WEBVTT
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Use linear approximation to estimate the square root of 100.5
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. So what we're going to hear, you define
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the function F X equals squared bags. And we're
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gonna find the linear approximation To this function at x
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equals 100 30s. We're going to find the equation
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of the line that is The linear approximations wave at
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x equals 100. For that we need a point
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where the line is going to pass through and the
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slope. This cases slope will be given by the
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derivative of the function. So the derivative of this
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function is derivative respect X Of X to the 1/2
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, which is the exponential representation of scores of eggs
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. You know, these derivative is 1/2 times x
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To the 1/2-1. And it ends here because
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there is no more calculation to do because we have
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the basis the variable X only that is we don't
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need to apply the change room. If we apply
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it In any way, the relative of the base
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X will be one. So in this case is
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not needed really. So we get one half times
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X to the negative one half And that's the same
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as 1/2 square it effects. So the first serve
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a tive of F at any point X is equal
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to one over to times square effects. Yeah.
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Yeah. And Dad, derivative at 100. Where
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is the point where we're gonna find these tangent line
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is equal to 1/2 squares of 100 Square to understand
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. So we get 1/20. Let's say this is
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a slow for the turn tonight. Now the tangent
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line of I've got X equals 100 has slope 1/20
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. And buses through. Mhm. The point 100
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f of 100 That is 100. The image of
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100 is the square root of 100. So it's
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a 1000.100 10. So we know a point of
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the line this case, attention line of to the
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care of F at X equals 100. And we
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know the slope with that. We can write the
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equation. So the equation of the tangent line two
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to graph of F At X Equal 100. He's
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given by then we know is why minus These.
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Follow here 10 equals slope. That is 1/20 Times
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6- This value here. 100. That is
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Y equals then Plus 1/20 X-100. That's the
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same expression we obtain If we develop the tailor paranormal
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at 100 of Degrees one. He's just that.
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Then we can say that the square root of X
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. That is the function. Let's say that that
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way function is Approximately equal to its tangent line at
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zero four eggs Appeative close to 100. That is
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when we are close to 100. This approximation is
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a good approximation. If we go away from that
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point, it's not the case. And that this
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means that square root of X is approximately equal to
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10 plus 1/20 Time 6-100 four X. Close
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to 100. And we used this idea to Express
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the fact that square to 100.5 Within approximately equal.
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Let's let's see that 100.5 fulfill these property because it's
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very close to 100, There is only a distant
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of 0.5. So this could be a good approximation
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to this, approximately equal to 10 plus 1/20 Times
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X. Which in this case is 104.5 minus 100
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. And with this calculation here is 10-plus 1/20.
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0.5. That's the same as 10 plus 0.5 Over
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20, that's equal to 10 plus 0.5 is 5/10
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. So it's five over 200. And we simplify
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this fraction divided both Stern by five, we get
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one over 40. And if we add up these
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two fractions, forget 401 over 40. That's a
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fraction we get to calculate and that's exactly equal in
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this case. Do it by hand. If you
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walk to 10 25 okay, that is The square
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root of 100 0.5 It's approximately equal to 10 0
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25. And thats approximation fine. Use leaner Approximations
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to the function through the tangent line at 100 and
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if you use a calculator we can verify that this
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is in fact a very good approximation