Contents

• Frontmatter
• INTRODUCTION: REVIEW OF ALGEBRA, GEOMETRY, AND TRIGONOMETRY
  • 0-1 The Real Numbers 1
  • 0-2 Inequalities 3
  • 0-3 Absolute Value 4
  • 0-4 Sets 5
  • 0-5 Plane and Solid Geometry 8
  • 0-6 Analytic Geometry 9
  • 0-7 Linear Equations in x and y 11
  • 0-8 Simultaneous Linear Equations 14
  • 0-9 Determinants 16
  • 0-10 Functions 19
  • 0-11 Real Functions of a Real Variable 21
  • 0-12 Real Functions of Several Real Variables 23
  • 0-13 Graph of a Second Degree Polynomial 24
  • 0-14 Circle, Ellipse, Hyperbola 27
  • 0-15 Trigonometry 31
  • 0-16 Polar Coordinates 33
  • 0-17 Complex Numbers 35
  • 0-18 Algebraic Equations 38
  • 0-19 Exponents and Logarithms 39
  • 0-20 Induction 41
  • 0-21 The Binomial Theorem, Permutations, and Combinations 43
• CHAPTER 1 TWO-DIMENSIONAL VECTOR GEOMETRY
  • 1-1 Introduction 46
  • 1-2 Directed Line Segments and Vectors 46
  • 1-3 Addition of Vectors 49
  • 1-4 Subtraction of Vectors 51
  • 1-5 Multiplication of Vectors by Scalars 52
  • †1-6 Geometric Applications 56
  • 1-7 Linear Independence, Basis 58
  • 1-8 Vectors as Number Pairs 61
  • 1-9 Angle between Vectors, Orthogonal Basis 64
  • 1-10 Inner Product 67
  • 1-11 Properties of the Inner Product 68
  • 1-12 Left Turn Operation, Directed Angle of Two Vectors, Area Formula 71
  • 1-13 Physical Applications, Statics 75
  • 1-14 Equation of Straight Line 78
  • 1-15 Parametric Equations for a Line 79
  • 1-16 Linear Equation for a Straight Line 81
• CHAPTER 2 LIMITS
  • 2-1 Concept of a Function, Terminology, Composition 84
  • 2-2 Qualitative Analysis of Functions of One Variable 87
  • 2-3 Operations on Functions of One Variable 88
  • 2-4 Inverse Functions 91
  • 2-5 Limits 95
  • 2-6 Continuity 101
  • 2-7 Theorems on Limits and Continuity 105
  • 2-8 Continuity of Polynomials and Other Common Functions 110
  • 2-9 Vector Spaces of Functions 114
  • 2-10 Limits as x Approaches + ∞ or - ∞ 118
  • 2-11 Infinite Limits of a Function 120
  • 2-12 Limits of Infinite Sequences 124
  • 2-13 The Least Upper Bound Axiom 129
  • ‡2-14 Proofs of Theorems on Limits and Continuity 132
• CHAPTER 3 DIFFERENTIAL CALCULUS
  • 3-1 Motivation for the Differential Calculus 139
  • 3-2 Definition of Derivative 144
  • 3-3 Fundamental Rules of Differentiation 153
  • †3-4 Proofs of Rules for Derivatives 155
  • 3-5 The Chain Rule 163
  • 3-6 Derivative of Inverse Functions 169
  • 3-7 Related Functions 175
  • 3-8 Implicit Functions 177
  • 3-9 Parametric Equations 183
  • 3-10 Vector Functions 187
  • 3-11 Differentiation of Vector Functions 189
  • 3-12 Differential Calculus Rules for Vector Functions 192
  • 3-13 Equation of Tangent and Normal Lines, Angle between Two Curves 196
  • 3-14 Second Derivative, Derivatives of Higher Order 200
  • 3-15 Geometric Meaning of the Derivatives of Higher Order 202
  • 3-16 Physical Meaning of the Higher Derivatives 205
  • 3-17 High Derivatives for Composite Functions, Inverse Functions, Functions Defined by Parametric Equations 208
  • 3-18 Higher Derivatives of Vector Functions 211
  • 3-19 Maxima and Minima 214
  • 3-20 Rolle's Theorem 219
  • 3-21 Mean Value Theorem 220
  • 3-22 The Differential 225
  • 3-23 Rules of Calculus in Terms of Differentials 228
  • 3-24 Numerical Applications of the Differential 230
  • 3-25 The Differential and Tangents 232
• CHAPTER 4 INTEGRAL CALCULUS
  • 4-1 Introduction 236
  • PART I. THE CONCEPTS OF INTEGRATION
    • 4-2 The Indefinite Integral 236
    • 4-3 The Definite Integral 240
    • 4-4 Area 245
  • PART II. THE INDEFINITE INTEGRAL
    • 4-5 Basic Properties of the Indefinite Integral 249
    • 4-6 Applications of Rules of Integration 252
    • 4-7 Substitution in Indefinite Integrals 254
    • †4-8 Theorems on Substitutions 260
    • 4-9 Integration by Parts 264
    • 4-10 Partial Fraction Expansion of Rational Functions (Case of Real Roots) 267
    • ‡4-11 Proof of Partial Fraction Expansion Theorem for the Case of Real Roots 270
    • †4-12 Partial Fraction Expansion (Case of Complex Roots and Quadratic Factors) 273
    • †4-13 Integration of Functions Given by Different Formulas in Adjoining Intervals 278
    • †4-14 Approximate Methods for Finding Indefinite Integrals 281
  • PART III. THE DEFINITE INTEGRAL
    • 4-15 The Definition of the Definite Integral 285
    • 4-16 Properties of the Definite Integral 290
    • 4-17 The Fundamental Theorem of Calculus 293
    • 4-18 Area 299
