Support Pack 2: Contents (continued) | 学习包 2：后续内容

📚 Support Pack 2: Contents (continued) | 学习包 2：后续内容

This article continues the AQA A-Level Mathematics Support Pack 2 by covering the later topics that often appear in Paper 1, Paper 2 and the applied papers. The focus is on consolidating core techniques, common exam traps, and the links between pure mathematics, statistics and mechanics.

本文继续 AQA A-Level 数学学习包 2 的内容，涵盖试卷 1、试卷 2 及应用卷中经常出现的后续主题。重点在于巩固核心方法、常见考试陷阱以及纯数学、统计和力学之间的联系。

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1. Exponentials and Natural Logarithms | 指数与自然对数

The exponential function y = eˣ the laws of logs and exponents, including ln(ab) = ln a + ln b and ln(a/b) = ln a - ln b.

指数函数 y = eˣ 及其反函数 y = ln x 是 AQA 纯数学的核心。你需要熟练运用对数和指数法则，包括 ln(ab) = ln a + ln b 以及 ln(a/b) = ln a - ln b。

eˣ = y ⇔ ln y = x

Common exam questions ask you to solve equations such as e²ˣ = 5 or to differentiate and integrate eᵏˣ. Remember that d/dx(eᵏˣ) = k eᵏˣ and ∫ eᵏˣ dx = (1/k)eᵏˣ + C. Always check whether the base e is involved before taking natural logarithms.

常见考题要求解如 e²ˣ = 5 的方程，或对 eᵏˣ 进行微分和积分。记住 d/dx(eᵏˣ) = k eᵏˣ，∫ eᵏˣ dx = (1/k)eᵏˣ + C。在取自然对数之前，始终先确认是否涉及底数 e。

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2. Trigonometric Identities and Equations | 三角恒等式与方程

cularly sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ.

AQA 经常考查两个勾股恒等式：sin² θ + cos² θ = 1 和 1 + tan² θ = sec² θ。你还应掌握倍角公式，特别是 sin 2θ = 2 sin θ cos θ 以及 cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ。

t values such as sin 30° = 1/2 and tan 45° = 1 are frequently needed without a calculator.

在给定区间内解三角方程时，画图或使用 CAST 图来找出所有解。始终检查题目要求使用弧度还是角度。像 sin 30° = 1/2 和 tan 45° = 1 这样的精确值经常需要在不使用计算器的情况下给出。

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3. Further Differentiation | 进阶微分

hain rule, product rule and quotient rule. For y = f(g(x)), the chain rule gives dy/dx = f'(g(x)) g'(x). For y = uv, the product rule is dy/dx = u'v + uv'. The quotient rule can be quoted as dy/dx = (vu' - uv') / v².

除了基本幂函数法则外，AQA 还要求熟练运用链式法则、乘法法则和除法法则。对于 y = f(g(x))，链式法则给出 dy/dx = f'(g(x)) g'(x)。对于 y = uv，乘法法则是 dy/dx = u'v + uv'。除法法则可写作 dy/dx = (vu' - uv') / v²。

implicit equation, differentiate both sides with respect to x, treating y as a function of x.

你还会遇到参数微分和隐函数微分。如果 x 和 y 都以参数 t 给出，则 dy/dx = (dy/dt) / (dx/dt)。对于隐函数方程，对两边关于 x 求导，并将 y 视为 x 的函数。

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4. Applications of Differentiation | 微分的应用

if d²y/dx² < 0 it is a local maximum, and if d²y/dx² = 0 further investigation is needed.

驻点出现在 dy/dx = 0 处。使用二阶导数来判断其类型：如果 d²y/dx² > 0，该点是局部极小值；如果 d²y/dx² < 0，该点是局部极大值；如果 d²y/dx² = 0，则需要进一步分析。

unction of one variable, then solve dy/dx = 0 and justify the nature of the turning point.

微分还可用于建立变化率和优化问题的模型。在实际问题中，要清楚地定义变量，将需要优化的量写成单一变量的函数，然后解 dy/dx = 0 并说明该驻点的性质。

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5. Integration Techniques | 积分技巧

egration by parts. Integration by parts uses the formula ∫ u dv/dx dx = uv - ∫ v du/dx dx.

积分是微分的逆运算。对 AQA 来说，最重要的技巧是反链式法则、换元积分法和分部积分法。分部积分法使用公式 ∫ u dv/dx dx = uv - ∫ v du/dx dx。

When integrating products such as x eˣ or x sin x, choose u as the algebraic factor and dv/dx as the exponential or trig factor. Always add the constant of integration for indefinite integrals, as omitting + C is a frequent loss of marks.

在积分如 x eˣ 或 x sin x 的乘积时，选择代数因子作为 u，把指数或三角因子作为 dv/dx。不定积分一定要加上积分常数，漏掉 + C 是常见的失分点。

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6. Definite Integration and Area | 定积分与面积

the signed area between the curve and the x-axis. Areas below the axis are negative. To find the total area enclosed, split the interval at the roots and take absolute values of each definite integral.

定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴之间的有符号面积。位于 x 轴下方的面积为负。要求围成的总面积，应在根处拆分区间，并取每个定积分的绝对值。

ch the graphs first to identify which curve is above the other, and solve for intersection points to fix the limits.

你可能还需要求两条曲线之间的面积。如果在区间 [a, b] 上 f(x) ≥ g(x)，则面积为 ∫ₐᵇ [f(x) - g(x)] dx。先画出图像以确定哪条曲线在上方，并解出交点以确定积分限。

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7. Numerical Methods | 数值方法

on, and the Newton-Raphson method. Newton-Raphson uses the iteration xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ).

AQA 包括在无法用代数精确求解时使用的数值方法。你应该知道如何使用变号法、迭代法和牛顿-拉弗森法。牛顿-拉弗森法使用迭代式 xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)。

When using a sign-change method, show that f(a) and f(b) have opposite signs and state that the function is continuous. For iteration, write down the rearrangement clearly and check convergence by ensuring the gradient of the iteration function is between -1 and 1 near the root.

使用变号法时，要说明 f(a) 和 f(b) 异号，并指出函数连续。对于迭代法，要清楚地写出重排后的方程，并通过确保迭代函数在根附近的梯度介于 -1 和 1 之间来检查收敛性。

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8. Vectors | 向量

mn vector or in terms of unit vectors i, j and k. The magnitude of a vector a = xi + yj is |a| =

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