Calculus: Single Variable, 2nd Edition

Cover......Page 1
Title Page......Page 7
Copyright......Page 8
Contents......Page 11
Preface......Page 15
Supplementary Resources......Page 19
Acknowledgments......Page 20
About the Authors......Page 23
Preview......Page 27
1 Number Systems......Page 28
Sets of Real Numbers......Page 29
Intervals......Page 30
Approximation......Page 33
EXERCISES......Page 35
2 Planar Coordinates and Graphing in the Plane......Page 37
The Distance Formula and Circles......Page 38
The Method of Completing the Square......Page 39
Parabolas, Ellipses, and Hyperbolas......Page 42
Regions in the Plane......Page 44
EXERCISES......Page 45
Slopes......Page 47
Equations of Lines......Page 49
Least Squares Lines......Page 54
EXERCISES......Page 57
4 Functions and Their Graphs......Page 60
Examples of Functions of a Real Variable......Page 61
Piecewise-Defined Functions......Page 62
Graphs of Functions......Page 63
Sequences......Page 65
Functions from Data......Page 66
EXERCISES......Page 71
Arithmetic Operations......Page 75
Polynomial Functions......Page 76
Composition of Functions......Page 77
Inverse Functions......Page 78
Vertical and Horizontal Translations......Page 83
Even and Odd Functions......Page 84
Pairing Functions—Parametric Curves......Page 85
Parameterized Curves and Graphs of Functions......Page 87
EXERCISES......Page 88
Sine and Cosine Functions......Page 91
Other Trigonometric Functions......Page 94
Trigonometric Identities......Page 96
Modeling with Trigonometric Functions......Page 97
EXERCISES......Page 98
Summary of Key Topics......Page 101
Review Exercises......Page 104
The Number π......Page 106
Analytic Geometry......Page 107
The Completeness Property of the Real Numbers......Page 108
Preview......Page 109
1 The Concept of Limit......Page 111
One-Sided Limits......Page 112
Specified Degrees of Accuracy......Page 113
Graphical Methods......Page 115
EXERCISES......Page 117
2 Limit Theorems......Page 120
One-Sided Limits......Page 121
Basic Limit Theorems......Page 123
A Rule That Tells When a Limit Does Not Exist......Page 124
The Pinching Theorem......Page 125
Some Important Trigonometric Limits......Page 126
EXERCISES......Page 128
The Definition of Continuity at a Point......Page 131
An Equivalent Formulation of Continuity......Page 132
Continuous Extensions......Page 134
One-Sided Continuity......Page 135
Some Theorems about Continuity—Arithmetic Operations and Composition......Page 136
Advanced Properties of Continuous Functions......Page 137
EXERCISES......Page 141
Infinite-Valued Limits......Page 144
Vertical Asymptotes......Page 145
Limits at Infinity......Page 147
Horizontal Asymptotes......Page 148
EXERCISES......Page 150
5 Limits of Sequences......Page 153
A Precise Discussion of Convergence and Divergence......Page 154
Some Special Sequences......Page 155
Limit Theorems......Page 157
Geometric Series......Page 159
Using Continuous Functions to Calculate Limits......Page 161
EXERCISES......Page 162
The Monotone Convergence Property of the Real Numbers......Page 164
Irrational Exponents......Page 166
Exponential and Logarithmic Functions......Page 168
The Number e......Page 171
An Application: Compound Interest......Page 173
The Natural Logarithm......Page 175
Exponential Decay......Page 176
EXERCISES......Page 178
Summary of Key Topics......Page 181
Review Exercises......Page 184
Bolzano and the Intermediate Value Theorem......Page 187
Cauchy and Weierstrass......Page 188
Preview......Page 189
1 Rates of Change and Tangent Lines......Page 190
The Definition of Instantaneous Velocity......Page 192
Instantaneous Rate of Change......Page 194
Sums of Functions......Page 196
The Concept of Tangent Line......Page 197
Normal Lines to Curves......Page 199
Corners and Vertical Tangent Lines......Page 200
EXERCISES......Page 201
2 The Derivative......Page 204
Other Notations for the Derivative......Page 206
The Derived Function......Page 207
Differentiability and Continuity......Page 208
Investigating Differentiability Graphically......Page 209
