Calculus With Analytic Geometry

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