AP Calculus AB & BC Important Formulas Summary | AP 微积分AB与BC重要公式总结

📚 AP Calculus AB & BC Important Formulas Summary | AP 微积分AB与BC重要公式总结

This article provides a comprehensive summary of the most important formulas for AP Calculus AB and BC. Mastering these formulas — from limits and derivatives to integrals and series — is essential for achieving a high score on the exam. Each section presents key results with clear notation, distinguishing between common AB material and BC-only topics.

本文为AP微积分AB与BC考试最重要的公式提供了全面总结。从极限、导数到积分与级数，掌握这些公式是取得高分的关键。每一节都以清晰的符号列出核心结论，并区分AB共同内容与BC专属考点。

1. Limits and Continuity | 极限与连续性

The limit of a function f(x) as x approaches a is the value L that f(x) gets arbitrarily close to. Formal properties of limits include the sum, product, quotient, and power rules.

函数 f(x) 当 x 趋近于 a 时的极限是 f(x) 无限接近的值 L。极限的基本性质包括加法、乘法、除法和幂运算规则。

lim_x→a [f(x) ± g(x)] = lim_x→a f(x) ± lim_x→a g(x)

lim_x→a [f(x) · g(x)] = lim_x→a f(x) · lim_x→a g(x)

Two fundamental trigonometric limits must be memorised: lim_x→0 sin x / x = 1 and lim_x→0 (1 − cos x) / x = 0. The Squeeze Theorem is often used to evaluate limits involving these forms.

必须牢记两个基本三角极限：lim_x→0 sin x / x = 1 和 lim_x→0 (1 − cos x) / x = 0。夹逼定理常用于求解涉及这些形式的极限。

A function f is continuous at x = a if lim_x→a f(x) = f(a). Continuity on an interval guarantees that the Intermediate Value Theorem holds: if f is continuous on [a, b] and N is between f(a) and f(b), then there exists c in (a, b) such that f(c) = N.

函数 f 在 x = a 处连续要求 lim_x→a f(x) = f(a)。在一个区间上连续保证了介值定理成立：若 f 在 [a, b] 上连续且 N 介于 f(a) 与 f(b) 之间，则存在 (a, b) 内的 c 使 f(c) = N。

2. Derivatives Rules | 导数基本公式

The derivative of a function measures the instantaneous rate of change. Basic differentiation rules are the foundation for all later topics.

函数的导数衡量瞬时变化率。基本求导规则是所有后续内容的基础。

Power Rule: d/dx (xⁿ) = n xⁿ⁻¹

Product Rule: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Quotient Rule: d/dx [u(x)/v(x)] = (u'(x)v(x) − u(x)v'(x)) / [v(x)]²

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

The derivatives of trigonometric functions, exponential functions, and logarithmic functions appear frequently:

三角函数、指数函数和对数函数的导数频繁出现：

d/dx sin x = cos x | d/dx sin x = cos x
d/dx cos x = −sin x
d/dx tan x = sec² x
d/dx e^x = e^x
d/dx a^x = a^x ln a
d/dx ln x = 1/x
d/dx logₐ x = 1/(x ln a)

Inverse trigonometric derivatives are required for both AB and BC: d/dx arcsin x = 1/√(1 − x²), d/dx arctan x = 1/(1 + x²).

反三角函数的导数在AB和BC中都要求掌握：d/dx arcsin x = 1/√(1 − x²)，d/dx arctan x = 1/(1 + x²)。

3. Applications of Derivatives | 导数的应用

The derivative facilitates the analysis of function behavior and the modelling of real-world situations.

导数有助于分析函数行为并建立现实情境的模型。

Tangent line at (a, f(a)): y = f(a) + f'(a)(x − a)

The Mean Value Theorem states that 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).

中值定理指出，若 f 在 [a, b] 上连续且在 (a, b) 上可导，则存在 (a, b) 内的 c 使得 f'(c) = (f(b) − f(a))/(b − a)。

Critical points occur where f'(x) = 0 or f'(x) does not exist. The First Derivative Test identifies local maxima and minima: if f’ changes from positive to negative, f has a local maximum; if from negative to positive, a local minimum.

临界点出现在 f'(x) = 0 或 f'(x) 不存在处。一阶导数判别法确定局部极大值和极小值：若 f’ 由正变负，f 有局部极大值；若由负变正，则有局部极小值。

The 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. Concavity is determined by the sign of f''.

二阶导数判别法：若 f'(c) = 0 且 f”(c) > 0，f 在 c 处有局部极小值；若 f”(c) < 0，则有局部极大值。函数的凹凸性由 f'' 的正负决定。

L’Hopital’s Rule is used to evaluate limits of indeterminate forms 0/0 or ∞/∞: lim x→a f(x)/g(x) = lim x→a f'(x)/g'(x), provided the limit of the quotient of derivatives exists.

