The Calculus 7 TOC

THE CALCULUS 7

Louis Leithold

HarperCollmsCollegePublisbers

CONTENTS

Preface

xiii

FUNCTIONS, LIMITS, AND CONTINUITY

1

1.1

FUNCTIONS AND THEIR GRAPHS

2

1.2

OPERATIONS ON FUNCTIONS AND TYPES OF FUNCTIONS

12

1.3

FUNCTIONS AS MATHEMATICAL MODELS

21

1.4

GRAPHICAL INTRODUCTION TO LIMITS OF FUNCTIONS

30

1.5

DEFINITION OF THE LIMIT OF A FUNCTION AND LIMIT THEOREMS

41

1.6

ONE-SIDED LIMITS

53

1.7

INFINITE LIMITS

59

1.8 1.9

CONTINUITY OF A FUNCTION AT A NUMBER CONTINUITY OF A COMPOSITE FUNCTION AND CONTINUITY ON AN INTERVAL CONTINUITY OF THE TRIGONOMETRIC FUNCTIONS AND THE SQUEEZE THEOREM

72

1.10

CHAPTER 1 REVIEW

THE DERIVATIVE AND DIFFERENTIATION

82 92 102

109

2.1

THE TANGENT LINE AND THE DERIVATIVE

110

2.2

DIFFERENTIABILITY AND CONTINUITY

118

2.3

THE NUMERICAL DERIVATIVE

128

2.4

THEOREMS ON DIFFERENTIATION OF ALGEBRAIC FUNCTIONS AND HIGHER-ORDER DERIVATIVES

132

2.5

RECTILINEAR MOTION

142

2.6

THE DERIVATIVE AS A RATE OF CHANGE

155

CONTENTS

2.7 2.8 2.9 2.10

DERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS 162 THE DERIVATIVE OF A COMPOSITE FUNCTION AND THE CHAIN RULE 172 THE DERIVATIVE OF THE POWER FUNCTION FOR RATIONAL EXPONENTS AND IMPLICIT DIFFERENTIATION 183 RELATED RATES 192 CHAPTER 2 REVIEW 201

BEHAVIOR OF FUNCTIONS AND THEIR GRAPHS, EXTREME FUNCTION VALUES, AND APPROXIMATIONS 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

MAXIMUM AND MINIMUM FUNCTION VALUES APPLICATIONS INVOLVING AN ABSOLUTE EXTREMUM ON A CLOSED INTERVAL ROLLE'S THEOREM AND THE MEAN VALUE THEOREM INCREASING AND DECREASING FUNCTIONS AND THE FIRST-DERIVATIVE TEST CONCAVITY, POINTS OF INFLECTION, AND THE SECOND-DERIVATIVE TEST SKETCHING GRAPHS OF FUNCTIONS AND THEIR DERIVATIVES LIMITS AT INFINITY SUMMARY OF SKETCHING GRAPHS OF FUNCTIONS ADDITIONAL APPLICATIONS OF ABSOLUTE EXTREMA APPROXIMATIONS BY NEWTON'S METHOD, THE TANGENT LINE, AND DIFFERENTIALS CHAPTER 3 REVIEW

THE DEFINITE INTEGRAL AND INTEGRATION 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

ANTIDIFFERENTIATION SOME TECHNIQUES OF ANTIDIFFERENTIATION DIFFERENTIAL EQUATIONS AND RECTILINEAR MOTION AREA THE DEFINITE INTEGRAL THE MEAN-VALUE THEOREM FOR INTEGRALS THE FUNDAMENTAL THEOREMS OF THE CALCULUS AREA OF A PLANE REGION

209 210 219 228 235 244 256 264 276 283 292 304

313 314 327 336 346 356 369 377 389

CONTENTS 4.9 4.10

VOLUMES OF SOLIDS BY SLICING, DISKS, AND WASHERS 398 VOLUMES OF SOLIDS BY CYLINDRICAL SHELLS 409 CHAPTER 4 REVIEW 415

LOGARITHMIC, EXPONENTIAL, INVERSE TRIGONOMETRIC, AND HYPERBOLIC FUNCTIONS 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

