📚 Year 11 Cambridge Further Mathematics: Quick Memorisation Guide to Key Terms | Year 11 剑桥进阶数学：关键术语词汇速记指南
This article provides a concise, bilingual guide to the essential terminology encountered in the Year 11 Cambridge Further Mathematics course. Mastering these terms is not just about memorising definitions; it is about building a solid conceptual foundation that allows you to read questions fluently and express solutions accurately. Each section pairs an English explanation with a Chinese equivalent, followed by key notations and examples where helpful.
本文为 Year 11 剑桥进阶数学课程中的核心术语提供了一份简明中英双语速记指南。掌握这些术语不仅仅是记住定义，更是为了建立扎实的概念基础，使你能够流畅地阅读题目并准确地表达解题过程。每个部分都安排了英文解释与中文对应，并在适当位置列出了关键符号和示例。
1. Functions and Relations | 函数与关系
Function (函数): A function f is a rule that assigns each input value x exactly one output value f(x). If any input produces more than one output, the relation is not a function.
函数 f 是一种规则，它为每个输入值 x 指定唯一的输出值 f(x)。若任一输入产生多个输出，此关系就不是函数。
Domain (定义域): The set of all possible input values (x-values) for which the function is defined. For a real function, we exclude values that cause division by zero or square roots of negative numbers.
定义域是所有可能输入值（x 值）的集合，函数在该集合上有定义。对于实函数，我们需要排除使分母为零或使平方根内为负数的值。
Range (值域): The set of all actual output values f(x) produced by the function when the domain is applied. It is the set of y-values the function can take.
值域是当定义域作用于函数时，函数产生的所有实际输出值 f(x) 的集合，即函数所能取到的 y 值集合。
One-to-one function (一一映射函数): A function where each y-value comes from exactly one x-value. It passes the horizontal line test and has an inverse function that is also a function.
一一映射函数是指每个 y 值仅由唯一一个 x 值产生。它通过水平线检验，且其反函数也是一个函数。
Composite function (复合函数): where the output of the first becomes the input of the second, written as fg(x) = f(g(x)).
复合函数是将两个函数组合，第一个函数的输出成为第二个函数的输入，记作 fg(x) = f(g(x))。
Inverse function (反函数): wapping x and y and then solving for y. Graphically, it is a reflection in the line y = x.
反函数 f⁻¹(x) 逆转 f(x) 的作用。可通过交换 x 和 y 后解出 y 来求得。图像上，它是关于直线 y = x 的反射。
2. Quadratic Equations and Inequalities | 二次方程与不等式
Quadratic equation (二次方程): An equation of the form ax² + bx + c = 0, where a ≠ 0. The solutions are called roots.
形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。其解称为根。
Discriminant (判别式): Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
表达式 Δ = b² - 4ac 决定根的性质。若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
Completing the square (配方法): Rewriting a quadratic as a(x + p)² + q, which reveals the vertex and helps solve equations.
将二次式改写为 a(x + p)² + q，可以揭示顶点坐标并用于解方程。
Quadratic inequality (二次不等式): An inequality such as ax² + bx + c > 0. Solve by finding the roots and testing intervals, or by sketching the parabola.
形如 ax² + bx + c > 0 的不等式。可通过求根并检验区间，或通过绘制抛物线来求解。
| Term (English) | 术语（中文） | Example |
|---|---|---|
| Vertex | 顶点 | For y = (x-2)² + 3, vertex (2,3) |
| Axis of symmetry | 对称轴 | x = 2 |
3. Indices and Surds | 指数与根式
Index laws (指数律): ifying expressions with powers: aᵐ×aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a⁰ = 1 (a ≠ 0).
指数律是简化幂表达式的规则：aᵐ×aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，以及 a⁰ = 1（a ≠ 0）。
Surd (根式): An irrational root of a rational number, such as √2 or ³√5. Surds are left in root form to maintain exactness.
根式是有理数的无理根，例如 √2 或 ³√5。保留根式形式是为了保持精确值。
Rationalising the denominator (分母有理化): The process of eliminating a surd from the denominator of a fraction, e.g. 1/√2 becomes √2/2.
