INTRODUCTION TO CALCULUS FOR BUSINESS AND ECONOMICS

RAMA SHANKER, Ph.D.

Associate Professor of Statistics

Department of Statistics Eritrea Institute of Technology MAINEFHI, ASMARA, ERITREA


RAVI SHANKER, Ph.D.

Assistant Professor of Mathematics & Head Department of Mathematics

G.L.A. College

Nilambar Pitambar University

Medininagar, Daltanganj, Jharkhand, India


CONTENTS

Chapter I : Functions Page 1-53
1.1. Introduction 1
1.2. Functions 1
1.3. Intervals 6
1.4. Types of Functions 6
1.4.1. Constant Function 6
1.4.2. Identity Function 7
1.4.3. Absolute Value Function 7
1.4.4. Linear Functions 7
1.4.5. Quadratic Functions 8
1.4.6. Polynomial Functions 9
1.4.7. Rational Functions 10
1.4.8. One-to-one Function 11
1.4.9. Odd and Even Functions 11
1.4.10. Monotonic Functions 13
1.4.11. Composite Functions 13
1.4.12. Inverse Functions 15
1.4.13. Explicit and Implicit Functions 17
1.4.14. Exponential Functions 17
1.4.15. Logarithmic Functions 21
1.4.16. Power Functions 23
1.4.17. Homogeneous Functions 25
1.4.18. Homothetic Functions 28
1.4.19. Concave and Convex Functions 31
1.4.20. Periodic Functions 32
1.5. Functions in Business and Economics 32
1.5.1. Demand Function 32
1.5.2. Supply Function 33
1.5.3. Cost Function 35
1.5.4. Revenue Function 35
1.5.5. Profit Function 35
1.5.6. Consumption Function 36
1.6. The algebra of Functions 36
1.7. Vertical and Horizontal Translation of Functions 37
1.8. Applications of Linear Functions in Business and Economics 39
1.9. Applications of Non-linear Functions in Business and Economics 45
Chapter II : Limits and Continuity of Functions 54-108
2.1. Introduction 54
2.2. The Concept of Limit 54
2.3. Properties of Limits and Evaluation of Limits 56
2.4. One Sided Limits 69
2.5. Limits at Infinity and Infinite Limit 73
2.6. The Concept of Continuity 78
2.7. One-Sided Continuity 84
.8. Continuity on Intervals 85
2.9. Types of Discontinuity 90
2.10. The Intermediate Value Theorem (IVT) 93
2.11. Asymptotes of Function 95
2.12. Applications of Limit and Continuity to Business and Economics. 100
Chapter III : Differentiation 109-186
3.1. Introduction 109
3.2. The Slope of a Graph 109
3.3. The Derivative of a Function 116
3.4. Differentiability and Continuity 120
3.5. Differentiability and Linear Approximations 122
3.6. Rules of Differentiation 125
3.7. The Chain Rule and Composite Functions 135
3.8. Derivatives of Logarithmic and Exponential Functions 143
3.9. Implicit Differentiation 152
3.10. Derivative of Inverse of a Function 159
3.11. Derivative of a Function in Parametric Form 161
3.12. Derivative of a Function with respect to another Function 165
3.13. Logarithmic Differentiation 166
3.14. Successive Differentiation 173
3.15. nth Derivative of some Standard Functions 179
3.16. Leibnitz's Theorem. 182
3.17. Second Derivative in Parametric Form. 185
Chapter IV : Further Topics in Differentiation 187-122
4.1. Introduction 187
4.2. Increasing and Decreasing Functions 187
4.3. Relative Maxima and Minima of a Function 193
4.4. Global Maxima and Minima of a Function 204
4.5. Concavity and Convexity of a Function 209
4.6. Rolle's and Lagrange's Mean Value Theorem 219
Chapter V : Applications of Differentiation to Business and Economics 223-316
5.1. Introduction 223
5.2. Average and Marginal Functions 223
.3. Marginal Revenue Product 232
5.4. Marginal Propensity to Consume and Save 233
5.5. The Investment Multiplier 236
5.6. The Money Multiplier 238
5.7. The Elasticity of Functions 239
5.8. Rate of Change, Relative Rate of Change and Related Rates 257
.9. Univariate Utility Function and Risk Aversion 264
5.10. Monopoly Problems 271
5.11. Effect of Taxation under Pure Competition 281
5.12. Effects of Taxation and Subsidy on Monopoly 284
5.13. Law of diminishing Returns 297
5.14. Applied Optimization Problems 300
Chapter VI : Partial Differentiation 317-358
6.1. Introduction 317
.2. Functions of Several Independent Variables 317
.3. Partial Differentiation 318
6.4. Rules of Partial Differentiation 319
6.5. Composite Functions and the Chain Rule 324
6.6. Second Order Partial Derivatives and Young's Theorem 327
6.7. Homogeneous Functions and Euler's Theorem 330
