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
- The differential equation governing exponential decay is $\dfrac{dy}{dt} = -ky$, where $k > 0$.
- The solution to this differential equation is $y(t) = y_0 e^{-kt}$, representing how the quantity changes over time.
- 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.
- In integration problems, exponential decay can be used to model phenomena such as radioactive decay and cooling processes.
- 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.
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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.
