本书编写的直接目的是为讲授高等数学的教师和学习高等数学的学生提供掌握相关内容的英文描述服务，进而使得学生学过本课程后，能够独立阅读相关的英文教材和文献。它可作为学生的配套教材和扩大知识领域的参考书，也可作为科技英语专业高等数学课程的参考书。

本书主要讲授一元函数的微积分学。

引子

1 Functions，Limits and Continuity函数、极限与连续

 1．0 Citing examples引例

 1．1 Functions函数

 Key points of this section本节重点

 单词和短语

 1．1．1 Concept of functions函数的概念

 1．1．2 Several common properties of functions函数的几种常见性态

 1．1．3 Composite functions and inverse functions复合函数与反函数

 1．1．4 Mappings映射

 1．1．5 Elementary functions and non-elementary functions初等函数与非初等函数

 1．1．6 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 1．2 Limits极限

 Key points of this section本节重点

 单词和短语

 1．2．1 Examples of concepts of limits极限概念引例

 1．2．2 Limits of functions for independent variable tending tO finite values

 自变量趋于有限值时函数的极限

 1．2．3 Limits of functions as independent variable tending tO infinity

 自变量趋于无穷大时函数的极限

 1．2．4 Limits of sequences of numbers数列的极限

 1．2．5 Infinitesimals and infinities无穷小与无穷大

 1．3 Properties and operations of limits极限的性质与运算

 Key points of this section本节重点

 单词和短语

 1．3．1 Several properties of limits极限的几个性质

 l．3．2 Four arithmetic operations rules of limits极限的四则运算法则

 1．3．3 Relations between limits of functions and limits of sequences of numbers

 函数极限与数列极限的关系

 1．3．4 Squeeze rule夹逼法则

 1．3．5 Operation rules of composite functions复合函数运算法则

 1．3．6 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 1．4 Monotone bounded principle and irrational number e

 单调有界原理和无理数e

 Kev points of this section本节重点

 单词和短语

 1．4．1 Monotonic bounded principle单调有界原理

 1．4．2 An important:limit：■重要极限：■

 1．4．3 Exoonential function er，logarithm function In x and hyperbolic functions

 指数函数er，对数函数Inz，双曲线

 1．5 Comparison between infinitesimals无穷小的比较

 Key points of this section本节重点

 单词和短语

 1．5．1 Order of infinitesimals无穷小的阶

 1．5．2 Compute the limit by the substitution of equivalent infinitesimals

 利用等价无穷小代换求极限

 1．5．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 1．6 Continuity and discontinuity of functions函数的连续与间断

 Kev points of this section本节重点

 单词和短语

 1．6．1 Continuity and discontinuity of functions函数的连续与间断

 1．6．2 Continuity of elementary functions初等函数的连续性

 1．6．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 1．7 Properties of continuous functions on closed interval

 闭区间上连续函数的性质

 Key points of this section本节重点

 1．7．1 Basic contents基本内容

 1．7．2 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 Exercises 习 题

2 Differential Calculus of one Variable Functions

 and its Applications一元函数微分学及其应用

 2．0 Cited examples引例

 2．1 Concepts of derivatives导数的概念

 Kev points of this section本节重点

 单词和短语

 2．1．1 Concepts of derivatives导数的概念

 2．1．2 Examples for finding derivative by definition用定义求导数举例

 2．1．3 Geometry interpretation of derivative导数的几何意义

 2．1．4 Relationship between continuity and differentiability连续性与可导性的关系

 2．1．5 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．2 Derivation rules求导法则

 Key points of this section本节重点

 单词和短语

 2．2．1 Derivation rules of sum，difference，product and quotient of functions

 函数的和、差、积、商的求导法则

 2．2．2 Derivation rules of composite functions复合函数的求导法则

 2．2．3 Derivation rules of inverse functions反函数的求导法则

 2．2．4 Some special derivation rules一些特殊的求导法则

 2．2．5 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．3 Derivatives of higher order and relative rates of change

 高阶导数与相关变化率

 Key points of this section本节重点

 单词和短语

 2．3．1 Derivatives of higher order高阶导数

 2．3．2 Relative rates of change相关变化率

 2．3．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．4 Differential and local I inear approximation of functions

 函数的微分与函数的局部线性逼近

 Key points of this section本节重点

 单词和短语

 2．4．1 Concept of differential微分的概念

 2．4．2 Formulas and operation rules of differentials微分公式与运算法则

 2．4．3 Geometric interpretation of differential微分的几何意义

 2．5 Computing limits by deriVatiVes——L’Hospital rule

 利用导数求极限——洛必达法则

 Key points of this section本节重点

 单词和短语

 2．5．1 LH spitaIrule f the indeterminate form百0罟型未定式的洛必达法则

 2．5．2 LHOSpitalrule f the indeterminate fornl o oo姜型未定式的洛必达法则

 2．5．3 Limits of other indeterminate forms其他类型未定式的极限

 2．5．4 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．6 Mean value theorems of differentials微分中值定理

