Introduction to Differential Calculus
Systematic Studies with Engineering Applications for Beginners

Introduction to Differential Calculus fully engages readers by presenting the fundamental theories and methods of differential calculus and then showcasing how the discussed concepts can be applied to real-world problems in engineering and the physical sciences. With its easy-to-follow style and accessible explanations, the book sets a solid foundation before advancing to specific calculus methods, demonstrating the connections between differential calculus theory and its applications.

The first five chapters introduce underlying concepts such as algebra, geometry, coordinate geometry, and trigonometry. Subsequent chapters present a broad range of theories, methods, and applications in differential calculus, including:

• Concepts of function, continuity, and derivative

• Properties of exponential and logarithmic function

• Inverse trigonometric functions and their properties

• Derivatives of higher order

• Methods to find maximum and minimum values of a function

• Hyperbolic functions and their properties

Readers are equipped with the necessary tools to quickly learn how to understand a broad range of current problems throughout the physical sciences and engineering that can only be solved with calculus. Examples throughout provide practical guidance, and practice problems and exercises allow for further development and fine-tuning of various calculus skills. Introduction to Differential Calculus is an excellent book for upper-undergraduate calculus courses and is also an ideal reference for students and professionals alike who would like to gain a further understanding of the use of calculus to solve problems in a simplified manner.

