📚 AP Calculus: Core Vocabulary for Scoring a 5 | AP微积分：5分必会核心词汇汇总
Mastering AP Calculus terminology is just as important as solving problems. Clear understanding of definitions, notations, and theorems enables you to interpret questions accurately, justify your answers on free-response sections, and link concepts across limits, derivatives, integrals, and series. This article compiles the essential vocabulary you must know to aim for a 5, organized by topic with concise English explanations followed by Chinese translations.
掌握AP微积分的术语与掌握解题技巧同样关键。准确理解定义、符号和定理能帮助你正确解读题目、在简答题中完整论证，并将极限、导数、积分和级数等知识融会贯通。本文按照专题整理了冲击5分必须掌握的核心词汇，每个术语均配以精简的英文解释和中文对照，方便你对照记忆。
1. Limits and Continuity | 极限与连续性
Limit: The value that a function f(x) approaches as the input x gets arbitrarily close to a specific value a. Notation: limx→a f(x) = L. The limit exists only if the left-hand limit and right-hand limit are equal.
极限: 当自变量 x 无限趋近于某个数值 a 时，函数 f(x) 趋近的值。记作 limx→a f(x) = L。极限存在的充要条件是左极限与右极限相等。
One-sided limits: limx→a⁻ f(x) (left-hand limit) and limx→a⁺ f(x) (right-hand limit). They describe behavior from one side only and are used to check continuity and identify jump discontinuities.
单侧极限: 左极限 limx→a⁻ f(x) 和右极限 limx→a⁺ f(x)。它们仅描述单侧趋近时的函数走向，常用于检验连续性或判定跳跃间断点。
Continuity at a point: A function f is continuous at x = a if limx→a ons must hold: f(a) is defined, the limit exists, and the limit equals the function value.
在某点连续: 若 limx→a f(x) = f(a)，则称函数在 x = a 处连续。必须满足三点：f(a) 有定义、极限存在、且极限值等于函数值。
Removable discontinuity: A "hole" in the graph where the limit exists but the function is either undefined at that point or has a different value. Can be "removed" by redefining the function value.
可去间断点: 图像上出现一个“洞”，该点极限存在但函数未定义或取值不同。可通过重新定义函数值“去除”这一间断。
Jump discontinuity: Occurs when the left-hand and right-hand limits exist but are not equal, causing a sudden jump in the function value. Common in piecewise functions.
跳跃间断点: 左右极限都存在但不相等，导致函数值发生跳跃突变。分段函数中常见此类间断。
Vertical asymptote: A line x = a where the function increases or decreases without bound, i.e., limx→a f(x) = ±∞. Often found where the denominator is zero but the numerator is non-zero.
竖直渐近线: 直线 x = a 处函数值趋向正无穷或负无穷，即 limx→a f(x) = ±∞。分母为零而分子不为零处常出现竖直渐近线。
2. Derivatives: Fundamentals | 导数基础
Derivative: The instantaneous rate of change of a function at a point, given by f'(x) = limh→0 [f(x+h) - f(x)] / h. It represents the slope of the tangent line.
导数: 函数在某点的瞬时变化率，定义为 f'(x) = limh→0 [f(x+h) - f(x)] / h。它表示函数图像在该点切线的斜率。
Difference quotient: The expression [f(x+h) - f(x)] / h, which approximates the slope of the secant line and forms the basis of the derivative definition.
差商: 表达式 [f(x+h) - f(x)] / h，用来近似割线斜率，也是导数定义的核心形式。
Tangent line: A line that touches the graph of a function at exactly one point locally and has slope equal to the derivative at that point.
切线: 与函数图像在某点局部仅有一个交点的直线，其斜率等于该点的导数值。
Differentiability: A function is differentiable at a point if the derivative exists there, meaning the graph is smooth (no corners, cusps, or vertical tangents). Differentiability implies continuity, but continuity does not guarantee differentiability.
可导性: 若函数在某点导数存在，则称该点可导，此时图像光滑（无尖点、折角或竖直切线）。可导必连续，但连续不一定可导。
Notation: Common forms include f'(x) (Lagrange), dy/dx (Leibniz), and y'. Leibniz notation is especially helpful for the chain rule and related rates.
