AP Calculus Formula Sheet | AP微积分公式手册

📚 AP Calculus Formula Sheet | AP微积分公式手册

This AP Calculus Formula Sheet is designed to provide a concise yet comprehensive reference for all key formulas you need to know for the AP Calculus AB and BC exams. From limits and derivatives to integrals, series, and parametric equations, this guide organizes essential results by topic, with clear notation and practical examples. Use it for quick review, homework help, or last-minute exam prep.

本AP微积分公式手册旨在为AP微积分AB和BC考试提供一份简洁而全面的公式参考。从极限、导数到积分、级数以及参数方程，本指南按主题整理了必备公式，采用清晰的表示法并附有实用示例。可用于快速复习、作业辅助或考前冲刺。

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1. Limits and Continuity | 极限与连续性

Limits describe the behavior of a function as the input approaches a particular value. Continuity requires that a function be defined at a point, its limit exist at that point, and the limit equal the function value.

极限描述函数在输入趋近于某个值时的行为。连续性要求在一点处函数有定义、极限存在且极限值等于函数值。

lim x→c f(x) = L

The limit of f(x) as x approaches c exists and equals L if f(x) can be made arbitrarily close to L by taking x sufficiently close to c.

当x趋近于c时，若通过让x充分靠近c可使f(x)任意接近L，则极限存在且等于L。

lim x→c⁺ f(x) = L (right-hand limit)   lim x→c⁻ f(x) = L (left-hand limit)

One-sided limits consider approach only from the right (positive side) or from the left (negative side). The two-sided limit exists if and only if both one-sided limits exist and are equal.

单侧极限只考虑从右侧（正方向）或左侧（负方向）趋近。双侧极限存在当且仅当两个单侧极限都存在且相等。

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

The Squeeze Theorem allows evaluation of a limit by bounding the function between two others that have the same limit.

夹逼定理通过将函数夹在两个具有相同极限的函数之间来计算极限。

Continuity at c: f(c) is defined, lim x→c f(x) exists, and lim x→c f(x) = f(c).

A function is continuous on an interval if it is continuous at every point. Discontinuities can be removable, jump, or infinite.

若函数在区间内每点都连续，则在该区间连续。不连续点可分为可去、跳跃或无穷间断点。

Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in (a,b) with f(c) = k.

介值定理：若f在闭区间[a,b]上连续，k介于f(a)与f(b)之间，则在开区间(a,b)内至少存在一点c使得f(c)=k。

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2. Definition of the Derivative | 导数的定义

The derivative measures the instantaneous rate of change of a function and is defined as the limit of the difference quotient. It can be expressed in multiple equivalent forms.

导数衡量函数的瞬时变化率，定义为差商的极限，可以有多种等价表示形式。

f ‘(x) = lim h→0 [f(x+h) – f(x)] / h

The derivative of f at x is the slope of the tangent line to the graph of f at that point.

函数f在x处的导数即为该点切线的斜率。

f ‘(a) = lim x→a [f(x) – f(a)] / (x – a)

An alternative form that directly gives the slope at a specific point a.

另一种形式，直接给出某点a处的斜率。

Notation: f ‘(x), dy/dx, d/dx [f(x)], y’

All represent the derivative of y with respect to x. The operator d/dx emphasizes the variable of differentiation.

均表示y关于x的导数。算子d/dx强调对x求导。

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3. Basic Differentiation Rules | 基本求导法则

These rules allow efficient calculation of derivatives without repeatedly using the limit definition.

这些法则使导数计算快捷而无需每次使用极限定义。

d/dx (c) = 0

The derivative of a constant is zero.

常数的导数为零。

Power Rule: d/dx (xⁿ) = n xⁿ⁻¹

Valid for any real number n. Bring down the exponent and reduce the exponent by 1.

幂法则对任意实数n成立。将指数移到前面，指数减1。

d/dx (c f(x)) = c f ‘(x)

The constant multiple rule: constants factor out.

常数倍法则：常数可提出。

d/dx [f(x) ± g(x)] = f ‘(x) ± g ‘(x)

The sum/difference rule: differentiate term by term.

和差法则：逐项求导。

Product Rule: d/dx [f(x) g(x)] = f ‘(x) g(x) + f(x) g ‘(x)

The derivative of a product is not the product of derivatives.

乘积的导数不等于导数的乘积，需使用此法则。

Quotient Rule: d/dx [f(x) / g(x)] = [f ‘(x) g(x) – f(x) g ‘(x)] / [g(x)]²

The denominator must not be zero. Remember “low d-high minus high d-low over low squared”.

分母不能为零。可记忆为“低微高减高微低，下方除以低平方”。

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4. Derivatives of Special Functions | 特殊函数的导数

Memorizing derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions is essential for the AP exam.

熟记指数函数、对数函数、三角函数和反三角函数的导数是AP考试的关键。

d/dx (eˣ) = eˣ

The natural exponential function is its own derivative.

自然指数函数的导数等于自身。

d/dx (aˣ) = aˣ ln a

For a > 0, a ≠ 1.

其中a>0，a≠1。

d/dx (ln x) = 1/x,   d/dx (logₐ x) = 1/(x ln a)

The derivative of the natural log and general log function.

自然对数和一般对数函数的导数。

d/dx (sin x) = cos x,   d/dx (cos x) = -sin x

Derivatives of sine and cosine are the start of the trigonometric cycle.

正弦和余弦的导数是三角求导循环的起点。

d/dx (tan x) = sec² x,   d/dx (cot x) = -csc² x

Derivatives of tangent and cotangent.

正切和余切的导数。

d/dx (sec x) = sec x tan x,   d/dx (csc x) = -csc x cot x

Derivatives of secant and cosecant.Published by TutorHao | AP Mathematics Revision Series | aleveler.com

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