A Preview of Calculus 2
1 Functions and Models 10
1.1 Four Ways to Represent a Function 11
1.2 Mathematical Models:A Catalog of Essential Functions 25
1.3 New Functions from Old Functions 38
1.4 Graphing Calculators and Computers 48
1.5 Exponential Functions 55
1.6 Inverse Functions and Logarithms 63
Review 77
Principles of Problem Solving 80
2 Limits and Derivatives 86
2.1 The Tangent and Velocity Problems 87
2.2 The Limit of a Function 92
2.3 Calculating Limits Using the Limit Laws 104
2.4 The Precise Definition of a Limit 114
2.5 Continuity 124
2.6 Limits at Infinity;Horizontal Asymptotes 135
2.7 Tangents,Velocities,and Other Rates of Change 149
2.8 Derivatives 158
Writing Project.Early Methods for Finding Tangents 164
2.9 The Derivative as a Function 165
Review 176
Problems Plus 180
3 Differentiation Rules 182
3.1 Derivatives of Polynomials and Exponential Functions 183
3.2 The Product and Quotient Rules 192
3.3 Rates of Change in the Natural and Social Sciences 199
3.4 Derivatives of Trigonometric Functions 211
3.5 The Chain Rule 217
3.6 Implicit Differentiation 227
3.7 Higher Derivatives 236
Applied Project.Where Should a Pilot Start Descent? 243
Applied Project.Building a Better Roller Coaster 243
3.8 Derivatives of Logarithmic Functions 244
3.9 Hyperbolic Functions 250
3.10 Related Rates 256
3.11 Linear Approximations and Differentials 262
Laboratory Project.Taylor Polynomials 269
Review 270
Problems Plus 274
4 Rpplications of Oifferentiation 278
4.1 Maximum and Minimum Values 279
Applied Project.The Calculus of Rainbows 288
4.2 The Mean Value Theorem 290
4.3 How Derivatives Affect the Shape of a Graph 296
4.4 Indeterminate Forms and L'Hospital's Rule 307
Writing Project.The Origins of L'Hospital's Rule 315
4.5 Summary of Curve Sketching 316
4.6 Graphing with Calculus and Calculators 324
4.7 Optimization Problems 331
Applied Project.The Shape of a Can 341
4.8 Applications to Business and Economics 342
4.9 Newton's Method 347
4.10 Antiderivatives 353
Review 361
Problems Plus 365
5 Integrals 368
5.1 Areas and Distances 369
5.2 The Definite Integral 380
Discovery Project.Area Functions 393
5.3 The Fundamental Theorem of Calculus 394
5.4 Indefinite Integrals and the Net Change Theorem 405
Writing Project.Newton,Leibniz,and the Invention of Calculus 413
5.5 The Substitution Rule 414
5.6 The Logarithm Defined as an Integral 422
Review 430
Problems Plus 434
6 Rpplications of Integration 436
6.1 Areas between Curves 437
6.2 Volumes 444
6.3 Volumes by Cylindrical Shells 455
6.4 Work 460
6.5 Average Value of a Function 464
Applied Project.Where to Sit at the Movies 468
Review 468
Problems Plus 470
7 Techniques of Integration 474
7.1 Integration by Parts 475
7.2 Trigonometric Integrals 482
7.3 Trigonometric Substitution 489
7.4 Integration of Rational Functions by Partial Fractions 496
7.5 Strategy for Integration 505
7.6 Integration Using Tables and Computer Algebra Systems 511
Discovery Project.Patterns in Integrals 517
7.7 Approximate Integration 518
7.8 Improper Integrals 530
Review 540
Problems Plus 543
8 Further Rpplications of Integration 546
8.1 Arc Length 547
Discovery Project.Arc Length Contest 554
8.2 Area of a Surface of Revolution 554
Discovery Project.Rotating on a Slant 560
8.3 Applications to Physics and Engineering 561
8.4 Applications to Economics and Biology 571
8.5 Probability 575
Review 582
Problems Plus 584
9 Differential Equations 586
9.1 Modeling with Differential Equations 587
9.2 Direction Fields and Euler's Method 592
9.3 Separable Equations 601
Applied Project.How Fast Does a Tank Drain? 609
Applied Project.Which Is Fasteer,Going Up or Coming Down? 610
9.4 Exponential Growth and Decay 611
Applied Project.Calculus and Baseball 622
9.5 The Logistic Equation 623
9.6 Linear Equations 632
9.7 Predator-Prey Systems 638
Review 644
Problems Plus 648
