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Zoom in toward the points $ (1, 0) $, $ (0, 1) $, and $ (-1, 0) $ on the graph of the function $ g(x) = (x^2 -1)^{2/3} $. What do you notice? Account for what you see in terms of the differentiability of $ g $.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri 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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That's problem. Number forty six of this tour Calculus a fetish in section two point eight. Zoom in towards the point one zero zero one and negative one zero on the graph of the function. G of X equals the quantity X squared minus one raised the two thirds power. What do you notice? Account for what you see in terms of the different ability of G. So he re applauded the function using our plotting function party in program or yep, he's a graphing calculator to blocked the function g of X. Ah, and then here we have identified the three points and they get one zero. Here is this point down here? Zero one is this point appear? And then one zero's At this point, with the three functions in question, we want to discuss the different ability of GS each of these points arm. Let's start with the point zero one dysfunction here we see that the function before and after is a smooth function. There is no dis continuity in the function, and in fact we can see exactly that. The slope of the tangent line there. If there's a tangent line here at X equals zero that it would be horizontal and that slippery p zero. So we know that the function is differential at the point zero one zero one. At the other two points one zero and negative one zero. The function has this custom shape. You call it a kink or a corner. But essentially the function, although continuous, has a sharp change in the in its derivative or in the slope of its tangent lines. Um, for both of the points coming from the left of X equals theta one and X equaled one, you are approaching with negative natively sloped attentions. And then afterwards you have positively slipped tension line and therefore there is a different that there is a discontinuity in the different ability of the function, meaning that the function is not defensible. Attic equals negative one, and that executes one So again, to recap the function G is not the French. Will it executed one or an X equals positive one. But it is a French politics equals one. Our X equals zero. So this point zero one

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