Comment by Animats
16 hours ago
Calculus is entry level math for all the hard sciences, engineering, and finance. It's how to think about how things change over time. Understanding what integrals and derivatives look like, especially as graphs, is very practical knowledge. Being able to think about compound interest and acceleration under gravity as driven by the same rules gives an understanding of what's happening underneath.
Whether the infinite-series summation limits are worth knowing is questionable, though. That's more of historical interest.
(But then, I once did a game physics engine and have a patent in that area. I spent a lot of time in the hell which is integration of stiff systems of nonlinear differential equations. Turned out that was the wrong way to approach the problem. Works, but there are better approaches.)
> Whether the infinite-series summation limits are worth knowing is questionable, though. That's more of historical interest.
A very common example is compound, continual interest accumulation. As a mathematician, I had trouble understanding how banks calculate it (as they would obviously not point me at the derivation of the formula), but it's actually a limit sum over infinitely small interest terms.
Similarly, these techniques work to explain and derive numerical methods for calculating many natural constants like e, pi etc.
The only intuitive definition of e I can remember is "how much $1 becomes after a year in a savings account with 100% interest compounded infinitely" funny enough.
I just call it “pi but for curves”
That definition also gives a nice introduction to e's taylor series!
> As a mathematician, I had trouble understanding how banks calculate it (as they would obviously not point me at the derivation of the formula)
This is not true? The formula is well known:
A = P*e^(rt)
where"
A = Final amount
P = Principal (initial investment/loan)
r = Annual interest rate (as a decimal)
t = Time in years
e = Euler's number
A loan is subdivided and sold in different allotments each year at a different interests rate, principal, and terms. You want to sell the entire debt at some point in the future. How do you calculate the price such that you don’t sell at a loss?
Why does that formula work? ;)
I did not say "derivation of the formula" by accident: this is about why would this produce something that makes sense as "interest-as-percentage"?
That is the integral
You're assuming banks use that formula and not just YOLO it, "haha fuck you pay me more".
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Where I live, the banks say that they compound interest daily, but pay interest monthly. So, I think the small interest term is just limited to a day where the error is small enough for the average person to not care.
My grad program had an advanced math sequence that was traditionally numerical methods for solving PDE’s and boundary value problems. Except my year, a visiting professor made it infinite series solutions using laplace transforms. Because why would you want an inexact numerical solution when you could have an exact solution to whatever degree of accuracy you need. Line integrals have their own special place in hell.
Right? Just because my professional career never took me in those directions doesn't mean I want to foreclose them for my kid! I did do electrical engineering classes in college wherein I needed calculus, I just never took a derivative/integral after graduation because I ended up on the coding side of a Computer Engineering degree.
Intentionally not learning Calculus will make your kid into a better AI researcher, as they will naturally study gradient free optimization that makes few assumptions about a problem space. Global Optimization as a concept is not dead guys!
i hope i don't get downvoted for this, but billions of people function just fine without the most rudimentary understanding of calculus. while i love maths, i am sympathetic to the view that it's not an essential life skill, and might even be less so in the future.
School shouldn't be teaching essential life skills. It should inspire and enable young people to go on to achieve great things -- at least some of them. How will we ever get another Newton, Einstein, von Neumann, Noether, etc. if we don't teach all young kids what is possible with math? Calculus arguably transformed our society and is a great showcase of the power of math. It can shape the way we think about things, because this level of precision is rarely needed when thinking about other disciplines.