微积分

CHAPTER 1 Functions， Limits and Continuity

1. 1 Mathematical Sign Language

1.1.1 Sets

1.1.2 Numbers

1.1.3 Intervals

1.1.4 Implication and Equivalence

1.1.5 Inequalities and Numbers

1.1.6 Absolute Value of a Number

1.1.7 Summation Notation

1.1.8 Factorial Notation

1.1.9 Binomial Coefficients

1 2 Functions

1.2.1 Definition of a Function

1.2.2 Properties of Functions

1.2. 3 Inverse and Composite Functions

1.2.4 Combining Functions

1. 2.5 Elementary Functions

1.3 Limits

I. 3.1 The Limit of a Sequence

1.3.2 The Limits of a Function

1.3.3 One-sided Limits

1.3.4 Limits Involving the Infinity Symbol

1.3.5 Properties of Limits of Functions

1.3.6 Calculating Limits Using Limit Laws

1.3.7 Two Important Limit Results

1.3.8 Asymptotic Functions and Small o Notation

1.4 Continuous and Discontinuous Functions

1.4.1 Definitions

1.4.2 Building Continuous Functions

1.4.3 Theorems on Continuous Functions

1.5 Further Results on Limits

1.5.1 The Precise Definition of a Limit

1.5.2 Limits at Infinity and Infinite Limits

1.5.3 Real Numbers and Limits

1.5.4 Asymptotes

1.5.5 Uniform Continuity

1.6 Additional Material

1.6.1 Cauchy

1.6.2 Heine

1.6.3 Weierstrass

1.7 Exercises

1.7.1 Evaluating Limits

1.7.2 Continuous Functions

1.7.3 Questions to Guide Your Revision

CHAPTER 2 Differential Calculus

2.1 The Derivative

2.1.1 The Tangent to a Curve

2.1.2 Instantaneous Velocity

2.1.3 The Definition of a Derivative

2.1.4 Notations for the Derivative

2.1.5 The Derivative as a Function

2.1.6 One sided Derivatives

2.1.7 Continuity of Differentiable Functions

2.1.8 Functions with no Derivative

2.2 Finding the Derivatives

2.2.1 Derivative Laws

2.2.2 .Derivative of an Inverse Function

2.2.3 Differentiating a Composite Function——The Chain Rule

2.3 Derivatives of Higher Orders a

2.4 Implicit Differentiation

2.4.1 Implicitly Defined Functions

2.4.2 Finding the Derivative of an Implicitly Defined：Function

2.4.3 Logarithmic Differentiation

2.4.4 Functions Defined by Parametric Equations

2.5 Related Rates of Change r

2.6 The Tangent Line Approximation and the Differential

2.7 Additional Material

2.7.1 Prellminary result needed to prove the Chain Rule

2.7.2 Proof of the Chain Rule

……

CHAPTER 3 The Mean Value Theorem and Applications of the Derivative

CHAPTER 4 Integral Calculus

Answers

Reference Books

《微积分1》是英文版大学数学微积分教材，分为上、下两册。上册为单变量微积分学，包括函数、极限和连续、导数、中值定理及导数的应用以及一元函数积分学等内容；下册为多变量微积分学，包括空间解析几何及向量代数、多元函数微分学、重积分、线积分与面积分、级数及微分方程初步等内容。
《微积分1》由两位国内作者和一位外籍教授共同完成，在内容体系安排上与国内主要微积分教材一致，同时也充分参考和借鉴了国外尤其是北美一些大学微积分教材的诸多特点，内容深入浅出，语言简洁通俗。