Calculus: Early Transcendentals

Contents......Page 5
Preface......Page 13
To the Student......Page 25
Calculators, Computers, and Other Graphing Devices......Page 26
Diagnostic Tests......Page 28
A Preview of Calculus......Page 33
Ch 1: Functions and Models......Page 41
1.1: Four Ways to Represent a Function......Page 42
1.2: Mathematical Models: A Catalog of Essential Functions......Page 55
1.3: New Functions from Old Functions......Page 68
1.4: Exponential Functions......Page 77
1.5: Inverse Functions and Logarithms......Page 87
Review......Page 100
Principles of Problem Solving......Page 103
Ch 2: Limits and Derivatives......Page 109
2.1: The Tangent and Velocity Problems......Page 110
2.2: The Limit of a Function......Page 115
2.3: Calculating Limits Using the Limit Laws......Page 127
2.4: The Precise Definition of a Limit......Page 136
2.5: Continuity......Page 146
2.6: Limits at Infinity; Horizontal Asymptotes......Page 158
2.7: Derivatives and Rates of Change......Page 172
2.8: The Derivative as a Function......Page 184
Review......Page 197
Problems Plus......Page 201
Ch 3: Differentiation Rules......Page 203
3.1: Derivatives of Polynomials and Exponential Functions......Page 204
3.2: The Product and Quotient Rules......Page 215
3.3: Derivatives of Trigonometric Functions......Page 222
3.4: The Chain Rule......Page 229
3.5: Implicit Differentiation......Page 240
3.6: Derivatives of Logarithmic Functions......Page 250
3.7: Rates of Change in the Natural and Social Sciences......Page 256
3.8: Exponential Growth and Decay......Page 269
3.9: Related Rates......Page 277
3.10: Linear Approximations and Differentials......Page 283
3.11: Hyperbolic Functions......Page 291
Review......Page 298
Problems Plus......Page 302
Ch 4: Applications of Differentiation......Page 307
4.1: Maximum and Minimum Values......Page 308
4.2: The Mean Value Theorem......Page 319
4.3: How Derivatives Affect the Shape of a Graph......Page 325
4.4: Indeterminate Forms and L'Hospital's Rule......Page 336
4.5: Summary of Curve Sketching......Page 347
4.6: Graphing with Calculus and Calculators......Page 355
4.7: Optimization Problems......Page 362
4.8: Newton's Method......Page 377
4.9: Antiderivatives......Page 382
Review......Page 390
Problems Plus......Page 395
Ch 5: Integrals......Page 397
5.1: Areas and Distances......Page 398
5.2: The Definite Integral......Page 410
5.3: The Fundamental Theorem of Calculus......Page 424
5.4: Indefinite Integrals and the Net Change Theorem......Page 434
5.5: The Substitution Rule......Page 444
Review......Page 453
Problems Plus......Page 457
Ch 6: Applications of Integration......Page 459
6.1: Areas between Curves......Page 460
6.2: Volumes......Page 470
6.3: Volumes by Cylindrical Shells......Page 481
6.4: Work......Page 487
6.5: Average Value of a Function......Page 493
Review......Page 498
Problems Plus......Page 500
Ch 7: Techniques of Integration......Page 503
7.1: Integration by Parts......Page 504
7.2: Trigonometric Integrals......Page 511
7.3: Trigonometric Substitution......Page 518
7.4: Integration of Rational Functions by Partial Fractions......Page 525
7.5: Strategy for Integration......Page 535
7.6: Integration Using Tables and Computer Algebra Systems......Page 540
7.7: Approximate Integration......Page 546
7.8: Improper Integrals......Page 559
Review......Page 569
Problems Plus......Page 572
Ch 8: Further Applications of Integration......Page 575
8.1: Arc Length......Page 576
8.2: Area of a Surface of Revolution......Page 583
8.3: Applications to Physics and Engineering......Page 590
8.4: Applications to Economics and Biology......Page 601
8.5: Probability......Page 605
Review......Page 613
Problems Plus......Page 615
Ch 9: Differential Equations......Page 617
9.1: Modeling with Differential Equations......Page 618
9.2: Direction Fields and Euler's Method......Page 623
9.3: Separable Equations......Page 631
9.4: Models for Population Growth......Page 642
9.5: Linear Equations......Page 652
9.6: Predator-Prey Systems......Page 659
Review......Page 666
Problems Plus......Page 669
