📚 AP Calculus: Pre-Exam Review Checklist | AP微积分：考前复习要点清单
As the AP Calculus exam approaches, a structured review checklist can boost your confidence and scores. This guide covers essential topics from limits to series, with key formulas, common pitfalls, and exam strategies for both AB and BC students.
随着AP微积分考试的临近，一份有条理的复习要点清单可以提升你的信心和得分。本指南涵盖了从极限到级数的核心知识点，包括关键公式、常见错误和考试策略，适用于AB和BC考生。
1. Limits and Continuity | 极限与连续性
Understand the intuitive and formal definition of a limit. Know that limₓ→ₐ f(x) = L if f(x) approaches L as x approaches a from both sides. Memorize limit laws: sum, product, quotient, and composition. Evaluate limits using direct substitution, factoring, rationalizing, and the Squeeze Theorem. One-sided limits limₓ→ₐ⁺ and limₓ→ₐ⁻ must be equal for the two-sided limit to exist.
理解极限的直观和正式定义。知道如果 x 从两侧趋近于 a 时 f(x) 趋近于 L，则 limₓ→ₐ f(x) = L。牢记极限法则：和、积、商与复合。使用直接代入、因式分解、有理化和夹逼定理求极限。左右极限 limₓ→ₐ⁺ 和 limₓ→ₐ⁻ 必须相等，双侧极限才存在。
Infinite limits give vertical asymptotes; limits at infinity describe horizontal asymptotes. Use dominant terms for rational functions. Continuity requires limₓ→ₐ f(x) = f(a). Know removable, jump, and infinite discontinuities. The Intermediate Value Theorem applies for continuous functions on a closed interval.
无穷极限对应垂直渐近线，无穷远处的极限描述水平渐近线。用主导项处理有理函数。连续要求 limₓ→ₐ f(x) = f(a)。了解可去间断、跳跃间断和无穷间断。闭区间上连续函数适用介值定理。
limₓ→₀ (sin x)/x = 1 ; limₓ→∞ (1 + 1/x)ˣ = e
2. Derivatives: Definition and Basic Rules | 导数：定义与基本法则
The derivative is defined as f′(x) = limₕ→₀ [f(x+h) – f(x)]/h. It represents the slope of the tangent line and the instantaneous rate of change. Memorize derivative rules: power rule d/dx (xⁿ) = n xⁿ⁻¹, product rule (uv)′ = u′v + uv′, quotient rule (u/v)′ = (u′v – uv′)/v², and chain rule (f∘g)′(x) = f′(g(x)) · g′(x).
导数定义为 f′(x) = limₕ→₀ [f(x+h) – f(x)]/h，表示切线的斜率和瞬时变化率。牢记求导法则：幂法则 d/dx (xⁿ) = n xⁿ⁻¹，积法则 (uv)′ = u′v + uv′，商法则 (u/v)′ = (u′v – uv′)/v²，以及链式法则 (f∘g)′(x) = f′(g(x)) · g′(x)。
Learn derivatives of trigonometric (sin x, cos x, tan x, etc.), exponential (eˣ, aˣ), logarithmic (ln x), and inverse trig functions. Implicit differentiation handles equations where y is not explicitly solved. Higher-order derivatives like f′′(x) measure acceleration or concavity.
学习三角（sin x, cos x, tan x 等）、指数（eˣ, aˣ）、对数（ln x）和反三角函数的导数。隐函数求导用于未解出 y 的方程。高阶导数如 f′′(x) 可衡量加速度或凹性。
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
| arcsin x | 1/√(1−x²) |
3. Applications of Derivatives | 导数的应用
Tangent and normal lines use point-slope form y−y₀ = m(x−x₀). Critical points occur where f′(x)=0 or f′(x) does not exist. First-derivative test identifies local maxima and minima based on sign changes. Second-derivative test uses f′′(c) to classify extrema. Absolute extrema on [a,b] require evaluating f at critical points and endpoints (Extreme Value Theorem).
切线与法线使用点斜式 y−y₀ = m(x−x₀)。临界点出现在 f′(x)=0 或 f′(x) 不存在处。一阶导数检验根据符号变化识别局部极大极小。二阶导数检验使用 f′′(c) 分类极值。闭区间 [a,b] 上的绝对极值需计算临界点和端点处的函数值（极值定理）。
Concavity is determined by f′′(x)>0 (concave up) and f′′(x)<0 (concave down). Inflection points occur where concavity changes. Curve sketching requires intercepts, asymptotes, increasing/decreasing intervals, and concavity. Optimization problems follow: define variables, write objective function, identify domain, find critical points, and check endpoints. Related rates involve implicit differentiation with respect to time t. Mean Value Theorem ensures f′(c) = [f(b)−f(a)]/(b−a) for some c in (a,b) if f is continuous on [a,b] and differentiable on (a,b).
