Single variable essential calculus

Cover......Page 1
Title Page......Page 6
Copyright......Page 8
Contents......Page 9
Preface......Page 13
To the Student......Page 20
Diagnostic Tests......Page 21
1.1 Functions and thier Representations......Page 27
REPRESENTATIONS OF FUNCTIONS......Page 29
PIECEWISE DEFINED FUNCTIONS......Page 31
SYMMETRY......Page 32
INCREASING AND DECREASING FUNCTIONS......Page 33
1.1 EXERCISES......Page 34
MATHEMATICAL MODELING......Page 37
TRANSFORMATIONS OF FUNCTIONS......Page 42
COMBINATIONS OF FUNCTIONS......Page 44
1.2 EXERCISES......Page 47
1.3 The Limit of a Function......Page 50
INTUITIVE DEFINITION OF A LIMIT......Page 51
ONE-SIDED LIMITS......Page 55
PRECISE DEFINITION OF A LIMIT......Page 57
1.3 EXERCISES......Page 59
1.4 Calculating Limits......Page 61
1.4 EXERCISES......Page 69
1.5 Continuity......Page 72
1.5 EXERCISES......Page 80
INFINITE LIMITS......Page 82
LIMITS AT INFINITY......Page 84
INFINITE LIMITS AT INFINITY......Page 89
PRECISE DEFINITIONS......Page 90
1.6 EXERCISES......Page 93
Review......Page 96
THE TANGENT PROBLEM......Page 99
THE VELOCITY PROBLEM......Page 102
DERIVATIVES......Page 103
RATES OF CHANGE......Page 104
2.1 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
2.2 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
2.3 EXERCISES......Page 131
THE PRODUCT RULE......Page 133
THE QUOTIENT RULE......Page 135
TRIGONOMETRIC FUNCTIONS......Page 137
2.4 EXERCISES......Page 138
2.5 The Chain Rule......Page 140
HOW TO PROVE THE CHAIN RULE......Page 145
2.5 EXERCISES......Page 146
2.6 Implicit Differentiation......Page 149
2.6 EXERCISES......Page 153
2.7 Related Rates......Page 154
2.7 EXERCISES......Page 158
2.8 Linear Approximations and Differentials......Page 161
APPLICATIONS TO PHYSICS......Page 162
DIFFERENTIALS......Page 163
2.8 EXERCISES......Page 164
Review......Page 166
3.1 Maximum and Minimum Values......Page 171
3.1 EXERCISES......Page 176
3.2 The Mean Value Theorem......Page 178
3.2 EXERCISES......Page 183
3.3 Derivatives and the Shapes of Graphs......Page 184
WHAT DOES f\' SAY ABOUT f?......Page 185
WHAT DOES f\' SAY ABOUT f?......Page 187
3.3 EXERCISES......Page 190
3.4 Curve Sketching......Page 192
GUIDELINES FOR SKETCHING A CURVE......Page 193
GRAPHING WITH TECHNOLOGY......Page 196
3.4 EXERCISES......Page 198
3.5 Optimization Problems......Page 199
APPLICATIONS TO BUSINESS AND ECONOMICS......Page 205
3.5 EXERCISES......Page 206
3.6 Newton’s Method......Page 210
3.6 EXERCISES......Page 213
3.7 Antiderivatives......Page 215
RECTILINEAR MOTION......Page 218
3.7 EXERCISES......Page 220
Review......Page 221
THE AREA PROBLEM......Page 225
THE DISTANCE PROBLEM......Page 232
4.1 EXERCISES......Page 234
4.2 The Definite Integral......Page 236
EVALUATING INTEGRALS......Page 240
THE MIDPOINT RULE......Page 243
PROPERTIES OF THE DEFINITE INTEGRAL......Page 244
4.2 EXERCISES......Page 247
4.3 Evaluating Definite Integrals......Page 249
INDEFINITE INTEGRALS......Page 251
APPLICATIONS......Page 254
4.4 The Fundamental Theorem of Calculus......Page 259
DIFFERENTIATION AND INTEGRATION AS INVERSE PROCESSES......Page 263
AVERAGE VALUE OF A FUNCTION......Page 264
4.4 EXERCISES......Page 266
4.5 The Substitution Rule......Page 268
DEFINITE INTEGRALS......Page 271
SYMMETRY......Page 272
4.5 EXERCISES......Page 273
Review......Page 275
5.1 Inverse Functions......Page 279
THE CALCULUS OF INVERSE FUNCTIONS......Page 283
5.1 EXERCISES......Page 285
5.2 The Natural Logarithmic Function......Page 287
LOGARITHMIC DIFFERENTIATION......Page 293
5.2 EXERCISES......Page 294
5.3 The Natural Exponential Function......Page 296
DIFFERENTIATION......Page 298
INTEGRATION......Page 299
5.3 EXERCISES......Page 300
GENERAL EXPONENTIAL FUNCTIONS......Page 302
EXPONENTIAL GRAPHS......Page 303
