Calculus: Single Variable, 5th Edition

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