WEBVTT
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All right. So we are first being asked to
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find the vertical and horizontal ask until so, um
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, dysfunction can be rewritten as one over e to
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the X Square. And now, if you haven't
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already noticed, um, this bottom function, I'm
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dysfunction denominator cannot equal zero because that's what defines vertical
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. I think that's one of the ways. And
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since exponential function never equal zero, there is no
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vertical Jacinto. So no vertical ascent toe. And
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so now we gotta find the horizontal axis until we
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find that I find the limit. So the limit
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is X goes to infinity, and this will be
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one over e to the X squared. And as
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you can see as, uh, this number guests
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, uh, the denominator gets exponentially higher. You
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get one of the infinity, which is a one
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of a really big numbers, just zero. And
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this is a case also for negative numbers because negative
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numbers squared. It's the same thing as positive number
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squared. So this was also a plus or minus
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infinity. So we have a horizontal attitude at y
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equals zero. Yeah, and then now, to
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find the intervals in which function increases or decreases,
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we apply the first derivative cath that I've been taking
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the first derivative. In this case, you will
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have to apply the chain rule, so I'll come
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out to be negative. Two X equals negative.
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Two x 10 e to the negative x squared.
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Uh, sorry about that. Just bring this down
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. I was just negative two x or eat the
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minus X squared and this can be rewritten at negative
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two X over E to the X squared. Then
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you set this equal to zero. The denominator cannot
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equal zero, so that's not one of the critical
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number. Negative two X equals zero when X is
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equal to zero. So we have a signed chart
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evaluation here, so it's going to be X,
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and you put it at zero, and then you
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bring it down and they were looking at the sign
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of a crime. But people are numbers less than
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zero. You get positive numbers and you fucking greater
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than zero. You get negative numbers, so we
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noticed increasing and decreasing. So we have a local
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max occurring at X equals zero. Uh, and
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we know that it is increasing from negative infinity to
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zero, and it is decreasing from zero to infinity
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. Now, to find where the funk says,
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uh, find the functions can cavity would take the
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second derivative tests that they have been trying in the
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second derivative. So again, this is a little
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bit more chain rule and, uh, combination of
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some protocol, and this will come out to be
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negative one plus two x squared statistical zero. Um
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, this part of the function cannot obviously equals zero
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. So we have to set this part of the
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pumpernickel zero. So this will be negative one plus
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two x squared because you have to find a critical
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number. You add one divide by Tuesday. You
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got X square called 1/2. This will be X
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equals. The square root of one is one.
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So we one over root two plus or minus because
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we took a square root. Now we can do
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a sign chart evaluation. This will be negative.
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1/2. It would be positive one over. Richer
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, not a line, but not a line.
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Looking at the sign of a double prime your fucking
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values less than negative one or two. You get
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positive numbers between these two negative and positive. Still
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can't give up. Down, up. So we
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have a con cave up interval occurring from negative infinity
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. Two negative 1/2. And from one over to
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to infinity. We have concrete down occurring between negative
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one over to positive 1/2. And we have inflection
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points occurring when sign change occurs. So that's occurred
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that both plus and minus route to I mean plus
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or minus, uh, 1/2. So inflection point
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point occurred that plus or minus 1/2. Now we
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have enough information to draw a graph. So it
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looks something like this. Uh, you know that
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there is a local max at zero, and we
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have a concave up shape from here from the negative
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. So we're kind of coming up like I use
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so this is like, like up you, and
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then I'll turn into a concave down shape. So
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now we start decreasing after zero, you come down
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and and then I'll go off to zero. And
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this this max occurring a gray here. Um,
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the peak is supposed we had zero Mhm. Sorry
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, if that wasn't clear enough, but this is
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the graph of F WebEx and it's symmetrical. That's
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it.