本书是在响应东南大学国际化需求, 根据国家教育部非数学专业数学基础课教学指导分委员会制定的工科类本科数学基础课程教学基本要求, 并结合东南大学多年教学改革实践经验编写的全英文教材。全书分为上、下两册, 此为上册, 主要包括极限, 一元函数微积分及其应用和常微分方程四部分。本书对基本概念的叙述清晰准确, 对基本理论的论述简明易懂, 例题习题的选配典型多样, 强调基本运算能力的培养及理论的实际应用。
Chapter 1 Limits
1.1 The Concept of Limits and its Properties
1.1.1 Limits of Sequence
1.1.2 Limits of Functions
1.1.3 Properties of Limits
Exercise 1.1
1.2 Limits Theorem
1.2.1 Rules for Finding Limits
1.2.2 The Sandwich Theorem
1.2.3 Monotonic Sequence Theorem
1.2.4 The Cauchy Criterion
Exercise 1.2
1.3 Two Important Special Limits
Exercise 1.3
1.4 Infinitesimal and Infinite Chapter 1 Limits
1.1 The Concept of Limits and its Properties
1.1.1 Limits of Sequence
1.1.2 Limits of Functions
1.1.3 Properties of Limits
Exercise 1.1
1.2 Limits Theorem
1.2.1 Rules for Finding Limits
1.2.2 The Sandwich Theorem
1.2.3 Monotonic Sequence Theorem
1.2.4 The Cauchy Criterion
Exercise 1.2
1.3 Two Important Special Limits
Exercise 1.3
1.4 Infinitesimal and Infinite
1.4.1 Infinitesimal
1.4.2 Infinite
Exercise 1.4
1.5 Continuous Function
1.5.1 Continuity
1.5.2 Discontinuity
Exercise 1.5
1.6 Theorems about Continuous Function on a Closed Interval
Exercise 1.6
Review and Exercise
Chapter 2 Differentiation
2.1 The Derivative
Exercise 2.1
2.2 Rules for Fingding the Derivative
2.2.1 Derivative of Arithmetic Combination
2.2.2 The Derivative Rule for Inverses
2.2.3 Derivative of Composition
2.2.4 Implicit Differentiation
2.2.5 Parametric Differentiation
2.2.6 Related Rates of Change
Exercise 2.2
2.3 Higher-Order Derivatives
Exercise 2.3
2.4 Differentials
Exercise 2.4
2.5 The Mean Value Theorem
Exercise 2.5
2.6 L'Hospital's Rule
Exercise 2.6
2.7 Taylor's Theorem
Exercise 2.7
2.8 Applications of Derivatives
2.8.1 Monotonicity
2.8.2 Local Extreme Values
2.8.3 Extreme Values
2.8.4 Concavity
2.8.5 Graphing Functions
Exercise 2.8
Review and Exercise
Chapter 3 The Integration
3.1 The Definite Integral
3.1.1 Two Examples
3.1.2 The Definition of Definite Integral
3.1.3 Properties of Definite Integrals
Exercise 3.1
3.2 The Indefinite Integral
Exercise 3.2
3.3 The Fundamental Theorem
3.3.1 First Fundamental Theorem
3.3.2 Second Fundamental Theorem
Exercise 3.3
3.4 Techniques of Indefinite Integration
3.4.1 Substitution in Indefinite Integrals
3.4.2 Indefinite Integration by Parts
3.4.3 Indefinite Integration of Rational Functions by
Partial Fractions
Exercise 3.4
3.5 Techniques of Definite Integration
3.5.1 Substitution in Definite Integrals
3.5.2 Definite Integration by Parts
Exercise 3.5
3.6 Applications of Definite Integrals
3.6.1 Lengths of Plane Curves
3.6.2 Area between Two Curves
3.6.3 Volumes of Solids
3.6.4 Areas of Surface of Revolution
3.6.5 Moments and Center of Mass
3.6.6 Work and Fluid Force
Exercise 3.6
3.7 Improper Integrals
3.7.1 Improper Integrals.Infinite Limits of Integration
3.7.2 Improper Integrals: Infinite Integrands
Exercise 3.7
Review and Exercise
Chapter 4 Differential Equations
4.1 The Concept of Differential Equations
Exercise 4.1
4.2 Differential Equations of the First Order
4.2.1 Equations with Variable Separable
4.2.2 Homogeneous Equation
Exercise 4.2
4.3 First-order Linear Differential Equations
Exercise 4.3
4.4 Equations Reducible to First Order
4.4.1 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 4.4
4.5 Linear Differential Equations
4.5.1 Basic Theory of Linear Differential Equations
4.5.2 Homogeneous Linear Differential Equations of the
Second Order with Constant Coefficients
4.5.3 Nonhomogeneous Linear Differential Equations of the
Second Order with Constant Coefficients
4.5.4 Euler Differential Equation
Exercise 4.5
4.6 Systems of Linear Differential Equations
with Constant Coefficients
Exercise 4.6
4.7 Applications
Exercise 4.7
Review and Exercise
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