📚 Year 11 Cambridge Further Maths: Core Concepts Summary | Year 11 Cambridge 进阶数学：核心知识点梳理
Year 11 Cambridge Further Mathematics builds on IGCSE topics, introducing advanced algebraic manipulation, calculus, and trigonometric functions that form the foundation for A-Level Maths and Further Maths. This article provides a concise yet comprehensive review of the essential topics, with clear explanations and key examples to support revision.
Year 11 剑桥进阶数学在 IGCSE 基础上拓展，引入了高等代数运算、微积分与三角函数，为 A-Level 数学和进阶数学奠定重要基础。本文将对核心知识点进行简明而全面的梳理，提供清晰的解释和关键示例，助力复习备考。
1. Functions | 函数
A function f is a mapping from a set of inputs (domain) to a set of outputs (range) such that each input relates to exactly one output. We write f : x ↦ f(x) or simply use the notation f(x). The domain must be specified, and the range consists of all possible output values.
函数 f 是将一组输入（定义域）映射到一组输出（值域）的规则，其中每个输入恰好对应一个输出。我们写作 f : x ↦ f(x) 或直接使用 f(x) 表示。定义域必须给定，值域则由所有可能的输出值构成。
n of f⁻¹ is the range of f, and the graphs of f and f⁻¹ are reflections in the line y = x.
求反函数 f⁻¹(x) 时，我们将 y = f(x) 中的 x 和 y 互换，再解出 y。f⁻¹ 的定义域就是 f 的值域，且 f 与 f⁻¹ 的图像关于直线 y = x 对称。
Composite functions such as gf(x) mean apply f first, then g. The order matters: gf(x) ≠ fg(x) generally. gf(x) is defined only when the range of f is a subset of the domain of g.
复合函数如 gf(x) 表示先作用 f，再作用 g。顺序至关重要：一般来说 gf(x) ≠ fg(x)。只有当 f 的值域是 g 的定义域的子集时，gf(x) 才有定义。
2. Quadratic Functions & Inequalities | 二次函数与不等式
ewrites it as a(x + p)² + q, revealing the vertex (−p, q) and the axis of symmetry x = −p.
二次函数的形式为 f(x) = ax² + bx + c，a ≠ 0，其图像是抛物线。通过配方法可化为 a(x + p)² + q，直接读出顶点 (−p, q) 和对称轴 x = −p。
< 0 gives no real roots. The quadratic formula x = (−b ± √Δ)/(2a) solves ax² + bx + c = 0.
判别式 Δ = b² − 4ac 决定了实根个数：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。求根公式 x = (−b ± √Δ)/(2a) 可解 ax² + bx + c = 0。
To solve a quadratic inequality like ax² + bx + c > 0, first find the roots and sketch the parabola. The solution set is the x-values where the graph lies above the x-axis, expressed as unions of intervals.
解一元二次不等式如 ax² + bx + c > 0 时，先求出根并画出抛物线草图。解集即为图像在 x 轴上方的 x 值，用区间的并表示。
3. Polynomials & Factor Theorem | 多项式与因式定理
ial case of the Remainder Theorem: when P(x) is divided by (x − a), the remainder is P(a).
因式定理指出，(x − k) 是多项式 P(x) 的因式当且仅当 P(k) = 0。这是余式定理的特例：P(x) 除以 (x − a) 时，余数为 P(a)。
We combine these theorems with polynomial division or synthetic division to factorise cubics and higher-degree polynomials. Once a factor is found, the quotient is a polynomial of lower degree that can be further factorised or solved.
我们将这些定理结合多项式除法或综合除法，对三次及更高次多项式进行因式分解。一旦找到一个因式，商式就是一个更低次的多项式，可继续分解或求解。
Sketching polynomial graphs involves identifying x-intercepts (roots), y-intercept, and end behaviour determined by the leading term. Repeated roots cause the graph to touch or cross the x-axis in specific ways.
绘制多项式图像时，需确定 x 轴截距（根）、y 轴截距以及由首项决定的远端走势。重根会使图像以特定方式触碰或穿过 x 轴。
4. Simultaneous Equations | 联立方程
Systems of linear equations can be solved by substitution, elimination, or graphically as the intersection of lines. For one linear and one quadratic equation, substitution reduces the system to a quadratic in one variable, which is then solved.
线性方程组可通过代入法、消元法或图像法（求直线交点）求解。对于一次线性和一个二次方程联立，代入后可化为单变量的一元二次方程再求解。
The geometrical interpretation is key: the intersection points of a line and a parabola can be two, one (tangent), or none. Discriminant analysis of the resulting quadratic tells us how many solutions exist without fully solving.
几何解释至关重要：直线与抛物线的交点可能有两个、一个（相切）或零个。对得到的二次方程进行判别式分析，可以在不解出全部坐标的情况下判断解的个数。
5. Logarithmic & Exponential Functions | 对数与指数函数
The function y = aˣ for a > 0, a ≠ 1 has domain x ∈ ℝ e logarithm x = logₐ y. Thus logₐ a = 1 and logₐ 1 = 0. The natural exponential function eˣ and natural log ln x = log_e x are particularly important.