洛必达法则用于求解 0/0 或 ∞/∞ 型不定式的极限：lim x→a f(x)/g(x) = lim x→a f'(x)/g'(x)，前提是导函数之比的极限存在。

Related rates problems use implicit differentiation with respect to time t, linking rates of change such as volume and radius.

相关变化率问题运用对时间 t 的隐函数求导，将体积、半径等变化率联系起来。

4. Integrals and Antiderivatives | 积分与反导数

Integration reverses differentiation. The indefinite integral ∫ f(x) dx gives a family of antiderivatives: F(x) + C, where F'(x) = f(x).

积分是微分的逆运算。不定积分 ∫ f(x) dx 给出反导数族：F(x) + C，其中 F'(x) = f(x)。

Basic integration formulas that every candidate must know:

每位考生必须掌握的基本积分公式：

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

The substitution method (u-substitution) is a fundamental technique for both AB and BC. Formally, ∫ f(g(x))g'(x) dx = ∫ f(u) du where u = g(x).

换元法是AB和BC共同的基础技巧。形式上，∫ f(g(x))g'(x) dx = ∫ f(u) du，其中 u = g(x)。

5. Fundamental Theorem of Calculus | 微积分基本定理

The Fundamental Theorem of Calculus (FTC) establishes the connection between differentiation and integration.

微积分基本定理建立了微分与积分之间的桥梁。

Part 1: If g(x) = ∫_a^x f(t) dt, then g'(x) = f(x). More generally, d/dx ∫_a^u(x) f(t) dt = f(u(x)) · u'(x).

第一部分：若 g(x) = ∫_a^x f(t) dt，则 g'(x) = f(x)。更一般地，d/dx ∫_a^u(x) f(t) dt = f(u(x)) · u'(x)。

Part 2: ∫_a^b f(x) dx = F(b) − F(a), where F is any antiderivative of f. This is used to evaluate definite integrals.

第二部分：∫_a^b f(x) dx = F(b) − F(a)，其中 F 是 f 的任意一个反导数。此部分用于计算定积分。

The FTC justifies finding total change from a rate of change and underpins the computation of area, volume, and net accumulation.

微积分基本定理证明了从变化率求总变化量的方法，为面积、体积和净累积量的计算奠定了理论基础。

6. Differential Equations | 微分方程

A differential equation relates a function to its derivatives. The simplest type is dy/dx = f(x), whose solution is y = ∫ f(x) dx + C.

微分方程将函数与其导数联系起来。最简单的一类为 dy/dx = f(x)，其解为 y = ∫ f(x) dx + C。

Separation of variables is a common method: rewrite dy/dx = g(x)h(y) as (1/h(y)) dy = g(x) dx and integrate both sides. The general solution includes an arbitrary constant; initial conditions determine a particular solution.

分离变量法是常用的方法：将 dy/dx = g(x)h(y) 改写为 (1/h(y)) dy = g(x) dx 并两边积分。通解包含任意常数；初始条件确定特解。

Exponential growth and decay models follow the form dy/dt = k y, yielding y = y₀ e^kt. Logistic growth (BC only) is modelled by dP/dt = kP(1 − P/L), where L is the carrying capacity.

指数增长与衰减模型遵循形式 dy/dt = k y，其解为 y = y₀ e^kt。逻辑斯蒂增长（仅BC）由 dP/dt = kP(1 − P/L) 建模，其中 L 为环境容纳量。

7. Applications of Integrals | 积分的应用

Definite integrals are used to calculate area, volume, and accumulated change.

定积分用于计算面积、体积和累积变化。

Area between curves y = f(x) and y = g(x) from a to b: A = ∫_a^b |f(x) − g(x)| dx

Volume by disks/washers about the x-axis: V = π ∫_a^b [R(x)]² dx or with a hole: V = π ∫_a^b ([R(x)]² − [r(x)]²) dx.

绕 x 轴旋转的圆盘/垫圈法体积：V = π ∫_a^b [R(x)]² dx，有洞时：V = π ∫_a^b ([R(x)]² − [r(x)]²) dx。

Volume by cylindrical shells about the y-axis: V = 2π ∫_a^b x f(x) dx (if rotating the region under f(x) and above the x-axis).

