CONTENTS

 

 

 

 

INTRODUCTION

                                                      xiii

 

1  REAL  AND  HYPERREAL  NUMBERS                                1

   1.1   The Real Line                                                      1

   1.2   Functions of Real Numbers                                           6

   1.3   Straight Lines                                                      16

   1.4   Slope and Velocity; The Hyperreal Line                                  21

   1.5   Infinitesimal, Finite, and Infinite Numbers                               27

   1.6   Standard Parts                                                     35

         Extra Problems for Chapter 1                                         41

 

 

2   DIFFERENTIATION                                                  43

   2.1   Derivatives                                                      43

   2.2   Differentials and Tangent Lines                                      53

   2.3   Derivatives of Rational Functions                                    60

   2.4   Inverse Functions                                                 70

   2.5   Transcendental Functions                                           78

   2.6   Chain Rule                                                       85

   2.7   Higher Derivatives                                                94

   2.8   Implicit Functions                                                 97

        Extra Problems for Chapter 2                                        103

 

 

3  CONTINUOUS FUNCTIONS                                              105

   3.1   How to Set Up a Problem                                             105

   3.2   Related Rates                                                       110

   3.3   Limits                                                              117

   3.4   Continuity                                                         124

   3.5   Maxima and Minima                                                 134

   3.6   Maxima and Minima- Applications                                       144

   3.7   Derivatives and Curve Sketching                                 151

   3.8   Properties of Continuous Functions                               159

         Extra Problems for Chapter 3                                    171

 

4   INTEGRATION                                                   175

   4.1   The Definite Integral                                            175

   4.2   Fundamental Theorem of Calculus                                 186

   4.3   Indefinite Integrals                                              198

   4.4   Integration by Change of Variables                                 209

   4.5   Area between Two Curves                                        218

   4.6   Numerical Integration                                           224

         Extra Problems for Chapter 4                                     234

 

5    LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS            237

   5.1   Infinite Limits                                                 237

   5.2   L’Hospital’s Rule                                               242

   5.3   Limits and Curve Sketching                                       248

   5.4   Parabolas                                                      256

   5.5   Ellipses and Hyperbolas                                           264

   5.6   Second Degree Curves                                            272

   5.7   Rotation of Axes                                                 276

   5.8   The  ε, δ Condition for Limits                                     282

   5.9   Newton’s Method                                                289

   5.10  Derivatives and Increments                                         294

         Extra Problems for Chapter 5                                       300

 

6   APPLICATIONS OF THE INTEGRAL                                302

   6.1   Infinite Sum Theorem                                         302

   6.2   Volumes of Solids of Revolution                                 308

   6.3   Length of a Curve                                             319

   6.4   Area of a Surface of Revolution                                  327

   6.5   Averages                                                    336

   6.6   Some Applications to Physics                                    341

   6.7   Improper integrals                                             351

         Extra Problems for Chapter 6                                    362

 

7   TRIGONOMETRIC  FUNCTIONS                                   365

   7.1   Trigonometry                                                 365

   7.2   Derivatives of Trigonometric Functions                            373

   7.3   Inverse Trigonometric Functions                                  381

   7.4   Integration by Parts                                             391

   7.5   Integrals of Powers of Trigonometric Functions                       397

   7.6   Trigonometric Substitutions                                       402

   7.7   Polar Coordinates                                            406

 

   7.8   Slopes and Curve Sketching in Polar Coordinates                   412

   7.9   Area in Polar Coordinates                                      420

   7.10  Length of a Curve in Polar Coordinates                           425

         Extra Problems for Chapter 7                                  428

 

8   EXPONENTIAL AND LOGARITHMIC FUNCTIONS                  431

   8.1      Exponential Functions                                      431

   8.2      Logarithmic Functions                                     436

   8.3      Derivatives of Exponential Functions and the Number e           441

   8.4      Some Uses of Exponential Functions                          449

   8.5      Natural Logarithms                                        454

   8.6      Some Differential Equations                                 461

   8.7      Derivatives and Integrals Involving In x                        469

   8.8      Integration of Rational Functions                             474

   8.9      Methods of Integration                                     481

           Extra Problems for Chapter 8                                 489

 

9   INFINITE SERIES                                               492

   9.1      Sequences                                               492

   9.2      Series                                                  501

   9.3      Properties of Infinite Series                                 507

   9.4      Series with Positive Terms                                  511

   9.5      Alternating Series                                         517

   9.6      Absolute and Conditional Convergence                        521

   9.7      Power Series                                             528

   9.8      Derivatives and Integrals of Power Series                      533

   9.9      Approximations by Power Series                             540

   9.10     Taylor’s Formula                                          547

   9.11     Taylor Series                                             554

            Extra Problems for Chapter 9                                561

 

10   VECTORS                                                      564

   10.1    Vector Algebra                                              564

   10.2    Vectors and Plane Geometry                                    576

   10.3    Vectors and Lines in Space                                     585

   10.4    Products of Vectors                                           593

   10.5    Planes in Space                                              604

   10.6    Vector Valued Functions                                       615

   10.7    Vector Derivatives                                            620

   10.8    Hyperreal Vectors                                            627

           Extra Problems for Chapter 10                                 635

                  

11   PARTIAL  DIFFERENTIATION                                      639

   11.1    Surfaces                                                     639

   11.2    Continuous Functions of Two or More Variables                      651

   11.3    Partial Derivatives                                              656

   11.4    Total Differentials and Tangent Planes                              662

   11.5    Chain Rule                                                    671

   11.6    Implicit Functions                                               678

   11.7    Maxima and Minima                                              688

   11.8    Higher Partial Derivatives                                          702

           Extra Problems for Chapter 11                                    708

 

12   MULTIPLE  INTEGRALS                                            711

   12.1    Double Integrals                                                711

   12.2    Iterated Integrals                                                724

   12.3    Infinite Sum Theorem and Volume                                  736

   12.4    Applications to Physics                                           743

   12.5    Double Integrals in Polar Coordinates                               749

   12.6    Triple Integrals                                                  757

   12.7    Cylindrical and Spherical Coordinates                                769

           Extra Problems for Chapter 12                                      783

 

13   VECTOR  CALCULUS                             785

   13.1    Directional Derivatives and Gradients                                785

   13.2    Line Integrals                                                    793

   13.3    Independence of Path                                              805

   13.4    Green’s Theorem                                                  815

   13.5    Surface Area and Surface Integrals                                    824

   13.6    Theorems of Stokes and Gauss                                       832

           Extra Problems for Chapter 13                                       842

 

14   DIFFERENTIAL  EQUATIONS                                           846

   14.1    Equations with Separable Variables                                     846

   14.2    First Order Homogeneous Linear Equations                              852

   14.3    First Order Linear Equations                                          857

   14.4    Existence and Approximation of Solutions                               864

   14.5    Complex Numbers                                                  874

   14.6    Second Order Homogeneous Linear Equations                            881

   14.7    Second Order Linear Equations                                        892

           Extra Problems for Chapter 14                                       900

EPILOGUE                                       902

APPENDIX: TABLES                              A1

       І    Trigonometric Functions                                  A1

       П   Greek Alphabet                                          A2

       Ⅲ   Exponential Functions                                 A3

       Ⅳ   Natural Logarithms                                    A3

       Ⅴ   Powers and Roots                                      A4

 

ANSWERS TO SELECTED PROBLEMS                         A5

INDEX                                                A57