Introduction to Differential Calculus

Foreword xiii Preface xvii Biographies xxv Introduction xxvii Acknowledgments xxix 1 From Arithmetic to Algebra (What must you know to learn Calculus?) 1 2 The Concept of a Function (What must you know to learn Calculus?) 19 3 Discovery of Real Numbers: Through Traditional Algebra (What must you know to learn Calculus?) 41 4 From Geometry to Coordinate Geometry (What must you know to learn Calculus?) 63 5 Trigonometry and Trigonometric Functions (What must you know to learn Calculus?) 97 6 More About Functions (What must you know to learn Calculus?) 129 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 9 The Idea of a Derivative of a Function 235 10 Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of Differentiable Functions 275 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 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 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 17 Derivatives and Differentials of Higher Order 511 18 Applications of Derivatives in Studying Motion in a Straight Line 535 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 21 The Generalized Mean Value Theorem (Cauchy's MVT), L' Hospital's Rule, and their Applications 625 infinity / infinity 638 22 Extending the Mean Value Theorem to Taylor's Formula: Taylor Polynomials for Certain Functions 653 23 Hyperbolic Functions and Their Properties 677 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