WEBVTT
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right. The way I want to do this question
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is uh using the by normal serum, our pascal's
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triangle, which would be uh knowing that when you
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take something to the third power, Your coefficients will
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be 1, 3, 3, 1 or X
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squared. So to be X cubed an a minus
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X squared and then plus X and then minus one
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. If you don't believe me, you can foil
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that out three times. But there is a shortcut
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and that's using pascal's triangle. So now what I
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would do is divide each one of these pieces.
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So I'm still not doing the entire derivative executed by
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by exporters. Just X. These X squares would
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just cancel the next woman would become three over X
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. And then the last thing I'm going to write
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is X to the negative second power. There's a
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reason I'm writing all of these like this because now
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when I do the anti derivative where we add one
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to the exponent, multiply by the reciprocal of that
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exponent, we have an issue when we get right
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here. And that issue is that it needs to
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be natural log that's the anti derivative of one over
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X. But then this one's fine, you add
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one to the expo and multiply by the reciprocal.
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Uh and they were going from 1-2. So as
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we do all this shortcut and all of this work
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. Now we're ready to plug in our upper bounds
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, which is two squared is four half of that
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is to minus six plus three. Natural log of
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two. And then to to the 91st power is
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the same thing as 1/2. And we need to
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subtract off Plugging in one and for all those to
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be 1/2 minus three plus three natural log of one
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plus one. Now, as you're looking through all
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of that, I hope you notice at least the
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one half and the minus one half of later cancelled
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Natural Log of one is 0. Um So all
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we're left with on the right side is that negative
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three plus one? Which would be negative too.
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That when we're subtracting turns into plus tube so they
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have to Plus two is 4-6 is-2.
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So if I've done all of my math, correct
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, this should be your correct answer. Um I'm
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happy with this answer. If you're curious how the
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answer key changed a little bit, you can use
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the power rule to change that to be an exponent
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. Uh and two cubed is eight. I'm not
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sure why anybody in their right mind would do this
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, but this is correct as well. Okay.