Beijing University of Chemical Technology 《Calculus》Syllabus 、Course Information Course Code:MAT138000T Course Name(in Chinese): Gao Deng Shu Xue Course Name(in Enalish): Calculus Course Category: Rasic theoretical course Target Students: Specialties of economic and management Term available:I Spring and autumn Total Credit Hours: 160 credit hours theoretical course:160 credit hours) Total Credits. 10 credits Prerequisites (Course Code): Parallels (Course Code): linear algebra The course is a basic mathematical course for the student of economic and management.The core content is calculus theory in the real number field.The course introduces the limit of univariable function and Course Description: infinite series,differential equation and spatial analytic geometry.The training abilities of logical reasoning. summarizing problem with mathematics,spatial imagine and self-study are the target of the course Textbooks Recommended: Wu Chuansheng,《Calculus》Beijing,.higher education press,2009,4,second edition. Department of applied mathematics of tongii Supplementary Materials university Calculus(I,II)(fifth edition) Beijing,higher education press,2002,7. ll Learning Goals and Objectives Training the abilities of Logical reasoning b) space imagine. c) Abstract summarize d) Self study, Skilled calculations, f Analysis problem and solve problem with calculus. Ill,Course Contents and Requirements The contents are divided into two parts.The one is higher demand with the low line mark and must be grasped with students.The other is lower demand,but this is not lack. Chapter 1.Function(4 Class hours)
Beijing University of Chemical Technology 《 Calculus 》Syllabus Ⅰ、Course Information Course Code： MAT138000T Course Name(in Chinese)： Gao Deng Shu Xue Course Name(in English)： Calculus Course Category： Basic theoretical course Target Students： Specialties of economic and management Term Available： Spring and autumn Total Credit Hours： 160 credit hours ( theoretical course : 160 credit hours) Total Credits： 10 credits Prerequisites（Course Code）： Parallels（Course Code）： Linear algebra Course Description： The course is a basic mathematical course for the students of economic and management . The core content is calculus theory in the real number field. The course introduces the limit of univariable function and multivariable function, continuity, derivatives, integration, infinite series, differential equation and spatial analytic geometry. The training abilities of logical reasoning, summarizing problem with mathematics, spatial imagine and self-study are the target of the course. Textbooks Recommended： Wu Chuansheng, 《Calculus》 Beijing, higher education press, 2009,4,second edition. Supplementary Materials: Department of applied mathematics of tongji university Calculus(Ⅰ,Ⅱ) (fifth edition), Beijing, higher education press, 2002,7. Ⅱ、Learning Goals and Objectives Training the abilities of a) Logical reasoning , b) Space imagine, c) Abstract summarize, d) Self study, e) Skilled calculations, f) Analysis problem and solve problem with calculus. Ⅲ、Course Contents and Requirements The contents are divided into two parts. The one is higher demand with the low line mark and must be grasped with students. The other is lower demand, but this is not lack. Chapter 1. Function (4 Class hours)
Concepts of functions;geometric characteristics;inverse function;composite function;primitive function;formation of simple function relationship. Basic Requirement 1. master the representation of functions;be able to establish a e practice 2. the following characteristics of a function:monotonicity,boundedness,odd/even, 3 mposite function,inverse function implicit function.and piecewise Master the properties and gtaph of a basic primitive function.understand the concent of basic functions Chapter 2.Limit and Continuity(16 Class hours) orhee for functions,concept and re esimal sma the concept of functionscontinuity:types of points of 2. IInderstand the concents and n erties of the infinitesimal small master the method of comparin the infinitesimal small:understand the concept of the infinitesimal large and its relationship with the infinitesimal small Comprehend the properties of limits and the two principles for existence of limits:master the law for four basic operations of limits:be able to utilize the two important limits. Understand the concept of continuity:be able to determine the types of points of discontinuity. Comprehend the properties of a continuous function and the continuity of primitive functions, understand the properties and application of continuous functions on a closed interval. Chapter 3.Derivative and Differentiation (14 Class hours) the relationship vative of com function:differentiation and its operation:high order derivative:the application derivative in economics. Basic Requirement 1.Understand the concept of derivatives and the relationship between differentiability and continuity be able to interpret derivatives from geometric and economic perspective(including the concept of margin and elasticity). 2. Master the derivative formula for primitive functions master the principles of four basic operations 3 the concept of high order able to obtain the high order derivative for 4 mple ehend the cor derivative and differentiation the uniformity on the formation of first-order differentiatio n:be able to derive differentiation. Chan n and the Differentiation Mean Value Theorem:Taylor Formula:L'hospital's Rule:functions'monotonicity. convexity and concavity;extreme value,minimum and maximum values:drawing a function. Basic Requirement Understand the Rolle's Theorem,Legrand's Mean Value Theorem,Cauchy's Mean Value Theorem: Master the od of d ng t of a fur nd its ster the e.minimum a nd ma im ole to lication 4. Be able to determine the convexity and concavity of a function using derivatives:be able to solve the inflection point and asymptotic lines Master the basic method and procedure of drawing a function;be able to make graphs for simple functions. Chapter 5.Indefinite Integration(10 class hours)
