Calculus: Early Transcendentals

Cover Page......Page 1
Title Page......Page 5
Copyright Page......Page 6
About the Author......Page 7
Dedication......Page 8
PREFACE......Page 9
SUPPLEMENTS......Page 12
ACKNOWLEDGMENTS......Page 14
CONTENTS......Page 16
DEFINITION OF A FUNCTION......Page 23
INDEPENDENT AND DEPENDENT VARIABLES......Page 25
THE VERTICAL LINE TEST......Page 26
THE ABSOLUTE VALUE FUNCTION......Page 27
PIECEWISE-DEFINED FUNCTIONS......Page 28
DOMAIN AND RANGE......Page 29
THE EFFECT OF ALGEBRAIC OPERATIONS ON THE DOMAIN......Page 30
DOMAIN AND RANGE IN APPLIED PROBLEMS......Page 31
ISSUES OF SCALE AND UNITS......Page 32
ARITHMETIC OPERATIONS ON FUNCTIONS......Page 37
COMPOSITION OF FUNCTIONS......Page 39
EXPRESSING A FUNCTION AS A COMPOSITION......Page 40
NEW FUNCTIONS FROM OLD......Page 41
TRANSLATIONS......Page 42
REFLECTIONS......Page 43
STRETCHES AND COMPRESSIONS......Page 44
EVEN AND ODD FUNCTIONS......Page 45
FAMILIES OF CURVES......Page 49
POWER FUNCTIONS; THE FAMILY y = xn......Page 50
INVERSE PROPORTIONS......Page 51
POWER FUNCTIONS WITH NONINTEGER EXPONENTS......Page 52
RATIONAL FUNCTIONS......Page 53
THE FAMILIES y = A sin Bx AND y = A cos Bx......Page 54
THE FAMILIES y = A sin(Bx – C) AND y = A cos(Bx – C)......Page 56
INVERSE FUNCTIONS......Page 60
CHANGING THE INDEPENDENT VARIABLE......Page 61
A METHOD FOR FINDING INVERSE FUNCTIONS......Page 62
EXISTENCE OF INVERSE FUNCTIONS......Page 63
INCREASING OR DECREASING FUNCTIONS ARE INVERTIBLE......Page 64
GRAPHS OF INVERSE FUNCTIONS......Page 65
INVERSE TRIGONOMETRIC FUNCTIONS......Page 66
EVALUATING INVERSE TRIGONOMETRIC FUNCTIONS......Page 68
IDENTITIES FOR INVERSE TRIGONOMETRIC FUNCTIONS......Page 69
IRRATIONAL EXPONENTS......Page 74
THE FAMILY OF EXPONENTIAL FUNCTIONS......Page 75
THE NATURAL EXPONENTIAL FUNCTION......Page 76
LOGARITHMIC FUNCTIONS......Page 77
SOLVING EQUATIONS INVOLVING EXPONENTIALS AND LOGARITHMS......Page 79
LOGARITHMIC SCALES IN SCIENCE AND ENGINEERING......Page 81
EXPONENTIAL AND LOGARITHMIC GROWTH......Page 82
1.1 Limits (An Intuitive Approach)......Page 89
TANGENT LINES AND LIMITS......Page 90
DECIMALS AND LIMITS......Page 91
LIMITS......Page 92
SAMPLING PITFALLS......Page 93
ONE-SIDED LIMITS......Page 94
THE RELATIONSHIP BETWEEN ONE-SIDED LIMITS AND TWO-SIDED LIMITS......Page 95
INFINITE LIMITS......Page 96
VERTICAL ASYMPTOTES......Page 98
SOME BASIC LIMITS......Page 102
LIMITS OF POLYNOMIALS AND RATIONAL FUNCTIONS AS x→a......Page 104
LIMITS INVOLVING RADICALS......Page 107
LIMITS OF PIECEWISE-DEFINED FUNCTIONS......Page 108
LIMITS AT INFINITY AND HORIZONTAL ASYMPTOTES......Page 111
INFINITE LIMITS AT INFINITY......Page 112
LIMITS OF POLYNOMIALS AS x→±∞......Page 113
LIMITS OF RATIONAL FUNCTIONS AS x→±∞......Page 114
A QUICK METHOD FOR FINDING LIMITS OF RATIONAL FUNCTIONS AS x→+∞ OR x→−∞......Page 115
LIMITS INVOLVING RADICALS......Page 116
END BEHAVIOR OF TRIGONOMETRIC, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS......Page 117
MOTIVATION FOR THE DEFINITION OF A TWO-SIDED LIMIT......Page 122
LIMITS AS x→±∞......Page 125
INFINITE LIMITS......Page 127
DEFINITION OF CONTINUITY......Page 132
CONTINUITY IN APPLICATIONS......Page 133
CONTINUITY ON AN INTERVAL......Page 134
CONTINUITY OF POLYNOMIALS AND RATIONAL FUNCTIONS......Page 135
CONTINUITY OF COMPOSITIONS......Page 136
