AP Calculus AB & BC Core Formula Compilation | AP微积分AB与BC核心公式汇编

📚 AP Calculus AB & BC Core Formula Compilation | AP微积分AB与BC核心公式汇编

Whether you are preparing for the AP Calculus AB or BC exam, having a solid command of essential formulas is key to success. This compilation covers limits, derivatives, integrals, differential equations, and BC‑only topics such as parametric calculus, series, and advanced integration techniques. Use this guide as a quick reference while practising past exam questions.

无论你正在备考AP微积分AB还是BC，牢固掌握核心公式都是取得高分的关键。本汇编涵盖了极限、导数、积分、微分方程以及BC专属内容（参数微积分、级数、高级积分方法），可作为刷题时的快速参考手册。

1. Limits and Continuity | 极限与连续性

The limit of a function f(x) as x approaches a is L if f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but x ≠ a). Symbolically, this is written as lim_x→a f(x) = L.

当x趋近于a时，如果f(x)可以无限接近L，则称L为f(x)在x→a时的极限，记作limₓ→ₐ f(x) = L。

lim_x→a f(x) = L   and   one‑sided limits: lim_x→a⁻ f(x), lim_x→a⁺ f(x)

For a limit to exist at a, the left‑hand and right‑hand limits must both exist and be equal. A function is continuous at a if lim_x→a f(x) = f(a). The Squeeze Theorem states that if g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L.

极限存在的充要条件是左右极限存在且相等。若limₓ→ₐ f(x) = f(a)，则函数在a点连续。夹逼定理：如果在a附近g(x) ≤ f(x) ≤ h(x)且lim g = lim h = L，则lim f = L。

Squeeze Theorem: g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L ⇒ lim f = L

Key special limits include lim_x→0 (sin x)/x = 1 and lim_x→0 (1 − cos x)/x = 0. Limits at infinity describe horizontal asymptotes: if lim_x→∞ f(x) = L, then y = L is a horizontal asymptote. Infinite limits signal vertical asymptotes.

两个重要极限：limₓ→₀ (sin x)/x = 1 和 limₓ→₀ (1 − cos x)/x = 0。无穷远处的极限刻画水平渐近线，无穷极限则对应垂直渐近线。

lim_x→0 sin x / x = 1,   lim_x→0 (1 − cos x)/x = 0,   lim_x→∞ (1 + 1/x)^x = e

2. Definition of the Derivative | 导数定义

The derivative of f at x is defined as the limit of the difference quotient. It represents the slope of the tangent line and the instantaneous rate of change. The alternative form is especially useful for finding the derivative at a specific point a.

导数被定义为差商的极限，它代表切线斜率与瞬时变化率。另一种等价形式常用于求某一点处的导数值。

f ‘(x) = lim_h→0 (f(x+h) − f(x)) / h

f ‘(a) = lim_x→a (f(x) − f(a)) / (x − a)

If a function is differentiable at a point, it must be continuous there; the converse is not always true. Common notations include f ‘(x), dy/dx, y’, and D_xf.

可导必定连续，但连续不一定可导。常见符号有 f ‘(x)、dy/dx、y’ 和 D_xf。

3. Differentiation Rules | 求导法则

Mastering basic derivative rules is essential for both AB and BC. The power rule, product rule, quotient rule, and chain rule form the foundation. Coupled with derivatives of elementary functions, these allow you to differentiate almost any expression encountered on the exam.

掌握基本求导法则是AB与BC的共同要求。幂法则、乘法法则、除法法则与链式法则构成核心，结合初等函数的导数公式即可处理考试中绝大多数函数。

Power: d/dx [xⁿ] = n xⁿ⁻¹   |   Product: (uv)’ = u’v + uv’   |   Quotient: (u/v)’ = (u’v − uv’) / v²

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

Derivatives of trigonometric and inverse trigonometric functions must be memorised.

三角函数与反三角函数的导数必须熟记。

sin x → cos x,   cos x → −sin x,   tan x → sec² x,   cot x → −csc<

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