Introduction of the question
I'm a student in mechanical engineering and we do have a lot of mathematic. We have algebra, calculus and numerical analysis classes. We are tending to learn how to approximate problems using numerical tools and we talked about many ways to actually do that with more or less good precision.
When I learned (via personnal research about differential geometry) that we can approximate a curved surface to be locally flat, and then, we can actually figure out what's happening at greater scales. Because of that, I started to compare this method with other methods and I concluded one thing, that, in order to study a problem, we can approximate it by "cutting" the problem into many little simple sub-problems.
I made this comparaison when thinking about how a deriative is fondamentally the search for the coefficient of a tangent which is, considering a "curve" to be locally flat and then work using that consideration and extending it to generalize the solution of the problem.
On this figure above, at the point A, the curve $f(x) = x^3 + 3x^2 - x - 1,5$ can be approximated flat when calculating the tangent.
This is how I see the approximation (not-to-scale) My idea behind that is that we are actually approximating the curve using infinitesimally small segments in order to recreate the curve. When doing this infini many times using infinitesimally small segments, we are basically left with the curve itself. But the concept remains the same.
Why can't we directly study a problem like geometry on a curved surface or directly study the rate of change of a function ?
Why do we use these kind of "approximations" to study a problem like finding the rate of change of a function ? And why can't we directly search for the rate of change of a function ?
I think I might have an idea of answer but I don't know if I'm on the right way.
I initially thought that this method is just the most intuitiv one to study a problem. And when thinking about it and doing research, I couldn't find any other tool or idea allowing the direct study of geometry on a curved surface for example. So I guessed its because of the lack of other solution that we discretize into many bits a problem. Also, I didn't knew if discretization and deriatives were related. I thought yes because they are on first sight based on the same concept. For me, when you are more or less cutting something $M$ into many little pieces, you are basially discretizing $M$. Again, I'm not sure about this idea.
This is because we are studying about Finite Element Method, and we are using that discretization in many application, mainly mechanical engineering. And because of that, I wanted to be a little clarified about these concepts.