THE INTERMEDIATE-VALUE THEOREM......Page 137
APPROXIMATING ROOTS USING THE INTERMEDIATE-VALUE THEOREM......Page 138
CONTINUITY OF INVERSE FUNCTIONS......Page 143
OBTAINING LIMITS BY SQUEEZING......Page 144
TANGENT LINES......Page 153
VELOCITY......Page 156
SLOPES AND RATES OF CHANGE......Page 159
RATES OF CHANGE IN APPLICATIONS......Page 161
DEFINITION OF THE DERIVATIVE FUNCTION......Page 165
DIFFERENTIABILITY......Page 168
THE RELATIONSHIP BETWEEN DIFFERENTIABILITY AND CONTINUITY......Page 170
OTHER DERIVATIVE NOTATIONS......Page 172
DERIVATIVE OF A CONSTANT......Page 177
DERIVATIVES OF POWER FUNCTIONS......Page 178
DERIVATIVE OF A CONSTANT TIMES A FUNCTION......Page 179
DERIVATIVES OF SUMS AND DIFFERENCES......Page 180
HIGHER DERIVATIVES......Page 181
DERIVATIVE OF A PRODUCT......Page 185
DERIVATIVE OF A QUOTIENT......Page 187
SUMMARY OF DIFFERENTIATION RULES......Page 189
2.5 Derivatives of Trigonometric Functions......Page 191
DERIVATIVES OF COMPOSITIONS......Page 196
AN ALTERNATIVE VERSION OF THE CHAIN RULE......Page 197
GENERALIZED DERIVATIVE FORMULAS......Page 198
DIFFERENTIATING USING COMPUTER ALGEBRA SYSTEMS......Page 200
FUNCTIONS DEFINED EXPLICITLY AND IMPLICITLY......Page 207
IMPLICIT DIFFERENTIATION......Page 209
DIFFERENTIABILITY OF FUNCTIONS DEFINED IMPLICITLY......Page 212
DERIVATIVES OF LOGARITHMIC FUNCTIONS......Page 214
LOGARITHMIC DIFFERENTIATION......Page 216
DERIVATIVES OF REAL POWERS OF x......Page 217
INCREASING OR DECREASING FUNCTIONS ARE ONE-TO-ONE......Page 219
DERIVATIVES OF EXPONENTIAL FUNCTIONS......Page 220
DERIVATIVES OF THE INVERSE TRIGONOMETRIC FUNCTIONS......Page 222
DIFFERENTIATING EQUATIONS TO RELATE RATES......Page 226
3.5 Local Linear Approximation; Differentials......Page 234
DIFFERENTIALS......Page 235
LOCAL LINEAR APPROXIMATION FROM THE DIFFERENTIAL POINT OF VIEW......Page 236
ERROR PROPAGATION......Page 237
MORE NOTATION; DIFFERENTIAL FORMULAS......Page 238
INDETERMINATE FORMS OF TYPE 0/0......Page 241
INDETERMINATE FORMS OF TYPE ∞/∞......Page 244
ANALYZING THE GROWTH OF EXPONENTIAL FUNCTIONS USING L’HÔPITAL’S RULE......Page 245
INDETERMINATE FORMS OF TYPE 0 · ∞......Page 246
INDETERMINATE FORMS OF TYPE 00,∞0, 1∞......Page 247
INCREASING AND DECREASING FUNCTIONS......Page 254
CONCAVITY......Page 257
INFLECTION POINTS......Page 258
INFLECTION POINTS IN APPLICATIONS......Page 260
LOGISTIC CURVES......Page 261
RELATIVE MAXIMA AND MINIMA......Page 266
FIRST DERIVATIVE TEST......Page 268
SECOND DERIVATIVE TEST......Page 269
GEOMETRIC IMPLICATIONS OF MULTIPLICITY......Page 271
ANALYSIS OF POLYNOMIALS......Page 272
PROPERTIES OF GRAPHS......Page 276
GRAPHING RATIONAL FUNCTIONS......Page 277
RATIONAL FUNCTIONS WITH OBLIQUE OR CURVILINEAR ASYMPTOTES......Page 280
GRAPHS WITH VERTICAL TANGENTS AND CUSPS......Page 281
GRAPHING OTHER KINDS OF FUNCTIONS......Page 283
GRAPHING USING CALCULUS AND TECHNOLOGY TOGETHER......Page 284
ABSOLUTE EXTREMA......Page 288
THE EXTREME VALUE THEOREM......Page 289
ABSOLUTE EXTREMA ON INFINITE INTERVALS......Page 291
ABSOLUTE EXTREMA OF FUNCTIONS WITH ONE RELATIVE EXTREMUM......Page 292
PROBLEMS INVOLVING FINITE CLOSED INTERVALS......Page 296
PROBLEMS INVOLVING INTERVALS THAT ARE NOT BOTH FINITE AND CLOSED......Page 301
AN APPLICATION TO ECONOMICS......Page 303
A BASIC PRINCIPLE OF ECONOMICS......Page 305
