📚 WJEC Year 13 Maths: Key Terminology Quick Memorisation Guide | WJEC 13年级数学：核心术语速记指南
In WJEC Year 13 Mathematics, understanding and memorising key terminology is crucial for interpreting exam questions accurately and providing precise solutions. This guide covers essential terms from pure mathematics, mechanics, and statistics, pairing each English definition with its Chinese equivalent to aid bilingual comprehension and rapid revision.
在 WJEC 13 年级数学中，理解和记忆核心术语对于准确解读试题、提供精确解答至关重要。本指南涵盖纯数、力学和统计的核心词汇，每个术语均以英文定义配中文解释，助力双语理解和快速复习。
1. Core Differentiation | 微分核心术语
The derivative of a function f(x) with respect to x, written as f' ge or the gradient of the tangent at any point on the curve. It is defined as the limit: f'(x) = lim (h→0) [f(x+h) - f(x)] / h.
导数 f'(x) 或 dy/dx 表示函数相对于 x 的瞬时变化率或曲线上任意一点切线的斜率，其定义为极限： f'(x) = lim (h→0) [f(x+h) - f(x)] / h。
A 'stationary point' stationary point is determined by the second derivative: if d²y/dx² > 0 it is a local minimum; if d²y/dx² < 0 it is a local maximum.
“驻点”出现在 dy/dx = 0 处。驻点的类型由二阶导数判定：若 d²y/dx² > 0 则为局部极小值；若 d²y/dx² < 0 则为局部极大值。
A 'point of inflection' Not all points where d²y/dx² = 0 are points of inflection; a sign change must be verified.
“拐点”是曲线凹凸性发生改变的点。其特征是 d²y/dx² = 0 并且二阶导数在该点两侧变号。并非所有 d²y/dx² = 0 的点都是拐点，必须验证符号变化。
The 'chain rule' (du/dx). For example, if y = (2x+1)⁵, set u = 2x+1 to obtain dy/dx = 5u⁴ × 2 = 10(2x+1)⁴.
“链式法则”用于复合函数求导。若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。例如 y = (2x+1)⁵，令 u = 2x+1 可得 dy/dx = 5u⁴ × 2 = 10(2x+1)⁴。
2. Core Integration | 积分核心术语
An 'indefinite integral' antiderivatives of f(x) and always includes an arbitrary constant of integration, + C. Differentiating the result recovers f(x).
“不定积分” ∫ f(x) dx 表示 f(x) 所有原函数的集合，必须包含任意积分常数 + C。对其求导即可还原 f(x)。
A 'definite integral' ∫ab to x = b. It is evaluated using the Fundamental Theorem of Calculus: F(b) - F(a), where F'(x) = f(x).
“定积分” ∫ab f(x) dx 计算曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有向面积。利用微积分基本定理计算：F(b) - F(a)，其中 F'(x) = f(x)。
'Integration by substitution' technique that simplifies an integral by changing the variable. Let u = g(x), then du = g'(x) dx, and the integral is transformed into ∫ f(u) du. Don't forget to revert to the original variable or change limits for definite integrals.
“换元积分法”通过变量代换简化积分。令 u = g(x)，则 du = g'(x) dx，积分变为 ∫ f(u) du。对于定积分，切记将积分限也相应转换，或者换回原变量。
'Integration by parts' en integrating products of functions such as x sin x or ln x. The choice of u and dv is guided by the LIATE rule (Log, Inverse trig, Algebraic, Trig, Exponential).
“分部积分法”源自乘法求导法则，公式为 ∫ u dv = uv - ∫ v du。适用于乘积函数积分，如 x sin x 或 ln x。通常按 LIATE 顺序（对数、反三角、代数、三角、指数）选择 u。
A 'differential equation' involves an unknown function and its derivatives. The general solution contains an arbitrary constant; a particular solution is obtained by applying given boundary or initial conditions.
“微分方程”包含未知函数及其导数。通解含有任意常数；通过代入给定的边界条件或初始条件可求得特解。
3. Functions and Graphs | 函数与图像
The 'domain' of a function is the set of all possible input values (x-values) for which the function is defined. The 'range' is the set of all possible output values (y-values) that the function can produce.
函数的“定义域”是使得函数有意义的所有输入值（x 值）的集合。“值域”是函数所有可能的输出值（y 值）的集合。
An 'inverse function' ection of y = f(x) in the line y = x. The domain of f⁻¹ is the range of f, and vice versa.
