How and why are these three graphs visually related? $y=ex^2 \sin\left(\frac{1}{x}\right)$, $y=ex^2$, $y=-ex^2$

Question

While graphing equations I come across, or find interesting, I found the relationship between graphs $$y=ex^2 \sin\left(\frac{1}{x}\right) \qquad y=ex^2 \qquad y=-ex^2$$

https://www.desmos.com/calculator/nhf6tphhbb

How are these three graphs related, and is there some pattern to their points of intersection?

It's nothing deep, it's just that the factor $\sin(1/x)$ oscillates between $-1$ and $+1$. — Hans Lundmark, Nov 05 '19 at 16:45

score 2 · Answer 1 · answered Nov 05 '19 at 16:45

2

As $-1\le\sin{(\frac{1}{x})}\le1$, we have $-ex^2\le ex^2\sin{(\frac{1}{x})}\le ex^2$.

So, the functions $ex^2$ and $-ex^2$ serve as upper bounds for $ex^2\sin(\frac{1}{x})$

answered Nov 05 '19 at 16:45

Ishan Deo

3,574

Goodness, I really am forgetting the basic high school maths! thank you for explaining something which should have been very simple! – Eva Nov 05 '19 at 16:49

How and why are these three graphs visually related? $y=ex^2 \sin\left(\frac{1}{x}\right)$, $y=ex^2$, $y=-ex^2$

1 Answers1