This book mainly describes the basic concepts, fundamental theories and standard methods in essential Calculus of one single variable. It is written in English. The content is compact, popular and concise. Various types of examples and exercises have been devised in each section and chapter. The emphasis is on the fundamental computational abilities, especially the normal computation methods used in Calculus and subsequent courses.
The main contents of this book includes Functions and Limits, Differentiation of Functions of One Single Variable, Integrals of Functions of One Single Variable and Applications of Integration, Differential Equations and Infinite Series. This book can serve as a textbook in English of essential Calculus taken by non-mathematics major students and international students, and is also suitable for other readers for reference purposes.

Chapter 1 Functions and Limits
1.1 Sets and Functions
1.1.1 Sets
1.1.2 Functions
1.1.3 Properties of Functions
1.1.4 Combining Functions and Inverse Functions
1.1.5 Elementary Functions
1.2 The Limits of Functions
1.2.1 Limits of Sequence
1.2.2 Limits of Functions at Infinities and a Point
1.2.3 Calculating Limits
1.3 Two Important Limits
1.4 Infinitesimal and Infinite
1.4.1 Infinitesimal
1.4.2 Infinite
1.5 Continuous Functions
1.5.1 Continuity of Functions
1.5.2 Discontinuity
1.5.3 Properties of Continuous Functions on a Closed Interval
Chapter Review
Chapter 2 Differentiation and Applications of Differentiation
2.1 The Definition of Derivatives
2.1.1 The Tangent Line Problem
2.1.2 The Derivative of a Function
2.1.3 One-Side Derivatives
2.1.4 Differentiability and Continuity
2.2 Basic Differentiation Rules and Derivatives of Inverse Functions
2.2.1 Basic Differentiation Rules
2.2.2 Derivatives of Inverse Functions
2.3 The Chain Rule and Higher-Order Derivatives
2.3.1 The Chain Rule
2.3.2 Higher-Order Derivatives
2.4 Derivatives of Implicit and Parametric Functions
2.4.1 Implicit Differentiation
2.4.2 Derivatives of Parametric Functions
2.5 Differentials
2.5.1 Tangent Line Approximations
2.5.2 Differentials Expression
2.6 The Mean Value Theorem
2.7 L' Hospital's Rule
2.8 Monotonicity of Functions and Convexity of Curves
2.8.1 Monotonicity of Functions
2.8.2 Convexity of Curves
2.9 Maximum and Minimum Values
2.9.1 Extreme and Local Values
2.9.2 The Closed Interval Method
2.9.3 Optimization Problems
Chapter Review
Chapter 3 InDefinite Integrals
3.1 The Definition and Properties of InDefinite Integrals
3.1.1 Antiderivatives and InDefinite Integrals
3.1.2 The Properties of InDefinite Integrals
3.2 The Substitution Rule for InDefinite Integrals
3.2.1 Integration by Substitution of the First Type
3.2.2 Integration by Substitution of the Second Type
3.3 Integration by Parts
3.4 Integration of Rational Functions by Partial Fractions
Chapter Review
Chapter 4 Definite Integrals
4.1 Introduction of the Definite Integrals
4.1.1 Areas and Distances
4.1.2 The Definition of Definite Integrals
4.1.3 Geometric Meaning of Definite Integrals
4.1.4 Properties of Definite Integrals
4.2 The Fundamental Theorems of Calculus
4.2.1 Variable Upper Bound Integral Function
4.2.2 Fundamental Theorems
4.3 The Calculation of Definite Integrals
4.3.1 The Methods of Substitution and Partial Integration for Definite Integrals
4.3.2 Integration by Parts
4.4 Improper Integrals on an Infinite Interval
4.5 An Application of Integration: Areas Between Curves
Chapter Review
Chapter 5 Differential Equations
5.1 General Differential Equations and Solutions
5.2 Separable Equations
5.3 First-Order Linear Equations
5.4 Homogeneous Second-Order Linear Equations with Constant Coefficients
5.4.1 Second-Order Linear Equations
5.4.2 Homogeneous Second-Order Linear Equations with Constant Coefficients
5.4.3 Summary
Chapter Review
Chapter 6 Infinite Series
6.1 Sequences
6.1.1 Introduction to Sequences
6.1.2 Convergence and Divergence
6.1.3 Calculating Limits of Sequences
6.1.4 Bounded Monotonic Sequences
6.2 Infinite Series
6.2.1 Partial Sums
6.2.2 Geometric Series
6.2.3 Telescoping Series
6.2.4 Harmonic Series
6.2.5 The n-th Term Test for a Divergent Series
6.2.6 Combining Series
6.2.7 Adding or Deleting Terms
6.2.8 Reindexing
6.3 The Comparison Tests
6.3.1 The Direct Comparison Test
6.3.2 The Limit Comparison Test
6.4 Alternating Series
6.5 Absolute Convergence and the Rat