Math for Business and Economics: Compendium of Essential Formulas, Third Edition
Franz W. Peren
Contents
List of Abbreviations XXI
1 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 … . . . . . . . . . . . 3
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
2 Logic 11
2.1 Mathematical Logic .. . . . . . . . . . . . . 11
2.2 Propositional Logic ..11
2.2.1 Propositional Variable .. . . . 11
2.2.2 Truth Tables .. . . . . . . . . . . . 12
3 Arithmetic 15
3.1 Sets … . . . . . . . . . . . . 15
3.1.1 General … . . 15
3.1.2 Set Relations .. . . . . . . . . . . . 16
3.1.3 Set Operations .. . . . . . . . . . 17
3.1.4 Relations, Laws, Rules of Calculation for Sets 19
3.1.5 Intervals … . 21
3.1.6 Numeral Systems .. . . . . . . . 22
3.1.6.1 Decimal System (Decadic System) 23
3.1.6.2 Dual System (Binary System) . . . . 23
3.1.6.3 Roman Numeral System . . . . . . . . . 24
3.2 Elementary Calculus .. . . . . . . . . . . . 24
3.2.1 Elementary Foundations .. 24
3.2.1.1 Axioms .. . . . . . . . 25
3.2.1.2 Factorisation .. . . . 25
3.2.1.3 Relations .. . . . . . . 26
3.2.1.4 Absolute Value, Signum . . . . . . . . . 26
3.2.1.5 Fractions .. . . . . . . 27
3.2.1.6 Polynomial Division . . . . . . . . . . . . . 27
3.2.1.7 Horner’s Scheme (Horner’s Method) 29
3.2.2 Conversions of Terms .. . . . 30
3.2.2.1 Binomial Formulas . . . . . . . . . . . . . . 30
3.2.2.2 Binomial Theorem . . . . . . . . . . . . . . 31
3.2.2.3 General Binomial Theorem for Natural
Exponents .. . 31
3.2.2.4 General Binomial Theorem for Real
Exponents .. . . . . . 31
3.2.2.5 Polynomial Terms . . . . . . . . . . . . . . . 32
3.2.3 Summation and Product Notation . . . . . . . . . 32
3.2.3.1 Summation Notation . . . . . . . . . . . . 32
3.2.3.2 Product Notation . . . . . . . . . . . . . . . 33
3.2.4 Powers, Roots .. . . . . . . . . . . 34
3.2.5 Logarithms ..37
3.2.6 Factorial … 39
3.2.7 Binomial Coefficient .. . . . . . 40
3.3 Sequences … . . . . . . 41
3.3.1 Definition … 41
3.3.2 Limit of a Sequence .. . . . . 44
3.3.3 Arithmetic and Geometric Sequences . . . . . 46
3.4 Series … . . . . . . . . . . . 47
3.4.1 Definition … 47
3.4.2 Arithmetic and Geometric Series . . . . . . . . . . 47
4 Algebra 51
4.1 Fundamental Terms..51
4.2 Linear Equations … . 53
4.2.1 Linear Equations with One Variable . . . . . . . 53
4.2.2 Linear Inequations with One Variable . . . . . 56
4.2.3 Linear Equations with Multiple Variables . . . 56
4.2.4 Systems of Linear Equations . . . . . . . . . . . . . 57
4.2.5 Linear Inequations with Multiple Variables . 61
4.3 Non-linear Equations .. . . . . . . . . . . . 62
4.3.1 Quadratic Equations with One Variable . . . . 62
4.3.2 Cubic Equations with One Variable . . . . . . . 65
4.3.3 Biquadratic Equations .. . . . 67
4.3.4 Equations of the nth Degree . . . . . . . . . . . . . . 68
4.3.5 Radical Equations .. . . . . . . 69
4.4 Transcendental Equations .. . . . . . . . 71
4.4.1 Exponential Equations .. . . 71
4.4.2 Logarithmic Equations .. . . 73
4.5 Approximation Methods .. . . . . . . . . . 75
4.5.1 Regula falsi (Secant Method) . . . . . . . . . . . . 75
4.5.2 Newton’s Method (Tangent Method) . . . . . . . 77
4.5.3 General Approximation Method (Fixed-point
Iteration) … 80
5 Linear Algebra 87
5.1 Fundamental Terms..87
5.1.1 Matrix … . . . 87
5.1.2 Equality/Inequality of Matrices . . . . . . . . . . . 88
5.1.3 Transposed Matrix .. . . . . . . 89
5.1.4 Vector … . . . 89
5.1.5 Special Matrices and Vectors . . . . . . . . . . . . 92
5.2 Operations with Matrices .. . . . . . . . . 94
