AP Calculus AB AP 微积分 AB
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· 8 units · key topics drilled across the official outline
· Next exam: 2026-05-11 · 2026-05-11
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Course outline
Unit 1 · Limits and Continuity 极限与连续性
Exam weight: 10-12%
1.1 — Introducing Calculus: Can Change Occur at an Instant? 微积分导论：变化能在瞬间发生吗？
· Instantaneous rate of change · Difference between average and instantaneous rates of change · Core ideas of calculus
1.2 — Defining Limits and Using Limit Notation 极限的定义与极限符号
· Intuitive definition of a limit · Using limit notation · Left-hand and right-hand limits
1.3 — Estimating Limit Values from Graphs 从图形估计极限值
· Graphical analysis · Reading limit values from graphs · Graphical determination of one-sided limits
1.4 — Estimating Limit Values from Tables 从表格估计极限值
· Numerical approximation · Table data analysis · Numerical estimation of limit values
1.5 — Determining Limits Using Algebraic Properties 利用代数性质求极限
· Limit properties and rules · Limits of sums, differences, products, and quotients · Constant multiple rule for limits
1.6 — Determining Limits Using Algebraic Manipulation 利用代数化简求极限
· Factoring · Rationalizing · Canceling common factors
1.7 — Selecting Procedures for Determining Limits 选择求极限的方法
· Direct substitution · Algebraic simplification · Strategy for selecting methods
1.8 — Determining Limits Using the Squeeze Theorem 利用夹逼定理求极限
· Squeeze Theorem · Bounding with inequalities · Trigonometric limits
1.9 — Connecting Multiple Representations of Limits 极限的多种表示方式的联系
· Graphical, numerical, and algebraic representations · Connecting multiple representations · Comprehensive understanding of limits
1.10 — Exploring Types of Discontinuities 探索不连续的类型
· Removable discontinuities · Jump discontinuities · Infinite discontinuities
1.11 — Defining Continuity at a Point 定义一点处的连续性
· Three conditions for continuity · Function value exists · Limit exists and equals the function value
1.12 — Confirming Continuity over an Interval 确认区间上的连续性
· Continuity over an interval · Endpoint continuity · Properties of continuous functions
1.13 — Removing Discontinuities 消除间断点
· Repairing removable discontinuities · Redefining function values · Continuous extension
1.14 — Connecting Infinite Limits and Vertical Asymptotes 无穷极限与垂直渐近线
· Infinite limits · Vertical asymptotes · One-sided infinite limits
1.15 — Connecting Limits at Infinity and Horizontal Asymptotes 无穷远处的极限与水平渐近线
· Limits at infinity · Horizontal asymptotes · End behavior
1.16 — Working with the Intermediate Value Theorem 介值定理的应用
· Intermediate Value Theorem · Existence of zeros · Intermediate values of continuous functions
Unit 2 · Differentiation: Definition and Fundamental Properties 微分：定义与基本性质
Exam weight: 10-12%
2.1 — Defining Average and Instantaneous Rates of Change 定义平均变化率与瞬时变化率
· Average rate of change · Instantaneous rate of change · Difference quotient
2.2 — Defining the Derivative of a Function and Using Derivative Notation 定义函数的导数与导数符号
· Limit definition of the derivative · Derivative notation · Slope of the tangent line
2.3 — Estimating Derivatives of a Function at a Point 估计函数在某点的导数
· Numerical estimation of derivatives · Graphical estimation of derivatives · Difference quotient approximation
2.4 — Connecting Differentiability and Continuity 可导性与连续性的关系
· Differentiability implies continuity · Continuity does not imply differentiability · Cases of non-differentiability
2.5 — Applying the Power Rule 幂函数求导法则
· Power Rule for differentiation · Positive integer exponents · General exponents
2.6 — Derivative Rules: Constant, Sum, Difference, and Constant Multiple 求导法则：常数、和、差与常数倍
· Derivative of a constant is zero · Sum and Difference Rules · Constant Multiple Rule
2.7 — Derivatives of cos x, sin x, eˣ, and ln x cos x、sin x、eˣ 和 ln x 的导数
· Derivatives of trigonometric functions · Derivative of exponential functions · Derivative of logarithmic functions
2.8 — The Product Rule 乘法求导法则
· Product Rule · Derivative of a product of two functions · Applications of the Product Rule
2.9 — The Quotient Rule 除法求导法则
· Quotient Rule · Derivative of rational functions · Applications of the Quotient Rule
2.10 — Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions 求正切、余切、正割和余割函数的导数
· Derivative of tan x · Derivative of sec x · Derivatives of other trigonometric functions
Unit 3 · Differentiation: Composite, Implicit, and Inverse Functions 微分：复合函数、隐函数与反函数
Exam weight: 9-13%
3.1 — The Chain Rule 链式法则
· Differentiating composite functions · Inner and outer functions · Applications of the Chain Rule