记法: 常用符号有 f'(x)（拉格朗日记法）、dy/dx（莱布尼茨记法）和 y'。莱布尼茨记法在处理链式法则和相关变化率时尤为有用。
3. Derivative Rules | 求导法则
Power rule [xⁿ] = n xⁿ⁻¹, valid for any real number n. This is the most fundamental derivative rule.
幂法则: d/dx [xⁿ] = n xⁿ⁻¹，对任意实数 n 成立。这是最基础的求导公式。
Product rule: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). The derivative of a product is not simply the product of the derivatives.
乘法法则: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)。两个函数乘积的导数不等于各自导数之积。
Quotient rule: d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². Remember the numerator order: "low d-high minus high d-low."
除法法则: d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]²。分子为“分母乘分子的导数减去分子乘分母的导数”。
Chain rule: d/dx [f(g(x))] = f'(g(x)) · g'). Used when differentiating composite functions. In Leibniz form: dy/dx = (dy/du)(du/dx).
链式法则: d/dx [f(g(x))] = f'(g(x)) · g'(x)，用于复合函数求导。莱布尼茨形式为 dy/dx = (dy/du)(du/dx)。
Implicit differentiation: A technique for finding dy/dx when y is defined implicitly by an equation in x and y. Differentiate both sides with respect to x, treating y as a function of x, and then solve for dy/dx.
隐函数求导: 当 y 由关于 x 和 y 的方程隐式定义时，通过对等式两边关于 x 求导（将 y 视作 x 的函数），然后解出 dy/dx 的方法。
Higher-order derivatives: The second derivative f''(x) gives the rate of change of the first derivative and relates to concavity. Notation: f''(x), d²y/dx². Third, fourth, etc., can be similarly defined.
高阶导数: 二阶导数 f''(x) 表示一阶导数的变化率，与凹凸性相关。记作 f''(x) 或 d²y/dx²。可类推三阶、四阶导数。
4. Applications of Derivatives | 导数的应用
Critical point: A point x = c in the domain of f where either f'(c) = 0 or f'(c) does not exist. Critical points are candidates for local extrema.
临界点: 在函数定义域内使得 f'(c) = 0 或 f'(c) 不存在的点 x = c。临界点是局部极值的可能位置。
Relative (local) maximum/minimum cal minimum if f(c) ≤ f(x) nearby. The first derivative test can classify critical points.
局部极大/极小值: 若在 x = c 附近恒有 f(c) ≥ f(x)，则 f 在 c 处取得局部极大值；反之若 f(c) ≤ f(x) 则为局部极小值。一阶导数判别法可对临界点进行分类。
Absolute (global) extrema: The highest (maximum) or lowest (minimum) value of f on a given interval. On a closed interval [a,b], compare critical point values and endpoint values.
绝对（全局）最值: 函数在给定区间上的最大值或最小值。在闭区间 [a,b] 上，需比较区间内临界点函数值与端点函数值。
Increasing/decreasing test: If f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing. Sign changes locate relative extrema.
单调性判别: 在某一区间内若 f'(x) > 0，则 f 递增；若 f'(x) < 0，则 f 递减。导数的符号变化可用于确定局部极值。
Concavity and inflection point: f is concave up where f''(x) > 0, concave down where f''(x) < 0. An inflection point occurs where f changes concavity and f''(x) = 0 or is undefined.
凹性与拐点: f''(x) > 0 时函数凹向上；f''(x) < 0 时凹向下。拐点是函数凹性发生变化的点，且在该点 f''(x) = 0 或不存在。
Mean Value Theorem (MVT) ous on [a,b] and differentiable on (a,b), there exists at least one c in (a,b) such that f'(c) = [f(b) - f(a)] / (b - a). Connects average rate of change to instantaneous rate.
中值定理: 若 f 在 [a,b] 上连续、在 (a,b) 内可导，则存在至少一个 c ∈ (a,b) 使得 f'(c) = [f(b) - f(a)] / (b - a)。它将平均变化率与瞬时变化率联系起来。
Optimization ximum or minimum values of a quantity subject to constraints. Typically involves setting f'(x) = 0 and checking endpoints.
最优化: 利用导数在约束条件下求某个量的最大值或最小值。通常要求令 f'(x) = 0 并检验端点和临界点。
Related rates: Problems where two or more quantities change with time and are related by an equation. Implicit differentiation with respect to time t yields relationships between their rates.
