📚 Glossary & Mnemonics Guide for Pre-U CAIE Mathematics | Pre-U CAIE 数学：词汇术语速记指南
Mastering the vocabulary of Pre-U Mathematics is half the battle. This guide presents essential terms grouped by topic, each accompanied by clear explanations and powerful memory aids. Whether you are grappling with the precise language of limits or untangling trigonometric identities, these mnemonics and etymological hints will help you recall definitions quickly and accurately under exam pressure.
掌握 Pre-U 数学的词汇是成功的一半。本指南按主题整理了核心术语，每个术语都配有清晰的解释和强大的记忆辅助。不论你是在与极限的精确语言搏斗，还是在梳理三角恒等式，这些速记法和词源提示都能帮助你在考试压力下快速、准确地回忆起定义。
1. Functions and Graphs | 函数与图像
A function f maps each input x from its domain to exactly one output f(x) in the codomain. The range is the set of all actual outputs. The vertical line test confirms a relation is a function: if any vertical line cuts the graph more than once, it fails. A one-one (injective) function passes the horizontal line test. Mnemonic: ‘One x, one y – that’s the function rule to apply.’ For domain and range, think: ‘Domain is x, as in the x-axis; Range is y, reflecting the height I spy.’
函数 f 将定义域中的每个输入 x 对应到值域中唯一确定的输出 f(x)。值域是所有实际输出的集合。垂直线测试可确认一个关系是否为函数：如果任意一条垂直线与图像相交多于一次，则不是函数。一一（单射）函数可通过水平线测试。记忆口诀：“一个 x，一个 y —— 这就是函数要遵守的定律。”关于定义域和值域，可以这样想：“定义域是 x，好比 x 轴；值域是 y，反映我瞄到的高度。”
An inverse function f⁻¹ reverses the mapping, swapping domain and range. Its graph is the reflection of y = f(x) in the line y = x. Only one-one functions have inverses. To remember: ‘Inverse flips x and y, swapped coordinates under the sky.’ Composite functions gf(x) mean apply f then g; think ‘f first, then g, like putting on socks f then shoes g.’
反函数 f⁻¹ 反转映射，交换定义域和值域。它的图像是 y = f(x) 关于直线 y = x 的反射。只有一一函数才有反函数。记忆：“反函翻转 x 和 y，坐标交换在天空下。”复合函数 gf(x) 表示先作用 f 再作用 g；想象“先 f 后 g，就像先穿袜子 f 再穿鞋 g。”
2. Differentiation Terminology | 微分术语
The derivative f'(x) or dy/dx measures the instantaneous rate of change, represented by the gradient of the tangent. The limit definition is lim(h→0) [f(x+h)−f(x)]/h. Mnemonic for the power rule: ‘Bring down the power, then subtract one hour.’ For the product rule: ‘First times derivative of second plus second times derivative of first.’ Quotient rule: ‘Low d high minus high d low, square the bottom and away we go.’
导数 f'(x) 或 dy/dx 衡量瞬时变化率，由切线的斜率表示。极限定义为 lim(h→0) [f(x+h)−f(x)]/h。幂法则的口诀：“把指数拽下来，然后指数减一。”乘法法则：“前乘后的导数加后乘前的导数。”除法法则：“分母乘分子导减去分子乘分母导，再除以下面的平方向前走。”
A stationary point occurs where dy/dx = 0. Use the second derivative d²y/dx² to classify: positive → minimum, negative → maximum. If d²y/dx² = 0, check sign change of the first derivative to identify a point of inflection. Think: ‘First derivative zero makes the tangent horizontal; second derivative tells which way the bowl is orientable.’ The chain rule: dy/dx = dy/du × du/dx. ‘Derivative of outside times derivative of inside,’ like peeling an onion.
驻点 出现在 dy/dx = 0 处。用二阶导数 d²y/dx² 进行分类：为正 → 极小值点，为负 → 极大值点。若 d²y/dx² = 0，则检查一阶导数的符号变化来确定拐点。记忆：“一阶导数为零，切线水平横；二阶导数告诉你碗口朝哪盛。”链式法则：dy/dx = dy/du × du/dx。“外层的导数乘内层的导数”，像剥洋葱一样。
3. Integration Terminology | 积分术语
Indefinite integration reverses differentiation: ∫ f(x) dx = F(x) + C, where C is the constant of integration. ‘Add one to the power, then divide by the new power, and don’t forget the constant tower.’ The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve and the x-axis. ‘Top minus bottom’ when applying limits: F(b) − F(a).
