Calculus BC · Unit 7: Differential Equations · 14 min read · Updated 2026-05-11
Exponential models with differential equations — AP Calculus BC
AP Calculus BC · Unit 7: Differential Equations · 14 min read
1. Core Exponential Growth and Decay Model★★☆☆☆⏱ 4 min
Exponential models describe quantities that change at a rate proportional to the current size of the quantity. They are one of the most frequently tested differential equation models on the AP Calculus BC exam, appearing in both multiple-choice and free-response questions. The core relationship is that the rate of change of $y(t)$ with respect to time $t$ is proportional to the current value of $y$.
Exam tip: Simplify $e^{kt}$ using logarithm exponent rules before plugging in a calculator; this avoids rounding errors and often gives an exact integer answer, which is what AP exam questions almost always expect.
2. Half-Life and Doubling Time★★☆☆☆⏱ 3 min
Half-life (for decay) and doubling time (for growth) are common special cases of exponential models. Instead of giving a second point to solve for $k$, you are given the time it takes for the quantity to halve or double. These relationships can be derived quickly on the exam if you forget them, but memorizing them saves time.
Half-life (decay): $k = -\frac{\ln 2}{T_{1/2}}$, where $T_{1/2}$ is the time to halve the quantity
Doubling time (growth): $k = \frac{\ln 2}{T_2}$, where $T_2$ is the time to double the quantity
Exam tip: After solving for $k$ in any decay problem, confirm $k$ is negative; if you get a positive $k$, you forgot the negative sign from $\ln(1/2)$ and need to correct your work before proceeding.
3. Newton's Law of Cooling★★★☆☆⏱ 4 min
Newton's Law of Cooling is a modified exponential model that describes the temperature change of an object relative to the constant temperature of its surroundings (called the ambient temperature). Unlike basic exponential models where rate is proportional to current amount, here the rate of change of the object's temperature is proportional to the difference between the object's temperature and the ambient temperature.
Solving this via separation of variables gives the general solution, where $T_0$ is the initial temperature of the object at $t=0$:
T(t) = T_s + (T_0 - T_s)e^{kt}
Exam tip: Never forget the shifted $T_s$ term; a common mistake is using the basic exponential solution $T(t) = T_0 e^{kt}$, which incorrectly predicts the object's temperature will approach 0 instead of the ambient temperature.
4. AP Style Practice Worked Examples★★★☆☆⏱ 3 min
Common Pitfalls
Why: Students mix up sign conventions for proportionality.
Why: Students forget that $\ln(1/2) = -\ln 2$ and drop the negative sign.
Why: Students rush to calculate a decimal value for $k$ instead of simplifying with logarithm rules.
Why: Students rush the separation of variables step.
Why: Students confuse discrete annual growth with continuous growth.