Derivatives of Sine and Cosine......Page 210
Summary of Differentiation Formulas......Page 212
EXERCISES......Page 213
3 Rules for Differentiation......Page 215
Addition, Subtraction, and Multiplication by a Constant......Page 216
Products and Quotients......Page 217
Numeric Differentiation......Page 220
EXERCISES......Page 223
Powers of x......Page 226
Trigonometric Functions......Page 229
The Derivative of the Natural Exponential Function......Page 230
EXERCISES......Page 233
A Rule for Differentiating the Composition of Two Functions......Page 236
Some Examples......Page 237
An Application......Page 240
Derivatives of Exponential Functions......Page 241
EXERCISES......Page 243
Continuity and Differentiability of Inverse Functions......Page 246
Derivatives of Logarithms......Page 249
Logarithmic Differentiation......Page 251
EXERCISES......Page 254
Notation for Higher Derivatives......Page 256
Velocity and Acceleration......Page 257
Approximation of Second Derivatives......Page 258
Leibniz’s Rule......Page 259
EXERCISES......Page 260
The Main Idea of Implicit Differentiation......Page 263
Some Examples......Page 264
Calculating Higher Derivatives......Page 267
Parametric Curves......Page 268
EXERCISES......Page 269
9 Differentials and Approximation of Functions......Page 272
Linearization......Page 274
Important Linearizations......Page 275
Differentials......Page 276
EXERCISES......Page 277
Inverse Trigonometric Functions......Page 279
Inverse Sine and Cosine......Page 280
The Inverse Tangent Function......Page 283
Other Inverse Trigonometric Functions......Page 284
The Hyperbolic Functions......Page 286
Derivatives of the Hyperbolic Functions......Page 288
The Inverse Hyperbolic Functions......Page 289
EXERCISES......Page 293
Summary of Key Topics......Page 294
Review Exercises......Page 298
The Solutions of Fermat and Descartes to the Tangent Problem......Page 301
Newton’s Method of Differentiation......Page 303
Proof of the Chain Rule......Page 304
Preview......Page 307
The Role of Implicit Differentiation......Page 308
Basic Steps for Solving a Related Rates Problem......Page 310
EXERCISES......Page 313
2 The Mean Value Theorem......Page 315
Maxima and Minima......Page 316
Locating Maxima and Minima......Page 317
Rolle’s Theorem and the Mean Value Theorem......Page 319
An Application of the Mean Value Theorem......Page 320
EXERCISES......Page 322
Using the Derivative to Tell When a Function Is Increasing or Decreasing......Page 325
Critical Points and the First Derivative Test for Local Extrema......Page 327
EXERCISES......Page 330
4 Applied Maximum-Minimum Problems......Page 333
Closed Intervals......Page 334
Examples with the Solution at an Endpoint......Page 337
Profit Maximization......Page 339
An Example Involving a Transcendental Function......Page 340
EXERCISES......Page 341
5 Concavity......Page 346
Using the Second Derivative to Test for Concavity......Page 347
Points of Inflection......Page 348
The Second Derivative Test at a Critical Point......Page 349
EXERCISES......Page 351
Basic Strategy of Curve Sketching......Page 353
Periodic Functions......Page 357
Skew-Asymptotes......Page 358
Graphing Calculators/Software......Page 360
EXERCISES......Page 362
l’Hôpital’s Rule for the Indeterminate Form 0/0......Page 364
l’Hôpital’s Rule for the Indeterminate Form ∞/∞......Page 367
The Indeterminate Form 0º......Page 368
The Indeterminate Form ∞º......Page 369
Putting Terms over a Common Denominator......Page 370
EXERCISES......Page 371
The Geometry of the Newton-Raphson Method......Page 374
Calculating with the Newton-Raphson Method......Page 375
Accuracy......Page 376
A Computer Implementation......Page 378
An Application in Economics: Bond Valuation......Page 379
EXERCISES......Page 380
Antidifferentiation......Page 383
Antidifferentiating Powers of x......Page 384
Antidifferentiation of other Functions......Page 386
Velocity and Acceleration......Page 387
EXERCISES......Page 389
Summary of Key Topics......Page 392
Review Exercises......Page 394
