Greg Egan on Nostr: Does this nice pattern continue? Only partly! If you plot y = A x^(N+2), then the Nth ...
Does this nice pattern continue?
Only partly!
If you plot y = A x^(N+2), then the Nth derivative of its curvature along the plotted curve will, at the origin, be A (N+2)!, which is the (N+2)th derivative of y.
However ...
The result is no longer unaffected if you add other terms, and it becomes nonlinear. For example, if you plot:
y = A x^2 + B x^3 + C x^4 + D x^5
then the 3rd derivative of the curvature, at the origin, is:
120 D - 456 A^2 B
Only partly!
If you plot y = A x^(N+2), then the Nth derivative of its curvature along the plotted curve will, at the origin, be A (N+2)!, which is the (N+2)th derivative of y.
However ...
The result is no longer unaffected if you add other terms, and it becomes nonlinear. For example, if you plot:
y = A x^2 + B x^3 + C x^4 + D x^5
then the 3rd derivative of the curvature, at the origin, is:
120 D - 456 A^2 B