What is the meaning of the third derivative of a function at a point

The rate of change of acceleration is studied in various situations in physics, mechanics and engineering design. From http://en.wikipedia.org/wiki/Jerk_%28physics%29: In physics, jerk, also known as jolt (especially in British English), surge and lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, the second derivative of velocity, or the third derivative of position. Jerk is defined by any of the following equivalent expressions: $$ \vec j=\frac {\mathrm{d} \vec a} {\mathrm{d}t}=\frac {\mathrm{d}^2 \vec v} {\mathrm{d}t^2}=\frac {\mathrm{d}^3 \vec s} {\mathrm{d}t^3}$$ where $\vec a$ is acceleration, $\vec v$ is velocity, $\vec s$ is position and $\mathit{t}$ is time. Jerk is a vector, and there is no generally used term to describe its scalar magnitude (e.g. "speed" as the scalar magnitude for velocity). Think of roller coaster designs. Indeed, in mechanics, the fourth derivative is also studied. It is called Jounce or Snap. From http://en.wikipedia.org/wiki/Jounce: In physics, jounce or snap is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively; in other words, the jounce is the rate of change of the jerk with respect to time. $$\vec s =\frac {d \vec j} {dt}=\frac {d^2 \vec a} {dt^2}=\frac {d^3 \vec v} {dt^3}=\frac {d^4 \vec r} {dt^4}$$ Sometimes even the higher derivatives are nontrivial and come into play. Think of a sudden impact, an earthquake, a http://en.wikipedia.org/wiki/Shock_%28mechanics%29, or a lightning effect in electrical systems. Constructs like http://en.wikipedia.org/wiki/Dirac_delta_function are very convenient for dealing with such situations. For instance, for a lightning, there is approximately a very high surge of current for a very brief instant and for any smooth approximation all the higher derivatives are nonzero. So you take the limit and manipulate it as if everything were concentrated at a single point.

It is a common theme in applied math that you can easily interpret first and second derivative or moment (in case of probability theory), but after that, trouble begins. That being said, the third derivative is used in calculating the http://en.wikipedia.org/wiki/Torsion_of_a_curve. Let's review an example (rather poor one, I admit). Let's work in $\mathbb{R}^3$ with a Cartesian coordinate system $x$, $y$, $z$ and the associated basis $\mathbf{e}_1$, $\mathbf{e}_2$, $\mathbf{e}_3$. Then let $\gamma$ be a unit circle given by $x = \cos \varphi$, $y = \sin \varphi$, $z = z_0$, parametrized naturally (by $\varphi$). Then $\dot{\gamma} \times \ddot{\gamma} = \mathbf{e}_3$, and then $\tau = {\dddot{\gamma}}^3 = 0$. I guess this particular curve is not very instructive, but I can't think of a better one off the top of my head :)

For the position function $p=f(t)$, you probably know that the first derivative $f'(t)$ gives the instantaneous velocity, and that the second derivative $f''(t)$ gives the instantaneous acceleration. The rate of change of the acceleration is called the http://en.wikipedia.org/wiki/Jerk_%28physics%29 (also known as the "surge", the "jolt", or the "lurch"). So the third derivative $f'''(t)$ would give the instantaneous jerk.

Suppose $y=f(x)$ is smooth on $\mathbb R$. The first derivative $y'$ represents the gradient of the curve. If $y'>0$ on $\mathbb R$, $y$ is strictly increasing. If $y'<0$ on $\mathbb R$, $y$ is strictly decreasing. The second derivative $y''$ represents the rate of change of the gradient. If $y'=0$ and $y''<0$, we have a local maximum. If $y'=0$ and $y''>0$, we have a local minimum. The third derivative $y'''$ represents the rate of change of gradient change. If $y''=0$ and $y'''\neq 0$, we have a point of inflection.

Since force is a constant scalar multiple of acceleration (at non-relativistic speeds), the third derivative of a position function, jerk, is a constant multiple of the rate of change of force. In other words the jerk of a unit mass object is equal to the rate of change of force, a quantity sometimes called "yank". This is analogous to the relationship between momentum and velocity. Note, force is the time-derivative of momentum. Then the snap (4'th derivative of position) of a unit mass object is equal to the second derivative of force, called "tug". Apparently, there's a whole heierarchy up to 6'th derivative of position! http://math.ucr.edu/home/baez/physics/General/jerk.html

By asking the question, you can interpret the 1st and 2nd derivatives in some meaningful ways. Consider the 3rd as the combination of them. In other words, consider the third derivative as the 'acceleration' if the velocity was in fact the displacement.

when you are in a car, and it is accelerating at a constant rate, the back of your seat is pushing on your back with a constant force, or a constant pressure if you like. if there is jerk, the pressure on your back will change. if the jerk is constant, the pressure on your back will change nicely, perhaps at a linear rate. if the jerk is not constant, the pressure on your back will change more erratically.