10 Parametric Equations and Polar Coordinates 650
10.1 Curves Defined by Parametric Equations 651
Laboratory Project.Running Circles around Circles 659
10.2 Calculus with Parametric Curves 660
Laboratory Project.Bézier Curves 669
10.3 Polar Coordinates 669
10.4 Areas and Lengths in Polar Coordinates 679
10.5 Conic Sections 684
10.6 Conic Sections in Polar Coordinates 692
Review 696
Problems Plus 699
11 Infinite Sequences and Series 700
11.1 Sequences 701
Laboratory Project.Logistic Sequences 713
11.2 Series 713
11.3 The Integral Test and Estimates of Sums 723
11.4 The Comparison Tests 730
11.5 Alternating Series 735
11.6 Absolute Convergence and the Ratio and Root Tests 740
11.7 Strategy for Testing Series 747
11.8 Power Series 749
11.9 Representations of Functions as Power Series 754
11.10 Taylor and Maclaurin Series 760
Laboratory Project.An Elusive Limit 772
11.11 The Binomial Series 772
Writing Project.How Newton Discovered the Binomial Series 776
11.12 Applications of Taylor Polynomials 776
Applied Project.Radiation from the Stars 785
Review 786
Problems Plus 789
12 Vectors and the Geometru of Space 792
12.1 Three-Dimensional Coordinate Systems 793
12.2 Vectors 798
12.3 The Dot Product 807
12.4 The Cross Product 814
Discovery Project.The Geometry of a Tetrahedron 822
12.5 Equations of Lines and Planes 822
Laboratory Project.Putting 3D in Perspective 832
12.6 Cylinders and Quadric Surfaces 832
12.7 Cylindrical and Spherical Coordinates 839
Laboratory Project.Families of Surfaces 844
Review 844
Problems Plus 847
13 Vector Functions 848
13.1 Vector Functions and Space Curves 849
13.2 Derivatives and Integrals of Vector Functions 856
13.3 Arc Length and Curvature 862
13.4 Motion in Space:Velocity and Acceleration 870
Applied Project.Kepler's Laws 880
Review 881
Problems Plus 884
14 Partial Derivatives 886
14.1 Functions of Several Variables 887
14.2 Limits and Continuity 902
14.3 Partial Derivatives 909
14.4 Tangent Planes and Linear Approximations 923
14.5 The Chain Rule 931
14.6 Directional Derivatives and the Gradient Vector 940
14.7 Maximum and Minimum Values 953
Applied Project.Designing a Dumpster 963
Discovery Project.Quadratic Approximations and Critical Points 964
14.8 Lagrange Multipliers 965
Applied Project.Rocket Science 972
Applied Project.Hydro-Turbine Optimization 973
Review 974
Problems Plus 978
15 Multiple Integrals 980
15.1 Double Integrals over Rectangles 981
15.2 Iterated Integrals 989
15.3 Double Integrals over General Regions 995
15.4 Double Integrals in Polar Coordinates 1003
15.5 Applications of Double Integrals 1009
15.6 Surface Area 1019
15.7 Triple Integrals 1023
Discovery Project.Volumes of Hyperspheres 1032
15.8 Triple Integrals in Cylindrical and Spherical Coordinates 1033
Applied Project.Roller Derby 1039
Discovery Project.The Intersection of Three Cylinders 1040
15.9 Change of Variables in Multiple Integrals 1041
Review 1049
Problems Plus 1052
16 Vector Calculus 1054
16.1 Vector Fields 1055
16.2 Line Integrals 1062
16.3 The Fundamental Theorem for Line Integrals 1074
16.4 Green's Theorem 1083
16.5 Curl and Divergence 1090
16.6 Parametric Surfaces and Their Areas 1098
16.7 Surface Integrals 1109
16.8 Stokes'Theorem 1121
Writing Project.Three Men and Two Theorems 1126
16.9 The Divergence Theorem 1127
16.10 Summary 1134
Review 1135
Problems Plus 1138
17 Second-Order Differential Equations 1140
17.1 Second-Order Linear Equations 1141
17.2 Nonhomogeneous Linear Equations 1147
17.3 Applications of Second-Order Differential Equations 1155
17.4 Series Solutions 1163
Review 1167
Appendixes R 1
A Numbers,Inequalities,and Absolute Values A 2
B Coordinate Geometry and Lines A 10
C Graphs of Second-Degree Equations A 16
D Trigonometry A 24
E Sigma Notation A 34
F Proofs of Theorems A 39
G Complex Numbers A 49
H Answers to Odd-Numbered Exercises A 57
Index A 125