Calculus: Single Variable, 5th Edition

calculus Teachers Recognize Calculus As The Leading Resource Among The Reform Projects That Employ The Rule Of Four And Streamline The Curriculum In Order To Deepen Conceptual Understanding. The Fifth Edition Uses All Strands Of The Rule Of Four - Graphical, Numeric, Symbolic/algebraic, And Verbal/applied Presentations - To Make Concepts Easier To Understand. The Book Focuses On Exploring Fundamental Ideas Rather Than Comprehensive Coverage Of Multiple Similar Cases That Are Not Fundamentally Unique. booknews calculus Can Be Taught As Nothing But Rules And Procedures--losing Sight Of Both The Mathematics And Its Inherent Practical Value. In 1989, The Calculus Consortium Based At Harvard Was Formed To Create A Completely New Calculus Curriculum. A Part Of Their Endeavor Is This Textbook, Which Presents A Radically Different Approach To The Teaching And Learning Of The Subject. The Two Guiding Principles: 1) Every Topic Should Be Presented Geometrically, Numerically, And Algebraically; And 2) Formal Definitions And Procedures Evolve From The Investigation Of Practical Problems (the Way Of Archimedes). Annotation C. Book News, Inc., Portland, Or (booknews.com) Cover Page......Page 1 Title Page......Page 5 Copyright Page......Page 6 PREFACE......Page 7 Table of Contents......Page 13 1 A LIBRARY OF FUNCTIONS......Page 17 1.1 FUNCTIONS AND CHANGE......Page 18 1.2 EXPONENTIAL FUNCTIONS......Page 26 1.3 NEW FUNCTIONS FROM OLD......Page 33 1.4 LOGARITHMIC FUNCTIONS......Page 40 1.5 TRIGONOMETRIC FUNCTIONS......Page 46 1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS......Page 54 1.7 INTRODUCTION TO CONTINUITY......Page 63 1.8 LIMITS......Page 67 REVIEW PROBLEMS......Page 76 CHECK YOUR UNDERSTANDING......Page 81 PROJECTS: MATCHING FUNCTIONS TO DATA, WHICH WAY IS THE WIND BLOWING?......Page 83 2 KEY CONCEPT: THE DERIVATIVE......Page 85 2.1 HOW DO WE MEASURE SPEED?......Page 86 2.2 THE DERIVATIVE AT A POINT......Page 92 2.3 THE DERIVATIVE FUNCTION......Page 101 2.4 INTERPRETATIONS OF THE DERIVATIVE......Page 109 2.5 THE SECOND DERIVATIVE......Page 114 2.6 DIFFERENTIABILITY......Page 120 REVIEW PROBLEMS......Page 125 CHECK YOUR UNDERSTANDING......Page 129 PROJECTS: HOURS OF DAYLIGHT AS A FUNCTION OF LATITUDE, US POPULATION......Page 130 3 SHORT-CUTS TO DIFFERENTIATION......Page 131 3.1 POWERS AND POLYNOMIALS......Page 132 3.2 THE EXPONENTIAL FUNCTION......Page 139 3.3 THE PRODUCT AND QUOTIENT RULES......Page 143 3.4 THE CHAIN RULE......Page 149 3.5 THE TRIGONOMETRIC FUNCTIONS......Page 156 3.6 THE CHAIN RULE AND INVERSE FUNCTIONS......Page 161 3.7 IMPLICIT FUNCTIONS......Page 167 3.8 HYPERBOLIC FUNCTIONS......Page 170 3.9 LINEAR APPROXIMATION AND THE DERIVATIVE......Page 174 3.10 THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS......Page 180 REVIEW PROBLEMS......Page 184 CHECK YOUR UNDERSTANDING......Page 188 PROJECTS: RULE OF 70, NEWTON’S METHOD......Page 189 4 USING THE DERIVATIVE......Page 191 4.1 USING FIRST AND SECOND DERIVATIVES......Page 192 4.2 OPTIMIZATION......Page 201 