Year 13 AQA Maths: Terminology Quick Memorisation Guide | AQA 数学词汇术语速记指南

📚 Year 13 AQA Maths: Terminology Quick Memorisation Guide | AQA 数学词汇术语速记指南

Mastering the precise terminology of A-level Mathematics is just as important as learning the techniques. This guide breaks down the essential vocabulary from Core Pure, Mechanics and Statistics into paired English–Chinese explanations, helping you memorise meanings, avoid common mix-ups and speak the language of an AQA mathematician.

掌握 A-level 数学中的精确术语与学习解题技巧同样重要。本指南将纯数、力学与统计的核心词汇分解为中英文配对解释，帮助你牢记含义、避免常见混淆，并熟练使用 AQA 数学的语言体系。

1. Core Algebra & Functions | 核心代数与函数

• Domain: The set of all possible input values (x-values) for which a function is defined. For f(x) = √x, the domain is x ≥ 0.

  定义域：函数有定义的所有可能输入值（x 值）的集合。如 f(x) = √x 的定义域为 x ≥ 0。

• Range: The set of all possible output values (f(x)-values) that the function actually produces. For f(x) = x², the range is f(x) ≥ 0.

  值域：函数真正产生的所有可能输出值（f(x) 值）的集合。f(x) = x² 的值域是 f(x) ≥ 0。

• One-to-one function: A function where each output comes from exactly one input. Passes the horizontal line test; only these have a true inverse function on their whole domain.

  一对一函数：每个输出值恰好来自一个输入值的函数。能通过水平线检验；只有这类函数在整个定义域上才有真正的反函数。

• Many-to-one function: A function where different inputs can give the same output, e.g. f(x) = x² gives f(2) = 4 and f(−2) = 4. Its inverse can only be defined if the domain is restricted.

  多对一函数：不同输入可以产生相同输出的函数，例如 f(x) = x² 满足 f(2) = 4 与 f(−2) = 4。限制定义域后才能定义其反函数。

• Inverse function (f⁻¹): Reverses the effect of f, so that f(f⁻¹(x)) = x. The domain of f⁻¹ is the range of f, and vice versa.

  反函数 (f⁻¹)：逆向完成 f 的作用，使得 f(f⁻¹(x)) = x。f⁻¹ 的定义域是 f 的值域，反之亦然。

• Composite function (gf(x)): A function applied inside another function, read as “g of f of x”. The range of f must be a subset of the domain of g for the composite to be valid.

  复合函数 (gf(x))：一个函数套在另一个函数内部，读作 “g of f of x”。f 的值域必须是 g 定义域的子集，复合函数才有意义。

• Modulus function |x|: Gives the absolute value (magnitude) of x, ignoring any negative sign. Its graph is V-shaped and it often creates piecewise definitions.

  模函数 |x|：给出 x 的绝对值（大小），忽略负号。图像呈 V 形，常被写成分段函数形式。

• Transformation of graphs: y = f(x + a) translates a units left, y = f(x) + a translates up, y = f(ax) stretches horizontally by factor 1/a, y = af(x) stretches vertically by factor a. Reflections include y = −f(x) in the x-axis and y = f(−x) in the y-axis.

  图像变换：y = f(x + a) 向左平移 a 个单位，y = f(x) + a 向上平移，y = f(ax) 水平方向伸缩 1/a 倍，y = af(x) 垂直方向伸缩 a 倍。反射包括关于 x 轴对称的 y = −f(x) 和关于 y 轴对称的 y = f(−x)。

2. Sequences & Series | 数列与级数

• Binomial expansion: An expression of the form (a + b)ⁿ can be expanded using the binomial theorem. For rational n the expansion is infinite and valid for |x| < 1 when written as (1 + x)ⁿ.

  二项展开式：形如 (a + b)ⁿ 的式子可用二项式定理展开。当 n 为有理数时，展开式为无穷级数，并在写成 (1 + x)ⁿ 形式且 |x| < 1 时收敛有效。

• nCr (binomial coefficient): The number of ways to choose r items from n, written as ⁿCᵣ or (n r). It appears in the coefficients of the binomial expansion and is calculated as n! / [r!(n − r)!].

  nCr（二项式系数）：从 n 项中选取 r 项的方法数，记作 ⁿCᵣ 或 (n r)。它出现在二项展开式的系数中，计算公式为 n! / [r!(n − r)!]。

• Arithmetic progression: A sequence with a constant difference d between terms. The nth term is a + (n−1)d and the sum of the first n terms is (n/2)(2a + (n−1)d).

