Enables readers to apply the fundamentals of differential
calculus to solve real-life problems in engineering and the
physical sciences



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.

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.

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

…