/sci/ - Science & Math

>>9735195
>He doesn't even know how to take a limit

>>9735197
I think he means the first principles definition of a derivative. Maybe he covered limits but missed the part where the derivative is defined as a limit. Frankly, it's no great loss. The shit is way too confusing.

>>9735333
>that shit is way too confusing

>>9735333
>it's no great loss.
>The shit is way too confusing.

>>9735333
lmao dude it is variation in y devided in variation of x

>>9735336
Newton and Leibniz never defined calculus in terms of limits. The concept of limits was later developed in order to rigorously prove the methods of calculus.

Limits are counterintuitive and first principals together with delta and epsilon are almost always responsible for causing confusion among students, and as a result numerous educators have criticized the orthodoxy of teaching calculus using these methods.

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>>9735379
variation in y? don't you mean /the change in/ y? an infinitesimally small change

>>9735460
>Limits are counterintuitive
What the fuck are you talking about? First if all, since Newton and Leibniz published their calculus, the notion of infinitesimal were harshly criticized, because that makes absolutely no sense from a traditional conceptions of numbers. Also, Newton, well aware of this, tried with the tools they had at the day to formalize his ideas. Notation and unification of mathematicak concepts was not well established, but even then Newton understood the idea of a limiting process that continued can lead to the proper result. https://www.sciencedirect.com/science/article/pii/S0315086000923012 this shouldn't be surprising since Achimedes already made used those ideas with his method of exhaustion which is literally a precursor to riemann sums. Leibniz had a philosophical conception based on monads so he justified it through this, but it became ridiculously complicated to properly define that and that's why analysis was born. What Leibniz has is a practical notation for efficient calculations, but most of it doesn't make sense, and it makes developing further notions much more difficult. I challenge you to teach an undergrad nonstandard analysis properly.

Secondly, why the hell does the idea of something approaching something is "unintuituve"? That's literally it, and for all the important cases, you can look at it graphically. Is the formal treatment tedious? Well yea, I suppose that in a standard course you shouldn't worry that much about proving everything, and let some things go. Hell at the end almost all you need is that 1/n tends to 0 as n tends to infinity and other limiting concepts, which end up being uses in many more things. Maybe shitty professors just write "for every epsilon>0 there exists delta
.." withouth showing it literally just means, you can get this two points close if you get this two other close enough because they are autists with no intuition, but in no way it makes less sense than "infinitesimal".

>>9735520
Infinitesimal is more intuitive because you just imagine an infinitely small difference. An infinitely small difference approaches an instantaneous slope.

>>9735527
>you just imagine an infinitely small difference
Sure thing. It's obvious there are numbers infinitely small and non 0. But maybe you could instead, idk, say find valid aproxiamtions and see how they converge.

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>>9735535
Yeah you're right, it's more intuitive to throw away h when you finish taking the derivative using first principles than to talk about infinitesimals.

>>9735551
Throw away? Don't make retardes general claims about pedagogical concepts in math if you yourself don't understand the subject. The only clarification you need to understand is that the limit is independent of the actual value at that point. The limit h/h as h tends to 0 doesn't care about the undefined division by 0, because for all points that are not 0 h/h is 1 so for all points arbitrarily close to 0 the function is 1 and so the limit is 0 itself by the definition thay depends on the behavior of the function and not on what the function does at that point. Thats why the notion of continuity is differenciated from the notion of the limit existing. For the derivative this is key, as you are not dividing by 0, but computing what's the value the secants are approaching.

>>9735562
The limit is 1*

>>9735333
It's not that confusing really, only poorly presented. The arbitrarily selected slope approaches the slope of the tangent line as h approaches x. It makes perfect sense when you fully understand that the limit definition of derivative that you've posted is just a slight variation on the slope formula.

>>9735520
Is that the same monad from category theory?