单变量微积分

CHAPTER 1 Prereqrusites for Calculus

1.1 Overview ofCalculus

1.2 Sets and Numbers

1.2.1 Sets

1.2.2 Numbers

1.2.3 The Least Upper Bound Property

1.2.4 The Extended Real Number System

1.2.5 1ntervals

1.3 Functions

1.3.1 Definition ofa Function

1.3.2 Graph ofa Function

1.3.3 Some Basic Functions and Their Graphs

1.3.4 Building New Functions

1.3.5 Fundamental Elementary Functions

1.3.6 Properties ofFunctions

1.4 Exercises

CHAPTER 2 Limits and Continruty

2.1 Rates ofChange and Derivatives

2.2 Limits of a Function

2.2.1 Definition of a Limit

2.2.2 Properties of Limits of Functions

2.2.3 Limit Laws

2.2.4 One-sided Liruits

2.2.5 Limits Involving Infinity and Asymptotes

2.3 Limits of Sequences

2.3.1 Definitions and Properties

2.3.2 Subsequences

2.4 Squeeze Theorem and Cauchy's Theorem

2.5 Infinitesimal Functions and Asymptotic Functions

2.6 Continuous and Discontinuous Functions

2.6.1 Continuity and Discontinuity

2.6.2 Continuous Functions

2.6.3 Theorems on Continuous Functions

2.6.4 Uniform Continuity

2.7 Some Proofs in Chapter

2.8 Exercises

CHAPTER 3 I'he Derivative

3.1 Derivative ofa Function at a Point

3.1.1 Instantaneous Rates of Change and Derivatives Revisited

3.1.2 One-sided Derivatives

3.1.3 A Function May Fail to Have a Derivative at a Point

3.2 Derivative as a Function

3.2.1 Graphing the Derivative of a Function

3.2.2 Derivatives of Some Basic Functions

3 .3 Derivative Laws

3.4 Derivative of an Jnverse Function

3.5 Differentiating a Composite Function - The Chain Rule

3.6 Derivatives ofHigher Orders

3.7 Implicit Differentiation

3.8 Functions Defined by Parametric and Polar Equations

3.8.1 Functions Defined by Parametric Equations

3.8.2 Polar Curves

3.9 Related Rates ofChange

3.10 The Tangent Line Approximation and the Differentia

3.10.1 Linearization

3.10.2 Differentials

3.11 Derivative Rules-Summary

3.12 Exercises

CHAPTER 4 Applications of the Derivative

4.1 Extreme Values and The Candidate Theorem

4.2 The Mean Value Theore

4.3 Monotonic Functions and The First Derivative Test

4.3.1 Monotonic Functions

4.3.2 The First Derivative Test

4.4 Extended Mean Value Theorem and the L'Hopital's Rules

4.4.1 Extended Mean Value Theorem

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CHAPTER 5 The Definite Integral

CHAPTER 6 Techniques for Integration and Improper Integrals

CHAPTER 7 Applications of the Definite Integral

CHAPTER 8 Infinite Series, Sequences, and Approximations

In recently years, more and more Chinese students are going overseas, either as exchange students, or to pursue degrees. At the same time, more and more students are coming from other countries to Chinese universities to further their studies. We've noticed that in terms of both knowledge transfer and intercultural communication, the English language has played an indispensable role. Furthermore, it is the interest of these students to have a smooth transition from one system to another, for their credits to transfer, and for them to immerse themselves in the new environment as quickly as possible. To this end, more and more Chinese schools are offering courses delivered bilingually or in English to enhance students' international outlook.
　　One of the greatest challenges in offering Chinese students a course in English is finding a suitable textbook: the textbook must seriously consider what students have done in their high schools; it must meet the national and local official course standards; and it should resonate with significant international flavors so as to benefit students.