“反函数” f⁻¹(x) 逆转 f 的作用，仅当 f 为一一映射时存在。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的反射。f⁻¹ 的定义域是 f 的值域，反之亦然。
A 'composite function' fg(x) or f(g(x)) applies one function to the result of another. The domain of fg is the set of x in the domain of g for which g(x) is in the domain of f.
“复合函数” fg(x) 或 f(g(x)) 将一个函数作用于另一个函数的结果。fg 的定义域是使得 g(x) 属于 f 的定义域的那些 x 值，且 x 在 g 的定义域内。
The 'modulus function' |x| ves the absolute value of x, defined as x if x ≥ 0 and -x if x < 0. Modulus equations like |x-2| = 3 lead to two branches: x-2 = 3 or x-2 = -3.
“模函数” |x| 给出 x 的绝对值，定义为若 x ≥ 0 则为 x，若 x < 0 则为 -x。形如 |x-2| = 3 的模方程需分两支：x-2 = 3 或 x-2 = -3。
s horizontally by -a; y = -f(x) reflects in the x-axis; y = af(x) stretches vertically by scale factor a.
图像变换包括平移、反射和伸缩。y = f(x) + a 竖直平移 a 个单位；y = f(x + a) 水平平移 -a 个单位；y = -f(x) 关于 x 轴反射；y = af(x) 竖直伸缩，伸缩因子为 a。
4. Sequences and Series | 数列与级数
An 'arithmetic sequence' has a common difference d between consecutive terms. The nth term is given by a + (n-1)d, and the sum of the first n terms is Sn = n/2 [2a + (n-1)d].
“等差数列”相邻两项有公差 d。第 n 项公式为 a + (n-1)d，前 n 项和 Sn = n/2 [2a + (n-1)d]。
A 'geometric sequence' has a common ratio r between consecutive terms. The nth term is arn-1. The sum of the first n terms is Sn = a(1 - rn)/(1 - r) for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 - r).
“等比数列”相邻两项有公比 r。第 n 项为 arn-1，前 n 项和 Sn = a(1 - rn)/(1 - r)（r ≠ 1）。当 |r| < 1 时，无穷项和存在：S∞ = a/(1 - r)。
The 'binomial expansion' generalises (a + b)n for rational n. For (1 + x)n where n is any rational, the expansion is 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ... . This is an infinite series, valid for |x| < 1.
“二项展开式”将 (a + b)n 推广到有理数指数 n。对于 (1 + x)n（n 为任意有理数），展开式为 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ... 。这是无穷级数，当 |x| < 1 时有效。
In a 'sequence', terms may be defined inductively using a recurrence relation such as un+1 = f(un r of a sequence as n → ∞, including whether it converges to a limit l satisfying l = f(l).
“数列”中，项可通过递推关系定义，如 un+1 = f(un)。需关注 n → ∞ 时数列的性态，包括是否收敛到满足 l = f(l) 的极限 l。
5. Trigonometry | 三角学
= 180°. Arc length of a sector: s = rθ; area of a sector: A = ½r²θ, where θ is in radians.
弧度是微积分中的核心角度度量。π 弧度 = 180°。扇形弧长 s = rθ；扇形面积 A = ½r²θ，其中 θ 采用弧度制。
'Small-angle approximations' θ²/2, tan θ ≈ θ. These are useful in deriving further results and solving limit problems.
“小角近似”在角度接近 0（弧度）时简化三角表达式：sin θ ≈ θ，cos θ ≈ 1 - θ²/2，tan θ ≈ θ。这在推导其他结果和极限问题中非常有用。
Key trigonometric identities include sin²θ + cos²θ ≡ 1, tan²θ + 1 ≡ sec²θ, and 1 + cot²θ ≡ und angle formulae are sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ formulae follow: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
基本三角恒等式包括 sin²θ + cos²θ ≡ 1，tan²θ + 1 ≡ sec²θ 以及 1 + cot²θ ≡ cosec²θ。和角公式为 sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B。倍角公式由此得出：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ。
and integrals must be memorised: d/dx(sec x) = sec x tan x, d/dx(cot x) = -cosec² x, etc.
