Calculus

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
Review......Page 81
Principles of Problem Solving......Page 83
2 LIMITS......Page 89
2.1 The Tangent and Velocity Problems......Page 90
2.2 The Limit of a Function......Page 95
2.3 Calculating Limits Using the Limit Laws......Page 106
2.4 The Precise Definition of a Limit......Page 116
2.5 Continuity......Page 126
Review......Page 137
Problems Plus......Page 139
3 DERIVATIVES......Page 141
3.1 Derivatives and Rates of Change......Page 142
3.2 The Derivative as a Function......Page 152
3.3 Differentiation Formulas......Page 164
3.4 Derivatives of Trigonometric Functions......Page 177
3.5 The Chain Rule......Page 184
3.6 Implicit Differentiation......Page 193
3.7 Rates of Change In The Natural And Social Sciences......Page 199
3.8 Related Rates......Page 211
3.9 Linear Approximations and Differentials......Page 218
Laboratory Project: Taylor Polynomials......Page 224
Review......Page 225
Problems Plus......Page 229
4 APPLICATIONS OF DIFFERENTIATION......Page 233
4.1 Maximum and Minimum Values......Page 234
Applied Project: The Calculus of Rainbows......Page 242
4.2 The Mean Value Theorem......Page 243
4.3 How Derivatives Affect the Shape of a Graph......Page 249
4.4 Limits At Infinity; Horizontal Asymptotes......Page 259
4.5 Summary Of Curve Sketching......Page 272
4.6 Graphing With Calculus And Calculators......Page 279
4.7 Optimization Problems......Page 285
Applied Project: The Shape Of A Can......Page 297
4.8 Newton’s Method......Page 298
4.9 Antiderivatives......Page 303
Review......Page 310
Problems Plus......Page 314
5 INTEGRALS......Page 317
5.1 Areas and Distances......Page 318
5.2 The Definite Integral......Page 329
Discovery Project: Area Functions......Page 341
5.3 The Fundamental Theorem Of Calculus......Page 342
5.4 Indefinite Integrals and the Net Change Theorem......Page 353
Writing Project: Newton, Leibniz, and the Invention of Calculus......Page 361
5.5 The Substitution Rule......Page 362
Review......Page 369
Problems Plus......Page 373
6 APPLICATIONS OF INTEGRATION......Page 375
6.1 Areas Between Curves......Page 376
6.2 Volumes......Page 383
6.3 Volumes by Cylindrical Shells......Page 394
6.4 Work......Page 399
6.5 Average Value Of A Function......Page 403
Review......Page 407
Problems Plus......Page 409
7 INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS......Page 413
7.1 Inverse Functions......Page 414
7.2 Exponential Functions and Their Derivatives......Page 421
7.3 Logarithmic Functions......Page 434
7.4 Derivatives of Logarithmic Functions......Page 440
7.2* The Natural Logarithmic Function......Page 450
7.3* The Natural Exponential Function......Page 459
7.4* General Logarithmic and Exponential Functions......Page 467
7.5 Exponential Growth and Decay......Page 476
7.6 Inverse Trigonometric Functions......Page 483
7.7 Hyperbolic Functions......Page 492
7.8 Indeterminate Forms and L’Hospital’s Rule......Page 499
Writing Project: The Origins Of L’hospital’s Rule......Page 510
Review......Page 511
Problems Plus......Page 515
8 TECHNIQUES OF INTEGRATION......Page 517
8.1 Integration by Parts......Page 518
8.2 Trigonometric Integrals......Page 525
8.3 Trigonometric Substitution......Page 532
8.4 Integration of Rational Functions by Partial Fractions......Page 538
8.5 Strategy for Integration......Page 548
8.6 Integration Using Tables and Computer Algebra Systems......Page 554
Discovery Project: Patterns in Integrals......Page 559
8.7 Approximate Integration......Page 560
8.8 Improper Integrals......Page 573
Review......Page 583
Problems Plus......Page 586
9 FURTHER APPLICATIONS OF INTEGRATION......Page 589
9.1 Arc Length......Page 590
9.2 Area of A Surface of Revolution......Page 597
Discovery Project: Rotating on a Slant......Page 603
9.3 Applications to Physics and Engineering......Page 604
9.4 Applications to Economics and Biology......Page 615
9.5 Probability......Page 620
Review......Page 627
Problems Plus......Page 629
10 DIFFERENTIAL EQUATIONS......Page 631
10.1 Modeling With Differential Equations......Page 632
10.2 Direction Fields and Euler’s Method......Page 637
10.3 Separable Equations......Page 645
Applied Project: How Fast Does a Tank Drain?......Page 653
Applied Project: Which Is Faster, Going Up or Coming Down?......Page 655
10.4 Models for Population Growth......Page 656
Applied Project: Calculus and Baseball......Page 666
10.5 Linear Equations......Page 667
