📚 AP Math: A Knowledge Review of Calculus BC | AP 数学：微积分BC知识点梳理
The AP Calculus BC exam tests a wide array of calculus topics, pushing beyond AB with additional techniques, sequences, and series. This knowledge review compiles the essential concepts, formulas, and methods to help you master the material efficiently. From limits to power series, every major area is summarised in a clear bilingual format.
AP 微积分 BC 考试涵盖广泛的微积分主题，比 AB 更深入，包括额外的积分技巧、数列与级数。本知识点梳理汇编了核心概念、公式和方法，帮助你高效掌握内容。从极限到幂级数，每个主要领域都以清晰的双语形式进行了总结。
1. Limits and Continuity | 极限与连续性
A limit describes the value a function approaches as the input nears a point. The intuitive definition leads to the formal ε-δ language, but for the exam you mainly need to evaluate limits using algebraic manipulation, tables, or graphs.
极限描述当输入趋近某一点时函数趋向的值。直观定义通向形式化的 ε-δ 语言，但在考试中你主要需要通过代数运算、表格或图像来求极限。
limₓ→ₐ f(x) = L if ∀ ε > 0, ∃ δ > 0 such that 0 < |x − a| < δ ⇒ |f(x) − L| < ε
When limits from the left and right exist and are equal, the two-sided limit exists. Limits at infinity describe end behaviour, and infinite limits indicate vertical asymptotes. The Squeeze Theorem is used for tricky oscillatory limits like (sin x)/x.
当左极限与右极限存在且相等时，双侧极限存在。无穷远处的极限描述末端行为，无穷极限指示垂直渐近线。夹逼定理用于处理如 (sin x)/x 那样的振荡极限。
A function is continuous at a point if the limit equals the function value. Types of discontinuities include removable, jump, and infinite. The Intermediate Value Theorem guarantees that a continuous function on a closed interval takes every value between f(a) and f(b).
若极限值等于函数值，则函数在该点连续。间断点类型包括可去、跳跃和无穷间断。介值定理保证闭区间上的连续函数能取到 f(a) 与 f(b) 之间的每一个值。
If f is continuous on [a, b] and k is between f(a) and f(b), then ∃ c ∈ (a, b) such that f(c) = k.
2. Derivatives: Definition and Rules | 导数：定义与求导法则
The derivative of a function at a point is the instantaneous rate of change, defined as the limit of the difference quotient. Geometrically, it gives the slope of the tangent line.
函数在某点的导数是瞬时变化率，定义为差商的极限。几何上，它给出切线的斜率。
f′(x) = limₕ→₀ [f(x+h) − f(x)] / h
Basic rules include the power rule (d/dx xⁿ = n xⁿ⁻¹), constant multiple, sum/difference, product, and quotient rules. The chain rule handles composite functions: if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x).
基本法则包括幂法则 (d/dx xⁿ = n xⁿ⁻¹)、常数倍法则、和差法则、乘积法则和商法则。链式法则处理复合函数：若 y = f(g(x))，则 dy/dx = f′(g(x)) · g′(x)。
Implicit differentiation allows you to find derivatives when y is not explicitly solved. Higher-order derivatives denote the rate of change of the rate of change, like acceleration as the second derivative of position.
隐函数求导允许你在未解出 y 的显式表达式时求导。高阶导数表示变化率的变化率，例如加速度是位置的二阶导数。
BC students also need derivatives of parametric, polar, and vector functions, as well as the derivative of inverses and exponential/logarithmic differentiation.
BC 学生还需要掌握参数方程、极坐标和向量函数的求导，以及反函数的导数和指数/对数求导。
3. Applications of Derivatives | 导数的应用
Derivatives enable us to find tangent and normal lines, related rates, and rectilinear motion. The Mean Value Theorem links average and instantaneous rates of change.
导数帮助我们找到切线和法线、相关变化率以及直线运动。中值定理将平均变化率与瞬时变化率联系起来。
If f is continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b) such that f′(c) = [f(b)−f(a)] / (b−a).
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 uses concavity.
