Modeling with Exponential Functions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| exponential model/ˌekspəˈnenʃl ˈmɒdl/ | mô hình mũ |
| initial value/ɪˈnɪʃl ˈvæljuː/ | giá trị ban đầu |
| exponential regression/ˌekspəˈnenʃl rɪˈɡreʃn/ | hồi quy mũ |
| growth factor/ɡrəʊθ ˈfæktə/ | y tố tăng trưởng |
| decay factor/dɪˈkeɪ ˈfæktə/ | y tố suy giảm |
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.
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 $b$.
To build $y = a\,b^x$ from data, the coefficient $a$ is the value when…
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At $x = 0$, $b^0 = 1$, so $y = a$: the coefficient is the initial value.
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 $y = a\,b^x$.
Tune the growth factor to fit the data
y = a·bˣ
Change a (the initial value) and b (the growth factor) until the curve passes through the data points.
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.
If a quantity grows by $20\%$ each period, the growth ____ (the base $b$) is $1.2$.
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A $20\%$ increase multiplies by $1 + 0.20 = 1.2$ — that multiplier is the growth factor.
A quantity that loses $15\%$ each period has a decay factor (base) of…
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Losing $15\%$ leaves $85\%$, so you multiply by $1 - 0.15 = 0.85$ each period — the decay factor.
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.
A colony follows $P = 100 \cdot 2^{\,t}$ (with $t$ in hours). What is $P$ at $t = 3$?
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$P = 100 \cdot 2^3 = 100 \cdot 8 = 800$. Plug the input into the model to predict.
Select all true statements about exponential models.
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Exponential growth extrapolated too far becomes unrealistic (nothing grows forever). The other three are correct.
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.
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.
An exponential model $y = a\,b^x$ has initial value $a$ and a base that is a growth factor ($b>1$) or decay factor ($0<b<1$). Find it from two points or an exponential regression, then predict within a sensible domain — never assume growth continues forever.
Write from a population of 1350
- P equals 50 times 3 to the power t.
- Predict the population at t equals 3.
A colony follows P equals 50 times 3 to the power t. What is P when t equals 3?
50 times 27 is 1350.
What is the predicted population?
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50 times 27 is 1350.