Math For Business And Economics: Compendium Of Essential Formulas

Preface......Page 6
Contents......Page 7
List of Abbreviations......Page 18
1.2 General Arithmetic Relations and Links......Page 22
1.3 Sets of Numbers......Page 23
1.4 Special Numbers and Links......Page 24
1.7 Trigonometric Functions, Hyperbolic Functions......Page 25
1.8 Vectors, Matrices......Page 26
1.9 Sets......Page 27
1.12 Order Structures......Page 28
1.13 SI Multiplying and Dividing Prefixes......Page 29
1.14 Greek Alphabet......Page 30
2.2.1 Propositional Variable......Page 31
2.2.2 Truth Tables......Page 32
3.1.1 General......Page 34
3.1.2 Set Relations......Page 35
3.1.3 Set Operations......Page 36
3.1.4 Relations, Laws, Rules of Calculation for Sets......Page 38
3.1.5 Intervals......Page 40
3.1.6 Numeral Systems......Page 41
3.1.6.2 Dual System (Binary System)......Page 42
3.2.1 Elementary Foundations......Page 43
3.2.1.2 Factorisation......Page 44
3.2.1.4 Absolute Value, Signum......Page 45
3.2.1.6 Polynomial Division......Page 46
3.2.2.2 Binomial Theorem......Page 48
3.2.3 Summation and Product Notation......Page 49
3.2.3.1 Summation Notation......Page 50
3.2.3.2 Product Notation......Page 51
3.2.4 Powers, Roots......Page 52
3.2.5 Logarithms......Page 55
3.2.6 Factorial......Page 57
3.2.7 Binomial Coefficient......Page 58
3.3.1 Definition......Page 60
3.3.2 Limit of a Sequence......Page 63
3.4.1 Definition......Page 65
3.4.2 Arithmetic and Geometric Series......Page 66
4.1 Fundamental Terms......Page 69
4.2.1 Linear Equations with One Variable......Page 71
4.2.3 Linear Equations with Multiple Variables......Page 74
4.2.4 Systems of Linear Equations......Page 75
4.2.5 Linear Inequalities with Multiple Variables......Page 79
4.3.1 Quadratic Equations with One Variable......Page 80
4.3.2 Cubic Equations with One Variable......Page 83
4.3.3 Biquadratic Equations......Page 85
4.3.4 Equations of the nth Degree......Page 86
4.3.5 Radical Equations......Page 87
4.4.1 Exponential Equations......Page 89
4.4.2 Logarithmic Equations......Page 91
4.5.1 Regula falsi (Secant Method)......Page 93
4.5.2 Newton’s Method (Tangent Method)......Page 95
4.5.3 General Approximation Method (Fixed-point Iteration)......Page 98
5.1.1 Matrix......Page 104
5.1.2 Equality/Inequality of Matrices......Page 105
5.1.4 Vector......Page 106
5.1.5 Special Matrices and Vectors......Page 109
5.2.1 Addition of Matrices......Page 111
5.2.2.1 Multiplication of a Matrix with a Scalar......Page 113
5.2.2.2 The Scalar Product of Two Vectors......Page 115
5.2.2.3 Multiplication of a Matrix by a Column Vector......Page 117
5.2.2.4 Multiplication of a Row Vector by a Matrix......Page 119
5.2.2.5 Multiplication of Two Matrices......Page 120
5.3.1 Introduction......Page 124
5.3.2 Determination of the Inverse with the Usage of the Gaussian Elimination Method......Page 126
5.4.2 Determination of the Rank of a Matrix......Page 130
5.5.1 Definition......Page 134
5.5.2 Calculation of Determinants......Page 135
5.5.3 Characteristics of Determinants......Page 141
5.6.1 Definition......Page 142
5.6.2 Determination of the Inverse with the Usage of the Adjoint......Page 144
6.1 Introduction......Page 146
6.2 Permutations......Page 150
6.3 Variations......Page 152
6.4 Combinations......Page 153
7.1.1 Fundamental Terms......Page 158
7.1.2.1 Simple Interest Calculation......Page 159
7.1.2.2 Compound Computation of Interest......Page 161
7.1.2.3 Composite Interest......Page 163
7.1.3.1 Simple Interest Calculation (linear)......Page 172
7.1.3.2 Simple Interest Using the Nominal Annual Interest Rate......Page 173
7.1.3.4 Interest with Compound Interest Using a Conforming Annual Interest Rate......Page 174
7.1.3.5 Mixed Interest......Page 176
7.1.3.6 Steady Interest Rate......Page 177
7.2 Annual Percentage Rate......Page 181
7.3.1.1 Linear Depreciation......Page 186
7.3.1.2 Arithmetic-Degressive Depreciation......Page 187
7.3.1.3 Geometric-Degressive Depreciation......Page 189
7.3.2 Units of Production Depreciation......Page 191
7.3.3 Extraordinary Depreciation......Page 192
7.4.1 Fundamental Terms......Page 193
7.4.2.1 Annual Annuity with Annual Interest......Page 196
7.4.2.2 Annual Annuity with Sub-Annual Interest......Page 200
7.4.2.3 Sub-Annual Annuity with Annual Interest......Page 203
7.4.2.4 Sub-Annual Annuity with Sub-Annual Interest......Page 207
7.4.3.1 Irregular Annuity Amount of Annuity......Page 209
7.4.3.2 Arithmetic Progressive Annuity......Page 210
7.4.3.3 Geometric Progressive Annuity......Page 212
7.4.4 Perpetuity......Page 215
7.5 Sinking Fund Calculation......Page 216
7.5.1 Fundamental Terms......Page 217
7.5.2 Annuity Repayment......Page 219
7.5.3 Repayment by Instalments......Page 221
7.5.4.1 Annuity Repayment with Premium......Page 223
7.5.4.2 Repayment of an Instalment Debt with Premium......Page 227