5.2.1 Addition of Matrices .. . . . . . 94
5.2.2 Multiplication of Matrices .. . 96
5.2.2.1 Multiplication of a Matrix with a
Scalar .. . . . . . . . . . 96
5.2.2.2 The Scalar Product of Two Vectors
.. . . . . . . . . . . 98
5.2.2.3 Multiplication of a Matrix by a Column
Vector .. . . . . 100
5.2.2.4 Multiplication of a Row Vector by
a Matrix .. . . . . . . . 102
5.2.2.5 Multiplication of Two Matrices . . . . 103
5.3 The Inverse of a Matrix .. . . . . . . . . . 107
5.3.1 Introduction .. . . . . . . . . . . . . 107
5.3.2 Determination of the Inverse with the Usage
of the Gaussian Elimination Method . . . . . . 109
5.4 The Rank of a Matrix .. . . . . . . . . . . . 113
5.4.1 Definition … 113
5.4.2 Determination of the Rank of a Matrix . . . . . 113
5.5 The Determinant of a Matrix .. . . . . . 117
5.5.1 Definition … 117
5.5.2 Calculation of Determinants . . . . . . . . . . . . . . 118
5.5.3 Characteristics of Determinants . . . . . . . . . . . 124
5.6 The Adjoint of a Matrix .. . . . . . . . . . . 125
5.6.1 Definition … 125
5.6.2 Determination of the Inverse with the Usage
of the Adjoint .. . . . . . . . . . . . 127
6 Combinatorics 129
6.1 Introduction … . . . . . . 129
6.2 Permutations … . . . . 133
6.3 Variations … . . . . . . . 135
6.4 Combinations … . . . 136
7 Financial Mathematics 141
7.1 Calculation of Interest .. . . . . . . . . . . 141
7.1.1 Fundamental Terms .. . . . . . 141
7.1.2 Annual Interest .. . . . . . . . . . 142
7.1.2.1 Simple Interest Calculation . . . . . . . 142
7.1.2.2 Compound Computation of Interest
.. . . . . . . . . . . . 144
7.1.2.3 Composite Interest . . . . . . . . . . . . . 146
7.1.3 Interest During the Period . . . . . . . . . . . . . . . 158
7.1.3.1 Simple Interest Calculation (linear) 159
7.1.3.2 Simple Interest Using the Nominal
Annual Interest Rate . . . . . . . . . 159
7.1.3.3 Compound Interest (exponential) . 160
7.1.3.4 Interest with Compound Interest
Using a Conforming Annual Interest
Rate .. . . . . . . 161
7.1.3.5 Mixed Interest .. . 162
7.1.3.6 Steady Interest Rate . . . . . . . . . . . . 163
7.2 Annual Percentage Rate .. . . . . . . . . 168
7.3 Depreciation … . . . . 173
7.3.1 Time Depreciation .. . . . . . . 173
7.3.1.1 Linear Depreciation . . . . . . . . . . . . . 173
7.3.1.2 Arithmetic-Degressive Depreciation 174
7.3.1.3 Geometric-Degressive Depreciation
.. . . . . . . . . . . . 176
7.3.2 Units of Production Depreciation . . . . . . . . . 178
7.3.3 Extraordinary Depreciation . . . . . . . . . . . . . . 179
7.4 Annuity Calculation ..180
7.4.1 Fundamental Terms .. . . . . . 180
7.4.2 Finite, Regular Annuity .. . . 183
7.4.2.1 Annual Annuity with Annual Interest 183
7.4.2.2 Annual Annuity with Sub-Annual
Interest .. . . . . . . . 187
7.4.2.3 Sub-Annual Annuity with Annual
Interest .. . . . . . . . 190
7.4.2.4 Sub-Annual Annuity with Sub-Annual
Interest .. . . . . . . . 194
7.4.3 Finite, Variable Annuity .. . . 213
7.4.3.1 Irregular Annuity . . . . . . . . . . . . . . . 213
7.4.3.2 Arithmetic Progressive Annuity . . . 220
7.4.3.3 Geometric Progressive Annuity . . . 231
7.4.4 Perpetuity ..234
7.5 Sinking Fund Calculation .. . . . . . . . . 235
7.5.1 Fundamental Terms .. . . . . . 236
7.5.2 Annuity Repayment .. . . . . . 238
7.5.3 Repayment by Instalments . . . . . . . . . . . . . . 241
7.5.4 Repayment with Premium . . . . . . . . . . . . . . . 243
7.5.4.1 Annuity Repayment with Premium 243
7.5.4.2 Repayment of an Instalment Debt
with Premium .. . . 248
7.5.5 Repayment with Discount (Disagio) . . . . . . . 249
