This book explores the differential calculus and its plentiful applications in engineering and the physical sciences. The first six chapters offer a refresher of algebra, geometry, coordinate geometry, trigonometry, the concept of function, etc. since these topics are vital to the complete understanding of calculus. The book then moves on to the concept of limit of a function. Suitable examples of algebraic functions are selected, and their limits are discussed to visualize all possible situations that may occur in evaluating limit of a function, other than algebraic functions. Also, applications and limitations of this definition, along with the algebra of limits (i.e. limit theorems) are discussed. Finally, Sandwich theorem, which is useful for evaluating limit(s) of trigonometric functions, is proved, and the concept of onesided limits is introduced. The methods for computing limits of algebraic functions are discussed, and the concept of continuity and related concepts are also featured at length. Suitable examples of functions and their graphs are selected carefully to prevent reader confusion. Classification of the points of discontinuity is explained, and the methods for checking continuity of functions involving trigonometric, exponential, and logarithmic functions are discussed through solved examples. Theorems on continuity of functions (i.e. algebra of continuous functions) are stated without proof. Also, very important theorems on continuity (without proof) are provided.
Through the use of examples and graphs, this book maintains a high level of precision in clarifying prerequisite materials such as algebra, geometry, coordinate geometry, trigonometry, and the concept of limits. The book explores concepts of limits of a function, limits of algebraic functions, applications and limitations for limits, and the algebra of limits. It also discusses methods for computing limits of algebraic functions, and explains the concept of continuity and related concepts in depth. This introductory submersion into differential calculus is an essential guide for engineering and the physical sciences students.
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 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
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 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
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