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: 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。

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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。

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2. Linear Models | 线性模型

Linear models have the form y = a + bx, where 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

The parameter b is the slope: when x increases by one unit, y 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。

Given data, the best linear model is often found by least-squares regression. The 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。

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3. Quadratic Models | 二次函数模型

Quadratic models have the form y = ax² + 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

The vertex occurs at x = −b/(2a). If a > 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

The coefficient −4.9 comes from half of the gravitational acceleration 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。

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4. Exponential Models | 指数模型

Exponential models describe quantities that grow or decay by a constant percentage in equal time intervals. The general 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。

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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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