Foreword xiii Preface xvii Biographies xxv Introduction xxvii Acknowledgments xxix 1 From Arithmetic to Algebra (What must you know to learnCalculus?) 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 Infinite10 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 learnCalculus?) 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 (Whatmust 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 tolearn 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 arePerpendicular 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 knowto 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 tolearn Calculus?) 149 7a.

1 Introduction 149 7a.2 Useful Notations 149 7a.3 The Concept of Limit of a Function: Informal Discussion151 7a.4 Intuitive Meaning of Limit of a Function 153 7a.5 Testing the Definition [Applications of the «, dDefinition 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 (Whatmust you know to learn Calculus?) 177 7b.

1 Introduction 177 7b.2 Methods for Evaluating Limits of Various AlgebraicFunctions 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 ofDiscontinuity (What must you know to learn Calculus?) 197 8.1 Introduction 197 8.2 Developing the Definition of Continuity "At aPoint" 204 8.3 Classification of the Points of Discontinuity: Types ofDiscontinuities 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 itsApplications 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 = x 1 and the Slope of its Graph at x = x 1 239 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 ofVarious 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 ComputingDerivatives 300 10.5 Derivatives of Inverse Functions 302 11a Basic Trigonometric Limits and Their Applications inComputing 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 Functions325 11b.

1 Introduction 325 11b.2 Limits of the Type (I) 328 11b.3 Limits of the Type (II) [ lim f ( x ), wherea&rae;0] 332 11b.4 Limits of Exponential and Logarithmic Functions 335 12 Exponential Form(s) of a Positive Real Number and itsLogarithm(s): Pre-Requisite for Understanding Exponential andLogarithmic 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 Functions362 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 10and 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 Functions378 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 andDecay 390 13b Methods for Computing Limits of Exponential and LogarithmicFunctions 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 TheirInverses 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 Function433 14.7 Formula for the Derivative of Inverse Cosecant Function436 14.8 Important Sets of Results and their Appli.