📚 Year 13 CCEA Mathematics: Quick Memorisation Guide to Key Terms | Year 13 CCEA 数学：词汇术语速记指南
Mastering the terminology in CCEA Year 13 Mathematics is half the battle — this guide groups essential AS-level terms from Pure Mathematics, Statistics, and Mechanics, providing quick bilingual explanations and memory hooks to boost your revision.
掌握 CCEA Year 13 数学的术语是成功的一半——本指南将 AS 阶段纯数、统计和力学必备词汇分类整理，提供中英双语快速释义和记忆点，助你高效复习。
1. Algebra and Functions | 代数与函数
These terms underpin everything from solving equations to sketching curves. A clear picture of domain, range, and function types will save you from common pitfalls.
这些术语是解方程和画图像的基础。清晰理解定义域、值域以及函数类型能帮你避开常见陷阱。
Domain: The set of all possible input values (x-values) for which a function is defined. Always check for division by zero or square roots of negatives.
定义域： 函数有定义的所有可能输入值（x 值）的集合。务必检查是否有除以零或负数开平方。
Range: The set of all possible output values (y-values) produced by a function over its domain. Think ‘what comes out’.
值域： 函数在其定义域内所有可能的输出值（y 值）的集合。想象成“会出来什么”。
One-to-one function: Each x-value maps to a unique y-value, and no two different x-values give the same y-value. Passes the horizontal line test.
一一映射函数： 每个 x 值对应唯一的 y 值，并且没有两个不同的 x 值产生相同的 y 值。通过水平线检验。
Inverse function f⁻¹: Reverses the effect of f; its domain is the range of f and vice versa. Remember to swap x and y then solve.
反函数 f⁻¹： 逆转 f 的作用；其定义域是 f 的值域，反之亦然。牢记交换 x 与 y 然后求解。
Composite function f(g(x)): Apply g first, then apply f to the result. The order matters — think ‘inside out’.
复合函数 f(g(x))： 先作用 g ，然后对结果作用 f。顺序很重要——按“由内向外”考虑。
Polynomial: An expression like anxn + … + a1x + a0, where n is a non-negative integer. The degree is the highest power.
多项式： 形如 anxn + … + a1x + a0 的表达式，其中 n 是非负整数。最高次幂为多项式的次数。
Remainder Theorem: When a polynomial f(x) is divided by (x – a), the remainder is f(a). A fast way to evaluate remainders without long division.
余数定理： 多项式 f(x) 除以 (x – a) 时，余数为 f(a)。无需长除法即可快速计算余数。
Factor Theorem: If f(a) = 0, then (x – a) is a factor of f(x). The reverse of the Remainder Theorem — use it to factorise cubics.
因式定理： 若 f(a) = 0，则 (x – a) 是 f(x) 的因式。余数定理的逆用——可用于三次多项式因式分解。
2. Coordinate Geometry | 坐标几何
The language of lines and circles on the Cartesian plane. Remember that gradients and perpendicularity unlock many exam questions.
笛卡尔平面上直线与圆的语言。记住，斜率和垂直关系是解答许多试题的关键。
Gradient (m): The steepness of a line, m = (y2 – y1) / (x2 – x1). Rise over run.
斜率 (m)： 直线的倾斜程度，m = (y2 – y1) / (x2 – x1)。纵差除以横差。
Midpoint: ((x1+x2)/2, (y1+y2)/2). The average of the coordinates.
中点： ((x1+x2)/2, (y1+y2)/2)。坐标的平均值。
Perpendicular lines: Two lines are perpendicular if the product of their gradients is -1, i.e. m1m2 = -1.
垂直直线： 若两直线斜率之积为 -1，即 m1m2 = -1，则它们互相垂直。
Equation of a circle: (x – a)² + (y – b)² = r², with centre (a, b) and radius r.
圆的方程： (x – a)² + (y – b)² = r²，圆心为 (a, b)，半径为 r。
Tangent to a circle: A line that touches the circle at exactly one point. The radius to the point of contact is perpendicular to the tangent.
圆的切线： 与圆恰好一个公共点的直线。切点处的半径与切线垂直。
Discriminant for intersection: Substituting line into circle gives a quadratic; the discriminant Δ = b² – 4ac tells you if the line cuts, touches, or misses the circle.
交点的判别式： 将直线代入圆得二次方程；判别式 Δ = b² – 4ac 可判断直线与圆相交、相切还是相离。
3. Trigonometry | 三角学
Radian measure and circular functions are central to calculus. Visualising the unit circle makes identities much easier to remember.
弧度制与圆函数是微积分的核心。想象单位圆会让恒等式好记很多。
Radian: The angle subtended by an arc equal in length to the radius. π radians = 180°. Think ‘arc over radius’.
弧度： 弧长等于半径的圆弧所对的圆心角。π 弧度 = 180°。联想“弧长除以半径”。
Unit circle: A circle of radius 1 centred at the origin. sin θ = y-coordinate, cos θ = x-coordinate.
单位圆： 以原点为心、半径为 1 的圆。sin θ 为 y 坐标，cos θ 为 x 坐标。
Pythagorean identity: sin²θ + cos²θ = 1. Derived directly from the unit circle equation.
