Pre-U Cambridge Mathematics: Vocabulary & Terminology Quick-Reference Guide | Pre-U Cambridge 数学：词汇术语速记指南

📚 Pre-U Cambridge Mathematics: Vocabulary & Terminology Quick-Reference Guide | Pre-U Cambridge 数学：词汇术语速记指南

Mastering the precise terminology of Pre-U Cambridge Mathematics is essential for interpreting exam questions correctly and constructing rigorous arguments. This guide provides key terms across algebra, calculus, vectors, probability, and mechanics, with memory aids to reinforce understanding. Each section pairs English definitions with Chinese translations and offers quick-reference tables where appropriate.

掌握 Pre-U Cambridge 数学的精确术语对于正确理解考题和构建严谨论证至关重要。本指南涵盖代数、微积分、向量、概率和力学等核心词汇，并提供记忆辅助以加深理解。每个部分均将英文定义与中文翻译配对，并在适当处提供速查表格。

1. Foundational Algebraic Terms | 基础代数术语

A polynomial is an expression built from variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents. Memory aid: ‘poly-‘ means many, and ‘-nomial’ comes from the Latin word for ‘name’ or ‘term’, so it is a many-termed expression.

多项式是由变量和系数仅通过加法、减法、乘法和非负整数指数构成的表达式。记忆：’poly-‘ 意为“多”，’-nomial’ 源于拉丁语“名称/项”，因此表示多项表达式。

The degree of a polynomial is the highest power of the variable. For example, in 5x³ + 2x − 1 the degree is 3. Think of degree as the ‘highest level’ of the variable.

多项式的次数指的是变量的最高指数。例如在 5x³ + 2x − 1 中，次数为 3。可以将次数想象成变量的“最高级别”。

A coefficient is the numerical factor in a term. In 7x² the coefficient is 7; in −4x the coefficient is −4. Remember: coefficient ‘co-operates’ with the variable.

系数是项中的数值因子。在 7x² 中系数为 7，在 −4x 中系数为 −4。记住：系数与变量“协同工作”。

Term	中文	Quick Memory Aid
Constant	常数	Stands alone, no variable
Variable	变量	Varies its value
Binomial	二项式	Bi- = two terms

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2. Functions and Their Properties | 函数及其性质

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Memory: domain = ‘do’ – the x-values you are allowed to put in; range = ‘reach’ – the resulting y-values.

函数的定义域是所有可能输入值（x值）的集合，而值域是所有可能输出值（y值）的集合。记忆：定义域是你“放入”的x值，值域是函数“达到”的y值。

An injective function (one-to-one) maps distinct inputs to distinct outputs; a horizontal line cuts at most once. Surjective (onto) means every element in the codomain is hit. A bijection is both injective and surjective, guaranteeing an inverse function.

单射函数（一一对应）将不同的输入映射到不同的输出；水平线最多与图像相交一次。满射意味着陪域中的每个元素都被取到。双射既是单射又是满射，从而保证存在反函数。

The inverse function f⁻¹(x) reverses the effect of f(x). Its graph is a reflection in the line y = x. Remember: inverse switches input and output.

反函数 f⁻¹(x) 逆转 f(x) 的作用。其图像关于直线 y = x 对称。记忆：反函数交换输入和输出。

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3. Differential Calculus Vocabulary | 微分学术语

The derivative f'(x) or dy/dx gives the gradient of the tangent to a curve at a point. It is defined as the limit of the difference quotient: dy/dx = limₕ→₀ [f(x+h) − f(x)] / h. Recall that ‘derivative’ comes from ‘derive’ – it is derived from the original function.

导数 f'(x) 或 dy/dx 给出曲线上某点切线的斜率。它定义为差商的极限：dy/dx = limₕ→₀ [f(x+h) − f(x)] / h。记住“导数”源自“推导” – 它从原函数推导而来。

A stationary point occurs where dy/dx = 0. It can be a maximum, minimum, or point of inflection. Use the second derivative test: if d²y/dx² > 0 it is a minimum; if d²y/dx² < 0 it is a maximum.

驻点出现在 dy/dx = 0 处。它可以是极大值、极小值或拐点。使用二阶导数判别：若 d²y/dx² > 0 则为极小；若 d²y/dx² < 0 则为极大。

The chain rule differentiates composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). Think of peeling an onion: differentiate the outer function, keep the inner, then multiply by derivative of the inner.

链式法则用于复合函数求导：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。想象剥洋葱：先对外层函数求导，保留内层，再乘以内层函数的导数。

d/dx (xⁿ) = n xⁿ⁻¹

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4. Integral Calculus Vocabulary | 积分学术语

Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx gives a family of functions plus the constant of integration C. Visualise integration as ‘anti-derivative’ – it undoes the derivative.

积分是微分的逆运算。不定积分 ∫ f(x) dx 给出一个函数族加上积分常数 C。将积分想象为“反导数” – 它撤销了导数的作用。

The definite integral ∫ₐᵇ f(x) dx calculates the net area between the curve and the x-axis from x = a to x = b. The fundamental theorem of calculus links differentiation and integration: if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).