不定积分 是微分的逆运算：∫ f(x) dx = F(x) + C，其中 C 是积分常数。“指数加一，除以新指数，别忘了常数塔。”定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴之间的有号面积。代入上下限时“上减下”：F(b) − F(a)。
Area between curves is found by integrating the difference (top function minus bottom function). When the curve crosses the x-axis, split the interval where f(x) becomes negative; total area requires absolute values. For volumes of revolution, use discs: V = π∫ y² dx. Think: ‘Square y, times π, integrate along the axis with glee.’ The trapezium rule approximates area using n strips: A ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]. Mnemonic: ‘Half the strip width times (first plus last plus twice the middle cast).’
曲线间面积 通过积分差值（上函数减下函数）求得。当曲线穿过 x 轴时，在 f(x) 变负的区间分开；总面积需要绝对值。对于旋转体体积，用圆盘法：V = π∫ y² dx。记忆：“y 平方，乘以 π，沿轴积分欢。”梯形法则 用 n 个条带近似面积：A ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]。口诀：“半条宽乘以（首末加两倍中间排）。”
4. Trigonometric Identities | 三角恒等式
The fundamental Pythagorean identity is sin²θ + cos²θ ≡ 1. From this, divide by cos²θ to get tan²θ + 1 ≡ sec²θ, or divide by sin²θ to get 1 + cot²θ ≡ cosec²θ. ‘Sin squared plus cos squared equals one, a trig foundation second to none.’ Reciprocal definitions: sec θ = 1/cos θ, csc/cosec θ = 1/sin θ, cot θ = 1/tan θ. Many students recall the order with ‘Some Old Hags Can Always Hide Their Old Age’ for SOH CAH TOA.
基本勾股恒等式 为 sin²θ + cos²θ ≡ 1。两边同除以 cos²θ 可得 tan²θ + 1 ≡ sec²θ，同除以 sin²θ 可得 1 + cot²θ ≡ cosec²θ。“正弦方加余弦方等于一，三角基础无人敌。”倒数定义：sec θ = 1/cos θ, csc/cosec θ = 1/sin θ, cot θ = 1/tan θ。许多学生用 “SOH CAH TOA” 记住正弦、余弦、正切的比对关系。
The compound angle formulae: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB). Memory aid: ‘Sine keeps the sign, cosine flips the sign.’ For sin(A+B) think ‘sine cosine cosine sine.’ For double angle identities: sin2θ = 2sinθ cosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. ‘Double angle formulas are versatile; pick the cos²θ or sin²θ form that suits your reversal.’
和角公式：sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。记忆技巧：“正弦符号相同，余弦符号相反。”对于 sin(A+B)，记“正弦余弦余弦正弦”。倍角恒等式：sin2θ = 2sinθ cosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。“倍角公式多用途，选 cos²θ 或 sin²θ 形式，哪边方便就转换。”
5. Vectors | 向量
A vector has magnitude and direction; a position vector is anchored at the origin. Unit vectors i, j, k point along Cartesian axes. The magnitude of a = xi + yj + zk is √(x² + y² + z²). ‘Magnitude is distance from start, square the components to get that part.’ The dot product a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂. It is a scalar. Perpendicular vectors have a·b = 0. Mnemonic: ‘Dot gives a scalar spot; if zero, they’re orthogonal and hot.’
向量 有大小和方向；位置向量 从原点出发。单位向量 i, j, k 分别指向坐标轴方向。向量 a = xi + yj + zk 的模为 √(x² + y² + z²)。“大小就是起点的距离，将各分量平方加起来就行。”《strong>点积 a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂。结果是一个标量。垂直向量满足 a·b = 0。口诀：“点积得到一个标量；若为零，它们正交又完美。”
The cross product a×b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. Its direction follows the right-hand rule. Only defined in 3D. Mnemonic for determinant expansion: ‘i times (skip row, column) minus j times (next block) plus k times (last flock).’ The vector equation of a line: r = a + t d, where a is a point on the line and d is the direction vector. ‘Point plus parameter times direction, the line’s linear connection.’ For planes: r·n = a·n, where n is normal.
