Introduction to Differential Calculus

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.

Foreword xiii 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 4.8 Parallel Lines 89 4.9 Relation Between the Slopes of (Nonvertical) Lines that are Perpendicular 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 know to 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 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 8.1 Introduction 197 8.2 Developing the Definition of Continuity “At a Point” 204 8.3 Classification of the Points of Discontinuity: Types of Discontinuities 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 its Applications 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 of Various 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 Computing Derivatives 300 10.5 Derivatives of Inverse Functions 302 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 a0] 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 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 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 14.1 Introduction 417 14.2 Trigonometric Functions (With Restricted Domains) and Their Inverses 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 Function 433 14.7 Formula for the Derivative of Inverse Cosecant Function 436 14.8 Important Sets of Results and their Applications 437 14.9 Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions 441 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 d y /d x 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 16.1 Introduction 487 16.2 Applying Differentials to Approximate Calculations 492 16.3 Differentials of Basic Elementary Functions 494 16.4 Two Interpretations of the Notation dy/dx 498 16.5 Integrals in Differential Notation 499 16.6 To Compute (Approximate) Small Changes and Small Errors Caused in Various Situations 503 17 Derivatives and Differentials of Higher Order 511 17.1 Introduction 511 17.2 Derivatives of Higher Orders: Implicit Functions 516 17.3 Derivatives of Higher Orders: Parametric Functions 516 17.4 Derivatives of Higher Orders: Product of Two Functions (Leibniz Formula) 517 17.5 Differentials of Higher Orders 521 17.6 Rate of Change of a Function and Related Rates 523 18 Applications of Derivatives in Studying Motion in a Straight Line 535 18.1 Introduction 535 18.2 Motion in a Straight Line 535 18.3 Angular Velocity 540 18.4 Applications of Differentiation in Geometry 540 18.5 Slope of a Curve in Polar Coordinates 548 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 20.1 Introduction 605 20.2 Rolle’s Theorem (A Theorem on the Roots of a Derivative) 608 20.3 Introduction to the Mean Value Theorem 613 20.4 Some Applications of the Mean Value Theorem 622 21 The Generalized Mean Value Theorem (Cauchy’s MVT), L’ Hospital’s Rule, and their Applications 625 21.1 Introduction 625 21.2 Generalized Mean Value Theorem (Cauchy’s MVT) 625 21.3 Indeterminate Forms and L’Hospital’s Rule 627 21.4 L’Hospital’s Rule (First Form) 630 21.5 L’Hospital’s Theorem (For Evaluating Limits(s) of the Indeterminate Form 0/0.) 632 21.6 Evaluating Indeterminate Form of the Type ∞/∞ 638 21.7 Most General Statement of L’Hospital’s Theorem 644 21.8 Meaning of Indeterminate Forms 644 21.9 Finding Limits Involving Various Indeterminate Forms (by Expressing them to the Form 0/0 or ∞/∞) 646 22 Extending the Mean Value Theorem to Taylor’s Formula: Taylor Polynomials for Certain Functions 653 22.1 Introduction 653 22.2 The Mean Value Theorem For Second Derivatives: The First Extended MVT 654 22.3 Taylor’s Theorem 658 22.4 Polynomial Approximations and Taylor’s Formula 658 22.5 From Maclaurin Series To Taylor Series 667 22.6 Taylor’s Formula for Polynomials 669 22.7 Taylor’s Formula for Arbitrary Functions 672 23 Hyperbolic Functions and Their Properties 677 23.1 Introduction 677 23.2 Relation Between Exponential and Trigonometric Functions 680 23.3 Similarities and Differences in the Behavior of Hyperbolic and Circular Functions 682 23.4 Derivatives of Hyperbolic Functions 685 23.5 Curves of Hyperbolic Functions 686 23.6 The Indefinite Integral Formulas for Hyperbolic Functions 689 23.7 Inverse Hyperbolic Functions 689 23.8 Justification for Calling sinh and cosh as Hyperbolic Functions Just as sine and cosine are Called Trigonometric Circular Functions 699 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