Calculus : with analytic geometry

목차 (Table of Contents)

• CONTENTS
• Ⅰ. Slopes and Rates of Change = 1
• 1-1. The Nature of Calculus = 1
• 1-2. Fundamentals of Plane Analytic Geometry = 3
• 1-3. The Slope of a Line = 11
• 1-4. Equations of Straight Lines = 19
• 1-5. Graphs and Equations = 26
• 1-6. Functions = 89
• 1-7. The Derivative of a Function. Velocity and Acceleration = 36
• 1-8. Functional Notation. Limits. Continuity = 45
• 1-9. Geometrical Meaning of the Derivative = 56
• 1-10. Increasing and Decreasing Functions = 61
• Ⅱ. The Inverse of Differentiation = 68
• 2-1. Some Functional Theory = 68
• 2-2. Antiderivatives = 73
• 2-3. Rectilinear Motion = 77
• 2-4. Parabolas = 82
• 2-5. Tangents to Parabolas = 91
• 2-6. The Definition of Area Under a Curve = 94
• 2-7. Finding Areas by Antidifferentiation = 100
• Review Questions and Problems for Chapters Ⅰ and Ⅱ = 103
• Ⅲ. Differentiation of Algebraic Functions = 107
• 3-1. The △-notation = 107
• 3-2. Sums, Products, and Quotients = 108
• 3-3. Composite Functions = 114
• 3-4. Second Derivatives = 119
• 3-5. Graphing Rational Functions = 123
• 3-6. Fractional Exponents = 131
• 3-7. Implicit Functions = 134
• 3-8. Circles and Ellipses = 137
• 3-9. Hyperbolas = 146
• 3-10. Maxima and Minima = 154
• 3-11. Extremal Problems with Side Conditions = 161
• 3-12. Related Rates = 165
• XIV. Further Study of Limits = 442
• 14-1. The Purposes of this Chapter = 442
• 14-2. A Study of Inequalities. Proofs of the Limit Theorems = 443
• 14-3. The Completeness Property of the Real Number, System = 446
• 14-4. Convergent Sequences = 450
• 14-5. L'Hospital's Rule = 455
• XV. Infinite Series and Taylor's Formula = 462
• 15-1. Sequences and Series = 462
• 15-2. Various Series Derived from Geometric Progressions = 466
• 15-3. Taylor's Formula with Integral Remainder = 470
• 15-4. Derivative Forms of the Remainder = 477
• 15-5. Absolute and Conditional Convergence = 481
• 15-6. Comparison Tests for Convergence = 482
• 15-7. Improper Integrals and the Integral Test = 485
• 15-8. Alternating Series = 488
• 15-9. The Ratio Test = 489
• 15-10. Power Series = 493
• Review Questions and Problems for Chapters XIII, XIV, XV = 498
• XVI. Methods of Approximation = 504
• 16-1. Approximation by Differentials = 504
• 16-2. The Intersection of Two Curves = 507
• 16-3. Newton's Method = 511
• 16-4. Approximating Definite Integrals = 514
• XVII. Determinants and Linear Systems = 519
• 17-1. Determinants of Order Two = 510
• 17-2. Determinants of Order Three = 524
• 17-3. Further Discussion of Third-Order Determinants = 528
• 17-4. The Solution of Linear Systems = 533
• 17-5. Determinants of Higher Order = 538
• XVIII. Analytic Geometry of Three Dimensions = 539
• 18-1. Fundamental Notions = 539
• 18-2. The Angle Between Two Vectors. The Scalar Product = 543
• 18-3. Planes and Linear Equations = 548
• 18-4. Planes and Straight Lines = 553
• 18-5. The Cross Product of Two Vectors = 557
• 18-6. Surfaces in Space = 559
• 18-7. Curves in Space = 564
• Review Questions and Problems for Chapters XVI, XVII, and XVIII = 569
• XIX. Partial Differentiation = 574
• 19-1. Functions of Several Variables = 574
• 19-2. Partial Derivatives = 578
• 19-3. The Differential of a Function of Several Variables = 582
• 19-4. Partial Derivatives of Higher Order = 587
• 19-5. The Chain Rule = 590
• 19-6. Extreme Value Problems = 595
• 19-7. Directional Derivatives. Gradients = 602
• 19-8. Implicitly Defined Functions = 608
• XX. Multiple Integrals = 615
• 20-1. Double Integrals = 615
• 20-2. Iterated Integrals = 62O
• 20-3. Iterated Integrals in Polar Coordinates = 626
• 20-4. Mass Systems and Newton's Law = 633
• 20-5. Surface Integrals = 640
• 20-6. Triple Integrals = 644
• 20-7. Threefold Iterated Integrals = 647
• 20-8. Cylindrical Coordinates = 650
• 20-9. Spherical Coordinates = 652
• XXI. Differential Equations = 658
• 21-1. Introductory Remarks = 658
• 21-2. First-Order Equations with Variables Separable = 660
• 21-3. First-Order Equations and One-Parameter Families = 665
• 21-4. Homogeneous First-Order Equations = 668
• 21-5. The General First-Order Linear Equation = 670
• 21-6. Miscellaneous Applications = 674
• 21-7. Equations of the Second Order. Some Special Types = 680
• 21-8. Linear Equations of the Second Order = 686
• 21-9. Linear Differential Equations with Constant Coefficients = 690
• 21-10. Oscillatory Systems = 693
• Review Questions and Problems for Chapters XIX, XX, and XXI = 697
• Appendices = 701
• The Greek Alphabet = 701
• Formulas from Geometry = 701
• Table of Integrals = 702
• Numerical Tables:
• Ⅰ. Natural Logarithms = 710
• Ⅱ. Exponential and Hyperbolic Functions = 712
• Ⅲ. Natural Functions for Angles in Radians = 713
• Ⅳ. Values of Trigonometric Functions = 715
• Ⅴ. Degrees and Minutes to Radians = 720
• Ⅵ. Radians to Degrees and Minutes = 720
• Answers to Odd-Numbered Problems = 721
• Index = 757