AP微积分课程设计

北京中加学校AP微积分课程的实行方案
正当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，不论是在管理，仍是在教育教课上都需要不停健全和完美体系体制。

但是，学科课程的建设和更新必然首
当其冲，火烧眉毛。

暑期在大连举行的《北京中加学校数学课程和教课多元化的研究》教
学商讨会为北京中加学校的学科建设开了先河，也确立了思想基础。

借此良机，对于北京
中加学校的特点学科之一，AP 微积分，我们借鉴过去的教课经验，整合国内外教课资源，依照美国大学理事会AP微积分的课程标准，制定了对于AP微积分课程的教课假想。

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，
在于以美国大学理事会规定的 AP?微积分课程标准为指导，依照理论与实践相联合的原则，经过对微积分基来源理及规律的讲解，使学生系统掌握极限、连续、导数和积分等知识的
基来源理、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楚的认识，提升
学生专业词汇量和阅读英语原版书本的能力，拓宽学生国际数学视线，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，快要两个学
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、成绩优异的学生有时机先修的美国大
学基础课程以获取美国大学学分专业的必修课。

要修业生在学完本课程后，掌握本课程的
基来源理、基本内容、基本方法及基本知识，并拥有对所学的微积分知识进行现实理解和
实质应用的能力，进而顺利经过AP考试。

据此，本课程查核侧重于基本知识的掌握、理解和应用剖析能力两个方面。

在各章的查核要求中，相关基本观点、基本理论、基本公式、应用剖析能力的内容按“识记、理解、简单应用和综合应用”四个层次要求。

三、教课进度
北京中加学校 AP微积分教课内容及其进度计划
普
学期国际班AP微积分课时分派
通
班
第第会合（ 4 课时）复合函数、反函数以高中课程： 6课
高一一
及作图计算器的使用时/ 周，共 54课
函数与基本初等函数
时；
一学模
（32 课时）
期块
AP 微积分： 2
课
分析几何（ 9 课时）
时/ 周，共 18课
时；
第直线与圆的方程（9极限高中课程： 6课
时/ 周，共 54课二课时）
极限的运算法例
时；
模
圆锥曲线（ 8 课时）
块
AP 微积分： 2课
三角函数（ 16 课时）
时/ 周，共 18课
时；
第第三角恒等变换导数高中课程： 2课
时/ 周，共 18课二三
反三角函数导数的基本公式
学模时；
期块
AP 微积分： 6课
时/ 周，共 54课
时；
第立体几何导数的运算法例高中课程： 2课
四时/ 周，共 18课
模时；
块
AP 微积分： 6课
时/ 周，共 54课
时；
第第常用逻辑用语导数的应用高中课程： 2 课
高五时/ 周，共18 课一平面向量
模时；
二
学块解三角形
AP 微积分： 6 课期时/ 周，共54 课
时；
第数列积分方程高中课程： 2 课
六不等式微分方程时/ 周，共 18课模时；
块
AP 微积分： 6课
时/ 周，共 54课
时；
第第复数AP微积分总复习高中课程： 2课
二七
统计AP微积分 AB考试
时/ 周，共 18课模时；
学块计数原理
AP 微积分： 6课期时/ 周，共 54课
时；
第参数方程AP微积分 BC高中课程： 4课
八时/ 周，共 36课极坐标
模时；
块
AP 微积分 BC：4
课时 / 周，共36
课时；
第总总复习AP微积分 BC高中课程： 4 课高时/ 周，共36 课一复毕业会考
三时；
习
学
AP 微积分 BC：4
期课时 / 周，共
课时；
36第微其余大学预修课程AP微积分 BC高中课程： 4课
二积
AP微积分 BC考试
时/ 周，共 36课分时；
学
AP 微积分 BC：4期课时 / 周，共36
课时；
四、课程内容
Chapter 2 Limits and Derivatives
第二章极限和导数
Teaching Teaching Requirements and Objectives Time
Content
教课要乞降目标学时教课内容
The Tangent The student will apply the derivative
and Velocity to solve problems,including tangent
Problems and normal lines to a curve,curve
切线和速率问题
sketching, velocity, acceleration.
The Limit of a Function The student will define and apply
properties of limits of functions.
the
函数的极限This will include limits of a constant, Calculating
sum,product,quotient,one-sided
limits,limits at infinity,infinite Limits Using
limits, and nonexistent limits.
the Limit Laws
利用极限法例计
算极限
Continuity The student will state the definition
of continuity and determine where a 连续性
function is continuous or
discontinuous.
This will include continuity at a
point;continuity over a closed
interval;and graphical interpretation
of continuity and discontinuity.
Limits at Infinity; Horizontal Asymptotes The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these
无量远处极限和
水平渐近线Tangents, Velocities,and Other Rates of Change
切线、速度和其
它的变化率functions using a graphing calculator. Properties of functions will include
domains, ranges, combinations, odd,
even, periodicity,
symmetry, 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 limits,and nonexistent limits.
Derivatives The student will find the derivative of
导数an algebraic function by using the
definition of a derivative.
The Derivative
as a Function This will include investigating and
describing the relationship between
导函数
differentiability and continuity.
Review
复习
Chapter 3 Differentiation Rules
第三章导数法例
Teaching Teaching Requirements and Objectives Time
Content
教课要乞降目标学时教课内容
Derivatives of The student will apply formulas to find
Polynomials and the derivative of algebraic, Exponential trigonometric,exponential,and Functions logarithmic functions and their
inverses.
多项式函数和指
数函数的导数
The Product The student will apply formulas to find
and Rules Quotient the derivative of the sum of elementary functions.
导数的乘法和除
法运算法例
Rates of Students will be able to understand the Change in the mathematical modeling process of Natural and derivatives(rates of changes) in the Social Sciences real world
自然科学和社会
科学中的变化率
of Trigonometric Students will
differentiation
be
rules
able
of
to use the
trigonometric
Functions functions
三角函数的导数
The Chain Rule The student will apply formulas to find
链式法例
the derivative of the sum, product,
quotient, inverse and composite (chain
rule) of elementary functions.
Implicit The student will find the derivative of
an implicitly defined function.
Differentiation
隐函数求导
Higher The student will find the higher order
derivatives of algebraic, Derivatives
trigonometric,exponential,and 高阶导数logarithmic functions.
of Logarithmic The student will use logarithmic Functions differentiation as a technique to
differentiate non-logarithmic
对数函数的导数
functions.
Hyperbolic The student will be able to understand Functions the definition of hyperbolic functions,
