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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 $.

differentiable at $(0,1), \quad$ not differentiable at $(-1,0)$ and $(1,0)$

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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