Calculus, Early Transcendentals

TABLE OF CONTENTS         About the AuthorsPreface 1 Functions, Graphs, and Models1.1 Functions and Mathematical ModelingInvestigation:   Designing a Wading Pool   1.2 Graphs of Equations and Functions1.3 Polynomials and Algebraic Functions1.4 Transcendental Functions1.5 Preview:  What Is Calculus?REVIEW - Understanding: Concepts and Definitions Objectives:  Methods and Techniques    2 Prelude to Calculus2.1 Tangent Lines and Slope PredictorsInvestigation:   Numerical Slope Investigations   2.2 The Limit ConceptInvestigation:   Limits, Slopes, and Logarithms   2.3 More About Limits Investigation:   Numerical Epsilon-Delta Limits   2.4 The Concept of Continuity REVIEW - Understanding: Concepts and DefinitionsObjectives:  Methods and Techniques   3 The Derivative3.1 The Derivative and Rates of Change3.2 Basic Differentiation Rules3.3 The Chain Rule  3.4 Derivatives of Algebraic Functions3.5 Maxima and Minima of Functions on Closed IntervalsInvestigation:   When Is Your Coffee Cup Stablest?   3.6 Applied Optimization Problems3.7 Derivatives of Trigonometric Functions3.8 Exponential and Logarithmic Functions  Investigation:   Discovering the Number  e  for Yourself  3.9 Implicit Differentiation and Related RatesInvestigation:   Constructing the Folium of Descartes  3.10 Successive Approximations and Newton's MethodInvestigation:   How Deep Does a Floating Ball Sink?  REVIEW - Understanding: Concepts, Definitions, and FormulasObjectives:  Methods and Techniques    4 Additional Applications of the Derivative4.1 Introduction4.2 Increments, Differentials, and Linear Approximation 4.3 Increasing and Decreasing Functions and the Mean Value Theorem4.4 The First Derivative Test and ApplicationsInvestigation:   Constructing a Candy Box With Lid   4.5 Simple Curve Sketching4.6 Higher Derivatives and Concavity4.7 Curve Sketching and AsymptotesInvestigation:   Locating Special Points on Exotic Graphs   4.8 Indeterminate Forms and L'Hopital's Rule 4.9 More Indeterminate Forms REVIEW - Understanding: Concepts, Definitions, and ResultsObjectives:  Methods and Techniques   5 The Integral   5.1 Introduction5.2 Antiderivatives and Initial Value Problems5.3 Elementary Area Computations5.4 Riemann Sums and the Integral Investigation:   Calculator/Computer Riemann Sums   5.5 Evaluation of Integrals5.6 The Fundamental Theorem of Calculus5.7 Integration by Substitution5.8 Areas of Plane Regions5.9 Numerical IntegrationInvestigation:   Trapezoidal and Simpson Approximations  REVIEW - Understanding: Concepts, Definitions, and ResultsObjectives:  Methods and Techniques   6 Applications of the Integral6.1 Riemann Sum Approximations6.2 Volumes by the Method of Cross Sections6.3 Volumes by the Method of Cylindrical ShellsInvestigation:   Design Your Own Ring!6.4 Arc Length and Surface Area of Revolution6.5 Force and Work6.6 Centroids of Plane Regions and Curves6.7 The Natural Logarithm as an IntegralInvestigation:   Natural Functional Equations6.8 Inverse Trigonometric Functions6.9 Hyperbolic FunctionsREVIEW - Understanding: Concepts, Definitions, and FormulasObjectives:  Methods and Techniques    7 Techniques of Integration  7.1 Introduction7.2 Integral Tables and Simple Substitutions7.3 Integration by Parts7.4 Trigonometric Integrals7.5 Rational Functions and Partial Fractions7.6 Trigonometric Substitutions7.7 Integrals Involving Quadratic Polynomials7.8 Improper IntegralsSUMMARY - Integration Strategies  REVIEW - Understanding: Concepts and TechniquesObjectives:  Methods and Techniques    8 Differential Equations   8.1 Simple Equations and Models8.2 Slope Fields and Euler's MethodInvestigation:   Computer-Assisted Slope Fields and Euler's Method  8.3 Separable Equations and