凹性由 f′′(x)>0（上凹）和 f′′(x)<0（下凹）决定。拐点出现在凹性改变处。函数作图需截距、渐近线、增减区间和凹性。优化问题遵循：定义变量，写出目标函数，确定定义域，找到临界点并检查端点。相关变化率涉及对时间 t 的隐函数求导。中值定理保证：若 f 在 [a,b] 连续且在 (a,b) 可导，则存在 c∈(a,b) 使得 f′(c) = [f(b)−f(a)]/(b−a)。
4. Integrals and the Fundamental Theorem of Calculus | 积分与微积分基本定理
Antiderivatives reverse differentiation; the indefinite integral ∫ f(x) dx represents the family of antiderivatives plus a constant C. Riemann sums approximate area under a curve: left, right, midpoint, and trapezoidal sums. The definite integral ∫ₐᵇ f(x) dx gives the net area between the graph and the x-axis from x=a to x=b. Properties include linearity, additivity over intervals, and reversal of limits.
反导数与求导互为逆运算；不定积分 ∫ f(x) dx 表示一族反导数，需加常数 C。黎曼和近似曲线下方面积：左和、右和、中点及梯形和。定积分 ∫ₐᵇ f(x) dx 给出从 x=a 到 x=b 的净面积。性质包括线性、区间可加性和上下限互换变号。
The Fundamental Theorem of Calculus has two parts: (1) F(x) = ∫ₐˣ f(t) dt implies F′(x)=f(x); (2) ∫ₐᵇ f(x) dx = F(b)−F(a), where F is any antiderivative of f. Learn to compute definite integrals using antiderivatives and interpret areas geometrically (circles, triangles, symmetries). The average value of f on [a,b] is 1/(b−a) ∫ₐᵇ f(x) dx.
微积分基本定理分为两部分：(1) 若 F(x)=∫ₐˣ f(t) dt，则 F′(x)=f(x)；(2) ∫ₐᵇ f(x) dx = F(b)−F(a)，其中 F 是 f 的任意反导数。学会用反导数计算定积分，并用几何意义解释面积（圆、三角形、对称性）。f 在 [a,b] 上的平均值是 1/(b−a) ∫ₐᵇ f(x) dx。
∫ₐᵇ f(x) dx = F(b)−F(a)
5. Integration Techniques and Applications | 积分技巧与应用
Substitution (u-substitution) reverses the chain rule: ∫ f(g(x))g′(x) dx = ∫ f(u) du. For BC students, integration by parts follows ∫ u dv = uv − ∫ v du. Partial fractions decompose rational functions into simpler fractions. Improper integrals have infinite limits or integrands with infinite discontinuities; they are evaluated using limits.
换元积分法（u-换元）是链式法则的逆运算：∫ f(g(x))g′(x) dx = ∫ f(u) du。对 BC 考生，分部积分公式为 ∫ u dv = uv − ∫ v du。部分分式将有理函数分解成更简单的分式。反常积分有无穷积分限或被积函数有无穷间断点；通过极限求值。
Applications: area between curves = ∫ₐᵇ (top − bottom) dx or ∫ (right−left) dy. Volumes of revolution: disc method V = π ∫ [R(x)]² dx, washer method V = π ∫ ([R(x)]² − [r(x)]²) dx. Volumes by cross-sections integrate known area A(x). BC also covers arc length L = ∫ₐᵇ √[1 + (dy/dx)²] dx and surface area of revolution (not always tested).
应用：曲线间面积 = ∫ₐᵇ (上−下) dx 或 ∫ (右−左) dy。旋转体体积：圆盘法 V = π ∫ [R(x)]² dx，垫圈法 V = π ∫ ([R(x)]² − [r(x)]²) dx。已知截面面积 A(x) 求体积用积分。BC 还涉及弧长 L = ∫ₐᵇ √[1 + (dy/dx)²] dx 和旋转体表面积（不一定常考）。
6. Differential Equations | 微分方程
A differential equation involves an unknown function and its derivatives. Slope fields graphically show solution curves by plotting short line segments with slope f(x,y). Match slope fields by checking at specific points. Separation of variables solves equations of the form dy/dx = g(x)h(y) by rewriting as (1/h(y)) dy = g(x) dx and integrating both sides.
微分方程包含未知函数及其导数。斜率场通过绘制斜率为 f(x,y) 的短线段直观显示解曲线。检查特定点匹配斜率场。分离变量法求解形如 dy/dx = g(x)h(y) 的方程，改写为 (1/h(y)) dy = g(x) dx 并两边积分。
Exponential growth and decay follow dy/dt = ky, solution y = Ceᵏᵗ, where doubling time or half-life can be found. BC includes logistic growth dP/dt = kP(1 − P/L), solution P = L/(1 + Ae⁻ᵏᵗ). Euler’s method approximates yₙ₊₁ ≈ yₙ + f(xₙ, yₙ) Δx to numerically solve initial value problems.
指数增长与衰减遵循 dy/dt = ky，解为 y = Ceᵏᵗ，可求倍增时间或半衰期。BC 包括逻辑斯谛增长 dP/dt = kP(1 − P/L)，解为 P = L/(1 + Ae⁻ᵏᵗ)。欧拉方法使用 yₙ₊₁ ≈ yₙ + f(xₙ, yₙ) Δx 数值求解初值问题。
7. Parametric Equations, Polar Coordinates, and Vectors (BC) | 参数方程、极坐标与向量（BC）
Parametric curves (x(t), y(t)) define motion. The derivative dy/dx = (dy/dt)/(dx/dt). Higher derivatives and tangent lines follow accordingly. Arc length is L = ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt. Speed is √[(dx/dt)² + (dy/dt)²].
参数曲线 (x(t), y(t)) 定义运动。导数 dy/d
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