THE INVERSE OF A FUNCTION THE NATURAL LOGARITHMIC FUNCTION LOGARITHMIC DIFFERENTIATION AND INTEGRALS YIELDING THE NATURAL LOGARITHMIC FUNCTION THE NATURAL EXPONENTIAL FUNCTION OTHER EXPONENTIAL AND LOGARITHMIC FUNCTIONS APPLICATIONS OF THE NATURAL EXPONENTIAL FUNCTION INVERSE TRIGONOMETRIC FUNCTIONS INTEGRALS YIELDING INVERSE TRIGONOMETRIC FUNCTIONS HYPERBOLIC FUNCTIONS CHAPTER 5 REVIEW

ADDITIONAL APPLICATIONS OF THE DEFINITE INTEGRAL 6.1 6.2 6.3 6.4 6.5

vii

LENGTH OF ARC OF THE GRAPH OF A FUNCTION CENTER OF MASS OF A ROD CENTER OF MASS OF A LAMINA AND CENTROID OF A PLANE REGION WORK FORCE DUE TO FLUID PRESSURE CHAPTER 6 REVIEW

423 424 439 451 458 469 477 491 507 512 526

533 534 541 548 557 564 569

TECHNIQUES OF INTEGRATION, INDETERMINATE FORMS, AND IMPROPER INTEGRALS 573 7.1 7.2 7.3 7.4

INTEGRATION BY PARTS TRIGONOMETRIC INTEGRALS INTEGRATION OF ALGEBRAIC FUNCTIONS BY TRIGONOMETRIC SUBSTITUTION INTEGRATION OF RATIONAL FUNCTIONS AND LOGISTIC GROWTH

574 583 594 601

viii

CONTENTS

7.5 7.6 7.7 7.8 7.9 7.10

INTEGRATION BY OTHER SUBSTITUTION TECHNIQUES AND TABLES NUMERICAL INTEGRATION THE INDETERMINATE FORM 0 / 0 AND CAUCHY'S MEAN-VALUE THEOREM OTHER INDETERMINATE FORMS IMPROPER INTEGRALS WITH INFINITE LIMITS OF INTEGRATION OTHER IMPROPER INTEGRALS CHAPTER 7 REVIEW

614 621 634 644 650 659 664

POLYNOMIAL APPROXIMATIONS, SEQUENCES, AND INFINITE SERIES 671 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

POLYNOMIAL APPROXIMATIONS BY TAYLOR'S FORMULA SEQUENCES INFINITE SERIES OF CONSTANT TERMS INFINITE SERIES OF POSITIVE TERMS INFINITE SERIES OF POSITIVE AND NEGATIVE TERMS A SUMMARY OF TESTS FOR CONVERGENCE OR DIVERGENCE OF AN INFINITE SERIES POWER SERIES DIFFERENTIATION AND INTEGRATION OF POWER SERIES TAYLOR SERIES POWER SERIES FOR NATURAL LOGARITHMS AND THE BINOMIAL SERIES CHAPTER 8 REVIEW

PARAMETRIC EQUATIONS, PLANE CURVES, AND POLAR GRAPHS 9.1 9.2 9.3 9.4 9.5

PARAMETRIC EQUATIONS AND PLANE CURVES LENGTH OF ARC OF A PLANE CURVE POLAR COORDINATES AND POLAR GRAPHS LENGTH OF ARC AND AREA OF A REGION FOR POLAR GRAPHS A UNIFIED TREATMENT OF CONIC SECTIONS AND POLAR EQUATIONS OF CONICS CHAPTER 9 REVIEW

672 681 693 707 720 732 735 744 755 765 773

777 778 785 790 804 813 822

CONTENTS

ix

VECTORS AND PLANES, LINES, AND SURFACES IN SPACE 825 10.1 10.2 10.3 10.4 10.5 10.6

VECTORS IN THE PLANE VECTORS IN THREE-DIMENSIONAL SPACE DOT PRODUCT PLANES AND LINES IN R3 CROSS PRODUCT SURFACES CHAPTER 10 REVIEW

