Single Variable Essential Calculus: Early Transcendentals

Cover......Page 1
Title Page......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 13
To the Student......Page 20
Diagnostic Tests......Page 21
1.1 Functions and Their Representations......Page 27
REPRESENTATIONS OF FUNCTIONS......Page 29
PIECEWISE DEFINED FUNCTIONS......Page 31
SYMMETRY......Page 32
INCREASING AND DECREASING FUNCTIONS......Page 33
EXERCISES......Page 34
MATHEMATICAL MODELING......Page 37
TRANSFORMATIONS OF FUNCTIONS......Page 42
COMBINATIONS OF FUNCTIONS......Page 44
EXERCISES......Page 47
1.3 The Limit of a Function......Page 50
INTUITIVE DEFINITION OF A LIMIT......Page 51
PRECISE DEFINITION OF A LIMIT......Page 57
EXERCISES......Page 59
1.4 Calculating Limits......Page 61
EXERCISES......Page 69
1.5 Continuity......Page 72
EXERCISES......Page 80
INFINITE LIMITS......Page 82
LIMITS AT INFINITY......Page 84
INFINITE LIMITS AT INFINITY......Page 89
PRECISE DEFINITIONS......Page 90
EXERCISES......Page 93
Review......Page 96
THE TANGENT PROBLEM......Page 99
THE VELOCITY PROBLEM......Page 102
DERIVATIVES......Page 103
RATES OF CHANGE......Page 104
EXERCISES......Page 106
2.2 The Derivative as a Function......Page 110
DIFFERENTIABLE FUNCTIONS......Page 113
HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?......Page 115
HIGHER DERIVATIVES......Page 116
EXERCISES......Page 118
2.3 Basic Differentiation Formulas......Page 121
POWER FUNCTIONS......Page 122
NEW DERIVATIVES FROM OLD......Page 124
THE SINE AND COSINE FUNCTIONS......Page 127
APPLICATIONS TO RATES OF CHANGE......Page 128
EXERCISES......Page 131
THE PRODUCT RULE......Page 133
THE QUOTIENT RULE......Page 135
TRIGONOMETRIC FUNCTIONS......Page 137
EXERCISES......Page 138
2.5 The Chain Rule......Page 140
HOW TO PROVE THE CHAIN RULE......Page 145
EXERCISES......Page 146
2.6 Implicit Differentiation......Page 149
EXERCISES......Page 153
2.7 Related Rates......Page 154
EXERCISES......Page 158
2.8 Linear Approximations and Differentials......Page 161
APPLICATIONS TO PHYSICS......Page 162
DIFFERENTIALS......Page 163
EXERCISES......Page 164
Review......Page 166
3.1 Exponential Functions......Page 171
THE NUMBER e AND THE NATURAL EXPONENTIAL FUNCTION......Page 174
EXERCISES......Page 176
3.2 Inverse Functions and Logarithms......Page 177
THE CALCULUS OF INVERSE FUNCTIONS......Page 181
LOGARITHMIC FUNCTIONS......Page 183
NATURAL LOGARITHMS......Page 184
EXERCISES......Page 187
DERIVATIVES OF LOGARITHMIC FUNCTIONS......Page 189
LOGARITHMIC DIFFERENTIATION......Page 193
DERIVATIVES OF EXPONENTIAL FUNCTIONS......Page 194
EXERCISES......Page 195
POPULATION GROWTH......Page 197
RADIOACTIVE DECAY......Page 199
NEWTON’S LAW OF COOLING......Page 200
CONTINUOUSLY COMPOUNDED INTEREST......Page 202
EXERCISES......Page 203
3.5 Inverse Trigonometric Functions......Page 205
EXERCISES......Page 209
3.6 Hyperbolic Functions......Page 210
INVERSE HYPERBOLIC FUNCTIONS......Page 213
EXERCISES......Page 215
3.7 Indeterminate Forms and l’Hospital’s Rule......Page 217
INDETERMINATE PRODUCTS......Page 220
INDETERMINATE DIFFERENCES......Page 221
INDETERMINATE POWERS......Page 222
EXERCISES......Page 223
Review......Page 225
4.1 Maximum and Minimum Values......Page 229
EXERCISES......Page 234
4.2 The Mean Value Theorem......Page 236
EXERCISES......Page 241
4.3 Derivatives and the Shapes of Graphs......Page 242
WHAT DOES f\' SAY ABOUT f ?\n......Page 243
WHAT DOES f\'\' SAY ABOUT f ?\n......Page 245
EXERCISES......Page 248
GUIDELINES FOR SKETCHING A CURVE......Page 251
GRAPHING WITH TECHNOLOGY......Page 255
EXERCISES......Page 256
4.5 Optimization Problems......Page 257
APPLICATIONS TO BUSINESS AND ECONOMICS......Page 263
EXERCISES......Page 264
4.6 Newton’s Method......Page 268
EXERCISES......Page 271
4.7 Antiderivatives......Page 273
