Calculus: A Complete Course

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 16
To the Student......Page 18
To the Instructor......Page 19
Acknowledgments......Page 20
What Is Calculus?......Page 22
P.1. Real Numbers and the Real Line......Page 24
Intervals......Page 26
The Absolute Value......Page 29
Equations and Inequalities Involving Absolute Values......Page 30
Axis Scales......Page 32
Increments and Distances......Page 33
Straight Lines......Page 34
Equations of Lines......Page 36
Circles and Disks......Page 38
Equations of Parabolas......Page 40
Shifting a Graph......Page 41
Ellipses and Hyperbolas......Page 42
P.4. Functions and Their Graphs......Page 44
The Domain Convention......Page 46
Graphs of Functions......Page 47
Even and Odd Functions; Symmetry and Reflections......Page 49
Reflections in Straight Lines......Page 50
Defining and Graphing Functions with Maple......Page 51
Sums, Differences, Products, Quotients, and Multiples......Page 54
Composite Functions......Page 56
Piecewise Defined Functions......Page 57
P.6. Polynomials and Rational Functions......Page 60
Roots, Zeros, and Factors......Page 62
Roots and Factors of Quadratic Polynomials......Page 63
Miscellaneous Factorings......Page 65
P.7. The Trigonometric Functions......Page 67
Some Useful Identities......Page 69
Some Special Angles......Page 70
The Addition Formulas......Page 72
Other Trigonometric Functions......Page 74
Maple Calculations......Page 75
Trigonometry Review......Page 76
Average Velocity and Instantaneous Velocity......Page 80
The Growth of an Algal Culture......Page 82
The Area of a Circle......Page 83
1.2. Limits of Functions......Page 85
One-Sided Limits......Page 89
The Squeeze Theorem......Page 90
Limits at Infinity......Page 94
Limits at Infinity for Rational Functions......Page 95
Infinite Limits......Page 96
Using Maple to Calculate Limits......Page 98
Continuity at a Point......Page 100
There Are Lots of Continuous Functions......Page 102
Continuous Extensions and Removable Discontinuities......Page 103
Continuous Functions on Closed, Finite Intervals......Page 104
Finding Roots of Equations......Page 106
1.5. The Formal Definition of Limit......Page 109
Other Kinds of Limits......Page 111
Chapter Review......Page 114
2.1. Tangent Lines and Their Slopes......Page 116
Normals......Page 120
2.2. The Derivative......Page 121
Some Important Derivatives......Page 123
Leibniz Notation......Page 125
Differentials......Page 127
Derivatives Have the Intermediate-Value Property......Page 128
2.3. Differentiation Rules......Page 129
Sums and Constant Multiples......Page 130
The Product Rule......Page 131
The Reciprocal Rule......Page 133
The Quotient Rule......Page 134
2.4. The Chain Rule......Page 137
Building the Chain Rule into Differentiation Formulas......Page 140
Proof of the Chain Rule (Theorem 6)......Page 141
Some Special Limits......Page 142
The Derivatives of Sine and Cosine......Page 144
The Derivatives of the Other Trigonometric Functions......Page 146
2.6. Higher-Order Derivatives......Page 148
Approximating Small Changes......Page 152
Average and Instantaneous Rates of Change......Page 154
Sensitivity to Change......Page 155
Derivatives in Economics......Page 156
2.8. The Mean-Value Theorem......Page 159
Increasing and Decreasing Functions......Page 161
Proof of the Mean-Value Theorem......Page 163
2.9. Implicit Differentiation......Page 166
Higher-Order Derivatives......Page 169
The General Power Rule......Page 170
Antiderivatives......Page 171
The Indefinite Integral......Page 172
Differential Equations and Initial-Value Problems......Page 174
Velocity and Speed......Page 177
Acceleration......Page 178
Falling Under Gravity......Page 181
