AP Calculus: Core Formulas and Typical Applications | AP微积分：核心公式梳理与典型应用

📚 AP Calculus: Core Formulas and Typical Applications | AP微积分：核心公式梳理与典型应用

This article organizes the essential formulas and classic applications for AP Calculus (AB & BC). It focuses on the concepts and equations that appear most frequently on the exam, with paired examples to help you understand when and how to use them.

本文系统梳理 AP 微积分（AB 与 BC）的核心公式与典型应用，紧扣考试中最常出现的考点，并通过配对的示例帮助你理解何时以及如何使用这些公式。

1. Limits and Continuity | 极限与连续性

The concept of a limit is the foundation of calculus. We write $\lim_{x \to a} f(x) = L$ to mean that $f(x)$ approaches $L$ as $x$ approaches $a$. The limit exists only if the left-hand and right-hand limits are equal.

极限是微积分的基础。我们记 $\lim_{x \to a} f(x) = L$，表示当 $x$ 趋近于 $a$ 时，$f(x)$ 趋近于 $L$。只有当左极限与右极限相等时，极限才存在。

limₓ→ₐ f(x) = L ⇔ limₓ→ₐ⁻ f(x) = limₓ→ₐ⁺ f(x) = L

For piecewise functions, always check both one-sided limits. A function is continuous at $x=a$ if $f(a)$ is defined, the limit exists, and they are equal.

对于分段函数，务必检查两侧的单侧极限。函数在 $x=a$ 处连续的条件是：$f(a)$ 有定义、极限存在且两者相等。

Typical limit techniques include direct substitution, factoring, rationalizing, and the squeeze theorem.

常见的求极限技巧包括直接代入、因式分解、有理化以及夹逼定理。

Direct substitution: if $f$ is continuous at $a$, then $\lim_{x\to a} f(x)=f(a)$.
Factoring: use when substitution gives $0/0$.
Squeeze theorem: if $g(x) \le f(x) \le h(x)$ and $\lim g = \lim h = L$, then $\lim f = L$.

直接代入：若 $f$ 在 $a$ 处连续，则 $\lim_{x\to a} f(x)=f(a)$。
因式分解：当代入后得到 $0/0$ 时使用。
夹逼定理：若 $g(x) \le f(x) \le h(x)$ 且 $\lim g = \lim h = L$，则 $\lim f = L$。

Example: $\lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} (x+2) = 4$.

示例：$\lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} (x+2) = 4$。

2. Derivatives: Definition and Basic Rules | 导数定义与基本法则

The derivative measures the instantaneous rate of change, defined as a limit of the difference quotient. It is also the slope of the tangent line to the curve at a point.

导数度量瞬时变化率，其定义是差商的极限。它也是曲线在某一点处切线的斜率。

f ′(x) = limₕ→₀ [f(x+h) − f(x)] / h

Essential differentiation rules:

基本求导法则如下：

Power rule: d/dx (xⁿ) = n xⁿ⁻¹	幂法则：d/dx (xⁿ) = n xⁿ⁻¹
Product rule: (uv)′ = u′v + uv′	乘积法则：(uv)′ = u′v + uv′
Quotient rule: (u/v)′ = (u′v − uv′)/v²	商法则：(u/v)′ = (u′v − uv′)/v²
Trigonometric derivatives: d/dx sin x = cos x, d/dx cos x = −sin x	三角函数导数：d/dx sin x = cos x，d/dx cos x = −sin x

Example: Differentiate $f(x) = x^3 \sin x$. Using the product rule, $f'(x) = 3x^2 \sin x + x^3 \cos x$.

示例：求 $f(x) = x^3 \sin x$ 的导数。利用乘积法则，$f'(x) = 3x^2 \sin x + x^3 \cos x$。

3. Chain Rule and Implicit Differentiation | 链式法则与隐函数求导

The chain rule is used when differentiating composite functions. If $y = f(g(x))$, then $dy/dx = f'(g(x)) \cdot g'(x)$. In Leibniz notation, $dy/dx = dy/du \cdot du/dx$.