Concepts of functions; geometric characteristics; inverse function; composite function; primitive function; formation of simple function relationship. Basic Requirement 1. Understand the concepts of functions, master the representation of functions; be able to establish a function relationship for simple practice. 2. Comprehend the following characteristics of a function: monotonicity, boundedness, odd/even, and periodicity. 3. Understand the concepts of composite function, inverse function, implicit function, and piecewise function. 4. Master the properties and graph of a basic primitive function, understand the concept of basic functions. Chapter 2. Limit and Continuity (16 Class hours) Definition and properties of limit for series; concepts, properties and basic operations principles of limits for functions; concept and relationship of the infinitesimal small and the infinitesimal large; the comparison among the infinitesimal small; two important limits; the concept of functions’ continuity; types of points of discontinuity; the property of continuity on a closed interval Basic Requirement 1. Comprehend the concepts of limits for series and for functions. 2. Understand the concepts and properties of the infinitesimal small; master the method of comparing the infinitesimal small; understand the concept of the infinitesimal large and its relationship with the infinitesimal small. 3. Comprehend the properties of limits and the two principles for existence of limits; master the law for four basic operations of limits; be able to utilize the two important limits. 4. Understand the concept of continuity; be able to determine the types of points of discontinuity. 5. Comprehend the properties of a continuous function and the continuity of primitive functions; understand the properties and application of continuous functions on a closed interval. Chapter 3. Derivative and Differentiation (14 Class hours) Definition of derivative; the geometric and economic interpretation of derivative; the relationship between differentiability and continuity; the formula for derivative operations; the derivative of composite function; differentiation and its operation; high order derivative; the application derivative in economics. Basic Requirement 1. Understand the concept of derivatives and the relationship between differentiability and continuity; be able to interpret derivatives from geometric and economic perspective (including the concept of margin and elasticity). 2. Master the derivative formula for primitive functions; master the principles of four basic operations with respect to derivatives and the method of obtaining derivatives for composite functions, implicit functions and inverse functions; master the method of obtaining derivatives through logarithm. 3. Comprehend the concept of high order derivatives; be able to obtain the high order derivative for simple functions. 4. Comprehend the concept of differentiation, the relationship between derivative and differentiation, the uniformity on the formation of first-order differentiation; be able to derive differentiation. Chapter 4. Mean value theorem and the application of derivatives Differentiation Mean Value Theorem; Taylor Formula; L'hospital's Rule; functions’ monotonicity, convexity and concavity; extreme value, minimum and maximum values; drawing a function. Basic Requirement 1. Understand the Rolle’s Theorem, Legrand’s Mean Value Theorem, Cauchy’s Mean Value Theorem; master the simple application of those theorems. 2. Be able to calculate the limit using L'hospital's Rule. 3. Master the method of determining the monotonicity of a function and its application; master the method of solving extreme, minimum and maximum value of a function, be able to solve simple problems for application. 4. Be able to determine the convexity and concavity of a function using derivatives; be able to solve the inflection point and asymptotic lines. Master the basic method and procedure of drawing a function; be able to make graphs for simple functions. Chapter 5. Indefinite Integration (10 class hours)