毕达哥拉斯恒等式： sin²θ + cos²θ = 1。直接由单位圆方程推导而来。
Periodic function: A function that repeats its values at regular intervals. sin θ and cos θ have period 2π; tan θ has period π.
周期函数： 函数值每隔一定区间重复出现的函数。sin θ 与 cos θ 的周期为 2π；tan θ 的周期为 π。
CAST diagram: A memory aid showing which trig ratios are positive in each quadrant. Starting in the 4th quadrant ‘C’ for Cos, then All, Sin, Tan moving anticlockwise.
CAST 图： 帮助记忆各象限中哪些三角函数为正值的示意图。逆时针方向依次为 Cos、All、Sin、Tan。
Trigonometric equations: Use principal values and symmetry to find all solutions in a given interval. Always sketch the graph to avoid missing solutions.
三角方程： 利用主值和对称性求给定区间内的所有解。始终画出草图以避免漏解。
4. Exponentials and Logarithms | 指数与对数
Exponential growth and log rules appear throughout modelling and calculus. The natural exponential ex is your best friend in differentiation.
指数增长和对数法则贯穿建模和微积分。自然指数 ex 是微分中最好的朋友。
Exponential function: f(x) = ax where a > 0 and a ≠ 1. The graph always passes through (0, 1).
指数函数： f(x) = ax，其中 a > 0 且 a ≠ 1。图像总经过点 (0, 1)。
Natural exponential ex: The exponential function with base e ≈ 2.718. Its derivative is itself, d/dx(ex) = ex.
自然指数 ex： 底数为 e ≈ 2.718 的指数函数。其导数等于它本身，d/dx(ex) = ex。
Logarithm: The inverse of an exponential. If ac = b, then logab = c. ‘Log’ asks for the power.
对数： 指数的逆运算。若 ac = b，则 logab = c。“对数”即问幂指数。
Natural log ln x: Log to base e, ln x = logex. Its derivative is 1/x, making it indispensable in integration.
自然对数 ln x： 以 e 为底的对数，ln x = logex。其导数为 1/x，在积分中不可或缺。
Laws of logs: log(xy) = log x + log y; log(x/y) = log x – log y; log xn = n log x. Combine or separate products, quotients, and powers.
对数法则： log(xy) = log x + log y；log(x/y) = log x – log y；log xn = n log x。可将乘、除、幂进行拆分或合并。
Change of base: logab = logcb / logca. Handy when you only have ln or log10 on your calculator.
换底公式： logab = logcb / logca。当计算器仅有 ln 或 log10 时非常方便。
5. Differentiation | 微分
Differentiation gives the instantaneous rate of change. Understanding the gradient function and its sign is vital for curve sketching and optimisation.
微分给出瞬时变化率。理解导数函数及其正负对曲线描绘与最优化至关重要。
Derivative f'(x): The limit of the difference quotient as h → 0: f'(x) = limh→0 [f(x+h) – f(x)] / h.
导数 f'(x)： 当 h → 0 时差商的极限：f'(x) = limh→0 [f(x+h) – f(x)] / h。
Power rule: If f(x) = xn, then f'(x) = nxn-1. Multiply by the power then drop the power by one.
幂法则： 若 f(x) = xn，则 f'(x) = nxn-1。乘以指数再将指数减一。
Stationary point: Where f'(x) = 0. The tangent is horizontal. Could be a maximum, minimum, or point of inflection.
驻点： f'(x) = 0 的点。切线水平。可能是极大值点、极小值点或拐点。
Second derivative: f”(x) gives the rate of change of the gradient. f”(x) > 0 implies the curve is convex (∪); f”(x) < 0 concave (∩).
二阶导数： f”(x) 表示斜率的变化率。f”(x) > 0 曲线下凸 (∪)；f”(x) < 0 曲线上凸 (∩)。
Chain rule: If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Differentiate the outer function, then multiply by derivative of the inner.
链式法则： 若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。先对外层函数求导，再乘以内层函数的导数。
Product rule: If y = uv, then dy/dx = u’v + uv’. Remember ‘first differentiate second leave, plus first leave second differentiate’.
积法则： 若 y = uv，则 dy/dx = u’v + uv’。记忆口诀“前导后不导，加前不导后导”。
Quotient rule: If y = u/v, then dy/dx = (vu’ – uv’) / v². The order matters — start with the denominator.
商法则： 若 y = u/v，则 dy/dx = (vu’ – uv’) / v²。顺序很重要——从分母开始。
6. Integration | 积分
Integration is the reverse of differentiation. It finds areas under curves and recovers original quantities from rates.
积分是微分的逆运算。用于求曲线下的面积，以及由变化率恢复原函数。
Indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x). Always add the constant of integration + C.
不定积分： ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。永远记得加上积分常数 + C。
Power rule for integration: ∫ xn dx = xn+1/(n+1) + C, provided n ≠ -1. Increase the power by one then divide by the new power.
积分幂法则： ∫ xn dx = xn+1/(n+1) + C，当 n ≠ -1。把指数加一，再除以新指数。
Definite integral:Published by TutorHao | Year 13 Mathematics Revision Series | aleveler.com
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