📚 AS Mathematics: End-of-Term Revision Checklist | AS 数学：期末复习提纲
A well-structured revision checklist is essential for AS Mathematics end-of-term success. This guide summarises the core topics across pure mathematics, statistics and mechanics, helping you identify key formulas, concepts and problem-solving strategies. Use it to track your progress and focus on areas that need extra practice. Each section pairs the English and Chinese explanation so that you can revise the mathematics and the vocabulary at the same time.
一份结构清晰的复习提纲对于 AS 数学期末成功至关重要。本指南总结了纯数学、统计与力学的核心专题，帮助你梳理关键公式、概念与解题策略。你可以用它来跟踪复习进度，并集中攻克薄弱环节。每个小节都采用中英对照的方式呈现，让你在复习数学知识的同时，也顺便熟悉专业术语的英文表达。
1. Algebra and Functions | 代数与函数
ⁿ = aᵐ⁻ⁿ, and expand products such as (2x + 3)(x − 5). Rearranging formulas and solving simultaneous linear equations should be completely automatic before you move on to harder topics.
代数是 AS 数学的核心。你必须能够化简根式、分母有理化、熟练运用指数律 aᵐ × aⁿ = aᵐ⁺ⁿ 与 aᵐ ÷ aⁿ = aᵐ⁻ⁿ，并能展开如 (2x + 3)(x − 5) 这样的乘积。在进入更难的专题之前，公式变形与解线性方程组应当已经是完全自动化的技能。
Function notation is a classic exam pitfall. Make sure you can state the domain and range of a function, and remember that the domain is the set of allowed inputs while the range is the set of possible outputs. For a composite function, apply the inner function first: fg(x) means f(g(x)).
函数记号是考试中的经典陷阱。务必能写出函数的定义域与值域，记住定义域是允许输入的集合，而值域是可能输出的集合。对于复合函数，先作用内层函数：fg(x) 表示 f(g(x))。
he range of the original function. A useful check is that f(f⁻¹(x)) = x for every x in the domain of f⁻¹.
求反函数 f⁻¹(x) 的方法是：先写 y = f(x)，交换 x 与 y，再整理成以 y 为主元的形式。一定要写出反函数的定义域，它等于原函数的值域。一个实用的检验方法是：对反函数定义域内的每个 x，都应有 f(f⁻¹(x)) = x。
- Expand and factorise confidently, including the difference of two squares.
- Simplify algebraic fractions and cancel common factors.
- Sketch graphs of y = f(x), y = f(x) + a and y = f(x + a).
- Solve equations involving algebraic fractions and check for extraneous roots.
- 熟悉平方差公式、提公因式与分组分解。
- 会化简代数分式并约去公因式。
- 能画出 y = f(x)、y = f(x) + a 与 y = f(x + a) 的草图。
- 会解含代数分式的方程，并检验是否产生增根。
2. Quadratics, Discriminant and Inequalities | 二次式、判别式与不等式
Solving quadratics is a fundamental skill. You should be able to solve them by factorisation, by completing the square, or by the quadratic formula. The discriminant decides the nature of the roots before you even solve the equation.
解二次方程是基本功。你应当能够用因式分解、配方法或求根公式来求解。判别式可以让你在动手解方程之前就判断出根的性质。
x = (−b ± √b² − 4ac) / 2a
Δ = b² − 4ac
oot and the curve touches the x-axis. If Δ < 0 there are no real roots and the curve does not meet the x-axis at all.
若 Δ > 0，有两个不相等的实根，抛物线与 x 轴交于两点；若 Δ = 0，有一个重根，抛物线与 x 轴相切；若 Δ < 0，没有实根，抛物线与 x 轴不相交。
Completing the square rewrites the expression in the form a(x + p)² + q, which immediately gives the vertex of the parabola and is often the fastest route to a maximum or minimum value. When solving quadratic inequalities, sketch the graph first, then read off the solution set.
配方把表达式改写为 a(x + p)² + q 的形式，可以直接读出抛物线的顶点，也常常是求最大值或最小值最快的途径。解二次不等式时，先画图，再从图上读出解集。
One rule that costs marks every year: whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip. So −2x > 6 gives x < −3, not x > −3. Write this reminder at the top of your rough work.
