AP Calculus AB & BC Core Formula Summary | AP微积分AB与BC核心公式汇总

📚 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。

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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。

lim_x→a f(x) = L

A function is continuous at x = a if lim_x→a f(x) = f(a).

函数在x = a处连续需要满足 lim_x→a f(x) = f(a)。

f is continuous at a ⇔ lim_x→a f(x) = f(a)

Special limits include lim_x→0 (sin x)/x = 1 and lim_x→0 (1 - cos x)/x = 0.

重要极限包括 lim_x→0 (sin x)/x = 1 与 lim_x→0 (1 - cos x)/x = 0。

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2. Basic Differentiation Rules | 基本求导法则

The derivative of a function f at x is defined as f '(x) = lim_h→0 [f(x+h) - f(x)]/h.

函数f在x处的导数定义为 f '(x) = lim_h→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)

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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)。

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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。

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)。

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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: ∫_a^b f(x) dx = lim_n→∞ ∑ f(x_i^*) Δx.

定积分是黎曼和的极限：∫_a^b f(x) dx = lim_n→∞ ∑ f(x_i^*) Δx。

Fundamental Theorem of Calculus, Part 1: if F(x) = ∫_a^x f(t) dt, then F '(x) = f(x).

微积分基本定理第一部分：若 F(x) = ∫_a^x f(t) dt，则 F '(x) = f(x)。

Part 2: ∫_a^b f(x) dx = F(b) - F(a), where F is any antiderivative of f.

第二部分：∫_a^b f(x) dx = F(b) - F(a)，其中F为f的任意一个原函数。

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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

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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 = ∫_a^b [f(x) - g(x)] dx, where f(x) ≥ g(x).

曲线间面积：A = ∫_a^b [f(x) - g(x)] dx，其中 f(x) ≥ g(x)。

Volume by disks/washers: V = π ∫_a^b [R(x)]² dx or π ∫_a^b ([R(x)]² - [r(x)]²) dx.

圆盘/垫圈法求体积：V = π ∫_a^b [R(x)]² dx 或 π ∫_a^b ([R(x)]² - [r(x)]²) dx。

Volume by shells (BC): V = 2π ∫_a^b r(x) h(x) dx.

壳层法求体积（BC）：V = 2π ∫_a^b r(x) h(x) dx。

Average value of f on [a,b]: f_avg = (1/(b-a)) ∫_a^b f(x) dx.

f在[a,b]上的平均值：f_avg = (1/(b-a)) ∫_a^b f(x) dx。

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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 e^kt.

指数增长/衰减模型：dy/dt = k y 的解为 y = y₀ e^kt。

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)。

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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 = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt.

参数曲线的弧长：L = ∫_a^b √[(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)|。

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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 ∫_α^β [f(θ)]² dθ.

极坐标曲线围成的面积：A = ½ ∫_α^β [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)。

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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/n^p converges if p > 1, diverges if p ≤ 1.

p-级数 ∑ 1/n^p 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: e^x = ∑ 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：e^x = ∑ xⁿ/n!，sin x = ∑ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = ∑ (-1)ⁿ x²ⁿ/(2n)!，1/(1-x) = ∑ xⁿ (|x|<1)。

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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）

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' 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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