7.5.5.1 Annuity Repayment with Discount
when Immediately Booked as Interest
Expense .. . . . 251
7.5.5.2 Annuity Repayment with Discount
when a Disagio is Included in Prepaid
Expenses .. . 253
7.5.5.3 Instalment Repayment with Discount
when Immediately Booked
as Interest Expense . . . . . . . . . . . . . 253
7.5.5.4 Instalment Repayment with Discount
when a Disagio is Included
in Prepaid Expenses . . . . . . . . . . . . 254
7.5.6 Grace Periods .. . . . . . . . . . 255
7.5.7 Rounded Annuities .. . . . . . 257
7.5.7.1 Percentage Annuity . . . . . . . . . . . . . 257
7.5.7.2 Repayment of Bonds . . . . . . . . . . . 260
7.5.8 Repayment During the Year . . . . . . . . . . . . . . 266
7.5.8.1 Annuity Repayment During the Year 266
7.5.8.2 Repayment by Instalments During
the Year .. . . . . . . . 274
7.6 Investment Calculation .. . . . . . . . . . . 279
7.6.1 Fundamental Terms .. . . . . . 280
7.6.2 Fundamentals of Financial Mathematics . . . 283
7.6.3 Methods of Static Investment Calculation . . 286
7.6.4 Methods of Dynamic Investment Calculation
… . . . . . 286
7.6.4.1 Net Present Value Method
(Net Present Value, Amount of
Capital, Final Asset Value) . . . . . . . 287
7.6.4.2 Annuity Method .. 290
7.6.4.3 Internal Rate of Return Method . . 293
8 Optimisation of Linear Models 297
8.1 Lagrange Method … 297
8.1.1 Introduction .. . . . . . . . . . . . . 297
8.1.2 Formation of the Lagrange Function . . . . . . . 297
8.1.3 Determination of the Solution . . . . . . . . . . . . . 298
8.1.4 Interpretation of λ .. . . . . . . . 301
8.1.5 Identification of the Type of Optimum . . . . . . 302
8.2 Linear Optimisation ..313
8.2.1 Introduction .. . . . . . . . . . . . . 313
8.2.2 The Linear Programming Approach . . . . . . . 313
8.2.3 Graphical Solution .. . . . . . . 314
8.2.4 Primal Simplex Algorithm .. 317
8.2.5 Simplex Tableau (Basic Structure) . . . . . . . . . 318
8.2.6 Dual Simplex Algorithm .. . . 324
9 Functions 335
9.1 Introduction … . . . . . . 335
9.1.1 Composition of Functions . . . . . . . . . . . . . . . 339
9.1.2 Inverse Function .. . . . . . . . 341
9.2 Classification of Functions .. . . . . . . 343
9.2.1 Rational Functions .. . . . . . 344
9.2.1.1 Polynomial Functions . . . . . . . . . . . 344
9.2.1.2 Broken Rational Functions . . . . . . . 344
9.2.2 Non-rational Functions .. . . 348
9.2.2.1 Power Functions .. 348
9.2.2.2 Root Function .. . 351
9.2.2.3 Transcendental Functions . . . . . . . . 352
9.2.2.3.1 Exponential Functions . 352
9.2.2.3.2 Logarithmic Functions . . 358
9.2.2.4 Trigonometric Functions (Angle Functions/
Circular Functions) . . . . . . . . . 364
9.3 Characteristics of Real Functions .. 392
9.3.1 Boundedness .. . . . . . . . . . . 392
9.3.2 Symmetry ..394
9.3.2.1 Axial Symmetry .. 394
9.3.2.2 Point Symmetry .. 396
9.3.3 Transformations .. . . . . . . . . 399
9.3.3.1 Vertex Form .. . . . 401
9.3.4 Continuity ..404
9.3.5 Infinite Discontinuities .. . . . 404
9.3.6 Removable Discontinuities . . . . . . . . . . . . . . . 406
9.3.7 Jump Discontinuities .. . . . . 407
9.3.8 Homogeneity .. . . . . . . . . . . 408
9.3.9 Periodicity ..409
9.3.10 Zeros … . . . 409
9.3.11 Local Extremes .. . . . . . . . . 410
9.3.12 Monotonicity .. . . . . . . . . . . . 411
9.3.13 Concavity and Convexity | Inflection Points . 412
9.3.14 Asymptotes .. . . . . . . . . . . . . 414
9.3.14.1 Horizontal Asymptotes . . . . . . . . . . 415
9.3.14.2 Vertical Asymptote . . . . . . . . . . . . . 417
9.3.14.3 Oblique Asymptote . . . . . . . . . . . . . 418
