INTRODUCTION TO DIFFERENTIAL CALCULUS: Systematic Studies with Engineering Applications for Beginners -- CONTENTS -- Foreword -- Preface -- Biographies -- Introduction -- Acknowledgments -- 1 From Arithmetic to Algebra (What must you know to learn Calculus?) -- 1.1 Introduction -- 1.2 The Set of Whole Numbers -- 1.3 The Set of Integers -- 1.4 The Set of Rational Numbers -- 1.5 The Set of Irrational Numbers -- 1.6 The Set of Real Numbers -- 1.7 Even and Odd Numbers -- 1.8 Factors -- 1.9 Prime and Composite Numbers -- 1.10 Coprime Numbers -- 1.11 Highest Common Factor (H.C.F.) -- 1.12 Least Common Multiple (L.C.M.) -- 1.13 The Language of Algebra -- 1.14 Algebra as a Language for Thinking -- 1.15 Induction -- 1.16 An Important Result: The Number of Primes is Infinite -- 1.17 Algebra as the Shorthand of Mathematics -- 1.18 Notations in Algebra -- 1.19 Expressions and Identities in Algebra -- 1.20 Operations Involving Negative Numbers -- 1.21 Division by Zero -- 2 The Concept of a Function (What must you know to learn Calculus?) -- 2.1 Introduction -- 2.2 Equality of Ordered Pairs -- 2.3 Relations and Functions -- 2.4 Definition -- 2.5 Domain, Codomain, Image, and Range of a Function -- 2.6 Distinction Between "f " and "f(x)" -- 2.7 Dependent and Independent Variables -- 2.8 Functions at a Glance -- 2.9 Modes of Expressing a Function -- 2.10 Types of Functions -- 2.11 Inverse Function f‾¹ -- 2.12 Comparing Sets without Counting their Elements -- 2.13 The Cardinal Number of a Set -- 2.14 Equivalent Sets (Definition) -- 2.15 Finite Set (Definition) -- 2.16 Infinite Set (Definition) -- 2.17 Countable and Uncountable Sets -- 2.18 Cardinality of Countable and Uncountable Sets -- 2.19 Second Definition of an Infinity Set -- 2.20 The Notion of Infinity -- 2.21 An Important Note About the Size of Infinity -- 2.22 Algebra of Infinity (∞).

3 Discovery of Real Numbers: Through Traditional Algebra (What must you know to learn Calculus?) -- 3.1 Introduction -- 3.2 Prime and Composite Numbers -- 3.3 The Set of Rational Numbers -- 3.3 The Set of Rational Numbers -- 3.4 The Set of Irrational Numbers -- 3.5 The Set of Real Numbers -- 3.6 Definition of a Real Number -- 3.7 Geometrical Picture of Real Numbers -- 3.8 Algebraic Properties of Real Numbers -- 3.9 Inequalities (Order Properties in Real Numbers) -- 3.10 Intervals -- 3.11 Properties of Absolute Values -- 3.12 Neighborhood of a Point -- 3.13 Property of Denseness -- 3.14 Completeness Property of Real Numbers -- 3.15 (Modified) Definition II (l.u.b.) -- 3.16 (Modified) Definition II (g.l.b.) -- 4 From Geometry to Coordinate Geometry (What must you know to learn Calculus?) -- 4.1 Introduction -- 4.2 Coordinate Geometry (or Analytic Geometry) -- 4.3 The Distance Formula -- 4.4 Section Formula -- 4.5 The Angle of Inclination of a Line -- 4.6 Solution(s) of an Equation and its Graph -- 4.7 Equations of a Line -- 4.8 Parallel Lines -- 4.9 Relation Between the Slopes of (Nonvertical) Lines that are Perpendicular to One Another -- 4.10 Angle Between Two Lines -- 4.11 Polar Coordinate System -- 5 Trigonometry and Trigonometric Functions (What must you know to learn Calculus?) -- 5.1 Introduction -- 5.2 (Directed) Angles -- 5.3 Ranges of sin θ and cos θ -- 5.4 Useful Concepts and Definitions -- 5.5 Two Important Properties of Trigonometric Functions -- 5.6 Graphs of Trigonometric Functions -- 5.7 Trigonometric Identities and Trigonometric Equations -- 5.8 Revision of Certain Ideas in Trigonometry -- 6 More About Functions (What must you know to learn Calculus?) -- 6.1 Introduction -- 6.2 Function as a Machine -- 6.3 Domain and Range -- 6.4 Dependent and Independent Variables -- 6.5 Two Special Functions -- 6.6 Combining Functions.

