Essential AP Calculus Formulas Summary | AP微积分必备公式汇总

📚 Essential AP Calculus Formulas Summary | AP微积分必备公式汇总

Mastering AP Calculus requires a solid grasp of key formulas spanning limits, derivatives, integrals, series, and more. This article compiles the essential formulas from both AB and BC curricula for quick revision and exam success.

掌握AP微积分需要牢固掌握涵盖极限、导数、积分、级数等关键公式。本文汇编了AB和BC课程中的必备公式，方便快速复习和迎考。

1. Limits and Continuity | 极限与连续性

The limit of f(x) as x approaches c is L if the values of f(x) become arbitrarily close to L when x is near c. Notation: lim_x→c f(x) = L.

当x趋近于c时，如果f(x)的值可以任意接近L，则极限为L。记作：lim_x→c f(x) = L。

For a limit to exist at c, both one‑sided limits must be equal: lim_x→c⁻ f(x) = lim_x→c⁺ f(x) = L.

极限存在的条件是左右极限必须相等：lim_x→c⁻ f(x) = lim_x→c⁺ f(x) = L。

Important fundamental limits:

重要基本极限：

lim_x→0 sin x / x = 1

lim_x→∞ (1 + 1/x)^x = e

The Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) near c and lim g = lim h = L, then lim f = L.

夹逼定理：如果在c附近有 g(x) ≤ f(x) ≤ h(x) 且 lim g = lim h = L，那么 lim f = L。

Continuity at x = c requires f(c) to be defined, lim_x→c f(x) to exist, and both to be equal. Types of discontinuities: removable, jump, infinite.

函数在x=c处连续需要 f(c) 有定义、极限存在且两者相等。间断点类型：可去、跳跃、无穷间断。

2. Definition of Derivative | 导数的定义

The derivative of f at x is the slope of the tangent line, defined by the difference quotient:

函数在x处的导数就是切线的斜率，由差商定义：

f'(x) = lim_h→0 [f(x+h) – f(x)] / h

An equivalent form using a specific point a: f'(a) = lim_x→a [f(x) – f(a)] / (x – a).

使用特定点a的等价形式：f'(a) = lim_x→a [f(x) – f(a)] / (x – a)。

Differentiability implies continuity, but continuity does not guarantee differentiability (e.g., corner, cusp, vertical tangent).

可微一定连续，但连续不一定可微（如尖点、折点、垂直切线）。

The derivative gives the instantaneous rate of change and can be interpreted as velocity when s(t) is position: v(t) = s'(t).

导数表示瞬时变化率；若 s(t) 是位置函数，则速度 v(t) = s'(t)。

3. Differentiation Rules | 基本微分法则

Power Rule: d/dx (xⁿ) = n x^n–1 for any real n.

幂规则：d/dx (xⁿ) = n x^n–1（n为任意实数）。

Constant Multiple & Sum/Difference: d/dx [c·f(x)] = c·f'(x); d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

常数倍与和差法则：d/dx [c·f(x)] = c·f'(x)；d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

Product Rule: (fg)’ = f’g + fg’

积法则：(fg)’ = f’g + fg’

d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule: (f/g)’ = [f’g – fg’] / g²

商法则：(f/g)’ = (f’g – fg’) / g²

d/dx [f(x)/g(x)] = [f'(x)g(x) – f(x)g'(x)] / [g(x)]²

Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx) = f'(g(x))·g'(x).

链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du)·(du/dx) = f'(g(x))·g'(x)。

Essential derivatives of elementary functions:

基本初等函数的导数公式：

d/dx (e^x) = e^x — d/dx (e^x) = e^x
d/dx (a^x) = a^x ln a — d/dx (a^x) = a^x ln a
d/dx (ln x) = 1/x — d/dx (ln x) = 1/x
d/dx (sin x) = cos x — d/dx (sin x) = cos x
d/dx (cos x) = –sin x — d/dx (cos x) = –sin x
d/dx (tan x) = sec² x — d/dx (tan x) = sec² x
d/dx (cot x) = –csc² x — d/dx (cot x) = –csc² x
d/dx (sec x) = sec x tan x — d/dx (sec x) = sec x tan x
d/dx (csc x) = –csc x cot x — d/dx (csc x) = –csc x cot x

4. Implicit and Inverse Differentiation | 隐函数与反函数微分

Implicit differentiation: When an equation defines y implicitly as a function of x, differentiate both sides with respect to x, applying the chain rule to terms involving y (e.g., d/dx[y²] = 2y·dy/dx).

隐函数求导：当方程隐含地定义y为x的函数时，方程两边同时对x求导，对含有y的项应用链式法则（例如 d/dx[y²] = 2y·dy/dx）。

Derivative of an inverse function:

反函数求导公式：

(f^–1)'(y) = 1 / f'(x)

where y = f(x) and f'(x) ≠ 0.