REVIEW OF TERMINOLOGY......Page 310
VELOCITY AND SPEED......Page 311
SPEEDING UP AND SLOWING DOWN......Page 312
ANALYZING THE POSITION VERSUS TIME CURVE......Page 313
NEWTON’S METHOD......Page 318
SOME DIFFICULTIES WITH NEWTON’S METHOD......Page 321
ROLLE’S THEOREM......Page 324
THE MEAN-VALUE THEOREM......Page 326
VELOCITY INTERPRETATION OF THE MEAN-VALUE THEOREM......Page 327
THE CONSTANT DIFFERENCE THEOREM......Page 328
THE AREA PROBLEM......Page 338
THE RECTANGLE METHOD FOR FINDING AREAS......Page 339
THE ANTIDERIVATIVE METHOD FOR FINDING AREAS......Page 341
THE RECTANGLE METHOD AND THE ANTIDERIVATIVE METHOD COMPARED......Page 343
ANTIDERIVATIVES......Page 344
THE INDEFINITE INTEGRAL......Page 345
INTEGRATION FORMULAS......Page 346
PROPERTIES OF THE INDEFINITE INTEGRAL......Page 347
INTEGRAL CURVES......Page 349
INTEGRATION FROM THE VIEWPOINT OF DIFFERENTIAL EQUATIONS......Page 350
SLOPE FIELDS......Page 351
u-SUBSTITUTION......Page 354
EASY TO RECOGNIZE SUBSTITUTIONS......Page 356
LESS APPARENT SUBSTITUTIONS......Page 358
INTEGRATION USING COMPUTER ALGEBRA SYSTEMS......Page 360
SIGMA NOTATION......Page 362
CHANGING THE LIMITS OF SUMMATION......Page 363
SUMMATION FORMULAS......Page 364
A DEFINITION OF AREA......Page 365
NET SIGNED AREA......Page 369
RIEMANN SUMS AND THE DEFINITE INTEGRAL......Page 375
PROPERTIES OF THE DEFINITE INTEGRAL......Page 378
DISCONTINUITIES AND INTEGRABILITY......Page 380
THE FUNDAMENTAL THEOREM OF CALCULUS......Page 384
THE RELATIONSHIP BETWEEN DEFINITE AND INDEFINITE INTEGRALS......Page 387
DUMMY VARIABLES......Page 389
THE MEAN-VALUE THEOREM FOR INTEGRALS......Page 390
PART 2 OF THE FUNDAMENTAL THEOREM OF CALCULUS......Page 391
INTEGRATING RATES OF CHANGE......Page 393
COMPUTING DISPLACEMENT AND DISTANCE TRAVELED BY INTEGRATION......Page 398
ANALYZING THE VELOCITY VERSUS TIME CURVE......Page 399
CONSTANT ACCELERATION......Page 400
FREE-FALL MODEL......Page 402
AVERAGE VALUE OF A CONTINUOUS FUNCTION......Page 407
AVERAGE VALUE AND AVERAGE VELOCITY......Page 409
TWO METHODS FOR MAKING SUBSTITUTIONS IN DEFINITE INTEGRALS......Page 412
5.10 Logarithmic and Other Functions Defined by Integrals......Page 418
THE CONNECTION BETWEEN NATURAL LOGARITHMS AND INTEGRALS......Page 419
ALGEBRAIC PROPERTIES OF ln x......Page 420
APPROXIMATING ln x NUMERICALLY......Page 421
DEFINITION OF ex......Page 422
IRRATIONAL EXPONENTS......Page 423
GENERAL LOGARITHMS......Page 424
FUNCTIONS DEFINED BY INTEGRALS......Page 425
EVALUATING AND GRAPHING FUNCTIONS DEFINED BY INTEGRALS......Page 426
INTEGRALS WITH FUNCTIONS AS LIMITS OF INTEGRATION......Page 427
A REVIEW OF RIEMANN SUMS......Page 435
AREA BETWEEN y = f (x) AND y = g(x)......Page 436
REVERSING THE ROLES OF x AND y......Page 440
VOLUMES BY SLICING......Page 443
VOLUMES BY DISKS PERPENDICULAR TO THE x-AXIS......Page 446
VOLUMES BY WASHERS PERPENDICULAR TO THE x-AXIS......Page 447
VOLUMES BY DISKS AND WASHERS PERPENDICULAR TO THE y-AXIS......Page 448
OTHER AXES OF REVOLUTION......Page 449
CYLINDRICAL SHELLS......Page 454
VARIATIONS OF THE METHOD OF CYLINDRICAL SHELLS......Page 456
ARC LENGTH......Page 460
FINDING ARC LENGTH BY NUMERICAL METHODS......Page 463
SURFACE AREA......Page 466
WORK DONE BY A CONSTANT FORCE APPLIED IN THE DIRECTION OF MOTION......Page 471
