an old math question clarified

[|ndußtrial]

if this grid's lines represent two dimensions, how would one describe the changing of the length of the vertical line segment drawn between points on curve x and line y (the one outlined by yellow bars?)

[Edward-san]

From the number of the vertical and horizontal lines, I suspect it's simple as describing the second quadrant of the circumference with radius 8 (the number of bars), translated 8 places to the right. Let me know if you need an explanation of this.

[cambertian]

My first instinct told me "integrals!" but those describe the full area underneath a curve and not segments from the curve.
Upon further thinking, this might have to do with the integral's best friend, derivatives (given that you're asking for the change in lengths over a distance.)

I can't really help you much further than that, I'm afraid...

[Gez]

So you don't want the length, but the changing of the length, right? This is the derivative.

You can then have a derivative to the derivative (the changing of the changing of the length), and so on.

An example of real-world application of derivatives: speed. The derivative of speed is acceleration. Then the derivative of acceleration is jerk. After that, you've got snap, crackle, and pop!