Fermat’s Investigation of Points of Inflection......Page 397
Fermat’s Principle of Least Time......Page 398
The Newton-Raphson Method......Page 399
Preview......Page 401
1 Introduction to Integration—The Area Problem......Page 402
Summation Notation......Page 403
Some Special Sums......Page 404
Approximation of Area......Page 405
A Precise Definition of Area......Page 407
Concluding Remarks......Page 410
EXERCISES......Page 411
Riemann Sums......Page 414
The Riemann Integral......Page 418
Calculating Riemann Integrals......Page 420
Using the Fundamental Theorem of Calculus to Compute Areas......Page 422
EXERCISES......Page 423
3 Rules for Integration......Page 425
Reversing the Direction of Integration......Page 428
Order Properties of Integrals......Page 429
The Mean Value Theorem for Integrals......Page 431
EXERCISES......Page 432
4 The Fundamental Theorem of Calculus......Page 434
Examples Illustrating the First Part of the Fundamental Theorem......Page 436
Examples of the Second Part of the Fundamental Theorem......Page 438
EXERCISES......Page 440
5 A Calculus Approach to the Logarithm and Exponential Functions......Page 443
Properties of the Natural Logarithm......Page 444
Graphing the Natural Logarithm Function......Page 446
Properties of the Exponential Function......Page 447
The Number e......Page 448
Logarithms and Powers with Arbitrary Bases......Page 449
Logarithms with Arbitrary Bases......Page 450
EXERCISES......Page 452
6 Integration by Substitution......Page 454
Some Examples of Indefinite Integration by Substitution......Page 455
The Method of Substitution for Definite Integrals......Page 456
The Role of the Chain Rule in the Method of Substitution......Page 458
Integral Tables......Page 459
Integrating Trigonometric Functions......Page 462
EXERCISES......Page 463
7 More on the Calculation of Area......Page 466
The Area between Two Curves......Page 467
Reversing the Roles of the Axes......Page 469
EXERCISES......Page 470
The Midpoint Rule......Page 472
The Trapezoidal Rule......Page 475
Simpson’s Rule......Page 478
Using Simpson’s Rule with Discrete Data......Page 480
EXERCISES......Page 481
Summary of Key Topics......Page 485
Review Exercises......Page 488
The Method......Page 491
Fermat and the Integral Calculus......Page 492
Notation......Page 493
Bernhard Riemann......Page 494
Preview......Page 495
Some Examples......Page 496
Advanced Examples......Page 498
Reduction Formulas......Page 501
EXERCISES......Page 503
2 Powers and Products of Trigonometric Functions......Page 505
Squares of Sine, Cosine, Secant, and Tangent......Page 506
Higher Powers of Sine, Cosine, Secant, and Tangent......Page 507
Odd Powers of Sine and Cosine......Page 508
Integrals That Involve Both Sine and Cosine......Page 509
Converting to Sines and Cosines......Page 511
EXERCISES......Page 512
3 Trigonometric Substitution......Page 514
Trigonometric Substitution......Page 515
General Quadratic Expressions That Appear Under a Radical......Page 518
Quadratic Expressions Not Under a Radical Sign......Page 519
EXERCISES......Page 521
The Method of Partial Fractions for Linear Factors......Page 524
The Method of Partial Fractions for Distinct Linear Factors......Page 525
Heaviside’s Method......Page 527
Repeated Linear Factors......Page 528
Summary of Basic Partial Fraction Forms......Page 529
EXERCISES......Page 530
Rational Functions with Quadratic Terms in the Denominator......Page 532
Checking for Irreducibility......Page 534
Calculating the Coefficients of a Partial Fraction Decomposition......Page 535
EXERCISES......Page 538
Integrals with Infinite Integrands......Page 540
Integrands with Interior Singularities......Page 541
Proving Convergence Without Evaluation......Page 543
EXERCISES......Page 545
The Integral on an Infinite Interval......Page 547
An Application to Finance......Page 549
Integrals Over the Entire Real Line......Page 550
Proving Convergence Without Evaluation......Page 551
EXERCISES......Page 552
Summary of Key Topics......Page 555
Review Exercises......Page 557