链式法则用于复合函数求导。若 $y = f(g(x))$，则 $dy/dx = f'(g(x)) \cdot g'(x)$。用莱布尼茨记号表示为 $dy/dx = dy/du \cdot du/dx$。

d/dx f(g(x)) = f ′(g(x)) · g ′(x)

Implicit differentiation is applied when $y$ is not explicitly defined in terms of $x$. Differentiate both sides with respect to $x$, then solve for $dy/dx$.

当 $y$ 未显式表示为 $x$ 的函数时，使用隐函数求导。对方程两边关于 $x$ 求导，然后解出 $dy/dx$。

Example: For $x^2 + y^2 = 25$, differentiating gives $2x + 2y \cdot dy/dx = 0$, so $dy/dx = -x/y$.

示例：对 $x^2 + y^2 = 25$ 求导得 $2x + 2y \cdot dy/dx = 0$，故 $dy/dx = -x/y$。

4. Applications of Derivatives | 导数的应用

Derivatives are used to analyze the behavior of functions: finding extrema, intervals of increase/decrease, concavity, and inflection points. They also model rates of change in related rates and optimization problems.

导数用于分析函数行为：寻找极值、增减区间、凹凸性及拐点。它还在相关变化率和优化问题中刻画变化率模型。

Critical points: $f'(x)=0$ or does not exist.
First derivative test: sign of $f’$ indicates increase (positive) or decrease (negative).
Second derivative test: $f”>0$ means local minimum; $f”<0$ means local maximum.
Inflection point: $f”$ changes sign.

临界点：$f'(x)=0$ 或不存在。
一阶导数判别法：$f’$ 的符号表示函数递增（正）或递减（负）。
二阶导数判别法：$f”>0$ 为局部极小值；$f”<0$ 为局部极大值。
拐点：$f”$ 改变符号。

Related rates: identify all rates, write an equation linking variables, differentiate with respect to time $t$, and substitute known values.

相关变化率：确定所有变化率，写出变量间的关系式，关于时间 $t$ 求导，再代入已知量。

Optimization: set up the objective function, find critical points on the relevant interval, and compare endpoints when necessary.

优化问题：建立目标函数，在相关区间内找临界点，必要时比较端点值。

5. Integrals: Antiderivatives and Fundamental Theorem | 积分：原函数与微积分基本定理

Integration is the inverse operation of differentiation. The set of all antiderivatives of $f$ is written as $\int f(x)\,dx = F(x) + C$, where $C$ is the constant of integration.

积分是微分的逆运算。函数 $f$ 的所有原函数记作 $\int f(x)\,dx = F(x) + C$，其中 $C$ 为积分常数。

The Fundamental Theorem of Calculus connects differentiation and integration. If $F$ is an antiderivative of $f$, then:

微积分基本定理将微分与积分联系起来。若 $F$ 是 $f$ 的原函数，则：

∫ₐᵇ f(x) dx = F(b) − F(a)

Also, the accumulation function $A(x) = \int_a^x f(t)\,dt$ has derivative $A'(x) = f(x)$.

同时，累积函数 $A(x) = \int_a^x f(t)\,dt$ 的导数为 $A'(x) = f(x)$。

Common antiderivatives include:

常见原函数包括：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）
∫ 1/x dx = ln\|x\| + C	∫ 1/x dx = ln\|x\| + C
∫ eˣ dx = eˣ + C	∫ eˣ dx = eˣ + C
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C	∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C

Example: $\int_0^1 x^2 dx = \left[ x^3/3 \right]_0^1 = 1/3$.

