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