Chapter Review......Page 184
3.1. Inverse Functions......Page 187
Derivatives of Inverse Functions......Page 191
Exponentials......Page 193
Logarithms......Page 194
The Natural Logarithm......Page 197
The Exponential Function......Page 200
General Exponentials and Logarithms......Page 202
Logarithmic Differentiation......Page 203
The Growth of Exponentials and Logarithms......Page 206
Exponential Growth and Decay Models......Page 207
Interest on Investments......Page 209
Logistic Growth......Page 211
The Inverse Sine (or Arcsine) Function......Page 213
The Inverse Tangent (or Arctangent) Function......Page 216
Other Inverse Trigonometric Functions......Page 218
3.6. Hyperbolic Functions......Page 221
Inverse Hyperbolic Functions......Page 224
Recipe for Solving ay” + by’ + cy = 0......Page 227
Simple Harmonic Motion......Page 230
Damped Harmonic Motion......Page 233
Chapter Review......Page 234
4.1. Related Rates......Page 237
Procedures for Related-Rates Problems......Page 238
4.2. Finding Roots of Equations......Page 243
Discrete Maps and Fixed-Point Iteration......Page 244
Newton’s Method......Page 246
“Solve” Routines......Page 250
4.3. Indeterminate Forms......Page 251
l’H^opital’s Rules......Page 252
Maximum and Minimum Values......Page 257
Critical Points, Singular Points, and Endpoints......Page 258
The First Derivative Test......Page 259
Functions Not Defined on Closed, Finite Intervals......Page 261
4.5. Concavity and Inflections......Page 263
The Second Derivative Test......Page 266
4.6. Sketching the Graph of a Function......Page 268
Asymptotes......Page 269
Examples of Formal Curve Sketching......Page 272
Numerical Monsters and Computer Graphing......Page 277
Floating-Point Representation of Numbers in Computers......Page 278
Machine Epsilon and Its Effect on Figure 4.45......Page 280
Determining Machine Epsilon......Page 281
4.8. Extreme-Value Problems......Page 282
Procedure for Solving Extreme-Value Problems......Page 284
4.9. Linear Approximations......Page 290
Approximating Values of Functions......Page 291
Error Analysis......Page 292
4.10. Taylor Polynomials......Page 296
Taylor’s Formula......Page 298
Big-O Notation......Page 301
Evaluating Limits of Indeterminate Forms......Page 303
Taylor Polynomials in Maple......Page 305
Persistent Roundoff Error......Page 306
Truncation, Roundoff, and Computer Algebra......Page 307
Chapter Review......Page 308
5.1. Sums and Sigma Notation......Page 312
Evaluating Sums......Page 314
5.2. Areas as Limits of Sums......Page 317
The Basic Area Problem......Page 318
Some Area Calculations......Page 319
Partitions and Riemann Sums......Page 323
The Definite Integral......Page 324
General Riemann Sums......Page 326
5.4. Properties of the Definite Integral......Page 328
A Mean-Value Theorem for Integrals......Page 331
Definite Integrals of Piecewise Continuous Functions......Page 332
5.5. The Fundamental Theorem of Calculus......Page 334
5.6. The Method of Substitution......Page 340
Trigonometric Integrals......Page 344
5.7. Areas of Plane Regions......Page 348
Areas Between Two Curves......Page 349
Chapter Review......Page 352
6.1. Integration by Parts......Page 355
Reduction Formulas......Page 359
6.2. Integrals of Rational Functions......Page 361
Linear and Quadratic Denominators......Page 362
Partial Fractions......Page 364
Completing the Square......Page 366
Denominators with Repeated Factors......Page 367
The Inverse Trigonometric Substitutions......Page 370
Inverse Hyperbolic Substitutions......Page 373
Other Inverse Substitutions......Page 374
The tan( /2) Substitution......Page 375
6.4. Other Methods for Evaluating Integrals......Page 377
The Method of Undetermined Coefficients......Page 378
Using Maple for Integration......Page 380