三角函数倒数：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ。必须记忆它们的导数和积分，如 d/dx(sec x) = sec x tan x，d/dx(cot x) = -cosec² x 等。
6. Vectors | 向量
A 'position vector' gives the location of a point relative to the origin. In 2D, a point P(x,y) has position vector p = xi + yj; in 3D, p = xi + yj + zk.
“位置向量”表示点相对于原点的位置。二维中点 P(x,y) 的位置向量为 p = xi + yj；三维中为 p = xi + yj + zk。
The 'magnitude' or length of a vector v = ai + bj is |v| = √a² + b². A 'unit vector' has magnitude 1; any non-zero vector can be normalised by dividing by its magnitude.
向量 v = ai + bj 的“模”或长度为 |v| = √a² + b²。“单位向量”的模为 1；任何非零向量除以其模即可单位化。
The 'scalar (dot) product' of two vectors a and b is defined as a · b = |a||b| cos θ, where θ is the angle between them. In component form, a · b = a₁b₁ + a₂b₂ + a₃b₃. It is used to find angles and to test orthogonality (a · b = 0).
两向量 a 与 b 的“标量积（点积）”定义为 a · b = |a||b| cos θ，其中 θ 为夹角。分量形式下 a · b = a₁b₁ + a₂b₂ + a₃b₃。点积用于求夹角和判断垂直（a · b = 0）。
The 'vector equation of a line' passing through a point with position vector a and parallel to direction vector b is r = a + λb, where λ is a scalar parameter. This describes all points on the line.
直线的“向量方程”通过已知位置向量 a 且平行于方向向量 b 的点，写作 r = a + λb，其中 λ 为标量参数。该方程描述直线上所有点。
7. Mechanics: Kinematics | 力学：运动学
'Displacement' is the vector distance from a fixed origin to the particle's position; 'velocity' is the rate of change of displacement; 'acceleration' is the rate of change of velocity. In calculus, v = ds/dt and a = dv/dt = d²s/dt².
“位移”是从固定原点到质点位置的矢量距离；“速度”是位移的变化率；“加速度”是速度的变化率。微积分表示中，v = ds/dt，a = dv/dt = d²s/dt²。
The 'suvat' nd s = (u+v)t/2. Here u is initial velocity, v final velocity, s displacement, and t time.
“匀加速运动公式”适用于加速度 a 恒定的运动：v = u + at，s = ut + ½at²，v² = u² + 2as，s = (u+v)t/2。其中 u 为初速度，v 为末速度，s 为位移，t 为时间。
'Projectile motion' . Vertically, constant acceleration -g: y = u sin θ t - ½gt². The trajectory is parabolic.
“抛体运动”将水平和竖直分量独立处理。水平方向为匀速运动（忽略空气阻力）：x = u cos θ t。竖直方向以加速度 -g 运动：y = u sin θ t - ½gt²。其轨迹为抛物线。
mum height, and range are found by analysing the vertical motion, often setting y = 0 or vy = 0.
当质点从某高度或地面抛出时，飞行时间、最大高度和射程均通过分析竖直运动求得，通常令 y = 0 或 vy = 0。
In variable acceleration problems, calculus is essential: velocity is the integral of acceleration with respect to time, and displacement is the integral of velocity. Boundary conditions determine constants of integration.
在变加速问题中，微积分必不可少：速度是加速度对时间的积分，位移是速度的积分。初始条件或边界条件确定积分常数。
8. Mechanics: Forces and Newton's Laws | 力学：力与牛顿定律
Newton's first law: a body remains at rest or in uniform motion unless acted upon by an external resultant force. Newton's second law: F = ma, where F is the resultant force, m mass, and a acceleration. Newton's third law: for every action, there is an equal and opposite reaction.
牛顿第一定律：物体保持静止或匀速直线运动状态，除非受到外来合外力作用。牛顿第二定律：F = ma，其中 F 为合外力，m 为质量，a 为加速度。牛顿第三定律：每个作用力都有一个等大反向的反作用力。
'Equilibrium' occurs when the resultant force on a particle is zero. For a particle in equilibrium under coplanar forces, the vector sum of forces is zero, and sums of resolved components in perpendicular directions are zero.
“平衡”指作用在质点上的合外力为零。对于共面力系下处于平衡的质点，力的矢量和为零，且各正交方向上的分力代数和为零。
'Friction' opposes motion. The maximum frictional force between two surfaces is Fmax = μR, where μ is the coefficient of friction and R is the
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