I’m familiar with the standard time-derivative interpretations in which the first time derivative of a position function is velocity, the second derivative is acceleration, and the third derivative is jerk. (Cue ‘snap, crackle, pop’ joke.)
Derivatives are also useful for sketching curves - the first derivative dy/dx takes x as input and produces slope as output. d[sup]2[/sup]y/dx[sup]2[/sup], the second derivative, takes slope as input and produces concavity as output.
What, if anything, is the curve-sketching interpretation of the third derivative d[sup]3[/sup]y/dx[sup]3[/sup]?
It’s the rate at which the concavity changes. Just as the concavity is the rate at which the slope changes, and the slope is the rate at which the height changes. Giving it a name (like we give the name “concavity” to, essentially, “the rate at which the slope changes”) doesn’t necessarily make it any clearer; it may even only provide an illusory sense of understanding. It is what it is; the rate of change of the rate of change of the rate of change of the height.
The third derivative is positive where the curve is moving towards greater upwards concavity/lesser downwards concavity and negative where the curve is moving towards lesser upwards concavity/greater downwards concavity, and the third derivative is larger in magnitude where this movement is faster.
Part of the problem: Can you tell, at a glance, the difference between the graphs of Y = X^2 and Y = X^3 on positive X? If you can’t, then you can’t hope to tell at a glance the difference between having a constantly zero third derivative and having a constant non-zero third derivative. And if you can’t even tell that difference at a glance, it’s perhaps not terribly useful to understand the third derivative in such pictorial terms.
Well, that could just mean there’s been no evolutionary pressure on us to sense that kind of difference — as there has been for noticing straight lines versus curves, for example.
I don’t remember the third derivative coming up in my (long ago) physics education, or in my occasional readings today, but in principle it could be important somewhere. Going by own limited knowledge though, Mother Nature does seem to be content to do everything with mere second-order differential equations.
The only time I’ve ever seen a third derivative specifically mentioned is in the notion of a self-concordant function. Other than that, the only place I remember seeing higher order derivatives is in various power series expansions.
Thanks, guys. I’ve just been thinking about linear regression models and wondered if there was any useful information to be gleaned from including a cubic term. It’s a silly question that had been knocking around my head for a little while.
The Abraham-Lorentz force is the force on an accelerating charged particle, and is proportional to the rate of change of acceleration. When they covered synchrotron radiation, where this comes up, the professor mentioned that this was about the only place where a third derivative showed up in physics.
I recall reading (sorry no cite) that NASA scientists in the design of the shuttle used up to the twelfth derivative in the design process. Can anyone confirm?
Personally, I found the statement a little difficult to accept at face value. I can cope with the concept of jerk as a measure of the smoothness of the ride and can concede that you may be interested in one derivative higher than that. Then you might want to differentiate another once or twice for some kind of optimisation process. That takes us out to the sixth derivative at a stretch. Beyoind that, I got nothing. (Maybe it is just as well I am not a rocket scientist.)
Sure, I suppose so. I certainly wasn’t claiming that third derivatives aren’t potentially useful or interesting; just that attempting to interpret them primarily through the lens of graphs isn’t going to usefully hijack in-built or already developed systems of pictorial intuition, the way we can get away with for some other concepts. I don’t think even zeroth, first, and second derivatives should be understood primarily in such pictorial terms, but at least with them you can at least pull it off to some extent.
Apparently the controlling measure for the wobble of tall buildings is the jerk, not the acceleration. For a constant acceleration if feels as though the floor is very slightly slanted, and we don’t have problems adjusting.
Possibly, but there are some major drawbacks as well. Cubic terms are very difficult to interpret, and for any model with even a moderate number of explanatory variables, there are so many cubic terms that you’re almost certain to see spurious significance. In general, you need a lot of data and a good reason to justify a model with superquadratic terms.
I can believe it, but twelfth derivatives of what? Probably not the twelfth derivative of position with respect to time. Rather, they probably had some function (which could be almost anything) for which they had a Taylor series approximation, and they took the Taylor series out to twelve terms.
I remember reading that Richard Nixon once said in a speech that the rate of increase of inflation was decreasing. This was described as the first (and probably only) time a sitting president has used the third derivative to advance his case for re-election.
I haven’t seen any claim of that sort, but I have seen indications that engineers considered the fourth derivative of position wrt time in building the Hubble. This seems believable, even though the page I linked to doesn’t cite its sources.
The 4th derivative turns up in the Euler–Bernoulli beam equation, which involves the second derivative of a second derivative (of deflection wrt distance along the beam). The 3rd derivative is related to the shear force.