分母有理化是消去分数分母中根式的过程，例如将 1/√2 化为 √2/2。
Fractional indices (分数指数): a1/n = ⁿ√a, and am/n = (ⁿ√a)ᵐ. These link roots and powers.
分数指数 a1/n = ⁿ√a，且 am/n = (ⁿ√a)ᵐ，将根与幂联系起来。
4. Logarithms and Exponentials | 对数与指数函数
Logarithm (对数): If aˣ = b, then logₐb = x, where a > 0, a ≠ 1. It is the inverse operation of exponentiation.
若 aˣ = b，则 logₐb = x，其中 a > 0，a ≠ 1。对数是取幂运算的逆运算。
Natural logarithm (自然对数): Written as ln x, it is the logarithm with base e ≈ 2.718. ln x = logₑx.
自然对数记作 ln x，是以 e ≈ 2.718 为底的对数，ln x = logₑx。
Laws of logarithms (对数运算律): logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx - logₐy; logₐ(xᵏ) = k logₐx.
对数运算律：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx - logₐy；logₐ(xᵏ) = k logₐx。
Change of base (换底公式): logₐb = logₓb / logₓa for any valid base c.
换底公式：logₐb = logₓb / logₓa，其中 c 为任意有效底数。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
Gradient (斜率): ss of a line, calculated as m = (y₂ - y₁) / (x₂ - x₁). It is constant for a straight line.
斜率表示直线的倾斜程度，计算公式为 m = (y₂ - y₁) / (x₂ - x₁)。对于直线，斜率为常数。
Distance between two points (两点间距离): d = √[(x₂ - x₁)² + (y₂ - y₁)²], derived from Pythagoras' theorem.
两点间距离公式 d = √[(x₂ - x₁)² + (y₂ - y₁)²]，由勾股定理导出。
Midpoint (中点): The point exactly halfway between two given points: ((x₁+x₂)/2, (y₁+y₂)/2).
中点即两给定点正中间的点：((x₁+x₂)/2, (y₁+y₂)/2)。
Perpendicular lines (垂直线): Two lines with gradients m₁ and m₂ are perpendicular if m₁ × m₂ = -1.
若两条直线的斜率 m₁ 和 m₂ 满足 m₁ × m₂ = -1，则两直线垂直。
Equation of a circle (圆的方程): (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，其中圆心为 (a, b)，半径为 r。
6. Trigonometry | 三角学
Primary trigonometric ratios (基本三角比): ngle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
在直角三角形中，基本三角比定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Reciprocal functions (倒数三角函数): sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.
倒数三角函数：sec θ = 1/cos θ，csc θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。
Fundamental identities (基本恒等式): sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
基本恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。
Radian measure (弧度制): π radians = 180°. Arc length s = rθ, and sector area A = ½r²θ, where θ is in radians.
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½r²θ，其中 θ 以弧度为单位。
CAST diagram (CAST 图): A mnemonic that shows which trigonometric ratios are positive in each quadrant: C (cos), A (all), S (sin), T (tan) in quadrants IV, I, II, III respectively.
CAST 图是一个助记工具，显示各象限中为正的三角比：第四象限 C（cos），第一象限 A（全正），第二象限 S（sin），第三象限 T（tan）。
7. Differentiation | 微分
Derivative (导数): The derivative dy/dx is the rate of change of y with respect to x, represented geometrically as the gradient of the tangent to the curve at a point.
导数 dy/dx 表示 y 关于 x 的变化率，几何上表示为曲线在某点切线的斜率。
Power rule (幂法则): y = xⁿ, then dy/dx = nxⁿ⁻¹. This is the fundamental rule for differentiating polynomials.
若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。这是微分多项式的基本法则。
Product rule (乘法法则): For y = uv, dy/dx = u (dv/dx) + v (du/dx).
对于 y = uv，dy/dx = u (dv/dx) + v (du/dx)。
Chain rule (链式法则): If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. Used for composite functions.
若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。用于复合函数。
Stationary points (驻点): ximum, minimum, or point of inflection) is determined by the second derivative or a sign test.
