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I was recently messing around with desmos by plotting random graphs. I came across this peculiar function, namely $y = \sin (\cos (e^x))$. I noticed that the graph is basically a sine wave whose period gets shorter and shorter. However, at a few $x$ values, this trend falters for a bit. The points I'm referring to are near <span class=$x = 8$, where there are white specks instead of the expected red.">

The points I'm referring to are near $x = 8,$ where there are white specks instead of the expected red. Can someone give me an insight into why this may be happening?

Any help would be greatly appreciated!

YuiTo Cheng
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John A.
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  • Have you also tried plotting the thing you are plugging into $\sin$ (i.e. plotting $y=\cos\left(e^x\right)$)? – Minus One-Twelfth Apr 13 '19 at 23:10
  • @MinusOne-Twelfth How is that supposed to help? It's a function that oscillates faster and faster between $-1$ and $1$, just like this one should oscillate faster and faster between $-\sin1$ and $\sin1$. –  Apr 13 '19 at 23:13
  • Maybe by visualising what you're plugging in to sine, you could better visualise the output. Anyway, doesn't hurt to try when exploring this. – Minus One-Twelfth Apr 13 '19 at 23:17

2 Answers2

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That’s simply a rendering error with the software. The graph continues to oscillate as you’d expect. Zoom in and the problem should resolve itself.

Bruno E
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The software may be performing some sampling on $x$ values to plot your function.

When the frequency of your function is low, for this case when $x$ is small, the effect of sampling is not noticeable.

But when the frequency of your function is too high relative to the sampling frequency, precisely when your function's frequency is higher than half the sampling frequency, aliasing occurs and your function is replaced by a lower-frequency function that has the same sample values.

In the extreme case, when the frequency of your function is around a multiple of the sampling frequency, the alias from the samples can appear to be a constant function, a function with lower frequency.

This may give the effect of the missing red around $x=8$, when the displayed frequency of the function is lower than expected.

peterwhy
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