Calculus : early transcendentals

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 4
Preface......Page 12
To the Student......Page 24
Diagnostic Tests......Page 25
A PREVIEW OF CALCULUS......Page 31
1 FUNCTIONS AND MODELS......Page 39
1.1 Four Ways to Represent a Function......Page 40
1.2 Mathematical Models: A Catalog of Essential Functions......Page 53
1.3 New Functions from Old Functions......Page 66
1.4 Graphing Calculators and Computers......Page 75
1.5 Exponential Functions......Page 81
1.6 Inverse Functions and Logarithms......Page 88
Review......Page 102
Principles of Problem Solving......Page 105
2 LIMITS AND DERIVATIVES......Page 111
2.1 The Tangent and Velocity Problems......Page 112
2.2 The Limit of a Function......Page 117
2.3 Calculating Limits Using the Limit Laws......Page 128
2.4 The Precise Definition of a Limit......Page 138
2.5 Continuity......Page 148
2.6 Limits at Infinity; Horizontal Asymptotes......Page 159
2.7 Derivatives and Rates of Change......Page 172
Writing Project: Early Methods for Finding Tangents......Page 182
2.8 The Derivative as a Function......Page 183
Review......Page 194
Problems Plus......Page 199
3 DIFFERENTIATION RULES......Page 201
3.1 Derivatives of Polynomials and Exponential Functions......Page 202
Applied Project: Building a Better Roller Coaster......Page 211
3.2 The Product and Quotient Rules......Page 212
3.3 Derivatives of Trigonometric Functions......Page 218
3.4 The Chain Rule......Page 226
Applied Project: Where Should a Pilot Start Descent?......Page 235
3.5 Implicit Differentiation......Page 236
3.6 Derivatives of Logarithmic Functions......Page 244
3.7 Rates of Change in the Natural and Social Sciences......Page 250
3.8 Exponential Growth and Decay......Page 262
3.9 Related Rates......Page 270
3.10 Linear Approximations and Differentials......Page 276
Laboratory Project: Taylor Polynomials......Page 282
3.11 Hyperbolic Functions......Page 283
Review......Page 290
Problems Plus......Page 294
4 APPLICATIONS OF DIFFERENTIATION......Page 299
4.1 Maximum and Minimum Values......Page 300
Applied Project: The Calculus of Rainbows......Page 308
4.2 The Mean Value Theorem......Page 309
4.3 How Derivatives Affect the Shape of a Graph......Page 316
4.4 Indeterminate Forms and L\'Hospital\'s Rule......Page 327
4.5 Summary of Curve Sketching......Page 336
4.6 Graphing with Calculus and Calculators......Page 344
4.7 Optimization Problems......Page 351
Applied Project: The Shape of a Can......Page 362
4.8 Newton\'s Method......Page 363
4.9 Antiderivatives......Page 369
Review......Page 376
Problems Plus......Page 380
5 INTEGRALS......Page 383
5.1 Areas and Distances......Page 384
5.2 The Definite Integral......Page 395
5.3 The Fundamental Theorem of Calculus......Page 408
5.4 Indefinite Integrals and the Net Change Theorem......Page 420
Writing Project: Newton, Leibniz, and the Invention of Calculus......Page 428
5.5 The Substitution Rule......Page 429
Review......Page 437
Problems Plus......Page 441
6 INTEGRALS......Page 443
6.1 Areas between Curves......Page 444
6.2 Volumes......Page 451
6.3 Volumes by Cylindrical Shells......Page 462
6.4 Work......Page 467
6.5 Average Value of a Function......Page 471
Review......Page 475
Problems Plus......Page 477
7 TECHNIQUES OF INTEGRATION......Page 481
7.1 Integration by Parts......Page 482
7.2 Trigonometric Integrals......Page 489
7.3 Trigonometric Substitution......Page 496
7.4 Integration of Rational Functions by Partial Fractions......Page 502
7.5 Strategy for Integration......Page 512
7.6 Integration Using Tables and Computer Algebra Systems......Page 518
Discovery Project: Patterns in Integrals......Page 523
7.7 Approximate Integration......Page 524
7.8 Improper Integrals......Page 537
Review......Page 547
Problems Plus......Page 550
8 FURTHER APPLICATIONS OF INTEGRATION......Page 553
8.1 Are Length......Page 554
8.2 Area of a Surface of Revolution......Page 561
Discovery Project: Rotating on a Slant......Page 567
8.3 Applications to Physics and Engineering......Page 568
8.4 Applications to Economics and Biology......Page 579
8.5 Probability......Page 584
Review......Page 591
Problems Plus......Page 593
9 DIFFERENTIAL EQUATIONS......Page 595
9.1 Modeling with Differential Equations......Page 596
9.2 Direction Fields and Euler\'s Method......Page 601
9.3 Separable Equations......Page 609
Applied Project: How Fast Does a Tank Drain?......Page 617
Applied Project: Which Is Faster, Going Up or Coming Down?......Page 619
9.4 Models for Population Growth......Page 620
Applied Project: Calculus and Baseball......Page 630