6.7 Raising a Function to a Power -- 6.8 Composition of Functions -- 6.9 Equality of Functions -- 6.10 Important Observations -- 6.11 Even and Odd Functions -- 6.12 Increasing and Decreasing Functions -- 6.13 Elementary and Nonelementary Functions -- 7a The Concept of Limit of a Function (What must you know to learn Calculus?) -- 7a.1 Introduction -- 7a.2 Useful Notations -- 7a.3 The Concept of Limit of a Function: Informal Discussion -- 7a.4 Intuitive Meaning of Limit of a Function -- 7a.5 Testing the Definition [Applications of the ε, δ Definition of Limit] -- 7a.6 Theorem (B): Substitution Theorem -- 7a.7 Theorem (C): Squeeze Theorem or Sandwich Theorem -- 7a.8 One-Sided Limits (Extension to the Concept of Limit) -- 7b Methods for Computing Limits of Algebraic Functions (What must you know to learn Calculus?) -- 7b.1 Introduction -- 7b.2 Methods for Evaluating Limits of Various Algebraic Functions -- 7b.3 Limit at Infinity -- 7b.4 Infinite Limits -- 7b.5 Asymptotes -- 8 The Concept of Continuity of a Function, and Points of Discontinuity (What must you know to learn Calculus?) -- 8.1 Introduction -- 8.2 Developing the Definition of Continuity "At a Point" -- 8.3 Classification of the Points of Discontinuity: Types of Discontinuities -- 8.4 Checking Continuity of Functions Involving Trigonometric, Exponential, and Logarithmic Functions -- 8.5 From One-Sided Limit to One-Sided Continuity and its Applications -- 8.6 Continuity on an Interval -- 8.7 Properties of Continuous Functions -- 9 The Idea of a Derivative of a Function -- 9.1 Introduction -- 9.2 Definition of the Derivative as a Rate Function -- 9.3 Instantaneous Rate of Change of y [=f(x)] at x=x1 and the Slope of its Graph at x=x1 -- 9.4 A Notation for Increment(s) -- 9.5 The Problem of Instantaneous Velocity -- 9.6 Derivative of Simple Algebraic Functions.

9.7 Derivatives of Trigonometric Functions -- 9.8 Derivatives of Exponential and Logarithmic Functions -- 9.9 Differentiability and Continuity -- 9.10 Physical Meaning of Derivative -- 9.11 Some Interesting Observations -- 9.12 Historical Notes -- 10 Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of Differentiable Functions -- 10.1 Introduction -- 10.2 Recalling the Operator of Differentiation -- 10.3 The Derivative of a Composite Function -- 10.4 Usefulness of Trigonometric Identities in Computing Derivatives -- 10.5 Derivatives of Inverse Functions -- 11a Basic Trigonometric Limits and Their Applications in Computing Derivatives of Trigonometric Functions -- 11a.1 Introduction -- 11a.2 Basic Trigonometric Limits -- 11a.3 Derivatives of Trigonometric Functions -- 11b Methods of Computing Limits of Trigonometric Functions -- 11b.1 Introduction -- 11b.2 Limits of the Type (I) -- 11b.3 Limits of the Type (II) [ lim/x→a f(x), where a ≠ 0] -- 11b.4 Limits of Exponential and Logarithmic Functions -- 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?) -- 12.1 Introduction -- 12.2 Concept of Logarithmic -- 12.3 The Laws of Exponent -- 12.4 Laws of Exponents (or Laws of Indices) -- 12.5 Two Important Bases: "10" and "e" -- 12.6 Definition: Logarithm -- 12.7 Advantages of Common Logarithms -- 12.8 Change of Base -- 12.9 Why were Logarithms Invented? -- 12.10 Finding a Common Logarithm of a (Positive) Number -- 12.11 Antilogarithm -- 12.12 Method of Calculation in Using Logarithm -- 13a Exponential and Logarithmic Functions and Their Derivatives (What must you know to learn Calculus?) -- 13a.1 Introduction -- 13a.2 Origin of e -- 13a.3 Distinction Between Exponential and Power Functions.

13a.4 The Value of e -- 13a.5 The Exponential Series -- 13a.6 Properties of e and Those of Related Functions -- 13a.7 Comparison of Properties of Logarithm(s) to the Bases 10 and e -- 13a.8 A Little More About e -- 13a.9 Graphs of Exponential Function(s) -- 13a.10 General Logarithmic Function -- 13a.11 Derivatives of Exponential and Logarithmic Functions -- 13a.12 Exponential Rate of Growth -- 13a.13 Higher Exponential Rates of Growth -- 13a.14 An Important Standard Limit -- 13a.15 Applications of the Function ex: Exponential Growth and Decay -- 13b Methods for Computing Limits of Exponential and Logarithmic Functions -- 13b.1 Introduction -- 13b.2 Review of Logarithms -- 13b.3 Some Basic Limits -- 13b.4 Evaluation of Limits Based on the Standard Limit -- 14 Inverse Trigonometric Functions and Their Derivatives -- 14.1 Introduction -- 14.2 Trigonometric Functions (With Restricted Domains) and Their Inverses -- 14.3 The Inverse Cosine Function -- 14.4 The Inverse Tangent Function -- 14.5 Definition of the Inverse Cotangent Function -- 14.6 Formula for the Derivative of Inverse Secant Function -- 14.7 Formula for the Derivative of Inverse Cosecant Function -- 14.8 Important Sets of Results and their Applications -- 14.9 Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions -- 15a Implicit Functions and Their Differentiation -- 15a.1 Introduction -- 15a.2 Closer Look at the Difficulties Involved -- 15a.3 The Method of Logarithmic Differentiation -- 15a.4 Procedure of Logarithmic Differentiation -- 15b Parametric Functions and Their Differentiation -- 15b.1 Introduction -- 15b.2 The Derivative of a Function Represented Parametrically -- 15b.3 Line of Approach for Computing the Speed of a Moving Particle.

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.