Concept of indefinite integration,basic integration formulas,the methods of change of variables, n by part. 2 Master the methods of change of variables and integtation by part for indefinite integrals. Chapter 6.definite Integration(16 class hours) Concept of definite integration,basic properties of definitive integration,basic integration formulas,the Newton-Leibniz formula,the methods of change of variables,integration by part.Application of definite equireme ot of definite and n of ir 9 al a Be famili正 n-Leibniz for eorem M erthemethodsofchangeofvariablesandintegrationby.partforhothindefiniteanddefinite nt Be able to compute the area of the plane graph and the volume of the rotation bodies.Be able to solve the simple applying problem about economics. 6. Comprehend the concept of general integration.Be able to compute the general integration. Chapter 7.Multivariate Differentiation and Integration (38 class hours) Conce vec curved surtace plane and multvarable function,directional derivatives, tion,partial derivative for c Basic tio high orders partia derivative,extreme value equirem +h。 etric idea of multi 2 Co and the erties of continuous function on a closed bounded set mprehend omrehend the concents of partial derivative and total differentiation Be able to compute 3 first order and second order partial derivative and total differentiation for composite functions.Be able to solve the partial derivative for implicit functions 4. Understand the concepts of extreme value and conditional extreme value of a multivariate function be able to solve the extreme value of a bi-variate function:know the Legrand's multiplier method:be able to solve some simple multivariate functions'maximum and minimum value problem and simple application problem in practice 5. omprehend the conc nstant series,positiv eseries,real number series,power series Comprehend the concepts of convergence diver gence and summation of an infinite series Master the basic properties of an infinite series and the necessary condition for its convergence Be familiar with geometric series and p-series and their convergence property Master its comparison test and ratio test 3. Understand the Leibniz theorem for alternate series:Know the concepts of absolute convergence and conditional convergence of an infinite series and their relationshin. Be able to calculate the convergence ratio.convergent interval and convergent fields Comprehend the basic properties of power series on its convergence interval (continuity of sum function.successive term differentiation and integration)Master the method to solve certain 6. urin expansion of to expand some basic functions in to power series. Chapter 9.Ordinary Differe s orde homogenous linear differential equation ith ic Reo quir al equation applications in the economic ent
Concept of indefinite integration, basic integration formulas, the methods of change of variables, integration by part. Basic Requirement 1. Understand the concept and properties of indefinite integrals. Master the basic properties and basic formula of indefinite integration. 2. Master the methods of change of variables and integration by part for indefinite integrals. Chapter 6. definite Integration (16 class hours) Concept of definite integration, basic properties of definitive integration, basic integration formulas, the Newton-Leibniz formula, the methods of change of variables, integration by part. Application of definite integration, generalized integration. Basic Requirement 1. Comprehend the concept of definite integration, basic properties and mean value of integration. 2. Understand the definite integral as a function of its integration limits and its derivative theorem. 3. Be familiar with the Newton-Leibniz formula. 4. Master the methods of change of variables and integration by part for both indefinite and definite integrals. 5. Be able to compute the area of the plane graph and the volume of the rotation bodies. Be able to solve the simple applying problem about economics. 6. Comprehend the concept of general integration. Be able to compute the general integration. Chapter 7. Multivariate Differentiation and Integration (38 class hours) Concept of vector Algebra, curved surface , plane and multivariable function, directional derivatives, partial derivative and total differentiation, partial derivative for composite functions and implicit functions, higher orders partial derivative, extreme value of a multivariate function, double integrals. Basic Requirement 1. Comprehend the concept and the geometric idea of multivariable function. 2. Comprehend concept of limit and continuity for bivariate functions.Comprehend properties of continuous function on a closed bounded set. 3. Comprehend the concepts of partial derivative and total differentiation. Be able to compute the first order and second order partial derivative and total differentiation for composite functions,. Be able to solve the partial derivative for implicit functions. 4. Understand the concepts of extreme value and conditional extreme value of a multivariate function; be able to solve the extreme value of a bi-variate function; know the Legrand’s multiplier method; be able to solve some simple multivariate functions' maximum and minimum value problem and simple application problem in practice. 5. Comprehend the concepts of double integrals and their properties. Master the method to calculate the double integrals (under Euclidean and polar coordinate system). Chapter 8. Series (16 class hours) Concepts and properties of constant series, positive series, real number series, power series. Basic Requirement 1. Comprehend the concepts of convergence, divergence and summation of an infinite series. 