Well, I started thinking about the relationship of my formula to the formula for radioactive decay. The formula they use currently says that if you start with 2 grams of something, and it decays by half every second, that the formula for how much you have after t seconds is:
But my problem with this is the integral of the above from 0 to infinity is 2.88, the original function is supposed to measure grams so the integral will be a certain number of grams*seconds. It's a little hard to explain but it's like they're saying they went through 2.88 grams when they actually only had 2 to start out with.
*Edit I think I found a better way to explain it, over all those seconds the total number of grams*seconds they had was 2.88, but what if you condensed that all to just one second, that would say they had 2.88 grams for one second, but they only started off with 2 so the most they could have for 1 second is 2.
It's kind of like it used 2.88 grams of space-time when it was only supposed to have a maximum of 2.
So I like my approximation to exponential decay better because it adds up to 2 as expected. I wouldn't end up with more when you add up how much I had over time.
*Further edit I don't know if this is going to help or hurt but a thought experiment I'm having... Say if you are able to get a certain amount of work out of a certain amount of mass proportional to the mass. Like it could be pressing down a lever or something. With a 2 gram mass you are able to get 2 units of work done for one second on the lever. What their formula is saying is if the 2 grams were radiosactive and decaying and you kept removing the decayed part, you could somehow do a total of 2.88 units of work over an infinite number of seconds. But this doesn't really make sense you shouldn't be able to do more work just because the mass is decaying, right? Right? I don't know I'm confusing myself.