What is the relationship between higher-order derivatives and their functions? When I say higher order I mean derivatives above the second derivative.
For example,
f(x) = x[sup]4[/sup]
f’(x) = 4x[sup]3[/sup] finds the slope of f(x) at x; it evaluates the rate of change of f. As f increases, the slope increases.
f’’(x) = 12x[sup]2[/sup] finds the slope of f’(x); it evaluates the rate of change of the slope of f. The slope changes more over shorter intervals and progressively slows down until x=0, where it changes progressively faster.
So, what is the relevance of f’’’(x) to f? f[sup]4/sup? f[sup]infinity/sup? In a similar note, what does the rate of change of acceleration called, and what does it measure?
The rate of change of acceleration is called jerk, and it represents…well, it represents the rate of change of acceleration.
IIRC, f’’(x) denotes the curvature of f at x. Basically, it’s the radius of the circle passing through f(x) which most closely approximates f. The sign indicates whether the circle is above or below the graph of the function (but I can’t remember which is which).
I don’t know of any geometric interpretations for higher-order derivatives.
I can’t even begin to imagine what the Hubble telescope would do with derivatives of that high an order (then again, I only started Calc 1&2 this year so I don’t know a whole lot about derivatives right now).