WORK DONE BY A VARIABLE FORCE APPLIED IN THE DIRECTION OF MOTION......Page 473
CALCULATING WORK FROM BASIC PRINCIPLES......Page 475
THE WORK–ENERGY RELATIONSHIP......Page 476
CENTER OF GRAVITY OF A LAMINA......Page 480
OTHER TYPES OF REGIONS......Page 485
THEOREM OF PAPPUS......Page 486
WHAT IS A FLUID?......Page 489
THE CONCEPT OF PRESSURE......Page 490
FLUID PRESSURE......Page 491
FLUID FORCE ON A VERTICAL SURFACE......Page 492
DEFINITIONS OF HYPERBOLIC FUNCTIONS......Page 496
HANGING CABLES AND OTHER APPLICATIONS......Page 497
WHY THEY ARE CALLED HYPERBOLIC FUNCTIONS......Page 498
DERIVATIVE AND INTEGRAL FORMULAS......Page 499
INVERSES OF HYPERBOLIC FUNCTIONS......Page 500
LOGARITHMIC FORMS OF INVERSE HYPERBOLIC FUNCTIONS......Page 502
DERIVATIVES AND INTEGRALS INVOLVING INVERSE HYPERBOLIC FUNCTIONS......Page 503
METHODS FOR APPROACHING INTEGRATION PROBLEMS......Page 510
A REVIEW OF FAMILIAR INTEGRATION FORMULAS......Page 511
THE PRODUCT RULE AND INTEGRATION BY PARTS......Page 513
GUIDELINES FOR INTEGRATION BY PARTS......Page 514
REPEATED INTEGRATION BY PARTS......Page 515
A TABULAR METHOD FOR REPEATED INTEGRATION BY PARTS......Page 517
INTEGRATION BY PARTS FOR DEFINITE INTEGRALS......Page 518
REDUCTION FORMULAS......Page 519
INTEGRATING POWERS OF SINE AND COSINE......Page 522
INTEGRATING PRODUCTS OF SINES AND COSINES......Page 523
INTEGRATING POWERS OF TANGENT AND SECANT......Page 525
INTEGRATING PRODUCTS OF TANGENTS AND SECANTS......Page 526
AN ALTERNATIVE METHOD FOR INTEGRATING POWERS OF SINE, COSINE, TANGENT, AND SECANT......Page 527
MERCATOR’S MAP OF THE WORLD......Page 528
THE METHOD OF TRIGONOMETRIC SUBSTITUTION......Page 530
INTEGRALS INVOLVING ax2 + bx + c......Page 534
PARTIAL FRACTIONS......Page 536
LINEAR FACTORS......Page 538
QUADRATIC FACTORS......Page 540
INTEGRATING IMPROPER RATIONAL FUNCTIONS......Page 542
CONCLUDING REMARKS......Page 543
PERFECT MATCHES......Page 545
MATCHES REQUIRING SUBSTITUTIONS......Page 546
MATCHES REQUIRING REDUCTION FORMULAS......Page 547
SPECIAL SUBSTITUTIONS......Page 548
INTEGRATING WITH COMPUTER ALGEBRA SYSTEMS......Page 550
COMPUTER ALGEBRA SYSTEMS HAVE LIMITATIONS......Page 551
A REVIEW OF RIEMANN SUM APPROXIMATIONS......Page 555
TRAPEZOIDAL APPROXIMATION......Page 556
COMPARISON OF THE MIDPOINT AND TRAPEZOIDAL APPROXIMATIONS......Page 557
SIMPSON’S RULE......Page 559
ERROR BOUNDS......Page 562
A COMPARISON OF THE THREE METHODS......Page 565
IMPROPER INTEGRALS......Page 569
INTEGRALS OVER INFINITE INTERVALS......Page 570
INTEGRALS WHOSE INTEGRANDS HAVE INFINITE DISCONTINUITIES......Page 573
ARC LENGTH AND SURFACE AREA USING IMPROPER INTEGRALS......Page 575
SOLUTIONS OF DIFFERENTIAL EQUATIONS......Page 583
INITIAL-VALUE PROBLEMS......Page 584
INHIBITED POPULATION GROWTH; LOGISTIC MODELS......Page 585
SPREAD OF DISEASE......Page 586
VIBRATIONS OF SPRINGS......Page 587
FIRST-ORDER SEPARABLE EQUATIONS......Page 590
EXPONENTIAL GROWTH AND DECAY MODELS......Page 593
DOUBLING TIME AND HALF-LIFE......Page 594
CARBON DATING......Page 595
FUNCTIONS OF TWO VARIABLES......Page 601
SLOPE FIELDS......Page 602
EULER’S METHOD......Page 603
ACCURACY OF EULER’S METHOD......Page 605
FIRST-ORDER LINEAR EQUATIONS......Page 608
MIXING PROBLEMS......Page 611
A MODEL OF FREE-FALL MOTION RETARDED BY AIR RESISTANCE......Page 612
DEFINITION OF A SEQUENCE......Page 618