RECTILINEAR MOTION......Page 276
EXERCISES......Page 278
Review......Page 279
THE AREA PROBLEM......Page 283
THE DISTANCE PROBLEM......Page 290
EXERCISES......Page 292
5.2 The Definite Integral......Page 294
EVALUATING INTEGRALS......Page 298
THE MIDPOINT RULE......Page 301
PROPERTIES OF THE DEFINITE INTEGRAL......Page 302
EXERCISES......Page 305
5.3 Evaluating Definite Integrals......Page 307
INDEFINITE INTEGRALS......Page 309
APPLICATIONS......Page 312
EXERCISES......Page 315
5.4 The Fundamental Theorem of Calculus......Page 317
DIFFERENTIATION AND INTEGRATION AS INVERSE PROCESSES......Page 321
AVERAGE VALUE OF A FUNCTION......Page 322
EXERCISES......Page 324
5.5 The Substitution Rule......Page 326
DEFINITE INTEGRALS......Page 330
SYMMETRY......Page 331
EXERCISES......Page 332
Review......Page 334
6.1 Integration by Parts......Page 337
EXERCISES......Page 342
TRIGONOMETRIC INTEGRALS......Page 343
TRIGONOMETRIC SUBSTITUTIONS......Page 347
EXERCISES......Page 352
6.3 Partial Fractions......Page 353
EXERCISES......Page 360
TABLES OF INTEGRALS......Page 361
COMPUTER ALGEBRA SYSTEMS......Page 363
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?......Page 365
EXERCISES......Page 366
6.5 Approximate Integration......Page 367
SIMPSON’S RULE......Page 371
EXERCISES......Page 376
TYPE 1: INFINITE INTERVALS......Page 379
TYPE 2: DISCONTINUOUS INTEGRANDS......Page 383
A COMPARISON TEST FOR IMPROPER INTEGRALS......Page 385
EXERCISES......Page 386
Review......Page 388
7.1 Areas Between Curves......Page 391
EXERCISES......Page 395
7.2 Volumes......Page 396
EXERCISES......Page 404
7.3 Volumes by Cylindrical Shells......Page 407
EXERCISES......Page 410
7.4 Arc Length......Page 412
THE ARC LENGTH FUNCTION......Page 416
EXERCISES......Page 417
7.5 Area of a Surface of Revolution......Page 419
EXERCISES......Page 423
WORK......Page 424
HYDROSTATIC PRESSURE AND FORCE......Page 427
MOMENTS AND CENTERS OF MASS......Page 430
EXERCISES......Page 434
SEPARABLE EQUATIONS......Page 438
LOGISTIC GROWTH......Page 441
MIXING PROBLEMS......Page 442
DIRECTION FIELDS......Page 443
EXERCISES......Page 444
Review......Page 447
8.1 Sequences......Page 451
EXERCISES......Page 460
8.2 Series......Page 462
EXERCISES......Page 469
TESTING WITH AN INTEGRAL......Page 472
TESTING BY COMPARING......Page 475
EXERCISES......Page 478
ALTERNATING SERIES......Page 480
ABSOLUTE CONVERGENCE......Page 484
THE RATIO TEST......Page 486
EXERCISES......Page 489
8.5 Power Series......Page 490
EXERCISES......Page 494
8.6 Representing Functions as Power Series......Page 496
DIFFERENTIATION AND INTEGRATION OF POWER SERIES......Page 497
EXERCISES......Page 500
8.7 Taylor and Maclaurin Series......Page 502
MULTIPLICATION AND DIVISION OF POWER SERIES......Page 511
PROOF OF TAYLOR’S FORMULA......Page 512
EXERCISES......Page 513
APPROXIMATING FUNCTIONS BY POLYNOMIALS......Page 515
APPLICATIONS TO PHYSICS......Page 519
EXERCISES......Page 520
Review......Page 523
9.1 Parametric Curves......Page 527
GRAPHING DEVICES......Page 529
THE CYCLOID......Page 530
EXERCISES......Page 531
TANGENTS......Page 534
ARC LENGTH......Page 536
EXERCISES......Page 539
9.3 Polar Coordinates......Page 541
POLAR CURVES......Page 543
TANGENTS TO POLAR CURVES......Page 546
GRAPHING POLAR CURVES WITH GRAPHING DEVICES......Page 547
EXERCISES......Page 548
9.4 Areas and Lengths in Polar Coordinates......Page 550
ARC LENGTH......Page 553
EXERCISES......Page 554
CONICS IN CARTESIAN COORDINATES......Page 555
CONICS IN POLAR COORDINATES......Page 556
EXERCISES......Page 560
Review......Page 561
A: Trigonometry......Page 563
B: Sigma Notation......Page 572
C: The Logarithm Defined as an Integral......Page 577
D: Proofs......Page 584
E: Answers to Odd-Numbered Exercises......Page 599
INDEX......Page 633