Modeling situations with differential equations — AP Calculus BC
1. Core Skill: Translating Verbal Rate Descriptions ★★☆☆☆ ⏱ 3 min
The core idea of translating context to differential equations is that any statement about how a quantity changes can be written as $\frac{dy}{dt}$ (or appropriate variables for your problem) equal to the net rate of change of the quantity. A key rule of thumb: if the problem states that the rate of change of $y$ is proportional to some quantity, you add a constant of proportionality $k$ to the right-hand side.
- Name all unknown functions and their independent variables
- Write the derivative of the unknown function of interest
- Translate proportionality relationships by adding a constant of proportionality
- Combine all increasing and decreasing rates to get the final net rate for the right-hand side
Exam tip: Always explicitly name the constant of proportionality in your DE for FRQs; AP exam graders require you to include the constant, even if you do not calculate its value, to earn full setup points.
2. Standard Common Contextual Models ★★☆☆☆ ⏱ 4 min
The AP exam frequently tests a set of standard contextual models that you should recognize immediately, though you should always be able to derive them from the problem description instead of relying solely on memorization. The most common standard models are:
- **Exponential growth/decay**: If a quantity changes at a rate proportional to the size of the quantity, $\frac{dy}{dt} = ky$, with $k>0$ for growth, $k<0$ for decay.
- **Newton's Law of Cooling/Heating**: The rate of change of an object's temperature is proportional to the difference between the object's temperature $T(t)$ and the constant ambient temperature $T_s$, so $\frac{dT}{dt} = k(T - T_s)$.
- **Logistic population growth**: For a population with carrying capacity $K$, the growth rate is proportional to both the population size and remaining capacity: $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ with $r>0$.
- **Rectilinear motion**: Acceleration is the rate of change of velocity, so $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ for motion along a line.
Exam tip: Always confirm that the sign of $\frac{dP}{dt}$ matches your expectation for the context: for logistic growth, $\frac{dP}{dt}$ must be positive when the population is between 0 and the carrying capacity.
3. Net Rate Models for Inflow/Outflow Systems ★★★☆☆ ⏱ 5 min
Many AP problems model mixing problems, bank accounts, or pollution systems where a quantity is both added and removed at different rates. The core rule for all these problems is:
\frac{dQ}{dt} = (\text{rate of } Q \text{ entering the system}) - (\text{rate of } Q \text{ leaving the system})
For mixing problems (the most common inflow/outflow problem), the inflow rate of solute is equal to the inflow concentration multiplied by the inflow flow rate. The outflow rate of solute is equal to the current concentration of solute in the well-mixed system multiplied by the outflow rate. Current concentration is $\frac{Q(t)}{V(t)}$, where $V(t)$ is the total volume at time $t$. If inflow and outflow rates are equal, $V(t)$ is constant; if not, $V(t) = V_0 + (r_{in} - r_{out})t$, where $V_0$ is the initial volume.
Exam tip: Always calculate $V(t)$ first in mixing problems; do not assume the volume is equal to the maximum tank capacity unless the problem says the tank is always full.
4. AP-Style Concept Check ★★☆☆☆ ⏱ 2 min
Common Pitfalls
Why: Students memorize a specific sign convention for Newton's Law and mark any other form incorrect, or get confused about what sign they need
Why: Students think the 'proportional' description is just context and forget that proportional means multiplied by an unknown constant
Why: Students assume the tank starts full, even when the problem says otherwise
Why: Students swap the order of the terms, leading to a negative growth rate when $P < K$
Why: Students confuse position, velocity, and acceleration derivatives