📚 Year 11 Edexcel Further Maths: Terminology Quick Reference | 词汇术语速记指南
This guide provides a rapid revision of essential terminology for the Year 11 Edexcel Further Mathematics course. Mastering these terms is crucial for understanding exam questions and articulating solutions clearly. Each key term is presented with a concise English definition followed by a Chinese explanation.
本指南为 Year 11 Edexcel 进阶数学课程提供必要的术语快速复习。掌握这些术语对于理解考试题目与清晰地表达解答至关重要。每个关键术语都附有简明的英文定义和中文解释。
1. Algebraic Fundamentals | 代数基础
Polynomial – an algebraic expression formed by summing terms of the form akxk, where each coefficient ak is a real number and the exponents are non‑negative integers. The highest exponent is the degree of the polynomial.
多项式 – 由若干形如 akxk 的项求和而成的代数式，其中各系数 ak 为实数，指数为非负整数。最高指数称为多项式的次数。
Leading coefficient – with the highest power in a polynomial. For p(x)=3x⁴+2x²−7, the leading coefficient is 3.
首项系数 – 多项式中最高次项所对应的系数。例如 p(x)=3x⁴+2x²−7 的首项系数为 3。
Factor theorem – lynomial f(x) if and only if f(a)=0. This is used to factorise polynomials and find roots.
因式定理 – 给出 (x−a) 是多项式 f(x) 的因式当且仅当 f(a)=0。该定理常用于因式分解多项式和求根。
Remainder theorem – when a polynomial f(x) is divided by (x−a), the remainder is equal to f(a). This provides a quick way to evaluate remainders without full long division.
余数定理 – 当多项式 f(x) 除以 (x−a) 时，余数等于 f(a)。利用它可以绕过完整的长除法直接求得余数。
Identity – an equation that holds for all permitted values of the variables, often denoted by the symbol '≡'. Unlike an ordinary equation, it does not need to be solved; it asserts two expressions are equivalent.
恒等式 – 对所有允许的变量值都成立的等式，通常用符号 '≡' 表示。与一般方程不同，恒等式不需要求解，它断言两个表达式等价。
2. Equations and Inequalities | 方程与不等式
Discriminant – : two distinct real roots if Δ>0, one repeated real root if Δ=0, and no real roots if Δ<0.
判别式 – 对于一元二次方程 ax²+bx+c=0，判别式为 Δ=b²−4ac。它决定了根的性质：Δ>0 时有两个不等实根，Δ=0 时有两个相等实根，Δ<0 时无实数根。
Quadratic inequality – an inequality involving a quadratic expression, e.g. x²−5x+6>0. Solutions are often expressed as intervals on the number line after finding critical values where the expression equals zero.
一元二次不等式 – 含有二次表达式的不等式，例如 x²−5x+6>0。求出表达式等于零的临界值后，解集通常用数轴上的区间表示。
Simultaneous equations – a set of two or more equations that share variables. Solutions can be found by elimination, substitution, or graphical methods. In Further Maths, one equation may be linear and the other quadratic.
联立方程组 – 含有两个或更多方程且共享变量的一组方程。可通过消元法、代入法或图像法求解。在进阶数学中，常出现一个线性方程与一个二次方程联立的情况。
Critical values – values of x that make an expression equal to zero or cause it to change sign. They partition the number line into intervals where the expression is either positive or negative.
临界值 – 使得表达式等于零或改变符号的 x 值。它们将数轴划分为若干区间，表达式在各区间内保持正或负。
3. Functions and Graphs | 函数与图像
Function – a mapping from a set of inputs (domain) to a set of outputs (range) such that each input is assigned exactly one output. Notation f(x) is read as 'f of x'.
函数 – 从输入集合（定义域）到输出集合（值域）的一种映射，每个输入对应唯一的输出。记号 f(x) 读作 'f of x'。
Domain and range – the domain is the set of all allowable input values (often x‑values) for which a function is defined. The range is the set of all output values (y‑values) the function can produce.
定义域与值域 – 定义域是使函数有意义的所有输入值（通常是 x 值）的集合；值域是函数能产生的所有输出值（y 值）的集合。
Composite function – output of another. Denoted by fg(x)=f(g(x)), meaning apply g first, then f. Order matters.