叉积 a×b 产生一个垂直于 a 和 b 的向量，大小为 |a||b| sin θ。方向遵循右手定则。仅在三维空间中定义。行列式展开口诀：“i 乘（跳过行列）减 j 乘（下一块）加 k 乘（最后一块）。”直线的向量方程：r = a + t d，其中 a 是直线上一点，d 是方向向量。“点加参数乘方向，直线线性连接通畅。”平面方程：r·n = a·n，其中 n 是法向量。
6. Complex Numbers | 复数
A complex number z = a + bi, where i² = −1. a is the real part Re(z), b is the imaginary part Im(z). The complex conjugate is z* = a − bi. It reflects across the real axis. Modulus |z| = √(a² + b²). Argument arg(z) = θ, with tan θ = b/a. Polar form: z = r(cos θ + i sin θ) = r cis θ. Euler’s formula: e^(iθ) = cos θ + i sin θ. ‘e to iθ gives cos plus i sin, a beautiful bridge within.’
复数 z = a + bi，其中 i² = −1。a 为实部 Re(z)，b 为虚部 Im(z)。复共轭为 z* = a − bi，它关于实轴对称。模 |z| = √(a² + b²)。辐角 arg(z) = θ，tan θ = b/a。极坐标形式：z = r(cos θ + i sin θ) = r cis θ。欧拉公式：e^(iθ) = cos θ + i sin θ。“e 的 iθ 次方给出 cos 加 i sin，一座美丽桥梁在中间。”
De Moivre’s theorem: (r cis θ)ⁿ = rⁿ cis(nθ) for any integer n. It is used to find powers and roots of complex numbers. For the nth roots, there are n distinct solutions spaced by 2π/n around the circle. Mnemonic: ‘Power goes to r and multiplies the angle; roots divide the angle, with 2πk entangled.’ The fundamental theorem of algebra states that a polynomial of degree n has n complex roots (counting multiplicity). ‘Degree n gives n roots, complex plane completes the boots.’
棣莫弗定理：(r cis θ)ⁿ = rⁿ cis(nθ)，对于任意整数 n 成立。它用于求复数的幂和根。n 次方根有 n 个不同的解，在圆上间隔 2π/n。口诀：“幂作用到 r 并乘角度；开根则除角度，带 2πk 缠绕。”代数基本定理 指出 n 次多项式有 n 个复数根（计重数）。“n 次得 n 根，复数平面完成理论深。”
7. Sequences and Series | 数列与级数
An arithmetic progression (AP) has constant difference d. nth term: uₙ = a + (n−1)d. Sum of n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term. Mnemonic: ‘Average of first and last times number of terms.’ A geometric progression (GP) has constant ratio r. nth term: uₙ = a rⁿ⁻¹. Sum to n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. ‘Ratio not one, sum is a one minus r to the n over one minus r.’
等差数列（AP）有常数公差 d。第 n 项：uₙ = a + (n−1)d。前 n 项和：Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 为末项。口诀：“首末平均乘项数。”等比数列（GP）有常数公比 r。第 n 项：uₙ = a rⁿ⁻¹。前 n 项和：Sₙ = a(1 − rⁿ)/(1 − r)，当 r ≠ 1。“公比不为一时，和等于 a 乘一减 r 的 n 次方除以一减 r。”
The sum to infinity of a GP exists if |r| < 1: S∞ = a/(1 − r). 'If ratio's magnitude is under one, infinite sum is a over one minus r, and done.' The binomial expansion for (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … converges for |x| < 1 when n is not a positive integer. 'n choose r' notation: C(n,r) or ⁿCᵣ. Pascal's triangle builds coefficients. 'Binomial expands in a row, powers ascend and descend as they go.'
等比无穷级数和 当 |r| < 1 时收敛：S∞ = a/(1 − r)。“若公比绝对值小于一，无穷和就是 a 除以一减 r，即刻完毕。”二项式展开 (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … 当 n 不是正整数时，|x| < 1 收敛。组合数记号：C(n,r) 或 ⁿCᵣ。帕斯卡三角形构建系数。“二项式展开依次排，幂次升降一起来。”
8. Probability and Statistics | 概率与统计
Probability P(A) ranges from 0 to 1. The sum of probabilities in a sample space is 1. For mutually exclusive events, P(A∪B) = P(A) + P(B). Independent events: P(A∩B) = P(A)×P(B). ‘Multiply for AND when independent; add for OR when disjoint, be diligent.’ Conditional probability P(A|B) = P(A∩B)/P(B). ‘Given B, the sample space shrinks, probability of A is the fraction that links.’
