7.8 Exponential Models with Differential Equations

2 min read•june 18, 2024

AP Calculus AB/BC ♾️

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What is the exponential growth model?

The exponential growth and decay model is a mathematical formula that helps us understand how something changes over time. It's commonly used to describe situations like population growth or the decay of radioactive substances.

Mathematical Representation:

In this model, we have an equation that looks like this:

k * (dy/dt) = y

Here, "y" represents the quantity we're interested in, which could be the population size or the amount of a substance. "t" represents time, and "k" is a constant that determines the rate of growth or decay.

Now, let's talk about the solutions of this equation. When we solve it, we find that the solutions have a specific form:

y = y₀ * e^(kt)

In this equation, "y₀" is the initial value of the quantity we're studying (the value of "y" at the beginning), and "e" is a mathematical constant called Euler's number, approximately equal to 2.718. The variable "k" determines how fast the quantity grows or decays.

Relationships Between Variables:

So, what does this equation tell us? Well, it tells us how the quantity "y" changes over time. If "k" is positive, the quantity will grow exponentially. If "k" is negative, the quantity will decay exponentially.

Variables Defined:

The "e^(kt)" part of the equation represents the exponential function, which describes the rapid increase or decrease of the quantity. The "y₀" value gives us the starting point of the quantity, allowing us to track its changes from the initial condition.

By using this equation, we can make predictions about the future values of the quantity based on the initial condition and the rate of growth or decay. It helps us understand the behavior of various phenomena, such as population growth, the decay of radioactive materials, or the spread of diseases.

In summary, the exponential growth and decay model helps us understand how a quantity changes over time. The equation provides solutions that describe the exponential growth or decay of the quantity, and it allows us to make predictions and analyze various real-world scenarios.

Information sourced from CED. Written with the help of ChatGPT.

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