LIMIT OF A SEQUENCE......Page 621
THE SQUEEZING THEOREM FOR SEQUENCES......Page 624
SEQUENCES DEFINED RECURSIVELY......Page 626
TERMINOLOGY......Page 629
TESTING FOR MONOTONICITY......Page 630
CONVERGENCE OF MONOTONE SEQUENCES......Page 632
SUMS OF INFINITE SERIES......Page 636
GEOMETRIC SERIES......Page 639
TELESCOPING SUMS......Page 641
HARMONIC SERIES......Page 642
THE DIVERGENCE TEST......Page 645
ALGEBRAIC PROPERTIES OF INFINITE SERIES......Page 646
THE INTEGRAL TEST......Page 648
p-SERIES......Page 649
PROOF OF THE INTEGRAL TEST......Page 650
THE COMPARISON TEST......Page 653
USING THE COMPARISON TEST......Page 654
THE LIMIT COMPARISON TEST......Page 655
THE RATIO TEST......Page 656
THE ROOT TEST......Page 657
ALTERNATING SERIES......Page 660
APPROXIMATING SUMS OF ALTERNATING SERIES......Page 661
ABSOLUTE CONVERGENCE......Page 663
THE RATIO TEST FOR ABSOLUTE CONVERGENCE......Page 665
SUMMARY OF CONVERGENCE TESTS......Page 666
LOCAL QUADRATIC APPROXIMATIONS......Page 670
MACLAURIN POLYNOMIALS......Page 671
TAYLOR POLYNOMIALS......Page 675
SIGMA NOTATION FOR TAYLOR AND MACLAURIN POLYNOMIALS......Page 676
THE nTH REMAINDER......Page 677
MACLAURIN AND TAYLOR SERIES......Page 681
POWER SERIES IN x......Page 683
FINDING THE INTERVAL OF CONVERGENCE......Page 684
POWER SERIES IN x – x0......Page 686
FUNCTIONS DEFINED BY POWER SERIES......Page 687
THE CONVERGENCE PROBLEM FOR TAYLOR SERIES......Page 690
ESTIMATING THE nTH REMAINDER......Page 691
APPROXIMATING TRIGONOMETRIC FUNCTIONS......Page 692
APPROXIMATING EXPONENTIAL FUNCTIONS......Page 694
APPROXIMATING π......Page 695
BINOMIAL SERIES......Page 696
SOME IMPORTANT MACLAURIN SERIES......Page 697
DIFFERENTIATING POWER SERIES......Page 700
INTEGRATING POWER SERIES......Page 701
POWER SERIES REPRESENTATIONS MUST BE TAYLOR SERIES......Page 703
SOME PRACTICAL WAYS TO FIND TAYLOR SERIES......Page 704
FINDING TAYLOR SERIES BY MULTIPLICATION AND DIVISION......Page 706
MODELING PHYSICAL LAWS WITH TAYLOR SERIES......Page 707
PARAMETRIC EQUATIONS......Page 714
EXPRESSING ORDINARY FUNCTIONS PARAMETRICALLY......Page 716
TANGENT LINES TO PARAMETRIC CURVES......Page 717
ARC LENGTH OF PARAMETRIC CURVES......Page 719
THE CYCLOID (THE APPLE OF DISCORD)......Page 720
POLAR COORDINATE SYSTEMS......Page 727
RELATIONSHIP BETWEEN POLAR AND RECTANGULAR COORDINATES......Page 728
GRAPHS IN POLAR COORDINATES......Page 729
SYMMETRY TESTS......Page 732
FAMILIES OF CIRCLES......Page 734
FAMILIES OF CARDIOIDS AND LIMAÇONS......Page 735
SPIRALS IN NATURE......Page 736
GENERATING POLAR CURVES WITH GRAPHING UTILITIES......Page 737
TANGENT LINES TO POLAR CURVES......Page 741
ARC LENGTH OF A POLAR CURVE......Page 743
AREA IN POLAR COORDINATES......Page 744
USING SYMMETRY......Page 746
INTERSECTIONS OF POLAR GRAPHS......Page 748
CONIC SECTIONS......Page 752
DEFINITIONS OF THE CONIC SECTIONS......Page 753
EQUATIONS OF PARABOLAS IN STANDARD POSITION......Page 754
A TECHNIQUE FOR SKETCHING PARABOLAS......Page 755
EQUATIONS OF ELLIPSES IN STANDARD POSITION......Page 756
A TECHNIQUE FOR SKETCHING ELLIPSES......Page 758
EQUATIONS OF HYPERBOLAS IN STANDARD POSITION......Page 759
A QUICK WAY TO FIND ASYMPTOTES......Page 760
A TECHNIQUE FOR SKETCHING HYPERBOLAS......Page 761
TRANSLATED CONICS......Page 762
REFLECTION PROPERTIES OF THE CONIC SECTIONS......Page 764
