📚 AP Calculus AB & BC Core Formula Summary | AP微积分AB与BC核心公式汇总
AP Calculus AB and BC are college-level courses that cover differential and integral calculus. AB corresponds roughly to one semester of college calculus, while BC extends into a second semester, adding more integration techniques, parametric and polar functions, and infinite series. This article provides a concise bilingual summary of the essential formulas you need to master for the AP exams. Use it as a quick reference during your final review.
AP微积分AB与BC是大学水平的课程，涵盖微分与积分。AB大致对应大学第一学期的微积分，BC则延伸到第二学期，增加了更多积分技巧、参数方程与极坐标函数以及无穷级数。本文为你提供必须掌握的核心公式的中英对照简明总结，助你在最后复习阶段快速查阅。
1. Limits & Continuity | 极限与连续性
The limit of a function f(x) as x approaches a exists if the left-hand and right-hand limits are equal.
当x趋近于a时，f(x)的极限存在要求左极限与右极限相等。
limx→a f(x) = L
A function is continuous at x = a if limx→a f(x) = f(a).
函数在x = a处连续需要满足 limx→a f(x) = f(a)。
f is continuous at a ⇔ limx→a f(x) = f(a)
Special limits include limx→0 (sin x)/x = 1 and limx→0 (1 – cos x)/x = 0.
重要极限包括 limx→0 (sin x)/x = 1 与 limx→0 (1 – cos x)/x = 0。
2. Basic Differentiation Rules | 基本求导法则
The derivative of a function f at x is defined as f ‘(x) = limh→0 [f(x+h) – f(x)]/h.
函数f在x处的导数定义为 f ‘(x) = limh→0 [f(x+h) – f(x)]/h。
Key rules: constant multiple, sum/difference, product, quotient, and chain rule.
核心法则：常数乘、和差、乘积、商以及链式法则。
| Rule | Formula |
|---|---|
| Constant | d/dx [c] = 0 |
| Power | d/dx [xn] = n xn-1 |
| Product | (fg)’ = f ‘g + f g’ |
| Quotient | (f/g)’ = (f ‘g – f g’) / g2 |
| Chain Rule | d/dx [f(g(x))] = f ‘(g(x)) g'(x) |
3. Derivatives of Trigonometric, Exponential & Logarithmic Functions | 三角、指数与对数函数的导数
Memorize these standard derivatives; they appear frequently throughout the exam.
牢记以下标准导数，它们在考试中出现频率极高。
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec2 x |
| sec x | sec x tan x |
| csc x | -csc x cot x |
| cot x | -csc2 x |
| ex | ex |
| ax | ax ln a |
| ln x | 1/x |
| loga x | 1 / (x ln a) |
Inverse trig derivatives: d/dx (arcsin x) = 1/√(1 – x2), d/dx (arccos x) = -1/√(1 – x2), d/dx (arctan x) = 1/(1 + x2).
反三角函数导数：d/dx (arcsin x) = 1/√(1 – x2)，d/dx (arccos x) = -1/√(1 – x2)，d/dx (arctan x) = 1/(1 + x2)。
4. Application of Derivatives: Related Rates & Optimization | 导数应用：相关变化率与优化问题
For related rates, differentiate an equation linking variables with respect to time t.
相关变化率问题上，对关联变量关于时间t的等式求导即可。
d/dt [A] = d/dt [πr2] → dA/dt = 2πr (dr/dt)
In optimization, find absolute extrema by checking critical points (f ‘(x)=0 or undefined) and endpoints.
在优化问题中，通过检验临界点（f ‘(x)=0 或无定义）及端点找到绝对极值。
Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) such that f ‘(c) = (f(b)-f(a))/(b-a).
中值定理：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)。
5. The Integral: Definition and Fundamental Theorem of Calculus | 积分定义与微积分基本定理
The definite integral of f from a to b is the limit of a Riemann sum: ∫ab f(x) dx = limn→∞ ∑ f(xi*) Δx.
