AP Calculus: Common Formulas Compilation | AP微积分常用公式汇编

📚 AP Calculus: Common Formulas Compilation | AP微积分常用公式汇编

This article collects the essential formulas you need to master for the AP Calculus AB and BC exams. Each section presents the most important limits, derivatives, integrals, and series formulas, with clear bilingual explanations to reinforce your understanding. Use this compilation as a quick reference during your revision, and make sure you can apply every formula in context.

本文汇编了AP微积分AB与BC考试必须掌握的核心公式。每个部分列出了最重要的极限、导数、积分和级数公式，并配有清晰的中英双语解释以加深理解。复习时可以将本文用作快速参考手册，务必确保能在具体问题中灵活运用每一个公式。

1. Limits & Continuity | 极限与连续性

The limit of a function describes its behavior as the input approaches a particular value. Key limit definitions and properties are fundamental to understanding derivatives and integrals.

函数的极限描述了当自变量趋近于某个特定值时函数的行为。关键的极限定义与性质是理解导数与积分的基础。

limₓ→ₐ f(x) = L

This means f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but x ≠ a).

这意味着只要让 x 充分接近 a（但 x ≠ a），f(x) 就可以无限趋近于 L。

limₓ→ₐ⁻ f(x) = limₓ→ₐ⁺ f(x) → limₓ→ₐ f(x) exists

For the two-sided limit to exist, the left-hand limit and the right-hand limit must be equal.

要使双侧极限存在，左极限与右极限必须相等。

limₓ→∞ f(x) = L

Describes the end behavior of f(x) as x grows without bound.

描述当 x 无限增大时 f(x) 的渐近行为。

Sum, product, quotient and composition rules for limits apply when the individual limits exist. Special limit formulas include the Squeeze Theorem and the limit of sin x / x.

当各极限存在时，可使用极限的四则运算和复合运算法则。特殊极限公式包括夹逼定理以及 sin x / x 的极限。

limₓ→₀ sin x / x = 1

This fundamental trigonometric limit is used to find derivatives of sine and cosine.

这一基本三角极限用于推导正弦和余弦的导数。

limₓ→₀ (1 – cos x) / x = 0

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2. Differentiation Rules | 求导法则

Differentiation rules allow us to compute the derivative of a function efficiently. The derivative f'(x) gives the instantaneous rate of change of f with respect to x.

求导法则让我们能够高效地计算函数的导数。导数 f'(x) 表示 f 关于 x 的瞬时变化率。

d/dx [c] = 0 (c is constant)

The derivative of a constant is zero.

常数的导数为零。

d/dx [xⁿ] = n xⁿ⁻¹ (Power Rule)

Multiply by the exponent and reduce the exponent by one.

幂函数求导：指数变系数，指数减一。

d/dx [c f(x)] = c f'(x) (Constant Multiple Rule)

A constant factor can be pulled out of the derivative.

常数因子可以提到导数符号外面。

d/dx [f(x) ± g(x)] = f'(x) ± g'(x) (Sum/Difference Rule)

The derivative of a sum is the sum of the derivatives.

和的导数等于导数的和。

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

For the product of two functions, differentiate one at a time.

两个函数乘积的导数：先导一个，另一个不变，然后加起来。

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

For a quotient, the denominator is squared, and the numerator uses a specific difference pattern.

商的导数：分母平方，分子为“上导下不导减去上不导下导”。

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

Multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

链式法则：外层函数在内部函数处的导数乘以内层函数的导数。

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3. Derivatives of Common Functions | 常见函数的导数

These are the derivatives you should have memorized without hesitation. They include trigonometric, inverse trigonometric, exponential, and logarithmic functions.

以下是你必须熟练记忆的导数公式，包括三角函数、反三角函数、指数函数和对数函数。

d/dx [sin x] = cos x

Derivative of sine is cosine.

正弦的导数是余弦。

d/dx [cos x] = -sin x

Derivative of cosine is negative sine.

余弦的导数是负正弦。

d/dx [tan x] = sec² x

Derivative of tangent is secant squared.

