7.8 Exponential Models

The solution

The differential equation dy/dt = ky has the solution y = y₀·e^(kt), where y₀ is the initial amount (the value at t = 0).

Growth vs. decay

If k > 0, the quantity grows exponentially; if k < 0, it decays. Used for population growth, radioactive decay, and continuously compounded interest.

Key terms

Exponential model

dy/dt = ky has solution y = y₀·e^(kt).

y₀

The initial amount (value at t = 0).

k > 0

Exponential growth.

k < 0

Exponential decay.

Uses

Population, radioactive decay, compound interest.

Try an AP-style question

A quantity satisfies dy/dt = ky, starts at 100, and doubles every 3 years. What is it at t = 6?

1. 600
2. 300
3. 200
4. 400