AP Calculus Formula Summary | AP 微积分公式汇总

📚 AP Calculus Formula Summary | AP 微积分公式汇总

This article provides a comprehensive collection of essential formulas for AP Calculus AB and BC. It covers limits, derivatives, integrals, series, and all key topics needed to excel on the exam.

本文整理了 AP 微积分 AB 和 BC 的核心公式汇总，涵盖极限、导数、积分、级数以及考试中所需的全部重要内容。

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1. Limits & Continuity | 极限与连续性

The limit of f(x) as x approaches a is L: lim_x→a f(x) = L, meaning f(x) can be made arbitrarily close to L by taking x sufficiently close to a (x ≠ a).

当 x 无限趋近于 a (但 x ≠ a) 时，f(x) 无限趋近于 L，则 lim_x→a f(x) = L。

Limit laws: lim [f(x) ± g(x)] = lim f(x) ± lim g(x); lim [f(x)·g(x)] = lim f(x) · lim g(x); lim [f(x)/g(x)] = lim f(x) / lim g(x) provided the denominator limit is not zero; lim [c·f(x)] = c·lim f(x); lim [f(x)]ⁿ = [lim f(x)]ⁿ.

极限运算法则：和、差、积、商 (分母极限非零)、常数倍以及幂的极限均可分别求极限再运算。

Squeeze Theorem: if f(x) ≤ g(x) ≤ h(x) near a, and lim f(x) = lim h(x) = L, then lim g(x) = L.

夹逼定理：如果在 a 附近 f(x) ≤ g(x) ≤ h(x)，且 f(x) 与 h(x) 的极限都是 L，则 g(x) 的极限也是 L。

Important special limits: lim_x→0 sin(x)/x = 1; lim_x→0 (1 − cos x)/x = 0; lim_x→∞ (1 + 1/x)^x = e; lim_x→0 (1 + x)^1/x = e.

重要特殊极限：lim_x→0 sin(x)/x = 1；lim_x→0 (1 − cos x)/x = 0；lim_x→∞ (1 + 1/x)^x = e；lim_x→0 (1 + x)^1/x = e。

Continuity: f is continuous at a if lim_x→a f(x) = f(a). Types of discontinuities: removable, jump, infinite.

连续性：若 lim_x→a f(x) = f(a)，则 f 在 a 点连续。间断点类型：可去型、跳跃型、无穷型。

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2. Derivatives Definition & Rules | 导数定义与运算法则

Definition of the derivative: f'(x) = lim_h→0 [f(x+h) − f(x)] / h, also denoted dy/dx.

导数的定义：f'(x) = lim_h→0 [f(x+h) − f(x)] / h，也可记作 dy/dx。

Sum/Difference Rule: (f ± g)’ = f’ ± g’.

和差法则：(f ± g)’ = f’ ± g’。

Product Rule: (f·g)’ = f’·g + f·g’.

乘法法则：(f·g)’ = f’·g + f·g’。

Quotient Rule: (f / g)’ = (f’·g − f·g’) / g².

除法法则：(f / g)’ = (f’·g − f·g’) / g²。

Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x).

链式法则：d/dx [f(g(x))] = f'(g(x)) · g'(x)。

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x; then solve for dy/dx.

隐函数求导：方程两边同时对 x 求导，将 y 视为 x 的函数，然后解出 dy/dx。

Higher-order derivatives: f”(x), f”'(x), f⁽ⁿ⁾(x).

高阶导数：f”(x), f”'(x), f⁽ⁿ⁾(x)。

Derivative of inverse function: (f⁻¹)'(x) = 1 / f'(f⁻¹(x)).

反函数求导：(f⁻¹)'(x) = 1 / f'(f⁻¹(x))。

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3. Common Derivatives | 常见函数导数

Basic derivatives: d/dx (c) = 0; d/dx (xⁿ) = n xⁿ⁻¹; d/dx (e^x) = e^x; d/dx (a^x) = a^x ln a; d/dx (ln x) = 1/x; d/dx (log_a x) = 1/(x ln a).

