TITLE: What is the meaning of the third derivative of a function at a point
QUESTION [80 upvotes]: (Originally asked on MO by AJAY.)
What is the geometric, physical, or other meaning of the third derivative of a function at a point?
If you have interesting things to say about the meaning of the first and second derivatives, please do so.

REPLY [10 votes]: An intuitive complement to Arturo Magidin's answer:
A good way to intuitively grasp the jerk (hence the third derivative of the position function) is to remember the last time you took a plane and realize that the following "equivalences" holds

No acceleration = constant speed = feels like when sitting in your chair at work = first derivative of the position function is zero.
Acceleration = speed increases = feels like someone is pushing you toward the back of your seat = second derivative is positive.
Increasing acceleration = the pace at which your speed increases gets higher and higher = feels like the guy who is pushing you toward the back of your seat is pushing harder and harder = jerk or third derivative is positive.

In a plane : 
1) Right before take-off, the plane is still, no acceleration, the derivative of the position function are zero.
2) Now the plane starts moving, you are not still anymore, and the first derivative of the position function is positive.
3) Not only are you moving, but the plane brutally accelerates. As a result of the acceleration, you feel like someone is pushing you toward the back of your seat: the second derivative of the position function is positive.
4) Quickly after the engines are on, not only do you feel like someone is pushing you toward the back or your seat but, in addition, it feels like this imaginary person is pushing harder and harder. This is because you accelerate more and more (the jerk is positive). During the first 2 second you went from 0 km/h to say 20 km/h, and during the 2 following ones, you went from, say, 20 km/h to 60 km/h: The third derivative of the position function is positive.
5) After some time, the plane still accelerates, but at a diminishing rate. It feels like the imaginary guy pushing you toward the back of your seat starts to release the pressure. He is still pushing (you would need a higher effort to stand up from your seat than if you were sitting in your office chair), but less and less intensely. The rate at which you accelerate is diminishing, hence the third derivative is negative. However, you are still accelerating so the second derivative is still positive.
6) Your plane eventually reaches its cruising altitude and maintain a constant speed of say 800 km/h. So now, your are not accelerating at all, the second and third derivative of the position function are zero. Only the first derivative remains positive.
7) When you land, the process is reversed. It feels like someone is pushing you in the back and you need the seatbelt to prevent you from falling forward. When it feels like the imaginary guy pushes you in the back stronger and stronger, then the jerk is negative, hence the third derivative of the position function is negative too.