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s_chanch
01/30/2022 03:54:57
(this is extremely cursed)
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Camilleone
01/30/2022 03:54:57
ah damn
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TeXit
01/30/2022 03:55:08
**Camilleone**
https://cdn.discordapp.c…697925898261.png
569167697925898261.png
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Camilleone
01/30/2022 03:55:11
there we go
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Ansh_
01/30/2022 03:55:28
ohhh right! I was at the point >_< but I took f'(x).dx to be ∆x instead of ∆y 🤦‍♂️
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Ansh_
01/30/2022 03:55:39
got it!
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abs_0
01/30/2022 03:55:44
I don’t really get this - how is adding infinitesimal increases equivalent to adding little rectangles under f’?
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Ansh_
01/30/2022 03:56:24
it's basically the rd(theta) argument lol
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Camilleone
01/30/2022 03:59:19
maybe we look at a simple picture to get the intuition
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Camilleone
01/30/2022 03:59:51
if that's alright with you i'll make a lot of simplifying assumptions, and then we'll see it extends generally?
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s_chanch
01/30/2022 04:00:47
ok uhh
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s_chanch
01/30/2022 04:00:55
i think i know what i don’t understand
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Ansh_
01/30/2022 04:01:07
hmm?
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s_chanch
01/30/2022 04:01:17
https://cdn.discordapp.c…686/IMG_0365.png
IMG_0365.png
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s_chanch
01/30/2022 04:01:29
so here, t and v multiply out to be s
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s_chanch
01/30/2022 04:01:42
this is why wiki had a whole section about ‘physical intuition’
1
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s_chanch
01/30/2022 04:02:14
is there some other intuition other than physics
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s_chanch
01/30/2022 04:02:27
that tells us ‘ftc1 = integral is inverse of differentiation’
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Ansh_
01/30/2022 04:03:06
what we do in physics is agreed by mathematics +_+ but mathematics is rigorous and physics only basically uses the definition and the intuition behind it
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s_chanch
01/30/2022 04:05:02
i want intuition, in general, not ‘physical intuition’
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s_chanch
01/30/2022 04:05:02
lol
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s_chanch
01/30/2022 04:05:18
that doesn’t only prove the theorem but tells us this too
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Camilleone
01/30/2022 04:06:02
well, the problem is kind of one that admits the physical intuition as the most convenient one
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Camilleone
01/30/2022 04:06:12
after all, calculus was invented based on physical intuition
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s_chanch
01/30/2022 04:06:16
it’s ‘convenient’ but not ‘general’
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s_chanch
01/30/2022 04:06:23
i’m looking for a general one
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abs_0
01/30/2022 04:06:54
Generality is a monster
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Ansh_
01/30/2022 04:06:58
:KEK:
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abs_0
01/30/2022 04:07:09
Our brains are wired for physical analogies
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abs_0
01/30/2022 04:07:16
Lol
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Camilleone
01/30/2022 04:07:19
so what, you want to study measure theory?
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s_chanch
01/30/2022 04:07:30
what’s that
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Camilleone
01/30/2022 04:07:30
powerful and lets you do integration like no one's business
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Camilleone
01/30/2022 04:07:34
but it's advanced
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s_chanch
01/30/2022 04:07:51
i’m just looking to understand why integration is reverse of differentiation
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Camilleone
01/30/2022 04:07:56
once you hit that level, you can basically differentiate functions by another function
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abs_0
01/30/2022 04:08:10
Generally speaking, continuous functions are differentiable nowhere
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abs_0
01/30/2022 04:08:14
That’s not fun is it
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Ansh_
01/30/2022 04:08:14
wait... why is the "general" intuition stronger than the "physical" one? (@_@;) I don't get it
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Ansh_
01/30/2022 04:08:26
Is there something that the physical intuition doesn't account for?
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Ansh_
01/30/2022 04:08:36
or is lacking in some way?
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s_chanch
01/30/2022 04:08:55
it’s so specific
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s_chanch
01/30/2022 04:09:04
and plays by some set of rule which we can conveniently exploit
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Ansh_
01/30/2022 04:09:11
wdym "it's so specific"
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s_chanch
01/30/2022 04:09:15
after all i get the physics intuition and still not mathematical one
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Ansh_
01/30/2022 04:09:15
what is it specific to?
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Camilleone
01/30/2022 04:09:24
it doesn't account for things like the popcorn function, but for sufficiently nice functions which everyone deals with _before real analysis_ it works
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Ansh_
01/30/2022 04:09:45
how do you even classify them as physics intuition and mathematical Intuition.. like what's the basis?
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s_chanch
01/30/2022 04:09:55
it’s easy to understand area under curve = distance, because the units in x y axes happen to cancel out into distance
1
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Ansh_
01/30/2022 04:10:25
huhhhh? wdym happens to cancel out...
