Calculus, 4th Edition

Cover......Page 1
Title Page......Page 3
Copyright......Page 4
DEDICATION......Page 5
Brief Table of Contents......Page 6
Table of Contents......Page 8
Seeing the Beauty and Power of Mathematics......Page 15
Applications Index......Page 26
CHAPTER 0 Preliminaries......Page 29
The Real Number System and Inequalities......Page 30
The Cartesian Plane......Page 34
Equations of Lines......Page 37
Functions......Page 40
0.3 Graphing Calculators and Computer Algebra Systems......Page 49
0.4 Trigonometric Functions......Page 55
0.5 Transformations of Functions......Page 64
1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve......Page 75
1.2 The Concept of Limit......Page 80
1.3 Computation of Limits......Page 87
1.4 Continuity and Its Consequences......Page 96
The Method of Bisections......Page 102
1.5 Limits Involving Infinity; Asymptotes......Page 106
Limits at Infinity......Page 109
1.6 Formal Definition of the Limit......Page 115
Exploring the Definition of Limit Graphically......Page 119
Limits Involving Infinity......Page 121
1.7 Limits and Loss-of-Significance Errors......Page 126
Computer Representation of Real Numbers......Page 127
2.1 Tangent Lines and Velocity......Page 135
The General Case......Page 137
Velocity......Page 139
2.2 The Derivative......Page 146
Alternative Derivative Notations......Page 149
Numerical Differentiation......Page 151
The Power Rule......Page 155
General Derivative Rules......Page 158
Higher Order Derivatives......Page 159
Acceleration......Page 160
Product Rule......Page 163
Quotient Rule......Page 165
Applications......Page 167
2.5 The Chain Rule......Page 170
2.6 Derivatives of Trigonometric Functions......Page 175
Applications......Page 180
2.7 Implicit Differentiation......Page 183
2.8 The Mean Value Theorem......Page 190
CHAPTER 3 Applications of Differentiation......Page 201
Linear Approximations......Page 202
Newton’s Method......Page 206
3.2 Maximum and Minimum Values......Page 213
3.3 Increasing and Decreasing Functions......Page 223
What You See May Not Be What You Get......Page 224
3.4 Concavity and the Second Derivative Test......Page 231
3.5 Overview of Curve Sketching......Page 240
3.6 Optimization......Page 251
3.7 Related Rates......Page 262
3.8 Rates of Change in Economics and the Sciences......Page 267
CHAPTER 4 Integration......Page 279
4.1 Antiderivatives......Page 280
4.2 Sums and Sigma Notation......Page 287
Principle of Mathematical Induction......Page 291
4.3 Area......Page 294
4.4 The Definite Integral......Page 301
Average Value of a Function......Page 307
4.5 The Fundamental Theorem of Calculus......Page 312
4.6 Integration by Substitution......Page 320
Substitution in Definite Integrals......Page 323
4.7 Numerical Integration......Page 326
Simpson’s Rule......Page 331
Error Bounds for Numerical Integration......Page 333
5.1 Area Between Curves......Page 343
5.2 Volume: Slicing, Disks and Washers......Page 352
Volumes by Slicing......Page 353
The Method of Disks......Page 356
The Method of Washers......Page 358
5.3 Volumes by Cylindrical Shells......Page 366
Arc Length......Page 373
Surface Area......Page 375
5.5 Projectile Motion......Page 380
5.6 Applications of Integration to Physics and Engineering......Page 389
6.1 The Natural Logarithm......Page 403
Logarithmic Differentiation......Page 409
6.2 Inverse Functions......Page 412
6.3 The Exponential Function......Page 419
Derivative of the Exponential Function......Page 421
6.4 The Inverse Trigonometric Functions......Page 427
6.5 The Calculus of the Inverse Trigonometric Functions......Page 433
Integrals Involving the Inverse Trigonometric Functions......Page 435
6.6 The Hyperbolic Functions......Page 439
The Inverse Hyperbolic Functions......Page 442
Derivation of the Catenary......Page 444
CHAPTER 7 Integration Techniques......Page 449
7.1 Review of Formulas and Techniques......Page 450
7.2 Integration by Parts......Page 454
Integrals Involving Powers of Trigonometric Functions......Page 461
Trigonometric Substitution......Page 465
7.4 Integration of Rational Functions Using Partial Fractions......Page 470
Brief Summary of Integration Techniques......Page 475
Using Tables of Integrals......Page 478
Integration Using a Computer Algebra System......Page 481
7.6 Indeterminate Forms and L’Hôpital’s Rule......Page 485
Other Indeterminate Forms......Page 489
Improper Integrals with a Discontinuous Integrand......Page 495
Improper Integrals with an Infinite Limit of Integration......Page 498