  等差数列：项与项之间保持常数公差 d 的数列。第 n 项为 a + (n−1)d，前 n 项和为 (n/2)(2a + (n−1)d)。

• Geometric progression: A sequence with a constant ratio r between terms. The nth term is arⁿ⁻¹ and the sum to n terms is a(1 − rⁿ)/(1 − r) when r ≠ 1.

  等比数列：项与项之间保持常数公比 r 的数列。第 n 项为 arⁿ⁻¹，前 n 项和为 a(1 − rⁿ)/(1 − r)，其中 r ≠ 1。

• Sum to infinity: For a geometric series with |r| < 1, the infinite sum converges to a/(1 − r).

  无穷和：对于满足 |r| < 1 的等比级数，其无穷和收敛于 a/(1 − r)。

• Maclaurin series: A Taylor series centred at zero, expressing a function as an infinite polynomial: f(x) = f(0) + f'(0)x + f”(0)x²/2! + … . AQA introduces standard expansions for eˣ, sin x, cos x and ln(1 + x).

  麦克劳林级数：以零为中心的泰勒级数，将函数表示为无穷多项式：f(x) = f(0) + f'(0)x + f”(0)x²/2! + …。AQA 要求掌握 eˣ、sin x、cos x 和 ln(1 + x) 的标准展开。

• Sigma notation (Σ): A compact way to write a sum: Σ (from r=1 to n) uᵣ means the sum of all terms u₁ + u₂ + … + uₙ.

  Σ 求和符号：求和的简写方式：Σ (从 r=1 到 n) uᵣ 表示所有项的和 u₁ + u₂ + … + uₙ。

• Recurrence relation: Defines a sequence by giving one or more starting terms and a rule to find subsequent terms. For example, uₙ₊₁ = 2uₙ + 1, u₁ = 3.

  递推关系：通过给出一个或多个起始项以及计算后续项的规则来定义数列。例如 uₙ₊₁ = 2uₙ + 1，u₁ = 3。

3. Trigonometry | 三角学

• Radian measure: The angle subtended by an arc of length equal to the radius of the circle. π radians = 180°; 1 radian ≈ 57.3°. All differentiation and integration of trig functions assumes radian mode.

  弧度制：弧长等于半径的弧所对的圆心角。π 弧度 = 180°；1 弧度约等于 57.3°。所有三角函数的微积分都以弧度制为基础。

• Arc length and sector area: For a sector of radius r and angle θ in radians: arc length = rθ, area = ½ r²θ.

  弧长与扇形面积：对于半径为 r、圆心角为 θ 弧度 的扇形，弧长 = rθ，面积 = ½ r²θ。

• Secant, cosecant and cotangent: sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ. These appear in identities and differentiation.

  正割、余割与余切：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。它们出现在恒等式和微分中。

• Pythagorean identities in extended form: sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. Remember the last two by dividing the first by cos²θ or sin²θ.

  扩展的勾股恒等式：sin²θ + cos²θ ≡ 1，1 + tan²θ ≡ sec²θ，1 + cot²θ ≡ cosec²θ。后两个可由第一个等式分别除以 cos²θ 或 sin²θ 得到。

• Compound angle formulae: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B. The sign for cos(A + B) flips.

  和角公式：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B。注意 cos(A + B) 中间的符号反转。

• Double angle identities: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A, tan 2A = 2 tan A/(1 − tan²A).

  二倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A，tan 2A = 2 tan A/(1 − tan²A)。

• R-alpha method: Expressing a sin θ ± b cos θ as R sin(θ ± α) or R cos(θ ∓ α), where R = √(a² + b²) and tan α = b/a. This is crucial for solving equations like a sin θ + b cos θ = c.

  R-α 法：将 a sin θ ± b cos θ 写成 R sin(θ ± α) 或 R cos(θ ∓ α) 的形式，其中 R = √(a² + b²)，tan α = b/a。这在求解形如 a sin θ + b cos θ = c 的方程时至关重要。

• Inverse trigonometric functions: arcsin x, arccos x and arctan x return angles in restricted domains (e.g. arcsin x gives values in [−π/2, π/2]). Their derivatives are standard results.

  反三角函数：arcsin x、arccos x 和 arctan x 返回限制值域内的角度（例如 arcsin x 的值域为 [−π/2, π/2]）。它们的导数都是标准结果。

4. Differentiation | 微分

• Derivative dy/dx: Measures the rate of change of y with respect to x, i.e. the gradient of the tangent to the curve. f'(x) = limₕ→₀ [f(x+h) − f(x)] / h.

