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# Local linearity

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.F (LO)
,
CHA‑3.F.1 (EK)

## Video transcript

let's say that we're interested in approximating what the square root of four point three six is equal to so we want to figure a problem what we want to figure out an approximation of this and we don't have a calculator at hand well one way to think about is we know what we know what the square root of four is we know that this is positive to the principal root of four is positive to say okay this is going to be a little bit more than two but let's say that we want to get a little bit more accurate and so I want to show you in this video is a method for doing that for approximating the value of a function near near a value where we already know the value so what am I talking about so let's just imagine that we had the function we have the function f of X is equal to the square root of x which is of course the same thing as X to the one-half power so we know what F of two is we know we know that F of 2 plus R we know that f of 4 is we know that f of 4 is the square root of 4 which is going to be equal to 2 or the principal root of 4 which is equal to positive 2 and what we want to approximate we want to figure out what F of we want to figure out what F of 4 0.36 is equal to this is just another way of framing the exact same question that we started off this video so let's just imagine our function let's just imagine it for a second so let me draw some axes this is my y-axis this is my this is my x-axis and let's graph y is equal to f of X so let's say it looks something like this y equals f of X looks something like that so that's pretty decent alright so that's that right there is y is equal to f of X and we know F of 4 is equal to 2 F of 4 is equal to 2 so this is when X is equal to 4 I haven't drawn it really to scale but hopefully this is clear enough so that right over here is going to be 2 to F of four and what we want to approximate is F of 4 point 3 6 so four point three six might be right around right around there and so we want to approximate we want to approximate this Y value right over here we want to approximate that right over here is f of four point three six and once again we're assuming we don't have a calculator at hand so how can we do that using what we know about derivatives well what if we were to figure out an equation for the line that is tangent to the point two tangent to this point right over here so the equation of the tangent line at X is equal to four and then we use that linearization that linearization to find to approximate values local to it and this technique is called local linearization so what I'm saying is let's figure out what this the equation of this line is let's call that L of X and then we can use that to a product that we can evaluate that at four point three six and hopefully that'll be a little bit easier to do than to try to figure out this right over here so how would we do that well one way to think about it obviously there's many ways to express a line but one way to think about it is okay it's going to L of X is going to be let it's going to be it's going to be F of four it's going to be F of four which is two it's going to be F of four plus the slope the slope at at x equals four which is of course the derivative F prime of 4 so that's going to be the slope of this line of L of X is f prime of four let me make that clear so this right over here is the slope the slope when X is at x equals four so it's the slope of this entire line and so any other point on this it's going to be f of 4 plus the slope times how far you are away from x equals four so it's going to be times X minus four let's just let's just validate that this makes sense when we put four point three six here when we put four point three six you actually let me zoom in this graph just to make things a little bit clearer so if this is so I'm going to do is zoom in I'm going to do the zoom in I'm going to try to zoom in into this region right over here so this is the point this is the point 4 comma F of 4 and we are going to graph L of X so let me do that so this right over here is L of X that's L of X and let's say this right over here this right over here is the point 4 point 3 6 comma F of 4 point 3 6 and the way we're going to approximate this value is to figure out what to figure out what this value is right over here and what is this one going to be this right over here is going to be this is going to be 4 point 3 6 comma L of 4 0.36 this line evaluated when X is equal to 4 point 3 6 and what is that going to be what is that going to be equal to well let's see let's just evaluate L of four point three six it's going to be F of four so it's going to be 2 plus the derivative so the slope of this line plus F prime of 4 times X minus four so four point three six minus four is going to be times 0.36 and that makes sense you're starting at two and you're saying okay my change in X my change in X is four point three six so my change in Y is going to be my slope times that change in X to get me that value to get you that value right over there so let's figure out let's figure out what this let's figure out what this thing what this thing actually is so to do that we need to figure out F prime of 4 so let's go back up here I'll try to leave actually I'll leave this little visualization here so let's see F Prime f prime of X is going to be 1/2 X to the negative 1/2 just using the power rule over here so f prime of 4 f prime of 4 is equal to 1/2 times 4 to the negative 1/2 which is of course equal to 1/2 times 1/2 4 to the 1/2 would be 2 4 to the negative 1/2 is going to be 1/2 so this is equal to 1/4 so L of we deserve a little bit of a drum roll now L of 4.36 is equal to F of 4 it's equal to F of 4 which is the let me just rewrite it it's F of 4 plus F prime of 4 plus G why am i switching to that color let me do the yellow plus F prime of 4 times x times four point three six four point three six let me make this actually the new color just so we see it so for 0.36 so times four point three six minus four minus four actually let me make all the four is one color too so you see is it's the same so just like that so what is this going to be well this we already established is positive two this we already established let me do this in the yellow color this we already established is 1/4 and this part right over here is 0.36 so this is going to be equal to 2 plus 1/4 times 0.36 is zero point naught or 0.09 so this is going to be equal to this is going to be equal to two point zero nine so that is our approximation and it should be at least based on how I graphed it a little bit higher than the actual value of the square root of four point three six but we could write that up here this is going to be approximately let me just write it this way the square root I'll just write down here so we could say the square root of four point three six which is the same thing as F of four point three six this is approximately equal to two point zero nine now let's just say we happen to find a calculator and just out of curiosity let's see how good of an approximation this actually is let's get a calculator out and so we want to do the square root of four point three six and we get two point zero eight so we actually if we were out of the nearest hundredths we got a pretty good approximation just like we saw in this in this indicative graph right over here it is our approximation was indeed a little bit higher than the actual value
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