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 CONTINUITY OF A FUNCTION AT A NUMBER 72
                       1.9 CONTINUITY OF A COMPOSITE FUNCTION AND
                            CONTINUITY ON AN INTERVAL 82
                       1.10 CONTINUITY OF THE TRIGONOMETRIC FUNCTIONS
                            AND THE SQUEEZE THEOREM 92
                            CHAPTER 1 REVIEW 102
                       THE DERIVATIVE AND DIFFERENTIATION 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 DERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS 162
                                2.8 THE DERIVATIVE OF A COMPOSITE FUNCTION AND
                                      THE CHAIN RULE 172
                                2.9 THE DERIVATIVE OF THE POWER FUNCTION FOR RATIONAL
                                      EXPONENTS AND IMPLICIT DIFFERENTIATION 183
                                2.10 RELATED RATES 192
                                      CHAPTER 2 REVIEW 201
                                BEHAVIOR OF FUNCTIONS AND THEIR
                                GRAPHS, EXTREME FUNCTION VALUES,
                                AND APPROXIMATIONS 209
                                3.1 MAXIMUM AND MINIMUM FUNCTION VALUES 210
                                3.2 APPLICATIONS INVOLVING AN ABSOLUTE EXTREMUM
                                      ON A CLOSED INTERVAL 219
                                3.3 ROLLE'S THEOREM AND THE MEAN VALUE THEOREM 228
                                3.4 INCREASING AND DECREASING FUNCTIONS AND
                                      THE FIRST-DERIVATIVE TEST 235
                                3.5 CONCAVITY, POINTS OF INFLECTION, AND THE
                                      SECOND-DERIVATIVE TEST 244
                                3.6 SKETCHING GRAPHS OF FUNCTIONS
                                      AND THEIR DERIVATIVES 256
                                3.7 LIMITS AT INFINITY 264
                                3.8 SUMMARY OF SKETCHING GRAPHS OF FUNCTIONS 276
                                3.9 ADDITIONAL APPLICATIONS OF ABSOLUTE EXTREMA 283
                                3.10 APPROXIMATIONS BY NEWTON'S METHOD, THE
                                      TANGENT LINE, AND DIFFERENTIALS 292
                                      CHAPTER 3 REVIEW 304
                                THE DEFINITE INTEGRAL AND INTEGRATION 313
                               4.1 ANTIDIFFERENTIATION 314
                                4.2 SOME TECHNIQUES OF ANTIDIFFERENTIATION 327
                                4.3 DIFFERENTIAL EQUATIONS AND RECTILINEAR MOTION 336
                                4.4 AREA 346
                                4.5 THE DEFINITE INTEGRAL 356
                                4.6 THE MEAN-VALUE THEOREM FOR INTEGRALS 369
                                4.7 THE FUNDAMENTAL THEOREMS OF THE CALCULUS 377
                                4.8 AREA OF A PLANE REGION 389
                                                 CONTENTS vii
                    4.9 VOLUMES OF SOLIDS BY SLICING, DISKS, AND WASHERS 398
                    4.10 VOLUMES OF SOLIDS BY CYLINDRICAL SHELLS 409
                         CHAPTER 4 REVIEW 415
                    LOGARITHMIC, EXPONENTIAL, INVERSE
                    TRIGONOMETRIC, AND HYPERBOLIC
                    FUNCTIONS 423
                    5.1 THE INVERSE OF A FUNCTION 424
                    5.2 THE NATURAL LOGARITHMIC FUNCTION 439
                    5.3 LOGARITHMIC DIFFERENTIATION AND INTEGRALS
                         YIELDING THE NATURAL LOGARITHMIC FUNCTION 451
                    5.4 THE NATURAL EXPONENTIAL FUNCTION 458
                    5.5 OTHER EXPONENTIAL AND LOGARITHMIC FUNCTIONS 469
                    5.6 APPLICATIONS OF THE NATURAL EXPONENTIAL
                         FUNCTION 477
                    5.7 INVERSE TRIGONOMETRIC FUNCTIONS 491
                    5.8 INTEGRALS YIELDING INVERSE TRIGONOMETRIC
                         FUNCTIONS 507
                    5.9 HYPERBOLIC FUNCTIONS 512
                         CHAPTER 5 REVIEW 526
                    ADDITIONAL APPLICATIONS OF THE
                    DEFINITE INTEGRAL 533
                    6.1 LENGTH OF ARC OF THE GRAPH OF A FUNCTION 534
                    6.2 CENTER OF MASS OF A ROD 541
                    6.3 CENTER OF MASS OF A LAMINA AND CENTROID
                         OF A PLANE REGION 548
                    6.4 WORK 557
                    6.5 FORCE DUE TO FLUID PRESSURE 564
                         CHAPTER 6 REVIEW 569
                    TECHNIQUES OF INTEGRATION, INDETERMINATE
                    FORMS, AND IMPROPER INTEGRALS 573
                    7.1 INTEGRATION BY PARTS 574
                    7.2 TRIGONOMETRIC INTEGRALS 583
                    7.3 INTEGRATION OF ALGEBRAIC FUNCTIONS
                         BY TRIGONOMETRIC SUBSTITUTION 594
                    7.4 INTEGRATION OF RATIONAL FUNCTIONS
                         AND LOGISTIC GROWTH 601