Rates of Change in Applied Contexts Other Than Motion
| English | Chinese | Pinyin |
|---|---|---|
| economics | 经济学 | jīng jì xué |
Rates are everywhere, not just motion
- A derivative measures how fast something changes — and that "something" needn't be position.
- Money, populations, temperature, chemical concentration: all change at rates you can model with a derivative.
- The same $f'(a)$ = rate idea applies; only the units and story change.
- This lesson practices reading derivatives across many fields.
Economics, biology, physics — same tool
- Economics 经济学: if $C(x)$ is cost, $C'(x)$ is marginal cost — the cost of one more unit ($\tfrac{\$}{\text{unit}}$).
- Biology: if $P(t)$ is a population, $P'(t)$ is the growth rate ($\tfrac{\text{organisms}}{\text{time}}$).
- Physics: if $Q(t)$ is charge, $Q'(t)$ is current ($\tfrac{\text{coulombs}}{\text{second}}$ = amperes).
- Whatever the field, the derivative's units are always output per input.
$C(x)$ is cost in dollars for $x$ items. The units of the marginal cost $C'(x)$ are...
Output per input: dollars per item.
In economics, the derivative of a cost function is called the ____ cost.
Marginal cost = cost of one more unit.
Match each quantity to the meaning of its derivative.
The derivative's units are always output-per-input for the field.
A rate function describes behavior over time
- Often the derivative is itself a function of time, $f'(t)$, describing how fast the quantity changes at each moment.
- Where $f'(t)$ is large, the quantity changes fast; where $f'(t)\approx0$, it is nearly steady.
- A graph of $f'(t)$ tells the story of the change — peaks are moments of fastest change.
- Reading that story is a core exam skill.
A draining tank's volume
y = ax² + c
This falling curve is volume $V(t)$; its (negative) slope is the drain rate $V'(t)$ — steeper means draining faster.
Sign and size, in context
- Positive rate: the quantity is growing (population rising, tank filling).
- Negative rate: shrinking (cooling, draining, a declining balance).
- The magnitude compares speeds: a rate of $-50$ is faster change than $-5$.
- Always pair the number with a units-carrying sentence — that's what earns the marks.
For $V(t)=100-4t^2$, the rate is $V'(t)=-8t$. Find $V'(3)$ (litres per minute).
$V'(3)=-8(3)=-24$; draining at $24$ L/min.
A negative rate of change means the modeled quantity is decreasing.
Negative derivative → the quantity falls.
Two balances change at rates $-5$ and $-50$ dollars/day. Which is changing faster?
Magnitude sets speed: $|-50|>|-5|$, so $-50$ changes faster.
Keep units straight, especially in economics. Marginal cost $C'(x)$ has units of dollars per item, not dollars — it's the cost of the next item, not the total. Reporting a rate as if it were an amount (or dropping the units entirely) loses the meaning the question is testing.
A tank drains so its volume is $V(t)=100-4t^2$ litres ($t$ in minutes).
- Rate of change: $V'(t)=-8t\ \tfrac{\text{L}}{\text{min}}$.
- At $t=3$: $V'(3)=-24\ \tfrac{\text{L}}{\text{min}}$.
- Interpretation: at $3$ minutes, the water is draining out at $24$ litres per minute (negative = decreasing).
A derivative models a rate of change in any field — economics (marginal cost), biology (growth rate), physics (current) — always with units of output per input. The sign says grow vs. shrink; the magnitude compares speeds. Interpret each rate in a full, units-carrying sentence.