WEBVTT
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It's like every were given the error function which is
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used in probability statistics and engineering. The error function
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is defined to be Two over the square root of
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pi times the integral from 0 to X of each
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. The negative T squared D T is I'm afraid
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of action. In part, they were asked to
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show that the integral from A to B of each
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of the negative T squared DT can be written in
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terms of error functions. So the integral from A
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to B. Eat a negative t squared DT.
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Well, who wants to the kids? If you
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wanted those two fellows for instance, Big favorite system
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Harleys. We can rewrite this as the integral from
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zero to be of E to the negative T squared
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DT plus the integral from a+20 of E to
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the negative T squared Bt. Using integral properties,
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we can rewrite this as everything I have Integral from
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0 to be. In fact, I'll write this
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as route high over to here times two of a
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root pie. And the integral from zero to be
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of E. To the negative T square D.
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T minus two of the Root Pie. Times.
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In the girl from zero to a. Of each
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of the negative T squared DT. The notice that
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by definition this is rude pi over two times error
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function evaluated at B minus the error function evaluated at
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a. Yeah, this is what we wanted to
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show for part egg. Mhm. Yeah. Then
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in part B we're trying to lose weight from china
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for us to show that the function Y equals E
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. To the X squared times the error function evaluated
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at X satisfies the differential equation. Why prime equals
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two X one Plus two over the square root of
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Pi. Well, all we have to do is
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simply plugs our function into this differential equation on both
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sides. So first let's find the left hand side
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. Wide prime. Buy the product rule. This
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is a derivative of E. V. X squared
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which is uh two X times E. To the
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X squared times the error function evaluated at X plus
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E. To the X squared times the derivative of
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the air function which this is going to be to
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overrule pie. And then by the fundamental theorem of
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calculus, the derivative of the error function is two
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of the route high times each of the negative X
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squared. Since the function E. To the negative
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T squared is continuous for all values of T.
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Yeah. Mhm. Yeah. As he Hard that's
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hard. eight are they hard up? Here's the
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thing bigger animals actually lost. And sure now that's
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the left hand side. The only way to see
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used. Do you do you do? I'm up
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party. Yeah. So you know that saying the
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house you can join now joined right now by partly
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my sister. So actually could simplify first and this
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is equal to two X times E. To the
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X squared air function of X plus. And this
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is simply two of her route pie Times each of
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the zero which is the same as one. Now
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we can rearrange since error function is Y over E
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. To the X squared. This is two X
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times not during not during the ship wow. Where
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we're just either the X squared error function. X
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. This is why This is two x. Y
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Plus two Over Group I. No and this is
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the right hand side of the equation. Therefore the
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function is a solution to this differential equation. That's
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probably enterprise. No that's not. Oh it's coming
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. Oh I can tell you.