📚 AP Calculus: Essential Vocabulary Summary | AP 微积分必考词汇汇总
Mastering the language of calculus is the first step toward success on the AP exam. This glossary presents the key terms you must know – from limits and derivatives to series – with concise definitions that clarify concepts and help you avoid common pitfalls. Each entry is given in English followed by its Chinese equivalent, ensuring bilingual clarity for learners across curricula.
掌握微积分的语言是在 AP 考试中取得成功的第一步。这份词汇表涵盖了从极限导数到级数的必考术语，提供简明定义，帮助厘清概念、避开常见误区。每个词条先给出英文解释，再给出对应中文，确保双语学习者一目了然。
1. Limits and Continuity | 极限与连续性
This section covers the vocabulary used to describe the behavior of functions as inputs approach a specific value or infinity, and the conditions for a function to be continuous.
本节涵盖用于描述当输入趋近某个值或无穷时函数行为以及函数连续条件的词汇。
Limit: The value that a function f(x) approaches as x gets arbitrarily close to a number a. Notation: limₓ→ₐ f(x) = L.
极限：当 x 无限趋近于 a 时，函数 f(x) 趋近的值。记作 limₓ→ₐ f(x) = L。
One-sided limit: The limit as x approaches a only from the left (x→a⁻) or only from the right (x→a⁺).
单侧极限：仅从左侧 (x→a⁻) 或仅从右侧 (x→a⁺) 趋近 a 时的极限。
Infinite limit: A limit where f(x) grows without bound (→ +∞ or –∞) as x approaches a finite value. It indicates a vertical asymptote.
无穷极限：当 x 趋近某个有限值时 f(x) 无限增大或减小（→ +∞ 或 –∞）的极限，表明存在垂直渐近线。
Continuity at a point: A function f is continuous at x = c if f(c) is defined, limₓ→꜀ f(x) exists, and the limit equals f(c).
点连续性：函数 f 在 x = c 处连续，若 f(c) 有定义，limₓ→꜀ f(x) 存在且等于 f(c)。
Removable discontinuity: A gap that could be “filled” by redefining a single point; the limit exists but does not equal the function value or the function is undefined there.
可去间断点：通过重新定义某一点就可以“填补”的间断；该点极限存在但不等于函数值或函数未定义。
Jump discontinuity: The left-hand and right-hand limits exist but are different finite numbers; the function “jumps” from one value to another.
跳跃间断点：左右极限均存在但为不同的有限值，函数从一个值“跳”到另一个值。
Infinite discontinuity: The limit from at least one side is infinite; typically occurs at a vertical asymptote.
无穷间断点：至少一侧的极限为无穷大；常出现在垂直渐近线处。
Squeeze theorem: If g(x) ≤ f(x) ≤ h(x) near a and both g and h have the same limit L, then f also has limit L.
夹逼定理：若在 a 附近有 g(x) ≤ f(x) ≤ h(x)，且 g 和 h 的极限均为 L，则 f 的极限也是 L。
Indeterminate form: An expression such as 0/0 or ∞/∞ that requires further analysis (factoring, L’Hôpital’s rule, etc.) to evaluate a limit.
未定式：形如 0/0 或 ∞/∞ 等需要进一步处理才能求极限的表达式。
2. Derivatives: Basics | 导数基础
The derivative measures how a function changes – its instantaneous rate of change. These terms form the foundation of differential calculus.
导数衡量函数的变化率——即瞬时变化率。这些术语是微分学的基础。
Derivative: The limit of the difference quotient: f'(x) = limₕ→₀ [f(x+h) – f(x)]/h. It gives the slope of the tangent line at each point.
导数：差商的极限：f'(x) = limₕ→₀ [f(x+h) – f(x)]/h。它给出各点切线的斜率。
Differentiability: A function is differentiable at x = a if the derivative f'(a) exists. Differentiability implies continuity, but not vice versa.
可导性：若 f'(a) 存在，则函数在 x = a 处可导。可导必连续，但连续不一定可导。
Tangent line: The straight line that just “touches” the curve at a point, having slope equal to the derivative at that point.
切线：刚好“触碰”曲线某点的直线，其斜率等于该点的导数值。
Secant line: A line that intersects a curve at two points; its slope is the average rate of change over an interval.
割线：与曲线交于两点的直线，其斜率是区间上的平均变化率。
Difference quotient: The expression [f(x+h) – f(x)]/h used in the limit definition of the derivative.
差商：用于导数极限定义的表达式 [f(x+h) – f(x)]/h。
Notation f'(x): Lagrange notation for the first derivative of f with respect to x.
记号 f'(x)：拉格朗日记号，表示 f 关于 x 的一阶导数。
dy/dx: Leibniz notation for the derivative of y with respect to x; useful for emphasizing the variable of differentiation.
dy/dx：莱布尼茨记号，表示 y 对 x 的导数，便于突出微分变量。
3. Differentiation Rules | 求导法则
Efficiently computing derivatives relies on a set of rules. These terms name the essential techniques you will use repeatedly.