其中 y = f(x) 且 f'(x) ≠ 0。

Inverse trigonometric derivatives:

反三角函数的导数：

d/dx (arcsin x) = 1 / √(1 – x²) — d/dx (arcsin x) = 1 / √(1 – x²)
d/dx (arccos x) = –1 / √(1 – x²) — d/dx (arccos x) = –1 / √(1 – x²)
d/dx (arctan x) = 1 / (1 + x²) — d/dx (arctan x) = 1 / (1 + x²)
d/dx (arccot x) = –1 / (1 + x²) — d/dx (arccot x) = –1 / (1 + x²)
d/dx (arcsec x) = 1 / (|x|√(x² – 1)) — d/dx (arcsec x) = 1 / (|x|√(x² – 1))

Logarithmic differentiation: Take ln of both sides and differentiate to handle complicated products, quotients, or powers.

对数求导法：两边取对数再求导，用于处理复杂的乘积、商或幂指函数。

5. Applications of Derivatives | 导数的应用

Critical points: where f'(x) = 0 or f'(x) does not exist. They are candidates for local extrema.

临界点：f'(x) = 0 或 f'(x) 不存在的点，它们是局部极值的候选点。

First Derivative Test: If f’ changes from positive to negative at c, f has a local maximum; from negative to positive, a local minimum.

一阶导数判别法：如果 f’ 在 c 处由正变负，则 f 有局部极大值；由负变正，则为局部极小值。

Second Derivative Test: If f'(c) = 0 and f”(c) > 0, f has a local minimum at c; if f”(c) < 0, a local maximum.

二阶导数判别法：若 f'(c)=0 且 f”(c)>0，则 f 在 c 处有局部极小值；若 f”(c)<0，则为局部极大值。

Concavity: f”(x) > 0 ⇒ concave up; f”(x) < 0 ⇒ concave down. Inflection points occur where concavity changes.

凹凸性：f”(x) > 0 表示凹向上；f”(x) < 0 表示凹向下。拐点出现在凹凸性改变的位置。

Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that

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)

Rolle’s Theorem is the special case where f(a) = f(b) ⇒ f'(c) = 0.

罗尔定理是中值定理的特例：当 f(a)=f(b) 时，存在 c 使 f'(c)=0。

L’Hôpital’s Rule: For indeterminate forms 0/0 or ∞/∞,

洛必达法则：对于 0/0 或 ∞/∞ 型不定式，

lim f/g = lim f’/g’

provided the limit on the right exists.

只要右边的极限存在。

Related rates involve differentiating an equation that relates two or more variables with respect to time t.

相关变化率问题是对联系多个变量的方程关于时间 t 求导。

Linear approximation (tangent line approximation) near x = a: f(x) ≈ f(a) + f'(a)(x – a).

线性近似（切线近似）在 x=a 附近：f(x) ≈ f(a) + f'(a)(x – a)。

6. Integrals and the Fundamental Theorem | 积分与基本定理

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

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

Definite integral: ∫_a^b f(x) dx represents the net signed area between the curve and the x‑axis.

定积分 ∫_a^b f(x) dx 表示曲线与x轴之间的带符号净面积。

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

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

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

第二部分：若 F 是 f 的任意原函数，则 ∫_a^b f(x) dx = F(b) – F(a)。

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

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

7. Integration Techniques | 积分技巧

Basic integration formulas:

基本积分公式：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1) — ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ –1）
∫ 1/x dx = ln |x| + C — ∫ 1/x dx = ln |x| + C
∫ e^x dx = e^x + C — ∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln a + C — ∫ a^x dx = a^x/ln a + C
∫ sin x dx = –cos x + C — ∫ sin x dx = –cos x + C
∫ cos x dx = sin x + C — ∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C — ∫ sec² x dx = tan x + C
∫ csc² x dx = –cot x + C — ∫ csc² x dx = –cot x + C
∫ sec x tan x dx = sec x + C — ∫ sec x tan x dx = sec x + C
∫ dx/√(1–x²) = arcsin x + C — ∫ dx/√(1–x²) = arcsin x + C
∫ dx/(1+x²) = arctan x + C — ∫ dx/(1+x²) = arctan x + C

u‑Substitution: For ∫ f(g(x))·g'(x) dx, let u = g(x), du = g'(x) dx, then integrate ∫ f(u) du. Adjust limits for definite integrals.

换元积分法：对于 ∫ f(g(x))·g'(x) dx，令 u=g(x), du=g'(x)dx，然后积分 ∫ f(u) du。定积分需要同时变换上下限。

Integration by Parts (BC):

分部积分法（BC）：

∫ u dv = uv – ∫ v du

Choose u according to LIATE (Log, Inverse trig, Algebraic, Trig, Exponential).

通常按 LIATE 顺序（对数、反三角、代数、三角、指数）选择 u。

Partial Fractions (BC): Decompose rational functions into simpler fractions and integrate term by term.

部分分式法（BC）：将有理函数分解为更简单的分式，然后逐项积分。

8. Applications of Integration | 积分的应用

Area between curves: Area = ∫_a^b [f(x) – g(x)] dx, where f(x) ≥ g(x) on [a, b].

曲线间的面积：面积 = ∫_a^b [f(x) – g(x)] dx，其中在 [a,b] 上 f(x) ≥ g(x)。

Volumes of revolution (disk method): V = π ∫_a^b [R(x)]² dx for rotation about the x‑axis.

旋转体体积（圆盘

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Essential AP Calculus Formulas Summary | AP微积分必备公式汇总