Preface xvii

Biographies xxv

Introduction xxvii

Acknowledgments xxix

1 From Arithmetic to Algebra (What must you know to learn Calculus?) 1

1.1 Introduction 1

1.2 The Set of Whole Numbers 1

1.3 The Set of Integers 1

1.4 The Set of Rational Numbers 1

1.5 The Set of Irrational Numbers 2

1.6 The Set of Real Numbers 2

1.7 Even and Odd Numbers 3

1.8 Factors 3

1.9 Prime and Composite Numbers 3

1.10 Coprime Numbers 4

1.11 Highest Common Factor (H.C.F.) 4

1.12 Least Common Multiple (L.C.M.) 4

1.13 The Language of Algebra 5

1.14 Algebra as a Language for Thinking 7

1.15 Induction 9

1.16 An Important Result: The Number of Primes is Infinite 10

1.17 Algebra as the Shorthand of Mathematics 10

1.18 Notations in Algebra 11

1.19 Expressions and Identities in Algebra 12

1.20 Operations Involving Negative Numbers 15

1.21 Division by Zero 16

2 The Concept of a Function (What must you know to learn Calculus?) 19

2.1 Introduction 19

2.2 Equality of Ordered Pairs 20

2.3 Relations and Functions 20

2.4 Definition 21

2.5 Domain, Codomain, Image, and Range of a Function 23

2.6 Distinction Between “f ” and “f(x)” 23

2.7 Dependent and Independent Variables 24

2.8 Functions at a Glance 24

2.9 Modes of Expressing a Function 24

2.10 Types of Functions 25

2.11 Inverse Function f 1 29

2.12 Comparing Sets without Counting their Elements 32

2.13 The Cardinal Number of a Set 32

2.14 Equivalent Sets (Definition) 33

2.15 Finite Set (Definition) 33

2.16 Infinite Set (Definition) 34

2.17 Countable and Uncountable Sets 36

2.18 Cardinality of Countable and Uncountable Sets 36

2.19 Second Definition of an Infinity Set 37

2.20 The Notion of Infinity 37

2.21 An Important Note About the Size of Infinity 38

2.22 Algebra of Infinity (1) 38

3 Discovery of Real Numbers: Through Traditional Algebra (What must you know to learn Calculus?) 41

3.1 Introduction 41

3.2 Prime and Composite Numbers 42

3.3 The Set of Rational Numbers 43

3.4 The Set of Irrational Numbers 43

3.5 The Set of Real Numbers 43

3.6 Definition of a Real Number 44

3.7 Geometrical Picture of Real Numbers 44

3.8 Algebraic Properties of Real Numbers 44

3.9 Inequalities (Order Properties in Real Numbers) 45

3.10 Intervals 46

3.11 Properties of Absolute Values 51

3.12 Neighborhood of a Point 54

3.13 Property of Denseness 55

3.14 Completeness Property of Real Numbers 55

3.15 (Modified) Definition II (l.u.b.) 60

3.16 (Modified) Definition II (g.l.b.) 60

4 From Geometry to Coordinate Geometry (What must you know to learn Calculus?) 63

4.1 Introduction 63

4.2 Coordinate Geometry (or Analytic Geometry) 64

4.3 The Distance Formula 69

4.4 Section Formula 70

4.5 The Angle of Inclination of a Line 71

4.6 Solution(s) of an Equation and its Graph 76

4.7 Equations of a Line 83

4.8 Parallel Lines 89

4.9 Relation Between the Slopes of (Nonvertical) Lines that are Perpendicular to One Another 90

4.10 Angle Between Two Lines 92

4.11 Polar Coordinate System 93

5 Trigonometry and Trigonometric Functions (What must you know to learn Calculus?) 97

5.1 Introduction 97

5.2 (Directed) Angles 98

5.3 Ranges of sin and cos 109

5.4 Useful Concepts and Definitions 111

5.5 Two Important Properties of Trigonometric Functions 114

5.6 Graphs of Trigonometric Functions 115

5.7 Trigonometric Identities and Trigonometric Equations 115

5.8 Revision of Certain Ideas in Trigonometry 120

6 More About Functions (What must you know to learn Calculus?) 129

6.1 Introduction 129

6.2 Function as a Machine 129

6.3 Domain and Range 130

6.4 Dependent and Independent Variables 130

6.5 Two Special Functions 132

6.6 Combining Functions 132

6.7 Raising a Function to a Power 137

6.8 Composition of Functions 137

6.9 Equality of Functions 142

6.10 Important Observations 142

6.11 Even and Odd Functions 143

6.12 Increasing and Decreasing Functions 144

6.13 Elementary and Nonelementary Functions 147

7a The Concept of Limit of a Function (What must you know to learn Calculus?) 149

7a.1 Introduction 149

7a.2 Useful Notations 149

7a.3 The Concept of Limit of a Function: Informal Discussion 151

7a.4 Intuitive Meaning of Limit of a Function 153

7a.5 Testing the Definition [Applications of the «, d Definition of Limit] 163

7a.6 Theorem (B): Substitution Theorem 174

7a.7 Theorem (C): Squeeze Theorem or Sandwich Theorem 175

7a.8 One-Sided Limits (Extension to the Concept of Limit) 175

7b Methods for Computing Limits of Algebraic Functions (What must you know to learn Calculus?) 177

7b.1 Introduction 177

7b.2 Methods for Evaluating Limits of Various Algebraic Functions 178

7b.3 Limit at Infinity 187

7b.4 Infinite Limits 190

7b.5 Asymptotes 192

8 The Concept of Continuity of a Function, and Points of Discontinuity (What must you know to learn Calculus?) 197

8.1 Introduction 197

8.2 Developing the Definition of Continuity “At a Point” 204

8.3 Classification of the Points of Discontinuity: Types of Discontinuities 214

8.4 Checking Continuity of Functions Involving Trigonometric, Exponential, and Logarithmic Functions 215

8.5 From One-Sided Limit to One-Sided Continuity and its Applications 224

8.6 Continuity on an Interval 224

8.7 Properties of Continuous Functions 225

9 The Idea of a Derivative of a Function 235

9.1 Introduction 235

9.2 Definition of the Derivative as a Rate Function 239

9.3 Instantaneous Rate of Change of y [=f(x)] at x=x₁ and the Slope of its Graph at x=x₁ 239

9.4 A Notation for Increment(s) 246

9.5 The Problem of Instantaneous Velocity 246

9.6 Derivative of Simple Algebraic Functions 259

9.7 Derivatives of Trigonometric Functions 263

9.8 Derivatives of Exponential and Logarithmic Functions 264

9.9 Differentiability and Continuity 264

9.10 Physical Meaning of Derivative 270

9.11 Some Interesting Observations 271

9.12 Historical Notes 273

10 Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of Differentiable Functions 275

10.1 Introduction 275

10.2 Recalling the Operator of Differentiation 277

10.3 The Derivative of a Composite Function 290

10.4 Usefulness of Trigonometric Identities in Computing Derivatives 300

10.5 Derivatives of Inverse Functions 302

11a Basic Trigonometric Limits and Their Applications in Computing Derivatives of Trigonometric Functions 307