 Key points of this section本节重点

 单词和短语

 2．6．1 Rolle theorem罗尔定理

 2．6．2 Lagrange mean value theorem拉格朗日中值定理

 2．6．3 Cauchy mean value theorem柯西中值定理

 2．6．4 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．7 Taylor formula Approximating functions by polynomials

 泰勒公式——用多项式逼近函数

 Key points of this section本节重点

 单词和短语

 2．7．1 Taylor polynomial and Taylor formula泰勒多项式与泰勒公式

 2．7．2 Maclaurin formulas in common use常用的麦克劳林公式

 2．7．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．8 Study properties of functions by derivatives利用导数研究函数的性质

 Key points of this section本节重点

 单词和短语

 2．8．1 Monotonicity of functions函数的单调性．

 2．8．2 Extremum of functions函数的极值

 2．8．3 Maximum and minimum of functions函数的最大值与最小值

 2．8．4 Convexity and inflection point of functions函数的凸性与拐点

 2．8．5 Asymptote lines of curves曲线的渐近线

 2．8．6 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 2．9 Curvature of plane curves平面曲线的曲率

 Key points of this section本节重点

 单词和短语

 2．9．1 Arc differential弧微分

 2．9．2 Curvature and its formula曲率和曲率公式

 Exercises 习 题

3 Integral Calculus of one Variable Functions and its Applications一元函数积分学及其应用

 3．0 Citing examples引例

 3．1 Concepts．properties and integrable rules of definite integrals

 定积分的概念、性质、可积准则

 Key points of this section本节重点

 单词和短语

 3．1．1 Examples for definite integrals定积分问题举例

 3．1．2 Concepts of definite integrals定积分的概念

 3．1．3 Geometric interpretation of definite integrals定积分的几何意义

 3．1．4 Integrable criterions可积准则

 3．1．5 Properties of definite integrals定积分的性质

 3．1．6 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 3．2 Fundamental theorem of calculus微积分基本定理

 Key points of this section本节重点

 单词和短语

 3．2．1 Newton-Leibniz formula牛顿一莱布尼兹公式

 3．2．2 Existence theorem of primitive function原函数存在定理

 3．2．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 3．3 Indefinite integrals不定积分

 Key points of this section本节重点

 单词和短语

 3．3．1 Concept and properties of indefinite integrals不定积分的概念及性质

 3．3．2 Fundamental integral formulas基本积分公式

 3．3．3 Integral rules积分法则

 3．3．4 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 3．4 Computation of definite integrals定积分的计算

 Key points of this section本节重点

 3．4．1 Integration by substitutions for definite integrals定积分的换元法

 3．4．2 Integration by parts for definite integrals定积分的分部积分法

 3．4．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 3．5 Applications of definite integrals定积分应用举例

 Key points of this section本节重点

 单词和短语

 3．5．1 Additivity of quanta and micro-element method总量的可加性与微元法

 3．5．2 Examples applying on geometry几何应用举例

 3．5．3 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

 3．5．3 Other applications其他应用

 3．6 Improper integral反常积分

 Key points of this section本节重点

 单词和短语

 3．6．1 Improper integrals on infinite intervals无穷区间上的反常积分

 3．6．2 Improper integrals of unbounded functions无界函数的反常积分

 3．6．3 Criterions of convergence of improper integrals反常积分的收敛判别法

 Exercises 习题

4 Differential Equations微分方程

 Key points of this chapter本章重点

 单词和短语

 4．0 Cited examples引例

 4．1 Basic concepts of differential equations微分方程的基本概念

 4．2 Elementary methods of integration for solving some simple

 differential equations某些简单微分方程的初等积分法

 4．3 Brief introduction for establ ishing differential equations

 建立微分方程方法简介

 4．4 Higher—order differential equations高阶微分方程

 4．4．1 Structures of general solutions of linear differential equations

 线性微分方程通解的结构

 4．4．2 Solutions of higher-order homogeneous linear differential equations

 with constant coefficients高阶常系数齐次线性微分方程的解法

 4．4．3 Solutions of higher-order nonhomogeneous linear differential equations

 with constant coefficients高阶常系数非齐次线性微分方程的解法

 4．5 Summarization of solving methods and typical examples

 解题方法归纳与典型例题

Exercises 习题