Year 11 Edexcel Further Maths: Vocabulary & Terminology Memorisation Guide | Year 11 Edexcel 进阶数学：词汇术语速记指南

📚 Year 11 Edexcel Further Maths: Vocabulary & Terminology Memorisation Guide | Year 11 Edexcel 进阶数学：词汇术语速记指南

Mastering the key terminology in Edexcel Year 11 Further Mathematics is essential for tackling complex problems with confidence. This guide provides clear definitions and memorable tricks for the most important terms, helping you build a solid foundation for exams.

掌握 Edexcel 爱德思 Year 11 进阶数学中的核心术语，是自信应对复杂问题的关键。本指南为你梳理最重要的定义和记忆技巧，帮助你在考试前打好扎实基础。

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1. Complex Numbers | 复数

Imaginary unit (i): The imaginary unit i is defined as i = √(-1), so that i² = –1. It forms the basis of complex numbers, written as a + bi.

虚数单位 (i)： 虚数单位 i 定义为 i = √(-1)，因此 i² = –1。它是复数 a + bi 的基石。

Memory tip: Think of i as a 'quarter turn' anticlockwise in the complex plane. Multiplying by i rotates a number by 90°.

记忆技巧： 把 i 想象成复平面上的逆时针四分之一旋转。乘以 i 相当于将数字旋转 90°。

Complex conjugate: The conjugate of z = a + bi is z* = a – bi. It reflects the point across the real axis.

共轭复数： 复数 z = a + bi 的共轭为 z* = a – bi。它在复平面上是关于实轴的镜像。

Memory tip: y : Simply flip the sign of the imaginary part. Use it to find the modulus squared: z z* = a² + b².

记忆技巧： 只需改变虚部的符号。可利用它求模的平方：z z* = a² + b²。

Modulus: The modulus |z| = √(a² + b²) gives the distance from the origin to the point representing z.

模： 模 |z| = √(a² + b²) 表示从原点到复数 z 对应点的距离。

Memory tip: Modulus is the 'length' of the complex number vector. Treat it like the hypotenuse of a right-angled triangle.

记忆技巧： 模就是复数向量的“长度”。可把它看作直角三角形的斜边。

Argument: ngle the line from the origin to z makes with the positive real axis. Principal argument usually lies in (–π, π].

辐角： 辐角 arg(z) = θ 是从原点到 z 的射线与正实轴之间的夹角。主值通常落在 (–π, π] 之间。

Memory tip: Argument answers 'what is the angle?' Use tan θ = b/a, but check the quadrant carefully.

记忆技巧： 辐角回答“角度是多少？”。使用 tan θ = b/a，但务必检查象限。

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2. Matrices | 矩阵

Order of a matrix: An m × n matrix has m rows and n columns. Order is given as 'rows by columns'.

矩阵的阶： m × n 矩阵具有 m 行和 n 列。阶数表示为“行数乘以列数”。

Memory tip: Always say rows first, then columns – like 'R × C'.

记忆技巧： 总是先说行，再说列——如同“R × C”。

Identity matrix (I): A square matrix with 1s on the main diagonal and 0s elsewhere. Multiplying any matrix by I (where defined) leaves it unchanged.

单位矩阵 (I)： 主对角线为 1、其余为 0 的方阵。任何矩阵左乘或右乘单位矩阵（可乘的条件下）都保持不变。

Memory tip: I behaves like the number 1 for matrices. Think 'I am the one'.

记忆技巧： I 就像数字 1 在矩阵中的作用。记住“我就是那个一”。

Determinant: For a 2 × 2 matrix M = [[a, b], [c, d]], det(M) = ad – bc. It determines whether a matrix has an inverse and scales areas.

行列式： 对于 2 × 2 矩阵 M = [[a, b], [c, d]]，det(M) = ad – bc。行列式决定矩阵是否有逆，并作为面积缩放因子。

Memory tip: : Cross-multiply the diagonals and subtract: down-right minus up-right. If det = 0, the matrix is singular.

记忆技巧： 对角线交叉相乘再相减：主对角线积减去副对角线积。如果 det = 0，则矩阵为奇异的。

Inverse matrix: For a 2 × 2 matrix M, M⁻¹ = (1/det(M)) [[d, –b], [–c, a]], provided det(M) = 0. It 'undoes' the transformation.

逆矩阵： 对于 2 × 2 矩阵 M，M⁻¹ = (1/det(M)) [[d, –b], [–c, a]]，前提是 det(M) ≠ 0。它用来“撤销”变换。

Memory tip: Swap a and d, change signs of b and c, then divide by the determinant. Remember: 'swap and negate'.

记忆技巧： 交换 a 和 d 的位置，改变 b 和 c 的符号，然后除以行列式。口诀：“交换并变号”。

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3. Functions | 函数

Domain: The set of all possible input values (x) for which a function f(x) is defined.

定义域： 函数 f(x) 所有可能输入值 x 的集合。

Memory tip: Domain = 'inputs allowed' k . Look out for square roots, denominators that cannot be zero, or logs of non-positive numbers.