11a.1 Introduction 307

11a.2 Basic Trigonometric Limits 308

11a.3 Derivatives of Trigonometric Functions 314

11b Methods of Computing Limits of Trigonometric Functions 325

11b.1 Introduction 325

11b.2 Limits of the Type (I) 328

11b.3 Limits of the Type (II) [ lim f(x), where a&rae;0] 332

11b.4 Limits of Exponential and Logarithmic Functions 335

12 Exponential Form(s) of a Positive Real Number and its Logarithm(s): Pre-Requisite for Understanding Exponential and Logarithmic Functions (What must you know to learn Calculus?) 339

12.1 Introduction 339

12.2 Concept of Logarithmic 339

12.3 The Laws of Exponent 340

12.4 Laws of Exponents (or Laws of Indices) 341

12.5 Two Important Bases: “10” and “e” 343

12.6 Definition: Logarithm 344

12.7 Advantages of Common Logarithms 346

12.8 Change of Base 348

12.9 Why were Logarithms Invented? 351

12.10 Finding a Common Logarithm of a (Positive) Number 351

12.11 Antilogarithm 353

12.12 Method of Calculation in Using Logarithm 355

13a Exponential and Logarithmic Functions and Their Derivatives (What must you know to learn Calculus?) 359

13a.1 Introduction 359

13a.2 Origin of e 360

13a.3 Distinction Between Exponential and Power Functions 362

13a.4 The Value of e 362

13a.5 The Exponential Series 364

13a.6 Properties of e and Those of Related Functions 365

13a.7 Comparison of Properties of Logarithm(s) to the Bases 10 and e 369

13a.8 A Little More About e 371

13a.9 Graphs of Exponential Function(s) 373

13a.10 General Logarithmic Function 375

13a.11 Derivatives of Exponential and Logarithmic Functions 378

13a.12 Exponential Rate of Growth 383

13a.13 Higher Exponential Rates of Growth 383

13a.14 An Important Standard Limit 385

13a.15 Applications of the Function ex: Exponential Growth and Decay 390

13b Methods for Computing Limits of Exponential and Logarithmic Functions 401

13b.1 Introduction 401

13b.2 Review of Logarithms 401

13b.3 Some Basic Limits 403

13b.4 Evaluation of Limits Based on the Standard Limit 410

14 Inverse Trigonometric Functions and Their Derivatives 417

14.1 Introduction 417

14.2 Trigonometric Functions (With Restricted Domains) and Their Inverses 420

14.3 The Inverse Cosine Function 425

14.4 The Inverse Tangent Function 428

14.5 Definition of the Inverse Cotangent Function 431

14.6 Formula for the Derivative of Inverse Secant Function 433

14.7 Formula for the Derivative of Inverse Cosecant Function 436

14.8 Important Sets of Results and their Applications 437

14.9 Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions 441

15a Implicit Functions and Their Differentiation 453

15a.1 Introduction 453

15a.2 Closer Look at the Difficulties Involved 455

15a.3 The Method of Logarithmic Differentiation 463

15a.4 Procedure of Logarithmic Differentiation 464

15b Parametric Functions and Their Differentiation 473

15b.1 Introduction 473

15b.2 The Derivative of a Function Represented Parametrically 477

15b.3 Line of Approach for Computing the Speed of a Moving Particle 480

15b.4 Meaning of dy/dx with Reference to the Cartesian Form y = f(x) and Parametric Forms x = f(t), y = g(t) of the Function 481

