📚 Year 13 OCR Maths: Key Terminology Quick-Memorisation Guide | Year 13 OCR 数学：词汇术语速记指南
Mastering the precise vocabulary of A Level Mathematics is just as vital as solving equations. This guide targets Year 13 OCR terminology across pure maths, mechanics and statistics, pairing each term with a memory hook so you can retrieve definitions and theorems confidently in the exam room.
掌握 A Level 数学的精确词汇与解方程同样重要。本指南聚焦 Year 13 OCR 纯数、力学和统计中的核心术语，为每个术语搭配记忆钩子，帮助你在考场上自信地回忆定义和定理。
1. Differentiation & Integration Hacks | 微分与积分速记窍门
Derivative from first principles: f'(x) = lim (h→0) [f(x+h) – f(x)] / h. Think of a secant line shrinking until its slope matches the tangent. The mnemonic: ‘Limit of the difference quotient reveals the local gradient.’
第一原理求导： f'(x) = lim (h→0) [f(x+h) – f(x)] / h。想像一条割线不断收缩，直到其斜率与切线一致。口诀：差商的极限揭示局部梯度。
Chain rule: dy/dx = dy/du × du/dx. Picture a chain of derivatives where the ‘du’ terms cancel as if they were fractions. Remind yourself: ‘Outside derivative times inside derivative, link by link.’
链式法则： dy/dx = dy/du × du/dx。把导数链想像成分数，中间的 du 似乎可以约掉。提示自己：「外层导数乘以内层导数，一环扣一环。」
Integration by parts: ∫ u dv = uv – ∫ v du. Use the memory trick ‘Ultraviolet Voodoo’ or simply remember that the product uv is corrected by subtracting the integral of v du. Always choose u that simplifies when differentiated.
分部积分法： ∫ u dv = uv – ∫ v du。记忆口诀「UV 减去 VDU」，或理解成先猜乘积 uv，再用减去的积分修正。优先选择求导后变得更简单的函数作为 u。
Improper integrals: These involve infinite limits or discontinuous integrands. Replace the infinite bound by a variable b and take the limit as b→∞. A quick check: if the limit exists and is finite, the integral converges.
反常积分： 涉及无穷限或被积函数不连续。用变量 b 代替无穷限，再取 b→∞ 时的极限。快速判断：若极限存在且有限，则积分收敛。
2. Trigonometry & Radian Vocabulary | 三角函数与弧度词汇
Radian measure: 1 rad ≈ 57.3°, π rad = 180°. To convert, multiply by π/180 or 180/π. The mnemonic: ‘Radians give a pure number for arc length — no units confusion when differentiating.’
弧度制： 1 弧度 ≈ 57.3°，π rad = 180°。换算时乘以 π/180 或 180/π。记忆要点：弧度让弧长变成纯数，求导时不再有单位混乱。
Pythagorean identities: sin²θ + cos²θ ≡ 1, and the derived forms 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. Memorise the second one by thinking ‘tan sitting on a sec’ and the third as ‘cot on a cosec’.
毕达哥拉斯恒等式： sin²θ + cos²θ ≡ 1，以及变形 1 + tan²θ ≡ sec²θ、1 + cot²θ ≡ cosec²θ。用联想记忆：tan 坐在 sec 上，cot 坐在 cosec 上。
Double angle formulas: sin2θ = 2 sinθ cosθ, cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ. The phrase ‘Sine two theta is twice the product’ helps recall sin2θ, while cos2θ is ‘cosine squared minus sine squared’.
二倍角公式： sin2θ = 2 sinθ cosθ，cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。口诀「正弦二倍角，两倍乘积」记 sin2θ；cos2θ 记「余弦方减正弦方」。
R–form expressions: a sinθ + b cosθ = R sin(θ + α), where R = √(a² + b²) and tan α = b/a. Picture a right triangle with legs a and b, hypotenuse R; α is the angle opposite b. This is your ‘amplitude-phase’ form.
R 公式： a sinθ + b cosθ = R sin(θ + α)，其中 R = √(a² + b²)，tan α = b/a。想像一个直角边为 a 和 b、斜边为 R 的三角形，α 是 b 的对角。这就是「幅值-相位」形式。
3. Vectors & 3D Space | 向量与三维空间
Unit vectors & position vectors: i, j, k denote unit vectors along the x, y, z axes. A position vector r = xi + yj + zk locates a point. Memorise: ‘i, j, k build the coordinate skeleton.’
单位向量与位置向量： i, j, k 分别表示沿 x, y, z 轴的单位向量。位置向量 r = xi + yj + zk 定位一个点。记忆：「i, j, k 搭建出坐标骨架。」
Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. Use the word ‘DOT = cos’: the dot product measures how aligned two vectors are. For perpendicular vectors, a·b = 0.
