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

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

03:54

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Oregon State University

Harvey Mudd College

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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.

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