AP Calculus: Ultimate Formula Compendium | AP 微积分：公式大全

📚 AP Calculus: Ultimate Formula Compendium | AP 微积分：公式大全

Whether you are preparing for the AP Calculus AB or BC exam, a clear and well-organized set of formulas is your essential tool. This compendium gathers the critical differentiation, integration, limit, series, and vector formulas you need to work efficiently under time pressure. Master them thoroughly, and you will approach both multiple-choice and free-response questions with confidence.

无论你在为 AP 微积分 AB 还是 BC 考试做准备，一份清晰、条理分明的公式合集都是不可或缺的工具。本大全汇集了极限、求导、积分、级数和向量等关键公式，帮助你在时间压力下高效解题。彻底掌握它们，你将能自信应对选择题和自由作答题。

1. Limits and Continuity | 极限与连续性

A function f(x) has a limit L as x approaches c if f(x) gets arbitrarily close to L from both sides. The limit exists only when the left-hand limit and right-hand limit are equal.

当 x 趋近于 c 时，若 f(x) 从两侧无限接近 L，则函数 f(x) 的极限为 L。极限存在的充要条件是左极限等于右极限。

lim_x→c f(x) = L ⇔ lim_x→c⁻ f(x) = lim_x→c⁺ f(x) = L

The formal definition (epsilon-delta) states that for every ε>0, there exists δ>0 such that 0<|x−c|<δ implies |f(x)−L|<ε. In AP calculus you will use this primarily to confirm limits rather than construct proofs.

形式化定义（ε-δ）：对任意 ε>0，存在 δ>0，使得当 0<|x−c|<δ 时，有 |f(x)−L|<ε。在 AP 微积分中，主要用于确认极限而非构造性证明。

Continuity at a point requires three conditions: f(c) is defined; lim_x→c f(x) exists; lim_x→c f(x) = f(c). Discontinuities can be removable (hole), jump, or infinite (vertical asymptote).

函数在一点连续需满足三个条件：f(c) 有定义；lim_x→c f(x) 存在；且极限值等于函数值 f(c)。间断点可分为可去间断（空洞）、跳跃间断和无穷间断（垂直渐近线）。

Key limit theorems include the Squeeze Theorem, and the special limit lim_x→0 (sin x)/x = 1, which is fundamental for trigonometric derivative proofs.

重要极限定理包括夹逼定理，以及特殊极限 lim_x→0 (sin x)/x = 1，它是推导三角函数导数的基石。

2. Derivative Rules | 导数规则

The derivative of f at x is defined as the limit of the difference quotient. The Leibniz notation dy/dx or Lagrange notation f′(x) are used interchangeably.

函数 f 在 x 处的导数定义为差商的极限。莱布尼茨记号 dy/dx 和拉格朗日记号 f′(x) 可互换使用。

f′(x) = lim_h→0 [f(x+h)−f(x)] / h

Basic power rule, sum rule, constant multiple rule, product rule, quotient rule, and chain rule constitute the core toolkit. Memorize them firmly.

基本的幂法则、和法则、常数倍法则、乘法法则、除法法则和链式法则构成了核心工具箱，必须牢牢记住。

Power Rule: d/dx (xⁿ) = n xⁿ⁻¹
幂法则： d/dx (xⁿ) = n xⁿ⁻¹
Product Rule: d/dx (uv) = u′v + uv′
乘法法则： d/dx (uv) = u′v + uv′
Quotient Rule: d/dx (u/v) = (u′v − uv′) / v²
除法法则： d/dx (u/v) = (u′v − uv′) / v²
Chain Rule: d/dx f(g(x)) = f′(g(x)) · g′(x)
链式法则： d/dx f(g(x)) = f′(g(x)) · g′(x)

Derivatives of exponential and logarithmic functions also appear frequently. The natural exponential eˣ is its own derivative, and the derivative of ln x is 1/x.

指数函数和对数函数的导数也频繁出现。自然指数 eˣ 的导数是其本身，ln x 的导数是 1/x。

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

Trigonometric derivatives must be at your fingertips. Notice the sign patterns: cos derivative gives −sin, while the integral has positive signs.

