AP Calculus: Exam Scope and Key Concepts Summary (with Practice Problems) | AP 微积分：考试范围与考点总结（附习题）

📚 AP Calculus: Exam Scope and Key Concepts Summary (with Practice Problems) | AP 微积分：考试范围与考点总结（附习题）

Welcome to the ultimate review guide for AP Calculus, designed to help you master the exam content—whether you are taking AB or BC. We break down every major topic, from limits to series, and include practice questions with step-by-step solutions.

欢迎阅读 AP 微积分终极复习指南，无论你参加的是 AB 还是 BC 考试，本指南都将助你掌握考试内容。我们从极限到级数，逐一剖析每个重要专题，并附有习题和详细解答。

1. Overview of AP Calculus Exams | AP 微积分考试概览

The AP Calculus program offers two courses: Calculus AB and Calculus BC. AB covers roughly one semester of college calculus, while BC covers two semesters and includes additional topics such as sequences, series, and polar functions. Both exams consist of multiple-choice and free-response sections, with a mix of calculator and non-calculator questions. Mastery of conceptual understanding, algebraic manipulation, and graphical analysis is essential.

AP 微积分项目提供两门课程：微积分 AB 和微积分 BC。AB 大约涵盖大学一个学期的微积分内容，而 BC 涵盖两个学期，并包括序列、级数和极坐标函数等额外专题。两门考试均包含选择题和自由回答题，以及允许使用计算器和不许使用计算器的部分。掌握概念理解、代数运算和图像分析能力至关重要。

The AB exam covers limits, derivatives, integrals, and the Fundamental Theorem of Calculus. BC adds advanced integration techniques, sequences, series, and parametric/polar/vector functions. The exam is 3 hours 15 minutes long, with 45 multiple-choice and 6 free-response questions. BC also reports an AB subscore to reflect your performance on AB-level topics.

AB 考试涵盖极限、导数、积分和微积分基本定理。BC 在此基础上增加了高级积分技巧、序列、级数以及参数/极坐标/向量函数。考试时长 3 小时 15 分钟，包含 45 道选择题和 6 道自由回答题。BC 还会给出 AB 子分数，反映你在 AB 级别专题上的表现。

2. Limits and Continuity | 极限与连续性

A limit describes the value a function approaches as the input approaches a given point. Notation: limₓ→ₐ f(x) = L. A limit exists only if both one-sided limits are equal. Infinite limits indicate vertical asymptotes, while limits at infinity describe end behavior.

极限描述了当输入趋近于某一点时函数所趋向的值。记作 limₓ→ₐ f(x) = L。只有当左右极限相等时极限才存在。无穷极限表明垂直渐近线，而无穷远处的极限描述函数的末端走势。

Key limit properties include the Squeeze Theorem: if g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L. A function is continuous at a if limₓ→ₐ f(x) = f(a). Important special limits: limₓ→₀ (sin x)/x = 1 and limₓ→₀ (1–cos x)/x = 0.

重要的极限性质包括夹逼定理：如果在 a 附近 g(x) ≤ f(x) ≤ h(x) 且 lim g = lim h = L，那么 lim f = L。函数在 a 点连续当且仅当 limₓ→ₐ f(x) = f(a)。重要特殊极限：limₓ→₀ (sin x)/x = 1 和 limₓ→₀ (1–cos x)/x = 0。

To evaluate limits, try direct substitution first. If indeterminate forms like 0/0 appear, use factoring, rationalization, or algebraic simplification. Limits can also be found graphically by observing the y-value a function approaches.

计算极限时首先尝试直接代入。如果出现 0/0 等不定式，则使用因式分解、有理化或代数化简。极限也可以通过观察函数趋近的 y 值从图像上读取。

3. Differentiation: Definition and Rules | 导数：定义与运算法则

The derivative of f at x is defined as f'(x) = limₕ→₀ (f(x+h)–f(x))/h, provided the limit exists. Geometrically, f'(a) gives the slope of the tangent line at x = a. Differentiability implies continuity, but the converse is false.

f 在 x 处的导数定义为 f'(x) = limₕ→₀ (f(x+h)–f(x))/h，只要该极限存在。从几何上看，f'(a) 给出 x = a 处切线的斜率。可导性蕴含连续性，但反之不成立。

Basic differentiation rules: constant rule: d/dx (c) = 0; power rule: d/dx (xⁿ) = n xⁿ⁻¹; sum rule: d/dx [f±g] = f’ ± g’; product rule: d/dx (fg) = f’g + fg’; quotient rule: d/dx (f/g) = (f’g–fg’)/g²; chain rule: d/dx f(g(x)) = f'(g(x)) · g'(x).

基本求导法则：常数法则 d/dx (c) = 0；幂法则 d/dx (xⁿ) = n xⁿ⁻¹；和差法则 d/dx [f±g] = f’ ± g’；乘积法则 d/dx (fg) = f’g + fg’；商法则 d/dx (f/g) = (f’g–fg’)/g²；链式法则 d/dx f(g(x)) = f'(g(x)) · g'(x)。

Derivatives of specific functions: d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (tan x) = sec² x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x. For implicit differentiation, differentiate both sides of an equation with respect to x, treating y as a function of x, and then solve for dy/dx.

特定函数的导数：d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (tan x) = sec² x；d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x。隐函数求导时，对等式两边关于 x 求导，将 y 视为 x 的函数，然后解出 dy/dx。

4. Applications of Derivatives | 导数的应用

Tangent line approximation (linearization): f(a+h) ≈ f(a) + f'(a)h. Related rates problems: write an equation linking variables, differentiate implicitly with respect to time, substitute known rates and solve for the unknown rate.

切线近似（线性化）：f(a+h) ≈ f(a) + f'(a)h。相关变化率问题：建立变量之间的关系方程，关于时间隐式求导，代入已知变化率并求解未知变化率。

Critical points occur where f'(x)=0 or f’ is undefined. The First Derivative Test: if f’ changes from positive to negative at a critical point, f has a local maximum; if from negative to positive, a local minimum. The Second Derivative Test: if f”(x) > 0, the graph is concave up and the critical point is a local minimum; if f”(x) < 0, concave down and local maximum.

临界点出现在 f'(x)=0 或 f’ 不存在处。一阶导数判别法：如果 f’ 在临界点由正变负，则 f 有局部极大值；如果由负变正则局部极小值。二阶导数判别法：若 f”(x) > 0，图像上凹且临界点为局部极小；若 f”(x) < 0，下凹且局部极大。

Optimization problems require finding absolute maximum or minimum values on a closed interval by evaluating f at critical points and endpoints. Motion along a line: position s(t), velocity v(t)=s'(t), acceleration a(t)=v'(t)=s”(t). Speed is |v(t)|.

优化问题需要在闭区间上通过计算临界点和端点处的函数值来求绝对最大值或最小值。直线运动：位置 s(t)，速度 v(t) = s'(t)，加速度 a(t)

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