一阶导数测试：若 f′ 在临界点由正转负，则 f 有局部极大值；由负转正则为局部极小值。二阶导数测试利用凹凸性。
Concavity is determined by f″: f″ > 0 ⇒ concave up; f″ < 0 ⇒ concave down. Inflection points occur where concavity changes. Optimization problems require identifying absolute extrema on closed intervals or modelling real‑world scenarios.
凹凸性由 f″ 决定：f″ > 0 表示向上凹；f″ < 0 表示向下凹。拐点出现在凹凸性改变之处。最优化问题需要在闭区间上找出绝对极值或建立真实场景模型。
L’Hopital’s Rule resolves indeterminate forms 0/0 or ∞/∞ by differentiating numerator and denominator separately. Repeated application may be needed.
洛必达法则通过分别对分子和分母求导，解决 0/0 或 ∞/∞ 的不定式，可能需反复应用。
4. Integrals and the Fundamental Theorem | 积分与微积分基本定理
Indefinite integrals represent antiderivatives. The definite integral gives the net area between a curve and the x‑axis over [a, b] and is defined as the limit of Riemann sums.
不定积分表示反导数。定积分给出曲线与 x 轴在 [a, b] 上的净面积，并定义为黎曼和的极限。
∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*) Δx
The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. Part 1: If F(x) = ∫ₐˣ f(t) dt, then F′(x) = f(x). Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.
微积分基本定理（FTC）联系了微分与积分。第一部分：若 F(x) = ∫ₐˣ f(t) dt，则 F′(x) = f(x)。第二部分：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的任一原函数。
The substitution method (u‑substitution) reverses the chain rule. For definite integrals, remember to change the limits accordingly. BC students also apply integration to accumulation functions and average value of a function.
换元积分法（u‑代换）是链式法则的逆运算。对于定积分，记得相应变换积分限。BC 学生还需将积分用于累积函数和函数的平均值。
5. Techniques of Integration | 积分技巧
BC requires sophisticated antiderivative techniques. Integration by parts stems from the product rule: ∫ u dv = uv − ∫ v du. Choose u and dv strategically, often using the LIATE guideline.
BC 要求掌握较复杂的反导数技巧。分部积分源自乘积法则：∫ u dv = uv − ∫ v du。策略性地选取 u 和 dv，常使用 LIATE 指导原则。
∫ x eˣ dx = x eˣ − eˣ + C (using u = x, dv = eˣ dx)
Integration of rational functions uses partial fractions. Factor the denominator and decompose into simpler fractions, then integrate term by term. Improper rational expressions require division first.
有理函数的积分使用部分分式。对分母进行因式分解，拆成简单分式，然后逐项积分。假分式需先做除法。
Trigonometric integrals involve powers of sine, cosine, secant, and tangent. Use identities like sin²x = (1−cos 2x)/2. Trigonometric substitution handles expressions like √(a² − x²), √(a² + x²), √(x² − a²) by substituting x = a sin θ, x = a tan θ, or x = a sec θ.
三角积分涉及正弦、余弦、正割、正切的幂次。应用恒等式如 sin²x = (1−cos 2x)/2。三角代换通过代换 x = a sin θ、x = a tan θ 或 x = a sec θ 处理形如 √(a² − x²)、√(a² + x²)、√(x² − a²) 的表达式。
Improper integrals have infinite limits or discontinuous integrands. Evaluate as limits of proper integrals: ∫ₐ∞ f(x) dx = limₜ→∞ ∫ₐᵇ f(x) dx. Convergence/divergence is determined by whether the limit exists.
反常积分有无穷积分限或被积函数不连续。通过求正常积分的极限来求值：∫ₐ∞ f(x) dx = limₜ→∞ ∫ₐᵇ f(x) dx。通过极限是否存在判定敛散性。
6. Applications of Integrals | 积分的应用
Definite integrals compute areas between curves, volumes of solids, arc length, and surface area. The area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ |f(x) − g(x)| dx.
定积分计算曲线间的面积、立体体积、弧长和表面积。两条曲线 y = f(x) 与 y = g(x) 在 a 到 b 间的面积是 ∫ₐᵇ |f(x) − g(x)| dx。
Volume by disc/washer method: rotating a region around an axis, V = π ∫ₐᵇ [R(x)² − r(x)²] dx. The shell method uses cylindrical shells: V = 2π ∫ₐᵇ (radius)(height) dx.
