📚 CCEA Year 13 Mathematics: Vocabulary Memorisation Guide | CCEA Year 13 数学词汇术语速记指南
Mastering mathematical vocabulary is the first step toward exam success in CCEA Year 13 Mathematics. This guide organises essential terminology from Pure Mathematics, Mechanics, and Statistics into clear bilingual tables, complete with memory tips and conceptual links. Whether you are preparing for AS Unit 1 (Pure Mathematics) or AS Unit 2 (Applied Mathematics), having these terms at your fingertips will sharpen your problem-solving accuracy and boost your confidence in written explanations.
掌握数学词汇是CCEA Year 13数学考试成功的第一步。本指南将纯数学、力学和统计学的核心术语整理成清晰的双语对照表，并配有记忆技巧和概念关联。无论你正在备考AS Unit 1（纯数学）还是AS Unit 2（应用数学），熟记这些术语都将提升你的解题准确性，并增强你在文字解释题中的信心。
1. Foundational Pure Mathematics Terms | 纯数学基础术语
These core terms underpin all of Pure Mathematics. Being able to define a polynomial, distinguish a coefficient from a constant, and identify the discriminant sets the stage for more advanced topics. Many exam questions begin by testing these fundamental definitions.
这些核心术语是所有纯数学内容的基础。能够定义多项式、区分系数与常数、识别判别式，将为更高阶的主题奠定基础。许多考试题目都是从测试这些基本定义开始的。
| English Term | Definition | 中文术语 | 释义 |
|---|---|---|---|
| Polynomial | An expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents | 多项式 | 由变量和系数组成，仅涉及加、减、乘和非负整数指数的表达式 |
| Coefficient | A numerical or constant factor multiplying a variable in an algebraic term | 系数 | 代数项中乘以变量的数值或常数因子 |
| Constant | A fixed value that does not change | 常数 | 不改变的固定值 |
| Discriminant | The expression b² - 4ac under the square root in the quadratic formula; determines the nature of roots | 判别式 | 二次公式中平方根下的表达式 b² - 4ac；决定根的性质 |
| Surd | An irrational number expressed using a root symbol that cannot be simplified to a rational number | 根式 | 使用根号表示且无法化简为有理数的无理数 |
| Asymptote | A line that a curve approaches arbitrarily closely but never touches | 渐近线 | 曲线无限接近但永不相交的直线 |
| Function | A relation where each input has exactly one output | 函数 | 每个输入恰好对应一个输出的关系 |
| Domain | The set of all possible input values for a function | 定义域 | 函数所有可能输入值的集合 |
| Range | The set of all possible output values of a function | 值域 | 函数所有可能输出值的集合 |
| Inverse Function | A function that reverses the effect of the original function; denoted f⁻¹(x) | 反函数 | 逆转原函数效果的函数；记作 f⁻¹(x) |
Memory tip: Think of the discriminant as the 'decision-maker' of a quadratic equation — positive means two distinct real roots, zero means one repeated root, negative means no real roots. The formula b² - 4ac sits right under the square root sign in the quadratic formula.
记忆技巧：把判别式想象成二次方程的"决策者"——正数意味着两个不同的实根，零意味着一个重复根，负数意味着没有实根。公式 b² - 4ac 正好位于二次公式的平方根符号下方。
2. Differentiation Terminology | 微分术语
Differentiation forms the backbone of calculus in Year 13. CCEA expects you to differentiate polynomials, use the chain/product/quotient rules, and apply derivatives to find tangents, normals, and stationary points on curves.
微分是Year 13微积分的核心。CCEA要求你能够对多项式进行微分，使用链式法则、乘积法则和商法则，并应用导数求解曲线的切线、法线和驻点。
| English Term | Definition | 中文术语 | 释义 |
|---|---|---|---|
| Derivative | The instantaneous rate of change of a function; denoted dy/dx or f'(x) | 导数 | 函数的瞬时变化率；记作 dy/dx 或 f'(x) |
| Chain Rule | A rule for differentiating composite functions: dy/dx = (dy/du) x (du/dx) | 链式法则 | 复合函数求导法则：dy/dx = (dy/du) x (du/dx) |
| Product Rule | If y = uv, then dy/dx = u(dv/dx) + v(du/dx) | 乘积法则 | 若 y = uv，则 dy/dx = u(dv/dx) + v(du/dx) |
| Quotient Rule | If y = u/v, then dy/dx = [v(du/dx) - u(dv/dx)] / v² | 商法则 | 若 y = u/v，则 dy/dx = [v(du/dx) - u(dv/dx)] / v² |
| Stationary Point | where the gradient is zero (dy/dx = 0); includes maxima, minima, and points of inflection | 驻点 | 曲线上梯度为零的点（dy/dx = 0）；包括极大值点、极小值点和拐点 |
| Second Derivative | The derivative of the derivative; denoted d²y/dx² or f''(x); used to determine concavity | 二阶导数 | 导数的导数；记作 d²y/dx² 或 f''(x)；用于判断凹凸性 |
| Tangent | A straight line that touches a curve at exactly one point without crossing it | 切线 | 与曲线恰好在一个点接触且不穿过曲线的直线 |
| Normal | A line perpendicular to the tangent at the point of contact | 法线 | 在切点处与切线垂直的直线 |
| Increasing Function | A function where f'(x) > 0 over an interval | 增函数 | 在某个区间内 f'(x) > 0 的函数 |
| Decreasing Function | A function where f'(x) < 0 over an interval | 减函数 | 在某个区间内 f'(x) < 0 的函数 |
Memory tip: For the quotient rule, remember the rhyme: 'low d-high minus high d-low, draw the line and square below.' Here 'low' is the denominator v, 'high' is the numerator u, and 'd' means derivative.
