Introduction to Differential Calculus

Ulrich L. Rohde, PhD, ScD, Dr-Ing, is Chairman of Synergy Microwave Corporation, President of Communications Consulting Corporation, and a Partner of Rohde & Schwarz. A Fellow of the IEEE, Professor Rohde holds several patents and has published more than 200 scientific papers. 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.

“The book is addressed mainly to students studying non-mathematical subjects. It will be also helpful for those who want to understand why it is important to study Calculus and how to apply it.”  (Zentralblatt MATH, 1 December 2012)

• Foreword xiii Preface xviiBiographies xxvIntroduction xxviiAcknowledgments xxix1 From Arithmetic to Algebra (What must you know to learn Calculus?) 11.1 Introduction 11.2 The Set of Whole Numbers 11.3 The Set of Integers 11.4 The Set of Rational Numbers 11.5 The Set of Irrational Numbers 21.6 The Set of Real Numbers 21.7 Even and Odd Numbers 31.8 Factors 31.9 Prime and Composite Numbers 31.10 Coprime Numbers 41.11 Highest Common Factor (H.C.F.) 41.12 Least Common Multiple (L.C.M.) 41.13 The Language of Algebra 51.14 Algebra as a Language for Thinking 71.15 Induction 91.16 An Important Result: The Number of Primes is Infinite 101.17 Algebra as the Shorthand of Mathematics 101.18 Notations in Algebra 111.19 Expressions and Identities in Algebra 121.20 Operations Involving Negative Numbers 151.21 Division by Zero 162 The Concept of a Function (What must you know to learn Calculus?) 192.1 Introduction 192.2 Equality of Ordered Pairs 202.3 Relations and Functions 202.4 Definition 212.5 Domain, Codomain, Image, and Range of a Function 232.6 Distinction Between “f ” and “f(x)” 232.7 Dependent and Independent Variables 242.8 Functions at a Glance 242.9 Modes of Expressing a Function 242.10 Types of Functions 252.11 Inverse Function f 1 292.12 Comparing Sets without Counting their Elements 322.13 The Cardinal Number of a Set 322.14 Equivalent Sets (Definition) 332.15 Finite Set (Definition) 332.16 Infinite Set (Definition) 342.17 Countable and Uncountable Sets 362.18 Cardinality of Countable and Uncountable Sets 362.19 Second Definition of an Infinity Set 372.20 The Notion of Infinity 372.21 An Important Note About the Size of Infinity 382.22 Algebra of Infinity (1) 383 Discovery of Real Numbers: Through Traditional Algebra (What must you know to learn Calculus?) 413.1 Introduction 413.2 Prime and Composite Numbers 423.3 The Set of Rational Numbers 433.4 The Set of Irrational Numbers 433.5 The Set of Real Numbers 433.6 Definition of a Real Number 443.7 Geometrical Picture of Real Numbers 443.8 Algebraic Properties of Real Numbers 443.9 Inequalities (Order Properties in Real Numbers) 453.10 Intervals 463.11 Properties of Absolute Values 513.12 Neighborhood of a Point 543.13 Property of Denseness 553.14 Completeness Property of Real Numbers 553.15 (Modified) Definition II (l.u.b.) 603.16 (Modified) Definition II (g.l.b.) 604 From Geometry to Coordinate Geometry (What must you know to learn Calculus?) 634.1 Introduction 634.2 Coordinate Geometry (or Analytic Geometry) 644.3 The Distance Formula 694.4 Section Formula 704.5 The Angle of Inclination of a Line 714.6 Solution(s) of an Equation and its Graph 764.7 Equations of a Line 834.8 Parallel Lines 894.9 Relation Between the Slopes of (Nonvertical) Lines that are Perpendicular to One Another 904.10 Angle Between Two Lines 924.11 Polar Coordinate System 935 Trigonometry and Trigonometric Functions (What must you know to learn Calculus?) 975.1 Introduction 975.2 (Directed) Angles 985.3 Ranges of sin and cos 1095.4 Useful Concepts and Definitions 1115.5 Two Important Properties of Trigonometric Functions 1145.6 Graphs of Trigonometric Functions 1155.7 Trigonometric Identities and Trigonometric Equations 1155.8 Revision of Certain Ideas in Trigonometry 1206 More About Functions (What must you know to learn Calculus?) 1296.1 Introduction 1296.2 Function as a Machine 1296.3 Domain and Range 1306.4 Dependent and Independent Variables 1306.5 Two Special Functions 1326.6 Combining Functions 1326.7 Raising a Function to a Power 1376.8 Composition of Functions 1376.9 Equality of Functions 1426.10 Important Observations 1426.11 Even and Odd Functions 1436.12 Increasing and Decreasing Functions 1446.13 Elementary and Nonelementary Functions 1477a The Concept of Limit of a Function (What must you know to learn Calculus?) 