AP Calculus: Essential Vocabulary Summary | AP 微积分：必备词汇汇总

📚 AP Calculus: Essential Vocabulary Summary | AP 微积分：必备词汇汇总

Mastering the terminology of AP Calculus is essential for success on the exam and in future mathematical studies. This comprehensive glossary covers key concepts from limits through series, presented with precise English definitions and their Chinese equivalents. Use this resource to strengthen your conceptual understanding and to ensure you can fluently communicate mathematical ideas in both languages.

掌握 AP 微积分的术语对于考试成功和未来的数学学习至关重要。这份全面的词汇表涵盖了从极限到级数的关键概念，以精确的英文定义和对应的中文释义呈现。利用此资源来加强概念理解，并确保能够用双语流畅地表达数学思想。

1. Limits and Continuity | 极限与连续性

Limit: The value L that f(x) approaches as x approaches a. Notation: limₓ→ₐ f(x) = L.

极限：当 x 趋近于 a 时，函数 f(x) 所趋近的值 L。记作 limₓ→ₐ f(x) = L。

One-sided limit: The limit as x approaches a from the right (x→a⁺) or from the left (x→a⁻).

单侧极限：x 从右侧 (x→a⁺) 或左侧 (x→a⁻) 趋近于 a 时的极限。

Continuity at a point: f is continuous at x = c if limₓ→c f(x) = f(c), f(c) is defined, and the limit exists.

点处的连续性：若 limₓ→c f(x) = f(c)，f(c) 有定义且极限存在，则 f 在 x = c 处连续。

Removable discontinuity: A discontinuity where the limit exists but does not equal the function value or the function is not defined.

可去间断点：极限存在但与函数值不相等或函数在该点无定义的间断点。

Intermediate Value Theorem (IVT): If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in (a,b) such that f(c)=k.

介值定理：若 f 在 [a,b] 上连续且 k 介于 f(a) 与 f(b) 之间，则存在 c∈(a,b) 使 f(c)=k。

Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) near a and lim g(x) = lim h(x) = L, then lim f(x) = L.

夹逼定理：若在 a 附近有 g(x) ≤ f(x) ≤ h(x) 且 lim g(x)=lim h(x)=L，则 lim f(x)=L。

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2. Derivatives: Definition and Interpretations | 导数：定义与解释

Derivative: f'(x) = limₕ→0 [f(x+h) – f(x)] / h, if the limit exists. It represents the instantaneous rate of change.

导数：f'(x) = limₕ→0 [f(x+h) – f(x)] / h，若极限存在。它表示瞬时变化率。

Tangent line: The line through (a, f(a)) with slope f'(a), representing the best linear approximation of f near x=a.

切线：过点 (a, f(a)) 且斜率为 f'(a) 的直线，代表 f 在 x=a 附近的最佳线性近似。

Secant line: A line passing through two points on the graph of a function; its slope gives the average rate of change.

割线：通过函数图像上两点的直线；其斜率给出平均变化率。

Differentiability: A function is differentiable at a point if its derivative exists there; implies continuity.

可微性：若函数在某点导数存在，则其在该点可微；可微一定连续。

Notation: Derivatives can be written as f'(x), dy/dx, y’, or d/dx[f(x)].

记号：导数可写作 f'(x)、dy/dx、y’ 或 d/dx[f(x)]。

Higher-order derivatives: second derivative f”(x) or d²y/dx² measures concavity and acceleration.

高阶导数：二阶导数 f”(x) 或 d²y/dx² 衡量凹凸性和加速度。

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3. Differentiation Rules | 微分法则

Power Rule: d/dx [xⁿ] = nxⁿ⁻¹ for any real constant n.

幂法则：对于任意实常数 n，d/dx [xⁿ] = nxⁿ⁻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)。

Derivative of eˣ: d/dx [eˣ] = eˣ.

eˣ 的导数：d/dx [eˣ] = eˣ。

Derivative of ln x: d/dx [ln x] = 1/x, for x > 0.

ln x 的导数：d/dx [ln x] = 1/x，x > 0。

Derivatives of trigonometric functions: d/dx [sin x] = cos x; d/dx [cos x] = -sin x; d/dx [tan x] = sec² x.

三角函数的导数：d/dx [sin x] = cos x；d/dx [cos x] = -sin x；d/dx [tan x] = sec² x。

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4. Applications of Derivatives | 导数的应用

Critical point: A point c in the domain of f where f'(c) = 0 or f'(c) does not exist.

临界点：f 定义域内满足 f'(c)=0 或 f'(c) 不存在的点 c。

Relative (local) maximum/minimum: f has a local max at c if f(c) ≥ f(x) for all x near c; local min similarly.

相对（局部）极大值/极小值：若对于 c 附近所有 x 有 f(c)≥f(x)，则 f 在 c 取局部极大值；极小值类似。

Absolute (global) extremum: The highest or lowest value of f on an interval.

