1
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There's one more aspect of iterative functions
2
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that I want to briefly mention, or remind you about,
3
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and that is that time is discrete in these iterated functions.
4
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So, we start with the seed: X nought (0) equals 2.
5
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We can think of that as the value at time zero (0).
6
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Then, here, the first iterate,
7
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this could be the value at time 1, and that's 4,
8
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the value at time 2, or the second iterate, is 5,
9
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and so on.
10
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As we did before,
11
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we can plot this in a time series plot.
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Here's a time series plot
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for the first time 4 or 5 iterates of this function 2, 4, 5
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and we can see the value starts at 2,
15
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it goes to 4, it goes to 5, just like we'd expect.
16
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Now, usually we draw a line between the dots,
17
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just because it makes it a more compelling looking graph,
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it's easier to read
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but this line shouldn't be taken literally.
20
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So, the value jumps from 2 to 4.
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It does not slide between values at 2 and 4.
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so, it has a value here; it has a value here;
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and it just jumps from one to the other.
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It doesn't have to go in between,
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it doesn't have to go through all these in between
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or intermediate values.
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We can draw a phase line
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so, it turns out, that this function
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has a single attracting fixed point at 6.
30
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So, we have arrows coming in
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to the fixed point at 6,
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and if we look at this phase line
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we might think that a point would start here,
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and it would just slide right in to 6,
35
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but actually it jumps, from 2, to 4, to 5, and so on.
36
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So, may be one should draw it like this:
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here's the first jump; the next jump;
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the next jump; and so on.
39
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Conventionally one doesn't draw it that way, but that might be a better picture.
40
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So, again, just to underscore: In these iterated functions,
41
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the number jumps from one to another,
42
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and it doesn't have to pass through intermediate values.
43
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Differential equations, which are the main topic of this unit,
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are different, in that they analyze a situation
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where a variable changes continuously.
46
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So, for example, the temperature of a cup of coffee:
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if the temperature starts at 40 degrees, and a little while later is 30 degrees,
48
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we can be sure that it didn't just instantly jump from 40 to 30,
49
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but it must have passed through all possible temperatures between 40 and 30.
50
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So, differential equations describe continuously changing phenomena,
51
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and these iterated functions describe phenomena that change in jumps.
52
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Differential equations is a topic in mathematics
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that's typically introduced using a lot of calculus.
54
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However, in this course, I'll introduce it using a bare minimum of calculus,
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almost no calculus at all.
56
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I think this way of introducing differential equations
57
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actually makes it easier to understand what differential equations are
58
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and what they mean.
59
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So, if you haven't had calculus before, don't worry.
60
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In the next several subunits,
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I'll only be using a few ideas and concepts from calculus,
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and I'll explain these along the way,
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and if you have had a differential equations class before,
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I think that there will still be plenty new for you in this unit.
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The techniques that I'll be introducing
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most likely you haven't seen in an introductory differential equations class
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particularly if it was taught in a traditional manner.
68
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So, in the next several subunits, I'll introduce differential equations
69
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and give you a number of different ways of thinking
70
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about how to solve them, and more importantly
71
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what those solutions mean.
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We'll start in the next subunit, where I'll introduce the idea of the derivative.