每年都会让考生丢分的一条规则：不等式两边同时乘以或除以一个负数时，不等号方向必须改变。因此 −2x > 6 的解是 x < −3，而不是 x > −3。把这条提醒写在草稿纸的最上方。
3. Coordinate Geometry | 坐标几何
Coordinate geometry rewards accuracy rather than cleverness. You need the gradient formula, the equation of a straight line, and the conditions for parallel and perpendicular lines. Parallel lines share the same gradient, while perpendicular lines have gradients that multiply to −1.
坐标几何考查的是准确而非聪明。你需要掌握斜率公式、直线方程，以及平行与垂直的条件。平行直线斜率相同，而垂直直线的斜率之积等于 −1。
m = (y₂ − y₁) / (x₂ − x₁)
y − y₁ = m(x − x₁)
ints and the distance between them are also frequently tested, so keep the formulas handy.
要熟记直线的三种形式：y = mx + c、y − y₁ = m(x − x₁) 以及 ax + by + c = 0。两点中点坐标与两点间距离也经常考，公式要随手可用。
(x − a)² + (y − b)² = r²
The equation of a circle with centre (a, b) and radius r appears in almost every paper. You may be asked to find the centre and radius by completing the square, or to prove that a given line is a tangent by substituting and checking that the discriminant is zero.
圆心为 (a, b)、半径为 r 的圆的方程几乎每张卷子都会出现。你可能需要配方求出圆心与半径，也可能需要用代入法并验证判别式是否为零，来证明某条直线是切线。
4. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are guaranteed exam content. For an arithmetic sequence with first term a and common difference d, the nth term and the sum of the first n terms are given by the standard formulas.
等差数列与等比数列是必考内容。对于首项为 a、公差为 d 的等差数列，第 n 项与前 n 项和由以下标准公式给出。
aₙ = a + (n − 1)d
Sₙ = n/2 × [2a + (n − 1)d]
is aₙ = arⁿ⁻¹ and the sum of the first n terms is Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. When |r| < 1 the series converges and the sum to infinity is S∞ = a / (1 − r).
对于公比为 r 的等比数列，第 n 项为 aₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ) / (1 − r)（r ≠ 1）。当 |r| < 1 时级数收敛，无穷项和为 S∞ = a / (1 − r)。
Binomial expansion is the other big topic here. Learn to expand (a + b)ⁿ using binomial coefficients, and remember that the general term lets you find a single coefficient without expanding everything.
二项式展开是这一章的另一个重点。要学会用二项式系数展开 (a + b)ⁿ，并记住通项公式可以让你不必全部展开就能求出某一项的系数。
- Know when a series converges and when it diverges.
- Use sigma notation to write sums compactly.
- Apply sequences to real contexts such as savings and depreciation.
- 会判断级数何时收敛、何时发散。
- 会用求和符号简洁地表示和式。
- 能把数列应用到储蓄、折旧等实际情境中。
5. Trigonometry | 三角学
Trigonometry in AS Mathematics moves beyond right-angled triangles. You must be fluent with sine, cosine and tangent for any angle, including angles in all four quadrants, and you must be able to work in both degrees and radians.
AS 数学的三角学已经超出了直角三角形的范围。你必须熟练掌握任意角的正弦、余弦与正切，包括四个象限中的角，并且能够同时在角度制与弧度制下运算。
sin²θ + cos²θ = 1
tanθ = sinθ / cosθ
. Remember that π radians = 180°, so arc length is rθ and sector area is ½r²θ when θ is in radians.
要记住 30°、45°、60° 的精确值，并理解正弦、余弦、正切图像的形状，包括周期与对称性。记住 π 弧度 = 180°，因此在 θ 为弧度时，弧长为 rθ，扇形面积为 ½r²θ。
For non-right-angled triangles you need the sine rule and the cosine rule. The area formula follows from the same setup and is often worth an easy mark.
对于非直角三角形，你需要正弦定理与余弦定理。面积公式来自同一套设定，往往能轻松拿到一分。
a / sin A = b / sin B = c / sin C
a² = b² + c² − 2bc cos A
Area = ½ ab sin C
Watch out for the ambiguous case of the sine rule, where two different triangles can satisfy the same data. Sketching the triangle before calculating usually prevents this error.