APPLICATIONS OF THE CONIC SECTIONS......Page 765
QUADRATIC EQUATIONS IN x AND y......Page 770
ROTATION OF AXES......Page 771
ELIMINATING THE CROSS-PRODUCT TERM......Page 772
THE FOCUS–DIRECTRIX CHARACTERIZATION OF CONICS......Page 776
POLAR EQUATIONS OF CONICS......Page 777
SKETCHING CONICS IN POLAR COORDINATES......Page 778
APPLICATIONS IN ASTRONOMY......Page 781
RECTANGULAR COORDINATE SYSTEMS......Page 789
DISTANCE IN 3-SPACE; SPHERES......Page 790
CYLINDRICAL SURFACES......Page 792
VECTORS IN PHYSICS AND ENGINEERING......Page 795
VECTORS VIEWED GEOMETRICALLY......Page 796
VECTORS IN COORDINATE SYSTEMS......Page 797
VECTORS WITH INITIAL POINT NOT AT THE ORIGIN......Page 798
RULES OF VECTOR ARITHMETIC......Page 799
UNIT VECTORS......Page 800
VECTORS DETERMINED BY LENGTH AND ANGLE......Page 801
RESULTANT OF TWO CONCURRENT FORCES......Page 802
ALGEBRAIC PROPERTIES OF THE DOT PRODUCT......Page 807
ANGLE BETWEEN VECTORS......Page 808
DIRECTION ANGLES......Page 809
DECOMPOSING VECTORS INTO ORTHOGONAL COMPONENTS......Page 810
ORTHOGONAL PROJECTIONS......Page 812
WORK......Page 813
DETERMINANTS......Page 817
CROSS PRODUCT......Page 818
ALGEBRAIC PROPERTIES OF THE CROSS PRODUCT......Page 819
GEOMETRIC PROPERTIES OF THE CROSS PRODUCT......Page 820
GEOMETRIC PROPERTIES OF THE SCALAR TRIPLE PRODUCT......Page 822
DOT AND CROSS PRODUCTS ARE COORDINATE INDEPENDENT......Page 823
MOMENTS AND ROTATIONAL MOTION IN 3-SPACE......Page 824
LINES DETERMINED BY A POINT AND A VECTOR......Page 827
VECTOR EQUATIONS OF LINES......Page 830
PLANES DETERMINED BY A POINT AND A NORMAL VECTOR......Page 835
DISTANCE PROBLEMS INVOLVING PLANES......Page 838
TRACES OF SURFACES......Page 843
THE QUADRIC SURFACES......Page 844
TECHNIQUES FOR GRAPHING QUADRIC SURFACES......Page 846
TRANSLATIONS OF QUADRIC SURFACES......Page 849
REFLECTIONS OF SURFACES IN 3-SPACE......Page 850
A TECHNIQUE FOR IDENTIFYING QUADRIC SURFACES......Page 851
CONSTANT SURFACES......Page 854
CONVERTING COORDINATES......Page 855
EQUATIONS OF SURFACES IN CYLINDRICAL AND SPHERICAL COORDINATES......Page 857
SPHERICAL COORDINATES IN NAVIGATION......Page 858
PARAMETRIC CURVES IN 3-SPACE......Page 863
PARAMETRIC EQUATIONS FOR INTERSECTIONS OF SURFACES......Page 864
VECTOR-VALUED FUNCTIONS......Page 865
GRAPHS OF VECTOR-VALUED FUNCTIONS......Page 866
VECTOR FORM OF A LINE SEGMENT......Page 867
LIMITS AND CONTINUITY......Page 870
DERIVATIVES......Page 871
DERIVATIVE RULES......Page 872
TANGENT LINES TO GRAPHS OF VECTOR-VALUED FUNCTIONS......Page 873
DERIVATIVES OF DOT AND CROSS PRODUCTS......Page 874
DEFINITE INTEGRALS OF VECTOR-VALUED FUNCTIONS......Page 875
ANTIDERIVATIVES OF VECTOR-VALUED FUNCTIONS......Page 876
SMOOTH PARAMETRIZATIONS......Page 880
ARC LENGTH FROM THE VECTOR VIEWPOINT......Page 881
ARC LENGTH AS A PARAMETER......Page 882
CHANGE OF PARAMETER......Page 883
FINDING ARC LENGTH PARAMETRIZATIONS......Page 884
PROPERTIES OF ARC LENGTH PARAMETRIZATIONS......Page 886
UNIT NORMAL VECTORS......Page 890
COMPUTING T AND N FOR CURVES PARAMETRIZED BY ARC LENGTH......Page 892
BINORMAL VECTORS IN 3-SPACE......Page 893
DEFINITION OF CURVATURE......Page 895
FORMULAS FOR CURVATURE......Page 897
AN INTERPRETATION OF CURVATURE IN 2-SPACE......Page 899
FORMULA SUMMARY......Page 900
VELOCITY, ACCELERATION, AND SPEED......Page 904