示例：$\int_0^1 x^2 dx = \left[ x^3/3 \right]_0^1 = 1/3$。

6. Integration Techniques | 积分技巧

AP Calculus BC requires more advanced techniques, while AB mainly uses basic antiderivatives and u-substitution. The most important methods are:

AP Calculus BC 要求更高级的技巧，而 AB 主要使用基本原函数和换元法。最重要的方法有：

u-substitution: Let $u = g(x)$, $du = g'(x)dx$, then $\int f(g(x))g'(x)dx = \int f(u)du$.
Integration by parts: $\int u\,dv = uv – \int v\,du$.
Partial fractions: decompose rational functions into simpler fractions.

换元法：令 $u = g(x)$，$du = g'(x)dx$，则 $\int f(g(x))g'(x)dx = \int f(u)du$。
分部积分法：$\int u\,dv = uv – \int v\,du$。
部分分式法：将有理函数分解为更简单的分式。

Example (u-substitution): Evaluate $\int 2x \cos(x^2) dx$. Let $u = x^2$, $du = 2x dx$, so the integral becomes $\int \cos u du = \sin u + C = \sin(x^2) + C$.

示例（换元法）：求 $\int 2x \cos(x^2) dx$。令 $u = x^2$，$du = 2x dx$，则积分化为 $\int \cos u du = \sin u + C = \sin(x^2) + C$。

Example (integration by parts): Evaluate $\int x e^x dx$. Choose $u=x$, $dv=e^x dx$, then $du=dx$, $v=e^x$, giving $x e^x – e^x + C$.

示例（分部积分法）：求 $\int x e^x dx$。取 $u=x$，$dv=e^x dx$，则 $du=dx$，$v=e^x$，结果为 $x e^x – e^x + C$。

7. Applications of Integrals | 积分的应用

Definite integrals are used to compute areas, volumes, average values, and accumulated change. The area between curves $y=f(x)$ and $y=g(x)$ on $[a,b]$ is $\int_a^b [f(x) – g(x)]\,dx$ when $f \ge g$.

定积分用于计算面积、体积、平均值和累积变化。曲线 $y=f(x)$ 与 $y=g(x)$ 在 $[a,b]$ 之间的面积为 $\int_a^b [f(x) – g(x)]\,dx$（当 $f \ge g$ 时）。

Volume of solids of revolution can be found using the disk method or the washer method:

旋转体的体积可以通过圆盘法或垫片法计算：

Disk: V = π ∫ₐᵇ [R(x)]² dx

Washer: V = π ∫ₐᵇ ([R(x)]² − [r(x)]²) dx

The average value of $f$ on $[a,b]$ is $\frac{1}{b-a}\int_a^b f(x)\,dx$.

函数 $f$ 在 $[a,b]$ 上的平均值为 $\frac{1}{b-a}\int_a^b f(x)\,dx$。

In AP BC, arc length is also tested: $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$.

在 AP BC 中还会考查弧长：$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$。

Example: Find the area bounded by $y = x^2$ and $y = x$ on $[0,1]$. The area is $\int_0^1 (x – x^2) dx = \left[ x^2/2 – x^3/3 \right]_0^1 = 1/2 – 1/3 = 1/6$.

示例：求 $y = x^2$ 与 $y = x$ 在 $[0,1]$ 上围成的面积。面积为 $\int_0^1 (x – x^2) dx = \left[ x^2/2 – x^3/3 \right]_0^1 = 1/2 – 1/3 = 1/6$。

8. Differential Equations | 微分方程

AP Calculus includes solving separable differential equations and modeling growth/decay. A separable equation can be written as $dy/dx = g(x)h(y)$, then integrated after separating variables.

AP 微积分包括求解可分离变量的微分方程以及增长/衰减模型。可分离方程可写成 $dy/dx = g(x)h(y)$，分离变量后两边积分。

∫ [1/h(y)] dy = ∫ g(x) dx

The solution curve can be found explicitly or implicitly, and initial conditions determine the constant $C$.

解曲线可以显式或隐式地找到，初始条件决定常数 $C$。

Exponential growth/decay model: $dy/dt = ky$, whose solution is $y = C e^{kt}$.

