📚 Applications to Modelling | 建模应用
Mathematical modelling is the process of translating a real-world situation into mathematical language, analysing the resulting equations or functions, and interpreting the solution in context. In Edexcel A Level Mathematics, modelling questions appear across pure, mechanics and statistics, requiring you to choose an appropriate model, use parameters, and discuss limitations.
数学建模是把现实情境翻译成数学语言、分析所得方程或函数、并在实际背景中解释解的过程。在 Edexcel A Level 数学中，建模题贯穿纯数、力学和统计，要求你选择适当的模型、使用参数并讨论局限性。
1. What is a Mathematical Model? | 什么是数学模型？
A mathematical model is a simplified representation of a real system using equations, functions, graphs or diagrams. It usually involves input variables, output variables, parameters and assumptions that make the problem tractable.
数学模型是使用方程、函数、图像或图形对真实系统进行的简化表示。它通常包含输入变量、输出变量、参数以及使问题可解的假设。
The modelling cycle is: formulate, solve, interpret, validate and refine. You should never treat a model as exact; it is useful only when its assumptions are reasonable for the context.
建模循环为：建立、求解、解释、验证和改进。绝不能把模型视为精确无误；只有当假设在相关背景下合理时，模型才有效。
2. Formulating a Model: Variables and Parameters | 建立模型：变量与参数
Begin by defining variables clearly, including units. For example, let t be time in seconds, and let h metres be the height of the ball above the ground. Parameters such as initial velocity u and acceleration due to gravity g are constants within one scenario but may change between scenarios.
首先清晰定义变量及其单位。例如，设 t 为时间（秒），设 h 米为球离地面的高度。参数如初速度 u 和重力加速度 g 在某一情境中是常数，但在不同情境中可能不同。
A good formulation identifies dependent and independent variables. In y = mx + c, x is the independent variable, y is the dependent variable, m is the gradient, and c is the intercept. These parameters are estimated from data or given by the context.
好的建模会区分因变量和自变量。在 y = mx + c 中，x 是自变量，y 是因变量，m 是斜率，c 是截距。这些参数由数据估计或由背景给定。
3. Linear Models and Direct Proportion | 线性模型与正比例
Linear models are used when a quantity changes at a constant rate. The equation y = mx + c gives a straight line; m represents the rate of change, and c is the initial value when x = 0.
当某个量以恒定速率变化时使用线性模型。方程 y = mx + c 表示一条直线；m 表示变化率，c 是 x = 0 时的初始值。
y = mx + c
Direct proportion is the special case y = kx, where c = 0. In mechanics, Hooke’s law F = kx models the extension of a spring up to its elastic limit, so the model is only valid within that range.
正比例是 c = 0 的特殊情况 y = kx。在力学中，胡克定律 F = kx 用于模拟弹簧在弹性限度内的伸长，因此该模型仅在该范围内有效。
F = kx
4. Quadratic and Projectile Models | 二次与抛体模型
Many modelling situations involve acceleration or area, giving quadratic functions. A projectile moving under constant gravitational acceleration has vertical displacement h = h₀ + ut − ½gt², where g ≈ 9.8 m s⁻².
许多建模情境涉及加速度或面积，从而产生二次函数。在恒定重力加速度下运动的抛体，其竖直位移为 h = h₀ + ut − ½gt²，其中 g ≈ 9.8 m s⁻²。
h = h₀ + ut − ½gt²
To find the maximum height, complete the square or set the derivative equal to zero. The roots of h = 0 give the time when the projectile returns to the initial level; negative times are usually discarded because they are outside the domain.
要求最大高度，可配方或令导数为零。h = 0 的根给出抛体回到初始高度的时间；负时间通常舍去，因为它们不在定义域内。
5. Exponential Growth and Decay | 指数增长与衰减
If a quantity grows or decays at a rate proportional to its current value, the model is exponential. The general forms are N = N₀eᵏᵗ for growth (k > 0) and N = N₀e⁻ᵏᵗ for decay (k > 0). Here N₀ is the initial amount and k is the growth/decay constant.