  导数 dy/dx：度量 y 相对于 x 的变化率，即曲线切线的斜率。f'(x) = limₕ→₀ [f(x+h) − f(x)] / h。

• Chain rule: For y = f(g(x)), dy/dx = f'(g(x)) × g'(x). Think “differentiate the outside, multiply by derivative of the inside”.

  链式法则：对于 y = f(g(x))，dy/dx = f'(g(x)) × g'(x)。记忆技巧：“外层求导，乘以内层导数”。

• Product rule: For y = uv, dy/dx = u dv/dx + v du/dx. Helpful mnemonic: “first times derivative of second plus second times derivative of first”.

  积法则：对于 y = uv，dy/dx = u dv/dx + v du/dx。常用口诀：“前导后不导加后导前不导”。

• Quotient rule: For y = u/v, dy/dx = (v du/dx − u dv/dx) / v². Remember the denominator is the square of the bottom, and the sign is minus.

  商法则：对于 y = u/v，dy/dx = (v du/dx − u dv/dx) / v²。注意分母是分母的平方，且分子中间是减号。

• Implicit differentiation: Used when y cannot easily be written as a function of x. Differentiate every term with respect to x, treating y as a function and using the chain rule, giving terms like dy/dx. Then solve for dy/dx.

  隐函数微分：当 y 难以显式写成 x 的函数时使用。对每一项关于 x 求导，将 y 视作函数并利用链式法则产生 dy/dx，然后解出 dy/dx。

• Parametric differentiation: When x = f(t) and y = g(t), the derivative dy/dx = (dy/dt) / (dx/dt). The second derivative d²y/dx² is found using d/dx(dy/dx) = d/dt(dy/dx) / (dx/dt).

  参数微分：当 x = f(t) 且 y = g(t) 时，导数 dy/dx = (dy/dt) / (dx/dt)。二阶导数 d²y/dx² 通过 d/dx(dy/dx) = d/dt(dy/dx) / (dx/dt) 求得。

• Stationary points: Points where dy/dx = 0. Use the second derivative test: d²y/dx² > 0 gives a minimum, d²y/dx² < 0 gives a maximum, d²y/dx² = 0 could be a point of inflection — check gradient on either side.

  驻点：满足 dy/dx = 0 的点。用二阶导数检验：d²y/dx² > 0 为极小值点，d²y/dx² < 0 为极大值点，d²y/dx² = 0 可能是拐点——需检查两侧梯度的符号。

• Differentiating standard functions: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x. Memorise these building blocks—they feed into chain, product and quotient rules.

  标准函数的微分：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。牢记这些基础导数——它们是链式、积法则和商法则的基石。

5. Integration | 积分

• Indefinite integral ∫ f(x) dx: Finds all antiderivatives of f(x), differing by a constant. Always write “+ c”.

  不定积分 ∫ f(x) dx：求出 f(x) 的所有原函数，彼此相差一个常数。务必写上“+ c”。

• Definite integral ∫ₐᵇ f(x) dx: Gives the net area between the curve and the x-axis from x = a to x = b. Evaluated as F(b) − F(a), where F is an antiderivative.

  定积分 ∫ₐᵇ f(x) dx：给出曲线与 x 轴之间从 x = a 到 x = b 所围的净值面积。计算为 F(b) − F(a)，其中 F 是某个原函数。

• Reverse chain rule: Recognising an integral of the form ∫ f'(g(x)) g'(x) dx = f(g(x)) + c. For example, ∫ (2x) e^(x²) dx = e^(x²) + c.

  反向链式法则：识别形如 ∫ f'(g(x)) g'(x) dx = f(g(x)) + c 的积分。例如 ∫ (2x) e^(x²) dx = e^(x²) + c。

• Integration by substitution: Replace a part of the integrand with u = g(x), find du/dx, rewrite dx as du / (du/dx), change limits if definite, then integrate. Works as the full formal version of the reverse chain rule.

  换元积分法：令 u = g(x) 替换被积函数的一部分，求出 du/dx，将 dx 替换为 du / (du/dx)，若为定积分则改变积分限，再进行积分。它是反向链式法则的完整形式化版本。

• Integration by parts: ∫ u dv = uv − ∫ v du. Choose u according to LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to simplify the repeated integral.

  分部积分法：∫ u dv = uv − ∫ v du。根据 LIATE 法则选择 u（对数、反三角、代数、三角、指数），使剩下的积分更简单。

• Standard integrals: Beyond power rule, know ∫ 1/x dx = ln |x| + c, ∫ eˣ dx = eˣ + c,

  Published by TutorHao | Year 13 Mathematics Revision Series | aleveler.com

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