In real life what does it mean? I remember d1 is the slope at a given point. D2… D3… D4 jerk? What do D5 and D6 mean? I took calc 2 in uni but I cant remember and have no idea anymore.
In: Mathematics
Physics major here.
So different levels of derivate are not only helpful in real life but help mathematically. It is impossible to imagine the 4th, 5th, 6th dimension but you can use those dimensions to solve higher level calculus and differential equation problems. Now back to your question. As for real life applications jerk has much use but the snap, crackle, and pop named higher power derivatives don’t. They do have a reason though. Think about the fundamentals of a derivative. A derivative is the rate at which something changes over time. We start with position and if it changes over time we get velocity. If velocity changes over the we get acceleration and so on. Well when an object moves from a stand still it increases in velocity which means it must have an acceleration. Acceleration cannot come from nowhere so it must have a jerk. Jerk must come from somewhere so it must have a snap etc. It exists but isn’t as useful. Jerk is useful. The physicality is expressed in the name. I jerk something out of your hand or you jerk to a stop. Jerk is the change in acceleration over time. My best way to describe it is 2 part. Mathematically and physically. Mathematically if we look at the force equation F=ma we rearrange to get a=F/m. To have a constant acceleration we must have just numerical values for force and mass, but if either of those change then the acceleration changes and provides a jerk value. That brings my to my physical explanation. I use a Rollercoaster for example, or a train. Everyone is familiar with the term g. If you go through a loop on the Rollercoaster you feel increased g forces on your body. A g is a unit based around gravity which is an acceleration value, 9.8m/s^2. Well let’s say at the peak of the loop you feel 3gs and normally you feel 1g. The rate at which the Rollercoaster gets you from normal g forces to peak g forces is the jerk. Engineers can use this to reduce the harmful effects of sudden acceleration on the body. One major use is the Euler spiral. This specific curve/spiral has a constant acceleration rate if traveled along it. This is used on highways and railroads in order to gradually accelerate the vehicle and the passengers inside so they don’t get thrown in their seat and only experience a gradual increase in the pressure their body applies to the car door for example. I’m sorry for the long winded response but I hope it answered your questions. If you have any more please feel free to ask.
Higher-order derivatives mostly just have their place as the derivative of the derivative. The lower-order derivatives can be interpreted as visible properties, but as the order gets larger, things become much more abstract. On an arbitrary curve, dy/dx is the slope, as you have mentioned. d²y/dx² tells you something about the curvature, but starting at d³y/dx³, it becomes much more abstract (“rate of change of curvature”? That hardly says more than “third derivative” does).
x(t) is something where a lot of derivatives can be felt. x is your location. dx/dt the change of location with respect to time – your velocity. d²x/dt² tells you how much your velocity changes, an acceleration which you immediately experience. d³x/dt³ is called jerk, and it tells you about a change in acceleration:
Think of moving through a perfectly circular curve at constant speed. The only accelration you feel on the straight is gravity. Now enter the curve. Your vehicle immediately produces a centripetal acceleration to follow the curve, which you feel as a sudden onset of centrifugal force (actually just your own inertia) slamming you against the side of the vehicle, a giant spike in jerk. If the curve eased in more gently from straight to curve, you would get a finite jerk and the same or even greater acceleration would not be as unpleasant.
Each level of derivative has a different purpose depending on context. For instance, the first derivative of location is velocity. The second is acceleration. The third, change in acceleration, has no name but us humans like when this value stays small.
That’s just one example tho. Going too many derivatives in has no significant meaning usually, beyond taylor approximations.
The first derivative is the rate of change. If you have a function that describes position, its rate of change is velocity.
The second derivative is the rate of change of the rate of change. In this case, it would be how quickly the velocity is changing, which is acceleration.
The third derivative is the rate of change of the rate of change of the rate of change. Harder to visualize, but still meaningful. Rate of change of acceleration is called jerk (or jolt), becasue that’s what you feel when you are in a vehicle that changes its rate of acceleration.
Higher-order derivatives are harder to visually and use whimsical names like jounce, flounce, snap, and yank. But that’s what math is for, way to precisely describe concepts we can’t hold in our brains all at once.
Also, derivatives don’t “start” anywhere, position is just a convenient example. Position itself is the first derivative of something called absement and the second derivative of absity.
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