Ch 10: Parametric Equations and Polar Coordinates......Page 671
10.1: Curves Defined by Parametric Equations......Page 672
10.2: Calculus with Parametric Curves......Page 681
10.3: Polar Coordinates......Page 690
10.4: Areas and Lengths in Polar Coordinates......Page 701
10.5: Conic Sections......Page 706
10.6: Conic Sections in Polar Coordinates......Page 714
Review......Page 721
Problems Plus......Page 724
Ch 11: Infinite Sequences and Series......Page 725
11.1: Sequences......Page 726
11.2: Series......Page 739
11.3: The Integral Test and Estimates of Sums......Page 751
11.4: The Comparison Tests......Page 759
11.5: Alternating Series......Page 764
11.6: Absolute Convergence and the Ratio and Root Tests......Page 769
11.7: Strategy for Testing Series......Page 776
11.8: Power Series......Page 778
11.9: Representations of Functions as Power Series......Page 784
11.10: Taylor and Maclaurin Series......Page 791
11.11: Applications of Taylor Polynomials......Page 806
Review......Page 816
Problems Plus......Page 819
Ch 12: Vectors and the Geometry of Space......Page 823
12.1: Three-Dimensional Coordinate Systems......Page 824
12.2: Vectors......Page 830
12.3: The Dot Product......Page 839
12.4: The Cross Product......Page 846
12.5: Equations of Lines and Planes......Page 855
12.6: Cylinders and Quadric Surfaces......Page 866
Review......Page 873
Problems Plus......Page 876
Ch 13: Vector Functions......Page 879
13.1: Vector Functions and Space Curves......Page 880
13.2: Derivatives and Integrals of Vector Functions......Page 887
13.3: Arc Length and Curvature......Page 893
13.4: Motion in Space: Velocity and Acceleration......Page 902
Review......Page 913
Problems Plus......Page 916
Ch 14: Partial Derivatives......Page 919
14.1: Functions of Several Variables......Page 920
14.2: Limits and Continuity......Page 935
14.3: Partial Derivatives......Page 943
14.4: Tangent Planes and Linear Approximations......Page 959
14.5: The Chain Rule......Page 969
14.6: Directional Derivatives and the Gradient Vector......Page 978
14.7: Maximum and Minimum Values......Page 991
14.8: Lagrange Multipliers......Page 1003
Review......Page 1013
Problems Plus......Page 1017
Ch 15: Multiple Integrals......Page 1019
15.1: Double Integrals over Rectangles......Page 1020
15.2: Double Integrals over General Regions......Page 1033
15.3: Double Integrals in Polar Coordinates......Page 1042
15.4: Applications of Double Integrals......Page 1048
15.5: Surface Area......Page 1058
15.6: Triple Integrals......Page 1061
15.7: Triple Integrals in Cylindrical Coordinates......Page 1072
15.8: Triple Integrals in Spherical Coordinates......Page 1077
15.9: Change of Variables in Multiple Integrals......Page 1084
Review......Page 1093
Problems Plus......Page 1097
Ch 16: Vector Calculus......Page 1099
16.1: Vector Fields......Page 1100
16.2: Line Integrals......Page 1107
16.3: The Fundamental Theorem for Line Integrals......Page 1119
16.4: Green's Theorem......Page 1128
16.5: Curl and Divergence......Page 1135
16.6: Parametric Surfaces and Their Areas......Page 1143
16.7: Surface Integrals......Page 1154
16.8: Stokes' Theorem......Page 1166
16.9: The Divergence Theorem......Page 1173
16.10: Summary......Page 1179
Review......Page 1180
Problems Plus......Page 1183
Ch 17: Second-Order Differential Equations......Page 1185
17.1: Second-Order Linear Equations......Page 1186
17.2: Nonhomogeneous Linear Equations......Page 1192
17.3: Applications of Second-Order Differential Equations......Page 1200
17.4: Series Solutions......Page 1208
Review......Page 1213
Appendixes......Page 1215
Appendix A: Numbers, Inequalities, and Absolute Values......Page 1216
Appendix B: Coordinate Geometry and Lines......Page 1224
Appendix C: Graphs of Second-Degree Equations......Page 1230
Appendix D: Trigonometry......Page 1238
Appendix E: Sigma Notation......Page 1248
Appendix F: Proofs of Theorems......Page 1253
Appendix G: The Logarithm Defined as an Integral......Page 1264
Appendix H: Complex Numbers......Page 1271
Appendix I: Answers to Odd-Numbered Exercises......Page 1279
Index......Page 1353
Concept Check Answers......Page 1371