    • 4-19 Area Under a Curve 301
    • 4-20 The Integral as an Accumulator 307
    • 4-21 Integration by Parts and Substitution 311
    • 4-22 Odd and Even Functions 313
    • 4-23 Inequalities for Integrals 317
    • 4-24 Mean Value Theorem for Integrals 318
    • 4-25 The Definite Integral as a Limit 320
    • ‡4-26 Proof of Existence of Riemann Integral of a Continuous Function 325
    • 4-27 Arc Length 329
    • 4-28 The Arc Length Function 333
    • †4-29 Change of Parameter 334
    • †4-30 Integration of Piecewise Continuous Functions 338
    • 4-31 Integration of Vector Functions 344
• CHAPTER 5 THE ELEMENTARY TRANSCENDENTAL FUNCTIONS
  • Introduction 350
  • †5-1 The Sine and Cosine Functions 350
  • †5-2 Extension of cos s and sin s to the Infinite Interval 354
  • †5-3 Identities 357
  • †5-4 Angle Functions 359
  • ‡5-5 Existence and Uniqueness of the Angle Function 363
  • †5-6 Integral of a Rational Function of sin x and cos x 365
  • ‡5-7 The Exponential and Logarithmic Functions 368
  • †5-8 The Complex Exponential Function 375
  • †5-9 Hyperbolic Functions 378
  • †5-10 Relation between Hyperbolic Functions and Trigonometric Functions 380
  • †5-11 Classification of Functions 383
• CHAPTER 6 APPLICATIONS OF DIFFERENTIAL CALCULUS
  • Introduction 386
  • 6-1 Tests for Maxima and Minima 386
  • 6-2 Maxima and Minima with Side Conditions. Lagrange Multiplier 395
  • 6-3 Concavity and Convexity, Inflection Points 400
  • 6-4 Remarks on Graphing 403
  • 6-5 Change of Coordinates 413
  • 6-6 Plane Curves: Vector Equations, Curvature 421
  • 6-7 Tangential and Normal Components of Acceleration. Circle of Curvature 427
  • 6-8 Curves in Polar Coordinates 432
  • 6-9 Acceleration and Curvature in Polar Coordinates 437
  • 6-10 Newton's Method 443
  • 6-11 Estimation of Error 448
  • 6-12 Taylor's Formula with Remainder 453
  • 6-13 Error in Newton's Method 457
  • 6-14 Indeterminate Forms, L'Hospital's Rule 459
  • †6-15 Proofs of L'Hospital's Rules 464
• CHAPTER 7 APPLICATIONS OF THE INTEGRAL CALCULUS
  • 7-1 Area between Two Curves 470
  • 7-2 Area in Polar Coordinates 473
  • 7-3 A General Area Formula 476
  • 7-4 A New Approach to Area 483
  • 7-5 Volume of Solid of Revolution 487
  • †7-6 Solids of Revolution: Polar Coordinates and Parametric Formula 492
  • 7-7 Volume of Other Solids 496
  • 7-8 Area of Surface of Revolution 499
  • 7-9 Distribution of Mass and Other Distributions 504
  • 7-10 Mass Distributions in the Plane 510
  • 7-11 Centroid 514
  • 7-12 Mass Distribution on Curves 515
  • 7-13 Other Applications of Integration 520
  • 7-14 Improper Integrals 528
  • 7-15 Differential Equations 536
  • 7-16 First Order Differential Equations 538
  • 7-17 Linear Differential Equations of Second Order 543
  • 7-18 The Homogeneous Linear Differential Equation of Second Order with Constant Coefficients 546
  • 7-19 The Nonhomogeneous Linear Equation of Second Order with Constant Coefficients 547
  • 7-20 Vibrations 550
  • 7-21 Numerical Evaluation of Integrals, Trapezoidal Rule 554
  • 7-22 Simpson's Rule 558
  • ‡7-23 Proofs of Expressions for Error in Trapezoidal and Simpson's Rules 561
• CHAPTER 8 INFINITE SERIES
  • 8-1 Introduction 566
  • 8-2 Infinte Sequences 568
  • ‡8-3 The Cauchy Condition for Sequences 571
  • 8-4 Infinite Series 574
  • 8-5 Properties of Infinite Series 580
  • ‡8-6 Cauchy Criterion for Infinite Series 583
  • 8-7 Comparison Tests for Series with Nonnegative Terms 584
  • 8-8 The Integral Test 587
  • 8-9 Absolute Convergence 591
  • 8-10 Ratio and Root Tests 592
  • 8-11 Alternating Series 595
  • ‡8-12 Rearrangement of Series 597
  • ‡8-13 Products of Series 598
  • 8-14 Sequences and Series of Functions 602
  • 8-15 Power Series 604
  • ‡8-16 Proof of Theorem on Radius of Convergence 606
  • 8-17 Properties of Power Series 608
  • ‡8-18 Proof of Theorem on Properties of Power Series 612
  • 8-19 Taylor's Formula with Remainder 616
  • 8-20 Taylor's Series 618
  • 8-21 Numerical Evaluation of Functions by Power Series 623
  • 8-22 Power Series Solution of Differential Equations 627
  • 8-23 Complex Power Series 630
  • 8-24 Fourier Series 633
• APPENDIXES A-1
  • 1. Table of Indefinite Integrals A-3
  • 2. Natural Trigonometric Functions for Angles in Radians A-11
  • 3. Exponential Function A-15
  • 4. Natural Logarithms A-19
  • 5. Trigonometric Formulas A-22
• ANSWERS TO SELECTED PROBLEMS A-25
• INDEX I-1

Top of page