正切的导数是正割的平方。

d/dx [cot x] = -csc² x

d/dx [sec x] = sec x tan x

d/dx [csc x] = -csc x cot x

For inverse trig functions, the derivatives involve square roots:

反三角函数的导数包含平方根：

d/dx [sin⁻¹ x] = 1 / √(1 – x²)

d/dx [cos⁻¹ x] = -1 / √(1 – x²)

d/dx [tan⁻¹ x] = 1 / (1 + x²)

d/dx [cot⁻¹ x] = -1 / (1 + x²)

d/dx [sec⁻¹ x] = 1 / (|x| √(x² – 1))

d/dx [csc⁻¹ x] = -1 / (|x| √(x² – 1))

Exponential and logarithmic derivatives:

指数与对数函数的导数：

d/dx [eˣ] = eˣ

The natural exponential function is its own derivative.

自然指数函数的导数等于它自身。

d/dx [aˣ] = aˣ ln a (a > 0)

d/dx [ln x] = 1/x

d/dx [logₐ x] = 1 / (x ln a)

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4. Implicit Differentiation & Related Rates | 隐函数求导与相关变化率

When a function is given implicitly or variables depend on time, the chain rule becomes essential.

当函数由隐式方程给出或变量依赖于时间时，链式法则至关重要。

If y is a function of x, then d/dx [yⁿ] = n yⁿ⁻¹ dy/dx, using the chain rule. Implicit differentiation involves differentiating both sides of an equation with respect to x and then solving for dy/dx.

若 y 是 x 的函数，则根据链式法则 d/dx [yⁿ] = n yⁿ⁻¹ dy/dx。隐函数求导即对方程两边关于 x 求导，再解出 dy/dx。

Equation: x² + y² = r² → d/dx [x² + y²] = d/dx [r²] → 2x + 2y dy/dx = 0 → dy/dx = -x/y

This technique is used when separating variables is difficult.

当难以分离变量时使用该方法。

For related rates, if a quantity A depends on time t, compute dA/dt by relating A to other time-dependent quantities and differentiating. A common formula arises from the Pythagorean theorem or similar geometry relationships.

在相关变化率问题中，若量 A 依赖于时间 t，可通过将 A 与其他随时间变化的量建立关系并求导来计算 dA/dt。常见公式来自勾股定理或类似几何关系。

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5. Integration Rules & Techniques | 积分法则与技巧

Integration reverses differentiation. The indefinite integral (antiderivative) includes a constant of integration C. Key rules and techniques must be automatic.

积分是微分的逆运算。不定积分（原函数）包含积分常数 C。必须熟练运用基本法则和积分技巧。

∫ k f(x) dx = k ∫ f(x) dx

Constants can be factored out of the integral.

常数可以提到积分号外。

∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Integrate term by term.

逐项积分。

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, n ≠ -1 (Power Rule for Integration)

Increase the exponent by one and divide by the new exponent, except when n = -1.

指数加一后除以新指数，但 n = -1 时例外。

∫ (1/x) dx = ln |x| + C

The integral of 1/x is the natural logarithm of the absolute value of x.

1/x 的积分是 ln|x| 加常数。

Basic integrals of common functions to memorize:

需记忆的常见函数积分：

∫ eˣ dx = eˣ + C

∫ aˣ dx = aˣ / ln a + C

∫ sin x dx = -cos x + C

∫ cos x dx = sin x + C

∫ sec² x dx = tan x + C

∫ csc² x dx = -cot x + C

∫ sec x tan x dx = sec x + C

∫ csc x cot x dx = -csc x + C

Techniques like u-substitution arise from the chain rule: ∫ f(g(x)) g'(x) dx = ∫ f(u) du with u = g(x).

换元积分法源自链式法则：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。

Integration by parts: ∫ u dv = uv – ∫ v du. Useful for products of functions like x eˣ or x sin x.

分部积分法：∫ u dv = uv – ∫ v du，常用于计算 x eˣ 或 x sin x 之类的乘积。

Partial fractions: a rational function can be decomposed into simpler fractions; for example, (2x+3)/(x²+x) = A/x + B/(x+1). Then integrate termwise.

部分分式分解：可将有理函数拆分为更简单的分式，如 (2x+3)/(x²+x) = A/x + B/(x+1)，然后逐项积分。

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6. Definite Integrals & The Fundamental Theorem of Calculus | 定积分与微积分基本定理

The definite integral ∫ₐᵇ f(x) dx gives the net area between f(x) and the x-axis from x = a to x = b. The Fundamental Theorem of Calculus connects differentiation and integration.

