Modeling with Exponential Functions · 用指数函数建模
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential model/ˌekspəˈnenʃl ˈmɒdl/ | 指数模型 | zhǐ shù mó xíng |
| initial value/ɪˈnɪʃl ˈvæljuː/ | 初始值 | chū shǐ zhí |
| exponential regression/ˌekspəˈnenʃl rɪˈɡreʃn/ | 指数回归 | zhǐ shù huí guī |
| growth factor/ɡrəʊθ ˈfæktə/ | 增长因子 | zēng zhǎng yīn zi |
| decay factor/dɪˈkeɪ ˈfæktə/ | 衰减因子 | shuāi jiǎn yīn zi |
Fitting growth to real data
- A colony of bacteria, a savings account, a viral video — all grow by a constant factor.
- Given a few measurements, we want the exact rule $y = a\,b^x$ that fits them.
- Then we can predict the future and interpret the growth rate.
- This is exponential modeling: from data to a formula to a forecast.
让增长拟合真实数据
- 一群细菌、一个储蓄账户、一段爆红视频——都按恒定的因子增长。
- 给定几个测量值,我们想要精确拟合它们的规则 $y = a\,b^x$。
- 然后就能预测未来,并解释增长率。
- 这就是指数建模:从数据到公式再到预测。
Building the model from data
- An exponential model 指数模型 has the form $y = a\,b^x$, just like any exponential function.
- The initial value 初始值 $a$ is the amount at $x = 0$ — read it straight from the data.
- The base $b$ is the multiplier per step — find it by dividing consecutive outputs.
- Two data points are enough to pin down both $a$ and $b$.
从数据构建模型
- 指数模型(exponential model)的形式是 $y = a\,b^x$,和任何指数函数一样。
- 初始值(initial value)$a$ 是 $x = 0$ 时的量——直接从数据里读出。
- 底数 $b$ 是每步的乘数——用相邻输出相除来求它。
- 两个数据点就足以确定 $a$ 和 $b$。
To build $y = a\,b^x$ from data, the coefficient $a$ is the value when… · 要从数据构建 $y = a\,b^x$,系数 $a$ 是……时的值。
At $x = 0$, $b^0 = 1$, so $y = a$: the coefficient is the initial value. · 在 $x = 0$ 时,$b^0 = 1$,所以 $y = a$:系数就是初始值。
Regression finds the best fit
- With many noisy points, use technology to run an exponential regression 指数回归.
- It finds the $a$ and $b$ whose curve comes closest to all the data at once.
- No point lands exactly on it, but the trend is captured.
- The output is a ready-to-use rule $y = a\,b^x$.
回归找到最佳拟合
- 面对许多带噪声的点,用技术做一次指数回归(exponential regression)。
- 它找到那组 $a$ 和 $b$,使曲线同时最接近所有数据。
- 没有哪个点正好落在上面,但趋势被抓住了。
- 输出是一条可以直接使用的规则 $y = a\,b^x$。
Tune the growth factor to fit the data · 调整增长因子来拟合数据
y = a·bˣ
Change a (the initial value) and b (the growth factor) until the curve passes through the data points. · 改变 a(初始值)和 b(增长因子),直到曲线穿过数据点。
Growth factor and decay factor
- If the base $b > 1$, it is a growth factor 增长因子: $b = 1.2$ means $+20\%$ each step.
- If $0 < b < 1$, it is a decay factor 衰减因子: $b = 0.85$ means $-15\%$ each step.
- Convert a percentage to a factor: growth $\to 1 + r$, decay $\to 1 - r$.
- Reading the factor tells you the story of the change.
增长因子与衰减因子
- 如果底数 $b > 1$,它是增长因子(growth factor):$b = 1.2$ 表示每步 $+20\%$。
- 如果 $0 < b < 1$,它是衰减因子(decay factor):$b = 0.85$ 表示每步 $-15\%$。
- 把百分比换成因子:增长 $\to 1 + r$,衰减 $\to 1 - r$。
- 读懂这个因子,就知道变化的故事。
If a quantity grows by $20\%$ each period, the growth ____ (the base $b$) is $1.2$. · 如果一个量每个周期增长 $20\%$,那么增长____(底数 $b$)是 $1.2$。
A $20\%$ increase multiplies by $1 + 0.20 = 1.2$ — that multiplier is the growth factor. · 增长 $20\%$ 就是乘以 $1 + 0.20 = 1.2$——这个乘数就是增长因子。
A quantity that loses $15\%$ each period has a decay factor (base) of… · 每个周期损失 $15\%$ 的量,其衰减因子(底数)是……
Losing $15\%$ leaves $85\%$, so you multiply by $1 - 0.15 = 0.85$ each period — the decay factor. · 损失 $15\%$ 就剩下 $85\%$,所以每个周期乘以 $1 - 0.15 = 0.85$——即衰减因子。
Predict, within limits
- Substitute an input into the model to predict an output.
- Choose a reasonable domain: negative time or absurdly large outputs are meaningless.
- Exponential growth cannot continue forever — resources, space, or money run out.
- A good model comes with a note about where it stops being trustworthy.
在限度内预测
- 把一个输入代入模型来预测输出。
- 选一个合理的定义域:负的时间或荒谬巨大的输出都没有意义。
- 指数增长不可能永远持续——资源、空间或金钱都会耗尽。
- 一个好模型会附带说明它在哪里不再可信。
A colony follows $P = 100 \cdot 2^{\,t}$ (with $t$ in hours). What is $P$ at $t = 3$? · 一个菌落遵循 $P = 100 \cdot 2^{\,t}$($t$ 以小时计)。$t = 3$ 时 $P$ 是多少?
$P = 100 \cdot 2^3 = 100 \cdot 8 = 800$. Plug the input into the model to predict. · $P = 100 \cdot 2^3 = 100 \cdot 8 = 800$。把输入代入模型来预测。
Select all · 所有 true statements about exponential models. · 选出关于指数模型的所有正确说法。
Exponential growth extrapolated too far becomes unrealistic (nothing grows forever). The other three are correct. · 指数增长外推太远会变得不现实(没有东西能永远增长)。其余三条正确。
Unlimited exponential growth is a fantasy. A model saying a pond's algae doubles daily will soon "predict" more algae than atoms on Earth. Every real exponential eventually slows — trust the model only over the range where its assumptions hold.
无限的指数增长是一种幻想。一个说池塘藻类每天翻倍的模型,很快就会"预测"出比地球上原子还多的藻类。每一个真实的指数增长最终都会变慢——只在其假设成立的范围内相信模型。
A sample of $100$ cells triples every hour: $P = 100 \cdot 3^{\,t}$.
- Initial value $a = 100$; growth factor $b = 3$ (tripling).
- After $2$ hours: $P = 100 \cdot 3^2 = 900$ cells.
- Reasonable domain: only while nutrients last — perhaps the first several hours.
一份 $100$ 个细胞的样本每小时变为三倍:$P = 100 \cdot 3^{\,t}$。
- 初始值 $a = 100$;增长因子 $b = 3$(变三倍)。
- $2$ 小时后:$P = 100 \cdot 3^2 = 900$ 个细胞。
- 合理的定义域:只在养分充足时——也许是最初的几个小时。
An exponential model $y = a\,b^x$ has initial value $a$ and a base that is a growth factor ($b>1$) or decay factor ($0
指数模型 $y = a\,b^x$ 有初始值 $a$,底数是增长因子($b>1$)或衰减因子($0