体积的圆盘/垫圈法：区域绕轴旋转，V = π ∫ₐᵇ [R(x)² − r(x)²] dx。壳层法使用圆柱壳：V = 2π ∫ₐᵇ (半径)(高) dx。
Arc length for a function y = f(x) over [a, b] is L = ∫ₐᵇ √(1 + [f′(x)]²) dx. For parametric curves, the formula adapts to L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt.
函数 y = f(x) 在 [a, b] 上的弧长为 L = ∫ₐᵇ √(1 + [f′(x)]²) dx。对于参数曲线，公式调整为 L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt。
Other applications include work, fluid pressure, and the average value of a function, all expressible as definite integrals.
其他应用包括做功、流体压强和函数的平均值，均可表示为定积分。
7. Differential Equations | 微分方程
A differential equation involves an unknown function and its derivatives. Separable equations can be written as g(y) dy = f(x) dx and solved by integrating both sides.
微分方程包含未知函数及其导数。可分离方程可写成 g(y) dy = f(x) dx，并通过两边积分求解。
Exponential growth/decay follows dy/dt = k y, yielding y = y₀ eᵏᵗ. Logistic growth introduces a carrying capacity: dy/dt = k y (1 − y/L), with solution y = L / (1 + a e⁻ᵏᵗ).
指数增长/衰减遵循 dy/dt = k y，得到 y = y₀ eᵏᵗ。逻辑斯谛增长引入承载能力：dy/dt = k y (1 − y/L)，解为 y = L / (1 + a e⁻ᵏᵗ)。
Slope fields provide a visualisation of solutions by drawing small line segments with slope f(x, y). You may be asked to sketch solution curves or match slope fields to differential equations.
斜率场通过绘制斜率为 f(x, y) 的短线段，为解提供可视化。你可能需要勾画解曲线或将斜率场与微分方程配对。
Euler’s method is a numerical approach to approximate solutions: yₙ₊₁ = yₙ + Δx · f(xₙ, yₙ). Step size Δx affects accuracy. BC students should know how to apply a few steps and interpret the error.
欧拉方法是一种数值近似解法：yₙ₊₁ = yₙ + Δx · f(xₙ, yₙ)。步长 Δx 影响准确度。BC 学生应会应用几步并解释误差。
8. Parametric, Polar, and Vector Functions | 参数方程、极坐标与向量函数
Parametric equations define x and y in terms of a third variable t. The derivative dy/dx = (dy/dt) / (dx/dt). Second derivative requires careful computation.
参数方程用第三个变量 t 定义 x 和 y。导数 dy/dx = (dy/dt) / (dx/dt)。二阶导数需仔细计算。
Arc length for parametric curves: L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt. Speed is √((dx/dt)² + (dy/dt)²). Vector-valued functions r(t) = ⟨x(t), y(t)⟩ have derivatives r′(t) = ⟨x′(t), y′(t)⟩ representing velocity.
参数曲线的弧长：L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt。速率为 √((dx/dt)² + (dy/dt)²)。向量值函数 r(t) = ⟨x(t), y(t)⟩ 的导数 r′(t) = ⟨x′(t), y′(t)⟩ 表示速度。
Polar coordinates (r, θ) represent points by distance and angle. Area in polar coordinates: A = (1/2) ∫ₐᵇ r² dθ. To find slope, use dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ).
极坐标 (r, θ) 用距离和角度表示点。极坐标面积：A = (1/2) ∫ₐᵇ r² dθ。求斜率使用 dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ)。
Common polar curves include circles, cardioids, and roses. Be able to set up integrals for area of a region bounded by polar graphs and find points of intersection.
常见极坐标曲线包括圆、心脏线和玫瑰线。要能列出极坐标图形所围区域面积的积分，并求交点。
9. Sequences and Series | 数列与级数
A sequence {aₙ} converges if limₙ→∞ aₙ exists; otherwise it diverges. Monotonic and bounded sequences have limits by the Monotone Convergence Theorem.