DISPLACEMENT AND DISTANCE TRAVELED......Page 906
NORMAL AND TANGENTIAL COMPONENTS OF ACCELERATION......Page 907
A MODEL OF PROJECTILE MOTION......Page 910
PARAMETRIC EQUATIONS OF PROJECTILE MOTION......Page 911
KEPLER’S LAWS......Page 917
NEWTON’S LAW OF UNIVERSAL GRAVITATION......Page 918
KEPLER’S THIRD LAW......Page 921
ARTIFICIAL SATELLITES......Page 922
NOTATION AND TERMINOLOGY......Page 928
GRAPHS OF FUNCTIONS OF TWO VARIABLES......Page 930
LEVEL CURVES......Page 931
LEVEL SURFACES......Page 933
GRAPHING FUNCTIONS OF TWO VARIABLES USING TECHNOLOGY......Page 934
LIMITS ALONG CURVES......Page 939
OPEN AND CLOSED SETS......Page 941
GENERAL LIMITS OF FUNCTIONS OF TWO VARIABLES......Page 942
CONTINUITY......Page 943
LIMITS AT DISCONTINUITIES......Page 945
EXTENSIONS TO THREE VARIABLES......Page 946
PARTIAL DERIVATIVES OF FUNCTIONS OF TWO VARIABLES......Page 949
PARTIAL DERIVATIVE NOTATION......Page 950
PARTIAL DERIVATIVES VIEWED AS RATES OF CHANGE AND SLOPES......Page 951
ESTIMATING PARTIAL DERIVATIVES FROM TABULAR DATA......Page 952
IMPLICIT PARTIAL DIFFERENTIATION......Page 953
PARTIAL DERIVATIVES OF FUNCTIONS WITH MORE THAN TWO VARIABLES......Page 954
HIGHER-ORDER PARTIAL DERIVATIVES......Page 955
EQUALITY OF MIXED PARTIALS......Page 956
THE WAVE EQUATION......Page 957
DIFFERENTIABILITY......Page 962
DIFFERENTIABILITY AND CONTINUITY......Page 965
DIFFERENTIALS......Page 966
LOCAL LINEAR APPROXIMATIONS......Page 968
CHAIN RULES FOR DERIVATIVES......Page 971
CHAIN RULES FOR PARTIAL DERIVATIVES......Page 974
OTHER VERSIONS OF THE CHAIN RULE......Page 975
IMPLICIT DIFFERENTIATION......Page 977
DIRECTIONAL DERIVATIVES......Page 982
THE GRADIENT......Page 985
PROPERTIES OF THE GRADIENT......Page 986
GRADIENTS ARE NORMAL TO LEVEL CURVES......Page 987
AN APPLICATION OF GRADIENTS......Page 988
TANGENT PLANES AND NORMAL VECTORS TO LEVEL SURFACES F(x, y, z) = c......Page 993
TANGENT PLANES TO SURFACES OF THE FORM z = f (x, y)......Page 994
TANGENT PLANES AND TOTAL DIFFERENTIALS......Page 995
USING GRADIENTS TO FIND TANGENT LINES TO INTERSECTIONS OF SURFACES......Page 996
EXTREMA......Page 999
THE EXTREME-VALUE THEOREM......Page 1000
FINDING RELATIVE EXTREMA......Page 1001
THE SECOND PARTIALS TEST......Page 1002
FINDING ABSOLUTE EXTREMA ON CLOSED AND BOUNDED SETS......Page 1004
EXTREMUM PROBLEMS WITH CONSTRAINTS......Page 1011
LAGRANGE MULTIPLIERS......Page 1012
THREE VARIABLES AND ONE CONSTRAINT......Page 1015
VOLUME......Page 1022
EVALUATING DOUBLE INTEGRALS......Page 1024
PROPERTIES OF DOUBLE INTEGRALS......Page 1028
DOUBLE INTEGRALS OVER NONRECTANGULAR REGIONS......Page 1031
SETTING UP LIMITS OF INTEGRATION FOR EVALUATING DOUBLE INTEGRALS......Page 1033
REVERSING THE ORDER OF INTEGRATION......Page 1035
AREA CALCULATED AS A DOUBLE INTEGRAL......Page 1036
SIMPLE POLAR REGIONS......Page 1040
DOUBLE INTEGRALS IN POLAR COORDINATES......Page 1041
EVALUATING POLAR DOUBLE INTEGRALS......Page 1042
FINDING AREAS USING POLAR DOUBLE INTEGRALS......Page 1044
CONVERTING DOUBLE INTEGRALS FROM RECTANGULAR TO POLAR COORDINATES......Page 1045
SURFACE AREA FOR SURFACES OF THE FORM z = f (x, y)......Page 1048
PARAMETRIC REPRESENTATION OF SURFACES......Page 1050
REPRESENTING SURFACES OF REVOLUTION PARAMETRICALLY......Page 1052
PARTIAL DERIVATIVES OF VECTOR-VALUED FUNCTIONS......Page 1053
TANGENT PLANES TO PARAMETRIC SURFACES......Page 1054
