the answer is $4(2+x)(1-x)$ so I have no idea how I get there
#help-23
faint glade
flat frigateBOT
Vortac
austere forge
the answer is $4(2+x)(1-x)$ so I have no idea how I get there
$9-(2x+1)^2 \ 3^2 - (2x+1)^2$
flat frigateBOT
c2b7_97705
vague phoenix
Expand the equation
austere forge
Expand the equation
no
vague phoenix
That's how I would do
faint glade
$3^2+4x^2+4x+1$?
flat frigateBOT
Vortac
austere forge
$3^2+4x^2+4x+1$?
no
faint glade
$(a-b)^2 = a^2-2ab-b^2$
flat frigateBOT
Vortac
faint glade
but I don't know these things by heart, I have to work them out
pallid magnet
there's a few identities that you need to memorize, then just practice until you can use them at will
pallid magnet
but I don't know these things by heart, I have to work them out
there's really no way around that
faint glade
can you apply that here?
I'm not making the connection because one is 3^2 and the other is some other identity (2x+1)^2. I don't see any common factors between the two
I get that together they make difference of squares
austere forge
I'm not making the connection because one is 3^2 and the other is some other ide...
look at the identity, a and b can be anything
a=3, b=2x+1
faint glade
yeah
faint glade
3+2x, 3-2x?
flat frigateBOT
c2b7_97705
austere forge
does that make sense?
faint glade
why does 3 become 4?
austere forge
because you have 3 + 1
3 + 2x + 1 = 4 + 2x
and 3 - (2x + 1) = 3 - 2x - 1 = 2 - 2x
faint glade
so I get rid of the square?
austere forge
what do you mean?
the difference of squares identity gets rid of the squares
a^2 - b^2 = (a-b)(a+b)
you have no squares on the right hand side
faint glade
$3^2-(2x+1)^2 = (a-b)^2 = (a-b)(a+b)$
austere forge
$3^2-(2x+1)^2 = (a-b)^2 = (a-b)(a+b)$
this is wrong
look at the difference of squares identity
a^2 - b^2 = (a-b)(a+b)
faint glade
oh whoops
flat frigateBOT
Vortac
austere forge
oh whoops
no, that first equality is completely wrong
you can't combine that way
$3^2 - (2x+1)^2 = a^2 - b^2 = (a-b)(a+b)$
flat frigateBOT
c2b7_97705
austere forge
this is the correct statement
faint glade
I still don't see where the 4 comes from in that
mighty mango
**Vortac**
this is not what "difference of squares" is
pallid magnet
I still don't see where the 4 comes from in that
you can factor
like for example
$4 = 2^2$
flat frigateBOT
IdelUser404
austere forge
I still don't see where the 4 comes from in that
you have (4+2x)(2-2x) right
can you factor that?
what do 4 and 2x share?
faint glade
a 2
austere forge
what do 2 and -2x share?
faint glade
a 2
faint glade
so I need to first work out (2x+1)^2?
faint glade
$3^2-(2x+1)^2$
flat frigateBOT
Vortac
faint glade
Turns into $3^2 - 4x^2+4x+1$
flat frigateBOT
Vortac
faint glade
That's where I'm lost. $3^2 - (2x+1)^2$ turns into $(4-2x)(4+2x)$ then?
flat frigateBOT
Vortac
faint glade
how many identities do I need to memorize?
worn fractal
The basic ones like (a+b)³, (a+b+c)² and many more like these
faint glade
So I get $(4+2x)(4-2x)$ which I can then factor into $2(2+x)2(2-x)$ which I think gives me $4(2+x)(2-x)$ but I'm supposed to get $4(2+x)(1-x)$?
flat frigateBOT
Vortac
worn fractal
I hope so , this helps
In the second step using the identity a²-b²=(a+b)(a-b)
The - sign opens with both the 2x and 1 so they both become -ve
faint glade
ahh okay
I was not subtracting in the second step
I'm going to need a lot of practice with this, they never showed us the special identities in class...
worn fractal
Ohh is it ?
safe radishBOT
safe radishBOT
Available help channel!
muted apex
i don't get how this u get this equation
safe radishBOT
i don't get how this u get this equation
safe radishBOT
safe radishBOT
✅ Original question: #help-23 message
safe radishBOT
muted apex
nvm
vague phoenix
Find dh/dt
Bro cant just open and close that fast
People need time to read your question
If you dont need help then close, dont just open/close
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Available help channel!
versed wave
safe radishBOT
versed wave
for part a, i want to model the number of defected boards as a normal distribution with mean = 12.5 and sd = 3.446, but plugging it into the cdf with x = 25 looks weird
is this answer correct?
mortal sandal
idk what your phi function is but it's definitely not what you want it to be
versed wave
phi is the cdf of a standard normal distribution
mortal sandal
If you expect 12.5 boards to be defective, do you think there's a 60% chance more than 25 of them will be?
versed wave
yeah exactly
mortal sandal
,calc 13/3.446
flat frigateBOT
Result:
3.7724898432966
pretty sure you don't get .3989 when you plug this into a normalcdf
safe radishBOT
@versed wave Has your question been resolved?
fathom jewel
,w phi(3.77)
flat frigateBOT
mortal sandal
oh neat
thin stream
3biii
winter whale
the combinatorics queen will help with this one
thin stream
70
thin stream
the combinatorics queen will help with this one
is slayla the combinatorics queen
YOOO
WASSUP FAM
lnmao
solar hazel
winter whale
carry on
mortal sandal
is it generating functions time
solar hazel
are the balls all distinguishable?
vague phoenix
Which balls
thin stream
are the balls all distinguishable?
