Exploring the Physical Significance of the Third Derivative in Function Analysis

[NATURE.M]

Does the third derivative have any physical significance in relation to the original function?
For instance, the first derivative is the slope function (and can be used to find the local max/min). And the second derivative demonstrates the concavity/curvature of the original function (or the rate at which the slope changes).

 

[lugita15]

Here's an old post of mine that may help answer your question:

Here's a few ways to think about it. If the first derivative is positive at x=x0, then that means that if you approximated f(x) with a line y=ax+b passing through x=x0, a would be positive, and we call that "increasing" (locally). If the second derivative is positive at x=x0, then if you approximated f(x) with a parabola y=ax^2 + bx + c passing through x=x0, then a would be positive, and we call that "concave up". If the third derivative is positive at x=x0, then if you approximate f(x) with the cubic function y=ax^3 + bx^2 + cx + d passing through x=x0, then a would be positive, and we call that ... unfortunately, we don't have a name for that. But you can see for yourself what it means for the leading coefficient of a cubic function to be positive.

Also, if the third derivative is positive, that means that the second derivative is increasing, which means that the first derivative is concave up. In other words, the slope of the original graph increases faster than just a constant rate of increase. So the slope is becoming steeper at an accelerating rate!

I hope that helps.

 

[SteamKing]

In kinematics, the third derivative of the position function represents the "jerk".

 

[NATURE.M]

Okay, I've heard the term jerk before used in a physics context, although it rarely (if ever) shows up in Physics Textbooks.

 

[SteamKing]

Things like 'jerk' and 'jounce' find application in some specialized areas of physics and engineering:
http://en.wikipedia.org/wiki/Jerk_(physics)

For a general introduction to physics, velocity and acceleration are more commonly encountered.

 

[cosmik debris]

Jerk and its higher orders are used in the design of camshafts where higher derivatives of position are optimised to keep mechanical stresses and strains within bounds, whilst providing the largest area under the curve. e.g. in a combustion engine you want to lift the valve as quickly as possible and keep it open as long as possible etc.