Attempting to intuitively understand the growth of the function `e^x`

Let’s say you have a unit. AT THE END OF 1 SECOND (or any given time interval), you have the unit grow by 3 (300%) (it becomes 3 times as large as it was before). You have this growth take place for 2 seconds. How would you represent this with an equation?

`1 * 3^2`

Let’s say you have a unit. THIS UNIT GROWS CONTINUOUSLY. You have it grow by 1 (100%) continuously. You have this growth take place for 2 seconds. How would you represent this with an equation?

`1 * e^2`

We can dissect each of these in time. For the first, at the end of the first second, we will have 3. Now, this growth is not continuous. It kind of spikes at the end of the interval. In fact, it could spike at the beginning of the interval, the middle, anywhere, really. I suppose it could even grow continuously within the interval, but, the overall growth within the interval would have to be three. That’s actually really interesting. We can represent this “spiky” growth with natural growth! We just use the ln(3).

`1 * 3^2 = 1 * (e^1.09861)^2`

We have e raised to the ln(3). This means that we have continuous growth within the interval, but, it amounts to 3. This means that, at every instant, the number is growing at slightly larger than 100% (exactly 100% would mean that e is raised to 1) (the important thing to note here is that the interval of recalculation alters the exponent. If we were to have the recalculation be every 1/100 intervals instead of 1/infinity, we would have 2.70^1.04. As the base gets smaller (closer to continuous) the exponent gets larger. It can be anywhere in between 3^1 and e^1.09861.

We contrast the continuous growth with a growth of exactly 3 at the end of the period.

Let’s say you have a unit. THIS UNIT GROWS CONTINUOUSLY. You have it grow by 3 (300%) continuously. You have this growth take place for two units of time. How would you represent this with an equation?

`1 * (e)^(3*2)`

It’s very temping to write 3*e, which is what I wrote originally. I think it can be thought of as you having three times the growth in each one unit of time, so you have to multiply the exponents.

An interesting thing to do would be to compare the rate of calculation with the resulting value for many rates of calculation (1+1/10)^10 and (1+1/11)^11, etc. What’s the approach of e? For example, as a base changes (goes down) how does the exponent change? What’s the relationship between the base and the exponent?

So, we’re still trying to get natural growth down. This is difficult, especially with so little sleep. Perhaps one way to understand it is by means of an example.

(I did not come up with this example, although, I’m recalling it and not copying it word for word). For some bacteria which doubles its amount every second, we could use a base other than e, but, it depends on how you want to view the growth of the bacteria. If you want to look at the population (in which only a fully grown bacteria would count as a unit), then you should use a base of 2. If you want to look at the biomass, you would want to use a base of e. This is because you’re counting even partially grown bacteria when you do biomass.

All right, I think that makes sense. It’s interesting that you can approximate with either discrete or continuos representation. Now, I’m realizing that, graphically, both are continuos. In fact, they are the same exact graph. The only difference is what they mean when you look at them. Perhaps, if you said that only after certain periods can the growth take place, it would be a ceiling or floor kind of deal. I don’t have any experience with that, but I think that’s what the graph would look like if you were to only count entirely grown bacteria. The thing is, 2^x is continuous. It’s only when you can only have discrete values of x where it matters.

All right, I think I understand that. One of the harder parts is likely understanding the relationship between time and growth rate. How you could have a sample growing at 100% growth rate for 6 units of time. Or, 600% growth rate for 1 unit of time and yield the same result. I may decide to leave this to be determined later.

Why are time and rate not treated separately. If you’re growing at a static rate of 100%, then e is raised to the first. If 300%, it’s raised to the 3rd. We need to think about what is really growing, or rather, what growth really is. I think the time thing works out in my head.

All right. I’ve seen based on the definition of e why one would take it to the r*t power, but I haven’t worked it out intuitively. If you have e to some power, you’re taking e that many times. We can split it up into the intervals of time so that we’re just left with rate. I don’t see myself getting anywhere on this at the moment, I’m going to go think about something else.