3.2 — Implicit Differentiation 隐函数微分
· Implicit differentiation · Differentiating both sides · Solving for dy/dx
3.3 — Differentiating Inverse Functions 反函数的微分
· Inverse function derivative formula · Derivative of an inverse function · Relationship between derivatives of inverse functions
3.4 — Differentiating Inverse Trigonometric Functions 反三角函数的微分
· Derivative of arcsin · Derivative of arctan · Inverse trigonometric derivative formulas
3.5 — Selecting Procedures for Calculating Derivatives 选择求导方法
· Selecting differentiation strategies · Combining multiple differentiation rules · Simplifying before differentiating
3.6 — Calculating Higher-Order Derivatives 求高阶导数
· Second derivative · Higher-order derivatives · Meaning of higher-order derivatives
Unit 4 · Contextual Applications of Differentiation 微分的实际应用
Exam weight: 10-15%
4.1 — Interpreting the Meaning of the Derivative in Context 在实际情境中理解导数的含义
· Practical meaning of the derivative · Interpreting rates of change · Unit analysis
4.2 — Straight-Line Motion: Connecting Position, Velocity, and Acceleration 直线运动：位置、速度与加速度
· Position function · Velocity is the derivative of position · Acceleration is the derivative of velocity
4.3 — Rates of Change in Applied Contexts Other Than Motion 非运动情境中的变化率
· Rates of change in applied contexts · Growth and decay rates · Applications of rates of change
4.4 — Introduction to Related Rates 相关变化率简介
· Related rates · Implicit differentiation with respect to time · Relationships between variables
4.5 — Solving Related Rates Problems 求解相关变化率问题
· Setting up equations · Differentiating with respect to time · Substituting known values to solve
4.6 — Approximating Values of a Function Using Local Linearity and Linearization 利用局部线性和线性化近似函数值
· Tangent line approximation · Linearization · Local linear approximation
4.7 — Using L'Hôpital's Rule for Determining Limits of Indeterminate Forms 利用洛必达法则求不定式极限
· L'Hopital's Rule · indeterminate form · infinity/infinity indeterminate form
Unit 5 · Analytical Applications of Differentiation 微分的分析应用
Exam weight: 15-18%
5.1 — Using the Mean Value Theorem 中值定理的应用
· Mean Value Theorem · Rolle's Theorem · Connection between average and instantaneous rates of change
5.2 — Extreme Value Theorem, Absolute vs. Relative Extrema 极值定理，绝对极值与相对极值
· Extreme Value Theorem · Absolute maximum and minimum · Relative extrema
5.3 — Determining Intervals on Which a Function Is Increasing or Decreasing 判断函数的递增或递减区间
· Sign of derivative and monotonicity · Increasing intervals · Decreasing intervals
5.4 — Using the First Derivative Test 一阶导数检验
· First Derivative Test · Determining extreme values · Sign changes of the derivative
5.5 — Using the Candidates Test to Determine Absolute Extrema 候选值法求绝对极值
· Absolute extrema on a closed interval · Comparing critical points and endpoints · Candidates Test
5.6 — Determining Concavity of Functions 判断函数的凹凸性
· Concavity · Second derivative and concavity · Inflection points
5.7 — Using the Second Derivative Test 二阶导数检验
· Second Derivative Test · Second-order determination of extrema · When the second derivative is zero
5.8 — Sketching Graphs of Functions and Their Derivatives 函数及其导数的图形描绘
· Relationship between function and derivative graphs · Inferring function graph from derivative graph · Curve sketching
5.9 — Connecting a Function, Its First Derivative, and Its Second Derivative 函数与其一阶、二阶导数的联系
· Combined analysis of function, first derivative, and second derivative · Information transfer among the three · Comprehensive graphical analysis
5.10 — Introduction to Optimization Problems 最优化问题简介
· Optimization problems · Setting up the objective function · Constraints
5.11 — Solving Optimization Problems 求解最优化问题
· Finding extreme values · Modeling real-world problems · Verifying optimal solutions
5.12 — Exploring Behaviors of Implicit Relations 探索隐函数关系的行为
· Graphical analysis of implicit relations · Extrema and concavity of implicit relations · Tangent lines of implicit relations
Unit 6 · Integration and Accumulation of Change 积分与变化的累积
Exam weight: 17-20%
6.1 — Exploring Accumulations of Change 探索变化的累积
· Accumulation of change · Relationship between area and accumulation · Intuitive understanding of integrals
6.2 — Approximating Areas with Riemann Sums 用黎曼和近似面积
· Riemann sums · Left, right, and midpoint approximations · Rectangular approximation
6.3 — Riemann Sums, Summation Notation, and Definite Integral Notation 黎曼和、求和符号与定积分符号
· Summation notation · Definite integral notation · Relationship between Riemann sums and definite integrals
6.4 — The Fundamental Theorem of Calculus and Accumulation Functions 微积分基本定理与累积函数
