Calculus
Calculus for all: derivatives.
On the first image, the Triangle-person is shown riding inline skates on a flat surface. Air is annotated as ‘UltraNull AirlessAir™’, ground with ‘SuperSlip ZeroFriction Surface™’, the surface with ‘f(x)=-(⅐⋅x)⁰’. The Triangle is yawning, subtitles says: ‘Constant speed; Feels like: commute’.
Second image shows the surface gently angled down. The Triangle rides happy, the surface is annotated ‘f(x)=-(⅐⋅x)¹’, and the subtitle says: ‘Rising speed; Constant acceleration; Feels like: fun’.
Third image shows a parabolically sloping surface. The Triangle is waving arms frantically, the surface is annotated ‘f(x)=-(⅐⋅x)²’, and the subtitle says: ‘Rising speed; Rising acceleration; Constant jerk; Feels like: exciting’.
Fourth image shows a hyperbolically sloping surface. The Triangle covers their eyes while the inline skates run out from under him. The surface is annotated ‘f(x)=-(⅐⋅x)³’, and the subtitle says: ‘Rising speed; Rising acceleration; Rising jerk; Constant snap; Feels like: extreme’.
On the final image, the surface is a sheer 90-degree cliff, to which the Triangle is holding on with dear life using hands and teeth, their legs dangling uselessly in the air. The surface is annotated: ‘f(x)=-(⅐⋅x)^∞’, and the subtitle says: ‘Rising speed; Rising acceleration; Rising jerk; Rising snap; Rising crackle; Rising pop; Rising bang; Rising boom; Rising crash … Feels like: doom’
Disclosure: I’m not a mathematician or a physicist, so this comic might not be entirely rigorous. Even so, I think it’s good enough to illustrate a deeper concept. Refer to your nearby mathematician for more info 😁
Also, physicists are such jokers - I swear I have not made up the names of successive derivatives such as jerk, snap, crackle and pop.
Today, I’m trying to visually show how derivatives work by relating them to an experience of free-wheeling down a hill. The key here is that we can interpret derivatives in a physical sense: acceleration is the same as the first derivative of speed, jerk is the second derivative of speed, snap is third, and so on.
In the comic, each successive surface “bends” more and more abruptly downwards. I hope you can feel the change is not just quantitative - if that was the case, we could just angle the surface down and be done with it. We are after something else - the qualitative change of the “increasing bend”, making each successive surface appear more and more dangerous.
If we look at this through calculus-tinted glasses, we will see that as we bend the surface, we add additional levels of derivatives. Flat surface is the most boring. On an angled surface the speed increases. On a bent surface, not only speed increases, but also the increase itself (acceleration) increases. And if we bend the surface even more, then not only speed and acceleration increase - the increase of the increase increases too.
We can keep that up indefinitely. The final image tries to blow your mind with mathematics - an abrupt cliff is infinite derivatives deep.
Note “adding levels of derivatives” is not a mathematical term. Further derivatives always exist, but are zero which makes them boring. “Bending” is also not very mathematical.
For completness: if you’d like to treat the illustrated surfaces as plots, then it has to be the plot of speed over time, not of position - the comic doesn’t work that well otherwise. I’ve supplied you with handy f(x) formulas if you’d like to play around with this.
As to why ⅐ - this is completely arbitrary, but it looked good on pictures. Also, primes are cool.