THE POWER RULE REVISITED......Page 304
GENERAL LOGARITHMIC FUNCTIONS......Page 305
5.4 EXERCISES......Page 308
5.5 Exponential Growth and Decay......Page 309
POPULATION GROWTH......Page 310
RADIOACTIVE DECAY......Page 311
NEWTON’S LAW OF COOLING......Page 313
CONTINUOUSLY COMPOUNDED INTEREST......Page 314
5.5 EXERCISES......Page 315
5.6 Inverse Trigonometric Functions......Page 317
5.6 EXERCISES......Page 323
5.7 Hyperbolic Functions......Page 324
INVERSE HYPERBOLIC FUNCTIONS......Page 326
5.7 EXERCISES......Page 329
5.8 Indeterminate Forms and l’Hospital’s Rule......Page 331
INDETERMINATE PRODUCTS......Page 334
INDETERMINATE DIFFERENCES......Page 335
INDETERMINATE POWERS......Page 336
5.8 EXERCISES......Page 337
Review......Page 339
6.1 Integration by Parts......Page 343
6.1 EXERCISES......Page 348
TRIGONOMETRIC INTEGRALS......Page 349
TRIGONOMETRIC SUBSTITUTIONS......Page 353
6.2 EXERCISES......Page 358
6.3 Partial Fractions......Page 359
6.3 EXERCISES......Page 366
TABLES OF INTEGRALS......Page 367
COMPUTER ALGEBRA SYSTEMS......Page 369
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?......Page 371
6.4 EXERCISES......Page 372
6.5 Approximate Integration......Page 373
SIMPSON’S RULE......Page 377
6.5 EXERCISES......Page 382
TYPE 1: INFINITE INTERVALS......Page 385
TYPE 2: DISCONTINUOUS INTEGRANDS......Page 389
A COMPARISON TEST FOR IMPROPER INTEGRALS......Page 391
6.6 EXERCISES......Page 392
Review......Page 394
7.1 Areas Between Curves......Page 397
7.1 EXERCISES......Page 401
7.2 Volumes......Page 402
7.2 EXERCISES......Page 410
7.3 Volumes by Cylindrical Shells......Page 413
7.3 EXERCISES......Page 416
7.4 Arc Length......Page 418
THE ARC LENGTH FUNCTION......Page 422
7.4 EXERCISES......Page 423
7.5 Area of a Surface of Revolution......Page 425
7.5 EXERCISES......Page 429
WORK......Page 430
HYDROSTATIC PRESSURE AND FORCE......Page 433
MOMENTS AND CENTERS OF MASS......Page 436
7.6 EXERCISES......Page 440
SEPARABLE EQUATIONS......Page 444
LOGISTIC GROWTH......Page 447
MIXING PROBLEMS......Page 448
DIRECTION FIELDS......Page 449
7.7 EXERCISES......Page 450
Review......Page 453
8.1 Sequences......Page 457
8.1 EXERCISES......Page 466
8.2 Series......Page 468
8.2 EXERCISES......Page 475
TESTING WITH AN INTEGRAL......Page 478
TESTING BY COMPARING......Page 481
8.3 EXERCISES......Page 484
ALTERNATING SERIES......Page 486
ABSOLUTE CONVERGENCE......Page 490
THE RATIO TEST......Page 492
8.4 EXERCISES......Page 495
8.5 Power Series......Page 496
8.5 EXERCISES......Page 500
8.6 Representing Functions as Power Series......Page 502
DIFFERENTIATION AND INTEGRATION OF POWER SERIES......Page 503
8.6 EXERCISES......Page 506
8.7 Taylor and Maclaurin Series......Page 508
MULTIPLICATION AND DIVISION OF POWER SERIES......Page 517
PROOF OF TAYLOR’S FORMULA......Page 518
8.7 EXERCISES......Page 519
APPROXIMATING FUNCTIONS BY POLYNOMIALS......Page 521
APPLICATIONS TO PHYSICS......Page 525
8.8 EXERCISES......Page 526
Review......Page 529
9.1 Parametric Curves......Page 533
GRAPHING DEVICES......Page 535
THE CYCLOID......Page 536
9.1 EXERCISES......Page 537
TANGENTS......Page 540
ARC LENGTH......Page 542
9.2 EXERCISES......Page 545
9.3 Polar Coordinates......Page 547
POLAR CURVES......Page 549
TANGENTS TO POLAR CURVES......Page 552
GRAPHING POLAR CURVES WITH GRAPHING DEVICES......Page 553
9.3 EXERCISES......Page 554
9.4 Areas and Lengths in Polar Coordinates......Page 556
ARC LENGTH......Page 559
9.4 EXERCISES......Page 560
CONICS IN CARTESIAN COORDINATES......Page 561
CONICS IN POLAR COORDINATES......Page 562
9.5 EXERCISES......Page 566
Review......Page 567
A: Trigonometry......Page 569
B: Sigma Notation......Page 578
C: Proofs......Page 583
D: Answers to Odd-Numbered Exercises......Page 595
INDEX......Page 629