WEBVTT
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Okay let's explain why the function is discontinuous that they
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give a number a let's sketch a graph. Okay
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let's sketch this graph. Right? So if X
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is less than zero then the F. Of X
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. Is equal to co sign next. So let's
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grab that. Two pi negative three pi halves negative
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pi negative pi house and zero. So on the
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left hand side when X is less than zero this
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is just co sign. Where does coastline start?
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Carson always starts at one drops down to negative,
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Dropls down to zero And that negative one and then
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one and then zero. Sorry listen this up then
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back up again. Nice little lotus like flower shape
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. You should know what the graph of coastline looks
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like. Okay now I missed a little tiny stuff
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that's pretty important here which is Co sign is less
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than zero when it's less than zero not less than
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or equal to. Which means there has to be
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an open circle here. This isn't even more important
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because the next piece of the piecewise is that f
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of X equals zero. If it's if X is
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equal to zero so at X equals zero we're going
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to have a point right here at zero. Okay
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if X is greater than zero and we're going to
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have the function one minus escort. So let's plug
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zero in for their if we plug zero in for
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one minus X squared We're going to get 1 0
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which is one. So let's not do that.
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Let's graph the function a little further on. Um
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mm. Black. Okay I don't really want to
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do this in terms of pie anymore because that's kind
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of a pain, right? And with this the
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scale of the graph doesn't matter as much. But
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let's let's try pie halves is about 1.7. So
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I take that, that's about one two, three
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for that should be enough. Okay let's do the
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right most side now one minus X squared. If
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I plug in one I'm going to get zero.
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If I plug in zero. This is important to
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I'm going to get one. But this is also
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an open circle because it says X is greater than
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zero, not X is greater than or equal to
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zero. The only place where I can place the
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only spot at zero where I can place a closed
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circle like a dot of value is zero because that
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is what the piece by says zero if X equals
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zero. The other functions don't let me do that
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. So now let's plug if we plug in the
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one we get zero. If we plug in the
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two we get negative three because two squared Is 4
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, 1-4 is-3. It'll just keep going
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down three Squared is nine. Yeah it's going to
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be way down there. So there will be something
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like we yep it's like a downward parabola. So
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it should actually look more like that. More of
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a bell shape than like that. Three should be
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somewhere there. Okay now let's talk about why it's
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discontinuous at equal zero. Let's take the limits.
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The limit As X approaches zero from the left of
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F of X is going to refer to a certain
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function in the piecewise zero from the left means less
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than zero. So this is cosine X As it
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approaches zero from the left. If we plug in
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zero further we get one. But if we plug
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in, yeah, zero from the right, we're
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going to get F of X greater than zero X
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plus of one minus X squared. We also get
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one. So in this situation this is a removable
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dis continuity because the limit from the left which is
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the co sign peace is equal to the limit from
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the right at one. Therefore the limit As X
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approaches zero of F of X exists however, And
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this is important here. The limit as X approaches
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zero of F of X is not equal to F
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of X. So the limit as X approaches zero
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from Alphabet's one. But the value as we see
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in the piecewise is zero. So it's discontinuous because
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of a removable this continuity. That's what we call
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this situation. Removable discontinuity when the limits from the
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left and the right exist, but they don't match
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the value of the function itself. That's a removable
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discontinuity. Yeah. Yeah