to base e; loge x = ln x

loga x

logarithm of x to base a; loga x = y , ay = x

log x

with x ; a > 0 and a 6= 1

common logarithm of x to base 10
log x = lg x = log10 x

lb x


binary (dyadic) logarithm of x to base 2
lb x = log2 x

1.7 Trigonometric Functions, Hyperbolic Functions
sin x

sine of x

cos x

cosine of x

tan x

tangent of x

cot x

cotangent of x

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