Calculus: Single and Multivariable

Cover......Page 1
Title Page......Page 5
Copyright Page......Page 6
Preface......Page 7
To Students: How to Learn from this Book......Page 13
Contents......Page 14
Chapter 1: A LIBRARY OF FUNCTIONS......Page 21
 1.1 FUNCTIONS AND CHANGE......Page 22
 1.2 EXPONENTIAL FUNCTIONS......Page 32
 1.3 NEW FUNCTIONS FROM OLD......Page 41
 1.4 LOGARITHMIC FUNCTIONS......Page 49
 1.5 TRIGONOMETRIC FUNCTIONS......Page 56
 1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS......Page 65
 1.7 INTRODUCTION TO CONTINUITY......Page 73
 1.8 LIMITS......Page 77
 REVIEW EXERCISES AND PROBLEMS......Page 88
 PROJECTS......Page 93
Chapter 2: KEY CONCEPT: THE DERIVATIVE......Page 95
 2.1 HOW DO WE MEASURE SPEED?......Page 96
 2.2 THE DERIVATIVE AT A POINT......Page 103
 2.3 THE DERIVATIVE FUNCTION......Page 110
 2.4 INTERPRETATIONS OF THE DERIVATIVE......Page 118
 2.5 THE SECOND DERIVATIVE......Page 124
 2.6 DIFFERENTIABILITY......Page 131
 REVIEW EXERCISES AND PROBLEMS......Page 136
 PROJECTS......Page 142
Chapter 3: SHORT-CUTS TO DIFFERENTIATION......Page 143
 3.1 POWERS AND POLYNOMIALS......Page 144
 3.2 THE EXPONENTIAL FUNCTION......Page 152
 3.3 THE PRODUCT AND QUOTIENT RULES......Page 156
 3.4 THE CHAIN RULE......Page 162
 3.5 THE TRIGONOMETRIC FUNCTIONS......Page 169
 3.6 THE CHAIN RULE AND INVERSE FUNCTIONS......Page 176
 3.7 IMPLICIT FUNCTIONS......Page 182
 3.8 HYPERBOLIC FUNCTIONS......Page 185
 3.9 LINEAR APPROXIMATION AND THE DERIVATIVE......Page 189
 3.10 THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS......Page 195
 REVIEW EXERCISES AND PROBLEMS......Page 200
 PROJECTS......Page 204
Chapter 4: USING THE DERIVATIVE......Page 205
 4.1 USING FIRST AND SECOND DERIVATIVES......Page 206
 4.2 OPTIMIZATION......Page 216
 4.3 OPTIMIZATION AND MODELING......Page 225
 4.4 FAMILIES OF FUNCTIONS AND MODELING......Page 236
 4.5 APPLICATIONS TO MARGINALITY......Page 244
 4.6 RATES AND RELATED RATES......Page 253
 4.7 L\'HOPITAL\'S RULE, GROWTH, AND DOMINANCE......Page 262
 4.8 PARAMETRIC EQUATIONS......Page 269
 REVIEW EXERCISES AND PROBLEMS......Page 280
 PROJECTS......Page 287
Chapter 5: KEY CONCEPT: THE DEFINITE INTEGRAL......Page 291
 5.1 HOW DO WE MEASURE DISTANCE TRAVELED?......Page 292
 5.2 THE DEFINITE INTEGRAL......Page 301
 5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS......Page 309
 5.4 THEOREMS ABOUT DEFINITE INTEGRALS......Page 318
 REVIEW EXERCISES AND PROBLEMS......Page 329
 PROJECTS......Page 336
Chapter 6: CONSTRUCTING ANTIDERIVATIVES......Page 339
 6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY......Page 340
 6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY......Page 346
 6.3 DIFFERENTIAL EQUATIONS AND MOTION......Page 352
 6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS......Page 360
 REVIEW EXERCISES AND PROBLEMS......Page 365
 PROJECTS......Page 370
Chapter 7: INTEGRATION......Page 373
 7.1 INTEGRATION BY SUBSTITUTION......Page 374
 7.2 INTEGRATION BY PARTS......Page 384
 7.3 TABLES OF INTEGRALS......Page 391
 7.4 ALGEBRAIC IDENTITIES AND TRIGONOMETRIC SUBSTITUTIONS......Page 396
