A Preview of Calculus 2
1 Functions and Models 10
1.1Four Ways to Represent a Function 11
1.2Mathematical Models：A Catalog of Essential Functions 25
1.3New Functions from Old Functions 38
1.4Graphing Calculators and Computers 48
1.5Exponential Functions 55
1.6Inverse Functions and Logarithms 63
Review 77
Principles of Problem Solving 80
2 Limits and Derivatives 86
2.1The Tangent and Velocity Problems 87
2.2The Limit of a Function 92
2.3Calculating Limits Using the Limit Laws 104
2.4The Precise Definition of a Limit 114
2.5Continuity 124
2.6Limits at Infinity；Horizontal Asymptotes 135
2.7Tangents，Velocities，and Other Rates of Change 149
2.8Derivatives 158
Writing Project。Early Methods for Finding Tangents 164
2.9The Derivative as a Function 165
Review 176
Problems Plus 180
3 Differentiation Aules 182
3.1Derivatives of Polynomials and Exponential Functions 183
3.2The Product and Quotient Rules 192
3.3Rates of Change in the Natural and Social Sciences 199
3.4Derivatives of Trigonometric Functions 211
3.5The Chain Rule 217
3.6Implicit Differentiation 227
3.7Higher Derivatives 236
Applied Project。Where Should a Pilot Start Descent? 243
Applied Project。Building a Better Roller Goaster 243
3.8Derivatives of Logarithmic Functions 244
3.9Hyperbolic Functions 250
3.10Related Rates 256
3.11Linear Approximations and Differentials 262
Laboratory Project。Taylor Polynomials 269
Review 270
Problems Plus 274
4 Applications of Differentiation 278
4.1Maximum and Minimum Values 279
Applied Project。The Calculus of Rainbows 288
4.2The Mean Value Theorem 290
4.3How Derivatives Affect the Shape of a Graph 296
4.4Indeterminate Forms and L’Hospital’s Rule 307
Writing Project。The Origins of L’Hospital’s Rule 315
4.5Summary of Curve Sketching 316
4.6Graphing 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
4.10Antiderivatives 353
Review 361
Problems PIus 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 Galculus 413
5.5 The Substitution Rule 414
5.6 The Logarithm Defined as an Integral 422
Review 430
Problems Plus 434
6 Applications of lntegration 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
7.1Integration by Parts 475
7.2Trigonometric Integrals 482
7.3Trigonometric Substitution 489
7.4Integration of Rational Functions by Partial Fractions 496
7.5Strategy for Integration 505
7.6Integration Using Tables and Computer Algebra Systems 511
Discovery Project 。 Patterns in Integrals 517
7.7Approximate Integration 518
7.8Improper Integrals 530
Review 540
Problems Plus 543
8 Further Applications of lntegration 546
8.1Arc Length 547
Discovery Project。Arc Length Contest 554
8.2Area of a Surface of Revolution 554
Discovery Project 。 Rotating on a 51ant 560
8.3Applications to Physics and Engineering 561
8.4Applications to Economics and Biology 571
8.5Probability 575
Review 582
Problems Plus 584
9 Differential Equations 586
9.1Modeling with Differential Equations 587
9.2Direction Fields and Euler’s Method 592
9.3Separable Equations 601
Applied Project。How Fast Does a Tank Drain? 609
Applied Project。Which Is Faster，Going Up or Coming Down? 610
9.4Exponential Growth and Decay 611
Applied Project。Calculus and Baseball 622
9.5The Logistic Equation 623
9.6Linear Equations 632
9.7Predator-Prey Systems 638
Review 644
Problems Plus 648
10 Parametric Equations and Poiar Coordinates 650
10.1Curves Defined by Parametric Equations 651
Laboratory Project。Running Circles around Circles 659
10.2Calculus with Parametric Curves 660
Laboratory Project。5ezier Curves 669
10.3Polar Coordinates 669
