高等数学

Chapter 1 Limits1.1 The Concept of Limits and its Properties1.1.1 Limits of Sequence1.1.2 Limits of Functions1.1.3 Properties of LimitsExercise 1.11.2 Limits Theorem1.2.1 Rules for Finding Limits1.2.2 The Sandwich Theorem1.2.3 Monotonic Sequence Theorem1.2.4 The Cauchy CriterionExercise 1.21.3 Two Important Special LimitsExercise 1.31.4 Infinitesimal and Infinite1.4.1 Infinitesimal1.4.2 InfiniteExercise 1.41.5 Continuous Function1.5.1 Continuity1.5.2 DiscontinuityExercise 1.51.6 Theorems about Continuous Function on a Closed IntervalExercise 1.6Review and ExerciseChapter 2 Differentiation2.1 The DerivativeExercise 2.12.2 Rules for Fingding the Derivative2.2.1 Derivative of Arithmetic Combination2.2.2 The Derivative Rule for Inverses2.2.3 Derivative of Composition2.2.4 Implicit Differentiation2.2.5 Parametric Differentiation2.2.6 Related Rates of ChangeExercise 2.22.3 Higher-Order DerivativesExercise 2.32.4 DifferentialsExercise 2.42.5 The Mean Value TheoremExercise 2.52.6 L'Hospital's RuleExercise 2.62.7 Taylor's TheoremExercise 2.72.8 Applications of Derivatives2.8.1 Monotonicity2.8.2 Local Extreme Values2.8.3 Extreme Values2.8.4 Concavity2.8.5 Graphing FunctionsExercise 2.8Review and ExerciseChapter 3 The Integration3.1 The Definite Integral3.1.1 Two Examples3.1.2 The Definition of Definite Integral3.1.3 Properties of Definite IntegralsExercise 3.13.2 The Indefinite IntegralExercise 3.23.3 The Fundamental Theorem3.3.1 First Fundamental Theorem3.3.2 Second Fundamental TheoremExercise 3.33.4 Techniques of Indefinite Integration3.4.1 Substitution in Indefinite Integrals3.4.2 Indefinite Integration by Parts3.4.3 Indefinite Integration of Rational Functions byPartial FractionsExercise 3.43.5 Techniques of Definite Integration3.5.1 Substitution in Definite Integrals3.5.2 Definite Integration by PartsExercise 3.53.6 Applications of Definite Integrals3.6.1 Lengths of Plane Curves3.6.2 Area between Two Curves3.6.3 Volumes of Solids3.6.4 Areas of Surface of Revolution3.6.5 Moments and Center of Mass3.6.6 Work and Fluid ForceExercise 3.63.7 Improper Integrals3.7.1 Improper Integrals.Infinite Limits of Integration3.7.2 Improper Integrals： Infinite IntegrandsExercise 3.7Review and ExerciseChapter 4 Differential Equations4.1 The Concept of Differential EquationsExercise 4.14.2 Differential Equations of the First Order4.2.1 Equations with Variable Separable4.2.2 Homogeneous EquationExercise 4.24.3 First-order Linear Differential EquationsExercise 4.34.4 Equations Reducible to First Order4.4.1 Equations of the Form y（n）=f（x）4.4，2 Equations of the Form y =y （x，y ）4.4.3 Equations of the Form y=f（y，y'）Exercise 4.44.5 Linear Differential Equations4.5.1 Basic Theory of Linear Differential Equations4.5.2 Homogeneous Linear Differential Equations of theSecond Order with Constant Coefficients4.5.3 Nonhomogeneous Linear Differential Equations of theSecond Order with Constant Coefficients4.5.4 Euler Differential EquationExercise 4.54.6 Systems of Linear Differential Equationswith Constant CoefficientsExercise 4.64.7 ApplicationsExercise 4.7Review and Exercise

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