 7.5 NUMERICAL METHODS FOR DEFINITE INTEGRALS......Page 407
 7.6 IMPROPER INTEGRALS......Page 415
 7.7 COMPARISON OF IMPROPER INTEGRALS......Page 423
 CHAPTER SUMMARY......Page 427
 REVIEW EXERCISES AND PROBLEMS......Page 428
 PROJECTS......Page 432
Chapter 8: USING THE DEFINITE INTEGRAL......Page 433
 8.1 AREAS AND VOLUMES......Page 434
 8.2 APPLICATIONS TO GEOMETRY......Page 442
 8.3 AREA AND ARC LENGTH IN POLAR COORDINATES......Page 451
 8.4 DENSITY AND CENTER OF MASS......Page 459
 8.5 APPLICATIONS TO PHYSICS......Page 469
 8.6 APPLICATIONS TO ECONOMICS......Page 479
 8.7 DISTRIBUTION FUNCTIONS......Page 486
 8.8 PROBABILITY, MEAN, AND MEDIAN......Page 493
 CHAPTER SUMMARY......Page 500
 REVIEW EXERCISES AND PROBLEMS......Page 501
 PROJECTS......Page 506
Chapter 9: SEQUENCES AND SERIES......Page 511
 9.1 SEQUENCES......Page 512
 9.2 GEOMETRIC SERIES......Page 518
 9.3 CONVERGENCE OF SERIES......Page 525
 9.4 TESTS FOR CONVERGENCE......Page 532
 9.5 POWER SERIES AND INTERVAL OF CONVERGENCE......Page 541
 REVIEW EXERCISES AND PROBLEMS......Page 549
 PROJECTS......Page 553
Chapter 10: APPROXIMATING FUNCTIONS USING SERIES......Page 557
 10.1 TAYLOR POLYNOMIALS......Page 558
 10.2 TAYLOR SERIES......Page 566
 10.3 FINDING AND USING TAYLOR SERIES......Page 572
 10.4 THE ERROR IN TAYLOR POLYNOMIAL APPROXIMATIONS......Page 580
 10.5 FOURIER SERIES......Page 585
 REVIEW EXERCISES AND PROBLEMS......Page 598
 PROJECTS......Page 602
Chapter11: DIFFERENTIAL EQUATIONS......Page 605
 11.1 WHAT IS A DIFFERENTIAL EQUATION?......Page 606
 11.2 SLOPE FIELDS......Page 611
 11.3 EULER\'S METHOD......Page 619
 11.4 SEPARATION OF VARIABLES......Page 624
 11.5 GROWTH AND DECAY......Page 629
 11.6 APPLICATIONS AND MODELING......Page 640
 11.7 THE LOGISTIC MODEL......Page 649
 11.8 SYSTEMS OF DIFFERENTIAL EQUATIONS......Page 659
 11.9 ANALYZING THE PHASE PLANE......Page 669
 REVIEW EXERCISES AND PROBLEMS......Page 675
 PROJECTS......Page 681
Chapter 12: FUNCTIONS OF SEVERAL VARIABLES......Page 685
 12.1 FUNCTIONS OF TWO VARIABLES......Page 686
 12.2 GRAPHS AND SURFACES......Page 694
 12.3 CONTOUR DIAGRAMS......Page 701
 12.4 LINEAR FUNCTIONS......Page 714
 12.5 FUNCTIONS OF THREE VARIABLES......Page 720
 12.6 LIMITS AND CONTINUITY......Page 725
 REVIEW EXERCISES AND PROBLEMS......Page 730
 PROJECTS......Page 734
Chapter 13: A FUNDAMENTAL TOOL: VECTORS......Page 737
 13.1 DISPLACEMENT VECTORS......Page 738
 13.2 VECTORS IN GENERAL......Page 746
 13.3 THE DOT PRODUCT......Page 754
 13.4 THE CROSS PRODUCT......Page 764
 REVIEW EXERCISES AND PROBLEMS......Page 772
 PROJECTS......Page 775
Chapter 14: DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES......Page 777
 14.1 THE PARTIAL DERIVATIVE......Page 778
 14.2 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY......Page 786
 14.3 LOCAL LINEARITY AND THE DIFFERENTIAL......Page 791
 14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE......Page 799
 14.5 GRADIENTS AND DIRECTIONAL DERIVATIVES IN SPACE......Page 809
 14.6 THE CHAIN RULE......Page 816
