AP Calculus AB — calculusStandard1 min read

Exponential Growth and Decay Models

Model exponential growth and decay for AP Calculus AB. Solve dy/dt = ky and apply to real-world contexts.

Tutor AI Team2026-02-25T02:14:57.097Z

The differential equation $\frac{dy}{dt} = ky$ models exponential growth ( $k > 0$ ) and decay ( $k < 0$ ). This is one of the most commonly tested DEs on AP Calculus AB.

The Solution

$y = y_0 e^{kt}$

where $y_0 = y(0)$ is the initial value.

Growth ($k > 0$)

Doubling time: $t_d = \frac{\ln 2}{k}$ .

Decay ($k < 0$)

Half-life: $t_h = \frac{\ln 2}{|k|}$ .

Worked Example: Population

Problem

Population doubles every 5 years. $P(0) = 1000$ .

$k = \frac{\ln 2}{5} \approx 0.1386$ . $P(t) = 1000e^{0.1386t}$ .

After 12 years: $P(12) = 1000e^{1.663} \approx 5278$ .

Solution

Worked Example: Radioactive Decay

Problem

Half-life 8 hours. 200g initially.

$k = \frac{-\ln 2}{8}$ . $N(t) = 200e^{-0.0866t}$ .

After 24 hours: $N(24) = 200e^{-2.079} \approx 25$ g.

Solution

Worked Example: Newton's Cooling

Problem

$\frac{dT}{dt} = k(T - T_s)$ → $T(t) = T_s + (T_0 - T_s)e^{kt}$ .

Solution

Practice Problems

1. $\frac{dy}{dt} = 0.03y$ , $y(0) = 500$ . Find $y(10)$ .
1. A substance decays: 100g → 60g in 4 hours. Find the half-life.
1. Coffee at 90°C cools in a 20°C room. After 5 min it's 70°C. Find $k$ .

Want to check your answers and get step-by-step solutions?

Key Takeaways

✓
$\frac{dy}{dt} = ky$ → $y = y_0 e^{kt}$ .
✓
$k > 0$ : growth. $k < 0$ : decay.
✓
Doubling time = $\frac{\ln 2}{k}$ . Half-life = $\frac{\ln 2}{|k|}$ .
✓
Newton's cooling: $\frac{dT}{dt} = k(T - T_s)$ .

Tags:

#ap#calculus#differential-equations#exponential#modelling

Ready to Ace Your AP Calculus AB calculus?

Get instant step-by-step solutions to any problem. Snap a photo and learn with Tutor AI — your personal exam prep companion.

The Solution

Growth ($k > 0$)

Decay ($k < 0$)

Worked Example: Population

Worked Example: Radioactive Decay

Worked Example: Newton's Cooling

Practice Problems

Key Takeaways

Related Topics

Volumes of Revolution

Slope Fields

Area Between Curves

Ready to Ace Your AP Calculus AB calculus?