📚 Mathematical Modelling: Common Mathematical Models and Their Applications | 数学建模：常见数学模型及应用解析
Mathematical modelling is the art of translating a real-world situation into mathematics, solving the resulting equations, and interpreting the results back into the original context. A good model focuses on the most important features, makes assumptions that can be tested, and gives predictions that are useful even when the world is messy.
数学建模是将现实世界的问题转化为数学语言，通过求解相应方程，再将结果还原到实际情境中的艺术。一个好的模型应抓住主要特征，提出可检验的假设，并能在数据不完美的情况下给出有实用价值的预测。
1. What Is a Mathematical Model? | 什么是数学模型？
A mathematical model is a simplified description of a system using equations, functions, tables, graphs, or algorithms. The simplification is deliberate: the model should include the variables that matter most and ignore factors that have little influence.
数学模型是用方程、函数、表格、图像或算法对系统进行的简化描述。这种简化是有意为之：模型应当包含最重要的变量，而忽略影响较小的因素。
The modelling process usually follows a cycle of steps.
数学建模通常遵循一个循环步骤。
Specify the question and identify the key variables.
明确问题，并识别关键变量。
Make assumptions and choose a suitable mathematical structure.
作出假设，并选择合适的数学结构。
Solve the mathematics and translate the solution back into the original context.
求解数学问题，并将解还原到原始情境中。
Test the model with data; revise the assumptions if the predictions are poor.
用数据检验模型；如果预测效果不佳，则修正假设。
In A-level mathematics, common models include linear, quadratic, exponential, logarithmic, logistic, matrix, differential equation, and statistical models. Each has a typical shape and a set of real-world applications.
在 A-level 数学中，常见模型包括线性、二次、指数、对数、逻辑斯蒂、矩阵、微分方程和统计模型。每种模型都有典型的形态和一组现实应用。
2. Linear Models | 线性模型
here x is the independent variable and y is the dependent variable. The graph is a straight line, so the rate of change is constant.
线性模型具有形式 y = a + bx，其中 x 是自变量，y 是因变量。其图像是一条直线，因此变化率是恒定的。
y = a + bx
changes by b units. The parameter a is the intercept: the predicted value of y when x = 0.
参数 b 是斜率：当 x 增加一个单位时，y 改变 b 个单位。参数 a 是截距：当 x = 0 时 y 的预测值。
Temperature conversion: F = 1.8C + 32.
温度转换：F = 1.8C + 32。
Currency exchange: total cost = exchange rate × amount + fixed commission.
货币兑换：总费用 = 汇率 × 金额 + 固定手续费。
Taxi fares: fare = starting charge + rate per kilometre × distance.
出租车计价：车费 = 起步价 + 每公里单价 × 距离。
e residual for a point is eᵢ = yᵢ − (a + bxᵢ), and we choose a and b to minimise S = Σeᵢ².
给定数据时，通常通过最小二乘回归找到最佳线性模型。每个数据点的残差为 eᵢ = yᵢ − (a + bxᵢ)，我们选择 a 和 b 使 S = Σeᵢ² 最小。
S = Σ(yᵢ − a − bxᵢ)²
Linear models are simple and easy to interpret, but they can only describe relationships with a constant rate of change.
线性模型简单且易于解释，但只能描述具有恒定变化率的关系。
3. Quadratic Models | 二次函数模型
+ bx + c, with a ≠ 0. Their graphs are parabolas, so they are ideal for problems involving a maximum or minimum value.
二次模型具有形式 y = ax² + bx + c，其中 a ≠ 0。它们的图像是抛物线，因此适合描述涉及最大值或最小值的问题。
y = ax² + bx + c, a ≠ 0
0, the parabola has a minimum; if a < 0, it has a maximum. The corresponding y-value is c − b²/(4a).
顶点出现在 x = −b/(2a) 处。如果 a > 0，抛物线有最小值；如果 a < 0，抛物线有最大值。对应的 y 值为 c − b²/(4a)。
x = −b/(2a), y = c − b²/(4a)
A classic example is projectile motion. For an object thrown upwards with initial speed v₀ from height h₀, its height after t seconds is approximately:
一个经典例子是抛体运动。对于以初速度 v₀ 从高度 h₀ 向上抛出的物体，t 秒后的高度近似为：
h(t) = −4.9t² + v₀t + h₀
on 9.8 m/s². Solving the equation h(t) = 0 gives the time when the object hits the ground.
系数 −4.9 来自重力加速度 9.8 m/s² 的一半。解方程 h(t) = 0 可得到物体落地的时间。
Profit models: P(n) = an² + bn + c can model profit against number of items sold.
利润模型：P(n) = an² + bn + c 可表示利润随销售数量的变化。
Area problems: given fixed perimeter, a quadratic gives the maximum rectangular area.
面积问题：在周长固定时，二次函数给出最大矩形面积。
Reaction path: the discriminant b² − 4ac tells whether the quadratic has real roots.
根的情况：判别式 b² − 4ac 可判断二次方程是否有实数根。
4. Exponential Models | 指数模型
form is N(t) = N₀eᵏᵗ, where N₀ is the initial amount and k is the continuous growth rate.
指数模型描述在相等时间间隔内按固定百分比增长或衰减的量。一般形式为 N(t) = N₀eᵏᵗ，其中 N₀ 是初始量，k 是连续增长率。
N(t) = N₀eᵏᵗ
If k > 0, the quantity grows exponentially; if k < 0, it decays exponentially. For example, if a population doubles every 3 years, it follows exponential growth.
如果 k > 0，该量指数增长；如果 k < 0，该量指数衰减。例如，如果一个种群每 3 年翻一番，它遵循指数增长。
Compound interest: A = P(1 + r/n)ⁿᵗ.
复利：A = P(1 + r/n)ⁿᵗ。
Continuous compounding: A = Peʳᵗ.
连续复利：A = Peʳᵗ。
Radioactive decay: mass after time t is m(t) = m₀(0.5)t/T, where T is the half-life.
放射性衰变：t 时刻的质量为 m(t) = m₀(0.5)t/T，其中 T 是半衰期。
Exponential models grow very quickly, so they are only valid over limited time intervals unless a carrying capacity is introduced.
指数模型增长极快，因此除非引入环境容量，否则只在有限时间区间内有效。
5. Logarithmic Models | 对数模型
Logarithmic models are useful when one quantity changes by multiplication while another changes by
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