Calculus AB · Unit 7: Differential Equations · 14 min read · Updated 2026-05-10
Exponential models with differential equations — AP Calculus AB
AP Calculus AB · Unit 7: Differential Equations · 14 min read
1. Core Concept: Exponential Models★★☆☆☆⏱ 3 min
Exponential models are the most common applied differential equation type on the AP Calculus AB exam, accounting for 2-3% of total exam weight, appearing in both multiple-choice and free-response sections. These models describe quantities whose rate of change is proportional to their current size, matching many real-world phenomena from population growth to radioactive decay.
2. Core Proportional Growth and Decay Model★★☆☆☆⏱ 4 min
The fundamental relationship for any exponential model translates to a simple differential equation, where $k$ is the constant of proportionality:
\frac{dy}{dt} = ky
If $k>0$, the quantity grows exponentially; if $k<0$, the quantity decays exponentially. We solve this via separation of variables to get the general solution with initial condition $y(0)=y_0$:
y(t) = y_0 e^{kt}
3. Doubling Time and Half-Life★★☆☆☆⏱ 3 min
Doubling time (for growth) and half-life (for decay) are special cases that let you find $k$ directly without a second measurement, or calculate time to reach a specific quantity.
For exponential growth with doubling time $T_{\text{double}}$ (time to double the initial quantity):
T_{\text{double}} = \frac{\ln 2}{k}, \quad k = \frac{\ln 2}{T_{\text{double}}}
For exponential decay with half-life $T_{\text{half}}$ (time to reduce the initial quantity by half):
T_{\text{half}} = -\frac{\ln 2}{k}, \quad k = -\frac{\ln 2}{T_{\text{half}}}
Since $k$ is negative for decay, $T_{\text{half}}$ is always positive, matching its physical meaning.
4. Newton's Law of Cooling★★★☆☆⏱ 4 min
Newton's Law of Cooling is a modified exponential model that describes temperature change of an object relative to a constant ambient (surrounding) temperature. The rate of change of the object's temperature is proportional to the difference between the object's temperature and the ambient temperature.
\frac{dT}{dt} = k(T - T_s)
Where $T(t)$ is the object's temperature at time $t$, $T_s$ is the constant ambient temperature, and $k<0$ always. Solving via separation of variables gives the general solution for initial temperature $T(0)=T_0$:
T(t) = T_s + (T_0 - T_s)e^{kt}
This model works for both cooling (hot object in a cool room) and warming (cool object in a warm room): the object's temperature always approaches the ambient temperature over time.
Common Pitfalls
Why: Confusing Newton's Law with the basic exponential growth/decay model; rate depends on temperature difference, not absolute temperature
Why: Mixing up growth and decay formulas and forgetting to check the sign of $k$
Why: Forgetting to fully isolate $T$ after integrating, leaving the constant ambient temperature term on the wrong side
Why: Forgetting that exponential models assume non-zero initial quantity; zero initial quantity is a trivial special case
Why: Forgetting the absolute value in $\ln|y|$ requires allowing $A$ to be negative to match the sign of the initial quantity