定积分 ∫ₐᵇ f(x) dx 计算从 x = a 到 x = b 之间曲线与 x 轴之间的净面积。微积分基本定理将微分与积分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。

Integration by substitution simplifies an integral by changing the variable, while integration by parts uses the formula ∫ u dv = uv − ∫ v du. Think of parts as ‘undoing the product rule’.

换元积分法通过变量替换简化积分，而分部积分法使用公式 ∫ u dv = uv − ∫ v du。可将分部积分视为“取消乘积法则”。

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5. Limits and Continuity | 极限与连续性

The limit of a function f(x) as x approaches a is the value that f(x) gets arbitrarily close to. We write limₓ→ₐ f(x) = L. A function is continuous at a if limₓ→ₐ f(x) = f(a). Remember: continuity means no holes or jumps.

函数 f(x) 当 x 趋近 a 时的极限是 f(x) 无限接近的值，记作 limₓ→ₐ f(x) = L。如果 limₓ→ₐ f(x) = f(a)，则函数在 a 处连续。记忆：连续意味着没有洞或跳跃。

A removable discontinuity occurs when a limit exists but the function is not defined or has a different value at the point. A jump discontinuity has left- and right-hand limits that differ. Use L’Hôpital’s rule to evaluate indeterminate forms like 0/0.

可去间断点出现在极限存在但函数在该点无定义或取不同值的情况。跳跃间断点的左右极限不同。使用洛必达法则可计算如 0/0 的不定式极限。

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6. Sequences, Series, and Summation | 数列、级数与求和

An arithmetic sequence has a common difference d: a, a+d, a+2d, … . A geometric sequence has a common ratio r: a, ar, ar², … . Memory: Arithmetic → add Difference; Geometric → multiply Ratio.

等差数列有公差 d：a, a+d, a+2d, … 。等比数列有公比 r：a, ar, ar², … 。记忆：等差加差，等比乘比。

The sum of the first n terms of an arithmetic series is Sₙ = n/2 [2a + (n−1)d]; for a geometric series Sₙ = a(1−rⁿ)/(1−r) when r ≠ 1. An infinite geometric series converges if |r| < 1, with sum S∞ = a/(1−r).

等差数列前 n 项和为 Sₙ = n/2 [2a + (n−1)d]；等比数列和 Sₙ = a(1−rⁿ)/(1−r)（r ≠ 1）。当 |r| < 1 时，无穷等比级数收敛，和为 S∞ = a/(1−r)。

Σₖ₌₁ⁿ k = n(n+1)/2

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7. Trigonometry and Circular Functions | 三角学与圆函数

Angles are often measured in radians; 180° = π rad. The primary trigonometric functions are sine, cosine, and tangent. Their reciprocal functions: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. Use SOH CAH TOA for right triangles.

角常用弧度度量；180° = π rad。基本三角函数为正弦、余弦和正切。它们的倒数函数：cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ。直角三角形记忆口诀：SOH CAH TOA。

Key identities: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ. Compound angle formulas: sin(A±B) = sin A cos B ± cos A sin B. Double-angle: sin 2θ = 2 sin θ cos θ.

核心恒等式：sin²θ + cos²θ = 1, tan θ = sin θ / cos θ。复合角公式：sin(A±B) = sin A cos B ± cos A sin B。倍角公式：sin 2θ = 2 sin θ cos θ。

The inverse trigonometric functions arcsin, arccos, arctan return the angle whose sine/cosine/tangent is a given value, with restricted domains to ensure they are functions.

反三角函数 arcsin, arccos, arctan 返回给定正弦/余弦/正切值所对应的角，其定义域受限以确保它们是函数。

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8. Exponentials and Logarithms | 指数与对数

The exponential function eˣ is its own derivative. The natural logarithm ln x is its inverse: ln(eˣ) = x, e^(ln x) = x. e ≈ 2.718 can be memorised as ‘Andrew Jackson’s expiry year’ (2.71828…).

指数函数 eˣ 的导数就是它自身。自然对数 ln x 是其反函数：ln(eˣ) = x, e^(ln x) = x。e ≈ 2.718 可记为 ‘Andrew Jackson’s expiry year’ (2.71828…)。

Logarithm laws: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln(aᵏ) = k ln a. These convert multiplication into addition, which is the original purpose of logs.

对数运算法则：ln(ab) = ln a + ln b；ln(a/b) = ln a − ln b；ln(aᵏ) = k ln a。这些法则将乘法化为加法，这正是对数的初衷。

Solving equations: to bring down an exponent, take logs of both sides, e.g. 2ˣ = 10 → x ln 2 = ln 10.

解方程：为将指数移下来，两边取对数，例如 2ˣ = 10 → x ln 2 = ln 10。

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9. Vectors in Pure and Applied Contexts | 向量在纯数与应用题中

A vector has both magnitude and direction, represented by a bold letter or an arrow. A scalar is just a number. The magnitude of v = ai + bj is |v| = √(a² + b²). A unit vector has magnitude 1.

向量既有大小又有方向，用粗体或箭头表示。标量只是一个数。向量 v = ai + bj 的模为 |v| = √(a² + b²)。

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