高效计算导数依赖于一系列法则。下面这些术语是你会反复使用的基本技巧。
Power rule: d/dx (xⁿ) = n xⁿ⁻¹ for any real constant n.
幂法则：d/dx (xⁿ) = n xⁿ⁻¹，n 为任意实常数。
Product rule: d/dx (u·v) = u’·v + u·v’.
乘积法则：d/dx (u·v) = u’·v + u·v’。
Quotient rule: d/dx (u/v) = (u’·v – u·v’) / v².
商法则：d/dx (u/v) = (u’·v – u·v’) / v²。
Chain rule: If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Used to differentiate compositions.
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。用于复合函数求导。
Implicit differentiation: A method for differentiating equations where y is not explicitly solved; differentiate both sides with respect to x and then solve for dy/dx.
隐函数求导：当方程未解出 y 时求导的方法；两边对 x 求导，再解出 dy/dx。
Derivative of inverse functions: If g = f⁻¹, then g'(x) = 1 / f'(g(x)).
反函数求导：若 g = f⁻¹，则 g'(x) = 1 / f'(g(x))。
Higher-order derivative: Derivatives of derivatives; notation f”(x), f”'(x), or d²y/dx² for the second derivative.
高阶导数：导数的导数；二阶导数记作 f”(x) 或 d²y/dx²。
4. Applications of Derivatives | 导数的应用
Derivatives reveal a function’s shape, extreme values, and real-world behavior. These terms are vital for the free-response section.
导数能揭示函数的形状、极值和实际行为。这些术语在自由回答部分至关重要。
Critical point: A point in the domain where f'(x) = 0 or f'(x) does not exist. Possible location of local extrema.
临界点：定义域内使得 f'(x) = 0 或 f'(x) 不存在的点，可能是局部极值点。
Relative (local) maximum/minimum: The highest/lowest point in an open interval around c.
相对（局部）极大/极小值：在 c 附近某个开区间内的最高/最低点。
Absolute (global) extremum: The overall highest or lowest value of f on a closed interval.
绝对（全局）极值：在闭区间上 f 的整体最大值或最小值。
First derivative test: Uses sign changes of f’ around a critical point to classify local maxima/minima.
一阶导数检验法：根据临界点附近 f’ 的符号变化判断局部极大/极小。
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.
二阶导数检验法：若 f'(c) = 0 且 f”(c) > 0，则 c 处有局部极小；若 f”(c) < 0，则局部极大。
Concavity: Describes the curvature of a graph. f”(x) > 0 means concave up (∪), f”(x) < 0 means concave down (∩).
凹凸性：描述图形弯曲方向。f”(x) > 0 为上凹 (∪)，f”(x) < 0 为下凹 (∩)。
Inflection point: A point where the function is continuous and the concavity changes (f” changes sign).
拐点：函数连续且凹凸性改变（f” 变号）的点。
Mean Value Theorem (MVT): For a differentiable function on [a,b], there exists c in (a,b) such that f'(c) = [f(b) – f(a)] / (b – a).
中值定理 (MVT)：对于在 [a,b] 上可导的函数，存在 c∈(a,b) 使得 f'(c) = [f(b) – f(a)]/(b – a)。
Rolle’s Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) where f'(c) = 0.
罗尔定理：若 f 在 [a,b] 连续、(a,b) 可导且 f(a)=f(b)，则存在 c∈(a,b) 使 f'(c)=0。
Optimization: Using derivatives to find maximum or minimum values of a quantity under given constraints.
最优化：利用导数在给定约束下求某个量的最大值或最小值。
Related rates: Problems where two or more quantities depending on time are linked by an equation; differentiate with respect to time.
相关变化率：两个或多个随时间变化的量通过方程联系，对时间求导以求解。
Linear approximation (local linearity): Using the tangent line at a point to estimate function values nearby: f(x) ≈ f(a) + f'(a)(x – a).
线性逼近（局部线性）：在一点用切线近似附近的函数值：f(x) ≈ f(a) + f'(a)(x – a)。
L’Hôpital’s rule: For limits of indeterminate forms 0/0 or ∞/∞, lim f/g = lim f’/g’ provided the limit of the derivatives exists.
洛必达法则：对未定式 0/0 或 ∞/∞，若导数比的极限存在，则 lim f/g = lim f’/g’。
5. Integrals: Basics | 积分基础
Integration reverses differentiation and computes accumulated quantities. Understanding the language of antiderivatives and definite integrals is crucial.
积分是微分的逆运算，用于计算累积量。理解原函数和定积分的语言十分关键。
Antiderivative: A function F whose derivative is f: F'(x) = f(x). The collection of all antiderivatives is the indefinite integral ∫ f(x) dx = F(x) + C.
原函数：导数为 f 的函数 F，即 F'(x) = f(x)。全体原函数构成不定积分 ∫ f(x) dx = F(x) + C。
Indefinite integral: The family of all antiderivatives of f; always includes an arbitrary constant C.
不定积分：f 的所有原函数的族；总是包含任意常数 C。
Definite integral: The net area between the graph of f and the x-axis from x = a to x = b, denoted ∫ₐᵇ f(x) dx.