三角函数的导数必须了如指掌。注意符号规律：cos 的导数是 −sin，而积分时符号为正。

Function	Derivative	函数	导数
sin x	cos x	sin x	cos x
cos x	−sin x	cos x	−sin x
tan x	sec² x	tan x	sec² x
cot x	−csc² x	cot x	−csc² x
sec x	sec x tan x	sec x	sec x tan x
csc x	−csc x cot x	csc x	−csc x cot x

Inverse trigonometric derivatives are tested especially on BC. For arcsin and arctan, memorization is highly recommended; others can be derived using implicit differentiation.

反三角函数导数尤其在 BC 考试中会考查。arcsin 和 arctan 建议背诵，其他可通过隐函数求导推导。

d/dx (arcsin x) = 1/√(1−x²), d/dx (arctan x) = 1/(1+x²)

3. Applications of Derivatives | 导数应用

The first derivative f′(x) tells where a function is increasing (f′>0) or decreasing (f′<0). Critical points occur where f′(x)=0 or f′ is undefined; they are candidates for local extrema.

一阶导数 f′(x) 指示函数的增减性：f′>0 递增，f′<0 递减。临界点出现在 f′(x)=0 或 f′ 不存在处，是局部极值的候选点。

The second derivative f″(x) reveals concavity: f″>0 means concave up, f″<0 means concave down. Inflection points occur where concavity changes, typically where f″=0 or undefined.

二阶导数 f″(x) 揭示函数的凹凸性：f″>0 凹向上，f″<0 凹向下。拐点出现在凹凸性改变处，通常 f″=0 或不存在。

L’Hôpital’s rule is a powerful tool for indeterminate limits of the form 0/0 or ∞/∞. Differentiate numerator and denominator separately and re-evaluate the limit.

洛必达法则是处理 0/0 或 ∞/∞ 型不定式极限的有力工具：分别对分子分母求导，再求极限。

lim_x→a f(x)/g(x) = lim_x→a f′(x)/g′(x) (if limit exists)

Optimization problems involve maximizing or minimizing a quantity. Set up a function, take its derivative, find critical points, and verify with first or second derivative test. Constraint equations are often used to reduce variables.

最优化问题寻求某量的最大值或最小值。建立函数，求导找到临界点，并用一阶或二阶导数检验验证。常通过约束方程减少变量个数。

Related rates link changing quantities through differentiation with respect to time. Identify the relation, differentiate implicitly, substitute known rates, and solve for the unknown rate.

相关变化率通过关于时间求导将变化的量联系起来。找出关系式，隐函数求导，代入已知变化率，解出未知变化率。

Mean Value Theorem (MVT) guarantees there exists at least one c in (a,b) such that f′(c) = [f(b)−f(a)]/(b−a), provided f is continuous on [a,b] and differentiable on (a,b).

中值定理保证：若 f 在 [a,b] 连续、(a,b) 可导，则至少存在一点 c∈(a,b)，使得 f′(c) = [f(b)−f(a)]/(b−a)。

4. Integrals and Antiderivatives | 积分与反导数

An antiderivative F is a function such that F′(x)=f(x). The indefinite integral ∫ f(x) dx = F(x) + C captures the family of all antiderivatives. Always add the constant C.

反导数 F 满足 F′(x)=f(x)。不定积分 ∫ f(x) dx = F(x) + C 表示全体反导数族，切勿忘记常数 C。

Basic integration formulas mirror differentiation rules. Start with the power rule for integration, which works for all n≠−1.

基本积分公式与求导规则对称。从幂函数积分开始，对所有 n≠−1 成立。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

For n = −1, the natural logarithm appears: ∫ 1/x dx = ln |x| + C. The absolute value ensures the domain covers all x≠0.

当 n=−1 时，出现自然对数：∫ 1/x dx = ln |x| + C。加绝对值确保定义域覆盖所有 x≠0。

Trigonometric integrals are best memorised in pairs. Notice that derivatives and integrals of sine/cosine alternate with sign changes.

三角函数的积分最好成对记忆。注意正弦、余弦的导数和积分交替出现，并伴随符号变化。

Function	Integral	函数	积分
∫ 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
∫ csc x cot x dx	−csc x + C	∫ csc x cot x dx	−csc x + C

Exponential integrals are straightforward. The integral of eˣ is eˣ, while aˣ integrates to aˣ/ln a.

指数积分很直接。eˣ 积分仍是 eˣ，aˣ 积分为 aˣ/ln a。

∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C

Inverse trigonometric integrals complete the family. These formulas often appear in BC partial fraction problems and arc length computations.

