How do I work with the exponential model #y = ae^(kt)#?
You work with the exponential model by learning its properties, and find an area for application.
The exponential models is one of the richest one in applications, it is unbelievable how it can be found in different areas, from biology to probability theory.
The first parameter is "a", this parameter tells us where it touches the y-axis when time is zero, just remember that any number powered 0 is 1, then:
The second parameter is "k", it tells you how fast it grows with time, the higher the value, the faster the function grows.
If you are familiar with calculus, take the limit to -infinity, then it touches x-axis at -infinity, in practical terms, never!
In general people need to make a regression (i.e. find the exponential from a set of experiments), then it is useful:
Just apply the logarithmic properties. Then you have a line.
Probability: exponential distribution
See for more: https://en.wikipedia.org/wiki/Exponential_distribution.
Where a is birth and b death rate.
Examples are plenty. This function is quite rich. you learn everyday from this simple function.