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An interesting everywhere-differentiable function

A graph has equation \[f(x)=-\frac{1}{n}x-\frac{1}{n}\sin x+\left|\frac{1}{n}\sin x-\frac{1}{n}x\right|\]

for a non-zero integer \(n\). Let's look at the graph for \(n=1\):

Imgur

Imgur

Hey look, no rough non-differentiable edges. Cool! Try other values of \(n\) to see what happens.

Can you create any other interesting graphs with the absolute value function that are differentiable everywhere?

Note by Daniel Liu
3 years, 6 months ago

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Sure, replace \(n\) with \({ n }^{ 2 }\) Another one, which is technically not a function, is \(\pm \sqrt { { x }^{ 2 }-\left| { x }^{ 3 } \right| } \), looks kind of like a lemniscate, is differentiable everywhere except at \(x=-1, x=1\), and has an area of \(16/15\). But the use of \(\pm \) makes this claim of differentiability at \(x=0\) iffy at best.

Michael Mendrin - 3 years, 6 months ago

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well, \(n\) is a dummy variable, so changing it doesn't really affect anything. If you meant \(x\), then the function isn't very clever nor differentiable everywhere; it just looks like a regular sine function with a little nick in the middle.

The second one is pretty cool. One can manage to graph the whole thing at once on desmos by using \[|y|=\sqrt{x^2-|x^3|}\]

Here is a comparison of the graph to the lemniscate: https://www.desmos.com/calculator/vfw1amglgn

Daniel Liu - 3 years, 6 months ago

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Yes, one can see that tiny difference in areas, the area of the lemniscate being exactly \(1\). As for the other, replace \(n\) with \(1+{ x }^{ 2 }\).

Michael Mendrin - 3 years, 6 months ago

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@Michael Mendrin Ooh, that graph looks pretty interesting. It might be convincing enough to pass as one non-piecewise graph, although people would have a hard time finding the equation for it.

Daniel Liu - 3 years, 6 months ago

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@Daniel Liu What program did you use to draw the graph? Thanks for posting this. I will post something related to graphs too.

Adrian Neacșu - 3 years, 6 months ago

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modulus functions !! they're awesome pieces very useful for manipulation ................ nice one !!............. seems like the graph changed it's mind after entering the positive x-axis!! :-p

Abhinav Raichur - 3 years, 6 months ago

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