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 \(n=1\):

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?

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestSure, 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.

Log in to reply

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

Log in to reply

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

Log in to reply

Log in to reply

Log in to reply

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

Log in to reply