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11 Infinite Sequences and Series 689

11.1 Sequences 690

Laboratory Project Logistic Sequences 703

11.2 Series 703

11.3 The Integral Test and Estimates of Sums 714

11.4 The Comparison Tests 722

11.5 Alternating Series 727

11.6 Absolute Convergence and the Ratio and Root Tests 732

11.7 Strategy for Testing Series 739

11.8 Power Series 741

11.9 Representations of Functions as Power Series 746

11.10 Taylor and Maclaurin Series 753

Laboratory Project An Elusive Limit 767

Writing Project How Newton Discovered the Binomial Series 767

11.11 Applications of Taylor Polynomials 768

Applied Project Radiation from the Stars 777

Review 778

Problems Plus 781

12 Vectors and the Geometry of Space 785

12.1 Three-Dimensional Coordinate Systems 786

12.2 Vectors 791

12.3 The Dot Product 800

12.4 The Cross Product 808

Discovery Project The Geometry of a Tetrahedron 816

12.5 Equations of Lines and Planes 816

Laboratory Project Putting 3D in Perspective 826

12.6 Cylinders and Quadric Surfaces 827

Review 834

Problems Plus 837

13 Vector Functions 839

13.1 Vector Functions and Space Curves 840

13.2 Derivatives and Integrals of Vector Functions 847

13.3 Arc Length and Curvature 853

13.4 Motion in Space:Velocity and Acceleration 862

Applied Project Kepler's Laws 872

Review 873

Problems Plus 876

14 Partial Derivatives 877

14.1 Functions of Several Variables 878

14.2 Limits and Continuity 892

14.3 Partial Derivatives 900

14.4 Tangent Planes and Linear Approximations 915

14.5 The Chain Rule 924

14.6 Directional Derivatives and the Gradient Vector 933

14.7 Maximum and Minimum Values 946

Applied Project Designing a Dumpster 956

Discovery Project Quadratic Approximations and Critical Points 956

14.8 Lagrange Multipliers 957

Applied Project Rocket Science 964

Applied Project Hydro-Turbine Optimization 966

Review 967

Problems Plus 971

15 Multiple Integrals 973

15.1 Double Integrals over Rectangles 974

15.2 Iterated Integrals 982

15.3 Double Integrals over General Regions 988

15.4 Double Integrals in Polar Coordinates 997

15.5 Applications of Double Integrals 1003

15.6 Surface Area 1013

15.7 Triple Integrals 1017

Discovery Project Volumes of Hyperspheres 1027

15.8 Triple Integrals in Cylindrical Coordinates 1027

Laboratory Project The Intersection of Three Cylinders 1032

15.9 Triple Integrals in Spherical Coordinates 1033

Applied Project Roller Derby 1039

15.10 Change of Variables in Multiple Integrals 1040

Review 1049

Problems Plus 1053

16 Vector Calculus 1055

16.1 Vector Fields 1056

16.2 Line Integrals 1063

16.3 The Fundamental Theorem for Line Integrals 1075

16.4 Green's Theorem 1084

16.5 Curl and Divergence 1091

16.6 Parametric Surfaces and Their Areas 1099

16.7 Surface Integrals 1110

16.8 Stokes'Theorem 1122

Writing Project Three Men and Two Theorems 1128

16.9 The Divergence Theorem 1128

16.10 Summary 1135

Review 1136

Problems Plus 1139

17 Second-Order Differential Equations 1141

17.1 Second-Order Linear Equations 1142

17.2 Nonhomogeneous Linear Equations 1148

17.3 Applications of Second-Order Differential Equations 1156

17.4 Series Solutions 1164

Review 1169