Unit 4.3 – Rates of Change in Applied Contexts Other Than Motion

AP® Calculus AB & BC | Real-World Rate Applications Beyond Straight-Line Motion

Not all rates of change are about position and velocity! In AP® problems, you'll find rates involving population, temperature, economics (cost/revenue/profit), chemical changes, and more. This section shows you how to recognize, set up, and interpret these real-world rates using derivatives.

📈 Fundamental Rate of Change Patterns

General Formulas

$ R'(a) = \text{instantaneous rate of change of } R \text{ at } x=a $
If $ y = f(x) $, then $ f'(x) = \text{rate of change of } y \text{ with respect to } x $
Units: The units of $ f'(x) $ are (units of y) per (units of x).
$ \Delta y \approx f'(a) \Delta x $ when $ \Delta x $ is small (this is a linear approximation).

🌍 Common AP® Rate Contexts (Not Motion)

Population Modeling:

If $ P(t) $ = population at time $ t $, then $ P'(t) $ = the instantaneous population growth rate (e.g., in people per year).

Temperature:

If $ T(h) $ = temperature at hour $ h $, then $ T'(2.5) $ is the rate the temperature is changing at $h=2.5$ hours (e.g., in °C per hour).

Economics – Marginal Cost, Revenue, or Profit:

If $ C(x) $ is the total cost to produce $ x $ items, then $ C'(x) $ is the marginal cost, which approximates the cost to produce one additional item at a production level of $ x $ (e.g., in dollars per item).

Chemistry:

If $ Q(t) $ = amount of a substance at $ t $ minutes, then $ Q'(t) $ is the rate at which the substance is being produced or consumed.

Area or Volume:

If $ A(t) $ is an area (in cm²) at time $ t $, then $ A'(t) $ is the rate the area is increasing. If $ V(t) $ is a volume at time $ t $, $ V'(t) $ is the rate of change of the volume.

🧮 Short Notes & Quick Tricks

Always attach units to your rate: "per year," "per item," "per second," etc.
Interpret negative rates as a decrease ("the temperature is dropping...").
The word "marginal" in economics is a keyword for the derivative (rate of change).
Use $ f'(a) $ for linear approximations to estimate changes for a small $ \Delta x $.
When using table values, always state what the variables and the rate represent in context.

📖 Worked Examples

Example 1: Population

If $ P(8)=12000 $ and $ P'(8)=350 $, it means the population is increasing at a rate of 350 people/year when t=8. We can estimate the population at t=8.2 using a linear approximation: $ P(8.2) \approx 12000 + (350)(0.2) = 12070 $.

Example 2: Marginal Cost

If $ C(100)=10000 $ and $ C'(100)=37 $, it means the cost to produce the 101st item is approximately $37.

Example 3: Temperature

If $ T'(3)=-1.2 $, it means the temperature is decreasing at a rate of 1.2°C per hour at t=3 hours.

📝 Practice: AP® Contexts & Written Interpretation

Try These:

If $ Q'(6) = -15 $ where $ Q $ is the volume in liters in a tank at time $t$ minutes, what is the meaning?
If the marginal revenue $ R'(250)=6.5 $, what does this mean for the sale of the 251st item?
If $ A'(2) = 4.9 $ where $A(t)$ is an area in cm² at time $t$ seconds, write a correct AP®-style interpretation.

Model Answers:

"At t=6 minutes, the volume in the tank is decreasing at a rate of 15 liters per minute."
"When 250 items have been sold, the revenue is increasing at a rate of $6.50 per item."
"At t=2 seconds, the area is increasing at a rate of 4.9 square centimeters per second."

✏️ AP® Exam Success for Context Applications

Write your answer in the context of the problem! Always say "rate of [quantity] with respect to [variable]" and use the actual names from the problem.
Include units: Forgetting units will cost you points!
Remember that "marginal" means the derivative at a specific production level.
For a negative rate, state that the quantity is "decreasing" at a positive rate.
Use complete sentences for interpretations, not just numbers or formulas.