相关变化率: 多个随时间变化的量之间存在等式关系，通过对时间 t 进行隐式求导，得到各变化率之间的关系。常用于几何或运动情境。
5. Integrals: Basics | 积分基础
Antiderivative: A function F whose derivative is f, i.e., F'(x) = f(x). The general antiderivative includes an arbitrary constant C: F(x) + C.
原函数: 导数等于 f 的函数 F，即 F'(x) = f(x)。一般原函数包含任意常数 C：F(x) + C。
Indefinite integral: The set of all antiderivatives of f, denoted ∫ f(x) dx. The answer always includes "+ C".
不定积分: 函数 f 的所有原函数的集合，记作 ∫ f(x) dx。结果务必加上积分常数 "+ C"。
Definite integral of f and the x-axis from x = a to x = b. Based on Riemann sums as the limit of summation.
定积分: ∫ₐᵇ f(x) dx 表示从 x = a 到 x = b 之间 f 的图像与 x 轴所围的有向面积。其定义为黎曼和当子区间宽度趋近于 0 时的极限。
Riemann sum: An approximation of a definite integral using rectangles. Left, right, and midpoint Riemann sums use different sample points. The definite integral is the limit of Riemann sums as the partition gets finer.
黎曼和: 用矩形面积近似定积分的方法。左黎曼和、右黎曼和及中点黎曼和分别选取左端点、右端点和中点作为高。当划分无限细密时，黎曼和的极限即为定积分。
Trapezoidal sum: An approximation of a definite integral using trapezoids instead of rectangles, typically more accurate than Riemann sums for the same number of subintervals.
梯形和: 用梯形面积代替矩形面积近似定积分，相同子区间数下通常比黎曼和更精确。
Fundamental Theorem of Calculus (FTC): FTC Part 1: If g(x) = ∫ₐˣ f(t) dt, then g' F(b) - F(a) where F is any antiderivative of f. Connects differentiation and integration.
微积分基本定理: 第一部分：若 g(x) = ∫ₐˣ f(t) dt，则 g'(x) = f(x)。第二部分：∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F 是 f 的任意一个原函数。沟通了微分与积分。
6. Integration Techniques | 积分技巧
u-substitution : The reverse of the chain rule. Choose u = g(x), rewrite the integral in terms of u and du = g'(x) dx, then integrate. Essential for composite functions.
换元积分法: 链式法则的逆运算。选取 u = g(x)，将被积表达式用 u 和 du = g'(x) dx 替换再积分。处理复合函数积分的基本方法。
Integration by parts (BC) dv = uv - ∫ v du. Useful for products of algebraic and transcendental functions, like x eˣ dx or x sin x dx.
分部积分法 (BC): 源自乘法法则：∫ u dv = uv - ∫ v du。适用于代数函数与超越函数之积，如 x eˣ dx 或 x sin x dx。
Partial fractions (BC): Decomposes a rational function into simpler fractions that can be integrated individually. Requires factoring the denominator and solving for constants.
部分分式积分 (BC): 将有理函数分解为几个较简单的分式之和，再逐项积分。需对分母进行因式分解并确定待定常数。
Improper integrals (BC) he integrand has a vertical asymptote within the interval. Evaluate using limits, e.g., ∫ₐ∞ f(x) dx = limb→∞ ∫ₐᵇ f(x) dx. Converges if the limit exists finite.
反常积分 (BC): 积分区间为无限，或被积函数在积分区间内有无穷间断点。用极限求值，如 ∫ₐ∞ f(x) dx = limb→∞ ∫ₐᵇ f(x) dx。若极限存在为有限值则收敛。
7. Applications of Integrals | 积分的应用
Area between curves ) on [a,b], the area is ∫ₐᵇ [f(x) - g(x)] dx. For functions of y, use ∫ [right - left] dy.
曲线间的面积: 若在 [a,b] 上恒有 f(x) ≥ g(x)，则面积等于 ∫ₐᵇ [f(x) - g(x)] dx。若以 y 为自变量，则用 ∫ [右函数 - 左函数] dy 计算。
Volume by disks/washers ng around an axis. Washer method subtracts inner radius: V = π ∫ₐᵇ ([R(x)]² - [r(x)]²) dx.