2. Master the basic properties of an infinite series and the necessary condition for its convergence. Be familiar with geometric series and p-series and their convergence property. Master its comparison test and ratio test. 3. Understand the Leibniz theorem for alternate series; Know the concepts of absolute convergence and conditional convergence of an infinite series and their relationship. 4. Be able to calculate the convergence ratio, convergent interval and convergent fields . 5. Comprehend the basic properties of power series on its convergence interval (continuity of sum function , successive term differentiation and integration ). Master the method to solve certain simple power series. 6. Be able to use Maclaurin expansion of to expand some basic functions in to power series. Chapter 9. Ordinary Differential Equation (16 class hours ) Basic concept, differential equation, the second order homogenous linear differential equation with constant coefficients, differential equation applications in the economics. Basic Requirement
1.Comprehend the concepts of differential equation,and its order,solution,general solution, particular solution and initial conditions. 2. Be able to solve the separation of variables.the homogenous and the first order linear differential 3. Master e method to solve the second order homogenous linear differential equation with constant 4. Be abcr olve the second order non-ho nogenous linear differential equation with constant able t h 6 D form of to u simple economical problems Chapter 10.Diff (10 cla c co ncept o th ence equa ,the first order linear difference equation with constant coefficients,the Basic rec rence equati 1. Comr hend the concepts of the difference,difference equation,general solution and particular 2. Master methods of the first order linear difference equation with constant coefficients be able to solve the second order linear difference equation with constant coefficients. Be able to solve the simple application problem about the economics. Practical Requirements 1. The exercises are divided into three parts according to the degree of difficulty: 1 the exercises after the every section in the teaching material. the exercises after the every chapter in the teaching material. 3 the exercise after the every chapter in the supplementary material The exercises are submitted in the period of the week The 20%in the course grade is the exercises. The exercises must be finished by oneself.The plagiarism will be strictly forbad V、Assignment There are two examinations in every school term.The one is held in the middle of the school term,but the other in the end of the school term.There are 120 minutes for ach exam very shoem a"scndfinal exam which has the came degree of diftculty of the final e M、Assessment Method 1.Midterm,final exam are both used hundred mark system 2.Exercises:20%,the examination in the middle of the school term:10%,the examination in the end of the school term:70%. Coordinator Yang Yongy 2010-12-19
1. Comprehend the concepts of differential equation, and its order, solution, general solution, particular solution and initial conditions. 2. Be able to solve the separation of variables, the homogenous and the first order linear differential equations. 3. Master the method to solve the second order homogenous linear differential equation with constant coefficients. 4. Be able to solve the second order non-homogenous linear differential equation with constant coefficients whose free term has a form of . 5. Be able to use differential equation to solve simple economical problems. Chapter 10 . Difference Equation (10 class hours ) Basic concept of the difference equation, the first order linear difference equation with constant coefficients, the application of the difference equation. Basic Requirement 1. Comprehend the concepts of the difference, difference equation, general solution and particular solution. 2. Master methods of the first order linear difference equation with constant coefficients. Be able to solve the second order linear difference equation with constant coefficients. 3. Be able to solve the simple application problem about the economics. Ⅳ、Practical Requirements i. The exercises are divided into three parts according to the degree of difficulty： 1. the exercises after the every section in the teaching material. 2. the exercises after the every chapter in the teaching material. 3. the exercise after the every chapter in the supplementary material. ii. The exercises are submitted in the period of the week. iii. The 20% in the course grade is the exercises. The exercises must be finished by oneself. The plagiarism will be strictly forbad. Ⅴ、Assignment There are two examinations in every school term. The one is held in the middle of the school term, but the other in the end of the school term. There are 120 minutes for each exam. Every school term has a "second final exam" which has the same degree of difficulty of the final exam, but voluntarily enroll. Ⅵ、Assessment Method 1. Midterm, final exam are both used hundred mark system . 2. Exercises: 20%, the examination in the middle of the school term: 10%, the examination in the end of the school term: 70%. Coordinator Yang Yongy 2010-12-19
