Calculus: Concepts and Contexts

Front Cover......Page 1
Title Page......Page 9
Copyright......Page 10
Contents......Page 11
Preface......Page 17
To the Student......Page 27
Diagnostic Tests......Page 28
A Preview of Calculus......Page 35
1 Functions and Models......Page 43
1.1 Four Ways to Represent a Function......Page 44
1.2 Mathematical Models: A Catalog of Essential Functions......Page 57
1.3 New Functions from Old Functions......Page 69
1.4 Graphing Calculators and Computers......Page 78
1.5 Exponential Functions......Page 84
1.6 Inverse Functions and Logarithms......Page 93
1.7 Parametric Curves......Page 103
Laboratory Project Running Circles Around Circles......Page 111
Review......Page 112
Principles of Problem Solving......Page 115
2 Limits and Derivatives......Page 121
2.1 The Tangent and Velocity Problems......Page 122
2.2 The Limit of a Function......Page 127
2.3 Calculating Limits Using the Limit Laws......Page 136
2.4 Continuity......Page 145
2.5 Limits Involving Infinity......Page 155
2.6 Derivatives and Rates of Change......Page 167
Writing Project Early Methods for Finding Tangents......Page 177
2.7 The Derivative as a Function......Page 178
2.8 What Does f\' Say about f?......Page 190
Review......Page 196
Focus on Problem Solving......Page 201
3 Differentiation Rules......Page 205
3.1 Derivatives of Polynomials and Exponential Functions......Page 206
3.2 The Product and Quotient Rules......Page 215
3.3 Derivatives of Trigonometric Functions......Page 222
3.4 The Chain Rule......Page 229
Laboratory Project Bézier Curves......Page 240
3.5 Implicit Differentiation......Page 241
3.6 Inverse Trigonometric Functions and Their Derivatives......Page 248
3.7 Derivatives of Logarithmic Functions......Page 253
Discovery Project Hyperbolic Functions......Page 259
3.8 Rates of Change in the Natural and Social Sciences......Page 260
3.9 Linear Approximations and Differentials......Page 272
Laboratory Project Taylor Polynomials......Page 279
Review......Page 280
Focus on Problem Solving......Page 283
4 Applications of Differentiation......Page 287
4.1 Related Rates......Page 288
4.2 Maximum and Minimum Values......Page 294
Applied Project The Calculus of Rainbows......Page 302
4.3 Derivatives and the Shapes of Curves......Page 303
4.4 Graphing with Calculus and Calculators......Page 314
4.5 Indeterminate Forms and l’Hospital’s Rule......Page 322
4.6 Optimization Problems......Page 331
Applied Project The Shape of a Can......Page 343
4.7 Newton’s Method......Page 344
4.8 Antiderivatives......Page 349
Review......Page 355
Focus on Problem Solving......Page 359
5 Integrals......Page 363
5.1 Areas and Distances......Page 364
5.2 The Definite Integral......Page 375
5.3 Evaluating Definite Integrals......Page 388
Discovery Project Area Functions......Page 398
5.4 The Fundamental Theorem of Calculus......Page 399
Writing Project Newton, Leibniz, and the Invention of Calculus......Page 406
5.5 The Substitution Rule......Page 407
5.6 Integration by Parts......Page 415
5.7 Additional Techniques of Integration......Page 421
5.8 Integration Using Tables and Computer Algebra Systems......Page 426
Discovery Project Patterns in Integrals......Page 432
5.9 Approximate Integration......Page 433
5.10 Improper Integrals......Page 445
Review......Page 455
Focus on Problem Solving......Page 460
6 Applications of Integration......Page 463
6.1 More About Areas......Page 464
6.2 Volumes......Page 470
Discovery Project Rotating on a Slant......Page 480
6.3 Volumes by Cylindrical Shells......Page 481
6.4 Arc Length......Page 487
6.5 Average Value of a Function......Page 492
6.6 Applications to Physics and Engineering......Page 496
Discovery Project Complementary Coffee Cups......Page 507
6.7 Applications to Economics and Biology......Page 508
6.8 Probability......Page 512
Review......Page 519
Focus on Problem Solving......Page 523
7 Differential Equations......Page 525
7.1 Modeling with Differential Equations......Page 526
7.2 Direction Fields and Euler’s Method......Page 531
7.3 Separable Equations......Page 540
Applied Project How Fast Does a Tank Drain?......Page 549
Applied Project Which Is Faster, Going Up or Coming Down?......Page 550
