0 Preliminaries 1
0.1 Real Numbers,Estimation,and Logic 1
0.2 Inequalities and Absolute Values 8
0.3 The Rectangular Coordinate System 16
0.4 Graphs of Equations 24
0.5 Functions and Their Graphs 29
0.6 Operations on Functions 35
0.7 Trigonometric Functions 41
0.8 Chapter Review 51
Review and Preview Problems 54
1 Limits 55
1.1 Introduction to Limits 55
1.2 Rigorous Study of Limits 61
1.3 Limit Theorems 68
1.4 Limits Involving Trigonometric Functions 73
1.5 Limits at Infinity;Infinite Limits 77
1.6 Continuity of Functions 82
1.7 Chapter Review 90
Review and Preview Problems 92
2 The Derivative 93
2.1 Two Problems with One Theme 93
2.2 The Derivative 100
2.3 Rules for Finding Derivatives 107
2.4 Derivatives of Trigonometric Functions 114
2.5 The Chain Rule 118
2.6 Higher-Order Derivatives 125
2.7 Implicit Differentiation 130
2.8 Related Rates 135
2.9 Differentials and Approximations 142
2.10 Chapter Review 147
Review and Preview Problems 150
3 Applications of the Derivative 151
3.1 Maxima and Minima 151
3.2 Monotonicity and Concavity 155
3.3 Local Extrema and Extrema on Open Intervals 162
3.4 Practical Problems 167
3.5 Graphing Functions Using Calculus 178
3.6 The Mean Value Theorem for Derivatives 185
3.7 Solving Equations Numerically 190
3.8 Antiderivatives 197
3.9 Introduction to Differential Equations 203
3.10 Chapter Review 209
Review and Preview Problems 214
4 The Definite Integral 215
4.1 Introduction to Area 215
4.2 The Definite Integral 224
4.3 The First Fundamental Theorem of Calculus 232
4.4 The Second Fundamental Theorem of Calculus and the Method of Substitution 243
4.5 The Mean Value Theorem for Integrals and the Use of Symmetry 253
4.6 Numerical Integration 260
4.7 Chapter Review 270
Review and Preview Problems 274
5 Applications of the Integral 275
5.1 The Area of a Plane Region 275
5.2 Volumes of Solids:Slabs,Disks,Washers 281
5.3 Volumes of Solids of Revolution:Shells 288
5.4 Length of a Plane Curve 294
5.5 Work and Fluid Force 301
5.6 Moments and Center of Mass 308
5.7 Probability and Random Variables 316
5.8 Chapter Review 322
Review and Preview Problems 324
6 Transcendental Functions 325
6.1 The Natural Logarithm Function 325
6.2 Inverse Functions and Their Derivatives 331
6.3 The Natural Exponential Function 337
6.4 General Exponential and Logarithmic Functions 342
6.5 Exponential Growth and Decay 347
6.6 First-Order Linear Differential Equations 355
6.7 Approximations for Differential Equations 359
6.8 The Inverse Trigonometric Functions and Their Derivatives 365
6.9 The Hyperbolic Functions and Their Inverses 374
6.10 Chapter Review 380
Review and Preview Problems 382
7 Techniques of Integration 383
7.1 Basic Integration Rules 383
7.2 Integration by Parts 387
7.3 Some Trigonometric Integrals 393
7.4 Rationalizing Substitutions 399
7.5 Integration of Rational Functions Using Partial Fractions 404
7.6 Strategies for Integration 411
7.7 Chapter Review 419
Review and Preview Problems 422
8 Indeterminate Forms and Improper Integrals 423
8.1 Indeterminate Forms of Type 0/0 423
8.2 Other Indeterminate Forms 428
8.3 Improper Integrals:Infinite Limits of Integration 433
8.4 Improper Integrals:Infinite Integrands 442
8.5 Chapter Review 446
Review and Preview Problems 448
9 Infinite Series 449
9.1 Infinite Sequences 449
9.2 Infinite Series 455
9.3 Positive Series:The Integral Test 463
9.4 Positive Series:Other Tests 468
9.5 Alternating Series,Absolute Convergence,and Conditional Convergence 474
9.6 Power Series 479
9.7 Operations on Power Series 484
9.8 Taylor and Maclaurin Series 489
9.9 The Taylor Approximation to a Function 497
9.10 Chapter Review 504
Review and Preview Problems 508
10 Conics and Polar Coordinates 509
10.1 The Parabola 509
10.2 Ellipses and Hyperbolas 513
10.3 Translation and Rotation of Axes 523
10.4 Parametric Representation of Curves in the Plane 530
10.5 The Polar Coordinate System 537
10.6 Graphs of Polar Equations 542
10.7 Calculus in Polar Coordinates 547
10.8 Chapter Review 552
Review and Preview Problems 554
11 Geometry in Space and Vectors 555
11.1 Cartesian Coordinates in Three-Space 555
11.2 Vectors 560
11.3 The Dot Product 566
11.4 The Cross Product 574
11.5 Vector-Valued Functions and Curvilinear Motion 579
11.6 Lines and Tangent Lines in Three-Space 589
11.7 Curvature and Components of Acceleration 593
11.8 Surfaces in Three-Space 603
11.9 Cylindrical and Spherical Coordinates 609
11.10 Chapter Review 613
Review and Preview Problems 616
12 Derivatives for Functions of Two or More Variables 617
12.1 Functions of Two or More Variables 617
12.2 Partial Derivatives 624
12.3 Limits and Continuity 629
12.4 Differentiability 635
12.5 Directional Derivatives and Gradients 641
12.6 The Chain Rule 647
12.7 Tangent Planes and Approximations 652
12.8 Maxima and Minima 657
12.9 The Method of Lagrange Multipliers 666
12.10 Chapter Review 672
Review and Preview Problems 674
13 Multiple Integrals 675
13.1 Double Integrals over Rectangles 675
13.2 Iterated Integrals 680
13.3 Double Integrals over Nonrectangular Regions 684
13.4 Double Integrals in Polar Coordinates 691
13.5 Applications of Double Integrals 696
13.6 Surface Area 700
13.7 Triple Integrals in Cartesian Coordinates 706
13.8 Triple Integrals in Cylindrical and Spherical Coordinates 713
13.9 Change of Variables in Multiple Integrals 718
13.10 Chapter Review 728
Review and Preview Problems 730
14 Vector Calculus 731
14.1 Vector Fields 731
14.2 Line Integrals 735
14.3 Independence of Path 742
14.4 Green's Theorem in the Plane 749
14.5 Surface Integrals 755
14.6 Gauss's Divergence Theorem 764
14.7 Stokes's Theorem 770
14.8 Chapter Review 773
Appendix A- 1
A.1 Mathematical Induction A- 1
A.2 Proofs of Several Theorems A- 3