6.8. Total Differentials and Implicit Differentiation 340
6.9. Second Order Total Differentials 344
6.10. Maxima and Minima of a Function of two Variables 347
6.11. Alternative Method of Maxima and Minima of Functions of two or more Variables. 354
Chapter VII : Applications of Partial Differentiation to Business and Economics 359-453
7.1. Introduction 359
7.2. Joint and Marginal Functions 359
7.3. Partial Elasticity of Demand and Nature of Commodities 362
7.4. The Production Function 375
7.5. Isoquants (or Constant Product Curves) 388
7.6. Elasticity of Substitution 392
7.7. The Bivariate Utility Function 398
7.8. Indifference Curve and Marginal Rate of Substitution 400
7.9. Multiple Production by Monopolist 406
7.10. Discriminating Monopoly Problems 415
7.11. Duopoly Problems 426
7.12. Means and Outliers 435
7.13. Multipliers 438
7.14. Applied Optimization Problems 445
Chapter VIII: Integration 454-539
8.1. Introduction 454
8.2. The Indefinite Integral 456
8.3. Rules of Indefinite Integrals 456
8.4. Methods of Integration 460
8.4.1. Integration by Substitution 461
8.4.2. Integration by Parts 465
8.4.3. Integration by Partial Fractions 468
8.5. Area under a Curve 481
8.6. The Definite Integral 483
8.7. The Area between two Curves 503
8.8. The Improper Integrals 521
8.9. Gamma and Beta Functions 531
8.10. Numerical Integration 532
8.11. Double Integrals 536
8.12. Differentiation of Integrals 539
Chapter IX : Applications of Integration to Business and Economics 540-647
9.1. Introduction 540
9.2. Total Cost and Average Cost Functions from Marginal Cost Functions 540
9.3. Total Revenue and Demand Functions from Marginal Revenue 546
9.4. Functions Maximum Profit from Marginal Revenue and Marginal Cost 552
9.5. Functions Maximum Profit over Time when Marginal Revenue and Marginal Cost are given as Functions of Time 556
9.6. Demand Function from the Price Elasticity of Demand 561
9.7. Consumption Function from Marginal Propensity to Consume 564
9.8. Consumer's and Producer's Surplus 566
9.9. The Learning Curves 590
9.10. Investment and Capital Formation 593
9.11. Continuous Income Stream (or A Cash Flow) 596
9.12. Rate of Sales 607
9.13. Bond Price Volatility and Bond Maturity 613
9.14. Measuring Income Inequality 614
9.15. Pareto Income Distribution 618
9.16. Capitalization 619
9.17. Continuous Probability 619
9.17.1. Probability Density Function 620
9.17.2. Cumulative Distribution Function 623
9.17.3. The Expected Value, The Variance, and The Median 627
.17.4. The Uniform Distribution 631
9.17.5. The Exponential Distribution 634
9.17.6. The Normal Distribution 637
9.17.7. Joint Probability Density Functions 638
Chapter X: Constrained Optimization 648-692
10.1. Introduction 648
10.2. The Substitution method 648
10.3. Lagrange Multiplier Method 650
10.3.1. Lagrange Multipliers in two Variables and Single Constraint 650
10.3.2. Lagrange Multipliers in n Variables and Single Constraint 653
10.3.3. Lagrange Multipliers in n Variables and m Constraints (m<n) 658
10.4. Economic Interpretation of Lagrange Multiplier 667
10.5. Constrained Optimization of Business and Economics problems 669
Chapter XI : Differential Equations 693-744
11.1. Introduction 693
11.2. Meaning of Differential Equation 693
11.3. Order and Degree of a Differential Equation 694
11.4. Formation of Differential Equations 695
11.5. Initial Value Problems 696
11.6. Solution of Differential Equation 699
11.7. Differential Equations of First Order and First Degree 710
11.7.1. Variable Separable Differential Equations 711
11.7.2. Homogeneous Differential Equations 719
11.7.3. Non-Homogeneous Differential Equations 724
11.7.4. Linear Differential Equations 729
11.7.5. Exact Differential Equations 740
Chapter XII : Applications of Differential Equations to Business and Economics 45-801
12.1. Introduction 745
12.2. Rates of Change 745
12.3. A Continuous Time Price Adjustment Model 754
12.4. National Income Determination 763
12.5. Malthusian Population Growth Model 768
12.6. The Logistic Growth Model 771
12.7. The Domar Growth Model 780
12.8. The Solow Growth Model 782
12.9. Continuous Income Stream 790
Answers :   802-853