4.3 FAMILIES OF FUNCTIONS......Page 209 4.4 OPTIMIZATION, GEOMETRY, AND MODELING......Page 216 4.5 APPLICATIONS TO MARGINALITY......Page 227 4.6 RATES AND RELATED RATES......Page 235 4.7 L’HOPITAL’S RULE, GROWTH, AND DOMINANCE......Page 244 4.8 PARAMETRIC EQUATIONS......Page 250 REVIEW PROBLEMS......Page 262 CHECK YOUR UNDERSTANDING......Page 267 PROJECTS: BUILDING A GREENHOUSE, FITTING A LINE TO DATA, FIREBREAKS......Page 268 5 KEY CONCEPT: THE DEFINITE INTEGRAL......Page 271 5.1 HOW DO WE MEASURE DISTANCE TRAVELED?......Page 272 5.2 THE DEFINITE INTEGRAL......Page 280 5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS......Page 287 5.4 THEOREMS ABOUT DEFINITE INTEGRALS......Page 298 REVIEW PROBLEMS......Page 306 CHECK YOUR UNDERSTANDING......Page 312 PROJECTS: THE CAR AND THE TRUCK, AN ORBITING SATELLITE......Page 313 6 CONSTRUCTING ANTIDERIVATIVES......Page 315 6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY......Page 316 6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY......Page 321 6.3 DIFFERENTIAL EQUATIONS......Page 328 6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS......Page 333 6.5 THE EQUATIONS OF MOTION......Page 338 REVIEW PROBLEMS......Page 341 CHECK YOUR UNDERSTANDING......Page 344 PROJECTS: DISTRIBUTION OF RESOURCES, YIELD FROM AN APPLE ORCHARD SLOPE FIELDS......Page 345 7 INTEGRATION......Page 347 7.1 INTEGRATION BY SUBSTITUTION......Page 348 7.2 INTEGRATION BY PARTS......Page 357 7.3 TABLES OF INTEGRALS......Page 363 7.4 ALGEBRAIC IDENTITIES AND TRIGONOMETRIC SUBSTITUTIONS......Page 368 7.5 APPROXIMATING DEFINITE INTEGRALS......Page 377 7.6 APPROXIMATION ERRORS AND SIMPSON’S RULE......Page 382 7.7 IMPROPER INTEGRALS......Page 387 7.8 COMPARISON OF IMPROPER INTEGRALS......Page 395 REVIEW PROBLEMS......Page 401 CHECK YOUR UNDERSTANDING......Page 405 PROJECTS: TAYLOR POLYNOMIAL INEQUALITIES......Page 406 8 USING THE DEFINITE INTEGRAL......Page 407 8.1 AREAS AND VOLUMES......Page 408 8.2 APPLICATIONS TO GEOMETRY......Page 414 8.3 AREA AND ARC LENGTH IN POLAR COORDINATES......Page 422 8.4 DENSITY AND CENTER OF MASS......Page 431 8.5 APPLICATIONS TO PHYSICS......Page 440 8.6 APPLICATIONS TO ECONOMICS......Page 449 8.7 DISTRIBUTION FUNCTIONS......Page 455 8.8 PROBABILITY, MEAN, AND MEDIAN......Page 462 REVIEW PROBLEMS......Page 470 CHECK YOUR UNDERSTANDING......Page 475 PROJECTS: VOLUME ENCLOSED BY TWO CYLINDERS, LENGTH OF A HANGING CABLE SURFACE AREA OF AN UNPAINTABLE CAN OF PAINT MAXWELL’S DISTRIBUTION OF MOLECULAR VELOCITIES......Page 476 9 SEQUENCES AND SERIES......Page 479 9.1 SEQUENCES......Page 480 9.2 GEOMETRIC SERIES......Page 486 9.3 CONVERGENCE OF SERIES......Page 492 9.4 TESTS FOR CONVERGENCE......Page 497 9.5 POWER SERIES AND INTERVAL OF CONVERGENCE......Page 506 REVIEW PROBLEMS......Page 513 CHECK YOUR UNDERSTANDING......Page 517 PROJECTS: A DEFINITION OF e, PROBABILITY OF WINNING IN SPORTS, PREDNISONE......Page 518 10 APPROXIMATING FUNCTIONS USING SERIES......Page 521 10.1 TAYLOR POLYNOMIALS......Page 522 10.2 TAYLOR SERIES......Page 530 10.3 FINDING AND USING TAYLOR