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Edexcel A-Level Pure Mathematics 4: Core Skills from Practice Set 200 | Edexcel A-Level Pure Mathematics 4：练习集 200 核心技能

2026-08-29 18:32 · 本文部分内容由AI辅助创作

📚 Edexcel A-Level Pure Mathematics 4: Core Skills from Practice Set 200 | Edexcel A-Level Pure Mathematics 4：练习集 200 核心技能

This revision guide consolidates the most frequently tested skills in Edexcel A-Level Pure Mathematics 4, including the techniques that appear in Practice Set 200. Each section pairs a concise explanation with the key formula or method you must apply under exam conditions, and every method is written in the exact form examiners expect to see in a final answer.

本复习指南整合 Edexcel A-Level Pure Mathematics 4 中最常考查的技能，包括练习集 200 中出现的方法。每一节将简明解释与考试条件下必须应用的关键公式或方法配对，并且所有写法都按照考官在最终答案中期望看到的形式给出。

1. Partial Fractions | 部分分式

When integrating a rational function, first express it as partial fractions if the denominator factorises. For distinct linear factors such as (x + a)(x + b), write the fraction in the form A/(x + a) + B/(x + b), where A and B are constants found by substitution or by equating coefficients.

对有理函数进行积分时，如果分母可以因式分解，应首先将其表示为部分分式。对于不同的线性因子，如 (x + a)(x + b)，将分式写成 A/(x + a) + B/(x + b) 的形式，其中 A 和 B 是通过代入或比较系数求得的常数。

P(x)/[(x + a)(x + b)] = A/(x + a) + B/(x + b)

If the denominator contains a repeated linear factor (x + a)², include both A/(x + a) and B/(x + a)². For an improper fraction where the degree of the numerator is greater than or equal to the degree of the denominator, perform polynomial long division first, then split the remainder term.

如果分母包含重复线性因子 (x + a)²，则应同时包含 A/(x + a) 和 B/(x + a)²。对于分子次数大于或等于分母次数的假分式，应首先进行多项式长除法，然后再对余项部分拆分。

Q(x)/[(x + a)²(x + b)] = A/(x + a) + B/(x + a)² + C/(x + b)

onstant quickly for distinct linear factors, but it fails for repeated factors and for irreducible quadratics. Substituting a value such as x = −a makes every other term vanish, which is usually the fastest route to a single unknown.

考试中有三条实用规则可以节省时间。写任何形式之前一定要把分母完全因式分解。掩盖法对不同的线性因子能快速给出各常数，但对重复因子和不可约二次因子无效。代入 x = −a 这样的值可以让其余项全部消失，这通常是求出单个未知数最快的途径。

2. Integrating with Partial Fractions | 用部分分式进行积分

Once the fraction is split, each distinct linear factor integrates to a natural logarithm. This is the single most common six-mark question in Practice Set 200, and marks are awarded for the correct coefficient as well as the correct logarithm.

分式拆分之后，每一个不同的线性因子积分后都得到一个自然对数。这是练习集 200 中最常见的六分题，分数同时授予正确的系数和正确的对数。

∫ [A/(x + a) + B/(x + b)] dx = A ln|x + a| + B ln|x + b| + C

A repeated factor behaves differently: it produces a negative power rather than a logarithm. Students who forget this lose marks even when the partial fractions themselves are correct.

重复因子的处理方式不同：它产生的是负幂项而不是对数。即使部分分式本身求对了，忘记这一点的学生仍会丢分。

∫ B/(x + a)² dx = −B/(x + a) + C

Always keep the modulus signs inside the logarithm and never omit the constant of integration. If the question gives a definite integral, substitute the limits only after the antiderivative is complete, and if an answer contains logarithms of several terms, check whether the question asks for a single combined logarithm.

对数内部一定要保留绝对值符号，并且绝不能漏掉积分常数。如果是定积分，要在求出原函数之后再代入上下限；如果答案包含若干项的对数，要检查题目是否要求合并成一个对数。

3. Parametric Equations | 参数方程

re t is the parameter. To convert to Cartesian form, eliminate t using substitution, or use a trigonometric identity when the equations involve sin t and cos t.