定积分：从 x = a 到 x = b 函数曲线与 x 轴之间的净面积，记作 ∫ₐᵇ f(x) dx。
Riemann sum: An approximation of a definite integral using rectangles: ∑ f(xᵢ*) Δx. Left, right, and midpoint sums use different sample points.
黎曼和：用矩形面积和近似定积分：∑ f(xᵢ*) Δx。左、右、中点黎曼和使用不同的取样点。
Trapezoidal sum: Approximation using trapezoids instead of rectangles; often more accurate than some Riemann sums.
梯形和：用梯形代替矩形的近似求和；常比某些黎曼和更准确。
Fundamental Theorem of Calculus (FTC): Connects differentiation and integration. Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x). Part 2: ∫ₐᵇ f(x) dx = F(b) – F(a) where F’ = f.
微积分基本定理 (FTC)：连接微分与积分。第一部分：若 F(x) = ∫ₐˣ f(t) dt，则 F'(x) = f(x)。第二部分：∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F’ = f。
Average value of a function: On [a,b], the average value is (1/(b – a)) ∫ₐᵇ f(x) dx.
函数平均值：在 [a,b] 上，平均值为 (1/(b – a)) ∫ₐᵇ f(x) dx。
Net change theorem: The integral of a rate of change gives total change: ∫ₐᵇ f'(x) dx = f(b) – f(a).
净变化定理：变化率的积分等于总变化量：∫ₐᵇ f'(x) dx = f(b) – f(a)。
6. Integration Techniques | 积分技巧
Many integrals cannot be solved directly; specialized techniques transform them into manageable forms. BC students must master additional methods.
许多积分无法直接求解，需要通过特定技巧转化。BC 学生需掌握更多方法。
Substitution method (u-substitution): Reverse of the chain rule; set u = g(x), du = g'(x) dx to rewrite ∫ f(g(x))g'(x) dx as ∫ f(u) du.
换元积分法（u 代换）：链式法则的逆运算；令 u = g(x), du = g'(x) dx，将原积分化为 ∫ f(u) du。
Integration by parts (BC): Reverse of the product rule: ∫ u dv = uv – ∫ v du.
分部积分法 (BC)：乘积法则的逆运算：∫ u dv = uv – ∫ v du。
Partial fraction decomposition (BC): Rewriting a rational function as a sum of simpler fractions to integrate each piece.
部分分式分解 (BC)：将有理函数表示为简单分式的和，以便逐项积分。
Improper integral (BC): An integral with an infinite limit of integration or an unbounded integrand; evaluated using limits.
反常积分 (BC)：积分限为无穷大或被积函数无界的积分；通过极限来求值。
Completing the square: Algebraically manipulating a quadratic so that substitution works, often for arctan or arcsin forms.
配方法：代数变形二次式以便使用代换，常用于反正切或反正弦形式的积分。
Trigonometric integrals (BC): Integrals involving powers of sine and cosine or secant and tangent, often solved using identites and substitution.
三角积分 (BC)：涉及正弦、余弦或正割、正切幂次的积分，常用恒等式和换元求解。
7. Applications of Integrals | 积分的应用
Definite integrals model area, volume, length, and other accumulations. Know the setups for disk, washer, and shell methods.
定积分可模拟面积、体积、弧长和其他累积量。务必掌握圆盘法、垫圈法和柱壳法的构造。
Area between curves: A = ∫ₐᵇ [f(x) – g(x)] dx where f(x) ≥ g(x) on [a,b]. Horizontal slices use dy.
曲线间面积：A = ∫ₐᵇ [f(x) – g(x)] dx，其中在 [a,b] 上 f(x) ≥ g(x)。水平切片使用 dy 积分。
Volume of revolution: disk method: V = π ∫ₐᵇ [R(x)]² dx, for solids obtained by rotating a region about an axis, perpendicular cross-section.
旋转体体积：圆盘法：V = π ∫ₐᵇ [R(x)]² dx，用于垂直旋转轴且截面垂直的情况。
Volume of revolution: washer method: V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx, for solids with a hole.
旋转体体积：垫圈法：V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx，用于有空心的旋转体。
Shell method: V = 2π ∫ₐᵇ (radius)·(height) dx or dy, used when parallel to the axis of rotation.
柱壳法：V = 2π ∫ₐᵇ (半径)·(高) dx 或 dy，用于切片平行于旋转轴的情形。
Arc length (BC): L = ∫ₐᵇ √(1 + [f'(x)]²) dx.
弧长 (BC)：L = ∫ₐᵇ √(1 + [f'(x)]²) dx。
Surface area of revolution (BC): S = 2π ∫ₐᵇ f(x) √(1 + [f'(x)]²) dx for rotation about the x-axis.
旋转曲面表面积 (BC)：绕 x 轴旋转时 S = 2π ∫ₐᵇ f(x) √(1 + [f'(x)]²) dx。
Accumulation function: A function defined by an integral with a variable upper limit; its derivative recovers the integrand (FTC).
累积函数：以变量为上限的积分定义的函数；其导数恢复被积函数（微积分基本定理）。
8. Differential Equations
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