Calculus with analytic geometry

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書誌事項

Calculus with analytic geometry

George F. Simmons

McGraw-Hill, c1996

2nd., International ed

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注記

Includes bibliographical references and index

内容説明・目次

内容説明

Written by acclaimed author and mathematician George Simmons, this revision is designed for the calculus course offered in two and four year colleges and universities. It takes an intuitive approach to calculus and focuses on the application of methods to real-world problems. Throughout the text, calculus is treated as a problem solving science of immense capability.

目次

CHAPTER 1: Numbers, Functions, and Graphs1-1 Introduction1-2 The Real Line and Coordinate Plane: Pythagoras1-3 Slopes and Equations of Straight Lines1-4 Circles and Parabolas: Descartes and Fermat1-5 The Concept of a Function1-6 Graphs of Functions1-7 Introductory Trigonometry 1-8 The Functions Sin O and Cos O CHAPTER 2: The Derivative of a Function2-0 What is Calculus ? 2-1 The Problems of Tangents2-2 How to Calculate the Slope of the Tangent2-3 The Definition of the Derivative2-4 Velocity and Rates of Change: Newton and Leibriz2-5 The Concept of a Limit: Two Trigonometric Limits2-6 Continuous Functions: The Mean Value Theorem and Other TheoremCHAPTER 3: The Computation of Derivatives3-1 Derivatives of Polynomials3-2 The Product and Quotient Rules3-3 Composite Functions and the Chain Rule3-4 Some Trigonometric Derivatives3-5 Implicit Functions and Fractional Exponents3-6 Derivatives of Higher OrderCHAPTER 4: Applications of Derivatives4-1 Increasing and Decreasing Functions: Maxima and Minima4-2 Concavity and Points of Inflection4-3 Applied Maximum and Minimum Problems4-4 More Maximum-Minimum Problems4-5 Related Rates4-6 Newtons Method for Solving Equations4-7 Applications to Economics: Marginal AnalysisCHAPTER 5: Indefinite Integrals and Differential Equations5-1 Introduction5-2 Differentials and Tangent Line Approximations5-3 Indefinite Integrals: Integration by Substitution5-4 Differential Equations: Separation of Variables5-5 Motion Under Gravity: Escape Velocity and Black HolesCHAPTER 6: Definite Integrals6-1 Introduction6-2 The Problem of Areas6-3 The Sigma Notation and Certain Special Sums6-4 The Area Under a Curve: Definite Integrals6-5 The Computation of Areas as Limits6-6 The Fundamental Theorem of Calculus6-7 Properties of Definite IntegralsCHAPTER 7: Applications of Integration7-1 Introduction: The Intuitive Meaning of Integration7-2 The Area between Two Curves7-3 Volumes: The Disk Method7-4 Volumes: The Method of Cylindrical Shells7-5 Arc Length7-6 The Area of a Surface of Revolution7-7 Work and Energy7-8 Hydrostatic ForcePART II CHAPTER 8: Exponential and Logarithm Functions8-1 Introduction8-2 Review of Exponents and Logarithms8-3 The Number e and the Function y = e <^>x8-4 The Natural Logarithm Function y = ln x8-5 ApplicationsPopulation Growth and Radioactive Decay8-6 More ApplicationsCHAPTER 9: Trigonometric Functions9-1 Review of Trigonometry9-2 The Derivatives of the Sine and Cosine9-3 The Integrals of the Sine and Cosine9-4 The Derivatives of the Other Four Functions9-5 The Inverse Trigonometric Functions 9-6 Simple Harmonic Motion9-7 Hyperbolic FunctionsCHAPTER 10 : Methods of Integration10-1 Introduction10-2 The Method of Substitution10-3 Certain Trigonometric Integrals10-4 Trigonometric Substitutions10-5 Completing the Square10-6 The Method of Partial Fractions10-7 Integration by Parts10-8 A Mixed Bag10-9 Numerical IntegrationCHAPTER 11: Further Applications of Integration11-1 The Center of Mass of a Discrete System11-2 Centroids 11-3 The Theorems of Pappus11-4 Moment of InertiaCHAPTER 12: Indeterminate Forms and Improper Integrals12-1 Introduction. The