SURFACE AREA OF PARAMETRIC SURFACES......Page 1056
DEFINITION OF A TRIPLE INTEGRAL......Page 1061
EVALUATING TRIPLE INTEGRALS OVER RECTANGULAR BOXES......Page 1062
EVALUATING TRIPLE INTEGRALS OVER MORE GENERAL REGIONS......Page 1063
VOLUME CALCULATED AS A TRIPLE INTEGRAL......Page 1064
INTEGRATION IN OTHER ORDERS......Page 1066
TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES......Page 1070
CONVERTING TRIPLE INTEGRALS FROM RECTANGULAR TO CYLINDRICAL COORDINATES......Page 1072
TRIPLE INTEGRALS IN SPHERICAL COORDINATES......Page 1073
CONVERTING TRIPLE INTEGRALS FROM RECTANGULAR TO SPHERICAL COORDINATES......Page 1077
CHANGE OF VARIABLE IN A SINGLE INTEGRAL......Page 1080
TRANSFORMATIONS OF THE PLANE......Page 1081
JACOBIANS IN TWO VARIABLES......Page 1083
CHANGE OF VARIABLES IN DOUBLE INTEGRALS......Page 1085
CHANGE OF VARIABLES IN TRIPLE INTEGRALS......Page 1087
DENSITY AND MASS OF AN INHOMOGENEOUS LAMINA......Page 1093
CENTER OF GRAVITY OF AN INHOMOGENEOUS LAMINA......Page 1094
CENTER OF GRAVITY AND CENTROID OF A SOLID......Page 1097
VECTOR FIELDS......Page 1106
GRAPHICAL REPRESENTATIONS OF VECTOR FIELDS......Page 1107
INVERSE-SQUARE FIELDS......Page 1108
CONSERVATIVE FIELDS AND POTENTIAL FUNCTIONS......Page 1109
DIVERGENCE AND CURL......Page 1110
THE LAPLACIAN ∇2......Page 1112
LINE INTEGRALS......Page 1116
EVALUATING LINE INTEGRALS......Page 1118
LINE INTEGRALS WITH RESPECT TO x, y, AND z......Page 1122
INTEGRATING A VECTOR FIELD ALONG A CURVE......Page 1125
WORK AS A LINE INTEGRAL......Page 1127
LINE INTEGRALS ALONG PIECEWISE SMOOTH CURVES......Page 1129
INDEPENDENCE OF PATH......Page 1133
THE FUNDAMENTAL THEOREM OF LINE INTEGRALS......Page 1134
LINE INTEGRALS ALONG CLOSED PATHS......Page 1135
A TEST FOR CONSERVATIVE VECTOR FIELDS......Page 1137
CONSERVATION OF ENERGY......Page 1141
GREEN’S THEOREM......Page 1144
FINDING AREAS USING GREEN’S THEOREM......Page 1146
GREEN’S THEOREM FOR MULTIPLY CONNECTED REGIONS......Page 1147
DEFINITION OF A SURFACE INTEGRAL......Page 1152
EVALUATING SURFACE INTEGRALS......Page 1153
SURFACE INTEGRALS OVER z = g(x, y), y = g(x, z), AND x = g(y, z)......Page 1154
ORIENTED SURFACES......Page 1160
FLUX......Page 1162
EVALUATING FLUX INTEGRALS......Page 1164
ORIENTATION OF NONPARAMETRIC SURFACES......Page 1166
ORIENTATION OF PIECEWISE SMOOTH CLOSED SURFACES......Page 1170
THE DIVERGENCE THEOREM......Page 1171
USING THE DIVERGENCE THEOREM TO FIND FLUX......Page 1173
SOURCES AND SINKS......Page 1176
GAUSS’S LAW FOR INVERSE-SQUARE FIELDS......Page 1177
GAUSS’S LAW IN ELECTROSTATICS......Page 1178
RELATIVE ORIENTATION OF CURVES AND SURFACES......Page 1180
STOKES’ THEOREM......Page 1181
USING STOKES’ THEOREM TO CALCULATE WORK......Page 1182
RELATIONSHIP BETWEEN GREEN’S THEOREM AND STOKES’ THEOREM......Page 1184
CURL VIEWED AS CIRCULATION......Page 1185
A GRAPHING FUNCTIONS USING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS......Page 1191
B TRIGONOMETRY REVIEW......Page 1203
C SOLVING POLYNOMIAL EQUATIONS......Page 1217
D SELECTED PROOFS......Page 1224
ANSWERS......Page 1235
WEB PROJECTS......Page 1289
ROBOTICS......Page 206
RAILROAD DESIGN......Page 582
ITERATION AND DYNAMICAL SYSTEMS......Page 713
COMET COLLISION......Page 788
BLAMMO THE HUMAN CANNONBALL......Page 927
HURRICANE MODELING......Page 1190