托马斯微积分Thomas Calculus

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Contents
iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
4.8 Antiderivatives 281
Questions to Guide Your Review 291
Practice Exercises 291
Additional and Advanced Exercises 295
5 Integrals 299
5.1 Area and Estimating with Finite Sums 299
5.2 Sigma Notation and Limits of Finite Sums 309
5.3 The Definite Integral 316
5.4 The Fundamental Theorem of Calculus 328
5.5 Indefinite Integrals and the Substitution Method 339
5.6 Definite Integral Substitutions and the Area Between Curves 347
Questions to Guide Your Review 357
Practice Exercises 357
Additional and Advanced Exercises 361
6 Applications of Definite Integrals 365
6.1 Volumes Using Cross-Sections 365
6.2 Volumes Using Cylindrical Shells 376
6.3 Arc Length 384
6.4 Areas of Surfaces of Revolution 390
6.5 Work and Fluid Forces 395
6.6 Moments and Centers of Mass 404
Questions to Guide Your Review 415
Practice Exercises 416
Additional and Advanced Exercises 417
7 Integrals and Transcendental Functions 420
7.1 The Logarithm Defined as an Integral 420
7.2 Exponential Change and Separable Differential Equations 430
7.3 Hyperbolic Functions 439
7.4 Relative Rates of Growth 448
Questions to Guide Your Review 453
Practice Exercises 453
Additional and Advanced Exercises 455
 Contents v
8 Techniques of Integration 456
8.1 Using Basic Integration Formulas 456
8.2 Integration by Parts 461
8.3 Trigonometric Integrals 469
8.4 Trigonometric Substitutions 475
8.5 Integration of Rational Functions by Partial Fractions 480
8.6 Integral Tables and Computer Algebra Systems 489
8.7 Numerical Integration 494
8.8 Improper Integrals 504
8.9 Probability 515
Questions to Guide Your Review 528
Practice Exercises 529
Additional and Advanced Exercises 531
9 First-Order Differential Equations 536
9.1 Solutions, Slope Fields, and Euler’s Method 536
9.2 First-Order Linear Equations 544
9.3 Applications 550
9.4 Graphical Solutions of Autonomous Equations 556
9.5 Systems of Equations and Phase Planes 563
Questions to Guide Your Review 569
Practice Exercises 569
Additional and Advanced Exercises 570
10 Infinite Sequences and Series 572
10.1 Sequences 572
10.2 Infinite Series 584
10.3 The Integral Test 593
10.4 Comparison Tests 600
10.5 Absolute Convergence; The Ratio and Root Tests 604
10.6 Alternating Series and Conditional Convergence 610
10.7 Power Series 616
10.8 Taylor and Maclaurin Series 626
10.9 Convergence of Taylor Series 631
10.10 The Binomial Series and Applications of Taylor Series 638
Questions to Guide Your Review 647
Practice Exercises 648
Additional and Advanced Exercises 650
11 Parametric Equations and Polar Coordinates 653
11.1 Parametrizations of Plane Curves 653
11.2 Calculus with Parametric Curves 661
11.3 Polar Coordinates 671
vi Contents
11.4 Graphing Polar Coordinate Equations 675
11.5 Areas and Lengths in Polar Coordinates 679
11.6 Conic Sections 683
11.7 Conics in Polar Coordinates 692
Questions to Guide Your Review 699
Practice Exercises 699
Additional and Advanced Exercises 701
12 Vectors and the Geometry of Space 704
12.1 Three-Dimensional Coordinate Systems 704
12.2 Vectors 709
12.3 The Dot Product 718
12.4 The Cross Product 726
12.5 Lines and Planes in Space 732
12.6 Cylinders and Quadric Surfaces 740
Questions to Guide Your Review 745
Practice Exercises 746
Additional and Advanced Exercises 748
13 Vector-Valued Functions and Motion in Space 751
13.1 Curves in Space and Their Tangents 751
13.2 Integrals of Vector Functions; Projectile Motion 759
13.3 Arc Length in Space 768
13.4 Curvature and Normal Vectors of a Curve 772
13.5 Tangential and Normal Components of Acceleration 778
13.6 Velocity and Acceleration in Polar Coordinates 784
Questions to Guide Your Review 788
Practice Exercises 788
Additional and Advanced Exercises 790
14 Partial Derivatives 793
14.1 Functions of Several Variables 793
14.2 Limits and Continuity in Higher Dimensions 801
14.3 Partial Derivatives 810
14.4 The Chain Rule 821
14.5 Directional Derivatives and Gradient Vectors 830
14.6 Tangent Planes and Differentials 839
14.7 Extreme Values and Saddle Points 848
14.8 Lagrange Multipliers 857
14.9 Taylor’s Formula for Two Variables 866
14.10 Partial Derivatives with Constrained Variables 870
Questions to Guide Your Review 875
Practice Exercises 876
Additional and Advanced Exercises 879
 Contents vii
15 Multiple Integrals 882
15.1 Double and Iterated Integrals over Rectangles 882
15.2 Double Integrals over General Regions 887
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 906
15.6 Moments and Centers of Mass 915
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 922
15.8 Substitutions in Multiple Integrals 934
Questions to Guide Your Review 944
Practice Exercises 944
Additional and Advanced Exercises 947
16 Integrals and Vector Fields 950
16.1 Line Integrals 950
16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 957
16.3 Path Independence, Conservative Fields, and Potential Functions 969
16.4 Green’s Theorem in the Plane 980
16.5 Surfaces and Area 992
16.6 Surface Integrals 1003
16.7 Stokes’ Theorem 1014
16.8 The Divergence Theorem and a Unified Theory 1027
Questions to Guide Your Review 1039
Practice Exercises 1040
Additional and Advanced Exercises 1042
17 Second-Order Differential Equations online
17.1 Second-Order Linear Equations
17.2 Nonhomogeneous Linear Equations
17.3 Applications
17.4 Euler Equations
17.5 Power Series Solutions
Appendices AP-1
A.1 Real Numbers and the Real Line AP-1
A.2 Mathematical Induction AP-6
A.3 Lines, Circles, and Parabolas AP-10
A.4 Proofs of Limit Theorems AP-19
A.5 Commonly Occurring Limits AP-22
A.6 Theory of the Real Numbers AP-23
A.7 Complex Numbers AP-26
A.8 The Distributive Law for Vector Cross Products AP-35
A.9 The Mixed Derivative Theorem and the Increment Theorem AP-36
Answers to Odd-Numbered Exercises A-1
Index I-1
Credits C-1
A Brief Table of Integrals T-1