高等数学一信科院计算机科学与技术

1、高等数学一信科院计算机科学与技术、电子信息科学与技术专业2+2教学大纲课程名称：高等数学一课程类别：必修课      总学时：108+108   周学时：6+6  学分：6+6编者姓名：顾会玲      单位：数学与计算科学学院  职称：讲师主审姓名：王学钦 单位：数学与计算科学学院 职称：教授授课对象：本科生      专业：计算机科学与技术、电子信息科学与技术2+2合作办学年级：一年级编写日期

2、：2009/07/03一、课程目的与教学基本要求（明确基本要求的三个档次，即牢固掌握、一般掌握和一般了解）高等数学是综合性大学各学科专业必修的数学课程，因专业背景和专业要求的差别,将有针对性地采用合适的教材,制订相应的教学大纲和教学计划。这门课程服务于中山大学信息科学与技术学院2+2联合办学班级的教学大纲，为大学一年级两个学期的课程而开设，分别包含6*17=102课时；主要讲授一元微积分、多元微积分与简单的常微分方程，使学生掌握经典微积分的基本概念，了解基本知识，训练基本技能，学会以微积分为基本工具理解、分析、处理或解决工程及相关领域中产生的微积分问题，为相关后继课程的学习提供知识准备。The

3、 objective is to master ideas and methods of calculus so that the students can use these to solve the questions they meet in studying other courses and in life. It is also very important to improve quality of students in all aspects to analyse questions and solve problems.需牢固掌握的内容：1. 数列极限与函数极限，极限运算法

4、则，函数的连续性2. 基本初等函数的导数，求导法则与初等函数的求导3. 极值问题，微分中值定理及其应用，极限未定式与洛必达法则4. 一元定积分与不定积分的基本性质，微积分基本定理与积分的计算技巧，积分的应用5. 一阶常微分方程与建模，变量分离的方程与线性方程的求解6. 曲线及其参数方程，极坐标与长度问题和面积问题7. 无穷序列与级数，函数项级数与幂级数的性质，泰勒级数与麦克劳林级数8. 三维空间的几何，点积与叉积（内积与外积），向量值函数的微积分，弧长与曲率9. 多元函数及偏导数，偏导数求导法则，方向导数与梯度，极值问题与拉格朗日乘子10. 二重积分与逐次积分，面积问题与体积问题，二重积分与变

5、量替换11. 向量场，线积分与格林公式，旋度与散度，面积分与斯托克斯公式，散度定理12. 二阶线性方程的解，非齐次方程的解，方程的级数解一般掌握内容：1. 极限概念的严格定义，函数的连续性与间断点分类2. 导数概念的严格定义与几何含义3. 定积分概念的严格定义4. 级数收敛及其判别法则，幂级数收敛半径，泰勒展开与麦克劳林展开5. 重积分概念的严格定义，多重积分的变量替换技巧二、课程内容（含学时分配，要落实到“章节”，并对各章节的重点、难点内容加以必要的说明）0. Introduction to Calculus, with basics of naive set theory (2 class

6、-hours)1. Limits and Derivatives : (12 class-hours)1) Tangent and velocity problem2) Limit of a function3) Calculating limits using the limit laws4) Precise definition of a limit5) Continuity6) Limits at infinite, horizontal asymptotes7) Tangents, velocities, and other rates of change8) Derivatives9

7、) Derivative as a function.2. Differentiation Rules: (14 class-hours)1) Derivatives of polynomials and exponential functions2) Product and quotient rules3) Rates of change in the natural and social sciences4) Derivatives of trigonometric functions5) Chain rule6) Implicit differentiation7) Higher der

8、ivatives8) Derivatives of logarithmic functions9) Hyperbolic functions10) Related rates11) Linear approximations and differentials.3. Applications of Differentiation: (12 class-hours)1) Maximum and minimum values2) The mean value theorem3) How derivatives affect the shape of a graph4) Indeterminate

9、forms and LHospitals Rule5) Summary of curve sketching6) Graphing with Calculus and Calculators7) Optimization problems8) Applications to business and economics9) Newtons method10) Antiderivatives.Mid-Term Exam (3 class-hours)4. Integrals: (10 class-hours)1) Areas and distances2) The definite integr

10、al3) The fundamental theorem of calculus4) Indefinite integrals and the Net Change Theorem5) The substitution6) The logarithm defined as an integral.5. Applications of integration: (5 class-hours)1) Areas between curves2) Volumes3) Volumes by cylindrical shells4) Work5) Average value of a function.6

11、. Techniques of Integration: (10 class-hours)1) Integration by parts2) Trigonometric integrals3) Trigonometric substitution4) Integration of rational functions by partial fractions5) Strategy for integration6) Integration using tables and computer algebra systems7) Approximate integration8) Improper

12、 integrals.7. Further applications of integration: (7 class-hours)1) Arc length2) Area of a surface of revolution3) Applications to physics and engineering4) Applications to economics and biology5) Probability.8. Differential Equations: (10 class-hours)1) Modeling with differential equations2) Direc

13、tion fields and Eulers Method3) Separable equations4) Exponential growth and decay5) The logistic equation6) Linear equations7) Predator-Prey systems.9. Parametric Equations and Polar Coordinates: (8 class-hours)1) Curves defined by parametric equations2) Calculus with parametric curves3) Polar coor

14、dinates4) Areas and lengths in polar coordinates5) Conic sections6) Conic sections in polar coordinates.-First Term ends hereBrief Review of Term-1 and Introduction to Term-2 (2 class-hours)10. Infinite sequences and series: (16 class-hours)1) Sequences2) Series3) The integral test and estimates of

15、sum4) Comparison tests5) Alternating series6) Absolute convergence and the ratio and root test7) Strategy of for testing series8) Power series9) Representation of functions as power series10) Taylor and Maclaurin series11) The binomial series12) Applications of Taylor polynomials.11. Vectors and the

16、 Geometry of Space: (12 class-hours)1) Three-dimensional coordinate systems2) Vectors3) The dot product4) The cross product5) Equations of lines and planes6) Cylinders and quadric surfaces7) Cylindrical and spherical coordinates.12. Vector Functions: (8 class-hours)1) Vector functions and space curv

17、es2) Derivatives and integrals of vector functions3) Arc length and curvature4) Motion in space: velocity and acceleration.13. Partial Derivatives: (13 class-hours)1) Functions of several variables2) Limits and continuity3) Partial derivatives4) Tangent planes and linear approximations5) The chain r

18、ule6) Directional derivatives and the gradient vector7) Maximum and minimum values8) Langrange multipliers.Mid-Term Exam (3 class-hours)14. Multiple Integrals: (15 class-hours)1) Double integral over rectangle2) Iterated integrals3) Double integral over general regions4) Double integrals in polar coordinates5) Applications of double integrals6) Surface area7) Triple integrals8) Triple integrals in cylindrical and spherical coordinates9) Change of variables in multiple integrals.15. Vector Calculus: (15 class-hours)1) Vector fields2) Line integrals3) The fundamental theorem for line inte