The aim of this book is to meet the requirements of teaching Calculus in English or inbilingual education according to the customs of teaching and the present domesticconditions. It is divided into two volumes. The first volume contains Calculus of singlevariable, simple differential equations, infinite series, and the second volume contains therest.The selection of the contents is in accordance with the fundamental requirements of teachingissued by the Ministry of Education of China, and is based on the accomplishments of reformin teaching during the past ten years. The arrangement and explanation of the main contentsin this book are approximately the same as the published Chinese version with the same titleand edited in chief by the first two authors. It may help readers to understand themathematics and to improve the level of their English by reading one of them and using theother one as a reference.This book may be used as a textbook for undergraduate students in the science andengineering schools whose majors are not mathematics, and may also be suitable to thereaders at the same level.

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Introduction
Chapter 1 Theoretical Basis of Calculus
  1.1 Sets and Functions
   1.1.1 Sets and their operations
   1.1.2 Concepts of mappings and functions
   1.1.3 Composition of mappings and composition of functions
   1.1.4 Inverse mappings and inverse functions
   1.1.5 Elementary functions and hyperbolic functions
   1.1.6 Some examples for modelling of functions in practical problems
  Exercises 1.1
  1.2 Limit of Sequence
   1.2.1 Concept of limit of a sequence
   1.2.2 Conditions for convergence of a sequence
   1.2.3 Rules of operations on convergent sequences
  Exercises 1.2
  1.3 Limit of Function
   1.3.1 The concept of limit of a function
   1.3.2 The properties and operation rules of functional limits
   1.3.3 Two important limits
  Exercises 1.3
  1.4 Infinitesimal and Infinite Quantities
   1.4.1 Infinitesimal quantities and their order
   1.4.2 Equivalence transformations of infinitesimals
   1.4.3 Infinite quantities
  Exercises 1.4
  1.5 Continuous Functions
   1.5.1 The concept of continuous function and classification of discontinuous points
   1.5.2 Operations on continuous functions and the continuity of elementary functions
   1.5.3 Properties of continuous functions on a closed interval
  Exercises 1.5
Chapter 2 The Differential Calculus and Its Applications
  2.1 Concept of Derivatives
   2.1.1 Definition of derivatives
   2.1.2 Relationship between derivability and continuity
   2.1.3 Some examples of derivative problems in science and technology
  Exercises 2.1
  2.2 Fundamental Derivation Rules
   2.2.1 Derivation rules for sum, difference, product and quotient of functions
   2.2.2 Derivation rule for composite functions
   2.2.3 The derivative of an inverse function
   2.2.4 Higherorder derivatives
  Exercises 2.2
  2.3 Derivation of Implicit Functions and Functions Defined by Parametric Equations
   2.3.1 Method of derivation of implicit functions
   2.3.2 Method of derivation of a function defined by parametric equations
   2.3.3 Related rates of change
  Exercises 2.3
  2.4 The Differential
   2.4.1 Concept of the differential
   2.4.2 Geometric meaning of the differential
   2.4.3 Rules of operations on differentials
   2.4.4 Application of the differential in approximate computation
  Exercises 2.4
  2.5 The Mean Value Theorem in Differential Calculus and L’Hospital‘s Rules
   2.5.1 Mean value theorems in differential calculus
   2.5.2 L'Hospital's rules
  Exercises 2.5
  2.6 Taylor's Theorem and Its Applications
   2.6.1 Taylor's theorem
   2.6.2 Maclaurin formulae for some elementary functions
   2.6.3 Some applications of Taylor‘s theorem
  Exercises 2.6
  2.7 Study of Properties of Functions
   2.7.1 Monotonicity of functions
   2.7.2 Extreme values of functions
   2.7.3 Global maxima and minima
   2.7.4 Convexity of functions
  Exercises 2.7
  Synthetic exercises
Chapter 3 The Integral Calculus and Its Applications
  3.1 Concept and Properties of Definite Integrals
   3.1.1 Examples of definite integral problems
   3.1.2 The definition of definite integral
   3.1.3 Properties of definite integrals
  Exercises 3.1
  3.2 The Newton Leibniz Formula and the Fundamental Theorems of Calculus
   3.2.1 Newton Leibniz formula
   3.2.2 Fundamental theorems of Calculus
  Exercises 3.2
  3.3 Indefinite Integrals and Integration
   3.3.1 Indefinite integrals
   3.3.2 Integration by substitutions
   3.3.3 Integration by parts
   3.3.4 Quadrature problems for elementary fundamental functions
  Exercises 3.3
  3.4 Applications of Definite Integrals
   3.4.1 Method of elements for setting up integral representations
   3.4.2 Some examples on the applications of the definite integral in geometry
   3.4.3 Some examples of applications of the definite integral in physics
  Exercises 3.4
  3.5 Some Types of Simple Differential Equations
   3.5.1 Some fundamental concepts
   3.5.2 First order differential equations with variables separable
   3.5.3 Linear equations of first order
   3.5.4 Equations of first order solvable by transformations of variables
   3.5.5 Differential equations of second order solvable by reduced order methods
   3.5.6 Some examples of application of differential equations
  Exercises 3.5
  3.6 Improper Integrals
   3.6.1 Integration on an infinite interval
   3.6.2 Integrals of unbounded functions
  Exercises 3.6
Chapter 4 Infinite Series
  4.1 Series of Constant Terms
   4.1.1 Concepts and properties of series with constant terms
   4.1.2 Convergence tests for series of positive terms
   4.1.3 Series with variation of signs and tests for convergence
  Exercises 4.1
  4.2 Power Series
   4.2.1 Concepts of series of functions
   4.2.2 Convergence of power series and operations on power series
   4.2.3 Expansion of functions in power series
   4.2.4 Some examples of applications of power series
   *4.2.5 Uniform convergence of series of functions
  Exercises 4.2
  4.3 Fourier Series
   4.3.1 Periodic functions and trigonometric series
   4.3.2 Orthogonality of the system of trigonometric functions and Fourier series
   4.3.3 Fourier expansions of periodic functions
   4.3.4 Fourier expansion of functions defined on the interval
   4.3.5 Complex form of Fourier series
  Exercises 4.3
  Synthetic exercises
Appendix Answers and Hints for Exercises