记忆技巧：对于商法则，记住这句口诀："分母乘分子的导减去分子乘分母的导，下面分母要平方。"这里"分母"是 v，"分子"是 u，"导"指的是导数。
3. Integration Terminology | 积分术语
Integration reverses differentiation and serves two key purposes in Year 13: finding antiderivatives (indefinite integrals) and calculating the area under a curve (definite integrals). CCEA also tests integration as the limit of a sum in applied contexts.
积分是微分的逆运算，在Year 13中有两个关键用途：求原函数（不定积分）和计算曲线下方面积（定积分）。CCEA还会在应用情境中考察积分作为求和极限的概念。
| English Term | Definition | 中文术语 | 释义 |
|---|---|---|---|
| Integral | The reverse operation of differentiation; represented by the symbol ∫ | 积分 | 微分的逆运算；用符号 ∫ 表示 |
| Indefinite Integral | An integral without limits; includes an arbitrary constant of integration +C | 不定积分 | 没有上下限的积分；包含任意积分常数 +C |
| Definite Integral | An integral with upper and lower limits; yields a numerical value representing area | 定积分 | 有上下限的积分；得出表示面积的数值 |
| Constant of Integration | The arbitrary constant '+C' added when finding an indefinite integral | 积分常数 | 求不定积分时添加的任意常数 '+C' |
| Area Under a Curve | The region bounded by a curve, the x-axis, and two vertical lines; calculated using a definite integral | 曲线下方面积 | 由曲线、x轴和两条垂直线围成的区域；使用定积分计算 |
| Fundamental Theorem of Calculus | The theorem linking differentiation and integration: if F'(x) = f(x), then ∫ f(x)dx = F(b) - F(a) over [a,b] | 微积分基本定理 | 联系微分与积分的定理：若 F'(x) = f(x)，则在 [a,b] 上 ∫ f(x)dx = F(b) - F(a) |
| Trapezium Rule | A numerical method for approximating a definite integral using trapezoids | 梯形法则 | 使用梯形近似计算定积分的数值方法 |
Memory tip: Never forget '+C' in an indefinite integral — CCEA examiners routinely deduct marks for this omission. Think of '+C' as the 'family identifier' since integration produces a whole family of curves differing only by a constant vertical shift.
记忆技巧：在不定积分中永远不要忘记 '+C' ——CCEA考官经常因遗漏此项而扣分。把 '+C' 想象成"家族标识"，因为积分会产生一整族曲线，它们之间仅相差一个常数垂直平移。
4. Trigonometric Functions | 三角函数
Trigonometry in Year 13 extends well beyond right-angled triangles to include the unit circle, radian measure, trigonometric identities, and the graphs of sine, cosine, and tangent. CCEA places strong emphasis on solving trigonometric equations within specified intervals.
Year 13的三角学远远超出直角三角形的范畴，涵盖了单位圆、弧度制、三角恒等式以及正弦、余弦和正切函数的图像。CCEA非常重视在指定区间内求解三角方程。
| English Term | Definition | 中文术语 | 释义 |
|---|---|---|---|
| Radian | A unit of angle measure where 2π radians equals 360°; the standard unit in calculus | 弧度 | 角度度量单位，2π 弧度等于360°；微积分中的标准单位 |
| Unit Circle | A circle of radius 1 centred at the origin; used to define trigonometric functions for all angles | 单位圆 | 以原点为圆心、半径为1的圆；用于定义所有角度的三角函数 |
| Period | The length of one complete cycle of a periodic function; for sin x and cos x the period is 2π | 周期 | 周期函数一个完整循环的长度；sin x 和 cos x 的周期为 2π |
| Amplitude | Half the distance between the maximum and minimum values of a periodic function | 振幅 | 周期函数最大值与最小值之间距离的一半 |
| Double-Angle Formula | Identities such as sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ - sin²θ | 倍角公式 | 如 sin 2θ = 2 sin θ cos θ 和 cos 2θ = cos²θ - sin²θ 等恒等式 |
| CAST Diagram | A mnemonic diagram showing which trigonometric ratios are positive in each quadrant | CAST图 | 一种助记图，显示各象限中哪些三角比为正 |
| Trigonometric Identity | An equation involving trigonometric functions that holds true for all values of the variable | 三角恒等式 | 对所有变量值都成立的包含三角函数的等式 |
Key identity: sin²θ + cos²θ = 1
关键恒等式：sin²θ + cos²θ = 1
5. Exponentials and Logarithms | 指数与对数
The exponential function eˣ and the natural logarithm ln x are inseparable companions in Year 13 calculus. CCEA examines their graphs, their differentiation and integration rules, and their role in modelling growth and decay problems.
指数函数 eˣ 和自然对数 ln x 是Year 13微积分中形影不离的伙伴。CCEA会考察它们的图像、微分和积分法则，以及它们在增长与衰减问题建模中的作用。
| English Term | Definition | 中文术语 | 释义 |
|---|---|---|---|
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