SERIES......Page 535 10.4 THE ERROR IN TAYLOR POLYNOMIAL APPROXIMATIONS......Page 541 10.5 FOURIER SERIES......Page 546 REVIEW PROBLEMS......Page 559 PROJECTS: SHAPE OF PLANETS, MACHIN’S FORMULA AND THE VALUE OF APPROXIMATING THE DERIVATIVE......Page 562 11 DIFFERENTIAL EQUATIONS......Page 565 11.1 WHAT IS A DIFFERENTIAL EQUATION?......Page 566 11.2 SLOPE FIELDS......Page 570 11.3 EULER’S METHOD......Page 577 11.4 SEPARATION OF VARIABLES......Page 580 11.5 GROWTH AND DECAY......Page 586 11.6 APPLICATIONS AND MODELING......Page 595 11.7 THE LOGISTIC MODEL......Page 603 11.8 SYSTEMS OF DIFFERENTIAL EQUATIONS......Page 614 11.9 ANALYZING THE PHASE PLANE......Page 623 11.10 SECOND-ORDER DIFFERENTIAL EQUATIONS: OSCILLATIONS......Page 628 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS......Page 635 REVIEW PROBLEMS......Page 644 CHECK YOUR UNDERSTANDING......Page 649 PROJECTS: SARS PREDICTIONS FOR HONG KONG, A S-I-R MODEL FOR SARS PARETO’S LAW, VIBRATIONS IN A MOLECULE......Page 650 APPENDIX......Page 653 A ROOTS, ACCURACY, AND BOUNDS......Page 654 B COMPLEX NUMBERS......Page 662 C NEWTON’S METHOD......Page 669 D VECTORS IN THE PLANE......Page 672 READY REFERENCE......Page 679 ANSWERS TO ODD-NUMBERED PROBLEMS......Page 691 INDEX......Page 725 Cover Page 1 Title Page 5 Copyright Page 6 PREFACE 7 Table of Contents 13 1 A LIBRARY OF FUNCTIONS 17 1.1 FUNCTIONS AND CHANGE 18 1.2 EXPONENTIAL FUNCTIONS 26 1.3 NEW FUNCTIONS FROM OLD 33 1.4 LOGARITHMIC FUNCTIONS 40 1.5 TRIGONOMETRIC FUNCTIONS 46 1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS 54 1.7 INTRODUCTION TO CONTINUITY 63 1.8 LIMITS 67 REVIEW PROBLEMS 76 CHECK YOUR UNDERSTANDING 81 PROJECTS: MATCHING FUNCTIONS TO DATA, WHICH WAY IS THE WIND BLOWING? 83 2 KEY CONCEPT: THE DERIVATIVE 85 2.1 HOW DO WE MEASURE SPEED? 86 2.2 THE DERIVATIVE AT A POINT 92 2.3 THE DERIVATIVE FUNCTION 101 2.4 INTERPRETATIONS OF THE DERIVATIVE 109 2.5 THE SECOND DERIVATIVE 114 2.6 DIFFERENTIABILITY 120 REVIEW PROBLEMS 125 CHECK YOUR UNDERSTANDING 129 PROJECTS: HOURS OF DAYLIGHT AS A FUNCTION OF LATITUDE, US POPULATION 130 3 SHORT-CUTS TO DIFFERENTIATION 131 3.1 POWERS AND POLYNOMIALS 132 3.2 THE EXPONENTIAL FUNCTION 139 3.3 THE PRODUCT AND QUOTIENT RULES 143 3.4 THE CHAIN RULE 149 3.5 THE TRIGONOMETRIC FUNCTIONS 156 3.6 THE CHAIN RULE AND INVERSE FUNCTIONS 161 3.7 IMPLICIT FUNCTIONS 167 3.8 HYPERBOLIC FUNCTIONS 170 3.9 LINEAR APPROXIMATION AND THE DERIVATIVE 174 3.10 THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS 180 REVIEW PROBLEMS 184 CHECK YOUR UNDERSTANDING 188 PROJECTS: RULE OF 70, NEWTON’S METHOD 189 4 USING THE DERIVATIVE 191 4.1 USING FIRST AND SECOND DERIVATIVES 192 4.2 OPTIMIZATION 201 4.3 FAMILIES OF FUNCTIONS 209 4.4 OPTIMIZATION, GEOMETRY, AND MODELING 216 4.5 APPLICATIONS TO MARGINALITY 227 4.6 RATES AND RELATED RATES 235 4.7 L’HOPITAL’S RULE, GROWTH, AND DOMINANCE 244 4.8 PARAMETRIC EQUATIONS 250 REVIEW PROBLEMS 262 CHECK YOUR UNDERSTANDING 267 PROJECTS: BUILDING A GREENHOUSE, FITTING A LINE TO DATA, FIREBREAKS 268 5 KEY CONCEPT: THE DEFINITE INTEGRAL 271 5.1 HOW DO WE MEASURE DISTANCE TRAVELED? 