Why does my calculator show a gap in the graph of $y=\left|\left(\sqrt[3]{x-3}+9\right)^5\right|$?

Question

I was learning a lesson in math about graphing, and I mashed a few random things together in my calculator (a TI-84 Plus CE) to create $y=\left|\left(\sqrt[3]{x-3}+9\right)^5\right|$ in a graph. Upon graphing, I saw nothing, so I used ZoomFit to find it. Upon graphing again, I saw this:

I would like to know why there is a gap in the line. I have never seen it before.

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Answer 1

Great question! This is a testament to finite precision of computational tools. The reason is because your function is not differentiable (in this case, the slope becomes infinite at X = 3). The calculator can only plot so many points, and so due to the steepness, there appears a large gap near this point.

Answer 2

This is more a matter of the TI-84 than math. At x=3, the line tangent to the curve is vertical. This means that as you get closer and closer to 3, the amount that y goes up for each step in x get larger and larger. On either side of the gap, the graph is indeed made up of vertical pixels. When the calculator gets to exactly x = 3, something happens with its graphing algorithm where the slope is so high that it just jumps to the next part of the graph. You should try playing around with your $\Delta X$ and TraceStep and see how they affect the gap.

You can also try Y = $\sqrt[3]{X}$ and see whether a similar thing happens at x = 0.

Answer 3

For the machine, the numbers are isolated points..

the number just after $3$ is some $3^+=3+\epsilon $ and just before $3$ is $3^-=3-\epsilon$.

so $$f (3^+)=(9+\epsilon^\frac13)^5$$ $$\approx 9^5 (1+5\frac {\epsilon^\frac13}{9})$$

and $$f (3^-)=(9-\epsilon^\frac13)^5$$ $$\approx 9^5 (1-5\frac {\epsilon^\frac13}{9})$$

the gap is $$f (3^+)-f (3^-)\approx 10.9^4.\epsilon^\frac13$$

which can be seen for some small computers.