[Figure: FiveStudy logo]
FiveStudy
Calculus BC · Unit 4 · 4.1
AP Calculus · Contextual Applications of Differentiation
Interpreting the Derivative in Context
Four things in every answer: what is changing, which direction, how fast with units, and when.
key idea$f'(a)=\text{rate of change of }f\text{, in }\frac{\text{output units}}{\text{input units}}\text{, at }x=a$
Setting up the model
In an applied question the derivative is never just a number — it is a rate with units and a moment attached. A complete interpretation names four things, and dropping any one of them loses the point.
The four-part sentence
- What is changing — the quantity $f$ measures, in words.
- Direction — increasing if $f'\gt 0$, decreasing if $f'\lt 0$.
- How fast — the numerical value, with its units: output units per input unit.
- When — the specific input value the derivative was evaluated at.
| Context | $f'$ means | Units |
|---|
| $P(t)$ population at time $t$ (years) | growth rate | people per year |
| $C(q)$ cost of $q$ items | marginal cost | dollars per item |
| $V(h)$ volume at depth $h$ (m) | how volume responds to depth | m$^{3}$ per metre |
| $T(t)$ temperature at time $t$ (min) | rate of heating or cooling | degrees per minute |
| $s(t)$ position at time $t$ (s) | velocity | metres per second |
| $v(t)$ velocity at time $t$ (s) | acceleration | m/s per second |
Units come from the ratio, always. If $f$ is in litres and $t$ in minutes, $f'$ is in litres per minute — no exceptions.
Worked examples
EX 1
The standard interpretation
$P'(5)=340$$P(t)$ is a town's population $t$ years after 2020, and $P'(5)=340$. Interpret this.
1
Name the quantity
$P$ measures population, so $P'$ measures how fast the population is changing.
2
Read the direction
$340$ is positive, so the population is increasing.
3
Attach the units
Population in people, time in years, so $P'$ is in people per year.
4
Locate it in time
$t=5$ means five years after 2020, i.e. during 2025.
5
Write the sentence
“In 2025 the town's population is increasing at a rate of about $340$ people per year.”
EX 2
Marginal cost
$C'(200)=13$$C(q)$ is the cost in dollars of producing $q$ widgets, and $C'(200)=13$. Interpret, and estimate the cost of the 201st widget.
1
Interpret the rate
Cost is increasing at about $13$ dollars per additional widget when $200$ are being produced.
2
Name it
Economists call this the marginal cost at $q=200$.
3
Estimate the next unit
$C(201)-C(200)\approx C'(200)\cdot1=13$ dollars.
4
Flag the approximation
$C'$ is an instantaneous rate; the actual 201st widget spans a whole unit, so “about $13” is the honest claim.
5
Why the approximation is good
Over a single unit the cost curve barely bends, so the tangent and the curve stay close.
EX 3
A negative rate
$T'(12)=-2.4$$T(t)$ is a cup of coffee's temperature in °C, $t$ minutes after pouring, and $T'(12)=-2.4$. Interpret.
1
Read the sign
Negative, so the temperature is decreasing — the coffee is cooling.
2
Give the magnitude
At a rate of $2.4$ degrees per minute.
3
Locate it
At $t=12$, i.e. twelve minutes after pouring.
4
Write the sentence
“Twelve minutes after pouring, the coffee's temperature is decreasing at about $2.4$ °C per minute.”
5
Do not drop the sign into the words
Writing “increasing at $-2.4$ degrees per minute” is clumsy and often marked wrong. Convert the sign into the word decreasing.
EX 4
Second derivative in context
$P''(5)\lt 0$For the same town, $P'(5)=340$ and $P''(5)=-45$. What does the pair tell you?
1
Read the first derivative
$P'\gt 0$: the population is still growing.
2
Read the second
$P''\lt 0$: the growth rate itself is decreasing.
3
Combine them
The town is gaining people, but more slowly each year — growth is slowing.
4
Give the units of $P''$
People per year, per year — people/year$^{2}$.
5
Avoid the classic error
$P''\lt 0$ does not mean the population is falling. It means the rate is falling. The population only falls when $P'\lt 0$.
Units & interpretation
Free-response readers look for the units and the context sentence. A bare number, however correct, is an incomplete answer.
| Weak answer | Why it loses marks | Complete answer |
|---|
| “$340$” | no units, no context, no time | “increasing at $340$ people per year in 2025” |
| “the population is $340$” | confuses $P'$ with $P$ | “the population is increasing at $340$ per year” |
| “increasing at $-2.4$” | sign left in the number | “decreasing at $2.4$ °C per minute” |
| “$P''\lt0$ so the population falls” | reads $P''$ as $P'$ | “the population is still growing, but more slowly” |
Each left-hand answer contains the right arithmetic. The marks are in the right-hand column.
Watch out
$f'$ is a rate, $f$ is an amount
“The population is $340$” describes $P$, not $P'$. Say increasing at to name the rate.
Units come from the ratio
Output units divided by input units, always. Litres and minutes give litres per minute; never guess.
Put the sign into the words
A negative derivative means decreasing. State the direction in English and quote the magnitude as a positive number.
$f''\lt0$ does not mean $f$ is decreasing
It means $f'$ is decreasing. A quantity can grow while its growth rate shrinks — Example 4.