Applications8.4 Linear Equations and Applications8.5 Population ModelsInvestigation:   Predator-Prey Equations and Your Own Game Preserve 8.6 Linear Second-Order Equations8.7 Mechanical VibrationsREVIEW - Understanding: Concepts, Definitions, and MethodsObjectives:  Methods and Techniques    9 Polar Coordinates and Parametric Curves  9.1 Analytic Geometry and the Conic Sections9.2 Polar Coordinates 9.3 Area Computations in Polar Coordinates9.4 Parametric CurvesInvestigation:   Trochoids Galore9.5 Integral Computations with Parametric CurvesInvestigation:   Moon Orbits and Race Tracks9.6 Conic Sections and Applications  REVIEW - Understanding: Concepts, Definitions, and FormulasObjectives:  Methods and Techniques   10 Infinite Series   10.1 Introduction10.2 Infinite SequencesInvestigation:   Nested Radicals and Continued Fractions10.3 Infinite Series and ConvergenceInvestigation:   Numerical Summation and Geometric Series10.4 Taylor Series and Taylor PolynomialsInvestigation:   Calculating Logarithms on a Deserted Island  10.5 The Integral TestInvestigation:   The Number  p, Once and for All10.6 Comparison Tests for Positive-Term Series10.7 Alternating Series and Absolute Convergence10.8 Power Series10.9 Power Series ComputationsInvestigation:   Calculating Trigonometric Functions on a Deserted Island   10.10 Series Solutions of differential EquationsREVIEW - Understanding: Concepts, and ResultsObjectives:  Methods and Techniques    11 Vectors, Curves, and Surfaces in Space 11.1 Vectors in the Plane11.2 Three-Dimensional Vectors11.3 The Cross Product of Two Vectors11.4 Lines and Planes in Space11.5 Curves and Motion in SpaceInvestigation:   Does a Pitched Baseball Really Curve?11.6 Curvature and Acceleration11.7 Cylinders and Quadric Surfaces11.8 Cylindrical and Spherical CoordinatesREVIEW - Understanding: Concepts, Definitions, and ResultsObjectives:  Methods and Techniques       12 Partial Differentiation12.1 Introduction12.2 Functions of Several Variables12.3 Limits and Continuity12.4 Partial Derivatives12.5 Multivariable Optimization Problems12.6 Increments and Linear Approximation12.7 The Multivariable Chain Rule12.8 Directional Derivatives and the Gradient Vector12.9 Lagrange Multipliers and Constrained OptimizationInvestigation:   Numerical Solution of Lagrange Multiplier Systems 12.10 Critical Points of Functions of Two VariablesInvestigation:   Critical Point Investigations REVIEW - Understanding: Concepts, Definitions, and ResultsObjectives:  Methods and Techniques    13 Multiple Integrals    13.1 Double IntegralsInvestigation:   Midpoint Sums Approximating Double Integrals  13.2 Double Integrals over More General Regions13.3 Area and Volume by Double Integration13.4 Double Integrals in Polar Coordinates13.5 Applications of Double IntegralsInvestigation:   Optimal Design of Race Car Wheels  13.6 Triple IntegralsInvestigation:   Archimedes' Floating Paraboloid 13.7 Integration in Cylindrical and Spherical Coordinates13.8 Surface Area13.9 Change of Variables in Multiple IntegralsREVIEW - Understanding: Concepts, Definitions, and ResultsObjectives:  Methods and Techniques    AppendicesA:    Real Numbers and InequalitiesB:    The Coordinate Plane and Straight LinesC:    Review of TrigonometryD:    Proofs of the Limit LawsE:    The Completeness of the Real Number SystemF:    Existence of the IntegralG:    Approximations and Riemann SumsH:    L'Hopital's Rule and Cauchy's Mean Value TheoremI: Proof of Taylor's FormulaJ: Conic Sections as Sections of a ConeK:   Proof of the Linear Approximation TheoremL:    Units of Measurement and Conversion FactorsM:   Formulas from Algebra, Geometry, and TrigonometryN:   The Greek Alphabet Answers to Odd-Numbered ProblemsReferences for Further StudyIndex