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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 02:52:04
i watched the animation.
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abs_0
01/30/2022 02:52:06
But it’s hard to get to a point where you just get it, without having to go “ok I’m adding up the rate of change of the area of the rectangle, which is the same as dadada” every time
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abs_0
01/30/2022 02:52:48
@s_chanch did you watch the 3b1b video
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701,895,792,763,600,900
s_chanch
01/30/2022 02:53:05
yea
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abs_0
01/30/2022 02:53:08
Personally it didn’t super help for me lol
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abs_0
01/30/2022 02:53:21
Nothing did, it’s just very unintuitive
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s_chanch
01/30/2022 02:54:06
ok
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abs_0
01/30/2022 02:54:11
Sorry that prob isn’t very helpful lol
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s_chanch
01/30/2022 02:54:14
this theorem is *intuitively true*
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s_chanch
01/30/2022 02:54:27
what isn’t is *how it implies integration, differentiation are opposites*
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abs_0
01/30/2022 02:54:27
You intuitively understand why they’re inverses?
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s_chanch
01/30/2022 02:54:54
no
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 02:54:57
this is my question
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abs_0
01/30/2022 02:54:57
Doesn’t that mean you get why they’re opposites?
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s_chanch
01/30/2022 02:55:11
i don’t
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s_chanch
01/30/2022 02:55:34
it should be obvious for you why this theorem implies differentiation reverses integration
1
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s_chanch
01/30/2022 02:55:36
not for me
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701,895,792,763,600,900
s_chanch
01/30/2022 02:55:46
(this is the motivation for ftc2 isn’t it)
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abs_0
01/30/2022 02:55:53
You mean the statement of the theorem, right?
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s_chanch
01/30/2022 02:55:59
no
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s_chanch
01/30/2022 02:56:04
i know the statement of the theorem
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s_chanch
01/30/2022 02:56:08
i know why it’s true
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701,895,792,763,600,900
s_chanch
01/30/2022 02:56:10
i can prove it
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abs_0
01/30/2022 02:56:13
Yeah
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abs_0
01/30/2022 02:56:22
But you don’t get *why* it’s true?
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s_chanch
01/30/2022 02:56:26
i can’t see why it has to do with ‘diff = inverse int’
1
0
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abs_0
01/30/2022 02:56:50
Oh
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d8b842f4e0a64afb952cef54909e6d8e
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abs_0
01/30/2022 02:56:59
Inverse function means f(g(x)) = x
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help-10
d8b842f4e0a64afb952cef54909e6d8e
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s_chanch
01/30/2022 02:57:01
this statement and ftc1 are complete aliens to one another
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 02:57:02
to me
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 02:57:33
We’re working with “functions of functions” (I’ll call it a functional)
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 02:57:51
composite functions you mean?
1
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:58:01
inverses act as undo buttons
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 02:58:04
So if f,g are functionals, then f(g(s(t)) = s(t) for any function s
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:58:30
we start at f
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:58:35
integrate
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:58:39
then differentiate
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:58:51
ftc1 says the result is f
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help-10
d8b842f4e0a64afb952cef54909e6d8e
420,244,167,726,333,950
kirby your mom
01/30/2022 02:59:08
Wow, how helpful
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help-10
d8b842f4e0a64afb952cef54909e6d8e
150,414,892,225,134,600
RokettoJanpu
01/30/2022 02:59:12
ie differentiation undoes integration
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help-10
d8b842f4e0a64afb952cef54909e6d8e
334,482,942,506,041,340
Zybikron
01/30/2022 02:59:22
Start with a function $f(x) = F'(x)$. Then $\frac{d}{dx}(\int_a^xf(t)\ dt) = \frac{d}{dx}(F(x) - F(a)) = F'(x) =f(x)$
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s_chanch
01/30/2022 02:59:24
it says F’(x) = f(x), where F(x) = int f(t) dt [a, x]
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d8b842f4e0a64afb952cef54909e6d8e
510,789,298,321,096,700
TeXit
01/30/2022 02:59:25
**Zybikron**
https://cdn.discordapp.c…942506041345.png
334482942506041345.png
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 02:59:40
so F is an antiderivative of f
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:00:20
but x is just some upper bound in F’s expression
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RokettoJanpu
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abs_0
01/30/2022 03:00:23
F is the antiderivative. In words, it’s literally saying “the derivative of the antiderivative is equal to the original function”
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:01:09
so basicallt
1
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:01:24
‘antiderivative just so happens to be an integral with x acting as an upper bound’
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:01:38
‘and that means integrals are heavily related to derivatives!’