参数曲线由一对方程 x = f(t) 与 y = g(t) 定义，其中 t 是参数。要转化为直角坐标形式，可以用代入法消去 t；当方程中出现 sin t 和 cos t 时，则使用三角恒等式。

x = t², y = 2t ⇒ t = y/2 ⇒ x = y²/4 ⇒ y² = 4x

The range of t matters. A restricted parameter range usually gives only part of the Cartesian curve, and questions frequently ask you to state the domain or range of the resulting equation. Always convert the restriction on t into a restriction on x and y.

t 的取值范围很重要。参数范围受限时，通常只对应直角坐标曲线的一部分，而题目常常要求你写出所得方程的定义域或值域。一定要把 t 的限制转换成对 x 和 y 的限制。

Sketching is also examined. Plot a table of values at convenient parameter values, mark the direction of increasing t with an arrow, and label any points where the curve crosses the axes or where the tangent is horizontal or vertical.

作图的考查同样常见。取几个方便的参数值列表描点，用箭头标出 t 增大时曲线的方向，并标注曲线与坐标轴的交点以及切线水平或垂直的点。

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4. Parametric Differentiation | 参数求导

Differentiating a parametric curve uses the chain rule in reverse. Differentiate each equation with respect to t first, then divide. This formula must be quoted accurately because it carries a method mark on its own.

对参数曲线求导使用的是链式法则的逆用。先分别对 t 求导，再相除。这个公式必须准确写出，因为它本身就占一个方法分。

dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0

For the second derivative, differentiate dy/dx with respect to t first and then divide by dx/dt again. A common error is to differentiate dy/dx with respect to x directly, which is not valid here.

求二阶导数时，先对 dy/dx 关于 t 求导，再除以 dx/dt。常见错误是直接对 dy/dx 关于 x 求导，在这里这是不成立的。

d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)

Tangents and normals follow immediately. Find the value of t at the given point, compute dy/dx there, and substitute into the straight-line equation. For the normal, use the negative reciprocal of the gradient.

切线方程和法线方程随之可得。先求该点对应的 t 值，算出该处的 dy/dx，再代入直线方程。法线用的是切线斜率的负倒数。

5. Implicit Differentiation | 隐函数求导

When y is not given explicitly as a function of x, differentiate every term with respect to x, remembering that differentiating any expression in y produces an extra factor of dy/dx through the chain rule.

当 y 没有明确写成 x 的函数时，需要对每一项关于 x 求导，并记住：对任何含 y 的表达式求导都会因链式法则而多出一个 dy/dx 因子。

d/dx [yⁿ] = n yⁿ⁻¹ (dy/dx)

Products such as xy require the product rule, giving x(dy/dx) + y. Rearrange the resulting equation to make dy/dx the subject, collecting all dy/dx terms on one side first.

像 xy 这样的乘积需要用到乘积法则，结果为 x(dy/dx) + y。整理所得方程，把 dy/dx 作为主元，先把所有含 dy/dx 的项集中到一边。

x² + y² = 25 ⇒ 2x + 2y(dy/dx) = 0 ⇒ dy/dx = −x/y

olve simultaneously with the original curve equation, since both conditions must hold at the same point.

主要应用有两个：求曲线上某点的斜率，以及求驻点。求驻点时令 dy/dx = 0，并与原曲线方程联立求解，因为两个条件必须在同一点上同时成立。

6. Binomial Expansion for Rational Indices | 有理指数的二项式展开

When the index is negative or fractional, the expansion becomes an infinite series. The formula is the same as the familiar binomial theorem, but the factorials never terminate, so the expansion is only valid for a restricted range of x.

当指数为负数或分数时，展开式成为无穷级数。公式与熟悉的二项式定理相同，但阶乘不会终止，因此展开式只在 x 的某一限定范围内成立。

(1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + … , valid for |x| < 1

For expressions of the form (a + bx)ⁿ, factor out aⁿ first so that the bracket starts with 1. The convergence condition must then be written in terms of the original variable.

对于形如 (a + bx)ⁿ 的表达式，首先要提出 aⁿ，使括号内以 1 开头。收敛条件随后必须用原变量表示。

(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ, valid for |bx/a| < 1

Signs deserve particular care. With n negative, all coefficients keep the sign of the corresponding binomial coefficient product, and with fractional n the coefficients decrease in a way that is easy to mis-multiply. Approximations are also examined: substitute a small value of x into the truncated series and state the accuracy in the required form.

符号尤其需要注意。当 n 为负数时，各项系数的符号由对应的二项系数乘积决定；当 n 为分数时，系数递减的方式很容易乘错。近似计算也是考点：把较小的 x 值代入截断的级数，并按题目要求的形式给出精度。