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DM
Numerade Educator

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Problem 4 Easy Difficulty

Use the given graph of $ f(x) = x^2 $ to find a number $ \delta $ such that
if $ \left| x - 1 \right| < \delta $ then $ \left| x^2 - 1 \right| < \frac{1}{2} $

Answer

$\delta=0.22474$ (or any smaller positive number)

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Video Transcript

So in this problem we're given this graph of F of X equals X squared and we're asked to find the number of delta so that her such that if x minus one you do this find delta So that X -1 less than delta, then x squared minus one is less than a half. Okay, so we have this kind of graph going on from this relationship which means we're looking for this delta and this delta and whichever is the minimum or the smallest of these two. Well, we can see that first of all that from from Y equals a half then as F of X was X squared. We just pronouncing confusion, this will make this F of X. Right, so that we're at this point here, F of X was x squared. Okay, So that means that X is approximately 0.7. Okay. And when F of X is 1 1/2. So now I'm up here or F of X was x squared then X is about 1.2. All right, so now what do we have here? We have that? The Absolute Value of 1-? Make sure got this. Right. Yeah, I didn't think so Of the value of 0.7 -1. 0.3 And the absolute value of 1.2 -1 Is 0.2. And so the minimum, let's see, let me phrase it this way, this one. This one Is Delta one and this one is Delta two. And so then the minimum of delta one and delta two Is well Delta two. Right at 0.2. So that is the delta, the number we're looking for. That makes all of this true up here, then.