反三角函数积分补全了积分族。这些公式常出现在 BC 的部分分式分解和弧长计算中。

∫ 1/√(1−x²) dx = arcsin x + C, ∫ 1/(1+x²) dx = arctan x + C

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

The Fundamental Theorem of Calculus (FTC) bridges differentiation and integration, forming the backbone of calculus. Part 1 states that if F is an antiderivative of f on [a,b], then ∫ₐᵇ f(x) dx = F(b)−F(a).

微积分基本定理（FTC）将求导与积分联系起来，是微积分的支柱。第一部分：若 F 是 f 在 [a,b] 上的反导数，则 ∫ₐᵇ f(x) dx = F(b)−F(a)。

Part 2 gives the derivative of an accumulation function: d/dx ∫ₐˣ f(t) dt = f(x). For an upper limit that is a function u(x), use the chain rule: d/dx ∫ₐᵘ⁽ˣ⁾ f(t) dt = f(u(x))·u′(x).

第二部分给出累积函数的导数：d/dx ∫ₐˣ f(t) dt = f(x)。若上限是函数 u(x)，则需用链式法则：d/dx ∫ₐᵘ⁽ˣ⁾ f(t) dt = f(u(x))·u′(x)。

If both limits are functions, split the integral or use the full formula: d/dx ∫ᵥ⁽ˣ⁾ᵘ⁽ˣ⁾ f(t) dt = f(u)·u′ − f(v)·v′. FTC questions frequently appear on the AP exam, both symbolically and graphically.

若上下限均为函数，可拆分积分或直接用全公式：d/dx ∫ᵥ⁽ˣ⁾ᵘ⁽ˣ⁾ f(t) dt = f(u)·u′ − f(v)·v′。FTC 问题在 AP 考试中频繁出现，包括符号形式和图形分析。

6. Integration Techniques | 积分技巧

Beyond basic antiderivatives, BC students must master several integration strategies. Substitution (u-substitution) is the most common: choose u=g(x), compute du=g′(x)dx, and rewrite the integral in terms of u.

除了基本反导数，BC 学生还需掌握多种积分策略。换元法（u 代换）最常用：设 u=g(x)，计算 du=g′(x)dx，将积分改写为关于 u 的形式。

Integration by parts derives from the product rule and is essential for products of algebraic and transcendental functions. The LIATE rule helps determine the choice of u.

分部积分法由乘法法则推导而来，对代数函数与超越函数的乘积至关重要。LIATE 规则有助于确定 u 的选择。

∫ u dv = uv − ∫ v du

For rational functions, partial fraction decomposition splits a complex fraction into simpler fractions that can be integrated individually. This technique requires factoring the denominator and solving for constants.

有理函数可通过部分分式分解，将复杂分式拆成简单分式逐个积分。这种方法需要分解分母并求解常数。

Trigonometric integrals use identities like sin²θ + cos²θ = 1, half-angle formulas sin²θ = (1−cos2θ)/2, and cos²θ = (1+cos2θ)/2 to reduce powers.

三角积分利用恒等式如 sin²θ + cos²θ = 1，以及半角公式 sin²θ = (1−cos2θ)/2，cos²θ = (1+cos2θ)/2 来降次。

Trigonometric substitution is used when you encounter √(a²−x²), √(a²+x²), or √(x²−a²). Let x = a sin θ, a tan θ, or a sec θ accordingly, and simplify using Pythagorean identities.

三角代换用于处理 √(a²−x²)、√(a²+x²) 或 √(x²−a²) 等形式。依情况令 x = a sin θ、a tan θ 或 a sec θ，并利用勾股恒等式化简。

Improper integrals involve infinite limits or integrands with vertical asymptotes. They are evaluated by taking limits. If the limit exists (finite), the integral converges; otherwise it diverges.

反常积分涉及无穷积分限或被积函数具有垂直渐近线，需通过求极限来计算。若极限存在（有限），积分收敛；否则发散。

7. Applications of Integrals | 积分应用

Definite integrals compute the net area between a curve and the x-axis. Total area requires attention to sign changes; integrate |f(x)| or split at x-intercepts.

