AP微积分课程设计.docx

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1、北京中加学校AP微积分课程的实施方案正值北京中加学校建校十五周年之际，为了实现百年名校的伟大目标，无论是在管理，还是在教育教学上都需要不断健全和完善体制机制。然而，学科课程的建设和更新势必首当其冲，迫在眉睫。暑假在大连举行的北京中加学校数学课程和教学多元化的探究教学研讨会为北京中加学校的学科建设开了先河，也奠定了思想基础。借此良机，对于北京中加学校的特色学科之一，AP微积分，我们借鉴过去的教学经验，整合国内外教学资源，依据美国大学理事会AP微积分的课程标准，拟定了关于AP微积分课程的教学设想。一、指导思想本课程是北京中加学校为学生开设的一门国际数学专业基础课。开设本课程的目的，在于以美国大学理

2、事会规定的AP微积分课程标准为指导，按照理论与实践相结合的原则，通过对微积分基本原理及规律的讲授，使学生系统掌握极限、连续、导数和积分等知识的基本原理、基本内容和基本方法，对微积分在经济活动中的应用有比较清晰的了解，提高学生专业词汇量和阅读英语原版书籍的能力，拓宽学生国际数学视野，使学生体验到数学的价值和美学认知。课内学时144，4学分，从高一第一学期开始开设，高二第二学期结束，将近两个学年授完。二、课程目标AP微积分是在高中学习阶段有余力、有能力、成绩优秀的学生有机会先修的美国大学基础课程以获得美国大学学分专业的必修课。要求学生在学完本课程后，掌握本课程的基本原理、基本内容、基本方法及基本知

3、识，并具有对所学的微积分知识进行现实理解和实际应用的能力，从而顺利通过AP考试。据此，本课程考核着重于基本知识的掌握、理解和应用分析能力两个方面。在各章的考核要求中，有关基本概念、基本理论、基本公式、应用分析能力的内容按“识记、理解、简单应用和综合应用”四个层次要求。三、教学进度北京中加学校AP微积分教学内容及其进度计划学期普通班国际班AP微积分课时分配高一第一学期第一模块集合（4课时）函数与基本初等函数（32课时）解析几何（9课时）复合函数、反函数以及作图计算器的使用高中课程：6课时/周，共54课时；AP微积分：2课时/周，共18课时；第二模块直线与圆的方程（9课时）圆锥曲线（8课时）三角函

4、数（16课时）极限极限的运算法则高中课程：6课时/周，共54课时；AP微积分：2课时/周，共18课时；第二学期第三模块三角恒等变换反三角函数导数导数的基本公式高中课程：2课时/周，共18课时；AP微积分：6课时/周，共54课时；第四模块立体几何导数的运算法则高中课程：2课时/周，共18课时；AP微积分：6课时/周，共54课时；高二第一学期第五模块常用逻辑用语平面向量解三角形导数的应用高中课程：2课时/周，共18课时；AP微积分：6课时/周，共54课时；第六模块数列不等式积分方程微分方程高中课程：2课时/周，共18课时；AP微积分：6课时/周，共54课时；第二学期第七模块复数统计计数原理AP微积

5、分总复习AP微积分AB考试高中课程：2课时/周，共18课时；AP微积分：6课时/周，共54课时；第八模块参数方程极坐标AP微积分BC高中课程：4课时/周，共36课时；AP微积分BC：4课时/周，共36课时；高三第一学期总复习总复习毕业会考AP微积分BC高中课程：4课时/周，共36课时；AP微积分BC：4课时/周，共36课时；第二学期微积分其它大学预修课程AP微积分BCAP微积分BC考试高中课程：4课时/周，共36课时；AP微积分BC：4课时/周，共36课时；四、课程内容Chapter 2 Limits and Derivatives第二章 极限和导数Teaching Content教学内容Te

6、aching Requirements and Objectives教学要求和目标Time学时2.1 The Tangent and Velocity Problems2.1切线和速率问题The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration.2.2 The Limit of a Function2.2 函数的极限The student will define an

7、d apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits.2.3 Calculating Limits Using the Limit Laws2.3利用极限法则计算极限2.5 Continuity2.5 连续性The student will state the definit

8、ion of continuity and determine where a function is continuous or discontinuous. This will include continuity at a point; continuity over a closed interval; and graphical interpretation of continuity and discontinuity.2.6 Limits at Infinity; Horizontal Asymptotes2.6 无穷远处极限和水平渐近线The student will defi

9、ne and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetr

10、y, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing.The student will also define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limi

