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Use a linear approximation (or differentials) to estimate the given number.$ \sqrt {100.5} $

$\approx 10.025$

00:44

Amrita B.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Derivatives

Differentiation

Harvey Mudd College

University of Nottingham

Idaho State University

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

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