There's one more aspect of iterative functions that I want to briefly mention, or remind you about, and that is that time is discrete in these iterated functions. So, we start with the seed: X nought (0) equals 2. We can think of that as the value at time zero (0). Then, here, the first iterate, this could be the value at time 1, and that's 4, the value at time 2, or the second iterate, is 5, and so on. As we did before, we can plot this in a time series plot. Here's a time series plot for the first time 4 or 5 iterates of this function 2, 4, 5 and we can see the value starts at 2, it goes to 4, it goes to 5, just like we'd expect. Now, usually we draw a line between the dots, just because it makes it a more compelling looking graph, it's easier to read but this line shouldn't be taken literally. So, the value jumps from 2 to 4. It does not slide between values at 2 and 4. so, it has a value here; it has a value here; and it just jumps from one to the other. It doesn't have to go in between, it doesn't have to go through all these in between or intermediate values. We can draw a phase line so, it turns out, that this function has a single attracting fixed point at 6. So, we have arrows coming in to the fixed point at 6, and if we look at this phase line we might think that a point would start here, and it would just slide right in to 6, but actually it jumps, from 2, to 4, to 5, and so on. So, may be one should draw it like this: here's the first jump; the next jump; the next jump; and so on. Conventionally one doesn't draw it that way, but that might be a better picture. So, again, just to underscore: In these iterated functions, the number jumps from one to another, and it doesn't have to pass through intermediate values. Differential equations, which are the main topic of this unit, are different, in that they analyze a situation where a variable changes continuously. So, for example, the temperature of a cup of coffee: if the temperature starts at 40 degrees, and a little while later is 30 degrees, we can be sure that it didn't just instantly jump from 40 to 30, but it must have passed through all possible temperatures between 40 and 30. So, differential equations describe continuously changing phenomena, and these iterated functions describe phenomena that change in jumps. Differential equations is a topic in mathematics that's typically introduced using a lot of calculus. However, in this course, I'll introduce it using a bare minimum of calculus, almost no calculus at all. I think this way of introducing differential equations actually makes it easier to understand what differential equations are and what they mean. So, if you haven't had calculus before, don't worry. In the next several subunits, I'll only be using a few ideas and concepts from calculus, and I'll explain these along the way, and if you have had a differential equations class before, I think that there will still be plenty new for you in this unit. The techniques that I'll be introducing most likely you haven't seen in an introductory differential equations class particularly if it was taught in a traditional manner. So, in the next several subunits, I'll introduce differential equations and give you a number of different ways of thinking about how to solve them, and more importantly what those solutions mean. We'll start in the next subunit, where I'll introduce the idea of the derivative.