· Fundamental Theorem of Calculus, Part 1 · Accumulation functions · Derivative of an integral with a variable upper limit
6.5 — Interpreting the Behavior of Accumulation Functions 解释累积函数的行为
· Increasing/decreasing behavior of accumulation functions · Extrema of accumulation functions · Relationship between integrand and accumulation function
6.6 — Applying Properties of Definite Integrals 定积分性质的应用
· Linearity properties of definite integrals · Additivity of integration intervals · Basic properties of definite integrals
6.7 — The Fundamental Theorem of Calculus and Definite Integrals 微积分基本定理与定积分
· Fundamental Theorem of Calculus, Part 2 · Evaluating definite integrals using antiderivatives · Newton-Leibniz formula
6.8 — Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 求原函数与不定积分：基本法则与符号
· Indefinite integrals · Basic integration formulas · Constant of integration
6.9 — Integrating Using Substitution 换元积分法
· u-substitution · Integration by substitution · Substitution in definite integrals
6.10 — Integrating Functions Using Long Division and Completing the Square 利用长除法和配方法积分
· Polynomial long division · Completing the square · Simplifying the integrand
6.14 — Selecting Techniques for Antidifferentiation 选择求原函数的方法
· Selecting integration methods · Identifying integral types · Combining integration techniques
Unit 7 · Differential Equations 微分方程
Exam weight: 6-12%
7.1 — Modeling Situations with Differential Equations 用微分方程建模
· Modeling with differential equations · Mathematical description of rates of change · Translating real-world problems into differential equations
7.2 — Verifying Solutions for Differential Equations 验证微分方程的解
· Verification by substitution · General and particular solutions · Methods for verifying solutions
7.3 — Sketching Slope Fields 绘制斜率场
· Slope fields · Drawing direction fields · Characteristics of slope fields
7.4 — Reasoning Using Slope Fields 利用斜率场推理
· Analyzing slope fields · Qualitative behavior of solution curves · Inferring solutions from slope fields
7.6 — Finding General Solutions Using Separation of Variables 用分离变量法求通解
· Separation of variables · Separable differential equations · Finding the general solution
7.7 — Finding Particular Solutions Using Initial Conditions 利用初始条件求特解
· Initial conditions · Particular solutions · Initial value problems
7.8 — Exponential Models with Differential Equations 微分方程的指数模型
· Exponential growth model · Exponential decay model · Solution to dy/dt = ky
Unit 8 · Applications of Integration 积分的应用
Exam weight: 10-15%
8.1 — Finding the Average Value of a Function 求函数的平均值
· Average value formula for a function · Average value using definite integrals · Mean Value Theorem for Integrals
8.2 — Connecting Position, Velocity, and Acceleration Using Integrals 用积分联系位置、速度与加速度
· Integrating velocity to find displacement · Integrating acceleration to find velocity · Total distance vs. displacement
8.3 — Using Accumulation Functions and Definite Integrals in Applied Contexts 在实际情境中使用累积函数与定积分
· Practical meaning of accumulated quantities · Applications of definite integrals · Net change
8.4 — Finding the Area Between Curves Expressed as Functions of x 求以 x 为自变量的曲线间面积
· Top minus bottom · Area of region enclosed by curves · Area using integrals
8.5 — Finding the Area Between Curves Expressed as Functions of y 求以 y 为自变量的曲线间面积
· Right minus left · Integration with respect to y · Horizontal slicing method
8.6 — Finding the Area Between Curves That Intersect at More Than Two Points 求交于两个以上点的曲线间面积
· Multiple intersection points · Piecewise integration · Absolute value integration
8.7 — Volumes with Cross Sections: Squares and Rectangles 截面体积：正方形和矩形截面
· Volumes with known cross sections · Square cross sections · Rectangular cross sections
8.8 — Volumes with Cross Sections: Triangles and Semicircles 截面体积：三角形和半圆截面
· Triangular cross sections · Semicircular cross sections · Cross-sectional area integration
8.9 — Volume with Disc Method: Revolving Around the x- or y-Axis 圆盘法求旋转体体积：绕 x 轴或 y 轴
· Disk method · Revolution about the x-axis · Revolution about the y-axis
8.10 — Volume with Disc Method: Revolving Around Other Axes 圆盘法求旋转体体积：绕其他轴
· Revolution about non-coordinate axes · Axis translation · Adjusting the radius
8.11 — Volume with Washer Method: Revolving Around the x- or y-Axis 垫圈法求旋转体体积：绕 x 轴或 y 轴
· Washer method · Outer and inner radii · Hollow solids of revolution
8.12 — Volume with Washer Method: Revolving Around Other Axes 垫圈法求旋转体体积：绕其他轴
· Washer method about non-coordinate axes · Adjusting outer and inner radii · Composite solids of revolution