11、ts, and nonexistent limits.2.7 Tangents, Velocities, and Other Rates of Change2.7切线、速度和其它的变化率2.8 Derivatives2.8导数The student will find the derivative of an algebraic function by using the definition of a derivative. This will include investigating and describing the relationship between differentiab

12、ility and continuity.2.9 The Derivative as a Function2.9 导函数Review复习Chapter 3 Differentiation Rules第三章 导数法则Teaching Content教学内容Teaching Requirements and Objectives教学要求和目标Time学时3.1 Derivatives of Polynomials and Exponential Functions3.1多项式函数和指数函数的导数The student will apply formulas to find the derivati

13、ve of algebraic, trigonometric, exponential, and logarithmic functions and their inverses.3.2 The Product and Quotient Rules3.2导数的乘法和除法运算法则The student will apply formulas to find the derivative of the sum of elementary functions.3.3 Rates of Change in the Natural and Social Sciences3.3自然科学和社会科学中的变化率

14、Students will be able to understand the mathematical modeling process of derivatives (rates of changes) in the real world3.4Derivatives of Trigonometric Functions3.4三角函数的导数Students will be able to use the differentiation rules of trigonometric functions 3.5 The Chain Rule3.5链式法则The student will appl

15、y formulas to find the derivative of the sum, product, quotient, inverse and composite (chain rule) of elementary functions.3.6 Implicit Differentiation3.6隐函数求导The student will find the derivative of an implicitly defined function.3.7 Higher Derivatives3.7高阶导数The student will find the higher order d

16、erivatives of algebraic, trigonometric, exponential, and logarithmic functions.3.8Derivative of Logarithmic Functions3.8对数函数的导数The student will use logarithmic differentiation as a technique to differentiate non-logarithmic functions.3.9 Hyperbolic Functions3.9 双曲函数The student will be able to unders

17、tand the definition of hyperbolic functions, and solve for its derivatives.3.11 Linear Approximations and Differentials3.11 线性逼近和微分The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change,

18、 Newtons method, differentials and linear approximations, and optimization problems.Review复习Chapter 4 Applications of Differentiation第四章 导数的应用Teaching Content教学内容Teaching Requirements and Objectives教学要求和目标Time学时4.1 Maximum and Minimum Values4.1 极大值和极小值The student will be able to understand extreme v

19、alues of a function, find critical values of a function and find extreme values of a function.4.2 The Mean Value Theorem4.2 中值定理The student will state (without proof) the Mean Value Theorem for derivatives and apply it both algebraically and graphically.4.3 How Derivative Affect the Shape of a Graph

20、4.3导数是如何改变图像的形状The student will graph these functions using a graphing calculator, including understanding and using the First Derivative Test and the Second Derivative Test to determine mins and maxs.4.4 Indeterminate Forms and L Hospitals Rules4.4 不定式和洛必达法则The student will use lHopitals rule to fi

21、nd the limit of functions whose limits yield the indeterminate forms: 0/0 and infinity/infinity4.7 Optimization Problems4.7最优化问题The student will be able to use derivatives to solve optimization problems4.9 Newtons Method4.9牛顿法则The student will be able to use Newtons method to approximate roots of an

22、 equation.4.10Antiderivatives4.10 原函数（反导数）The student will be able to understand the concept of an antiderivative, the geometry of the antiderivative and that of slope fields and also work rectilinear motion problems with antiderivativesReview复习Chapter 5 Integrals第五章 积分Teaching Content教学内容Teaching R

23、equirements and Objectives教学要求和目标Time学时5.1 Areas and Distances5.1 面积和距离The student will identify the properties of the definite integral. This will include the Fundamental Theorem of Calculus and the definite integral as an area and as a limit of a sum as well as the fundamental theorem.5.2 The Defi

24、nite Integral5.2不定积分The student will compute an approximate value for a definite integral. This will include numerical calculations using Riemann Sums and the Trapezoidal Rule.5.3 The Fundamental Theorem of Calculus5.3微积分基本定理The student will identify the properties of the definite integral. This wil

25、l include the Fundamental Theorem of Calculus and the definite integral as an area and as a limit of a sum as well as the fundamental theorem. The integral from a to x of f(t)d(t) dt/dx = f(x)5.4 Indefinite Integrals and the Net Change Theorem5.4 不定积分和原函数定理The student will find the indefinite integr

26、al of algebraic, exponential, logarithmic, and trigonometric functions. 5.5 The Substitution Rule5.5 换元积分法The student will find the indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions. The special integration techniques of substitution (change of variables) and in

27、tegration by parts will be included.Review复习Chapter 6 Applications of Integration 第六章 积分的应用Teaching Content教学内容Teaching Requirements and Objectives教学要求和目标Time学时6.1 Areas Between Curves6.1 曲边面积The student will apply the definite integral to solve problems. These problems will include finding distance

