WEBVTT
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Okay, we want to find a route of this
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equation. Um where we want to show that there
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is one in the interval from 2-3 using the intermediate
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value there. So root usually means the value of
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some function is zero. We actually don't have that
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. So we have to make that. Let's make
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the function natural log of X minus x plus route
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X. Because if that were zero then this equation
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would be true. Right? We can get from
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there to that equal zero by adding or subtracting stuff
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from both sides. Okay. Um So we have
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that. Uh let's look at uh this function is
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defined on the endpoints. Natural log is defined for
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two and three. So let's look at the values
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of FF two and Ffs and because we have a
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natural log we need a calculator, I'm not going
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to be able to show that on the screen because
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I don't have a good computer calculator. But whenever
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you're typing into a calculator, just make sure all
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the parentheses work out correctly. So FF two,
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Let's see plus the root of two um Is approximately
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and we just need an approximate value 0.107 And then
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f of three is approximately negative 0.169. And this
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negative is very important. So what's going to happen
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, let's sketch this. I don't know what this
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function looks like, but between two and three The
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value at two is some positive number. The value
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at three is some negative number and the function is
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continuous continuous In the interval 2- three. so
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there's no way to get from 2-3 without lifting your
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Pencil or pen without passing through zero. So that
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means the intermediate value theorem says there exists Some value
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of x in the interval 2-3 such that FFX is
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zero where zero is just, I can pick any
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value, so I could have picked.05. That's
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also a value between.107, a negative.169.
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And there would have to be a value because I
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can't get from one to the other continuously without going
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through all the values in the middle.