指数增长/衰减模型：$dy/dt = ky$，其解为 $y = C e^{kt}$。

Example: Solve $dy/dx = 2xy$ with $y(0)=3$. Separating gives $dy/y = 2x dx$, so $\ln|y| = x^2 + C$, hence $y = 3 e^{x^2}$.

示例：解 $dy/dx = 2xy$，且 $y(0)=3$。分离变量得 $dy/y = 2x dx$，故 $\ln|y| = x^2 + C$，因此 $y = 3 e^{x^2}$。

Slope fields are used to visualize differential equations. At each point $(x,y)$, draw a short segment with slope $f(x,y)$.

斜率场用于可视化微分方程。在每个点 $(x,y)$ 处，画一段斜率为 $f(x,y)$ 的短线段。

9. Series and Sequences (AP BC) | 级数与数列（AP BC）

AP Calculus BC includes infinite series, convergence tests, and power series. A sequence is a list of numbers; a series is the sum of a sequence.

AP Calculus BC 包括无穷级数、收敛性检验和幂级数。数列是一列数字；级数是数列的和。

Geometric series: $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ if $|r| < 1$; diverges otherwise.

几何级数：$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$（当 $|r| < 1$ 时）；否则发散。

Important convergence tests:

重要收敛性检验：

nth-term test: if $\lim_{n\to\infty} a_n \neq 0$, the series diverges.
Ratio test: $\lim_{n\to\infty} |a_{n+1}/a_n| = L$; converges if $L<1$, diverges if $L>1$.
Integral test: for positive decreasing $f(n)=a_n$, $\sum a_n$ converges iff $\int_1^\infty f(x) dx$ converges.

第 $n$ 项检验：若 $\lim_{n\to\infty} a_n \neq 0$，则级数发散。
比值检验：$\lim_{n\to\infty} |a_{n+1}/a_n| = L$；若 $L<1$ 收敛，$L>1$ 发散。
积分检验：对正且递减的 $f(n)=a_n$，$\sum a_n$ 收敛当且仅当 $\int_1^\infty f(x) dx$ 收敛。

Power series centered at $x=a$: $\sum_{n=0}^{\infty} c_n (x-a)^n$. The radius of convergence is found using the ratio test.

以 $x=a$ 为中心的幂级数：$\sum_{n=0}^{\infty} c_n (x-a)^n$。收敛半径通过比值检验求得。

Taylor series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$. For $a=0$, it is a Maclaurin series.

泰勒级数：$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$。当 $a=0$ 时为麦克劳林级数。

Example: $e^x = \sum_{n=0}^{\infty} x^n/n!$ for all $x$.

示例：$e^x = \sum_{n=0}^{\infty} x^n/n!$ 对所有 $x$ 成立。

10. Common Formulas and Tables | 常用公式表

The following table summarizes the most frequently needed formulas on the AP Calculus exam.

下表总结了 AP 微积分考试中最常用的公式。

Derivatives	Integrals
d/dx (xⁿ) = n xⁿ⁻¹	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
d/dx (eˣ) = eˣ	∫ eˣ dx = eˣ + C
d/dx (ln x) = 1/x	∫ 1/x dx = ln\|x\| + C
d/dx (sin x) = cos x	∫ cos x dx = sin x + C
d/dx (cos x) = −sin x	∫ sin x dx = −cos x + C

For the AP exam, memorize the derivative of inverse trig functions and the integration formulas for $\sqrt{a^2-x^2}$ and $\frac{1}{x^2+a^2}$.

对于 AP 考试，还需要记住反三角函数的导数，以及 $\sqrt{a^2-x^2}$ 和 $\frac{1}{x^2+a^2}$ 的积分公式。

Linearity of integrals: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$; $\int c f(x) dx = c \int f(x) dx$.

积分的线性性质：$\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$；$\int c f(x) dx = c \int f(x) dx$。

Always remember to include the constant $+C$ for indefinite integrals.

求不定积分时始终记得加上常数 $+C$。

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AP Calculus: Core Formulas and Typical Applications | AP微积分：核心公式梳理与典型应用