Thomas Calculus in SI Units

Cover......Page 1
Thomas’ Calculus: Thirteenth Edition in SI Units......Page 2
 Copyright......Page 3
Contents......Page 4
Preface......Page 10
 Functions and Their Graphs......Page 16
 Combining Functions; Shifting and Scaling Graphs......Page 29
 Trigonometric Functions......Page 36
 Graphing with Software......Page 44
  Practice Exercises......Page 51
  Additional and Advanced Exercises......Page 53
 Rates of Change and Tangents to Curves......Page 56
 Limit of a Function and Limit Laws......Page 63
 The Precise Definition of a Limit......Page 74
 One-Sided Limits......Page 83
 Continuity......Page 90
 Limits Involving Infinity; Asymptotes of Graphs......Page 101
  Questions to Guide Your Review......Page 114
  Practice Exercises......Page 115
  Additional and Advanced Exercises......Page 117
 Tangents and the Derivative at a Point......Page 120
 The Derivative as a Function......Page 125
 Differentiation Rules......Page 133
 The Derivative as a Rate of Change......Page 142
 Derivatives of Trigonometric Functions......Page 152
 The Chain Rule......Page 159
 Implicit Differentiation......Page 166
 Related Rates......Page 171
 Linearization and Differentials......Page 180
  Practice Exercises......Page 192
  Additional and Advanced Exercises......Page 197
 Extreme Values of Functions......Page 200
 The Mean Value Theorem......Page 208
 Monotonic Functions and the First Derivative Test......Page 214
 Concavity and Curve Sketching......Page 219
 Applied Optimization......Page 230
 Newton’s Method......Page 242
 Antiderivatives......Page 247
  Questions to Guide Your Review......Page 257
  Practice Exercises......Page 258
  Additional and Advanced Exercises......Page 260
 Area and Estimating with Finite Sums......Page 264
 Sigma Notation and Limits of Finite Sums......Page 274
 The Definite Integral......Page 281
 The Fundamental Theorem of Calculus......Page 293
 Indefinite Integrals and the Substitution Method......Page 304
 Definite Integral Substitutions and the Area Between Curves......Page 311
  Practice Exercises......Page 321
  Additional and Advanced Exercises......Page 324
 Volumes Using Cross-Sections......Page 328
 Volumes Using Cylindrical Shells......Page 339
 Arc Length......Page 346
 Areas of Surfaces of Revolution......Page 352
 Work and Fluid Forces......Page 357
 Moments and Centers of Mass......Page 366
  Practice Exercises......Page 377
  Additional and Advanced Exercises......Page 379
 Inverse Functions and Their Derivatives......Page 381
 Natural Logarithms......Page 389
 Exponential Functions......Page 397
 Exponential Change and Separable Differential Equations......Page 408
 Indeterminate Forms and L’Hôpital’s Rule......Page 418
 Inverse Trigonometric Functions......Page 426
 Hyperbolic Functions......Page 439
 Relative Rates of Growth......Page 448
  Questions to Guide Your Review......Page 453
  Practice Exercises......Page 454
  Additional and Advanced Exercises......Page 457
 Using Basic Integration Formulas......Page 459
 Integration by Parts......Page 464
 Trigonometric Integrals......Page 472
 Trigonometric Substitutions......Page 478
 Integration of Rational Functions by Partial Fractions......Page 483
 Integral Tables and Computer Algebra Systems......Page 492
 Numerical Integration......Page 497
 Improper Integrals......Page 507
 Probability......Page 518
  Questions to Guide Your Review......Page 531
  Practice Exercises......Page 532
  Additional and Advanced Exercises......Page 534
 Solutions, Slope Fields, and Euler’s Method......Page 539
 First-Order Linear Equations......Page 547
 Applications......Page 553
 Graphical Solutions of Autonomous Equations......Page 559
 Systems of Equations and Phase Planes......Page 566
  Practice Exercises......Page 572
  Additional and Advanced Exercises......Page 573
 Sequences......Page 575
 Infinite Series......Page 587
 The Integral Test......Page 596
 Comparison Tests......Page 603
 Absolute Convergence; The Ratio and Root Tests......Page 607