驻点即满足 dy/dx = 0 的点。其性质（极大值、极小值或拐点）通过二阶导数或符号检验判定。
8. Integration | 积分
Integration as anti-differentiation (积分作为微分的逆运算): . The indefinite integral ∫ f(x) dx gives a family of functions differing by a constant C.
积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族相差常数 C 的函数。
Basic integration rule (基本积分法则): ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1.
基本积分法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。
Definite integral (定积分): ea under the curve y = f(x) from x = a to x = b. Use the fundamental theorem: F(b) - F(a).
定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x) 下从 x = a 到 x = b 的带号面积。利用微积分基本定理：F(b) - F(a)。
Area under a curve (曲线下的面积): For regions where f(x) is negative, the integral gives a negative value; take the absolute value when finding total area. Use separate integrals for parts above and below the x-axis.
当 f(x) 为负时，积分给出负值；求总面积时需取绝对值。对 x 轴上方和下方的部分分别积分。
9. Vectors | 向量
Vector (向量): A quantity with both magnitude and direction, represented as a or in column form [x, y]. It is depicted by a directed line segment.
向量是既有大小又有方向的量，表示作 a 或列向量 [x, y]，用有向线段图示。
Magnitude (模): The length of a vector. For a = (x, y), the magnitude |a| = √x² + y².
向量的模即其长度。对于 a = (x, y)，模 |a| = √x² + y²。
Unit vector (单位向量): A vector with magnitude 1. The unit vector in the direction of a is a/|a|.
单位向量是模为 1 的向量。沿 a 方向的单位向量为 a/|a|。
Position vector (位置向量): The vector from the origin to a point P, giving the coordinates of P as components.
位置向量是从原点到点 P 的向量，其分量即为 P 的坐标。
Vector addition and scalar multiplication (向量加法与数乘): Add component-wise; multiply each component by the scalar. These obey commutative and associative laws.
向量加法按分量相加；数乘将每个分量乘以标量。运算满足交换律和结合律。
10. Sequences and Series | 数列与级数
Arithmetic progression (AP) (等差数列): e nth term is uₙ = a + (n-1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n-1)d].
等差数列是具有公差 d 的数列。第 n 项 uₙ = a + (n-1)d，前 n 项和 Sₙ = n/2 [2a + (n-1)d]。
Geometric progression (GP) (等比数列): term is uₙ = arⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
等比数列是具有公比 r 的数列。第 n 项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 - rⁿ)/(1 - r)（r ≠ 1）。
Sum to infinity (无穷和): For a GP with |r| < 1, the series converges and S∞ = a/(1 - r).
对于 |r| < 1 的等比数列，级数收敛，且 S∞ = a/(1 - r)。
Sigma notation (西格玛记号): Σ from k=1 to n of uₖ denotes the sum u₁ + u₂ + ... + uₙ.
Σ（从 k=1 到 n）uₖ 表示和 u₁ + u₂ + ... + uₙ。
11. Permutations, Combinations and Binomial Theorem | 排列组合与二项式定理
Factorial (阶乘): n! = n × (n-1) × ... × 1, with 0! defined as 1. It counts the number of ways to arrange n distinct items.
阶乘 n! = n × (n-1) × ... × 1，定义 0! = 1。它用于计算 n 个不同物品的排列数。
Permutation (排列): ⁿPᵣ = n!/(n-r)! gives the number of ordered arrangements of r objects chosen from n distinct objects.
排列 ⁿPᵣ = n!/(n-r)! 给出从 n 个不同对象中选取 r 个进行有序排列的方式数。
Combination (组合): ⁿCᵣ = n! / (r!(n-r)! oose r objects from n without regard to order. Also written as binomial coefficient (n r).
组合 ⁿCᵣ = n! / (r!(n-r)!) 给出从 n 个对象中无序选取 r 个的方式数，也记作二项式系数 (n r)。
Binomial expansion (二项展开式): (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, for r = 0 to n. The coefficients come from Pascal's triangle or the formula.
二项展开式 (a
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