点积（标量积）： a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ。口诀「点积得余弦」，点积衡量两向量对齐的程度。若垂直，则 a·b = 0。
Angle between vectors: cos θ = (a·b) / (|a||b|). This rearranges the dot product. Think ‘angle comes from dividing dot by magnitudes’. Always check your calculator is in degrees or radians as required.
向量夹角： cos θ = (a·b) / (|a||b|)。这不过是点积的变形。记成「夹角由点积除以模长获得」。注意根据题目要求确认计算器处于角度或弧度模式。
Vector equation of a line: r = a + λb, where a is a point on the line and b is the direction vector. The λ scales the direction. Remind yourself: ‘Start at a, travel along b.’
直线的向量方程： r = a + λb，其中 a 是直线上一点，b 是方向向量。λ 沿方向伸缩。提示自己：「从 a 出发，沿 b 行走。」
4. Exponentials & Logarithms | 指数与对数函数
Natural exponential and log: eˣ and ln x are inverse functions: ln(eˣ) = x, e^(ln x) = x. The number e ≈ 2.718 is the unique base where the gradient of eˣ equals its value at every point. Think ‘e is the natural growth engine’.
自然指数与对数： eˣ 与 ln x 互为反函数：ln(eˣ) = x, e^(ln x) = x。e ≈ 2.718 是唯一使 eˣ 的斜率处处等于自身的底数。记成「e 是自然增长的引擎」。
Derivative of ln x: d/dx (ln x) = 1/x. For ln(kx) the derivative is still 1/x because ln(kx) = ln k + ln x. A handy trick: ‘Log gives you a reciprocal.’
ln x 的导数： d/dx (ln x) = 1/x。对于 ln(kx) 导数仍为 1/x，因为 ln(kx) = ln k + ln x。技巧：「对数函数给出倒数。」
Change of base: logₐ b = ln b / ln a. This formula swaps any base into natural logs. Remember it by thinking ‘log b over log a, new base on the bottom.’
换底公式： logₐ b = ln b / ln a。此公式将任意底数换为自然对数。记忆：「新底跑到底下，log b 除以 log a。」
Logarithmic differentiation: For functions like y = xˣ, take ln of both sides, differentiate implicitly, then solve for dy/dx. The hint: ‘When the variable climbs into the power, bring down the logs.’
对数求导法： 对于 y = xˣ 这类函数，两边取 ln，隐函数求导，再解出 dy/dx。提示：「当变量爬到指数上时，请出对数把它拽下来。」
5. Numerical Methods & Iteration | 数值方法与迭代
Simple iteration: Rearrange f(x)=0 to x = g(x), then use xₙ₊₁ = g(xₙ). Convergence requires |g'(x)| < 1 near the root. Picture a staircase or cobweb diagram: if the gradient is small, steps shrink towards the root.
简单迭代： 将 f(x)=0 改写为 x = g(x)，然后迭代 xₙ₊₁ = g(xₙ)。收敛要求根附近 |g'(x)| < 1。想像楼梯图或蛛网图：若斜率较小，步长会收缩到根。
Newton–Raphson method: xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ). The geometric idea is following the tangent to the x‑axis. A quick check: the method fails if f'(xₙ) ≈ 0. Mnemonic: ‘New x is old x minus the ratio.’
牛顿–拉弗森方法： xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ)。几何图像是沿着切线与 x 轴相交。快速检测：若 f'(xₙ) ≈ 0 则方法失效。口诀：「新 x 等于旧 x 减去比值。」
Trapezium rule: Approx = h/2 [y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ], where h = (b-a)/n. Visualise fitting trapeziums under the curve. Remember: ‘Half the strip width times (first + last + twice the middles).’
梯形法则： 近似值 = h/2 [y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ]，其中 h = (b-a)/n。想像在曲线下放置梯形。记成：「半条宽乘（首项+末项+两倍中间项）。」
Sign–change method: If f(a) and f(b) have opposite signs and f is continuous, a root lies between a and b. Use this to start iteration. Caution: there may be an even number of roots if the signs are the same.
符号变换法： 若 f(a) 与 f(b) 异号且 f 连续，则 a 与 b 之间存在根。用此定位迭代起点。注意：同号时不代表无根，可能有偶数个根。
6. Mechanics: Kinematics & Forces | 力学：运动学与力
SUVAT equations: For constant acceleration a: v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as, s = vt – ½at². The mnemonic: ‘Silly Unicorns Vacuum All Tarmac’ can help list the five quantities.
SUVAT 方程： 对于匀加速度 a：v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as, s = vt – ½at²。可用「SUVAT」五个字母直接列出五个量。
Variable acceleration calculus: v = ds/dt, a = dv/dt = d²s/dt². To recover displacement from velocity, integrate: s = ∫ v dt. Think: ‘Differentiate downwards, integrate upwards.’