 Alternating Series and Conditional Convergence......Page 613
 Power Series......Page 619
 Taylor and Maclaurin Series......Page 629
 Convergence of Taylor Series......Page 634
 The Binomial Series and Applications of Taylor Series......Page 641
  Questions to Guide Your Review......Page 650
  Practice Exercises......Page 651
  Additional and Advanced Exercises......Page 653
 Parametrizations of Plane Curves......Page 656
 Calculus with Parametric Curves......Page 664
 Polar Coordinates......Page 674
 Graphing Polar Coordinate Equations......Page 678
 Areas and Lengths in Polar Coordinates......Page 682
 Conic Sections......Page 686
 Conics in Polar Coordinates......Page 695
  Practice Exercises......Page 702
  Additional and Advanced Exercises......Page 704
 Three-Dimensional Coordinate Systems......Page 707
 Vectors......Page 712
 The Dot Product......Page 721
 The Cross Product......Page 729
 Lines and Planes in Space......Page 735
 Cylinders and Quadric Surfaces......Page 743
  Questions to Guide Your Review......Page 748
  Practice Exercises......Page 749
  Additional and Advanced Exercises......Page 751
 Curves in Space and Their Tangents......Page 754
 Integrals of Vector Functions; Projectile Motion......Page 762
 Arc Length in Space......Page 771
 Curvature and Normal Vectors of a Curve......Page 775
 Tangential and Normal Components of Acceleration......Page 781
 Velocity and Acceleration in Polar Coordinates......Page 787
  Practice Exercises......Page 791
  Additional and Advanced Exercises......Page 793
 Functions of Several Variables......Page 796
 Limits and Continuity in Higher Dimensions......Page 804
 Partial Derivatives......Page 813
 The Chain Rule......Page 824
 Directional Derivatives and Gradient Vectors......Page 833
 Tangent Planes and Differentials......Page 842
 Extreme Values and Saddle Points......Page 851
 Lagrange Multipliers......Page 860
 Taylor’s Formula for Two Variables......Page 869
 Partial Derivatives with Constrained Variables......Page 873
  Questions to Guide Your Review......Page 878
  Practice Exercises......Page 879
  Additional and Advanced Exercises......Page 882
 Double and Iterated Integrals over Rectangles......Page 885
 Double Integrals over General Regions......Page 890
 Area by Double Integration......Page 899
 Double Integrals in Polar Form......Page 903
 Triple Integrals in Rectangular Coordinates......Page 909
 Moments and Centers of Mass......Page 918
 Triple Integrals in Cylindrical and Spherical Coordinates......Page 925
 Substitutions in Multiple Integrals......Page 937
  Practice Exercises......Page 947
  Additional and Advanced Exercises......Page 950
 Line Integrals......Page 953
 Vector Fields and Line Integrals: Work, Circulation, and Flux......Page 960
 Path Independence, Conservative Fields, and Potential Functions......Page 972
 Green’s Theorem in the Plane......Page 983
 Surfaces and Area......Page 995
 Surface Integrals......Page 1006
 Stokes’ Theorem......Page 1017
 The Divergence Theorem and a Unified Theory......Page 1030
  Questions to Guide Your Review......Page 1042
  Practice Exercises......Page 1043
  Additional and Advanced Exercises......Page 1045
 Second-Order Linear Equations......Page 1048
 Nonhomogeneous Linear Equations......Page 1055
 Applications......Page 1064
 Euler Equations......Page 1070
 Power Series Solutions......Page 1073
 Real Numbers and the Real Line......Page 1081
 Mathematical Induction......Page 1086
 Lines, Circles, and Parabolas......Page 1090
 Proofs of Limit Theorems......Page 1099
 Commonly Occurring Limits......Page 1102
 Theory of the Real Numbers......Page 1103
 Complex Numbers......Page 1106
 The Distributive Law for Vector Cross Products......Page 1115
 The Mixed Derivative Theorem and the Increment Theorem......Page 1116
Answers to Odd-Numbered Exercises......Page 1121
Credits......Page 1192
Index......Page 1193
A Brief Table of Integrals......Page 1209
Basic Formulas and Rules......Page 1215