Using Integral Tables......Page 381
Special Functions Arising from Integrals......Page 382
Improper Integrals of Type I......Page 384
Improper Integrals of Type II......Page 386
Estimating Convergence and Divergence......Page 389
6.6. The Trapezoid and Midpoint Rules......Page 392
The Trapezoid Rule......Page 393
The Midpoint Rule......Page 395
Error Estimates......Page 396
6.7. Simpson’s Rule......Page 399
6.8. Other Aspects of Approximate Integration......Page 403
Using Taylor’s Formula......Page 404
Romberg Integration......Page 405
The Importance of Higher-Order Methods......Page 408
Other Methods......Page 409
Chapter Review......Page 410
7.1. Volumes by Slicing—Solids of Revolution......Page 414
Volumes by Slicing......Page 415
Solids of Revolution......Page 416
Cylindrical Shells......Page 419
7.2. More Volumes by Slicing......Page 423
Arc Length......Page 427
The Arc Length of the Graph of a Function......Page 428
Areas of Surfaces of Revolution......Page 431
Mass and Density......Page 434
Moments and Centres of Mass......Page 437
Two- and Three-Dimensional Examples......Page 438
7.5. Centroids......Page 441
Pappus’s Theorem......Page 444
7.6. Other Physical Applications......Page 446
Hydrostatic Pressure......Page 447
Work......Page 448
Potential Energy and Kinetic Energy......Page 451
7.7. Applications in Business, Finance, and Ecology......Page 453
The Economics of Exploiting Renewable Resources......Page 454
7.8. Probability......Page 457
Discrete Random Variables......Page 458
Expectation, Mean, Variance, and Standard Deviation......Page 459
Continuous Random Variables......Page 461
The Normal Distribution......Page 465
Heavy Tails......Page 468
Separable Equations......Page 471
First-Order Linear Equations......Page 475
Chapter Review......Page 479
8.1. Conics......Page 483
Parabolas......Page 484
The Focal Property of a Parabola......Page 485
Ellipses......Page 486
The Focal Property of an Ellipse......Page 487
Hyperbolas......Page 488
The Focal Property of a Hyperbola......Page 490
Classifying General Conics......Page 491
8.2. Parametric Curves......Page 494
General Plane Curves and Parametrizations......Page 496
Some Interesting Plane Curves......Page 497
8.3. Smooth Parametric Curves and Their Slopes......Page 500
The Slope of a Parametric Curve......Page 501
Sketching Parametric Curves......Page 503
Arc Lengths and Surface Areas......Page 504
Areas Bounded by Parametric Curves......Page 506
8.5. Polar Coordinates and Polar Curves......Page 508
Some Polar Curves......Page 510
Polar Conics......Page 513
8.6. Slopes, Areas, and Arc Lengths for Polar Curves......Page 515
Areas Bounded by Polar Curves......Page 517
Arc Lengths for Polar Curves......Page 518
Chapter Review......Page 519
9.1. Sequences and Convergence......Page 521
Convergence of Sequences......Page 523
9.2. Infinite Series......Page 529
Geometric Series......Page 530
Telescoping Series and Harmonic Series......Page 532
Some Theorems About Series......Page 533
The Integral Test......Page 536
Using Integral Bounds to Estimate the Sum of a Series......Page 538
Comparison Tests......Page 539
The Ratio and Root Tests......Page 542
Using Geometric Bounds to Estimate the Sum of a Series......Page 544
9.4. Absolute and Conditional Convergence......Page 546
The Alternating Series Test......Page 547
Rearranging the Terms in a Series......Page 550
9.5. Power Series......Page 552
Algebraic Operations on Power Series......Page 555
Differentiation and Integration of Power Series......Page 557
Maple Calculations......Page 562
9.6. Taylor and Maclaurin Series......Page 563
Maclaurin Series for Some Elementary Functions......Page 564
Other Maclaurin and Taylor Series......Page 567
