Anonymous at Sun, 12 May 2024 14:38:54 UTC No. 16172353

>>16172344
>Integrate function
>Calculate its derivative
>Hmm, I got back my original function. There might be a connection here

Anonymous at Sun, 12 May 2024 14:54:58 UTC No. 16172371

>>16172344
Most mathematical operations have some inverse, and even those that are not fully-inversible still allow for the recovery of at least some information about the original input.

If you can add something to an input, you can subtract something.
If you multiply an input by something, you can divide by something.
If you take an input to some power, you can recover some possible roots.
If you can take a derivative to find the rate at which an input is changing, there must be an inverse to this process which can recover the input, within some constant offset.

Anonymous at Sun, 12 May 2024 15:03:54 UTC No. 16172382

>>16172344
In original formulations of calculus it had to do with the physical notions of displacement, velocity and acceleration.
The derivative was how one got from positional displacement to velocity, and the integral was how one recovered positional displacement (up to, possibly, an unknown additive constant).

In terms of the later developments, it comes down to the notion of an invertible function and the linearity of the derivative/integral.

Anonymous at Sun, 12 May 2024 15:11:55 UTC No. 16172395

>>16172344
It was fucking obvious. If you add all the differences (function going up and down) up to point x, you get f at x. If you subtract all the "slices" of f from all the slices plus the f(x+dx)dx one and divide by dx, you get the same.

Anonymous at Sun, 12 May 2024 15:19:00 UTC No. 16172402

>>16172353
integration is finding area bounded by a curve and x-axis. now find the connection to antiderivative

Anonymous at Sun, 12 May 2024 20:10:31 UTC No. 16172757

>>16172344
Derivatives extract speed from distances
Integral merge speed into distances

Anonymous at Sun, 12 May 2024 20:13:43 UTC No. 16172764

>>16172402
For each dX that area changes depending on the current value of Y and this rate of change is changing with the current rate of change of the value of Y.

Anonymous at Sun, 12 May 2024 20:14:04 UTC No. 16172768

>>16172344
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

Sage at Sun, 12 May 2024 22:30:16 UTC No. 16172949

>>16172402
>How DID mathematicians know
>Integration IS about
Are you retarded? Do you truly think your modern freshman is taught about integration and differentiation the same way it was treated hundreds of years ago?

Anonymous at Mon, 13 May 2024 03:07:21 UTC No. 16173240

>>16172402
big curve mean fast area increase, negative curve mean area decrease

Anonymous at Mon, 13 May 2024 05:08:02 UTC No. 16173345

>>16172344
How mathematicians knew there was a connection between addition and subtraction?

Anonymous at Mon, 13 May 2024 06:04:29 UTC No. 16173409

>>16172382
Your problem here is thinking that anon has any ability to have any objective reasoning.

Anonymous at Mon, 13 May 2024 06:14:17 UTC No. 16173428

>>16172344
From looking at the most convenient discrete analog: telescoping series.

Anonymous at Mon, 13 May 2024 11:17:53 UTC No. 16173690

>>16173345
How do you know there's a connection between the slope of a curve and the area under the curve?