概率 P(A) 取值范围从 0 到 1。样本空间中所有概率之和为 1。互斥事件满足 P(A∪B) = P(A) + P(B)。独立事件：P(A∩B) = P(A)×P(B)。“独立时 AND 用乘法；互斥时 OR 用加法，一定要勤奋检查。”条件概率 P(A|B) = P(A∩B)/P(B)。“给定 B，样本空间缩小，A 的概率是相关联的分数。”
A random variable X has expected value E(X) = Σ x P(X=x) or integral for continuous. Variance Var(X) = E[(X−μ)²] = E(X²) − μ². ‘Mean of squares minus square of mean.’ For the binomial distribution X ~ B(n, p): E(X) = np, Var(X) = np(1−p). ‘n times p for mean, n p q for variance, a neat machine.’ The normal distribution N(μ, σ²) is bell-shaped; standardise to Z = (X − μ)/σ ~ N(0,1). ‘Z is the number of standard deviations from the mean.’
随机变量 X 的期望值 E(X) = Σ x P(X=x) 或连续情形下的积分。方差 Var(X) = E[(X−μ)²] = E(X²) − μ²。“平方的均值减去均值的平方。”二项分布 X ~ B(n, p)：E(X) = np，Var(X) = np(1−p)。“n 乘 p 为均值，n p q 为方差，整洁就像一部机器。”正态分布 N(μ, σ²) 呈钟形；标准化为 Z = (X − μ)/σ ~ N(0,1)。“Z 是距离均值多少个标准差。”
9. Exponentials and Logarithms | 指数与对数
An exponential function aˣ has base a > 0. The natural base e ≈ 2.718 is defined via limit (1 + 1/n)ⁿ. The derivative of eˣ is eˣ itself, and ∫eˣ dx = eˣ + C. ‘e to the x stays the same, differentiation’s simplest game.’ The natural logarithm ln x is the inverse of eˣ; ln(eˣ) = x. ‘ln and e undo each other like a lock and key.’
指数函数 aˣ 的底 a > 0。自然底数 e ≈ 2.718 定义为极限 (1 + 1/n)ⁿ。eˣ 的导数就是它自身，∫eˣ dx = eˣ + C。“e 的 x 次方不变样，微分最简单的模样。”自然对数 ln x 是 eˣ 的反函数；ln(eˣ) = x。“ln 和 e 互相解开，就像锁和钥匙一样。”
Logarithm laws: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln aᵏ = k ln a. ‘Log of a product equals sum of logs; log of a quotient equals difference of logs; log of a power is exponent times log.’ The change-of-base formula: log_b a = log_c a / log_c b. ‘Different base? Just divide logs with the same new base.’ Exponential growth and decay models: A = A₀ e^(kt). k > 0 for growth, k < 0 for decay. 'k positive means gain, k negative means drain.'
对数运算法则：ln(ab) = ln a + ln b；ln(a/b) = ln a − ln b；ln aᵏ = k ln a。“乘积的对数等于对数的和；商的对数等于对数的差；幂的对数是指数乘以对数。”换底公式：log_b a = log_c a / log_c b。“底数不同？只需用同样的新底数做对数相除。”指数增长与衰减模型：A = A₀ e^(kt)。k > 0 为增长，k < 0 为衰减。“k 正表增长，k 负示衰减。”
10. Key Exam Command Words | 关键考试指令词
Pre-U questions use precise verbs: State means write down a concise answer without justification. Show that or Prove requires a step-by-step logical argument leading to a given result. Hence or Hence or otherwise suggests using the previous result, but ‘otherwise’ allows an alternative method. Determine or Find expects a calculation or reasoning to obtain a specific value or expression. Sketch a graph demands key features: intercepts, turning points, asymptotes, and general shape, not precise plotting.
Pre-U 考题使用精确的动词：State 意思是写下简洁答案，无需论证。Show that 或 Prove 要求一步步的逻辑推理导出给定结果。Hence 或 Hence or otherwise 建议使用上一问的结论，但 ‘otherwise’ 允许替代方法。Determine 或 Find 期望通过计算或推理得出特定值或表达式。Sketch 图像要求标注关键特征：截距、转折点、渐近线和大致形状，而不是精确描点。
Exact value means leave in surd or π form, not decimal. Verify means check that a given value or expression satisfies a condition, often by substitution. Justify or Explain requires a written reason or interpretation. Mnemonic for command words: ‘State is straight; Show needs flow; Verify – check it’s so; Sketch – key features to and fro.’ Remember that Evaluate often implies a numerical answer after calculation.
Exact value 意思是保留根号或 π 的形式，而非小数。Verify 指检查给定的值或表达式是否满足条件，常通过代值验证。Justify 或 Explain 要求书面给出理由或解释。指令词口诀：“State 直接答；Show 需推导流；Verify 验证确如此；Sketch 关键特征来回描。”记住 Evaluate 常暗示计算后得到一个数值答案。
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