定积分 ∫ₐᵇ f(x) dx 表示 f(x) 与 x 轴在 x = a 到 x = b 之间的净面积。微积分基本定理将微分与积分联系起来。

FTC Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x)

The derivative of the accumulation function recovers the integrand.

变上限积分函数的导数等于被积函数。

FTC Part 2: ∫ₐᵇ f(x) dx = F(b) – F(a), where F is any antiderivative of f

Evaluate the antiderivative at the upper limit and subtract the value at the lower limit.

用任一原函数在上限的值减去下限的值。

Properties of definite integrals include additivity over intervals, reversal of limits changes sign, and comparison properties.

定积分的性质包括区间可加性、颠倒上下限改变符号以及比较性质。

Average value of a function on [a, b]: (1/(b-a)) ∫ₐᵇ f(x) dx.

函数在 [a, b] 上的平均值： (1/(b-a)) ∫ₐᵇ f(x) dx。

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7. Applications of Integration | 积分的应用

Integration is used to compute areas, volumes, arc lengths, and accumulated change. Be familiar with both rectangular and polar setups.

积分可用于计算面积、体积、弧长和累积变化。需熟悉直角坐标与极坐标下的设置。

Area between curves: For functions of x, A = ∫ₐᵇ (top – bottom) dx; for functions of y, A = ∫ₑᶠ (right – left) dy.

曲线间面积：对于 x 的函数，A = ∫ₐᵇ (上曲线 – 下曲线) dx；对于 y 的函数，A = ∫ₑᶠ (右曲线 – 左曲线) dy。

Volume by disks/washers: Rotating around x-axis: V = π ∫ₐᵇ [R(x)]² dx; with a hole, V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx.

圆盘/垫圈法求体积：绕 x 轴旋转：V = π ∫ₐᵇ [R(x)]² dx；有空心时，V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx。

Volume by cylindrical shells: Rotating around y-axis: V = 2π ∫ₐᵇ x f(x) dx (vertical shells).

柱壳法求体积：绕 y 轴旋转：V = 2π ∫ₐᵇ x f(x) dx（垂直薄壳）。

Arc length: L = ∫ₐᵇ √(1 + [f'(x)]²) dx for y = f(x).

弧长：对于 y = f(x)，L = ∫ₐᵇ √(1 + [f'(x)]²) dx。

Surface area (parametric or Cartesian): not always tested on AP but useful.

表面积（参数或直角坐标）：AP不常考，但有用。

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8. Parametric, Polar & Vector-Valued Functions | 参数方程、极坐标与向量值函数

For AP Calculus BC, you must handle derivatives and integrals in parametric and polar forms, as well as motion with vectors.

AP微积分BC要求掌握参数方程与极坐标下的导数与积分，以及向量的运动分析。

Parametric curves: Given x = f(t), y = g(t), the derivative dy/dx = (dy/dt) / (dx/dt) provided dx/dt ≠ 0.

参数曲线：给定 x = f(t), y = g(t)，导数 dy/dx = (dy/dt) / (dx/dt)，且 dx/dt ≠ 0。

Second derivative: d²y/dx² = (d/dt [dy/dx]) / (dx/dt).

二阶导数：d²y/dx² = (d/dt [dy/dx]) / (dx/dt)。

Arc length in parametric form: L = ∫ₐᵝ √((dx/dt)² + (dy/dt)²) dt.

参数形式下的弧长：L = ∫ₐᵝ √((dx/dt)² + (dy/dt)²) dt。

Polar coordinates: r = f(θ). The area enclosed by a polar curve from θ = α to θ = β is (1/2) ∫ₐᵝ [f(θ)]² dθ.

极坐标：r = f(θ)。曲线在 θ = α 到 θ = β 之间围成的面积为 (1/2) ∫ₐᵝ [f(θ)]² dθ。

Slope in polar: dy/dx = (r’ sin θ + r cos θ) / (r’ cos θ – r sin θ) where r’ = dr/dθ.

