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1 Limits and Continuity 1
1.1 Limits (An Intuitive Approach) 1
1.2 Computing Limits 13
1.3 Limits at Infinity; End Behavior of a Function 22
1.4 Limits (Discussed More Rigorously) 31
1.5 Continuity 40
1.6 Continuity of Trigonometric Functions 51
1.7 Inverse Trigonometric Functions 56
1.8 Exponential and Logarithmic Functions 63
2 The Derivative 79
2.1 Tangent Lines and Rates of Change 79
2.2 The Derivative Function 89
2.3 Introduction to Techniques of Differentiation 100
2.4 The Product and Quotient Rules 108
2.5 Derivatives of Trigonometric Functions 113
2.6 The Chain Rule 118
3 Topics in Differentiation 129
3.1 Implicit Differentiation 129
3.2 Derivatives of Logarithmic Functions 135
3.3 Derivatives of Exponential and Inverse Trigonometric Functions 140
3.4 Related Rates 146
3.5 Local Linear Approximation; Differentials 153
3.6 L’Hôpital’s Rule; Indeterminate Forms 160
4 The Derivative in Graphing and Applications 172
4.1 Analysis of Functions I: Increase, Decrease, and Concavity 172
4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 183
4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 192
4.4 Absolute Maxima and Minima 204
4.5 Applied Maximum and Minimum Problems 212
4.6 Rectilinear Motion 225
4.7 Newton’s Method 233
4.8 Rolle’s Theorem; Mean-Value Theorem 239
5 Integration 253
5.1 An Overview of the Area Problem 253
5.2 The Indefinite Integral 258
5.3 Integration by Substitution 268
5.4 The Definition of Area as a Limit; Sigma Notation 275
5.5 The Definite Integral 285
5.6 The Fundamental Theorem of Calculus 294
5.7 Rectilinear Motion Revisited Using Integration 306
5.8 Average Value of a Function and its Applications 314
5.9 Evaluating Definite Integrals by Substitution 319
5.10 Logarithmic and Other Functions Defined by Integrals 325
6 Applications of the Definite Integral in Geometry, Science, and Engineering 341
6.1 Area Between Two Curves 341
6.2 Volumes by Slicing; Disks and Washers 349
6.3 Volumes by Cylindrical Shells 358
6.4 Length of a Plane Curve 364
6.5 Area of a Surface of Revolution 370
6.6 Work 375
6.7 Moments, Centers of Gravity, and Centroids 383
6.8 Fluid Pressure and Force 392
6.9 Hyperbolic Functions and Hanging Cables 398
7 Principles of Integral Evaluation 412
7.1 An Overview of Integration Methods 412
7.2 Integration by Parts 415
7.3 Integrating Trigonometric Functions 423
7.4 Trigonometric Substitutions 431
7.5 Integrating Rational Functions by Partial Fractions 437
7.6 Using Computer Algebra Systems and Tables of Integrals 445
7.7 Numerical Integration; Simpson’s Rule 454
7.8 Improper Integrals 467
8 Mathematical Modeling with Differential Equations 481
8.1 Modeling with Differential Equations 481
8.2 Separation of Variables 487
8.3 Slope Fields; Euler’s Method 498
8.4 First-Order Differential Equations and Applications 504
9 Infinite Series 514
9.1 Sequences 514
9.2 Monotone Sequences 524
9.3 Infinite Series 531
9.4 Convergence Tests 539
9.5 The Comparison, Ratio, and Root Tests 547
9.6 Alternating Series; Absolute and Conditional Convergence 553
9.7 Maclaurin and Taylor Polynomials 563
9.8 Maclaurin and Taylor Series; Power Series 573
9.9 Convergence of Taylor Series 582
9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 591
10 Parametric and Polar Curves; Conic Sections 605
10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 605
10.2 Polar Coordinates 617
10.3 Tangent Lines, Arc Length, and Area for Polar Curves 630
10.4 Conic Sections 639
10.5 Rotation of Axes; Second-Degree Equations 656
10.6 Conic Sections in Polar Coordinates 661
11 Three-Dimensional Space; Vectors 674
11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 674
11.2 Vectors 680
11.3 Dot Product; Projections 691
11.4 Cross Product 700
11.5 Parametric Equations of Lines 710
11.6 Planes in 3-Space 717
11.7 Quadric Surfaces 725
11.8 Cylindrical and Spherical Coordinates 735
12 Vector-Valued Functions 744
12.1 Introduction to Vector-Valued Functions 744
12.2 Calculus of Vector-Valued Functions 750
12.3 Change of Parameter; Arc Length 759
12.4 Unit Tangent, Normal, and Binormal Vectors 768
12.5 Curvature 773
12.6 Motion Along a Curve 781
12.7 Kepler’s Laws of Planetary Motion 794
13 Partial Derivatives 805
13.1 Functions of Two or More Variables 805
13.2 Limits and Continuity 815
13.3 Partial Derivatives 824
13.4 Differentiability, Differentials, and Local Linearity 837
13.5 The Chain Rule 845
13.6 Directional Derivatives and Gradients 855
13.7 Tangent Planes and Normal Vectors 866
13.8 Maxima and Minima of Functions of Two Variables 872
13.9 Lagrange Multipliers 883
14 Multiple Integrals 894
14.1 Double Integrals 894
14.2 Double Integrals over Nonrectangular Regions 902
14.3 Double Integrals in Polar Coordinates 910
14.4 Surface Area; Parametric Surfaces 918
14.5 Triple Integrals 930
14.6 Triple Integrals in Cylindrical and Spherical Coordinates 938
14.7 Change of Variables in Multiple Integrals; Jacobians 947
14.8 Centers of Gravity Using Multiple Integrals 959
15 Topics in Vector Calculus 971
15.1 Vector Fields 971
15.2 Line Integrals 980
15.3 Independence of Path; Conservative Vector Fields 995
15.4 Green’s Theorem 1005
15.5 Surface Integrals 1013
15.6 Applications of Surface Integrals; Flux 1021
15.7 The Divergence Theorem 1030
15.8 Stokes’ Theorem 1039
A Appendices
A Trigonometry Review (Summary) A1
B Functions (Summary) A8
C New Functions from Old (Summary) A11
D Families of Functions (Summary) A16
E Inverse Functions (Summary) A23
Answers to Odd-Numbered Exercises A28
Index I-1
