Building and Applying a Model · 构建与应用模型
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| function model/ˈfʌŋkʃn ˈmɒdl/ | 函数模型 | hán shù mó xíng |
| regression/rɪˈɡreʃn/ | 回归 | huí guī |
| interpolation/ɪnˌtɜːpəˈleɪʃn/ | 内插 | nèi chā |
| extrapolation/ekˈstræpəleɪʃn/ | 外推 | wài tuī |
| parameters/pəˈræmɪtəz/ | 参数 | cān shù |
From data to a formula
- Selecting a model family is only half the job; now you build the actual rule.
- A good model turns a table of measurements into a formula you can compute with.
- Once you have the formula, you can predict values you never measured.
- This is how precalculus becomes a tool for the real world.
从数据到公式
- 选定模型族只是任务的一半;现在你要构建真正的规则。
- 一个好模型把一张测量数据表,变成你能拿来计算的公式。
- 一旦有了公式,你就能预测那些你从未测量过的值。
- 这正是微积分预备成为现实世界工具的方式。
Build the model
- A function model 函数模型 is a rule that captures how one quantity depends on another.
- You can build it from a verbal description, a table of values, or a graph.
- "Costs $\$5$ to start plus $\$2$ per item" becomes $C = 5 + 2n$ straight away.
- Match the family to the shape (Lesson 1.13), then pin down the exact numbers.
构建模型
- 函数模型(function model)是一条规则,刻画一个量如何依赖另一个量。
- 你可以从一段文字描述、一张数值表,或一张图来构建它。
- "启动花 5 元,加上每件 2 元"可以直接写成 $C = 5 + 2n$。
- 让模型族与形状匹配(第 1.13 课),再确定具体的数字。
Regression finds the best fit
- With many scattered points, regression 回归 finds the best-fit line or curve using technology.
- It chooses the model that comes closest to all the data at once.
- The result gives you the numbers in the model rule, ready to use.
- No single point sits on it perfectly — it captures the overall trend.
回归找到最佳拟合
- 面对许多散开的点,回归(regression)用技术找到最佳拟合的直线或曲线。
- 它选出一个同时最接近所有数据的模型。
- 结果给出模型规则里的数字,可以直接使用。
- 没有哪个点会完美地落在它上面——它抓住的是整体趋势。
A model line — read its rate and starting value · 一条模型直线——读出它的速率和起始值
y = ax + b
Here a is the rate of change and b is the starting value. Adjust them to fit data, then plug in an input to make a prediction. · 这里 a 是变化率,b 是起始值。调整它们来拟合数据,再代入一个输入来做预测。
A regression finds… · 回归能找到……
Regression fits the model that comes closest to all · 所有 the points at once — the best-fit model, using technology. · 回归拟合出一个最接近所有点的模型——用技术求得的最佳拟合模型。
Interpolate versus extrapolate
- Interpolation 内插 predicts between known data points — usually safe.
- Extrapolation 外推 predicts beyond the data — riskier the further you go.
- A model with no data to lean on can drift far from reality.
- Trust interpolation more than extrapolation, and say which one you used.
内插与外推
- 内插(interpolation)预测已知数据点之间的值——通常安全。
- 外推(extrapolation)预测数据之外的值——走得越远越冒险。
- 一个没有数据可依靠的模型,可能严重偏离现实。
- 相信内插多于外推,并说明你用的是哪一种。
Predicting a value between · 之间 known data points is called . · 预测已知数据点之间的值,叫做。
Interpolation stays inside the data range, so it is usually reliable. · 内插留在数据范围之内,所以通常是可靠的。
Predicting far outside the range of the data (extrapolation) is usually… · 预测数据范围之外很远处(外推)通常是……
Far from the data the model has no evidence to lean on, so extrapolation is the riskiest prediction. · 在远离数据处,模型没有证据可依靠,所以外推是风险最大的预测。
Interpret the parameters
- The parameters 参数 (the numbers in the rule) carry real-world meaning.
- In $C = 5 + 2n$, the $5$ is the fixed start-up cost and the $2$ is the cost per item.
- Reading parameters in context turns a formula back into a story about the situation.
- A good answer always says what each number means, with its units.
解释参数
- 参数(parameters)(规则里的那些数字)带有真实世界的含义。
- 在 $C = 5 + 2n$ 中,$5$ 是固定的启动成本,$2$ 是每件的成本。
- 在情境中解读参数,能把公式重新变回关于这个情境的故事。
- 一个好答案总会说出每个数字代表什么,并带上单位。
A model gives cost $= 2x + 1$ (in thousands). Predict the cost when $x = 10$. · 一个模型给出 成本 $= 2x + 1$(单位:千)。预测 $x = 10$ 时的成本。
Substitute: $2(10) + 1 = 21$ thousand. Plugging an input into the model is how you apply it. · 代入:$2(10) + 1 = 21$ 千。把一个输入代入模型,就是应用它的方式。
The parameters of a model (like the slope) can be interpreted in terms of the real situation. · 模型的参数(比如斜率)可以用真实情境来解释。
In cost $= 2x + 1$, the $2$ is the cost per unit and the $1$ is the fixed start-up cost — real meanings. · 在 成本 $= 2x + 1$ 中,$2$ 是每单位的成本,$1$ 是固定的启动成本——都有真实含义。
Put the steps of using a model in order. · 把使用一个模型的步骤按顺序排列。
Data → fit → predict → check. The final check catches a model that has drifted from reality. · 数据 → 拟合 → 预测 → 检查。最后的检查能发现偏离现实的模型。
Extrapolating too far is a classic trap. A linear sales model may predict billions of sales in twenty years — clearly nonsense. Always sanity-check a prediction against the real situation before trusting it.
外推得太远是一个经典的陷阱。一个线性销售模型可能预测二十年后有数十亿的销量——显然是胡说。在相信一个预测之前,永远要拿它和真实情境核对一下。
A phone battery drops from $100\%$ by about $8\%$ per hour.
- Model: $B = 100 - 8h$, a linear function model.
- Interpolate at $h = 3$: $B = 100 - 24 = 76\%$ — inside the data, reliable.
- Extrapolate at $h = 20$: $B = -60\%$ — impossible, so the model has broken down.
一部手机电池从 $100\%$ 起,大约每小时下降 $8\%$。
- 模型:$B = 100 - 8h$,一个线性函数模型。
- 在 $h = 3$ 处内插:$B = 100 - 24 = 76\%$——在数据之内,可靠。
- 在 $h = 20$ 处外推:$B = -60\%$——不可能,所以模型已经失效。
Build a function model from words, a table, or a graph, using regression for the best fit. Use it to interpolate (safe, inside the data) or extrapolate (risky, beyond it), interpret its parameters in context, and always check the prediction makes real-world sense.
从文字、表格或图构建一个函数模型,用回归求最佳拟合。用它来内插(安全,在数据之内)或外推(冒险,在数据之外),在情境中解读它的参数,并始终检查预测在现实世界中是否说得通。