绝对（全局）极值：函数在区间上的最大值或最小值。

First Derivative Test: If f’ changes from positive to negative at c, f has a local max at c; if from negative to positive, a local min.

一阶导数测试：若 f’ 在 c 处由正变负，则 f 在 c 取局部极大值；若由负变正，则取局部极小值。

Concavity: If f”(x) > 0 on an interval, f is concave up; if f”(x) < 0, f is concave down.

凹凸性：若在区间上 f”(x) > 0，则 f 为凹向上；若 f”(x) < 0，则为凹向下。

Inflection point: A point where the graph changes concavity and f” changes sign.

拐点：图像凹凸性改变且 f” 变号的位置。

Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) such that f'(c) = [f(b)-f(a)]/(b-a).

中值定理：若 f 在 [a,b] 连续、在 (a,b) 可导，则存在 c∈(a,b) 使 f'(c)=[f(b)-f(a)]/(b-a)。

Related rates: Technique for finding the rate of change of one quantity in terms of the rate of change of another quantity that is related geometrically or physically.

相关变化率：通过几何或物理上相关联的量，用一个量的变化率求另一个量的变化率的方法。

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5. Integrals and Antiderivatives | 积分与反导数

Antiderivative: A function F is an antiderivative of f if F'(x) = f(x).

反导数：若 F'(x)=f(x)，则 F 是 f 的一个反导数。

Indefinite integral: The family of all antiderivatives of f, written ∫ f(x) dx = F(x) + C.

不定积分：f 的所有反导数构成的族，记作 ∫ f(x) dx = F(x) + C。

Definite integral: The signed area between the graph of f and the x-axis from a to b, denoted ∫ₐᵇ f(x) dx.

定积分：f 的图像与 x 轴之间从 a 到 b 的带符号面积，记作 ∫ₐᵇ f(x) dx。

Riemann sum: Approximation of a definite integral using sums of rectangle areas: Σ f(xᵢ*) Δx.

黎曼和：使用矩形面积之和近似定积分：Σ f(xᵢ*) Δx。

Left, Right, Midpoint Riemann sums: Rectangles are evaluated at the left endpoint, right endpoint, or midpoint of each subinterval.

左、右、中点黎曼和：在每个子区间的左端点、右端点或中点处计算矩形高度。

Trapezoidal Rule: Approximation using trapezoids: Δx/2 [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)].

梯形法则：使用梯形近似：Δx/2 [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]。

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

FTC Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x), provided f is continuous.

FTC 第一部分：若 F(x)=∫ₐˣ f(t) dt 且 f 连续，则 F'(x)=f(x)。

FTC Part 2: If F is any antiderivative of f on [a,b], then ∫ₐᵇ f(x) dx = F(b) – F(a).

FTC 第二部分：若 F 是 f 在 [a,b] 上的任一反导数，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。

Accumulation function: A function defined by an integral with a variable upper limit, e.g., g(x)=∫ₐˣ f(t) dt.

累积函数：由带有可变上限的积分定义的函数，例如 g(x)=∫ₐˣ f(t) dt。

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7. Techniques of Integration (BC) | 积分技巧（BC）

u-Substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, the reverse of the chain rule.

u-代换法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，链式法则的逆运算。

Integration by parts: ∫ u dv = uv – ∫ v du, derived from the product rule.

分部积分法：∫ u dv = uv – ∫ v du，由乘法法则导出。

Partial fractions: Decomposing a rational function into simpler fractions to integrate.

部分分式法：将有理函数分解为更简单的分式以便积分。

Improper integrals: Integrals with infinite limits or unbounded integrands; evaluated as limits of proper integrals.

反常积分：积分区间无限或被积函数无界的积分；通过正常积分的极限计算。

Trigonometric integrals: Integrals involving powers of sin x, cos x, sec x, etc., often simplified with identities.

三角积分：涉及 sin x、cos x、sec x 等幂次的积分，常借助三角恒等式化简。

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8. Applications of Integrals | 积分的应用

Area between curves: A = ∫ₐᵇ [f(x) – g(x)] dx, where f(x) ≥ g(x) on [a,b].

曲线间的面积：A = ∫ₐᵇ [f(x) – g(x)] dx，其中在 [a,b] 上 f(x) ≥ g(x)。

Volume by disk method: V = π ∫ₐᵇ [R(x)]² dx for solids of revolution with solid cross sections.

圆盘法求体积：对于截面为实心的旋转体，V = π ∫ₐᵇ [R(x)]² dx。

Volume by washer method: V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx, for hollowed solids.

垫圈法求体积：对于空心旋转体，V = π ∫ₐᵇ ([R(x)]² – [r(x)]²) dx。

Volume by shell method: V = 2π ∫ₐᵇ r(y) h(y) dy for cylindrical shells (typically vertical axis).

柱壳法求体积：对于柱形薄壳，V =

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