A Comparison Test......Page 502
7.8 Probability......Page 507
Growth and Decay Problems......Page 519
Compound Interest......Page 524
8.2 Separable Differential Equations......Page 529
Logistic Growth......Page 533
8.3 Direction Fields and Euler’s Method......Page 538
Predator-Prey Systems......Page 549
CHAPTER 9 Infinite Series......Page 559
9.1 Sequences of Real Numbers......Page 560
9.2 Infinite Series......Page 572
9.3 The Integral Test and Comparison Tests......Page 582
Comparison Tests......Page 586
9.4 Alternating Series......Page 593
Estimating the Sum of an Alternating Series......Page 596
9.5 Absolute Convergence and the Ratio Test......Page 599
The Ratio Test......Page 601
Summary of Convergence Tests......Page 604
9.6 Power Series......Page 607
Representation of Functions as Power Series......Page 615
Proof of Taylor’s Theorem......Page 623
9.8 Applications of Taylor Series......Page 627
The Binomial Series......Page 632
9.9 Fourier Series......Page 635
Functions of Period Other Than 2π......Page 641
Fourier Series and Music Synthesizers......Page 645
10.1 Plane Curves and Parametric Equations......Page 653
10.2 Calculus and Parametric Equations......Page 662
10.3 Arc Length and Surface Area in Parametric Equations......Page 669
10.4 Polar Coordinates......Page 677
10.5 Calculus and Polar Coordinates......Page 688
Parabolas......Page 696
Ellipses......Page 699
Hyperbolas......Page 701
10.7 Conic Sections in Polar Coordinates......Page 705
CHAPTER 11 Vectors and the Geometry of Space......Page 715
11.1 Vectors in the Plane......Page 716
11.2 Vectors in Space......Page 725
Vectors in R³......Page 727
11.3 The Dot Product......Page 732
Components and Projections......Page 736
11.4 The Cross Product......Page 742
11.5 Lines and Planes in Space......Page 754
Planes in R³......Page 756
Cylindrical Surfaces......Page 762
Quadric Surfaces......Page 763
An Application......Page 769
CHAPTER 12 Vector-Valued Functions......Page 777
12.1 Vector-Valued Functions......Page 778
Arc Length in R³......Page 781
12.2 The Calculus of Vector-Valued Functions......Page 786
12.3 Motion in Space......Page 797
Equations of Motion......Page 801
12.4 Curvature......Page 807
12.5 Tangent and Normal Vectors......Page 814
Tangential and Normal Components of Acceleration......Page 818
Kepler’s Laws......Page 822
12.6 Parametric Surfaces......Page 827
13.1 Functions of Several Variables......Page 837
13.2 Limits and Continuity......Page 850
13.3 Partial Derivatives......Page 861
13.4 Tangent Planes and Linear Approximations......Page 872
Increments and Differentials......Page 876
13.5 The Chain Rule......Page 882
Implicit Differentiation......Page 888
13.6 The Gradient and Directional Derivatives......Page 892
13.7 Extrema of Functions of Several Variables......Page 902
Proof of the Second Derivatives Test......Page 911
13.8 Constrained Optimization and Lagrange Multipliers......Page 915
14.1 Double Integrals......Page 929
Double Integrals over a Rectangle......Page 931
Double Integrals over General Regions......Page 936
14.2 Area, Volume and Center of Mass......Page 944
Moments and Center of Mass......Page 948
14.3 Double Integrals in Polar Coordinates......Page 954
14.4 Surface Area......Page 961
14.5 Triple Integrals......Page 966
Mass and Center of Mass......Page 972
14.6 Cylindrical Coordinates......Page 976
14.7 Spherical Coordinates......Page 984
Triple Integrals in Spherical Coordinates......Page 985
14.8 Change of Variables in Multiple Integrals......Page 990
15.1 Vector Fields......Page 1005
15.2 Line Integrals......Page 1018
15.3 Independence of Path and Conservative Vector Fields......Page 1031
15.4 Green’s Theorem......Page 1042
15.5 Curl and Divergence......Page 1050
15.6 Surface Integrals......Page 1060
Parametric Representation of Surfaces......Page 1063
15.7 The Divergence Theorem......Page 1072
15.8 Stokes’ Theorem......Page 1081
15.9 Applications of Vector Calculus......Page 1089
CHAPTER 16 Second-Order Differential Equations......Page 1101
16.1 Second-Order Equations with Constant Coefficients......Page 1102
16.2 Nonhomogeneous Equations: Undetermined Coefficients......Page 1110
16.3 Applications of Second-Order Equations......Page 1118
16.4 Power Series Solutions of Differential Equations......Page 1126
Appendix A: Proofs of Selected Theorems......Page 1139
Appendix B: Answers to Odd-Numbered Exercises......Page 1151
Credits......Page 1213
Index......Page 1215