and solve for its derivatives.
双曲函数
Linear The student will apply the derivative Approximations to solve problems,including tangent and and normal lines to a curve, curve Differentials sketching,velocity,acceleration,
related rates of change,Newton's 线性迫近和微分
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 Teaching Requirements and Objectives Time
Content
教课要乞降目标学时教课内容
Maximum and The student will be able to understand
Minimum Values extreme values of a function, find
极大值和极小值
critical values of a function and find
extreme values of a function.
The Mean Value The student will state (without proof) Theorem the Mean Value Theorem for derivatives
中值定理
and apply it both algebraically and
graphically.
How Derivative Affect the Shape of a The student will graph these functions using a graphing calculator, including understanding and using the First
Graph Derivative Test and the Second
导数是怎样改变
Derivative Test to determine min’s and
max’s.
图像的形状
Indeterminate The student will use l'Hopital's rule Forms and L ’to find the limit of functions whose Hospital ’s limits yield the indeterminate forms: Rules0/0 and infinity/infinity
不定式和洛必达
法例
Optimization Problems The student will be able to use derivatives to solve optimization problems
最优化问题
Newton ’s The student will be able to use Method Newton’s method to approximate roots 牛顿法例
of an equation.
The student will be able to understand
the concept of an antiderivative, the
原函数（反导
geometry of the antiderivative and that 数）
of slope fields and also work
rectilinear motion problems with
antiderivatives
Review
复习
Chapter 5 Integrals
第五章积分
Teaching Teaching Requirements and Objectives Time
Content
教课要乞降目标学时教课内容
Areas and Distances The student will identify properties of
the definite integral.
the
面积和距离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.
The Definite Integral The student will compute an approximate value for a definite integral.
不定积分This will include numerical
calculations using Riemann Sums and the
Trapezoidal Rule.
The The student will identify the Fundamental properties of the definite integral. Theorem of
This will include the Fundamental Calculus
Theorem of Calculus and the definite
微积分基本定理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)
Indefinite Integrals and the Net Change Theorem The student will find the indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions.
不定积分和原函数定理
The Substitution Rule The student will find the indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions.
换元积分法
The special integration techniques of
substitution (change of variables) and
integration by parts will be included.
Review
复习
Chapter 6 Applications of Integration
第六章积分的应用
Teaching Teaching Requirements and Objectives Time Content
教课要乞降目标学时教课内容
Areas Between Curves The student will apply
integral to solve problems.
the definite
曲边面积These problems will include finding
distance 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 and shell methods,and
volumes of solids with known cross-
sectional areas.
Volumes The student will apply the definite
integral to solve problems.
体
These problems will include area
between curves, volumes of solids of
revolution about the axes or lines
parallel to the axes using disc/washer
and shell methods, and volumes of
solids with known cross-sectional
areas.
Volumes by The student will apply the definite Cylindrical integral to solve problems.
Shells
These problems will include⋯ area 柱体体between curves, volumes of solids of
revolution about the axes or lines
parallel to the axes using disc/washer
and shell methods, and volumes of
solids with known cross-sectional
areas.
Work The student will apply the definite
integral to solve problems.
物体功
These problems will include finding
distance traveled on a line and
velocity from acceleration with initial
conditions, growth and decay problems,
work done.
Average Value The student will apply the definite
of a Function integral to solve problems.
实函数均值These problems will include the average
value of a function.
Density The student will apply the definite Function integral to solve problems.
密度函数These problems 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 Teaching Requirements and Objectives Time Content
教课要乞降目标学时教课内容
Integration by The student will find the definite and
Parts indefinite integral of algebraic,
exponential, logarithmic, and
分部积分法
trigonometric functions.
Trigonometric
The special integration techniques of
Integrals
substitution (change of variables) and
三角函数积分integration by parts will be included.
Trigonometric
Substitution
三角函数替代
Integration of
Rational
Functions by
Partial
Fractions
有理函数的分部
积分
Review
复习
五、查核方式
为了彰显我校“多一把尺子，多一位人材”的教育教课理念，本课程采纳形成性查核与终结性查核相联合的方式。