Taylor’s Formula Revisited......Page 570
Approximating the Values of Functions......Page 572
Indeterminate Forms......Page 574
9.8. The Binomial Theorem and Binomial Series......Page 576
The Binomial Series......Page 577
The Multinomial Theorem......Page 579
Periodic Functions......Page 581
Fourier Series......Page 582
Convergence of Fourier Series......Page 583
Fourier Cosine and Sine Series......Page 585
Chapter Review......Page 586
10. Vectors and Coordinate Geometry in 3-Space......Page 590
10.1. Analytic Geometry in Three Dimensions......Page 591
Describing Sets in the Plane, 3-Space, and n-Space......Page 594
10.2. Vectors......Page 596
Vectors in 3-Space......Page 598
Hanging Cables and Chains......Page 600
The Dot Product and Projections......Page 602
Vectors in n-Space......Page 604
10.3. The Cross Product in 3-Space......Page 606
Determinants......Page 608
The Cross Product as a Determinant......Page 610
Applications of Cross Products......Page 612
Planes in 3-Space......Page 614
Lines in 3-Space......Page 616
Distances......Page 618
10.5. Quadric Surfaces......Page 621
10.6. Cylindrical and Spherical Coordinates......Page 624
Cylindrical Coordinates......Page 625
Spherical Coordinates......Page 626
Matrices......Page 629
Determinants and Matrix Inverses......Page 631
Linear Equations......Page 634
Quadratic Forms, Eigenvalues, and Eigenvectors......Page 637
10.8. Using Maple for Vector and Matrix Calculations......Page 639
Vectors......Page 640
Matrices......Page 644
Linear Equations......Page 645
Eigenvalues and Eigenvectors......Page 646
Chapter Review......Page 648
11.1. Vector Functions of One Variable......Page 650
Differentiating Combinations of Vectors......Page 654
Motion Involving Varying Mass......Page 657
Circular Motion......Page 658
Rotating Frames and the Coriolis Effect......Page 659
11.3. Curves and Parametrizations......Page 664
Parametrizing the Curve of Intersection of Two Surfaces......Page 666
Arc Length......Page 667
The Arc-Length Parametrization......Page 669
The Unit Tangent Vector......Page 671
Curvature and the Unit Normal......Page 672
Torsion and Binormal, the Frenet-Serret Formulas......Page 675
11.5. Curvature and Torsion for General Parametrizations......Page 679
Tangential and Normal Acceleration......Page 681
Evolutes......Page 682
An Application to Track (or Road) Design......Page 683
Maple Calculations......Page 684
11.6. Kepler’s Laws of Planetary Motion......Page 686
Ellipses in Polar Coordinates......Page 687
Polar Components of Velocity and Acceleration......Page 689
Central Forces and Kepler’s Second Law......Page 690
Derivation of Kepler’s First and Third Laws......Page 691
Conservation of Energy......Page 693
Chapter Review......Page 695
12.1. Functions of Several Variables......Page 699
Graphs......Page 700
Level Curves......Page 701
Using Maple Graphics......Page 704
12.2. Limits and Continuity......Page 707
12.3. Partial Derivatives......Page 711
Tangent Planes and Normal Lines......Page 714
Distance from a Point to a Surface: A Geometric Example......Page 716
12.4. Higher-Order Derivatives......Page 718
The Laplace and Wave Equations......Page 721
12.5. The Chain Rule......Page 724
Higher-Order Derivatives......Page 729
12.6. Linear Approximations, Differentiability, and Differentials......Page 734
Proof of the Chain Rule......Page 736
Differentials......Page 737
Functions from n-Space to m-Space......Page 738
Differentials in Applications......Page 740
Differentials and Legendre Transformations......Page 742
12.7. Gradients and Directional Derivatives......Page 745
Directional Derivatives......Page 746
Rates Perceived by a Moving Observer......Page 750
The Gradient in Three and More Dimensions......Page 751