28、 traveled on a line and velocity from acceleration with initial conditions, growth and decay problems, solutions of separable differential equations, the average value of a function, area between curves, volumes of solids of revolution about the axes or lines parallel to the axes using disc/washer a

29、nd shell methods, and volumes of solids with known cross-sectional areas.6.2 Volumes6.2 体积The student will apply the definite integral to solve problems. These problems will include area between curves, volumes of solids of revolution about the axes or lines parallel to the axes using disc/washer an

30、d shell methods, and volumes of solids with known cross-sectional areas.6.3 Volumes by Cylindrical Shells6.3 圆柱体体积The student will apply the definite integral to solve problems. These problems will include area between curves, volumes of solids of revolution about the axes or lines parallel to the a

31、xes using disc/washer and shell methods, and volumes of solids with known cross-sectional areas.6.4 Work6.4 物体功The student will apply the definite integral to solve problems. These problems will include finding distance traveled on a line and velocity from acceleration with initial conditions, growt

32、h and decay problems, work done.6.5 Average Value of a Function6.5 实函数均值The student will apply the definite integral to solve problems. These problems will include the average value of a function.6.6 Density Function6.6 密度函数The student will apply the definite integral to solve problems. These proble

33、ms will include finding distance traveled on a line and velocity from acceleration with initial conditions, growth and decay problems.Review复习Chapter 7 Techniques of Integration第七章 积分技巧Teaching Content教学内容Teaching Requirements and Objectives教学要求和目标Time学时7.1 Integration by Parts7.1 分部积分法The student w

34、ill find the definite and indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions. The special integration techniques of substitution (change of variables) and integration by parts will be included.7.2 Trigonometric Integrals7.2 三角函数积分7.3 Trigonometric Substitution7.3

35、 三角函数替换7.4 Integration of Rational Functions by Partial Fractions7.4 有理函数的分部积分Review复习五、考核方式为了彰显我校“多一把尺子，多一位人才”的教育教学理念，本课程采用形成性考核与终结性考核相结合的方式。（一）形成性考核内容 本课程的形成性考核具体内容分为学习内容考核和学习过程考核。学习内容考核：1 平时作业：任课教师课堂上集中布置的随堂作业和课后作业。同时，不同年级的教师在布置作业时也可根据各班的实际情况适当加以细化、加入一些限制性条件或进一步的要求。2阶段性测验：是根据课程教学安排布置的阶段性综合测验。阶段性测

36、验主要考查学生在一个单元内的学习状态，其测查内容属于教学中的重点，涉及到讲过的大部分基本概念、基本原则和基本方法。3课堂讨论：在课程教学过程中，在指定时间，围绕一定的主题，对课程的重点、难点内容，集中进行若干次课堂讨论。课堂讨论的题目提前一周告知学生，让学生以学习小组为单位进行讨论前的准备，每学习小组推选一人做代表性发言并提供以小组名义提交的讨论提纲。指导教师根据各小组学生参与程度、发言情况及讨论提纲给予评价并以小组形式给予评定成绩。学习过程考核：1建立学习小组：参与试点的班级应该以5-6名学生组成学习小组，指定学习组长并报班主任及指导教师。根据学校和任课老师的要求，有计划有目的地开展学习活动

37、，完成并按时上交形成性考核中要求以小组为单位进行的作业。小组学习应该有尽可能详细的学习过程记录，反映学生在学习小组活动中的内容、体会及存在问题，期末交到指导教师处，作为指导教师对学生形成性考核成绩评定的依据之一。2. 课堂出勤：学生是否按时上课，在很大程度反映了该学生对待学习的态度；学生在课堂上是否听从教师的指令也都应该考虑在考勤的范围内。（二）终结性考试终结性考试主要考核学生对AP微积分的基本理论、基本知识、基本概念的理解与把握，总分100分。终结性考试的题型严格参照AP考试的题型和设置。 （三）成绩评定形成性考核成绩占总成绩的60%。形成性考核成绩由任课教师根据学生实际表现情况评定，由教学处责任教师评审，最终确定学生的课程形成性考核成绩。北京中加学校对形成性考核进行抽样检查。终结性考核占课程总成绩的40%，见表2，然后通过计算机在线登分并汇总。表2 北京中加学校学生学业成绩评定分配表项目形成性考核内容终结性考核内容作业测验讨论小组出勤比例15%15%10%10%10%40%学期成绩为期中成绩和期末成绩各占50%。根据北京中加学校的规定，本课程实行形成性考核成绩和终结性考试成绩的总综合成绩达到60分及以上（及格），即可获得本课程相应学分。