1497a.1 Introduction 1497a.2 Useful Notations 1497a.3 The Concept of Limit of a Function: Informal Discussion 1517a.4 Intuitive Meaning of Limit of a Function 1537a.5 Testing the Definition [Applications of the «, d Definition of Limit] 1637a.6 Theorem (B): Substitution Theorem 1747a.7 Theorem (C): Squeeze Theorem or Sandwich Theorem 1757a.8 One-Sided Limits (Extension to the Concept of Limit) 1757b Methods for Computing Limits of Algebraic Functions (What must you know to learn Calculus?) 1777b.1 Introduction 1777b.2 Methods for Evaluating Limits of Various Algebraic Functions 1787b.3 Limit at Infinity 1877b.4 Infinite Limits 1907b.5 Asymptotes 1928 The Concept of Continuity of a Function, and Points of Discontinuity (What must you know to learn Calculus?) 1978.1 Introduction 1978.2 Developing the Definition of Continuity “At a Point” 2048.3 Classification of the Points of Discontinuity: Types of Discontinuities 2148.4 Checking Continuity of Functions Involving Trigonometric, Exponential, and Logarithmic Functions 2158.5 From One-Sided Limit to One-Sided Continuity and its Applications 2248.6 Continuity on an Interval 2248.7 Properties of Continuous Functions 2259 The Idea of a Derivative of a Function 2359.1 Introduction 2359.2 Definition of the Derivative as a Rate Function 2399.3 Instantaneous Rate of Change of y [=f(x)] at x=x1 and the Slope of its Graph at x=x1 2399.4 A Notation for Increment(s) 2469.5 The Problem of Instantaneous Velocity 2469.6 Derivative of Simple Algebraic Functions 2599.7 Derivatives of Trigonometric Functions 2639.8 Derivatives of Exponential and Logarithmic Functions 2649.9 Differentiability and Continuity 2649.10 Physical Meaning of Derivative 2709.11 Some Interesting Observations 2719.12 Historical Notes 27310 Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of Differentiable Functions 27510.1 Introduction 27510.2 Recalling the Operator of Differentiation 27710.3 The Derivative of a Composite Function 29010.4 Usefulness of Trigonometric Identities in Computing Derivatives 30010.5 Derivatives of Inverse Functions 30211a Basic Trigonometric Limits and Their Applications in Computing Derivatives of Trigonometric Functions 30711a.1 Introduction 30711a.2 Basic Trigonometric Limits 30811a.3 Derivatives of Trigonometric Functions 31411b Methods of Computing Limits of Trigonometric Functions 32511b.1 Introduction 32511b.2 Limits of the Type (I) 32811b.3 Limits of the Type (II) [ lim f(x), where a&rae;0] 33211b.4 Limits of Exponential and Logarithmic Functions 33512 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?) 33912.1 Introduction 33912.2 Concept of Logarithmic 33912.3 The Laws of Exponent 34012.4 Laws of Exponents (or Laws of Indices) 34112.5 Two Important Bases: “10” and “e” 34312.6 Definition: Logarithm 34412.7 Advantages of Common Logarithms 34612.8 Change of Base 34812.9 Why were Logarithms Invented? 35112.10 Finding a Common Logarithm of a (Positive) Number 35112.11 Antilogarithm 35312.12 Method of Calculation in Using Logarithm 35513a Exponential and Logarithmic Functions and Their Derivatives (What must you know to learn Calculus?) 35913a.1 Introduction 35913a.2 Origin of e 36013a.3 Distinction Between Exponential and Power Functions 36213a.4 The Value of e 36213a.5 The Exponential Series 36413a.6 Properties of e and Those of Related Functions 36513a.7 Comparison of Properties of Logarithm(s) to the Bases 10 and e 36913a.8 A Little More About e 37113a.9 Graphs of Exponential Function(s) 37313a.10 General Logarithmic Function 37513a.11 Derivatives of Exponential and Logarithmic Functions 37813a.12 Exponential Rate of Growth 38313a.13 Higher Exponential Rates of Growth 38313a.14 An Important Standard Limit 38513a.15 Applications of the Function ex: Exponential Growth and Decay 39013b Methods for Computing Limits of Exponential and Logarithmic Functions 40113b.1 Introduction 40113b.2 Review of Logarithms 40113b.3 Some Basic Limits 40313b.4 Evaluation of Limits Based on the Standard Limit 41014 Inverse Trigonometric Functions and Their Derivatives 41714.1 Introduction 41714.2 Trigonometric Functions (With Restricted Domains) and Their Inverses 42014.3 The Inverse Cosine Function 42514.4 The Inverse Tangent Function 42814.5 Definition of the Inverse Cotangent Function 43114.6 Formula for the Derivative of Inverse Secant Function 43314.7 Formula for the Derivative of Inverse Cosecant Function 43614.8 Important Sets of Results and their Applications 43714.9 Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions 44115a Implicit Functions and Their Differentiation 45315a.1 Introduction 45315a.2 Closer Look at the Difficulties Involved 45515a.3 The Method of Logarithmic Differentiation 46315a.4 Procedure of Logarithmic Differentiation 46415b Parametric Functions and Their Differentiation 47315b.1 Introduction 47315b.2 The Derivative of a Function Represented Parametrically 47715b.3 Line of Approach for Computing the Speed of a Moving Particle 48015b.