数列 {aₙ} 若 limₙ→∞ aₙ 存在则收敛；否则发散。单调有界数列由单调收敛定理保证极限存在。
Infinite series Σ aₙ converge if the sequence of partial sums approaches a finite limit. The first test: if limₙ→∞ aₙ ≠ 0, the series diverges (nth term test for divergence).
无穷级数 Σ aₙ 若部分和数列趋近有限极限则收敛。首要检验：若 limₙ→∞ aₙ ≠ 0，则级数发散（n 项发散检验）。
Geometric series Σ arⁿ converges to a/(1−r) if |r| < 1. The p‑series Σ 1/nᵖ converges if p > 1. Use the integral test for positive decreasing functions, comparing Σ f(n) with ∫₁∞ f(x) dx.
几何级数 Σ arⁿ 当 |r| < 1 时收敛于 a/(1−r)。p‑级数 Σ 1/nᵖ 在 p > 1 时收敛。对正项递减函数使用积分检验，比较 Σ f(n) 与 ∫₁∞ f(x) dx。
Comparison tests: compare term‑wise with a known convergent or divergent series. Limit Comparison Test: if limₙ→∞ aₙ/bₙ = c, 0 < c < ∞, then Σ aₙ and Σ bₙ share the same convergence behaviour.
比较检验：逐项与已知收敛或发散级数比较。极限比较检验：若 limₙ→∞ aₙ/bₙ = c，且 0 < c < ∞，则 Σ aₙ 与 Σ bₙ 敛散性相同。
Alternating series Σ (−1)ⁿ⁺¹ bₙ converge if bₙ > 0, bₙ decreasing, and lim bₙ = 0 (Alternating Series Test). A series converges absolutely if Σ |aₙ| converges; conditional convergence means the series converges but not absolutely.
交错级数 Σ (−1)ⁿ⁺¹ bₙ 若 bₙ > 0、递减且 lim bₙ = 0 则收敛（交错级数检验）。若 Σ |aₙ| 收敛，则该级数绝对收敛；条件收敛表示级数收敛但不绝对收敛。
Ratio and Root Tests are used for series with factorials or powers. For Σ aₙ, ratio: lim |aₙ₊₁/aₙ| = L. If L < 1, absolute convergence; L > 1 (or ∞), divergence; L = 1, inconclusive.
比值与根值检验用于含阶乘或幂次的级数。对于 Σ aₙ，比值：lim |aₙ₊₁/aₙ| = L。若 L < 1，绝对收敛；L > 1（或 ∞），发散；L = 1 时失效。
10. Power Series and Taylor Series | 幂级数与泰勒级数
A power series centred at c has the form Σ aₙ (x − c)ⁿ. The radius of convergence R is found using the Ratio Test (or Root Test). The interval of convergence is checked at endpoints.
以 c 为中心的幂级数形如 Σ aₙ (x − c)ⁿ。收敛半径 R 用比值检验（或根值检验）求得。收敛区间需检验端点。
Taylor series: f(x) = Σ ₙ₌₀∞ [f⁽ⁿ⁾(a) / n!] (x − a)ⁿ. A Maclaurin series is a Taylor series centred at a = 0; it expands a function into an infinite sum of polynomial terms.
泰勒级数：f(x) = Σ ₙ₌₀∞ [f⁽ⁿ⁾(a) / n!] (x − a)ⁿ。麦克劳林级数是 a = 0 的泰勒级数；它将函数展开为无限多项式项之和。
Important Maclaurin expansions for the exam: eˣ = Σ xⁿ/n!, sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σ (−1)ⁿ x²ⁿ/(2n)!, and 1/(1−x) = Σ xⁿ for |x| < 1.
考试重要的麦克劳林展开：eˣ = Σ xⁿ/n!，sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = Σ (−1)ⁿ x²ⁿ/(2n)!，以及 1/(1−x) = Σ xⁿ (|x| < 1)。
To build new series, use substitution, differentiation, or integration of known series term‑by‑term within the interval of convergence. The Lagrange error bound estimates the remainder when a Taylor polynomial approximates a function.
构建新级数时，可在收敛区间内对已知级数逐项代换、求导
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