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In this sub unit I'll introduce the idea of the derivative,
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which is the central idea of differential calculus.
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If you've had calculus recently
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or feel pretty comfortable with it
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you can probably skip this video
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since it's pretty standard material.
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So, we'll start by thinking about rates of change.
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So, as an example,
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suppose you take a bicycle ride for 30 minutes
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and at the end of those 30 minutes you've travelled 21 kilometres,
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and you might want to know: on average how fast was I going?
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what was my average speed during this bicycle ride?
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Well we can calculate this as follows:
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So, average speed - how fast you were going during your thirty minute bicycle ride
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well, that's the change in position,
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how far you've travelled, over the change in time,
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and that means how much time has elapsed,
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how long were you bicycling for.
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In symbol, that's often written delta s over delta t,
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'delta', this triangle thing, means 'change in'.
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s here will stand for: change in position - how far you went,
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and so you went 21 kilometres in 30 minutes,
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and so that would be the .7,
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and the units are kilometres per minute.
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Now, this doesn't mean that at
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every instant you were going at exactly .7 kilometres per minute
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most likely, sometimes, you're were going faster and sometimes slower,
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faster down a hill, slower up the hill, faster when you feel more energetic,
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slower when you're tired,
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so, this is an average. You weren't going this speed the entire time,
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but on average, this was your speed,
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but then you might ask another question: How fast were you going, say,
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10 minutes, at exactly 10 minutes into your ride?
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So, this is a different sort of question.
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It's asking your speed - how fast you're going
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at a particular instant, not over an entire 30 minutes.
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So, this is a harder and more subtle question.
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So, one way you might think to answer this is:
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look at your average speed for a time interval close
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to this 10 minute point.
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So, let me write out how you might think about that.
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So, you might say: the average speed
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from t equals 9 to t equals 11 minutes,
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so, same idea - it's an average speed, but now the time interval is smaller,
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and notice that 10 minutes is right in between 9 and 11,
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and then that would be the distance travelled in this time,
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in this case, not during your entire journey,
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but just the journey from t equals 9 to 11,
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and you'll divide that by the time that has elapsed
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and that would just be 2 minutes.
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So, that could be,
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and just to make up a number for the sake of concreteness,
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maybe in those 2 minutes, from t equals 9 to t equals 11 in your journey
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let's say you went 1.2 kilometres,
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divide by 2 minutes,
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and let's say that would be 0.6 kilometres per minute,
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so, your average speed from
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t equals 9 to 11, in this case, and I'm just making up these numbers so we have
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something to talk about,
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is a little bit less than your average speed.
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You're going a little bit slower than average here.