272 5.2 THE DEFINITE INTEGRAL 280 5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 287 5.4 THEOREMS ABOUT DEFINITE INTEGRALS 298 REVIEW PROBLEMS 306 CHECK YOUR UNDERSTANDING 312 PROJECTS: THE CAR AND THE TRUCK, AN ORBITING SATELLITE 313 6 CONSTRUCTING ANTIDERIVATIVES 315 6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY 316 6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY 321 6.3 DIFFERENTIAL EQUATIONS 328 6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS 333 6.5 THE EQUATIONS OF MOTION 338 REVIEW PROBLEMS 341 CHECK YOUR UNDERSTANDING 344 PROJECTS: DISTRIBUTION OF RESOURCES, YIELD FROM AN APPLE ORCHARD SLOPE FIELDS 345 7 INTEGRATION 347 7.1 INTEGRATION BY SUBSTITUTION 348 7.2 INTEGRATION BY PARTS 357 7.3 TABLES OF INTEGRALS 363 7.4 ALGEBRAIC IDENTITIES AND TRIGONOMETRIC SUBSTITUTIONS 368 7.5 APPROXIMATING DEFINITE INTEGRALS 377 7.6 APPROXIMATION ERRORS AND SIMPSON’S RULE 382 7.7 IMPROPER INTEGRALS 387 7.8 COMPARISON OF IMPROPER INTEGRALS 395 REVIEW PROBLEMS 401 CHECK YOUR UNDERSTANDING 405 PROJECTS: TAYLOR POLYNOMIAL INEQUALITIES 406 8 USING THE DEFINITE INTEGRAL 407 8.1 AREAS AND VOLUMES 408 8.2 APPLICATIONS TO GEOMETRY 414 8.3 AREA AND ARC LENGTH IN POLAR COORDINATES 422 8.4 DENSITY AND CENTER OF MASS 431 8.5 APPLICATIONS TO PHYSICS 440 8.6 APPLICATIONS TO ECONOMICS 449 8.7 DISTRIBUTION FUNCTIONS 455 8.8 PROBABILITY, MEAN, AND MEDIAN 462 REVIEW PROBLEMS 470 CHECK YOUR UNDERSTANDING 475 PROJECTS: VOLUME ENCLOSED BY TWO CYLINDERS, LENGTH OF A HANGING CABLE SURFACE AREA OF AN UNPAINTABLE CAN OF PAINT MAXWELL’S DISTRIBUTION OF MOLECULAR VELOCITIES 476 9 SEQUENCES AND SERIES 479 9.1 SEQUENCES 480 9.2 GEOMETRIC SERIES 486 9.3 CONVERGENCE OF SERIES 492 9.4 TESTS FOR CONVERGENCE 497 9.5 POWER SERIES AND INTERVAL OF CONVERGENCE 506 REVIEW PROBLEMS 513 CHECK YOUR UNDERSTANDING 517 PROJECTS: A DEFINITION OF e, PROBABILITY OF WINNING IN SPORTS, PREDNISONE 518 10 APPROXIMATING FUNCTIONS USING SERIES 521 10.1 TAYLOR POLYNOMIALS 522 10.2 TAYLOR SERIES 530 10.3 FINDING AND USING TAYLOR SERIES 535 10.4 THE ERROR IN TAYLOR POLYNOMIAL APPROXIMATIONS 541 10.5 FOURIER SERIES 546 REVIEW PROBLEMS 559 CHECK YOUR UNDERSTANDING 562 PROJECTS: SHAPE OF PLANETS, MACHIN’S FORMULA AND THE VALUE OF APPROXIMATING THE DERIVATIVE 562 11 DIFFERENTIAL EQUATIONS 565 11.1 WHAT IS A DIFFERENTIAL EQUATION? 566 11.2 SLOPE FIELDS 570 11.3 EULER’S METHOD 577 11.4 SEPARATION OF VARIABLES 580 11.5 GROWTH AND DECAY 586 11.6 APPLICATIONS AND MODELING 595 11.7 THE LOGISTIC MODEL 603 11.8 SYSTEMS OF DIFFERENTIAL EQUATIONS 614 11.9 ANALYZING THE PHASE PLANE 623 11.10 SECOND-ORDER DIFFERENTIAL EQUATIONS: OSCILLATIONS 628 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS 635 REVIEW PROBLEMS 644 CHECK YOUR UNDERSTANDING 649 PROJECTS: SARS PREDICTIONS FOR HONG KONG, A S-I-R MODEL FOR SARS PARETO’S LAW, VIBRATIONS IN A MOLECULE 650 APPENDIX 653 A ROOTS, ACCURACY, AND BOUNDS 654 B COMPLEX NUMBERS 662 C NEWTON’S METHOD 669 D VECTORS IN THE PLANE 672 READY REFERENCE 679 ANSWERS TO ODD-NUMBERED PROBLEMS 691 INDEX 725