7.5.5 Grace Periods......Page 229
7.5.6.1 Percentage Annuity......Page 231
7.5.6.2 Repayment of Bonds......Page 234
7.5.7.1 Annuity Repayment During the Year......Page 239
7.5.7.2 Repayment by Instalments During the Year......Page 246
7.6 Investment Calculation......Page 251
7.6.1 Fundamental Terms......Page 252
7.6.2 Fundamentals of Financial Mathematics......Page 255
7.6.3 Methods of Static Investment Calculation......Page 257
7.6.4.1 Net Present Value Method (Net Present Value, Amount of Capital, Final Asset Value)......Page 258
7.6.4.2 Annuity Method......Page 261
7.6.4.3 Internal Rate of Return Method......Page 263
8.1.2 Formation of the Lagrange Function......Page 266
8.1.3 Determination of the Solution......Page 267
8.1.4 Interpretation of λ......Page 268
8.2.2 The Linear Programming Approach......Page 271
8.2.3 Graphical Solution......Page 272
8.2.4 Simplex Algorithm......Page 275
8.2.5 Simplex Tableau (Basic Structure)......Page 276
9.1 Introduction......Page 282
9.1.1 Composition of Functions......Page 286
9.1.2 Inverse Function......Page 288
9.2 Classification of Functions......Page 290
9.2.1.2 Broken Rational Functions......Page 291
9.2.2.1 Power Functions......Page 295
9.2.2.2 Root Function......Page 298
9.2.2.3.1 Exponential Functions......Page 299
9.2.2.3.2 Logarithmic Functions......Page 305
9.2.2.4 Trigonometric Functions (Angle Functions/Circular Functions)......Page 311
9.3.1 Boundedness......Page 339
9.3.2.1 Axial Symmetry......Page 341
9.3.2.2 Point Symmetry......Page 343
9.3.3 Transformations......Page 346
9.3.3.1 Vertex Form......Page 348
9.3.5 Infinite Discontinuities......Page 351
9.3.6 Removable Discontinuities......Page 353
9.3.7 Jump Discontinuities......Page 354
9.3.8 Homogeneity......Page 355
9.3.10 Zeros......Page 356
9.3.11 Local Extremes......Page 357
9.3.12 Monotonicity......Page 358
9.3.13 Concavity and Convexity | Inflection Points......Page 359
9.3.14 Asymptotes......Page 361
9.3.14.1 Horizontal Asymptotes......Page 362
9.3.14.2 Vertical Asymptote......Page 364
9.3.14.3 Oblique Asymptote......Page 365
9.3.14.4 Asymptotic Curve......Page 366
9.3.15 Tangent Lines to a Curve......Page 367
9.3.16 Normal Lines to a Curve......Page 368
9.4 Exercises......Page 369
10.1.1 General......Page 374
10.1.2 First Derivative of Elementary Functions......Page 377
10.1.3 Derivation Rules......Page 379
10.1.4 Higher Derivations......Page 381
10.1.6 Curve Sketching......Page 382
10.2.1 Partial Derivatives (1st Order)......Page 392
10.2.2 Partial Derivatives (2nd Order)......Page 395
10.2.3.1 Relative Extrema without Constraint of the Function f = f (x, y)......Page 397
10.2.3.2 Relative Extrema with m Constraints of the Function f = f (x1, ..., xn) with m < n......Page 401
10.2.4 Differentials of the Function f = f (x1, ..., xn)......Page 405
10.3.1 Mean Value Theorem for Differential Calculus......Page 407
10.3.3 Rolle’s Theorem......Page 408
10.3.4 L’Hospital’s Rule......Page 409
10.3.5 Bounds Theorem for Differential Calculus......Page 410
11.1 Introduction......Page 411
11.2.1 Definition/Determining the Antiderivative......Page 412
11.2.2 Elementary Calculation Rules for the Indefinite Integral......Page 415
11.3.1 Introduction......Page 416
11.3.2 Relationship between the Definite and the Indefinite Integral......Page 420
11.3.3.1 Partial Integration......Page 425
11.3.3.2 Integration by Substitution......Page 427
11.4 Multiple Integrals......Page 428
11.5.1 Cost Functions......Page 429
11.5.2 Revenue Function (= Sales Function)......Page 431
11.5.3 Profit Functions......Page 432
12.1 Definition of Elasticity......Page 436
12.2 Arc Elasticity......Page 437
12.3 Point Elasticity......Page 442
12.4 Price Elasticity of Demand εxp......Page 445
12.5 Cross Elasticity of Demand εxAρB......Page 450
12.6 Income Elasticity of Demand εxy......Page 451
13.1 Supply Function......Page 453
13.2 Demand Function / Inverse Demand Function......Page 455
13.3 Market Equilibrium......Page 457
13.4 Buyer’s Market and Seller’s Market......Page 458
13.6 Demand Gap......Page 459
13.7 Revenue Function......Page 460
13.8 Cost Functions......Page 466
13.9 Neoclassical Cost Function......Page 474
13.10 Cost Function According to the Law of Diminishing Returns......Page 481
13.11 Direct Costs versus Indirect Costs......Page 492
13.11.1 One-Dimensional Cost Allocation Principles......Page 495
13.11.2 Multi-Dimensional Cost Allocation Principles......Page 497
13.12 Profit Function......Page 500
The Current Human Lifestyle Cannot be Continued......Page 508
The Peren Theorem......Page 509
Options for Securing Human Livelihood......Page 511
Individual Prosperity Effects......Page 512
Appendix A Financial Mathematical Factors......Page 515
Appendix B Bibliography......Page 561
Index......Page 565