15b.5 Derivative of One Function with Respect to the Other 483

16 Differentials “dy” and “dx”: Meanings and Applications 487

16.1 Introduction 487

16.2 Applying Differentials to Approximate Calculations 492

16.3 Differentials of Basic Elementary Functions 494

16.4 Two Interpretations of the Notation dy/dx 498

16.5 Integrals in Differential Notation 499

16.6 To Compute (Approximate) Small Changes and Small Errors Caused in Various Situations 503

17 Derivatives and Differentials of Higher Order 511

17.1 Introduction 511

17.2 Derivatives of Higher Orders: Implicit Functions 516

17.3 Derivatives of Higher Orders: Parametric Functions 516

17.4 Derivatives of Higher Orders: Product of Two Functions (Leibniz Formula) 517

17.5 Differentials of Higher Orders 521

17.6 Rate of Change of a Function and Related Rates 523

18 Applications of Derivatives in Studying Motion in a Straight Line 535

18.1 Introduction 535

18.2 Motion in a Straight Line 535

18.3 Angular Velocity 540

18.4 Applications of Differentiation in Geometry 540

18.5 Slope of a Curve in Polar Coordinates 548

19a Increasing and Decreasing Functions and the Sign of the First Derivative 551

19a.1 Introduction 551

19a.2 The First Derivative Test for Rise and Fall 556

19a.3 Intervals of Increase and Decrease (Intervals of Monotonicity) 557

19a.4 Horizontal Tangents with a Local Maximum/Minimum 565

19a.5 Concavity, Points of Inflection, and the Sign of the Second Derivative 567

19b Maximum and Minimum Values of a Function 575

19b.1 Introduction 575

19b.2 Relative Extreme Values of a Function 576

19b.3 Theorem A 580

19b.4 Theorem B: Sufficient Conditions for the Existence of a Relative Extrema—In Terms of the First Derivative 584

19b.5 Sufficient Condition for Relative Extremum (In Terms of the Second Derivative) 588

19b.6 Maximum and Minimum of a Function on the Whole Interval (Absolute Maximum and Absolute Minimum Values) 593

19b.7 Applications of Maxima and Minima Techniques in Solving Certain Problems Involving the Determination of the Greatest and the Least Values 597

20 Rolle’s Theorem and the Mean Value Theorem (MVT) 605

20.1 Introduction 605

20.2 Rolle’s Theorem (A Theorem on the Roots of a Derivative) 608

20.3 Introduction to the Mean Value Theorem 613

20.4 Some Applications of the Mean Value Theorem 622

21 The Generalized Mean Value Theorem (Cauchy’s MVT), L’ Hospital’s Rule, and their Applications 625

21.1 Introduction 625

21.2 Generalized Mean Value Theorem (Cauchy’s MVT) 625

21.3 Indeterminate Forms and L’Hospital’s Rule 627

21.4 L’Hospital’s Rule (First Form) 630

21.5 L’Hospital’s Theorem (For Evaluating Limits(s) of the Indeterminate Form 0/0.) 632

21.6 Evaluating Indeterminate Form of the Type

∞/∞ 638

21.7 Most General Statement of L’Hospital’s Theorem 644

21.8 Meaning of Indeterminate Forms 644

21.9 Finding Limits Involving Various Indeterminate Forms (by Expressing them to the Form 0/0 or ∞/∞) 646

22 Extending the Mean Value Theorem to Taylor’s Formula: Taylor Polynomials for Certain Functions 653

22.1 Introduction 653

22.2 The Mean Value Theorem For Second Derivatives: The First Extended MVT 654

22.3 Taylor’s Theorem 658

22.4 Polynomial Approximations and Taylor’s Formula 658

22.5 From Maclaurin Series To Taylor Series 667

22.6 Taylor’s Formula for Polynomials 669

22.7 Taylor’s Formula for Arbitrary Functions 672

23 Hyperbolic Functions and Their Properties 677

23.1 Introduction 677

23.2 Relation Between Exponential and Trigonometric Functions 680

23.3 Similarities and Differences in the Behavior of Hyperbolic and Circular Functions 682

23.4 Derivatives of Hyperbolic Functions 685

23.5 Curves of Hyperbolic Functions 686

23.6 The Indefinite Integral Formulas for Hyperbolic Functions 689

23.7 Inverse Hyperbolic Functions 689

23.8 Justification for Calling sinh and cosh as Hyperbolic Functions Just as sine and cosine are Called Trigonometric Circular Functions 699

Appendix A (Related To Chapter-2) Elementary Set Theory 703

Appendix B (Related To Chapter-4) 711

Appendix C (Related To Chapter-20) 735

Index 739

G. C. Jain, BSc, is a retired scientist from the Defense Research and Development Organization in India.

Ajay K. Poddar, PhD, is Chief Scientist at Synergy Microwave Corporation. A Senior Member of the IEEE, Dr. Poddar holds several dozen patents and has published more than 180 scientific papers.

A. K. Ghosh, PhD, is Professor in the Department of Aerospace Engineering at IIT Kanpur, India. He has published more than 120 scientific papers.