定积分计算曲线与 x 轴之间的净面积。求总面积需注意符号变化：积分 |f(x)| 或在 x 截断处拆分。

The area between two curves y=f(x) and y=g(x) from a to b is ∫ₐᵇ [f(x)−g(x)] dx, where f(x)≥g(x) on [a,b]. For functions of y, use ∫ [right−left] dy.

两条曲线 y=f(x) 与 y=g(x) 在 [a,b] 间的面积是 ∫ₐᵇ [f(x)−g(x)] dx，假设 f(x)≥g(x)。若关于 y 积分，用 ∫ [右−左] dy。

Volume by disks and washers: revolving around x-axis produces V = π ∫ₐᵇ [f(x)]² dx. For region between two curves, use washer method: V = π ∫ₐᵇ ([R(x)]²−[r(x)]²) dx.

圆盘法和垫圈法求体积：绕 x 轴旋转得 V = π ∫ₐᵇ [f(x)]² dx。两曲线间区域用垫圈法：V = π ∫ₐᵇ ([R(x)]²−[r(x)]²) dx。

Volume by cylindrical shells is used when revolving around a vertical axis using y bounds: V = 2π ∫ₐᵇ (radius)(height) dy. Both methods are interchangeable; choose the one that eases integration.

柱壳法用于绕垂直轴旋转、用 y 界积分：V = 2π ∫ₐᵇ (半径)(高) dy。圆盘/垫圈与柱壳法可互换，选择使积分简便的方法。

Arc length for y=f(x) on [a,b] is L = ∫ₐᵇ √(1 + [f′(x)]²) dx. For parametric curves, see the parametric section.

函数 y=f(x) 在 [a,b] 上的弧长 L = ∫ₐᵇ √(1 + [f′(x)]²) dx。参数曲线弧长见参数部分。

Average value of f on [a,b] is f_avg = (1/(b−a)) ∫ₐᵇ f(x) dx. This represents the constant height giving the same integral.

函数 f 在 [a,b] 上的平均值：f_avg = (1/(b−a)) ∫ₐᵇ f(x) dx。这表示产生相同积分的常数高度。

8. Differential Equations | 微分方程

Differential equations involve an unknown function and its derivatives. AP focuses on first-order separable equations: dy/dx = g(x)h(y). Separate variables: (1/h(y)) dy = g(x) dx, then integrate both sides.

微分方程涉及未知函数及其导数。AP 考试聚焦一阶可分离方程：dy/dx = g(x)h(y)。分离变量： (1/h(y)) dy = g(x) dx，两边积分。

Slope fields visualize solutions by drawing short line segments with slope f(x,y) at lattice points. The particular solution follows the field and passes through a given initial condition.

斜率场通过在格点上绘制短线段（斜率为 f(x,y)）来可视化解。特解沿着场的方向并通过给定的初始条件。

Exponential growth and decay are modeled by dy/dt = ky, whose solution is y = y₀ eᵏᵗ. If k>0 it’s growth; k<0 it's decay. Half-life and doubling time can be derived from this model.

指数增长与衰减模型为 dy/dt = ky，解为 y = y₀ eᵏᵗ。k>0 增长，k<0 衰减。半衰期和倍增时间可由此模型推导。

Logistic growth, specific to BC, is dy/dt = ky(1 − y/L) where L is carrying capacity. The solution is y = L / (1 + Ce⁻ᵏᵗ). The inflection point occurs at y = L/2.

BC 特定的逻辑斯谛增长：dy/dt = ky(1 − y/L)，L 为容纳量。解为 y = L / (1 + Ce⁻ᵏᵗ)。拐点出现在 y = L/2 处。

9. Parametric, Polar, and Vector Functions | 参数方程、极坐标与向量函数

Parametric equations x = f(t), y = g(t) define a curve in the plane. The derivative dy/dx is computed as (dy/dt)/(dx/dt). The second derivative d²y/dx² = d(dy/dx)/dt ÷ dx/dt.

参数方程 x = f(t), y = g(t) 定义平面上一条曲线。导数 dy/dx = (dy/dt)/(dx/dt)。二阶导 d²y/dx² = d(dy/dx)/dt ÷ dx/dt。

Arc length for parametric curve: L = ∫ₜ₁ᵗ₂ √( (dx/dt)² + (dy/dt)² ) dt. Speed is the magnitude of the velocity vector ⟨dx/dt, dy/dt⟩.

