WEBVTT
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Okay, This question wants us to find a function
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. Such the trap is I'd rule is a better
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approximation in the mid point, which usually isn't the
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case sort of do this. Let's find a function
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that totally messes with the midpoint approximation. So to
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do that, we gotta find a function that does
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it not represent itself at the mid points, which
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means acts totally different at the mid points. And
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it does everywhere else, but it still has to
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be continuous. So let's say we have something like
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this. So it starts it one, then it
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hit zero at the midpoint goes back up. Then
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it goes back down a hits 1.5 again. And
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then you see the pattern so we can see for
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sure that area from midpoint, it's just gonna be
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Delta X plus the sum of f of mid points
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, which is just Delta X Times zero, which
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equals zero. Because the only points were using the
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sample from are the zeros right here. And if
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you're wondering what this function is, I just picked
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absolute value of co sign of Pi X just to
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get the period and positive definite nous that we wanted
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. So for the trap is oId. On the
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other hand, we get 1/2 times f of zero
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plus two F of one plus f of two,
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which is 1/2 times one plus one plus one.
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So area from the trap is oId is too,
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and actual area. Oh, and just for reference
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, we should write again. The area from midpoint
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is equal to zero, so we know the actual
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area is positive. So we should already have an
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intuition that the midpoint is a terrible approximation. But
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if we actually calculate this integral out, it's approximately
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1.27 So the area of the trap is oId is
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closer.