变加速度微积分： v = ds/dt, a = dv/dt = d²s/dt²。从速度求位移则积分：s = ∫ v dt。记住：「向下求导，向上积分。」
Newton’s second law: F = ma, where F is the resultant force in newtons, m in kg, a in m s⁻². Break it into components: ΣF_x = ma_x, ΣF_y = ma_y. ‘Resultant force produces acceleration in the same direction.’
牛顿第二定律： F = ma，F 为合力（单位牛顿），m 为质量（kg），a 为加速度（m s⁻²）。分解为分量：ΣF_x = ma_x，ΣF_y = ma_y。「合力产生同向加速度。」
Moments: Moment = Force × perpendicular distance. For equilibrium, total clockwise moments = total anticlockwise moments about any pivot. ‘Force times lever arm turns objects; balance them to zero rotation.’
力矩： 力矩 = 力 × 垂直距离。平衡时，对任意支点的顺时针力矩之和等于逆时针力矩之和。「力乘力臂使物体转动，达到平衡则总力矩为零。」
7. Statistics: Probability Distributions | 统计：概率分布
Binomial distribution: X ~ B(n, p) with P(X=r) = ⁿCᵣ p^r (1-p)^(n-r). Mean = np, variance = np(1-p). To recall the variance, say ‘np times (1-p)’ — just the mean times failure probability.
二项分布： X ~ B(n, p)，P(X=r) = ⁿCᵣ p^r (1-p)^(n-r)。均值 = np，方差 = np(1-p)。记忆方差：均值乘以失败概率。
Normal distribution: X ~ N(μ, σ²). The standardised variable Z = (X – μ)/σ ~ N(0,1). Using Z turns any normal problem into a standard normal table lookup. ‘Subtract the mean, divide by standard deviation.’
正态分布： X ~ N(μ, σ²)，标准化变量 Z = (X – μ)/σ ~ N(0,1)。借助 Z 值可将任意正态问题转化为查标准正态表。「减均值，除以标准差。」
Binomial approximation to normal: When n is large and p is close to 0.5, X ~ B(n, p) can be approximated by N(np, np(1-p)). Apply continuity correction (±0.5) to improve accuracy. ‘Large n, mid p → bell curve.’
二项分布的正态近似： 当 n 很大且 p 接近 0.5 时，X ~ B(n, p) 可近似为 N(np, np(1-p))。使用连续性校正 (±0.5) 提高精度。「n 大 p 中等，变成钟形曲线。」
Mean and variance of a sample mean: If X̄ is the mean of a random sample size n from N(μ, σ²), then X̄ ~ N(μ, σ²/n). The distribution tightens as n grows. ‘Sample means vary less: divide variance by n.’
样本均值的分布： 若 X̄ 是来自 N(μ, σ²) 容量为 n 的随机样本均值，则 X̄ ~ N(μ, σ²/n)。随着 n 增大分布更集中。「样本均值波动更小：方差除以 n。」
8. Hypothesis Testing & Critical Regions | 假设检验与临界域
Null and alternative hypotheses: H₀ is the default statement (e.g., μ = 5), H₁ is what you suspect (μ ≠ 5 or μ > 5). The mnemonic: ‘H₀ is innocent until proven guilty; H₁ is the challenger.’
零假设与备择假设： H₀ 是默认陈述（如 μ = 5），H₁ 是怀疑的陈述（μ ≠ 5 或 μ > 5）。口诀：「H₀ 是无辜的，直到证明有罪；H₁ 是挑战者。」
Significance level and p‑value: α (e.g., 0.05) is the probability of rejecting H₀ when true. The p‑value is the probability of obtaining a result at least as extreme. Rule: ‘If p is low, H₀ must go.’
显著性水平与 p 值： α（如 0.05）是 H₀ 为真时拒绝它的概率。p 值是得到至少如此极端结果的概率。法则：「p 值低，H₀ 弃。」
Critical region: The set of values of the test statistic that leads to rejecting H₀. For a two‑tail test at 5%, the critical region is divided into two 2.5% tails. Draw a bell curve and shade the tails.
临界域： 检验统计量中导致拒绝 H₀ 的取值集合。对于 5% 双尾检验，临界域分成两个 2.5% 的尾部。画一个钟形曲线并涂黑尾部。
One‑tail vs two‑tail: Use one‑tail when H₁ specifies a direction (greater or less), two‑tail for ‘not equal’. ‘Directional claim → one tail; any difference → two tails.’ This halves the critical significance in each tail.
单尾与双尾检验： 若 H₁ 指明方向（大于或小于）则用单尾；若只是不等于则用双尾。「方向性断言→单尾；任何差异→双尾。」双尾每侧显著性减半。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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