复合函数 – 将一个函数的输出作为另一个函数的输入而得到的函数。记作 fg(x)=f(g(x))，表示先作用 g 再作用 f。顺序至关重要。
Inverse function – fies f(f⁻¹(x))=x and f⁻¹(f(x))=x. The graph of an inverse is a reflection in the line y=x.
反函数 – 能反转原函数作用的函数，记作 f⁻¹(x)，满足 f(f⁻¹(x))=x 和 f⁻¹(f(x))=x。反函数的图像是原函数关于直线 y=x 的对称图形。
Modulus function – defined as |x|, which gives the absolute value of x, effectively removing any negative sign. Its graph is a V‑shape and it always yields a non‑negative output.
绝对值函数 – 定义为 |x|，表示 x 的绝对值，即去掉负号后的数值。其图像呈 V 形，输出总是非负的。
4. Coordinate Geometry | 坐标几何
Gradient – a measure of the steepness of a line, calculated as the change in y divided by the change in x. A positive gradient slopes upward, a negative gradient slopes downward.
斜率 – 度量直线陡峭程度的量，计算公式为 y 的变化量除以 x 的变化量。斜率为正时直线向上倾斜，斜率为负时向下倾斜。
Perpendicular lines – two lines are perpendicular if the product of their gradients is −1. If a line has gradient m, a line perpendicular to it has gradient −1/m (provided m ≠ 0).
垂直直线 – 若两条直线的斜率乘积为 −1，则它们互相垂直。若一直线斜率为 m，则垂直于它的直线斜率为 −1/m（m≠0）。
Circle equation – a,b) and radius r. Expanding gives x²+y²+2fx+2gy+c=0, from which centre (−f,−g) and radius √f²+g²−c can be found.
圆的方程 – 标准形式 (x−a)²+(y−b)²=r² 表示圆心为 (a,b)、半径为 r 的圆。展开得一般式 x²+y²+2fx+2gy+c=0，从中可得圆心 (−f,−g) 及半径 √f²+g²−c。
Tangent to a circle – a straight line that touches the circle at exactly one point. A tangent is perpendicular to the radius drawn to the point of contact.
圆的切线 – 与圆仅有一个公共点的直线。切线垂直于经过切点的半径。
Distance formula – ce between two points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]. Derived from Pythagoras' theorem.
距离公式 – 两点 (x₁,y₁) 与 (x₂,y₂) 之间的距离为 √[(x₂−x₁)²+(y₂−y₁)²]。该公式源于勾股定理。
5. Trigonometry | 三角学
Radian – subtended at the centre of a circle by an arc equal in length to the radius. π rad = 180°.
弧度 – 角度度量单位；当弧长等于半径时，该弧所对的圆心角大小为 1 弧度。π 弧度 = 180°。
Arc length and sector area – for a circle of radius r, the arc length spanning θ radians is rθ, and the area of the corresponding sector is ½ r²θ.
弧长与扇形面积 – 半径为 r 的圆，θ 弧度角所对的弧长为 rθ，对应扇形的面积为 ½ r²θ。
Trigonometric identities – key identities include sin²θ+cos²θ≡1 and tanθ≡sinθ/cosθ. These are used to simplify expressions and solve equations.
三角恒等式 – 常用恒等式包括 sin²θ+cos²θ≡1 以及 tanθ≡sinθ/cosθ。它们用于化简表达式和求解方程。
CAST diagram – a mnemonic to determine the sign of sin, cos and tan in the four quadrants. Starting from the first quadrant (All positive), then moving anticlockwise: Sine positive, Tangent positive, Cosine positive.
CAST 图 – 帮助记忆三角函数在各象限符号的图表。从第一象限（全部为正）出发，逆时针依次为：正弦正、正切正、余弦正。
General solution – have infinitely many solutions. General solutions express all these using integer multiples of the period (e.g. n·360° or n·2π).
通解 – 由于三角函数具有周期性，形如 sinθ=k 的方程有无穷多解。通解利用周期的整数倍（如 n·360° 或 n·2π）来表示全部解。
6. Sequences and Series | 序列与级数
Arithmetic progression (AP) – a sequence where the difference between consecutive terms is constant, called the common difference d. The nth term is a+(n−1)d and the sum of the first n terms is Sn=n/2[2a+(n−1)d].
等差数列 – 相邻两项之差为常数的数列，该常数称为公差 d。第 n 项为 a+(n−1)d，前 n 项和为 Sn=n/2[2a+(n−1)d]。
Geometric progression (GP) – a sequence where each term is found by multiplying the previous term by a fixed common ratio r. The nth term is arⁿ⁻¹ and the sum of the first n terms is Sn=a(1−rⁿ)/(1−r) for r≠1.
