 So I wanted to make a couple comments today about how exponents end up in so many of our equations when we're dealing with things like environmental science and environmental engineering and why they continue to appear and become sort of very prominent. So let's talk first about a sort of simple system. Let's say I have a population, a population of individuals, a simple biology sort of population here. And if I want to plot that population over time, I might start with the population and I might measure that we have a certain number of individuals at a certain point in time. And I might look for a little bit in time and I say, okay, over the course of a period of time, every so often if that population is a healthy population, that population will grow. And usually that grow rate, how fast the population grows, will add some fraction of the population. For example, let's say this is a fairly healthy population and the population itself grows by about 10%, well let's say 20%, by about 20% in a year. So if I mark some marks off here on the vertical axis, we could say that after some period of time, my population grows from its current amount up some amount, we'll say about 20%. Well, now it's 20% bigger. If we wait for another year and we grow again by that 20%, notice we're not only going to add the same amount we did before, but we're going to add in a little additional amount that represents 20% of the 20% that we added. So in this case we go up another 20%, but this is 20% of our original length plus 20%. And if we increase that, if we continue that pattern, take the original length and add 20%, take the original length and add 20%, we get a particular curve. And notice that curve starts getting very high very quickly. Well, this is a concept known as exponential growth. The idea here though, what creates exponential growth is our idea that we had at the beginning, that we have a population, but we want to figure out how much to add to the population, how much that population changes over time. And if that population changes over time, then we know that our population later will be equal to our population now, plus that change in population. So the population is going to grow, but if we go and figure out how much it grows, if we take and multiply the current population by some number, in this case if I do something like a value k here, actually usually the way we look at it with populations is we sort of say that there's an existing population, one plus some fraction, there's the 20%, if we were doing the 20% in this case, times the existing population. And if I wanted a number that k was a number of 0.2, then I'd be adding 20% each time. If the number was 0.4, I'd be adding 40% each time. Well, if you actually create this kind of relationship where the change of the population depends on the population itself, that's how you end up with something called exponential growth. That's pretty much characteristic of exponential growth where the change of something depends upon the thing itself. As long as that change is a positive change, notice that we're actually have some sort of positive value here. If instead I consider something else, maybe for example instead of my population I'm considering, I don't know, something evaporating from a tank and there's an amount of volume of water that's in a tank and that volume water is evaporating over time so the change in that volume over time is going to be equal to the volume, but we're going to take our original volume and subtract some fraction. Let me write that a little bit more, 1 minus k times the volume itself. Now again, the change in the volume is depending upon the volume itself. It depends on how much is in there. Vaporation may not sort of be the best example here, but let's assume that some of it's leaking out or pouring out, but the amount that's going away depends upon it. Well, if I look, if I move forward some period of time, let's say that that k is 0.2 again. Well, that means after my first period of time I only have 80% of my original. So I go about 80% of the original after one period of time. Well then I consider that amount and I subtract 20% of that amount following the same basic idea and I end up with 80% of that original and continue 80% of that and 80% of that. And notice if we continue to take 80% we're always going to have some tiny fraction left so we never kind of go off entirely, but we eventually get to smaller and smaller and smaller and smaller amounts. This relationship here is what we call exponential decay and it's characterized again by the relationship that the change of what we're talking about depends upon the thing that we're talking about itself, that the change in my volume depends upon the initial volume and that that change has sort of a negative value associated with it. Well, there are many reasons. There are a couple different ways of looking at this but one of the ways that we sort of represent exponential growth is with the letter E that if we look here our relationship, if we want to figure out the relationship of our population in time this is a pretty standard relationship. P equals P0E to the kT where P0 is my initial population, the starting population down here and E is a number value. It's represented to... it's actually a number very much like pi 2.718 and not repeating but it continues. It's an irrational number that continues there. And k is the same percentage fraction here and T is a period of time that isn't explicitly listed. It's listed as change in T over here in sort of our original relationship but if we want to sort of express our population as a function of time, this is what it looks like. Now, there are many calculus reasons for demonstrating what this value is and you will see that eventually in your calculus class but I sort of wanted to introduce the concept here that what creates an exponential function is this relationship where the change depends on the value itself. For our decay, same idea. My volume here would be equal to the initial volume E to the kT but in this case there is a negative value in there for our exponential function. Now, you can look for exponential functions in any place where you can kind of see where the rate of change of something might depend upon the thing itself. Consider for example, diffusion. If we have a little blob, we'll consider it to be sort of a spherical blob of pollutant, some sort of pollutant, some sort of concentrated pollutant and we're interested in how much pollutant there is at another point in time. One of the things you'll see with diffusion is that the blob itself tends to spread out and so we can have a measure of the size of the blob, the radius of the blob for example and the radius why the radius of the blob is useful is because the volume depends on the radius of the blob. Let me make that a small r. The volume of the blob depends on the radius of the blob and if the density is equal to the mass over the volume then that means that the density depends on the radius of the blob, depends on the size of the blob. Okay? And notice the density depends on the size of the blob with an inverse here that the V is at the bottom. Okay? But it has a dependency upon the side and or if we're talking about density we can also sort of talk about concentration, the same idea where concentration is the amount inside the amount of the solution, the same sort of idea that the concentration depends on the size of this little blob. Well, how does the blob expand? How does the blob sort of move? Well if we're thinking about it getting expanding a little bit bigger it's represented by how much stuff is in there and how much stuff is in there. The stuff that's going to expand is going to be all the stuff along the edge. So everything along the edge of the blob is going to move out and sort of expand that size. Well how much stuff moves? How much stuff changes that concentration? Well that's going to be dependent upon the area, the surface area of the blob in the first place and that surface area is a function of the blob. How much that spreads and how fast it spreads, the bigger it is the slower it's going to spread out because it has more, there's less stuff to sort of spread out. So if how fast it spreads depends on how far it's already spread now you have a relationship that the change in that radius over time is going to be some function of the radius itself. And as soon as you recognize that you can see why there is some sort of exponential if we kind of describe this in time there's going to be some sort of exponential function to this relationship. And in this case it's going to be a negative exponential function with some sort of rate. Because as we get larger the concentration is actually going to get smaller or the rate of change of the concentration is going to get smaller because there's less stuff to spread out. When you're very compact there's lots of stuff over a small area. Once you get spread out you have the same amount of stuff but spread out over a bigger area so the amount that's available for further spreading is going to be less. So it's not surprising that diffusion for example is an exponential decaying function.