A Preview of Calculus 2
1 Functions and Models 10
1.1 Four Ways to Represent a Function 11
1.2 Mathematical Models 24
1.3 New Functions from Old Functions 38
1.4 Graphing Calculators and Computers 49
1.5 Exponential Functions 56
1.6 Inverse Functions and Logarithms 64
1.7 Parametric Curves 75
Laboratory Project Running Circles around Circles 83
Review 84
Principles of Problem Solving 88
2 Limits and Derivatives 94
2.1 The Tangent and Velocity Problems 95
2.2 The Limit of a Function 100
2.3 Calculating Limits Using the Limit Laws 110
2.4 Continuity 119
2.5 Limits Involving Infinity 130
2.6 Tangents，Velocities，and Other Rates of Change 142
2.7 Derivatives 150
Writing Project Early Methods for Finding Tangents 157
2.8 The Derivative as a Function 157
2.9 Linear Approximations 171
2.10 What Does f’ Say about f？ 175
Review 181
Focus on Problem Solving 185
3 Differentiation Rules 188
3.1 Derivatives of Polynomials and Exponential Functions 189
Applied Project Building a Better Roller Coaster 198
3.2 The Product and Quotient Rules 199
3.3 Rates of Change in the Natural and Social Sciences 206
3.4 Derivatives of Trigonometric Functions 218
3.5 The Chain Rule 225
Laboratory Project Bezier Curves 236
Applied Project Where Should a Pilot Start Descent？ 237
3.6 Implicit Differentiation 237
3.7 Derivatives of Logarithmic Functions 245
Discovery Project Hyperbolic Functions 251
3.8 Linear Approximations and Differentials 252
Laboratory Project Taylor Polynomials 257
Review 258
Focus on Problem Solving 261
4Applications of Differentiation 264
4.1 Related Rates 265
4.2 Maximum and Minimum Values 271
Applied Project The Calculus of Rainbows 279
4.3 Derivatives and the Shapes of Curves 280
4.4 Graphing with Calculus and Calculators 291
4.5 Indeterminate Forms and l’Hospital’s Rule 298
Writing Project The Origins of l’Hospital’s Rule 307
4.6 Optimization Problems 307
Applied Project The Shape of a Can 318
4.7 Applications to Economics 319
4.8 Newton’s Method 324
4.9 Antiderivatives 329
Review 336
Focus on Problem Solving 340
5 Integrals 344
5.1 Areas and Distances 345
5.2 The Definite Integral 357
5.3 Evaluating Definite Integrals 369
Discovery Project Area Functions 379
5.4 The Fundamental Theorem of Calculus 380
Writing Project Newton，Leibniz，and the Invention of Calculus 388
5.5 The Substitution Rule 389
5.6 Integration by Parts 396
5.7 Additional Techniques of Integration 403
5.8 Integration Using Tables and Computer Algebra Systems 409
Discovery Project Patterns in Integrals 415
5.9 Approximate Integration 416
5.10 Improper Integrals 428
Review 438
Focus on Problem Solving 442
6 Applications of Integration 446
6.1 More about Areas 447
6.2 Volumes 453
Discovery Project Rotating on a Slant 466
6.3 Arc Length 467
Discovery Project Arc Length Contest 472
6.4 Average Value of a Function 473
Applied Project Where to Sit at the Movies 476
6.5 Applications to Physics and Engineering 476
6.6 Applications to Economics and Biology 487
6.7 Probability 492
Review 499
Focus on Problem Solving 502
7 Differential Equations 506
7.1 Modeling with Differential Equations 507
7.2 Direction Fields and Euler’s Method 512
7.3 Separable Equations 522
Applied Project Which Is Faster，Going Up or Coming Down？ 530
7.4 Exponential Growth and Decay 531
Applied Project Calculus and Baseball 540
7.5 The Logistic Equation 541
7.6 Predator-Prey Systems 550
Review 557
Focus on Problem Solving 560
8 Infinite Sequences and Series 562
8.1 Sequences 563
Laboratory Project Logistic Sequences 573
8.2 Series 573
8.3 The Integral and Comparison Tests； Estimating Sums 583
8.4 Other Convergence Tests 592
8.5 Power Series 600
8.6 Representations of Functions as Power Series 605
8.7 Taylor and Maclaurin Series 611
8.8 The Binomial Series 622
Writing Project How Newton Discovered the Binomial Series 626
8.9 Applications of Taylor Polynomials 626
Applied Project Radiation from the Stars 634
8.10 Using Series to Solve Differential Equations 635
Review 640
Focus on Problem Solving 643
9 Vectors and the Geometry of Space 646
9.1 Three-Dimensional Coordinate Systems 647
9.2 Vectors 652
9.3 The Dot Product 661
9.4 The Cross Product 667
Discovery Project The Geometry of a Tetrahedron 675
9.5 Equations of Lines and Planes 676
9.6 Functions and Surfaces 685
9.7 Cylindrical and Spherical Coordinates 694
Laboratory Project Families of Surfaces 699
Review 700
Focus on Problem Solving 703
10 Vector Functions 704
10.1 Vector Functions and Space Curves 705
10.2 Derivatives and Integrals of Vector Functions 711
10.3 Arc Length and Curvature 717
10.4 Motion in Space 725
Applied Project Kepler’s Laws 735
10.5 Parametric Surfaces 736
Review 742
Focus on Problem Solving 745
11 Partial Derivatives 748
11.1 Functions of Several Variables 749
11.2 Limits and Continuity 760
11.3 Partial Derivatives 766
11.4 Tangent Planes and Linear Approximations 779
11.5 The Chain Rule 790
11.6 Directional Derivatives and the Gradient Vector 798
11.7 Maximum and Minimum Values 811
Applied Project Designing a Dumpster 820
Discovery Project Quadratic Approximations and Critical Points 821
11.8 Lagrange Multipliers 822
Applied Project Rocket Science 829
Applied Project Hydro-Turbine Optimization 830
Review 831
Focus on Problem Solving 836
12Multiple Integrals 838
12.1 Double Integrals over Rectangles 839
12.2 Iterated Integrals 849
12.3 Double Integrals over General Regions 854
12.4 Double Integrals in Polar Coordinates 863
12.5 Applications of Double Integrals 868
12.6 Surface Area 878
12.7 Triple Integrals 883
Discovery Project Volumes of Hyperspheres 893
12.8 Triple Integrals in Cylindrical and Spherical Coordinates 893
Applied Project Roller Derby 900
Discovery Project The Intersection of Three Cylinders 901
12.9 Change of Variables in Multiple Integrals 901
Review 910
Focus on Problem Solving 914
13 Vector Calculus 916
13.1 Vector Fields 917
13.2 Line Integrals 924
13.3 The Fundamental Theorem for Line Integrals 936
13.4 Green’s Theorem 945
13.5 Curl and Divergence 952
13.6 Surface Integrals 960
13.7 Stokes’ Theorem 971
Writing Project Three Men and Two Theorems 977
13.8 The Divergence Theorem 978
13.9 Summary 985
Review 986
Focus on Problem Solving 989