12.8. Implicit Functions......Page 755
Systems of Equations......Page 756
Choosing Dependent and Independent Variables......Page 758
The Implicit Function Theorem......Page 760
12.9. Taylor’s Formula, Taylor Series, and Approximations......Page 765
Approximating Implicit Functions......Page 769
Chapter Review......Page 771
13.1. Extreme Values......Page 773
Classifying Critical Points......Page 775
13.2. Extreme Values of Functions Defined on Restricted Domains......Page 781
Linear Programming......Page 784
The Method of Lagrange Multipliers......Page 787
Problems with More than One Constraint......Page 792
13.4. Lagrange Multipliers in n-Space......Page 795
Using Maple to Solve Constrained Extremal Problems......Page 800
Significance of Lagrange Multiplier Values......Page 802
Nonlinear Programming......Page 803
13.5. The Method of Least Squares......Page 804
Linear Regression......Page 806
Applications of the Least Squares Method to Integrals......Page 808
Differentiating Integrals with Parameters......Page 811
Envelopes......Page 815
Equations with Perturbations......Page 818
13.7. Newton’s Method......Page 820
Implementing Newton’s Method Using a Spreadsheet......Page 822
Solving Systems of Equations......Page 823
Finding and Classifying Critical Points......Page 825
Boltzmann Entropy......Page 828
Shannon Entropy......Page 829
Information Theory......Page 830
Chapter Review......Page 833
14.1. Double Integrals......Page 836
Properties of the Double Integral......Page 839
Double Integrals by Inspection......Page 840
14.2. Iteration of Double Integrals in Cartesian Coordinates......Page 842
Improper Integrals of Positive Functions......Page 849
A Mean-Value Theorem for Double Integrals......Page 851
14.4. Double Integrals in Polar Coordinates......Page 854
Change of Variables in Double Integrals......Page 858
14.5. Triple Integrals......Page 864
14.6. Change of Variables in Triple Integrals......Page 870
Cylindrical Coordinates......Page 871
Spherical Coordinates......Page 873
The Surface Area of a Graph......Page 877
The Gravitational Attraction of a Disk......Page 879
Moments and Centres of Mass......Page 880
Moment of Inertia......Page 882
Chapter Review......Page 886
15.1. Vector and Scalar Fields......Page 888
Field Lines (Integral Curves, Trajectories, Streamlines)......Page 890
Vector Fields in Polar Coordinates......Page 892
Nonlinear Systems and Liapunov Functions......Page 893
15.2. Conservative Fields......Page 895
Equipotential Surfaces and Curves......Page 897
Sources, Sinks, and Dipoles......Page 901
15.3. Line Integrals......Page 904
Evaluating Line Integrals......Page 905
15.4. Line Integrals of Vector Fields......Page 909
Connected and Simply Connected Domains......Page 911
Independence of Path......Page 912
15.5. Surfaces and Surface Integrals......Page 916
Parametric Surfaces......Page 917
Surface Integrals......Page 918
Smooth Surfaces, Normals, and Area Elements......Page 919
Evaluating Surface Integrals......Page 922
The Attraction of a Spherical Shell......Page 925
Oriented Surfaces......Page 928
The Flux of a Vector Field Across a Surface......Page 929
Calculating Flux Integrals......Page 931
Chapter Review......Page 933
16.1. Gradient, Divergence, and Curl......Page 935
Interpretation of the Divergence......Page 937
Distributions and Delta Functions......Page 939
Interpretation of the Curl......Page 941
16.2. Some Identities Involving Grad, Div, and Curl......Page 944
Scalar and Vector Potentials......Page 946
Maple Calculations......Page 948
16.3. Green’s Theorem in the Plane......Page 950
The Two-Dimensional Divergence Theorem......Page 953
16.4. The Divergence Theorem in 3-Space......Page 954
Variants of the Divergence Theorem......Page 958