定积分是黎曼和的极限：∫ab f(x) dx = limn→∞ ∑ f(xi*) Δx。
Fundamental Theorem of Calculus, Part 1: if F(x) = ∫ax f(t) dt, then F ‘(x) = f(x).
微积分基本定理第一部分：若 F(x) = ∫ax f(t) dt，则 F ‘(x) = f(x)。
Part 2: ∫ab f(x) dx = F(b) – F(a), where F is any antiderivative of f.
第二部分：∫ab f(x) dx = F(b) – F(a)，其中F为f的任意一个原函数。
6. Integration Techniques: Substitution, Integration by Parts, Partial Fractions | 积分技巧：换元法、分部积分、部分分式
u-substitution reverses the chain rule: ∫ f(g(x)) g'(x) dx = ∫ f(u) du.
u-换元法是链式法则的逆用：∫ f(g(x)) g'(x) dx = ∫ f(u) du。
Integration by parts (BC essential, AB may see simple cases): ∫ u dv = uv – ∫ v du.
分部积分法（BC必备，AB可能涉及简单情形）：∫ u dv = uv – ∫ v du。
Partial fractions are used to integrate rational functions by decomposing into simpler fractions.
部分分式法用于将有理函数分解为更简单的分式再积分。
∫ (1/(x2 – 1)) dx = ∫ [1/(2(x-1)) – 1/(2(x+1))] dx
7. Applications of Integrals: Area, Volume, and Average Value | 积分应用：面积、体积与平均值
Area between curves: A = ∫ab [f(x) – g(x)] dx, where f(x) ≥ g(x).
曲线间面积：A = ∫ab [f(x) – g(x)] dx，其中 f(x) ≥ g(x)。
Volume by disks/washers: V = π ∫ab [R(x)]2 dx or π ∫ab ([R(x)]2 – [r(x)]2) dx.
圆盘/垫圈法求体积：V = π ∫ab [R(x)]2 dx 或 π ∫ab ([R(x)]2 – [r(x)]2) dx。
Volume by shells (BC): V = 2π ∫ab r(x) h(x) dx.
壳层法求体积（BC）：V = 2π ∫ab r(x) h(x) dx。
Average value of f on [a,b]: favg = (1/(b-a)) ∫ab f(x) dx.
f在[a,b]上的平均值：favg = (1/(b-a)) ∫ab f(x) dx。
8. Differential Equations: Separation of Variables & Exponential Models | 微分方程：分离变量与指数模型
For a separable DE dy/dx = g(x) h(y), rewrite as (1/h(y)) dy = g(x) dx and integrate.
对于可分离微分方程 dy/dx = g(x) h(y)，重写为 (1/h(y)) dy = g(x) dx 再积分。
Exponential growth/decay model: dy/dt = k y → y = y0 ekt.
指数增长/衰减模型：dy/dt = k y 的解为 y = y0 ekt。
Logistic growth (BC): dP/dt = kP (1 – P/K), solution P(t) = K / (1 + (K-P0)/P0 e-kt).
逻辑斯蒂增长模型（BC）：dP/dt = kP (1 – P/K)，解为 P(t) = K / (1 + (K-P0)/P0 e-kt)。
9. Parametric Equations and Vector-Valued Functions (BC) | 参数方程与向量值函数 (BC)
For x = f(t), y = g(t), the derivative dy/dx = (dy/dt) / (dx/dt) = g'(t)/f ‘(t).
对于参数方程 x = f(t), y = g(t)，导数 dy/dx = (dy/dt) / (dx/dt) = g'(t)/f ‘(t)。
Second derivative: d2y/dx2 = (d/dt [dy/dx]) / (dx/dt).
二阶导数：d2y/dx2 = (d/dt [dy/dx]) / (dx/dt)。
Arc length of a parametric curve: L = ∫ab √[(dx/dt)2 + (dy/dt)2] dt.