Genesis & Development 6......Page 559
Preview......Page 563
Volumes by Slicing—The Method of Disks......Page 564
Solids of Revolution......Page 566
The Method of Washers......Page 568
Rotation about a Line that Is Not a Coordinate Axis......Page 570
The Method of Cylindrical Shells......Page 571
A Final Remark......Page 574
EXERCISES......Page 575
The Basic Method for Calculating Arc Length......Page 578
Some Examples of Arc Length......Page 579
Parametric Curves......Page 580
Surface Area......Page 582
EXERCISES......Page 585
3 The Average Value of a Function......Page 587
The Basic Technique......Page 588
Random Variables......Page 591
Average Values in Probability Theory......Page 593
Population Density Functions......Page 594
EXERCISES......Page 595
Moments (Two Point Systems)......Page 598
Center of Mass......Page 600
EXERCISES......Page 604
Using Integrals to Calculate Work......Page 606
Examples with Weights That Vary......Page 607
An Example Involving a Spring......Page 609
Examples that Involve Pumping a Fluid from a Reservoir......Page 610
EXERCISES......Page 611
6 First Order Differential Equations—Separable Equations......Page 614
Slope Fields......Page 615
Initial Value Problems......Page 616
Separable Equations......Page 617
Equations of the Form dy/dx = g(x)......Page 618
Examples from the Physical Sciences......Page 619
Logistic Growth......Page 622
EXERCISES......Page 625
Solving Linear Differential Equations......Page 632
An Application: Mixing Problems......Page 635
An Application: Electric Circuits......Page 636
Linear Equations with Constant Coefficients......Page 637
Newton’s Law for Temperature Change......Page 638
The Linear Drag Law......Page 639
EXERCISES......Page 641
Summary of Key Topics......Page 645
Review Exercises......Page 647
Arc Length......Page 651
The Catenary......Page 652
The Catenoid......Page 654
Preview......Page 655
Limits of Infinite Sequences—A Review......Page 657
The Definition of an Infinite Series......Page 659
Convergence of Infinite Series......Page 660
A Telescoping Series......Page 661
The Harmonic Series......Page 662
Basic Properties of Series......Page 663
Series of Powers (Geometric Series)......Page 664
EXERCISES......Page 666
The Divergence Test......Page 669
Series with Nonnegative Terms......Page 670
The Tail End of a Series......Page 671
The Integral Test......Page 672
p-Series......Page 674
An Extension......Page 675
EXERCISES......Page 676
The Comparison Test for Convergence......Page 679
The Comparison Test for Divergence......Page 681
An Advanced Example......Page 682
The Limit Comparison Test......Page 683
EXERCISES......Page 685
The Alternating Series Test......Page 687
Some Examples......Page 688
Absolute Convergence......Page 689
EXERCISES......Page 691
The Ratio Test......Page 693
Examples......Page 696
EXERCISES......Page 698
Radius and Interval of Convergence......Page 700
Power Series about an Arbitrary Base Point......Page 705
Addition and Scalar Multiplication of Power Series......Page 707
Differentiation and Antidifferentiation of Power Series......Page 708
EXERCISES......Page 709
Power Series Expansions of Some Standard Functions......Page 712
The Relationship between the Coefficients and Derivatives of a Power Series......Page 716
An Application to Differential Equations......Page 718
Taylor Series and Polynomials......Page 719
EXERCISES......Page 721
Taylor’s Theorem......Page 724
Estimating the Error Term......Page 727
Achieving a Desired Degree of Accuracy......Page 728
Taylor Series Expansions of the Common Transcendental Functions......Page 730
The Binomial Series......Page 733
Using Taylor Polynomials to Evaluate Indeterminate Forms......Page 734
EXERCISES......Page 735
Summary of Key Topics......Page 738
Review Exercises......Page 741
Infinite Series in the 17th Century......Page 744
James Gregory and Sir Isaac Newton......Page 745
Colin Maclaurin (1698–1746)......Page 746
Table of Integrals......Page 747
Answers to Selected Exercises......Page 761
Index......Page 817