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Problem 3 Medium Difficulty

Use the given graph of $ f(x) = \sqrt{x} $ to find a number $ \delta $ such that
if $ \left| x - 4 \right| < \delta $ then $ \left| \sqrt{x} - 2 \right| < 0.4 $



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Emanuel J.

February 5, 2021

Use the given graph of f(x)to find

Video Transcript

Yeah. Hello. We have one of these obnoxious questions with epsilon delta is regarding the graph of a function. In this case F of X equals root acts. So we're told to use the graph, I don't have access to that right away. But we can sketch a quick graph. Got 1234567891. I have a couple of points where I can do square roots readily James. So let's plot a couple of points that we know we're going to be here at four. We know we're going to be two at nine. We know we're gonna be three and the graph is going to do kind of like that. So what we're interested in actually is this point right here at X equals four. Because we are asked given that Square Root of X -2. So we're giving the absolute value Less than .4. We want to find a delta. Don't listen very I'm gonna find a delta. Such that we have computer indentation here. We want to dealt us such that if I can write to uh huh, Absolute Value X -4. Less than delta. So looking at the graph what this means is this right here we rewrite that as saying we want F of x -F of four because 2 is the square root of four to be in this range. Right? That's what we're saying. And so now we need to find Something so that X -4 being inside of delta being inside of delta And negative delta. Whose forgot to write my negative science. Those are important. Negative 0.4 negative delta. So this is kind of an epsilon delta thing. We have specified an epsilon of 0.4 around the value that we're interested in. So we're saying we are within this range, we now want to find what range on X. We'll meet the lower and upper end. If you can see, I've got this band on the Y axis surrounding Y equals two and that gives us a band on the X axis surrounding X equals four right there. So because my graph is crudely drawn at best, I can't actually estimate that from the graph, but fortunately a little bit of algebra can give us an exact answer. So if you wanted to know how to do this algebraic lee, you're about to learn. We're looking for a condition on X. So let's start with the condition we want to meet, we want to meet the condition That square root X -2 is greater than negative .4, But less than positive .4. So let's start by adding to everywhere. Because this is like saying we have an inequality here and then we have another inequality here. We could work with them separately, but rather We can do it at the same time. You add to everywhere. The left side becomes 1.6 less than route X, less than 2.4 because we added to over here. So now we can square everything because nothing ugly is going to happen when we do that. We get 2.56 over here, less than X. Less than I want to say. It was 5.76 when I did that earlier. Yeah. 5.76. Okay, so those are conditions, if X is in that range then it will meet the original condition. But the problem is we can't develop a delta off of this because these are not equidistant. So this is like saying for minus 1.44. And this is like saying four plus 1.76. So we can't get any farther away from four Than 1.44 before squaring it causes us to fall outside of this range right here. We have to be in this range. But if we try to choose plus one, if we try to choose a delta of 1.76 then we end up below this endpoint right here. So our delta then must be given by 1.44. And if you want to you can check it, try to in four minus 1.44, squaring it, You'll get where you need to be. Try doing try taking the square root of that and you'll end up within this range. Do four plus 1.44. Well that's 5.44 which falls inside this range. And so we have found our delta. Thank you folks have a good day