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Camilleone
01/30/2022 04:10:35
honestly i prefer to think of it as the ideas we have before introducing measure theory, and after redefining the whole thing using lebesgue
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Ansh_
01/30/2022 04:10:54
ohhhh
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Ansh_
01/30/2022 04:11:12
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Camilleone
01/30/2022 04:11:23
HAHAHAHA oh my god
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Ansh_
01/30/2022 04:11:25
~~perfect sticker moment~~
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Camilleone
01/30/2022 04:12:18
what i just talked about is unit independent though???
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Camilleone
01/30/2022 04:12:24
i didn't specify velocity
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s_chanch
01/30/2022 04:12:33
wait where lol
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Camilleone
01/30/2022 04:12:40
i just used 'rate of change' which works in general
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Ansh_
01/30/2022 04:12:43
Were you following right now?
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Camilleone
01/30/2022 04:14:04
.
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s_chanch
01/30/2022 04:15:29
is your explanation the one from this
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s_chanch
01/30/2022 04:15:36
https://youtu.be/FnJqaIESC2s
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Camilleone
01/30/2022 04:16:36
i have no idea, i invented it on the spot
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Camilleone
01/30/2022 04:16:44
i just woke up from a nap after all
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s_chanch
01/30/2022 04:17:14
can i get some visualisation..?
1
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s_chanch
01/30/2022 04:17:23
i’m not getting it
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s_chanch
01/30/2022 04:17:25
;-;
1
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abs_0
01/30/2022 04:18:58
Physical, intuitive ideas precede the abstract generalizations
0
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abs_0
01/30/2022 04:19:12
That’s the most natural way to learn it, in my opinion
0
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abs_0
01/30/2022 04:20:09
Because it includes the original motivation and context, which is very important
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Ansh_
01/30/2022 04:20:51
Well it's the $\Delta y = \Delta x \theta$ argument tbh
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abs_0
01/30/2022 04:21:30
What is that argument
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Camilleone
01/30/2022 04:21:31
Ignore the scratch, you want the graphs
https://cdn.discordapp.c…20130_172057.jpg
20220130_172057.jpg
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Ansh_
01/30/2022 04:21:32
where you use the limit definition of derivative of f(x) for the d(theta)
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Camilleone
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TeXit
01/30/2022 04:22:53
**Ansh**
https://cdn.discordapp.c…233012371577.png
706934233012371577.png
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s_chanch
01/30/2022 04:23:27
i still don’t get it
1
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706,934,233,012,371,600
Ansh_
01/30/2022 04:25:37
All I see r probability ;-;
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
569,167,697,925,898,240
Camilleone
01/30/2022 04:25:57
_ignore that scratch lol_
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
706,934,233,012,371,600
Ansh_
01/30/2022 04:26:11
that's SCRATCH? >_<
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
706,934,233,012,371,600
Ansh_
01/30/2022 04:26:50
i feel like the actual argument in concern is the scratch here xD
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
569,167,697,925,898,240
Camilleone
01/30/2022 04:27:04
HAHAHAHA
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
569,167,697,925,898,240
Camilleone
01/30/2022 04:27:15
i'm sorry, but i don't want to get out of bed to get a clean paper
0
1
569,167,697,925,898,240
Camilleone
1596
Camilleone
false
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:28:11
alright uhh
1
0
569,167,697,925,898,240
Camilleone
1596
Camilleone
false
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:28:16
what you mean is this right?
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:28:17
Makes sense
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:28:31
Hahaha this is possibly the most simple function you could start with
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:29:15
Ok but
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:29:27
https://cdn.discordapp.c…754/IMG_0366.png
IMG_0366.png
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:29:29
if you add up
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:29:33
all the tangents between a and b
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:29:38
you get the average rate of change?
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:29:49
It almost seems coincidental that the area from 0 to 2 = 2c is the same as f(2)
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:29:58
I don’t think I’m thinking hard enough
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:30:08
this doesn’t make sense to me
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 04:30:17
at least intuitively
1
0
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:30:41
Not avg rate of change
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
569,167,697,925,898,240
Camilleone
01/30/2022 04:30:43
that's because it's not, it's strictly speaking f(2) - f(0)
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 04:30:58
You add up the slopes of the tangent lines, and you get the value of the function where you stopped
0
1
null
null
null
null
null
365
help-10
d8b842f4e0a64afb952cef54909e6d8e
706,934,233,012,371,600
Ansh_
01/30/2022 04:31:07
btw am I right with my understanding? >>> We use the limit definition of derivative to define f' = theta as the slope of the secant from (a,f(a)) to (a+h,f(a+h)) and use the "for h small" to conclude ∆y = (∆x)( theta ) so we're ultimately left with int(f'(x)dx) = f(x) ?
0
1
null
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null