要小心正弦定理的歧义情形：同一组已知条件可能对应两个不同的三角形。计算前先画草图，通常就能避免这个错误。
6. Exponentials and Logarithms | 指数与对数
Exponentials and logarithms are inverse operations, and almost every question in this topic depends on that relationship. Converting between the two forms is the single most useful skill.
指数与对数互为逆运算，这一章几乎所有题目都依赖这一关系。在对数式与指数式之间互相转换，是最有用的一项技能。
logₐ b = c ⇔ aᶜ = b
g(x / y) = log x − log y, and log(xⁿ) = n log x. They apply to any base, including base e.
三条对数运算法则承担了大部分工作：log(xy) = log x + log y，log(x / y) = log x − log y，log(xⁿ) = n log x。它们适用于任何底数，包括底数 e。
The number e ≈ 2.718 appears constantly in growth and decay models. The function y = eˣ rse y = ln x should be sketched automatically, and you should know that the graph of y = eˣ is always above the x-axis.
常数 e ≈ 2.718 在增长与衰减模型中反复出现。函数 y = eˣ 及其反函数 y = ln x 应当能自动画出草图，你还要知道 y = eˣ 的图像始终位于 x 轴上方。
- Solve equations of the form aˣ = b by taking logarithms of both sides.
- Use the change of base formula to evaluate logarithms on a calculator.
- Model exponential growth and decay problems, including half-life questions.
- 会用两边取对数的方法解形如 aˣ = b 的方程。
- 会用换底公式在计算器上求对数值。
- 能建立指数增长与衰减模型，包括半衰期问题。
7. Differentiation | 微分
Differentiation is the largest single topic in AS pure mathematics. Start with the power rule, which applies to every polynomial you will meet.
微分是 AS 纯数学中分值最高的单一专题。先从幂法则开始，它会适用于你遇到的所有多项式。
y = axⁿ → dy/dx = naxⁿ⁻¹
Rewrite roots and fractions as powers before differentiating. For example, √x s x^½ and 1/x² becomes x⁻². Marks are frequently lost by differentiating before rewriting.
求导之前，先把根式和分式改写为幂的形式。例如 √x 写成 x^½，1/x² 写成 x⁻²。很多失分都是因为没有先改写就直接求导。
onary point using the second derivative: if d²y/dx² > 0 it is a minimum, if d²y/dx² < 0 it is a maximum.
你必须会求某点处的切线或法线方程，并通过令 dy/dx = 0 找到驻点。用二阶导数判断驻点的性质：若 d²y/dx² > 0 为极小值，若 d²y/dx² < 0 为极大值。
Increasing and decreasing functions, rates of change and optimisation problems all rely on the same toolkit. In optimisation, define the variable, form an expression, differentiate, solve, and always answer the question that was actually asked.
函数的增减性、变化率与最优化问题都依赖同一套工具。解最优化题时，先设变量、列出表达式、求导、求解，最后一定要回答题目真正问的问题。
8. Integration | 积分
Integration is differentiation in reverse, and at AS level you mainly need the power rule for integration together with definite integrals for areas.
积分是微分的逆运算，在 AS 阶段你主要需要积分的幂法则，以及用于求面积的定积分。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1
Never forget the constant of integration for indefinite integrals. It is a mark that is lost far more often than it should be. For definite integrals the constant cancels, so the notation brackets are used instead.
求不定积分时千万不要忘记积分常数。这一分丢得远比应该丢的多。定积分中常数会相消，所以改用方括号代入上下限的写法。
To find the area between a curve and the x-axis, integrate between the limits where the curve crosses the axis. If part of the curve lies below the axis, that part produces a negative value, so split the integral and take the absolute value of each piece.
求曲线与 x 轴之间的面积时，在曲线与轴的交点作为上下限进行积分。如果曲线有一部分位于 x 轴下方，那部分积分结果为负，因此要把积分拆开，并对每一段取绝对值。
- Evaluate definite integrals accurately with a calculator check where allowed.
- Find areas between a curve and a straight line by subtracting.
- Interpret the result of an integral in context, including its units.
- 会用计算器复核定积分的结果（在允许的情况下）。
- 会用相减的方法求曲线与直线之间的面积。
- 能在实际情境中解释积分结果的意义，包括单位。
9. Statistics: Data, Probability and Distributions | 统计：数据、概率与分布
Statistics questions are often the easiest marks on the paper, provided you read carefully. Know the difference between mean, median and mode, and understand when each is the appropriate average to quote.