VECTOR-VALUED FUNCTIONS 11.1 11.2 11.3 11.4 11.5

VECTOR-VALUED FUNCTIONS AND CURVES IN R3 CALCULUS OF VECTOR-VALUED FUNCTIONS THE UNIT TANGENT AND UNIT NORMAL VECTORS AND ARC LENGTH AS PARAMETER CURVATURE CURVILINEAR MOTION CHAPTER 11 REVIEW

DIFFERENTIAL CALCULUS OF FUNCTIONS OF MORE THAN ONE VARIABLE 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

FUNCTIONS OF MORE THAN ONE VARIABLE LIMITS AND CONTINUITY OF FUNCTIONS OF MORE THAN ONE VARIABLE PARTIAL DERIVATIVES DIFFERENTIABILITY AND THE TOTAL DIFFERENTIAL THE CHAIN RULE FOR FUNCTIONS OF MORE THAN ONE VARIABLE DIRECTIONAL DERIVATIVES AND GRADIENTS TANGENT PLANES AND NORMALS TO SURFACES EXTREMA OF FUNCTIONS OF TWO VARIABLES LAGRANGE MULTIPLIERS CHAPTER 12 REVIEW

MULTIPLE INTEGRATION 13.1 13.2

CYLINDRICAL AND SPHERICAL COORDINATES DOUBLE INTEGRALS

826 838 850 861 873 886 903

907 908 916 925 932 941 953

957 958 970 986 999 1011 1021 1031 1036 1051 1061

1069 1070 1076

CONTENTS

13.3 13.4 13.5 13.6

APPLICATIONS OF DOUBLE INTEGRALS DOUBLE INTEGRALS IN POLAR COORDINATES TRIPLE INTEGRALS TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES CHAPTER 13 REVIEW

INTRODUCTION TO THE CALCULUS OF VECTOR FIELDS 14.1 14.2 14.3 14.4 14.5 14.6

VECTOR FIELDS LINE INTEGRALS LINE INTEGRALS INDEPENDENT OF THE PATH GREEN'S THEOREM SURFACE INTEGRALS GAUSS'S DIVERGENCE THEOREM AND STOKES'S THEOREM CHAPTER 14 REVIEW

APPENDIX PRECALCULUS TOPICS A.1 REAL NUMBERS AND INEQUALITIES A.2 COORDINATES AND GRAPHS OF EQUATIONS A.3 LINES A.4 PARABOLAS A.5 CIRCLES A.6 TRANSLATION OF AXES A.7 ELLIPSES A.8 HYPERBOLAS A.9 THE TRIGONOMETRIC FUNCTIONS A . 10 THE GENERAL EQUATION OF THE SECOND DEGREE IN TWO VARIABLES AND ROTATION OF AXES A . 1 1 PARTIAL FRACTIONS SUPPLEMENTARY SECTIONS 1.5 SUPPLEMENT 1.7 SUPPLEMENT 1.10 SUPPLEMENT 2.8 SUPPLEMENT 4.5 SUPPLEMENT

1090 1102 1111 1117 1124

1129 1130 1141 1150 1161 1174 1182 1189 A-l A-l A-l 3 A-22 A-32, A-37 A-42 A-48 A-58 A-67

CONTENTS

5.1 8.2 8.5 8.8

SUPPLEMENT SUPPLEMENT SUPPLEMENT SUPPLEMENT

12.3 SUPPLEMENT 12.4 SUPPLEMENT 12.8 SUPPLEMENT FORMULAS FROM TRIGONOMETRY FORMULAS FROM GEOMETRY THE GREEK ALPHABET ANSWERS TO ODD-NUMBERED EXERCISES INDEX

xi

A-104 A-l08 A-l09 A-l 10 A-l 14 A-l 16 A-l 18 A-120 A-l 21 A-l 21 A-l 23 1-1