Exponential decay

Definition

Exponential decay describes a process where the quantity decreases at a rate proportional to its current value. It is often modeled by the function $y(t) = y_0 e^{-kt}$, where $k$ is the decay constant.

5 Must Know Facts For Your Next Test

1. The differential equation governing exponential decay is $\dfrac{dy}{dt} = -ky$, where $k > 0$.
2. The solution to this differential equation is $y(t) = y_0 e^{-kt}$, representing how the quantity changes over time.
3. Half-life is a key concept in exponential decay and represents the time it takes for a quantity to reduce to half its initial amount.
4. In integration problems, exponential decay can be used to model phenomena such as radioactive decay and cooling processes.
5. The integral of an exponential decay function can be used to calculate total quantities over time, such as remaining substance or energy.

Review Questions

• What is the general form of the function that models exponential decay?
• How would you set up and solve the differential equation for a substance undergoing exponential decay?
• Explain what half-life means in the context of exponential decay.

Related terms

Exponential Growth: A process where a quantity increases at a rate proportional to its current value, typically modeled by $y(t) = y_0 e^{kt}$.

Half-Life: The time required for a quantity undergoing exponential decay to decrease to half its initial amount.

Decay Constant: $k$ in the exponential decay formula $y(t) = y_0 e^{-kt}$, representing the rate at which the quantity decays.