只要仔细读题，统计题往往是卷面上最容易拿分的部分。要清楚平均数、中位数与众数的区别，并理解在什么情况下应当引用哪一个作为代表值。
Measures of spread matter just as much as measures of location. The range is simple but sensitive to outliers, while the interquartile range describes the middle half of the data and is far more robust.
离散程度的度量与集中趋势的度量同样重要。极差计算简单，但对异常值敏感；四分位距描述中间一半数据，稳健得多。
| Measure | 度量 | Formula | 公式 | Use | 用途 |
| Variance | 方差 | σ² = Σ(x − x̄)² / n | Spread in squared units | 以平方单位衡量离散 |
| Standard deviation | 标准差 | σ = √(Σ(x − x̄)² / n) | Spread in original units | 与原数据同单位 |
| Interquartile range | 四分位距 | IQR = Q₃ − Q₁ | Robust to outliers | 不受异常值影响 |
For probability, use Venn diagrams and tree diagrams to organise the information. The addition rule and the conditional probability formula are both essential.
概率部分，用韦恩图与树状图来整理信息。加法法则与条件概率公式都不可或缺。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(B | A) = P(A ∩ B) / P(A)
The binomial distribution models the number of successes in n independent trials with constant probability p. Learn to recognise the setting, quote X ~ B(n, p), and use the formula for a single value.
二项分布用来描述 n 次独立试验中成功次数，每次成功概率恒为 p。要学会识别适用情境，写出 X ~ B(n, p)，并用公式求单个取值对应的概率。
P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ
10. Mechanics: Kinematics and Forces | 力学：运动学与力
Mechanics questions are best approached with a clear diagram and a named list of the quantities you know. The five constant-acceleration formulas are the backbone of kinematics.
解力学题最好的办法是先画清晰的示意图，再列出已知量的名称清单。五个匀加速公式是运动学的骨架。
v = u + at
s = ut + ½at²
v² = u² + 2as
Always write down which quantity is missing, because that tells you which formula to use. Choose a positive direction at the start and keep it consistently for the whole question.
先写下哪个量是未知或不涉及的，因为这决定了该用哪个公式。开始时就选定正方向，并在整道题中保持一致。
Velocity-time graphs carry a great deal of information: the gradient gives acceleration and the area under the graph gives displacement. If the graph drops below the time axis, the motion has reversed direction.
速度-时间图包含大量信息：图像的斜率给出加速度，图像下方的面积给出位移。如果图像落到时间轴下方，说明运动方向已经反向。
For forces, resolve in two perpendicular directions and apply Newton' on of motion. On a slope, the weight component along the plane is mg sinθ and perpendicular to the plane it is mg cosθ.
处理力的问题时，沿两个互相垂直的方向分解，并沿运动方向应用牛顿第二定律 F = ma。在斜面上，重力沿斜面的分量为 mg sinθ，垂直斜面的分量为 mg cosθ。
- Draw free-body diagrams showing every force acting on the object.
- Remember that friction opposes motion and satisfies F ≤ μR.
- Handle connected particles by treating the system as a whole, then isolating one body.
- 画出受力分析图，标出作用在物体上的每一个力。
- 记住摩擦力与运动方向相反，且满足 F ≤ μR。
- 处理连接体时，先整体分析，再隔离其中一个物体。
11. Exam Technique and a Four-Week Revision Plan | 考试技巧与四周复习计划
Content knowledge alone is not enough. The final weeks should combine topic revision with timed practice, so that accuracy and speed improve together. A simple four-week structure works well.
只有知识储备是不够的。最后几周应当把专题复习与限时训练结合起来，让准确率与速度同步提升。一个简单的四周安排就很有效。
| Week | 周次 | Focus | 重点 |
| 1 | 第 1 周 | Pure algebra, quadratics, coordinate geometry | 纯数学：代数、二次式、坐标几何 |
| 2 | 第 2 周 | Trigonometry, exponentials, calculus | 三角、指数对数、微积分 |
| 3 | 第 3 周 | Statistics and mechanics, mixed papers | 统计与力学，混合卷 |
| 4 | 第 4 周 | Timed past papers, error log, formula recall | 限时真题、错题本、公式默写 |
Keep an error log
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