That video is calculus and is about finding the minimum and maximum using the second directive. I don't have much to say about how the function works, mostly because I'm still processing it myself, but mostly what it is used for.
When I took trigonometry years ago, one of the many things that bothered me about mathematics was their examples of "real world" applications of using trig. The problem was something like "There's a guy who sets up tents for the circus, He needs so much rope, and if the pole is this length and the angle needs to that angle, how much rope does he need?" The problem was dumb I thought because you would figure out how much rope you need the first time, add a little extra, then always keep that length of rope. Now, what it was TRYING to do was make math accessible, and showing the student how to translate this "real world" problem into mathematical language to solve. I still maintain it was dumb.
A great example of trig in the real world? Carpentry, or most forms of civil engineering. Triangles are everywhere in construction, and the formons I've met not only know SOHCAHTOA by heart, but they know what it does and how to use it properly.
Anything that has to be done by enough people repetitively needs to be or has been turned into a tool for maximum efficacy. I need second derivatives at my job. There is a device called a Near-Infrared Machine, or NIR, that is awesome because it analyzes the heat waves from the bonds of molecules to determine what kind of molecule it is. It prints out a graph that is the molecular make-up of the material in question. I like to tell people that I've seen molecules, but I like to tell people all sorts of crazy shit. This machine, though, is data analysis. To get anything useful from this machine, you need a ton of points of data and then you need the second derivative.
The second derivative is useful for data analysis, which he briefly mentions for statisticians. Then he talks about his drive to work. It's a great tool for data analysis because have you ever been given a list of numbers and tried to find the minimum and maximum by looking at it, then tried to do a bunch of predictions on it? I have, because sometimes I don't have the right tools, and anyone who tells you I'm smart is a liar. It's horrible, but the thing is, if every time I need a second derivative I did the whole thing out by hand, then I would slowly go insane and I wouldn't have time for anything else and I couldn't store anything else in my brain because I would have to devote part of my brain to just doing the second goddamn derivative. So I have a machine do it for me, or if I want to get fancy, Python or R or Perl can do it faster than I can.
Knowing how it works is cool and important because of zen, really. Simply put, by understanding the makeup of something, what you can do with it is awesome. Failing that, just simply knowing where to use it and when to use it is more important for people. In my Utopian dream world, everyone can work on their own vehicles, make road warrior constructs from crap they find in the junk yard, and know how to make a 20 year old computer work by running the right version of linux. They also know how to program. This world is the least of my eccentricities, I assure you. This is a crazy fantasy land, so I would be happy with people just knowing how to drive a car in a good way and learning how to properly maintain it. Math is as much a tool as a car is, and can be just as dangerous sometimes, so learning how to use the right tools for the right job would be enough for. Please math teachers of the world: I know you're a math nerd that likes to see the patterns in everyday life, but you're car ride to work may not be the demonstration of a principle.