圆盘/垫圈法求体积: 旋转体体积，圆盘法：绕轴旋转时 V = π ∫ₐᵇ [R(x)]² dx。垫圈法需减去内半径的平方：V = π ∫ₐᵇ ([R(x)]² - [r(x)]²) dx。
Volume by cross sections x-axis, volume = ∫ₐᵇ A(x) dx. Common shapes: squares, semicircles, equilateral triangles.
已知截面求体积: 对于垂直于 x 轴的截面积为 A(x) 的立体，其体积为 ∫ₐᵇ A(x) dx。常见截面形状有正方形、半圆、等边三角形等。
Average value of a function This is the height of a rectangle with base [a,b] having the same area as under the curve.
函数的平均值: 在 [a,b] 上，函数的平均值为 (1/(b-a)) ∫ₐᵇ f(x) dx。相当于与曲线下方面积相等的同底矩形的“高”。
Arc length (BC): For a smooth function y = f(x) from x = a to x = b, length L = ∫ₐᵇ √(1 + [f'(x)]²) dx. For parametric curves, L = ∫ √[(dx/dt)² + (dy/dt)²] dt.
弧长 (BC): 光滑曲线 y = f(x) 在 x = a 到 b 间的长度 L = ∫ₐᵇ √(1 + [f'(x)]²) dx。对于参数曲线，弧长为 ∫ √[(dx/dt)² + (dy/dt)²] dt。
8. Differential Equations | 微分方程
Differential equation n : An equation that involves an unknown function and its derivatives, e.g., dy/dx = ky. Order refers to the highest derivative appearing.
微分方程: 包含未知函数及其导数的方程，如 dy/dx = ky。方程的阶由出现的最高阶导数决定。
Slope field g small line segments with slope f(x,y) at sample points. Helps visualize solution curves.
斜率场: 对一阶微分方程 dy/dx = f(x,y)，在样本点画出斜率为 f(x,y) 的短线段的图形。可直观描绘解的走势。
Separable equation written as g(y) dy = h(x) dx. Solved by integrating both sides after separating variables.
可分离方程: 能写成 g(y) dy = h(x) dx 形式的微分方程。解法为分离变量后对两边积分。
General solution vs. particular solution icular solution is found by applying an initial condition (e.g., y(0) = 2) to determine C.
通解与特解: 通解含有任意常数 C。利用初始条件（如 y(0) = 2）确定 C 的值，可得到一个特解。
Exponential growth/decay: Solutions to dy/dt = ky are y = Cekt. If k > 0, exponential growth; if k < 0, decay. The doubling time or half-life can be derived.
指数增长/衰减: 微分方程 dy/dt = ky 的解为 y = Cekt。k > 0 时指数增长，k < 0 时衰减。可由解给出倍增时间或半衰期。
Logistic growth (BC) L is the carrying capacity. The solution is a sigmoid curve approaching L asymptotically.
逻辑斯蒂增长 (BC): 模型为 dP/dt = kP(1 - P/L)，其中 L 为环境承载力。解曲线为 S 形，渐近地趋于 L。
9. Parametric, Polar, and Vector Functions (BC) | 参数方程、极坐标与向量函数 (BC)
Parametric equations g(t), where t is a parameter. The derivative dy/dx = (dy/dt) / (dx/dt) provided dx/dt ≠ 0.
参数方程: 通过 x = f(t), y = g(t) 定义曲线，t 为参数。导数 dy/dx = (dy/dt) / (dx/dt)，要求 dx/dt ≠ 0。
Arc length for parametric curves: L = ∫αβ √[(dx/dt)² + (dy/dt)²] dt over t from α to β.
参数曲线的弧长: L = ∫αβ √[(dx/dt)² + (dy/dt)²] dt，t 从 α 到 β。
Polar coordinates igin and θ is the angle from the positive x-axis. Relationships: x = r cos θ, y = r sin θ.
极坐标: 用 (r, θ) 表示点，r 为到原点的距离，θ 为极角。转换关系：x = r cos θ, y = r sin θ。
Area in polar coordinates: The area bounded by the polar curve r = f(θ) from θ = α to θ = β is (1/2) ∫αβ [f(θ)]² dθ.
极坐标下的面积: 由极坐标曲线 r = f
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