绕 y 轴旋转的柱壳法体积：V = 2π ∫_a^b x f(x) dx（旋转 f(x) 下方、x 轴上方的区域时）。

The average value of f on [a, b] is (1/(b − a)) ∫_a^b f(x) dx. For BC only, the length of a curve y = f(x) from a to b is L = ∫_a^b √(1 + [f'(x)]²) dx.

f 在 [a, b] 上的平均值是 (1/(b − a)) ∫_a^b f(x) dx。仅限BC：曲线 y = f(x) 从 a 到 b 的弧长为 L = ∫_a^b √(1 + [f'(x)]²) dx。

8. Advanced Integration Techniques (BC Only) | 高级积分技巧（仅BC）

BC students must be proficient with additional integration methods beyond u-substitution.

BC考生必须熟练掌握u换元法之外的其他积分方法。

Integration by Parts: ∫ u dv = u v − ∫ v du. It is typically used when the integrand is a product of functions from different families, such as polynomial times exponential or logarithm.

分部积分法：∫ u dv = u v − ∫ v du。通常在被积函数是多项式与指数函数或对数的乘积等不同函数族的乘积时使用。

Integration by Partial Fractions: A rational function P(x)/Q(x) where degree(P) < degree(Q) is decomposed into a sum of simpler fractions, e.g., 1/((x−a)(x−b)) = A/(x−a) + B/(x−b).

部分分式积分法：将分子次数低于分母的有理函数 P(x)/Q(x) 分解为较简单分式的和，例如 1/((x−a)(x−b)) = A/(x−a) + B/(x−b)。

Improper Integrals: Integrals with infinite limits or infinite discontinuities are evaluated as limits of proper integrals. For example, ∫_a^∞ f(x) dx = lim_b→∞ ∫_a^b f(x) dx. Convergence means this limit exists as a finite number.

反常积分：具有无穷积分限或无穷间断点的积分作为正常积分的极限来计算。例如，∫_a^∞ f(x) dx = lim_b→∞ ∫_a^b f(x) dx。收敛是指该极限存在且为有限值。

9. Parametric Equations and Polar Coordinates (BC Only) | 参数方程与极坐标（仅BC）

Curves can be defined parametrically, where x = x(t) and y = y(t). The derivative dy/dx is found by the chain rule: dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

曲线可以用参数形式定义，x = x(t) 且 y = y(t)。导数 dy/dx 通过链式法则求出：dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。

The second derivative is d²y/dx² = (d/dt [dy/dx]) / (dx/dt). The length of a parametric curve from t = α to t = β is L = ∫_α^β √((dx/dt)² + (dy/dt)²) dt.

二阶导数为 d²y/dx² = (d/dt [dy/dx]) / (dx/dt)。参数曲线从 t = α 到 t = β 的弧长为 L = ∫_α^β √((dx/dt)² + (dy/dt)²) dt。

In polar coordinates (r, θ), the area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = ½ ∫_α^β [f(θ)]² dθ. The slope of the polar curve is computed using x = r cos θ, y = r sin θ.

在极坐标 (r, θ) 中，极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积为 A = ½ ∫_α^β [f(θ)]² dθ。极坐标曲线的斜率通过 x = r cos θ, y = r sin θ 来求导。

10. Vector-Valued Functions (BC Only) | 向量值函数（仅BC）

A vector-valued function r(t) = ⟨x(t), y(t)⟩ describes position in the plane. The velocity vector is v(t) = r'(t) = ⟨x'(t), y'(t)⟩ and acceleration is a(t) = r”(t). Speed is the magnitude ‖v(t)‖ = √((x'(t))² + (y'(t))²).

向量值函数 r(t) = ⟨x(t), y(t)⟩ 描述平面上的位置。速度向量为 v(t) = r'(t) = ⟨x'(t), y'(t)⟩，加速度为 a(t) = r”(t)。速率是其模长 ‖v(t)‖ = √((x'(t))² + (y'(t))²)。

The displacement (net change in position) from t = a to t = b is ∫_a^b v(t) dt, and the total distance travelled is ∫_a^b ‖v(t)‖ dt.

从 t = a 到 t = b 的位移（位置的净变化）为 ∫_a^b v(t) dt，总路程为 ∫_a^b ‖v(t)‖ dt。

11. Infinite Series (BC Only) | 无穷级数（仅BC）

An infinite series Σ a_n converges if the sequence of partial sums approaches a finite limit. The nth term test states that if lim a_n ≠ 0, the series diverges.

若部分和序列趋近于一个有限极限，则无穷级数 Σ a_n 收敛。第 n 项判别法指出，若 lim a_n ≠ 0，则级数发散。

Key convergence tests:

关键的收敛判别法：

Geometric series: Σ arⁿ⁻¹ converges to a/(1 − r) if |r| < 1.
p-series: Σ 1/n^p converges if p > 1, diverges if p ≤ 1.

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AP Calculus AB & BC Important Formulas Summary | AP 微积分AB与BC重要公式总结