9.3.14.4 Asymptotic Curve . . . . . . . . . . . . . . . 419
9.3.15 Tangent Lines to a Curve .. 420
9.3.16 Normal Lines to a Curve .. 421
9.4 Exercises … . . . . . . . . 422
10 Differential Calculus 427
10.1 Differentiation of Functions with One Independent
Variable … . . . . . . . . . 427
10.1.1 General … . . 427
10.1.2 First Derivative of Elementary Functions . . . 430
10.1.3 Derivation Rules .. . . . . . . . 432
10.1.4 Higher Derivations .. . . . . . . 434
10.1.5 Differentiation of Functions with Parameters
… . . . . . 435
10.1.6 Curve Sketching .. . . . . . . . 435
10.2 Differentiation of Functions with More Than One Independent
Variable ..445
10.2.1 Partial Derivatives (1st Order) . . . . . . . . . . . . . 445
10.2.2 Partial Derivatives (2nd Order) . . . . . . . . . . . . 448
10.2.3 Local Extrema of the Function f = f (x, y) . . 450
10.2.3.1 Relative Extrema without Constraint
of the Function f = f (x, y) . . . . . . . 450
10.2.3.2 Relative Extrema with m Constraints
of the Function f = f (x1, . . . , xn)
with m < n .. . . . . . 459
10.2.4 Differentials of the Function f = f (x1, …, xn) 463
10.3 Theorems of Differentiable Functions . . . . . . . . . . . . . 465
10.3.1 Mean Value Theorem for Differential Calculus
… . . . 465
10.3.2 Generalized Mean Value Theorem for Differential
Calculus .. . . . . . . . 466
10.3.3 Rolle’s Theorem .. . . . . . . . . 466
10.3.4 L’Hospital’s Rule .. . . . . . . . . 467
10.3.5 Bounds Theorem for Differential Calculus . . 468
11 Integral Calculus 469
11.1 Introduction … . . . . . . 469
11.2 The Indefinite Integral .. . . . . . . . . . . . 470
11.2.1 Definition/Determining the Antiderivative . . . 470
11.2.2 Elementary Calculation Rules for the Indefinite
Integral .. . . . . . . . . . . . . 473
11.3 The Definite Integral .. . . . . . . . . . . . . 474
11.3.1 Introduction .. . . . . . . . . . . . . 474
11.3.2 Relationship between the Definite and the
Indefinite Integral .. . . . . . . . 478
11.3.3 Special Techniques of Integration . . . . . . . . . 483
11.3.3.1 Partial Integration . . . . . . . . . . . . . . 483
11.3.3.2 Integration by Substitution . . . . . . . 485
11.4 Multiple Integrals … . 486
11.5 Integral Calculus and Economic Problems . . . . . . . . . 487
11.5.1 Cost Functions .. . . . . . . . . . 487
11.5.2 Revenue Function (= Sales Function) . . . . . . 489
11.5.3 Profit Functions .. . . . . . . . . . 490
12 Elasticities 495
12.1 Definition of Elasticity .. . . . . . . . . . . . 495
12.2 Arc Elasticity … . . . . 496
12.3 Point Elasticity … . . . 501
12.4 Price Elasticity of Demand εxp .. . . . 504
12.5 Cross Elasticity of Demand εxA pB .. . 509
12.6 Income Elasticity of Demand εxy .. . . 511
13 Economic Functions 513
13.1 Supply Function … . 513
13.2 Demand Function / Inverse Demand Function . . . . . . 515
13.3 Market Equilibrium ..517
13.4 Buyer’s Market and Seller’s Market . . . . . . . . . . . . . . . 518
13.5 Supply Gap … . . . . . 519
13.6 Demand Gap … . . . . 519
13.7 Revenue Function … 521
13.8 Cost Functions … . . 527
13.9 Neoclassical Cost Function .. . . . . . . 535
13.10 Cost Function According to the Law of Diminishing
Returns … . . . . . . . . . 543
13.11 Direct Costs versus Indirect Costs .. 556
13.11.1 One-Dimensional Cost Allocation Principles 559
13.11.2 Multi-Dimensional Cost Allocation Principles 561
13.12 Profit Function … . . . 564
14 The Peren Theorem
The Mathematical Frame in Which We Live 573
A Financial Mathematical Factors 581
B Bibliography 627
Index 635