基本导数：d/dx (c) = 0；d/dx (xⁿ) = n xⁿ⁻¹；d/dx (e^x) = e^x；d/dx (a^x) = a^x ln a；d/dx (ln x) = 1/x；d/dx (log_a x) = 1/(x ln a)。

Trigonometric derivatives: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x; d/dx (cot x) = −csc² x; d/dx (sec x) = sec x tan x; d/dx (csc x) = −csc x cot x.

三角函数的导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x；d/dx (cot x) = −csc² x；d/dx (sec x) = sec x tan x；d/dx (csc x) = −csc x cot x。

Inverse trigonometric derivatives: d/dx (arcsin x) = 1/√(1−x²); d/dx (arccos x) = −1/√(1−x²); d/dx (arctan x) = 1/(1+x²); d/dx (arccot x) = −1/(1+x²); d/dx (arcsec x) = 1/(|x|√(x²−1)); d/dx (arccsc x) = −1/(|x|√(x²−1)).

反三角函数的导数：d/dx (arcsin x) = 1/√(1−x²)；d/dx (arccos x) = −1/√(1−x²)；d/dx (arctan x) = 1/(1+x²)；d/dx (arccot x) = −1/(1+x²)；d/dx (arcsec x) = 1/(|x|√(x²−1))；d/dx (arccsc x) = −1/(|x|√(x²−1))。

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4. Applications of Derivatives | 导数的应用

Equation of tangent line at x = a: y − f(a) = f'(a)(x − a). Normal line: slope is −1/f'(a).

切线方程（在 x=a 处）：y − f(a) = f'(a)(x − a)。法线斜率：−1/f'(a)。

Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), then 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，在 (a,b) 可导，则存在 c∈(a,b) 使得 f'(c) = [f(b)−f(a)]/(b−a)。

L’Hopital’s Rule: For indeterminate forms 0/0 or ∞/∞, lim f(x)/g(x) = lim f'(x)/g'(x) provided the limit exists.

洛必达法则：若极限为 0/0 或 ∞/∞ 不定型，则 lim f(x)/g(x) = lim f'(x)/g'(x)（若极限存在）。

Critical points: where f'(x) = 0 or f'(x) does not exist. First derivative test for relative extrema; second derivative test: if f'(c)=0 and f”(c) > 0 → local min, f”(c) < 0 → local max.

临界点：f'(x)=0 或不可导的点。一阶导数判定极值；二阶导数判定：若 f'(c)=0 且 f”(c)>0 则为局部极小，f”(c)<0 为局部极大。

Concavity: f”(x) > 0 implies concave up; f”(x) < 0 implies concave down. Inflection point: where concavity changes sign.

凹性：f”(x)>0 凹向上；f”(x)<0 凹向下。拐点：凹性改变符号的点。

Rectilinear motion: position s(t), velocity v(t) = s'(t), speed = |v(t)|, acceleration a(t) = v'(t) = s”(t).

直线运动：位置 s(t)，速度 v(t)=s'(t)，速率 |v(t)|，加速度 a(t)=v'(t)=s”(t)。

Related rates: relate rates of change of variables using differentiation with respect to time.

相关速率：对各变量关于时间求导，建立变化率之间的关系。

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5. Integrals & Fundamental Theorem | 积分与微积分基本定理

Indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x).

不定积分：∫ f(x) dx = F(x) + C，其中 F'(x)=f(x)。

Fundamental Theorem of Calculus, Part 1: If f is continuous on [a,b], then g(x) = ∫_a^x f(t) dt is differentiable and g'(x) = f(x).

微积分基本定理第一部分：若 f 在 [a,b] 连续，则 g(x)=∫_a^x f(t) dt 可导，且 g'(x)=f(x)。

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

微积分基本定理第二部分：∫_a^b f(x) dx = F(b) − F(a)，其中 F 是 f 的任意一个原函数。

Properties of definite integrals: ∫_a^b k·f(x) dx = k ∫_a^b f(x) dx; ∫_a^b [f ± g] dx = ∫_a^b f dx ± ∫_a^b g dx; ∫_a^b f(x) dx = −∫_b^a f(x) dx; ∫_a^c f = ∫_a^b f + ∫_b^c f.

定积分性质：常数因子可提；和差可拆；区间反向变号；积分区间可加性。

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

函数在 [a

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