记忆技巧： 定义域 = “允许的输入”。注意根号内非负、分母不为零、对数真数必须为正。

Range: The set of all possible output values (f(x)) that a function can produce when x runs through the domain.

值域： 当 x 取遍定义域时，函数所有可能的输出值 f(x) 的集合。

Memory tip: Range = 'outputs reached'. Sketch the graph to see minimum and maximum y-values.

记忆技巧： 值域 = “能达到的输出”。画出图像观察 y 的最小和最大值。

Inverse function (f⁻¹): fect of f. If f(a) = b, then f⁻¹(b) = a. Its graph is a reflection of f in the line y = x.

反函数 (f⁻¹)： 逆转 f 的作用的函数。若 f(a) = b，则 f⁻¹(b) = a。其图像是 f 关于直线 y = x 的反射。

Memory tip: Swap x and y, then solve for y. Composition f(f⁻¹(x)) = x serves as a check.

记忆技巧： 交换 x 和 y，再解出 y。用 f(f⁻¹(x)) = x 来验证。

Composite function (f ∘ g): A function formed by applying g first, then f: (f ∘ g)(x) = f(g(x)).

复合函数 (f ∘ g)： 先作用 g，再作用 f 所构成的函数：(f ∘ g)(x) = f(g(x))。

Memory tip: Work from inside out – the right-hand function operates first. The order matters: f ∘ g is usually not the same as g ∘ f.

记忆技巧： 从内向外运算——右边的函数先作用。注意顺序：f ∘ g 和 g ∘ f 通常不同。

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4. Differentiation | 微分

Derivative (dy/dx): The rate of change of y with respect to x. It gives the gradient of the tangent to the curve at any point.

导数 (dy/dx)： y 关于 x 的变化率。它给出了曲线在任意点处切线的斜率。

Memory tip: Think 'steepness' – the derivative tells how quickly y climbs or falls. For a power xⁿ, multiply by the exponent and reduce it by 1.

记忆技巧： 想象“陡峭程度”——导数表示 y 上升或下降的快慢。对于幂函数 xⁿ，乘以指数再将指数减 1。

Chain rule: Used to differentiate composite functions: dy/dx = (dy/du) × (du/dx). Also called 'function of a function' rule.

链式法则： 用于求复合函数的导数：dy/dx = (dy/du) × (du/dx)。也称为“复合函数求导法则”。

Memory tip: Peel the layers – differentiate the outer function, keeping the inner unchanged, then multiply by the derivative of the inner function.

记忆技巧： 像剥洋葱——先对外层函数求导、内层不变，再乘以内层函数的导数。

Stationary point: he curve has zero gradient here, so it could be a maximum, minimum or point of inflection.

驻点： 满足 dy/dx = 0 的点。曲线在此处的斜率为零，可能是极大点、极小点或拐点。

Memory tip: Stationary = 'stopped moving' (gradient is zero). Use the second derivative or a sign table to classify.

记忆技巧： 驻点 = “静止不动”（斜率为零）。用二阶导数或符号表来判定类型。

Second derivative (d²y/dx²): The derivative of the derivative. It tells you about the concavity of the curve and helps classify stationary points.

二阶导数 (d²y/dx²)： 导数的导数。它描述曲线的凹凸性，并帮助判断驻点类型。

Memory tip: Positive d²y/dx² means the curve is bowl-shaped (minimum), negative means it is cap-shaped (maximum).

记忆技巧： 二阶导数为正表示曲线呈碗状（极小值），为负呈帽状（极大值）。

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5. Integration | 积分

Indefinite integral: tive of a function, written as ∫ f(x) dx = F(x) + C, where C is an arbitrary constant.

不定积分： 函数的一般反导数，记作 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。

Memory tip: Integration is 'anti-differentiation'. Always add +C to show there are infinitely many antiderivatives.

记忆技巧： 积分是“微分逆运算”。永远不要忘记加上 +C，表示无穷多个原函数。

Definite integral: Evaluated between limits a and b: ∫_a^b f(x) dx = F(b) – F(a). It gives the net area under the curve.

定积分： 在上下限 a、b 之间求值：∫_a^b f(x) dx = F(b) – F(a)。它给出曲线下的净面积。

Memory tip: Top minus bottom – substitute the upper limit, subtract the value from the lower limit. C cancels out.

记忆技巧： 上限代入减去下限代入。积分常数 C 在相减时抵消。

Area under a curve: The area bounded by y = f(x), the x-axis and lines x = a, x = b is given by ∫_a^b f(x) dx, provided f(x) ≥ 0.

曲线下方面积： 由 y = f(x)、x 轴及直线 x = a、x = b 围成的面积等于 ∫_a^b f(x) dx，前提是 f(x) ≥ 0。

Memory tip: If the curve dips below the x-axis, the integral gives a negative number. Use absolute values or split the integral.

记忆技巧： 若曲线落到 x 轴下方，积分会得出负值。此时需用绝对值或将积分分段

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