Mean Value Theorem Revisited12-2 The Interminate Form 0/0. L'Hospital's Rule12-3 Other Interminate Forms12-4 Improper Integrals 12-5 The Normal DistributionCHAPTER 13: Infinite Series of Constants13-1 What is an Infinite Series ?13-2 Convergent Sequences13-3 Convergent and Divergent Series13-4 General Properties of Convergent Series13-5 Series on Non-negative Terms: Comparison Tests13-6 The Integral Test13-7 The Ratio Test and Root Test13-8 The Alternating Series TestCHAPTER 14: Power Series14-1 Introduction14-2 The Interval of Convergence14-3 Differentiation and Integration of Power Series14-4 Taylor Series and Taylor's Formula14-5 Computations Using Taylor's Formula14-6 Applications to Differential Equations14. 7 (optional) Operations on Power Series14. 8 (optional) Complex Numbers and Euler's FormulaPART IIICHAPTER 15: Conic Sections15-1 Introduction15-2 Another Look at Circles and Parabolas15-3 Ellipses15-4 Hyperbolas15-5 The Focus-Directrix-Eccentricity Definitions15-6 (optional) Second Degree EquationsCHAPTER 16: Polar Coordinates16-1 The Polar Coordinate System16-2 More Graphs of Polar Equations16-3 Polar Equations of Circles, Conics, and Spirals16-4 Arc Length and Tangent Lines16-5 Areas in Polar CoordinatesCHAPTER 17: Parametric Equations17-1 Parametric Equations of Curves17-2 The Cycloid and Other Similar Curves17-3 Vector Algebra17-4 Derivatives of Vector Function17-5 Curvature and the Unit Normal Vector17-6 Tangential and Normal Components of Acceleration17-7 Kepler's Laws and Newton's Laws of GravitationCHAPTER 18: Vectors in Three-Dimensional Space18-1 Coordinates and Vectors in Three-Dimensional Space18-2 The Dot Product of Two Vectors18-3 The Cross Product of Two Vectors18-4 Lines and Planes18-5 Cylinders and Surfaces of Revolution18-6 Quadric Surfaces18-7 Cylindrical and Spherical CoordinatesCHAPTER 19: Partial Derivatives19-1 Functions of Several Variables19-2 Partial Derivatives19-3 The Tangent Plane to a Surface19-4 Increments and Differentials19-5 Directional Derivatives and the Gradient19-6 The Chain Rule for Partial Derivatives19-7 Maximum and Minimum Problems19-8 Constrained Maxima and Minima19-9 Laplace's Equation, the Heat Equation, and the Wave Equation19-10 (optional) Implicit FunctionsCHAPTER 20: Multiple Integrals20-1 Volumes as Iterated Integrals20-2 Double Integrals and Iterated Integrals20-3 Physical Applications of Double Integrals20-4 Double Integrals in Polar Coordinates20-5 Triple Integrals20-6 Cylindrical Coordinates20-7 Spherical Coordinates20-8 Areas of curved SurfacesCHAPTER 21: Line and Surface Integrals21-1 Green's Theorem, Gauss's Theorem, and Stokes' Theorem21-2 Line Integrals in the Plane21-3 Independence of Path21-4 Green's Theorem21-5 Surface Integrals and Gauss's Theorem21-6 Maxwell's Equations : A Final ThoughtAppendicesA: The Theory of CalculusA-1 The Real Number SystemA-2 Theorems About LimitsA-3 Some Deeper Properties of Continuous FunctionsA-4 The Mean Value theoremA-5 The Integrability of Continuous FunctionsA-6 Another Proof of the Fundamental Theorem of CalculusA-7 Continuous Curves With No LengthA-8 The Existence of e = lim h->0 (1 + h) <^>1/hA-9 Functions That Cannot Be IntegratedA-10 The Validity of Integration by Inverse SubstitutionA-11 Proof of the Partial fractions TheoremA-12 The Extended Ratio Tests of Raabe and GaussA-13 Absolute vs Conditional ConvergenceA-14 Dirichlet's TestA-15 Uniform Convergence for Power SeriesA-16 Division of Power SeriesA-17 The Equality of Mixed Partial DerivativesA-18 Differentiation Under the Integral SignA-19 A Proof of the Fundamental LemmaA-20 A Proof of the Implicit Function TheoremA-21 Change of Variables in Multiple IntegralsB: A Few Review TopicsB-1 The Binomial TheoremB-2 Mathematical Induction

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