 14.7 SECOND-ORDER PARTIAL DERIVATIVES......Page 826
 14.8 DIFFERENTIABILITY......Page 835
 CHAPTER SUMMARY......Page 841
 REVIEW EXERCISES AND PROBLEMS......Page 842
 PROJECTS......Page 847
Chapter 15: OPTIMIZATION: LOCAL AND GLOBAL EXTREMA......Page 849
 15.1 CRITICAL POINTS: LOCAL EXTREMA AND SADDLE POINTS......Page 850
 15.2 OPTIMIZATION......Page 859
 15.3 CONSTRAINED OPTIMIZATION: LAGRANGE MULTIPLIERS......Page 868
 REVIEW EXERCISES AND PROBLEMS......Page 880
 PROJECTS......Page 884
Chapter 16: INTEGRATING FUNCTIONS OF SEVERAL VARIABLES......Page 887
 16.1 THE DEFINITE INTEGRAL OF A FUNCTION OF TWO VARIABLES......Page 888
 16.2 ITERATED INTEGRALS......Page 895
 16.3 TRIPLE INTEGRALS......Page 904
 16.4 DOUBLE INTEGRALS IN POLAR COORDINATES......Page 911
 16.5 INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES......Page 916
 16.6 APPLICATIONS OF INTEGRATION TO PROBABILITY......Page 926
 REVIEW EXERCISES AND PROBLEMS......Page 931
 PROJECTS......Page 935
Chapter 17: PARAMETERIZATION AND VECTOR FIELDS......Page 937
 17.1 PARAMETERIZED CURVES......Page 938
 17.2 MOTION, VELOCITY, AND ACCELERATION......Page 947
 17.3 VECTOR FIELDS......Page 957
 17.4 THE FLOW OF A VECTOR FIELD......Page 963
 CHAPTER SUMMARY......Page 969
 REVIEW EXERCISES AND PROBLEMS......Page 970
 PROJECTS......Page 973
Chapter 18: LINE INTEGRALS......Page 977
 18.1 THE IDEA OF A LINE INTEGRAL......Page 978
 18.2 COMPUTING LINE INTEGRALS OVER PARAMETERIZED CURVES......Page 987
 18.3 GRADIENT FIELDS AND PATH-INDEPENDENT FIELDS......Page 994
 18.4 PATH-DEPENDENT VECTOR FIELDS AND GREEN\'S THEOREM......Page 1005
 CHAPTER SUMMARY......Page 1016
 REVIEW EXERCISES AND PROBLEMS......Page 1017
 PROJECTS......Page 1022
Chapter 19: FLUX INTEGRALS AND DIVERGENCE......Page 1025
 19.1 THE IDEA OF A FLUX INTEGRAL......Page 1026
 19.2 FLUX INTEGRALS FOR GRAPHS, CYLINDERS, AND SPHERES......Page 1036
 19.3 THE DIVERGENCE OF A VECTOR FIELD......Page 1045
 19.4 THE DIVERGENCE THEOREM......Page 1054
 REVIEW EXERCISES AND PROBLEMS......Page 1060
 PROJECTS......Page 1064
Chapter 20: THE CURL AND STOKES\' THEOREM......Page 1067
 20.1 THE CURL OF A VECTOR FIELD......Page 1068
 20.2 STOKES\' THEOREM......Page 1076
 20.3 THE THREE FUNDAMENTAL THEOREMS......Page 1082
 REVIEW EXERCISES AND PROBLEMS......Page 1087
 PROJECTS......Page 1091
Chapter 21: PARAMETERS, COORDINATES, AND INTEGRALS......Page 1093
 21.1 COORDINATES AND PARAMETERIZED SURFACES......Page 1094
 21.2 CHANGE OF COORDINATES IN A MULTIPLE INTEGRAL......Page 1104
 21.3 FLUX INTEGRALS OVER PARAMETERIZED SURFACES......Page 1109
 REVIEW EXERCISES AND PROBLEMS......Page 1113
 PROJECTS......Page 1114
APPENDICES......Page 1115
 A: ROOTS, ACCURACY, AND BOUNDS......Page 1116
 B: COMPLEX NUMBERS......Page 1124
 C: NEWTON\'S METHOD......Page 1131
 D: VECTORS IN THE PLANE......Page 1134
 E: DETERMINANTS......Page 1140
READY REFERENCE......Page 1143
ANSWERS TO ODD-NUMBERED PROBLEMS......Page 1161
INDEX......Page 1225
Helpful Quick References......Page 1241
 The Binomial Theorem......Page 1242
 A Short Table of Indefinite Integrals......Page 1243