10.4Areas and Lengths in Polar Coordinates 679
10.5Conic Sections 684
10.6Conic Sections in Polar Coordinates 692
Review 696
Problems Plus 699
11 Infinite Sequences and Series 700
11.1Sequences 701
Laboratory Project。Logistic Sequences 713
11.2Series 713
11.3The Integral Test and Estimates of Sums 723
11.4The Comparison Tests 730
11.5Alternating Series 735
11.6Absolute Convergence and the Ratio and Root Tests 740
11.7Strategy for Testing Series 747
11.8Power Series 749
11.9Representations of Functions as Power Series 754
11.10Taylor and Maclaurin Series 760
Laboratory Project。An Elusive Limit 772
11.11The Binomial Series 772
Writing Project。How Newton Discovered the Binomial 5eriee 776
11.12Applications of Taylor Polynomials 776
Applied Project。Radiation from the Stars 785
Review 786
Problems Plue 789
12 Vectors and the Geometrq of Space 792
12.1Three-Dimensional Coordinate Systems 793
12.2Vectors 798
12.3The Dot Product 807
12.4The Cross Product 814
Discovery Project。The Geometry of a Tetrahedron 822
12.5Equations of Lines and Planes 822
Laboratory Project。Putting 3D in Perspective 832
12.6Cylinders and Quadric Surfaces 832
12.7Cylindrical and Spherical Coordinates 839
Laboratory Project。Families of Surfaces 844
Review 844
Problems Plus 847
13 Vector Functions 848
13.1Vector Functions and Space Curves 849
13.2Derivatives and Integrals of Vector Functions 856
13.3Arc Length and Curvature 862
13.4Motion in Space：Velocity and Acceleration 870
Applied Project。Kepler’s Laws 880
Review 881
Problems Plus 884
14 Partial DeriVatiVeS 886
14.1Functions of Several Variables 887
14.2Limits and Continuity 902
14.3Partial Derivatives 909
14.4Tangent Planes and Linear Approximations 923
14.5The Chain Rule 931
14.6Directional Derivatives and the Gradient Vector 940
14.7Maximum and Minimum Values 953
Applied Project。Designing a Dumpster 963
Discovery Project。Quadratic Approximations and Critical Points 964
14.8Lagrange Multipliers 965
Applied ProjectRocket Science 972
Applied ProjectHydro-Turbine Optimization 973
Review 974
Problems Plus 978
15 Multiple Integrals 980
15.1Double Integrals over Rectangles 981
15.2Iterated Integrals 989
15.3Double Integrals over General Regions 995
15.4Double Integrals in Polar Coordinates 1003
15.5Applications of Double Integrals 1009
15.6Surface Area 1019
15.7Triple Integrals 1023
Discovery Project。Volumes of Hyperspheres 1032
15.8Triple Integrals in Cylindrical and Spherical Coordinates 1033
Applied Project。Roller Derby 1039
Discovery Project。The Intersection of Three Cylinders 1040
15.9Change of Variables in Multiple Integrals 1041
Review 1049
Problems Plus 1052
16VeCtor CalCUlUS 1054
16.1Vector Fields 1055
16.2Line Integrals 1062
16.3The Fundamental Theorem for Line Integrals 1074
16.4Green’s Theorem 1083
16.5Curl and Divergence 1090
16.6Parametric Surfaces and Their Areas 1098
16.7Surface Integrals 1109
16.8Stokes’Theorem 1121
Writing Project。Three Men and Two Theorems 1126
16.9The Divergence Theorem 1127
16.10Summary 1134
Review 1135
Problems Plus 1138
17 Second-Drder Differential Equations 1140
17.1Second-Order Linear Equations 1141
17.2Nonhomogeneous Linear Equations 1147
17.3Applications of Second-Order Differential Equations 1155
17.4Series Solutions 1163
Review 1167
AppendixesAl 2
ANumbers，Inequalities，and Absolute ValuesA 2
BCoordinate Geometry and LinesA 10
CGraphs of Second-Degree EquationsA 16
DTrigonometryA 24
ESigma NotationA 34
FProofs of TheoremsA 39
GComplex NumbersA 49
HAnswers to Odd-Numbered ExercisesA 57
IndexA 125