1
0
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help-10
d8b842f4e0a64afb952cef54909e6d8e
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abs_0
01/30/2022 03:01:47
Oh you’re wondering why it has those bounds
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:02:04
Instead of saying $$F(x) = \int f(x)dx$$
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help-10
d8b842f4e0a64afb952cef54909e6d8e
510,789,298,321,096,700
TeXit
01/30/2022 03:02:07
**abs\_0**
https://cdn.discordapp.c…375735459842.png
307070375735459842.png
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:02:19
?
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:02:36
no
1
0
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:02:48
from this
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0
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:02:52
to this
1
0
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:02:54
i don’t understand
1
0
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help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:03:06
Do you understand this
0
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:03:26
yea
1
0
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help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:03:29
Okay
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:04:07
The thing that links > antiderivative is integral with upper bound stuff to > integrals are inverses of derivatives is precisely the Fundamental Theorem of Calculus
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s_chanch
01/30/2022 03:04:21
????
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abs_0
01/30/2022 03:04:26
Sorry
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abs_0
01/30/2022 03:04:28
Editing rn
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abs_0
01/30/2022 03:05:59
@s_chanch ?
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701,895,792,763,600,900
s_chanch
01/30/2022 03:06:11
i don’t get this
1
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abs_0
01/30/2022 03:06:22
You don’t get how those could be linked?
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abs_0
01/30/2022 03:06:51
(This is rhetorical, you’re not supposed to)
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:07:05
(That’s why it’s called the fundamental theorem of calculus, why it’s so powerful)
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abs_0
01/30/2022 03:07:36
(It links these two crazy things that seem totally unrelated)
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Ray
01/30/2022 03:09:22
integral of dy = y + C. do u understand how that's intuitive? dy/dx = slope. do u understand how that's intuitive? dy/dx * dx = dy. do u understand how that's intuitive? integral of dy/dx * dx = y + C. do u understand how that's intuitive?
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s_chanch
01/30/2022 03:10:16
how
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abs_0
01/30/2022 03:10:16
Bro ok hold up
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abs_0
01/30/2022 03:10:40
@Ray the dy/dx * dx = dy thing always felt so hand wavy and sketchy to me
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abs_0
01/30/2022 03:10:53
When dy/dx is presented in calculus, it’s basically a symbol
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d8b842f4e0a64afb952cef54909e6d8e
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abs_0
01/30/2022 03:11:05
When did it become concrete values you can multiply by
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abs_0
01/30/2022 03:11:28
How? That’s what the proof explains
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:11:46
they’re differentials?
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:12:19
the theorem says ‘if we have f, take the integral of it from a to x, then differentiate it, we get f back’
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:12:37
Exactly
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239,501,588,589,182,980
Ray
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:12:41
how does it become
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
239,501,588,589,182,980
Ray
01/30/2022 03:12:44
this is basically like doing u sub. it's hand wavy yes but it's legal b/c there is rigorous math behind it that lets u do that. dy = dy * (dx/dx) = dy/dx * dx. this kinda thing u see in physics a lot
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:12:48
How do you get f back?
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s_chanch
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:12:50
‘integrals are opposite of derivatives’
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:13:12
This doesn’t sound like opposites to you?
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:13:31
You take a function, integrate it up to x, differentiate with respect to x, and you get the original function back
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s_chanch
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:13:32
no
1
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:13:35
They undo each other
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:13:52
one you differentiate with respect to x
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0
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:14:03
one you integrate with respect to a and x
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0
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:14:15
Broooo I hate that a lot lol
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s_chanch
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d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:14:21
For applications I understand though
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d8b842f4e0a64afb952cef54909e6d8e
239,501,588,589,182,980
Ray
01/30/2022 03:14:28
if u do A, and then do B, and doing B reverses when u did A isn't that opposites
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1
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null
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:14:34
You can’t go through 10 hoops to avoid small logical problems
0
1
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:15:03
It sounds like what you’re really asking is what these bounds are about
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help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:15:23
Would you still be confused if I said $$F(x) = \int f(t) dt$$ instead?
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help-10
d8b842f4e0a64afb952cef54909e6d8e
510,789,298,321,096,700
TeXit
01/30/2022 03:15:25
**abs\_0**
https://cdn.discordapp.c…375735459842.png
307070375735459842.png
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d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:15:26
i guess?
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help-10
d8b842f4e0a64afb952cef54909e6d8e
701,895,792,763,600,900
s_chanch
01/30/2022 03:15:43
isn’t boundless integration introduced after ftc2?
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0
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365
help-10
d8b842f4e0a64afb952cef54909e6d8e
307,070,375,735,459,840
abs_0
01/30/2022 03:15:53
Yes, that’s precisely the issue
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