9.5 Linear Equations......Page 631
9.6 Predator-Prey Systems......Page 637
Review......Page 643
Problems Plus......Page 647
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES......Page 649
10.1 Curves Defined by Parametric Equations......Page 650
Laboratory Project: Running Circles around Circles......Page 658
10.2 Calculus with Parametric Curves......Page 659
10.3 Polar Coordinates......Page 668
10.4 Areas and Lengths in Polar Coordinates......Page 679
10.5 Conic Sections......Page 683
10.6 Conic Sections in Polar Coordinates......Page 691
Review......Page 698
Problems Plus......Page 701
11 INFINITE SEQUENCES AND SERIES......Page 703
11.1 Sequences......Page 704
11.2 Series......Page 716
11.3 The Integral Test and Estimates of Sums......Page 726
11.4 The Comparison Tests......Page 734
11.5 Alternating Series......Page 739
11.6 Absolute Convergence and the Ratio and Root Tests......Page 743
11.7 Strategy for Testing Series......Page 750
11.8 Power Series......Page 752
11.9 Representations of Functions as Power Series......Page 757
11.10 Taylor and Maclaurin Series......Page 763
Writing Project: How Newton Discovered the Binomial Series......Page 777
11.11 Applications of Taylor Polynomials......Page 778
Applied Project: Radiation from the Stars......Page 786
Review......Page 787
Problems Plus......Page 790
12 VECTORS AND THE GEOMETRY OF SPACE......Page 793
12.1 Three-Dimensional Coordinate Systems......Page 794
12.2 Vectors......Page 799
12.3 The Dot Product......Page 808
12.4 The Cross Product......Page 815
12.5 Equations of Lines and Planes......Page 823
12.6 Cylinders and Quadric Surfaces......Page 833
Review......Page 841
Problems Plus......Page 844
13 VECTOR FUNCTIONS......Page 845
13.1 Vector Functions and Space Curves......Page 846
13.2 Derivatives and Integrals of Vector Functions......Page 853
13.3 Arc Length and Curvature......Page 859
13.4 Motion in Space: Velocity and Acceleration......Page 867
Applied Project: Kepler\'s Laws......Page 877
Review......Page 878
Problems Plus......Page 881
14 PARTIAL DERIVATIVES......Page 883
14.1 Functions of Several Variables......Page 884
14.2 Limits and Continuity......Page 899
14.3 Partial Derivatives......Page 907
14.4 Tangent Planes and Linear Approximations......Page 921
14.5 The Chain Rule......Page 930
14.6 Directional Derivatives and the Gradient Vector......Page 939
14.7 Maximum and Minimum Values......Page 951
Discovery Project: Quadratic Approximations and Critical Points......Page 962
14.8 Lagrange Multipliers......Page 963
Applied Project: Rocket Science......Page 970
Applied Project: Hydro-Turbine Optimization......Page 972
Review......Page 973
Problems Plus......Page 977
15 MULTIPLE INTEGRALS......Page 979
15.1 Double Integrals over Rectangles......Page 980
15.2 Iterated Integrals......Page 988
15.3 Double Integrals over General Regions......Page 994
15.4 Double Integrals in Polar Coordinates......Page 1003
15.5 Applications of Double Integrals......Page 1009
15.6 Triple Integrals......Page 1019
15.7 Triple Integrals in Cylindrical Coordinates......Page 1029
Discovery Project: The Intersection of Three Cylinders......Page 1034
15.9 Change of Variables in Multiple Integrals......Page 1041
Review......Page 1050
Problems Plus......Page 1053
16 VECTOR CALCULUS......Page 1055
16.1 Vector Fields......Page 1056
16.2 Line Integrals......Page 1063
16.3 The Fundamental Theorem for Line Integrals......Page 1075
16.4 Green\'s Theorem......Page 1084
16.5 Curl and Divergence......Page 1090
16.6 Parametric Surfaces and Their Areas......Page 1099
16.7 Surface Integrals......Page 1110
16.8 Stokes\' Theorem......Page 1121
Writing Project: Three Men and Two Theorems......Page 1127
16.9 The Divergence Theorem......Page 1128
16.10 Summary......Page 1134
Review......Page 1135
Problems Plus......Page 1138
17 SECOND-ORDER DIFFERENTIAL EQUATIONS......Page 1139
17.1 Second-Order Linear Equations......Page 1140
17.2 Nonhomogeneous Linear Equations......Page 1146
17.3 Applications of Second-Order Differential Equations......Page 1154
17.4 Series Solutions......Page 1162
Review......Page 1166
APPENDIXES......Page 1168
A: Numbers, Inequalities, and Absolute Values......Page 1169
B: Coordinate Geometry and Lines......Page 1177
C: Graphs of Second-Degree Equations......Page 1183
D: Trigonometry......Page 1191
E: Sigma Notation......Page 1201
F: Proofs of Theorems......Page 1206
G: The Logarithm Defined as an Integral......Page 1217
H: Complex Numbers......Page 1224
I: Answers to Odd-Numbered Exercises......Page 1232
INDEX......Page 1298