Key Definitions, Properties, and Theorems 

xvii  

11 Parametric Equations and Polar Coordinates 


674  (52) 

11.1 Curves Defined by Parametric Equations 


675  (7) 

Laboratory Project Families of Hypocycloids 


682  (1) 


682  (7) 

Laboratory Project Bezier Curves 


689  (1) 

11.3 Arc Length and Surface Area 


689  (5) 


694  (10) 

11.5 Areas and Lengths in Polar Coordinates 


704  (5) 


709  (7) 

11.7 Conic Sections in Polar Coordinates 


716  (6) 


722  (2) 


724  (2) 

12 Infinite Sequences and Series 


726  (90) 


727  (11) 

Laboratory Project Logistic Sequences 


738  (1) 


738  (10) 

12.3 The Integral Test and Estimates of Sums 


748  (7) 

12.4 The Comparison Tests 


755  (5) 


760  (5) 

12.6 Absolute Convergence and the Ratio and Root Tests 


765  (7) 

12.7 Strategy for Testing Series 


772  (2) 


774  (5) 

12.9 Representations of Functions as Power Series 


779  (6) 

12.10 Taylor and Maclaurin Series 


785  (11) 

12.11 The Binomial Series 


796  (4) 

Writing Project How Newton Discovered the Binomial Series 


799  (1) 

12.12 Applications of Taylor Polynomials 


800  (9) 

Applied Project Radiation from the Stars 


808  (1) 


809  (3) 


812  (4) 

13 Vectors and the Geometry of Space 


816  (54) 

13.1 ThreeDimensional Coordinate Systems 


817  (5) 


822  (8) 


830  (7) 


837  (9) 

Discovery Project The Geometry of a Tetrahedron 


845  (1) 

13.5 Equations of Lines and Planes 


846  (9) 

13.6 Cylinders and Quadric Surfaces 


855  (6) 

13.7 Cylindrical and Spherical Coordinates 


861  (5) 

Laboratory Project Families of Surfaces 


866  (1) 


866  (3) 


869  (1) 


870  (36) 

14.1 Vector Functions and Space Curves 


871  (6) 

14.2 Derivatives and Integrals of Vector Functions 


877  (6) 

14.3 Arc Length and Curvature 


883  (8) 

14.4 Motion in Space: Velocity and Acceleration 


891  (10) 

Applied Project Kepler's Laws 


900  (1) 


901  (3) 


904  (2) 


906  (94) 

15.1 Functions of Several Variables 


907  (14) 

15.2 Limits and Continuity 


921  (8) 


929  (13) 

15.4 Tangent Planes and Linear Approximations 


942  (9) 


951  (9) 

15.6 Directional Derivatives and the Gradient Vector 


960  (13) 

15.7 Maximum and Minimum Values 


973  (12) 

Applied Project Designing a Dumpster 


983  (1) 

Discovery Project Quadratic Approximations and Critical Points 


984  (1) 

15.8 Lagrange Multipliers 


985  (9) 

Applied Project Rocket Science 


992  (1) 

Applied Project HydroTurbine Optimization 


993  (1) 


994  (4) 


998  (2) 


1000  (74) 

16.1 Double Integrals over Rectangles 


1001  (9) 


1010  (5) 

16.3 Double Integrals over General Regions 


1015  (8) 

16.4 Double Integrals in Polar Coordinates 


1023  (6) 

16.5 Applications of Double Integrals 


1029  (10) 


1039  (3) 


1042  (10) 

Discovery Project Volumes of Hyperspheres 


1052  (1) 

16.8 Triple Integrals in Cylindrical and Spherical Coordinates 


1052  (8) 

Applied Project Roller Derby 


1058  (1) 

Discovery Project The Intersection of Three Cylinders 


1059  (1) 

16.9 Change of Variables in Multiple Integrals 


1060  (8) 


1068  (4) 


1072  (2) 


1074  (84) 


1075  (6) 


1081  (12) 

17.3 The Fundamental Theorem for Line Integrals 


1093  (9) 


1102  (7) 


1109  (8) 

17.6 Parametric Surfaces and Their Areas 


1117  (10) 


1127  (12) 


1139  (6) 

Writing Project Three Men and Two Theorems 


1144  (1) 

17.9 The Divergence Theorem 


1145  (7) 


1152  (1) 


1153  (3) 


1156  (2) 

18 SecondOrder Differential Equations 


1158  

18.1 SecondOrder Linear Equations 


1159  (6) 

18.2 Nonhomogeneous Linear Equations 


1165  (7) 

18.3 Applications of SecondOrder Differential Equations 


1172  (8) 


1180  (4) 


1184  
Appendixes 

A1  (38) 
F Proofs of Theorems 

A2  (3) 
G Complex Numbers 

A5  (8) 
H Answers to OddNumbered Exercises 

A13  (26) 
Index 

A39  