7.4 Exponential Growth and Decay......Page 551
Applied Project Calculus and Baseball......Page 561
7.5 The Logistic Equation......Page 562
7.6 Predator-Prey Systems......Page 572
Review......Page 579
Focus on Problem Solving......Page 583
8 Infinite Sequences and Series......Page 585
8.1 Sequences......Page 586
Laboratory Project Logistic Sequences......Page 596
8.2 Series......Page 597
8.3 The Integral and Comparison Tests; Estimating Sums......Page 607
8.4 Other Convergence Tests......Page 617
8.5 Power Series......Page 624
8.6 Representations of Functions as Power Series......Page 630
8.7 Taylor and Maclaurin Series......Page 636
Writing Project How Newton Discovered the Binomial Series......Page 650
8.8 Applications of Taylor Polynomials......Page 651
Applied Project Radiation from the Stars......Page 659
Review......Page 660
Focus on Problem Solving......Page 663
9 Vectors and the Geometry of Space......Page 665
9.1 Three-Dimensional Coordinate Systems......Page 666
9.2 Vectors......Page 671
9.3 The Dot Product......Page 680
9.4 The Cross Product......Page 686
Discovery Project The Geometry of a Tetrahedron......Page 694
9.5 Equations of Lines and Planes......Page 695
Laboratory Project Putting 3D in Perspective......Page 704
9.6 Functions and Surfaces......Page 705
9.7 Cylindrical and Spherical Coordinates......Page 714
Laboratory Project Families of Surfaces......Page 719
Review......Page 720
Focus on Problem Solving......Page 723
10 Vector Functions......Page 725
10.1 Vector Functions and Space Curves......Page 726
10.2 Derivatives and Integrals of Vector Functions......Page 733
10.3 Arc Length and Curvature......Page 739
10.4 Motion in Space: Velocity and Acceleration......Page 748
Applied Project Kepler’s Laws......Page 758
10.5 Parametric Surfaces......Page 759
Review......Page 765
Focus on Problem Solving......Page 767
11 Partial Derivatives......Page 769
11.1 Functions of Several Variables......Page 770
11.2 Limits and Continuity......Page 781
11.3 Partial Derivatives......Page 788
11.4 Tangent Planes and Linear Approximations......Page 802
11.5 The Chain Rule......Page 812
11.6 Directional Derivatives and the Gradient Vector......Page 821
11.7 Maximum and Minimum Values......Page 834
Applied Project Designing a Dumpster......Page 843
Discovery Project Quadratic Approximations and Critical Points......Page 844
11.8 Lagrange Multipliers......Page 845
Applied Project Rocket Science......Page 852
Applied Project Hydro-Turbine Optimization......Page 853
Review......Page 854
Focus on Problem Solving......Page 859
12 Multiple Integrals......Page 861
12.1 Double Integrals over Rectangles......Page 862
12.2 Iterated Integrals......Page 870
12.3 Double Integrals over General Regions......Page 876
12.4 Double Integrals in Polar Coordinates......Page 885
12.5 Applications of Double Integrals......Page 890
12.6 Surface Area......Page 900
12.7 Triple Integrals......Page 905
12.8 Triple Integrals in Cylindrical and Spherical Coordinates......Page 915
Applied Project Roller Derby......Page 921
Discovery Project The Intersection of Three Cylinders......Page 922
12.9 Change of Variables in Multiple Integrals......Page 923
Review......Page 931
Focus on Problem Solving......Page 935
13 Vector Calculus......Page 937
13.1 Vector Fields......Page 938
13.2 Line Integrals......Page 945
13.3 The Fundamental Theorem for Line Integrals......Page 957
13.4 Green’s Theorem......Page 966
13.5 Curl and Divergence......Page 973
13.6 Surface Integrals......Page 981
13.7 Stokes’ Theorem......Page 992
Writing Project Three Men and Two Theorems......Page 998
13.8 The Divergence Theorem......Page 999
13.9 Summary......Page 1005
Review......Page 1006
Focus on Problem Solving......Page 1009
Appendixes......Page 1011
A: Intervals, Inequalities, and Absolute Values......Page 1012
B: Coordinate Geometry......Page 1017
C: Trigonometry......Page 1027
D: Precise Definitions of Limits......Page 1036
E: A Few Proofs......Page 1046
F: Sigma Notation......Page 1051
G: Integration of Rational Functions by Partial Fractions......Page 1057
H: Polar Coordinates......Page 1065
I: Complex Numbers......Page 1081
J: Answers to Odd-Numbered Exercises......Page 1090
Index......Page 1145