参数曲线弧长：L = ∫ₜ₁ᵗ₂ √( (dx/dt)² + (dy/dt)² ) dt。速率是速度向量 ⟨dx/dt, dy/dt⟩ 的大小。

Polar coordinates (r, θ) relate to Cartesian by x = r cos θ, y = r sin θ. Area bounded by r = f(θ) from α to β is A = ½ ∫ₐᵝ [f(θ)]² dθ. Pay close attention to radial boundaries and symmetry.

极坐标 (r, θ) 与直角坐标关系：x = r cos θ, y = r sin θ。曲线 r = f(θ) 在 α 到 β 范围内的面积：A = ½ ∫ₐᵝ [f(θ)]² dθ。注意径向边界和对称性。

Area between two polar curves r₁(θ) and r₂(θ) is ½ ∫ (r_outer² − r_inner²) dθ. Determine intersection points by solving r₁(θ)=r₂(θ).

两条极坐标曲线间的面积：½ ∫ (r_outer² − r_inner²) dθ。通过解 r₁(θ)=r₂(θ) 确定交点。

Arc length in polar: L = ∫ √( r² + (dr/dθ)² ) dθ. This formula is derived from the parametric representation using θ as parameter.

极坐标弧长：L = ∫ √( r² + (dr/dθ)² ) dθ。该公式由以 θ 为参数的参数表示推导而来。

Vector-valued functions r(t) = ⟨x(t), y(t)⟩ describe motion. The velocity is v(t)=r′(t), acceleration a(t)=v′(t). Speed is ‖v(t)‖, and the displacement vector is ∫ v(t) dt.

向量值函数 r(t) = ⟨x(t), y(t)⟩ 描述运动。速度 v(t)=r′(t)，加速度 a(t)=v′(t)。速率是 ‖v(t)‖，位移向量为 ∫ v(t) dt。

10. Infinite Series | 无穷级数

Infinite series Σ aₙ converge if the sequence of partial sums approaches a finite limit. A necessary (but not sufficient) condition is lim aₙ = 0. The nth-term test for divergence: if lim aₙ≠0, the series diverges.

无穷级数 Σ aₙ 收敛，若部分和序列趋于有限极限。必要（非充分）条件是 lim aₙ = 0。第 n 项发散判别法：若 lim aₙ≠0，级数发散。

Geometric series Σ arⁿ⁻¹ converges to a/(1−r) if |r|<1, and diverges otherwise. This is the most frequently applied series in AP.

几何级数 Σ arⁿ⁻¹ 当 |r|<1 时收敛于 a/(1−r)，否则发散。这是 AP 中最常用的级数。

p-series Σ 1/nᵖ converges for p>1 and diverges for p≤1. The harmonic series Σ 1/n diverges (p=1).

p 级数 Σ 1/nᵖ 当 p>1 收敛，p≤1 发散。调和级数 Σ 1/n（p=1）发散。

Convergence tests for non-negative series: Integral Test, Comparison Test, Limit Comparison Test, Ratio Test, and Root Test. For alternating series, the Alternating Series Test confirms convergence if aₙ decreases to 0.

非负项级数的收敛判别法：积分判别法、比较判别法、极限比较判别法、比值判别法和根值判别法。交错级数用交错级数审敛法，若 aₙ 递减趋于 0，则收敛。

Absolute vs conditional convergence: Σ aₙ converges absolutely if Σ |aₙ| converges. If Σ aₙ converges but Σ |aₙ| diverges, it converges conditionally.

绝对收敛与条件收敛：若 Σ |aₙ| 收敛，则 Σ aₙ 绝对收敛。若 Σ aₙ 收敛但 Σ |aₙ| 发散，则条件收敛。

Taylor and Maclaurin series expand a function into an infinite polynomial. Maclaurin series (centered at 0) for key functions must be memorized.

泰勒级数和麦克劳林级数将函数展开为无穷多项式。以 0 为中心的麦克劳林级数必须记住。

eˣ = Σ xⁿ/n! sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)! cos x = Σ (−1)ⁿ x²ⁿ/(2n)!
1/(1−x) = Σ xⁿ (|x|<1) ln(1+x) = Σ (−1)ⁿ⁻¹ xⁿ/n

Lagrange error bound provides an upper bound for the remainder when using a Taylor polynomial of degree n. Know

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AP Calculus: Ultimate Formula Compendium | AP 微积分：公式大全