等比数列 – 每一项都是前一项乘以固定公比 r 所得到的数列。第 n 项为 arⁿ⁻¹，前 n 项和为 Sn=a(1−rⁿ)/(1−r)，其中 r≠1。
Series – the sum of the terms of a sequence. Finite series have a defined number of terms; infinite series may converge to a limit or diverge.
级数 – 序列各项之和。有限级数的项数确定；无穷级数可能收敛于某个极限，也可能发散。
Sigma notation (Σ) – a concise way to write a series. For example, Σi=1n i² represents the sum of squares from 1 to n.
西格玛记法 (Σ) – 表示级数的简洁方式。例如 Σi=1n i² 表示从 1 到 n 的平方和。
7. Binomial Expansion | 二项式展开
Binomial theorem – for a positive integer n, (a+b)ⁿ = Σk=0n nCk aⁿ⁻kbk, where nCk = n!/(k!(n−k)!) are the binomial coefficients.
二项式定理 – 对于正整数 n，(a+b)ⁿ = Σk=0n nCk aⁿ⁻kbk，其中 nCk = n!/(k!(n−k)!) 为二项式系数。
Pascal's triangle – a triangular array where each row gives the binomial coefficients for a given power. The kth entry in row n corresponds to nCk.
帕斯卡三角形 – 一种三角形数组，每行给出对应次幂的二项式系数。第 n 行的第 k 个数即为 nCk。
Term independent of x – in an expansion, the term where the variable x has an exponent of zero. To find it, set up the general term and solve for the power of x to be zero.
与 x 无关的项（常数项） – 展开式中变量 x 的指数为零的项。可通过写出通项并令 x 的指数为零来求得。
nCr notation – also written as nCr or binomnr (we use nCr here), it calculates the number of ways to choose r items from n distinct items without regard to order.
nCr 符号 – 也写作 nCr（或组合数符号），表示从 n 个不同元素中无序选取 r 个的方法数。
8. Calculus: Differentiation | 微积分：微分
Derivative – the rate of change of a function, denoted f' dy/dx. Geometrically, it gives the gradient of the tangent to the curve y=f(x) at a point.
导数 – 函数的变化率，记作 f'(x) 或 dy/dx。几何上，它表示曲线 y=f(x) 上某点切线的斜率。
Power rule – if f(x)=xⁿ, then f' s rule applies to any real exponent n and is the foundation of polynomial differentiation.
幂法则 – 若 f(x)=xⁿ，则 f'(x)=nxⁿ⁻¹。该法则适用于任意实数指数 n，是多项式微分的基础。
Stationary point – a point on a curve where the gradient is zero (f' cal minimum, or a point of inflection. The nature is determined by the second derivative or by examining sign changes of f'(x).
驻点 – 曲线上梯度为零（f'(x)=0）的点。它可能是局部极大值、局部极小值或拐点。其性质可通过二阶导数或 f'(x) 的符号变化来判断。
Tangent and normal – the tangent to a curve at a point (x₁,y₁) has gradient f'(x₁). The normal is perpendicular to the tangent, so its gradient is −1/f'(x₁) provided f'(x₁)≠0.
切线与法线 – 曲线在 (x₁,y₁) 处的切线斜率为 f'(x₁)。法线垂直于切线，其斜率为 −1/f'(x₁)（假设 f'(x₁)≠0）。
Second derivative – the derivative of the derivative, f''(x) or d²y/dx². It describes the rate of change of the gradient and indicates concavity. Used in classifying stationary points.
二阶导数 – 导数的导数，记作 f''(x) 或 d²y/dx²。它描述斜率的变化率并反映曲线的凹凸性，常用于驻点分类。
9. Calculus: Integration | 微积分：积分
Indefinite integral – the reverse process of differentiation. If F'(x)=f(x), then ∫ f(x) dx = F(x)+C, where C is the constant of integration.
不定积分 – 微分的逆运算。若 F'(x)=f(x)，则 ∫ f(x) dx = F(x)+C，其中 C 为积分常数。
Power rule for integration – ∫ xⁿ dx = xⁿ⁺¹/(n+1)+C, for n≠−1. This mirrors the differentiation power rule in reverse
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