函数 y = aˣ（a > 0, a ≠ 1）的定义域为 x ∈ ℝ，值域为 y > 0。其反函数是对数 x = logₐ y，因此 logₐ a = 1，logₐ 1 = 0。自然指数函数 eˣ 和自然对数 ln x = log_e x 尤为重要。
. These are indispensable for solving equations where the unknown appears in the exponent.
关键对数运算法则：logₐ (MN) = logₐ M + logₐ N，logₐ (M/N) = logₐ M − logₐ N，logₐ (Mᵏ) = k logₐ M。这些法则对于求解未知数在指数位置的方程不可或缺。
Exponential growth and decay are modelled by functions of the form y = Aeᵏˣ or y = Abˣ nstants A (initial value) and k or b (growth/decay factor) are determined from given data.
指数增长与衰减可用形如 y = Aeᵏˣ 或 y = Abˣ 的函数建模。常数 A（初始值）和 k 或 b（增长/衰减因子）可根据给定数据确定。
6. Trigonometry | 三角学
60° and their radian equivalents must be memorised. For example, sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3. These values are derived from special right-angled triangles.
必须熟记 30°、45°、60° 及其弧度制对应值的精确三角比。例如 sin 30° = 1/2，cos 45° = √2/2，tan 60° = √3。这些值可由特殊直角三角形导出。
like sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ extend this for reciprocal trig functions.
基本恒等式：tanθ = sinθ/cosθ，以及勾股恒等式 sin²θ + cos²θ = 1。进一步的恒等式如 sec²θ = 1 + tan²θ 和 cosec²θ = 1 + cot²θ 则是倒数三角函数的扩展。
Solving trigonometric equations within a given interval often requires using these identities to reduce the equation to a single trig function, then applying the CAST diagram or graph to find all solutions.
在给定区间内求解三角方程通常需要利用上述恒等式将方程化为单一三角函数，再借助 CAST 图或函数图像求出所有解。
7. Circular Measure (Radians) | 弧度制
≈ 57.3°. Formulae are simplified in radian mode: arc length s = rθ, sector area A = ½ r²θ.
一弧度是指圆心角所对的弧长等于半径时的角度。π 弧度 = 180°，因此 1 rad ≈ 57.3°。在弧度制下公式得以简化：弧长 s = rθ，扇形面积 A = ½ r²θ。
he sector area. The segment area is given by A = ½ r²(θ − sinθ), provided θ is in radians.
计算弓形的周长和面积时需从扇形面积中减去三角形面积。若 θ 以弧度为单位，弓形面积公式为 A = ½ r²(θ − sinθ)。
8. Differentiation | 微分
The derivative f' The power rule: d/dx (xⁿ) = nxⁿ⁻¹. Constant multiples and sums differentiate term by term.
导数 f'(x) 表示 f 对 x 的变化率，由差商的极限定义。幂函数求导法则：d/dx (xⁿ) = nxⁿ⁻¹。常数倍与多项式可逐项求导。
The chain rule: dy/dx = dy/du × du/dx. For products, use the product rule: d/dx (uv) = u'v + uv'. The quotient rule: d/dx (u/v) = (u'v − uv')/v². These handle composite, product and rational functions.
链式法则：dy/dx = dy/du × du/dx。乘积法则：d/dx (uv) = u'v + uv'。商法则：d/dx (u/v) = (u'v − uv')/v²。这些法则分别处理复合函数、乘积函数和有理函数。
Stationary points occur where f'(x) = 0. The second derivative f''(x) determines their nature: f''(x) > 0 indicates a local minimum, f''(x) < 0 a maximum, and f''(x) = 0 may be a point of inflection.
驻点出现在 f'(x) = 0 处。二阶导数 f''(x) 决定驻点性质：f''(x) > 0 为局部极小值点，f''(x) < 0 为局部极大值点，f''(x) = 0 可能为拐点。
9. Integration | 积分
reverse process of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). The power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。幂函数积分法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。
x = b. It is evaluated as F(b) − F(a). Areas below the x-axis give negative contributions.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有向面积。其计算为 F(b) − F(a)。x 轴下方区域的面积贡献为负值。
To find the area between a curve and a line, or between two curves, integrate the difference of the functions over the intersection interval. Remember to take absolute values or split the region if curves cross.
求曲线与直线之间或两条曲线之间的面积时，需在交点区间内对函数之差积分。若曲线相交，务必取绝对值或分段积分。
10. Vectors in Two Dimensions | 二维向量
A vector has magnitude and direction, represented as (x, y) or xi + yj. The magnitude |v| = √x² + y². Addition, subtraction, and scalar multiplication are performed component-wise.
向量具有大小和方向，可表示为 (x, y) 或 xi + yj。模长 |v| = √x² + y²。向量的加减法和数乘运算均按分量进行。
ative to origin O is OA. The vector AB = OB − OA. A unit vector in the direction of v is v̂ = v / |v|. Parallel vectors are scalar multiples of each other.
点 A 相对于原点 O 的位置向量为 OA。向量 AB = OB − OA。沿 v 方向的单位向量为 v̂ = v / |v|。平行的向量彼此成标量倍数关系。
ty motion: r = r₀ + vt. The magnitude of the velocity vector is speed. Relative velocity vA relative to B = vA − vB.
速度和位移问题可用向量描述匀速运动：r = r₀ + vt。速度向量的模长即为速率。相对速度 vA相对于B = vA − vB。
Published by TutorHao | Further Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图