若某量以与其当前值成比例的速率增长或衰减，模型就是指数型。增长 (k > 0) 的一般形式为 N = N₀eᵏᵗ，衰减 (k > 0) 为 N = N₀e⁻ᵏᵗ。其中 N₀ 是初始数量，k 是增长/衰减常数。
N = N₀eᵏᵗ
Doubling time for growth is ln 2 / k, and half-life for decay is ln 2 / k. In exam questions, you often take logarithms of both sides to linearise the equation: ln N = ln N₀ + kt.
增长的倍增时间为 ln 2 / k，衰减的半衰期也是 ln 2 / k。在考试题中，常对等式两边取对数使其线性化：ln N = ln N₀ + kt。
ln N = ln N₀ + kt
6. Logarithmic Transformations in Modelling | 建模中的对数变换
When data appear to follow a power law y = axⁿ or an exponential law y = abˣ, logarithms convert the relationship into a straight line. For y = axⁿ, taking logs gives log y = log a + n log x, so plotting log y against log x yields gradient n and intercept log a.
当数据似乎遵循幂律 y = axⁿ 或指数律 y = abˣ 时，取对数可把关系转化为直线。对于 y = axⁿ，取对数得 log y = log a + n log x，因此绘制 log y 对 log x 的图像，斜率为 n，截距为 log a。
log y = log a + n log x
For y = abˣ, taking natural logs gives ln y = ln a + x ln b. Thus if ln y is plotted against x, the gradient is ln b and the vertical intercept is ln a. This technique is commonly examined in Edexcel applied questions.
对于 y = abˣ，取自然对数得 ln y = ln a + x ln b。因此若绘制 ln y 对 x 的图像，斜率为 ln b，纵截距为 ln a。该技巧在 Edexcel 应用题中经常考查。
ln y = ln a + x ln b
7. Trigonometric Models for Periodic Data | 周期数据的三角函数模型
Periodic phenomena such as tide height, temperature over a day, or rotation can be modelled by sine or cosine functions. A general model is h = a sin(b(t − c)) + d, where a is the amplitude, 2π/b is the period, c is the phase shift, and d is the vertical shift.
周期性现象，如潮汐高度、一天中的温度变化或旋转，可用正弦或余弦函数建模。一般模型为 h = a sin(b(t − c)) + d，其中 a 是振幅，2π/b 是周期，c 是相移，d 是垂直平移。
h = a sin(b(t − c)) + d
The amplitude is half the difference between maximum and minimum values, and the vertical shift is the average of the maximum and minimum. The period is the time taken for one complete cycle.
振幅是最大值与最小值之差的一半，垂直平移是最大值和最小值的平均值。周期是完成一个完整循环所需的时间。
8. Differential Equations in Modelling | 微分方程建模
Many natural laws are expressed as differential equations. The equation dP/dt = kP models population growth or radioactive decay, while dT/dt = −k(T − E) models Newton’s law of cooling, where E is the surrounding temperature.
许多自然定律用微分方程表示。方程 dP/dt = kP 用于模拟人口增长或放射性衰变，而 dT/dt = −k(T − E) 表示牛顿冷却定律，其中 E 是环境温度。
dP/dt = kP
To solve separable equations, separate variables and integrate. For dP/dt = kP, separate to ∫ (1/P) dP = ∫ k dt, giving ln P = kt + c, hence P = Aeᵏᵗ. You must use initial conditions to find the constant A.
解可分离变量方程时，分离变量并积分。对于 dP/dt = kP，分离得 ∫ (1/P) dP = ∫ k dt，从而 ln P = kt + c，因此 P = Aeᵏᵗ。必须用初始条件求常数 A。
P = Aeᵏᵗ
9. Kinematics and Calculus in Motion Modelling | 运动学与微积分建模
In mechanics modelling, displacement s, velocity v and acceleration a are linked by calculus: v = ds/dt and a = dv/dt = d²s/dt². Conversely, s = ∫ v dt and v = ∫ a dt. Constant acceleration gives the suvat equations, such as v = u + at and s = ut + ½at².
在力学建模中，位移 s、速度 v 和加速度 a 通过微积分联系：v = ds/dt，a = dv/dt = d²s/dt²。反过来，s = ∫ v dt，v = ∫ a dt。恒定加速度给出 suvat 方程，如 v =
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