参数曲线的弧长：L = ∫ab √[(dx/dt)2 + (dy/dt)2] dt。
Position, velocity, acceleration vectors: v(t) = r'(t), a(t) = v'(t); speed = |v(t)|.
位置、速度、加速度向量：v(t) = r'(t)，a(t) = v'(t)；速率为 |v(t)|。
10. Polar Coordinates and Area (BC) | 极坐标与面积计算 (BC)
Conversion: x = r cos θ, y = r sin θ, r2 = x2 + y2, tan θ = y/x.
转换公式：x = r cos θ，y = r sin θ，r2 = x2 + y2，tan θ = y/x。
Area enclosed by a polar curve r = f(θ) from θ = α to β: A = ½ ∫αβ [f(θ)]2 dθ.
极坐标曲线围成的面积：A = ½ ∫αβ [f(θ)]2 dθ。
Slope of a polar curve: dy/dx = (r’ sin θ + r cos θ) / (r’ cos θ – r sin θ).
极坐标曲线的切线斜率：dy/dx = (r’ sin θ + r cos θ) / (r’ cos θ – r sin θ)。
11. Infinite Sequences and Series (BC) | 无穷数列与级数 (BC)
A geometric series ∑ arn converges to a/(1 – r) if |r| < 1, diverges otherwise.
几何级数 ∑ arn 当 |r| < 1 时收敛于 a/(1 - r)，否则发散。
p-series ∑ 1/np converges if p > 1, diverges if p ≤ 1.
p-级数 ∑ 1/np 若 p > 1 收敛，若 p ≤ 1 发散。
Tests for convergence: nth term, integral, comparison, limit comparison, alternating series, ratio, and root tests.
收敛性判别法：第n项检验、积分检验、比较检验、极限比较检验、交错级数检验、比值检验和根值检验。
Taylor series centered at a: f(x) = ∑ [f(n)(a) / n!] (x – a)n. Maclaurin series: a = 0.
以a为中心的泰勒级数：f(x) = ∑ [f(n)(a) / n!] (x – a)n。麦克劳林级数即a=0的情形。
Key Maclaurin series: ex = ∑ xn/n!, sin x = ∑ (-1)n x2n+1/(2n+1)!, cos x = ∑ (-1)n x2n/(2n)!, 1/(1-x) = ∑ xn (|x|<1).
重要麦克劳林级数：ex = ∑ xn/n!，sin x = ∑ (-1)n x2n+1/(2n+1)!，cos x = ∑ (-1)n x2n/(2n)!，1/(1-x) = ∑ xn (|x|<1)。
12. Important Theorems (MVT, IVT, L’Hôpital’s Rule) | 重要定理（中值定理、介值定理、洛必达法则）
Intermediate Value Theorem: if f is continuous on [a,b] and k is between f(a) and f(b), there exists c in (a,b) with f(c) = k.
介值定理：若f在[a,b]上连续且k介于f(a)与f(b)之间，则存在c∈(a,b)使得f(c)=k。
Extreme Value Theorem: a continuous function on a closed interval attains an absolute max and an absolute min.
极值定理：闭区间上的连续函数必定能取到绝对最大值和绝对最小值。
L’Hôpital’s Rule: for indeterminate forms 0/0 or ∞/∞, lim f(x)/g(x) = lim f ‘(x)/g'(x) if the limit exists.
洛必达法则：对于0/0或∞/∞型未定式，lim f(x)/g(x) = lim f ‘(x)/g'(x)，只需后者的极限存在。
Rolle’s Theorem: if f(a)=f(b) and f is continuous on [a,b], differentiable on (a,b), then there exists c in (a,b) with f ‘(c)=0.
罗尔定理：若f(a)=f(b)且f在[a,b]连续、(a,b)可导，则存在c∈(a,b)使f ‘(c)=0。
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