16.5. Stokes’s Theorem......Page 960
Fluid Dynamics......Page 965
Electrostatics......Page 967
Magnetostatics......Page 968
Maxwell’s Equations......Page 970
16.7. Orthogonal Curvilinear Coordinates......Page 972
Coordinate Surfaces and Coordinate Curves......Page 974
Scale Factors and Differential Elements......Page 975
Grad, Div, and Curl in Orthogonal Curvilinear Coordinates......Page 979
Chapter Review......Page 982
Differentials and Vectors......Page 985
17.1. k-Forms......Page 986
Bilinear Forms and 2-Forms......Page 987
k-Forms......Page 989
Forms on a Vector Space......Page 991
17.2. Differential Forms and the Exterior Derivative......Page 992
The Exterior Derivative......Page 993
1-Forms and Legendre Transformations......Page 996
Closed and Exact Forms......Page 997
Smooth Manifolds......Page 999
Integration in n Dimensions......Page 1001
Parametrizing and Integrating over a Smooth Manifold......Page 1002
Oriented Manifolds......Page 1005
Pieces-with-Boundary of a Manifold......Page 1007
Integrating a Differential Form over a Manifold......Page 1010
17.5. The Generalized Stokes Theorem......Page 1012
Proof of Theorem 4 for a k-Cube......Page 1013
Completing the Proof......Page 1015
The Classical Theorems of Vector Calculus......Page 1016
18. Ordinary Differential Equations......Page 1020
18.1. Classifying Differential Equations......Page 1022
Separable Equations......Page 1025
First-Order Homogeneous Equations......Page 1026
Exact Equations......Page 1027
Integrating Factors......Page 1028
18.3. Existence, Uniqueness, and Numerical Methods......Page 1030
Existence and Uniqueness of Solutions......Page 1031
Numerical Methods......Page 1032
Equations Reducible to First Order......Page 1038
Second-Order Linear Equations......Page 1039
18.5. Linear Differential Equations with Constant Coefficients......Page 1041
Constant-Coefficient Equations of Higher Order......Page 1042
Euler (Equidimensional) Equations......Page 1044
18.6. Nonhomogeneous Linear Equations......Page 1046
Resonance......Page 1049
Variation of Parameters......Page 1050
Maple Calculations......Page 1052
18.7. The Laplace Transform......Page 1053
Some Basic Laplace Transforms......Page 1055
More Properties of Laplace Transforms......Page 1056
The Heaviside Function and the Dirac Delta Function......Page 1058
18.8. Series Solutions of Differential Equations......Page 1062
18.9. Dynamical Systems, Phase Space, and the Phase Plane......Page 1066
A Differential Equation as a First-Order System......Page 1067
Existence, Uniqueness, and Autonomous Systems......Page 1068
Second-Order Autonomous Equations and the Phase Plane......Page 1069
Fixed Points......Page 1071
Linear Systems, Eigenvalues, and Fixed Points......Page 1072
Implications for Nonlinear Systems......Page 1075
Predator–Prey Models......Page 1077
Chapter Review......Page 1080
Appendix I: Complex Numbers......Page 1082
Graphical Representation of Complex Numbers......Page 1083
Complex Arithmetic......Page 1085
Roots of Complex Numbers......Page 1089
Appendix II: Complex Functions......Page 1092
Limits and Continuity......Page 1093
The Complex Derivative......Page 1094
The Exponential Function......Page 1096
The Fundamental Theorem of Algebra......Page 1098
Limits of Functions......Page 1102
Continuous Functions......Page 1103
Completeness and Sequential Limits......Page 1104
Continuous Functions on a Closed, Finite Interval......Page 1105
Appendix IV: The Riemann Integral......Page 1108
Uniform Continuity......Page 1111
Appendix V: Doing Calculus with Maple......Page 1114
List of Maple Examples and Discussion......Page 1115
Answers to Odd-Numbered Exercises......Page 1116
Index......Page 1158
Back Cover......Page 1176