（一）形成性查核内容
本课程的形成性查核详细内容分为学习内容查核和学习过程查核。

学习内容查核：
1．平常作业：任课教师讲堂上集中部署的随堂作业和课后作业。

同时，不一样年级的教师在部署作业时也可依据各班的实质状况适合加以细化、加入一些限制性条件或进一步的要求。

2．阶段性测试：是依据课程教课安排部署的阶段性综合测试。

阶段性测试主要考察学生在一个单元内的学习状态，其测查内容属于教课中的要点，波及到讲过的大多数基本观点、基来源则和基本方法。

3．讲堂议论：在课程教课过程中，在指准时间，环绕必定的主题，对课程的要点、难点内容，集中进行若干次讲堂议论。

讲堂议论的题目提早一周见告学生，让学生以学习小组为单位进行议论前的准备，每学习小组选举一人做代表性讲话并供给以小组名义提交的议论纲要。

指导教师依据各小组学生参加程度、讲话状况及议论纲要赐予评论并以小组形式赐予评定成绩。

学习过程查核：
1．成立学习小组：参加试点的班级应当以5-6 名学生构成学习小组，指定学习组长并报班主任及指导教师。

依据学校和任课老师的要求，有计划有目的地展开学习活动，完
成并准时上交形成性查核中要求以小组为单位进行的作业。

小组学习应当有尽可能详
尽的学习过程记录，反应学生在学习小组活动中的内容、领会及存在问题，期末交到
指导教师处，作为指导教师对学生形成性查核成绩评定的依照之一。

2.讲堂出勤：学生能否准时上课，在很大程度反应了该学生对待学习的态度；学生在讲
堂上能否遵从教师的指令也都应当考虑在考勤的范围内。

（二）终结性考试
终结性考试主要查核学生对 AP 微积分的基本理论、基本知识、基本观点的理解与掌握，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
形成性查核成绩占总成绩的60%。

形成性查核成绩由任课教师依据学生实质表现状况评定，由教课处责任教师评审，最
后确立学生的课程形成性查核成绩。

北京中加学校正形成性查核进行抽样检查。

终结性查
核占课程总成绩的 40%，见表 2，而后经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派表
项目形成性查核内容终结性
作业测试议论小组出勤查核内容
AP微积分课程设计
比率15%15%10%10%10%40%学期成绩为期中成绩和期末成绩各占50%。

依据北京中加学校的规定，本课程推行形成性查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），即可获取本课程相应学分。