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Use the given graph of $ f $ to find a number $ \delta $ such thatif $ 0 < \left| x - 3 \right| < \delta $ then $ \left| f(x) - 2 \right| < 0.5 $
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Calculus 1 / AB
Limits and Derivatives
The Precise Definition of a Limit
Missouri State University
Harvey Mudd College
University of Michigan - Ann Arbor
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 were given this graph we're asked to find the number delta. Such that If zero is less than the absolute value of X -3, less than delta then F of x minus two sorry minus two is less than 0.5. Okay, so You can see from the graph right that when F of X -2 is less than .5? Well then we are in this range aren't we? Somewhere between 2.5 and 1.5 for the F of X. Okay. And that means then that We have these two deltas, right? Delta one Will be 3 -2.6 which is 0.4 and Delta two which will be 3.8 -3 which is me fix that up just a little bit right there because that's an absolute value sign there we go. 0.8. And so then that means that the delta we're looking for is the minimum Of these two Delta 1 and delta two. The minimum of those two is .4. So this is 0.4. So we noticed that If .4 is delta and everything less than .4 away from three, Right from 2.6 up to 3.4 being that range right there wouldn't It is all within 0.5 of two on the f. Of X, is it not? It is okay. Yeah so there's our answer. Um Right there
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