Exponential Models

Exponential Models

Part 1 of 7 — Exponential Growth and Decay

The Differential Equation

$\frac{dy}{dt} = ky \implies y = y_0 e^{kt}$

• $k > 0$: exponential growth
• $k < 0$: exponential decay

Worked Example

A bacteria culture starts with 500 and grows to 1500 in 2 hours.

$1500 = 500e^{2k}$

$3 = e^{2k}$

$k = \frac{\ln 3}{2} \approx 0.549$

Population at time $t$: $P(t) = 500e^{0.549t}$

Exponential Growth 🎯

Key Takeaways — Part 1

1. $\frac{dy}{dt} = ky$ always has solution $y = y_0 e^{kt}$

Exponential Models

Part 2 of 7 — Newton's Law of Cooling

The Model

$\frac{dT}{dt} = k(T - T_s)$

Exponential Models

Part 3 of 7 — Compound Interest & Continuous Growth

Compound Interest

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Exponential Models

Part 4 of 7 — Derivatives and Integrals of Exponentials

Key Rules

$\frac{d}{dx}[e^{kx}] = ke^{kx} \qquad \int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$

Exponential Models

Part 5 of 7 — Logistic Growth Preview

The Logistic Model

$\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$

Exponential Models

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Exponential Models — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Exponential Models — Complete! ✅