No matter the particular letters used, the green variable stands for the ending amount, the blue variable stands for the beginning amount, the red variable stands for the growth or decay constant, and the purple variable stands for time.
Get comfortable with this formula; you'll be seeing a lot of it.
(I might want to check this value quickly in my calculator, to make sure that this growth constant is positive, as it should be.
If I have a negative value at this stage, I need to go back and check my work.), try to do the calculations completely within the calculator in order to avoid round-off error.
From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature.
In this section, we examine exponential growth and decay in the context of some of these applications. These systems follow a model of the form \(y=y_0e^,\) where \(y_0\) represents the initial state of the system and \(k\) is a positive constant, called the growth constant.
Notice that in an exponential growth model, we have \[ y′=ky_0e^=ky.
\label\] That is, the rate of growth is proportional to the current function value. Equation \ref involves derivatives and is called is a common example of exponential growth. It seems plausible that the rate of population growth would be proportional to the size of the population.
If something decreases in value at a constant rate, you may have exponential decay on your hands.
In this tutorial, learn how to turn a word problem into an exponential decay function.