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 48115b.5 Derivative of One Function with Respect to the Other 48316 Differentials “dy” and “dx”: Meanings and Applications 48716.1 Introduction 48716.2 Applying Differentials to Approximate Calculations 49216.3 Differentials of Basic Elementary Functions 49416.4 Two Interpretations of the Notation dy/dx 49816.5 Integrals in Differential Notation 49916.6 To Compute (Approximate) Small Changes and Small Errors Caused in Various Situations 50317 Derivatives and Differentials of Higher Order 51117.1 Introduction 51117.2 Derivatives of Higher Orders: Implicit Functions 51617.3 Derivatives of Higher Orders: Parametric Functions 51617.4 Derivatives of Higher Orders: Product of Two Functions (Leibniz Formula) 51717.5 Differentials of Higher Orders 52117.6 Rate of Change of a Function and Related Rates 52318 Applications of Derivatives in Studying Motion in a Straight Line 53518.1 Introduction 53518.2 Motion in a Straight Line 53518.3 Angular Velocity 54018.4 Applications of Differentiation in Geometry 54018.5 Slope of a Curve in Polar Coordinates 54819a Increasing and Decreasing Functions and the Sign of the First Derivative 55119a.1 Introduction 55119a.2 The First Derivative Test for Rise and Fall 55619a.3 Intervals of Increase and Decrease (Intervals of Monotonicity) 55719a.4 Horizontal Tangents with a Local Maximum/Minimum 56519a.5 Concavity, Points of Inflection, and the Sign of the Second Derivative 56719b Maximum and Minimum Values of a Function 57519b.1 Introduction 57519b.2 Relative Extreme Values of a Function 57619b.3 Theorem A 58019b.4 Theorem B: Sufficient Conditions for the Existence of a Relative Extrema—In Terms of the First Derivative 58419b.5 Sufficient Condition for Relative Extremum (In Terms of the Second Derivative) 58819b.6 Maximum and Minimum of a Function on the Whole Interval (Absolute Maximum and Absolute Minimum Values) 59319b.7 Applications of Maxima and Minima Techniques in Solving Certain Problems Involving the Determination of the Greatest and the Least Values 59720 Rolle’s Theorem and the Mean Value Theorem (MVT) 60520.1 Introduction 60520.2 Rolle’s Theorem (A Theorem on the Roots of a Derivative) 60820.3 Introduction to the Mean Value Theorem 61320.4 Some Applications of the Mean Value Theorem 62221 The Generalized Mean Value Theorem (Cauchy’s MVT), L’ Hospital’s Rule, and their Applications 62521.1 Introduction 62521.2 Generalized Mean Value Theorem (Cauchy’s MVT) 62521.3 Indeterminate Forms and L’Hospital’s Rule 62721.4 L’Hospital’s Rule (First Form) 63021.5 L’Hospital’s Theorem (For Evaluating Limits(s) of the Indeterminate Form 0/0.) 63221.6 Evaluating Indeterminate Form of the Type∞/∞ 63821.7 Most General Statement of L’Hospital’s Theorem 64421.8 Meaning of Indeterminate Forms 64421.9 Finding Limits Involving Various Indeterminate Forms (by Expressing them to the Form 0/0 or ∞/∞) 64622 Extending the Mean Value Theorem to Taylor’s Formula: Taylor Polynomials for Certain Functions 65322.1 Introduction 65322.2 The Mean Value Theorem For Second Derivatives: The First Extended MVT 65422.3 Taylor’s Theorem 65822.4 Polynomial Approximations and Taylor’s Formula 65822.5 From Maclaurin Series To Taylor Series 66722.6 Taylor’s Formula for Polynomials 66922.7 Taylor’s Formula for Arbitrary Functions 67223 Hyperbolic Functions and Their Properties 67723.1 Introduction 67723.2 Relation Between Exponential and Trigonometric Functions 68023.3 Similarities and Differences in the Behavior of Hyperbolic and Circular Functions 68223.4 Derivatives of Hyperbolic Functions 68523.5 Curves of Hyperbolic Functions 68623.6 The Indefinite Integral Formulas for Hyperbolic Functions 68923.7 Inverse Hyperbolic Functions 68923.8 Justification for Calling sinh and cosh as Hyperbolic Functions Just as sine and cosine are Called Trigonometric Circular Functions 699Appendix A (Related To Chapter-2) Elementary Set Theory 703Appendix B (Related To Chapter-4) 711Appendix C (Related To Chapter-20) 735Index 739