极坐标系下的斜率：dy/dx = (r’ sin θ + r cos θ) / (r’ cos θ – r sin θ)，其中 r’ = dr/dθ。

Vector-valued functions: r(t) = ⟨x(t), y(t)⟩. Velocity v(t) = r'(t), speed = |v| = √((x’)² + (y’)²), acceleration a(t) = r”(t).

向量值函数：r(t) = ⟨x(t), y(t)⟩。速度 v(t) = r'(t)，速率 = |v| = √((x’)² + (y’)²)，加速度 a(t) = r”(t)。

Total distance traveled from t = a to t = b is ∫ₐᵇ |v(t)| dt.

从 t = a 到 t = b 的总路程为 ∫ₐᵇ |v(t)| dt。

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9. Sequences & Series | 数列与级数

BC students must work with infinite sequences and series, convergence tests, power series, and Taylor/Maclaurin polynomials.

BC考生必须掌握无穷数列与级数、收敛性判别法、幂级数以及泰勒/麦克劳林多项式。

Sequence convergence: A sequence aₙ converges to L if limₙ→∞ aₙ = L. Otherwise it diverges.

数列收敛：如果 limₙ→∞ aₙ = L，则数列 aₙ 收敛于 L，否则发散。

Geometric series: Σₙ₌₀∞ arⁿ = a / (1 – r) if |r| < 1. For finite sum: Sₙ = a(1 - rⁿ) / (1 - r).

几何级数：Σₙ₌₀∞ arⁿ = a / (1 – r) 当 |r| < 1。前 n 项和：Sₙ = a(1 - rⁿ) / (1 - r)。

p-series: Σ 1/nᵖ converges if p > 1, diverges if p ≤ 1.

p-级数：Σ 1/nᵖ 当 p > 1 时收敛，当 p ≤ 1 时发散。

Harmonic series: Σ 1/n diverges.

调和级数：Σ 1/n 发散。

Telescoping series: Terms cancel, limit of partial sums determines convergence.

裂项相消级数：项相互抵消，由部分和的极限确定收敛性。

Convergence tests: Ratio test, Root test, Comparison test, Limit Comparison test, Integral test, Alternating series test. Formulas for error bounds: Alternating series remainder |Rₙ| ≤ aₙ₊₁; Integral test remainder bound Rₙ ≤ ∫ₙ∞ f(x) dx.

收敛判别法：比值判别法、根值判别法、比较判别法、极限比较判别法、积分判别法、交错级数判别法。误差界公式：交错级数余项 |Rₙ| ≤ aₙ₊₁；积分判别余项界 Rₙ ≤ ∫ₙ∞ f(x) dx。

Power series: Σ cₙ (x – a)ⁿ. Radius of convergence R is found using ratio or root test. Interval of convergence must be tested with endpoints.

幂级数：Σ cₙ (x – a)ⁿ。收敛半径 R 由比值法或根值法求得，收敛区间需检验端点。

Taylor series: f(x) = Σₙ₌₀∞ (f⁽ⁿ⁾(a) / n!) (x – a)ⁿ. Maclaurin series is the case a = 0.

泰勒级数：f(x) = Σₙ₌₀∞ (f⁽ⁿ⁾(a) / n!) (x – a)ⁿ。麦克劳林级数是 a = 0 的特例。

Important Maclaurin series to memorize:

必须记忆的重要麦克劳林级数：

eˣ = Σₙ₌₀∞ xⁿ / n! = 1 + x + x²/2! + x³/3! + …

sin x = Σₙ₌₀∞ (-1)ⁿ x²ⁿ⁺¹ / (2n+1)! = x – x³/3! + x⁵/5! – …

cos x = Σₙ₌₀∞ (-1)ⁿ x²ⁿ / (2n)! = 1 – x²/2! + x⁴/4! – …

1/(1 – x) = Σₙ₌₀∞ xⁿ, |x| < 1

ln(1 + x) = Σₙ₌₁∞ (-1)ⁿ⁻¹ xⁿ / n, -1 < x ≤ 1

arctan x = Σₙ₌₀∞ (-1)ⁿ x²ⁿ⁺¹ / (2n+1), |x| ≤ 1

Lagrange error bound for Taylor polynomials: |Rₙ(x)| ≤ max|f⁽ⁿ⁺¹⁾(z)| / (n+1

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