10.6 Predator-prey Systems......Page 673
Review......Page 679
Problems Plus......Page 683
11 PARAMETRIC EQUATIONS AND POLAR COORDINATES......Page 685
11.1 Curves Defined by Parametric Equations......Page 686
Laboratory Project: Running Circles Around Circles......Page 694
11.2 Calculus With Parametric Curves......Page 695
11.3 Polar Coordinates......Page 704
11.4 Areas and Lengths in Polar Coordinates......Page 715
11.5 Conic Sections......Page 719
11.6 Conic Sections in Polar Coordinates......Page 727
Review......Page 734
Problems Plus......Page 737
12 INFINITE SEQUENCES AND SERIES......Page 739
12.1 Sequences......Page 740
12.2 Series......Page 752
12.3 The Integral Test and Estimates of Sums......Page 762
12.4 The Comparison Tests......Page 770
12.5 Alternating Series......Page 775
12.6 Absolute Convergence and The Ratio and Root Tests......Page 779
12.7 Strategy for Testing Series......Page 786
12.8 Power Series......Page 788
12.9 Representations of Functions as Power Series......Page 793
12.10 Taylor and Maclaurin Series......Page 799
Writing Project: How Newton Discovered The Binomial Series......Page 813
12.11 Applications of Taylor Polynomials......Page 814
Applied Project: Radiation From The Stars......Page 822
Review......Page 823
Problems Plus......Page 826
13 VECTORS AND THE GEOMETRY OF SPACE......Page 829
13.1 Three-Dimensional Coordinate Systems......Page 830
13.2 Vectors......Page 835
13.3 The Dot Product......Page 844
13.4 The Cross Product......Page 851
13.5 Equations of Lines and Planes......Page 859
13.6 Cylinders and Quadric Surfaces......Page 869
Review......Page 877
Problems Plus......Page 880
14 VECTOR FUNCTIONS......Page 881
14.1 Vector Functions and Space Curves......Page 882
14.2 Derivatives and Integrals of Vector Functions......Page 889
14.3 Arc Length and Curvature......Page 895
14.4 Motion in Space: Velocity and Acceleration......Page 903
Applied Project: Kepler’s Laws......Page 913
Review......Page 914
Problems Plus......Page 917
15 PARTIAL DERIVATIVES......Page 919
15.1 Functions of Several Variables......Page 920
15.2 Limits and Continuity......Page 935
15.3 Partial Derivatives......Page 943
15.4 Tangent Planes and Linear Approximations......Page 957
15.5 The Chain Rule......Page 966
15.6 Directional Derivatives and the Gradient Vector......Page 975
15.7 Maximum and Minimum Values......Page 987
Discovery Project: Quadratic Approximations and Critical Points......Page 998
15.8 Lagrange Multipliers......Page 999
Applied Project: Rocket Science......Page 1006
Applied Project: Hydro-turbine Optimization......Page 1008
Review......Page 1009
Problems Plus......Page 1013
16 MULTIPLE INTEGRALS......Page 1015
16.1 Double Integrals over Rectangles......Page 1016
16.2 Iterated Integrals......Page 1024
16.3 Double Integrals over General Regions......Page 1030
16.4 Double Integrals in Polar Coordinates......Page 1039
16.5 Applications of Double Integrals......Page 1045
16.6 Triple Integrals......Page 1055
16.7 Triple Integrals In Cylindrical Coordinates......Page 1065
16.8 Triple Integrals in Spherical Coordinates......Page 1070
16.9 Change of Variables In Multiple Integrals......Page 1077
Review......Page 1086
Problems Plus......Page 1089
17 VECTOR CALCULUS......Page 1091
17.1 Vector Fields......Page 1092
17.2 Line Integrals......Page 1099
17.3 The Fundamental Theorem for Line Integrals......Page 1111
17.4 Green’s Theorem......Page 1120
17.5 Curl and Divergence......Page 1126
17.6 Parametric Surfaces and Their Areas......Page 1135
17.7 Surface Integrals......Page 1146
17.8 Stokes’ Theorem......Page 1157
Writing Project: Three Men and Two Theorems......Page 1163
17.9 The Divergence Theorem......Page 1164
17.10 Summary......Page 1170
Review......Page 1171
Problems Plus......Page 1174
18 SECOND-ORDER DIFFERENTIAL EQUATIONS......Page 1175
18.1 Second-Order Linear Equations......Page 1176
18.2 Nonhomogeneous Linear Equations......Page 1182
18.3 Applications of Second-Order Differential Equations......Page 1190
18.4 Series Solutions......Page 1198
Review......Page 1202
APPENDIXES......Page 1204
A: Numbers, Inequalities, and Absolute Values......Page 1205
B: Coordinate Geometry and Lines......Page 1213
C: Graphs of Second-degree Equations......Page 1219
D: Trigonometry......Page 1227
E: Sigma Notation......Page 1237
F: Proofs of Theorems......Page 1242
G: Complex Numbers......Page 1251
H: Answers To Odd-numbered Exercises......Page 1260
INDEX......Page 1328