i mean the balls are numbered
solar hazel
also... 3000? there are only two 0 balls
mortal sandal
probably yeah
balls of the same number should still be distinguishable I'm guessing
mortal sandal
otherwise these problems get weird
thin stream
Which balls
thin stream
3biii
the cases are only 2100 and 1110 then
because there is only 2 zero ballz
the only diff in the balls are they numbers
mortal sandal
What was the answer?
thin stream
im only now getting to iii
oh wait
holup
let me do i and ii again
6C4 + 5C4 + 7C4
ye its correct
55
the order in which u ick up the balls matters
mortal sandal
But the balls are distinguishable
Even the ones with the same number
Otherwise the answer would be 3
thin stream
But the balls are distinguishable
oh yeah u right
wait
oes that mean
2100 and 2100 are two diff
thignis?
two diff... permutations?
zat makes no sense
mortal sandal
It means 2100 isn't enough information to determine a specific outcome
Which 2? Which 1?
thin stream
wait since the ballsa re distuingishable fro meach other
mortal sandal
Which 0 was first.
thin stream
is it just 4!+4!+4!
solar hazel
,w (6 choose 3) * 2 + 5*6
thin stream
62
flat frigateBOT
solar hazel
solar hazel
that is the intended interpretation i guess
mortal sandal
Order doesn't matter?
thin stream
solar hazel
who wrote this 
mortal sandal
Well if you want to calculate the probability of a certain outcome that's the best way to do it
But they probably should have mentioned that
thin stream
who wrote this <:ducky_skull:1121451613224173689>
its from this paper
solar hazel
it's still completely unclear what the intended interpretation is without already knowing the answer
solar hazel
every ball is distinguishable. how many ways can you pick 4 of them (order does not matter) so that they sum to 3
solar hazel
let's do ts with generating functions just for fun
thin stream
what dat
solar hazel
,w expand (1 + y)^2(1 + yx)^6(1 + yx^2)^5
flat frigateBOT
mortal sandal
fun
solar hazel
hm wolfram hid the answer inside the + ... +
thin stream
what IS THAT
thin stream
2100 and 1110
so 4! + 4!
oh wait here order matters
solar hazel
ok there it is. coefficient of x^3y^4 is 70
thin stream
ok there it is. coefficient of x^3y^4 is 70
solar hazel
i never miss an opportunity to shill multivariable generating functions
winter whale
the ipad is whatever u sent and the kid with the book is me
true
solar hazel
y^4 signifies picking 4 balls, and x^3 signifies the 4 balls picked sum to 3. due to my artful generating function creation, the coefficient is the number of ways to pick
thin stream
y^4 signifies picking 4 balls, and x^3 signifies the 4 balls picked sum to 3. du...
truly a work of art boss man
winter whale
bro doesnt miss the opportunity to glaze herself 😭
thin stream
😭
how was i supposed to know order does not matter in 3biii?
wait if order idd matter
mortal sandal
thin stream
it would be 4! + 4! right
solar hazel
how was i supposed to know order does not matter in 3biii?
i would not have known without being able to confirm it from the answer
mortal sandal
No
solar hazel
y^4 signifies picking 4 balls, and x^3 signifies the 4 balls picked sum to 3. du...
the coefficient of x^m * y^n is the number of ways to pick n balls so that their sum is m
thin stream
the coefficient of x^m * y^n is the number of ways to pick n balls so that their...
yes
this is
💯
massive
massive understanding
solar hazel
it's just basic generating function theory
mortal sandal
Well
thin stream
and thats saying something
mortal sandal
the coefficient of x^m * y^n is the number of ways to pick n balls so that their...
If we ignore the balls with 3s on them anyways
thin stream
2100 -> 5P1 * 6P1 * 2P2
1110 -> 6P3 * 2P1
and u add them
if order did matter
and if im not trippin
winter whale
https://klipy.com/gifs/ninja-ninjaishyper
focus
solar hazel
If we ignore the balls with 3s on them anyways
oh. yea i excluded those for the sake of 3biii to make the computation smaller since they didn’t affect that
thin stream
nah you also have like 0012 case
isnt that accounted for in um 5P1 * 6P1 * 2P2
doesnt that basically translate to
select one two and for every selection of two select 1 one and for every one of those selet two zeros
i gtg
afk
for like an hour
can i keep this open
mortal sandal
If the bot will let you sure
mortal sandal
isnt that accounted for in um 5P1 * 6P1 * 2P2
doesnt that basically translate to...
No it isn't accounted for
The only part that's accounted for is swapping the two 0s
solar hazel
a few minutes ago i tried to send a message about how i’m kinda just trolling and you can ignore what i’m saying about generating functions. but my internet cut out
mortal sandal
rip
winter whale
what if his life was on the line
solar hazel
then i would hope he wouldn't be on discord asking about ways to pick balls that sum to 3
safe radishBOT
@thin stream Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
high cliff
i already opened a similar help request yesterday but i was absolutely baked so ill reask: how do i verify a generic limit of a succession? (example in the photo). I've always been under the impression (high school calculus, functions rather than successions) that we start with the hypothesis that our expression is indeed between l - epsilon and l + epsilon and then work our way into a ni of epsilon that we can provably say exists. But apparently this isnt the process, its the other way around, we estimate a possible ni(epsilon) and then work from n > ni(epsilon) into the "initial" assumption about the limit (l-epsilon < a(n) < l + epsilon). So like genuinely and just materially what am i supposed to do?
safe radishBOT
i already opened a similar help request yesterday but i was absolutely baked so ...
high cliff
My impression is that ni can literally be anything as long as it makes the succession EVENTUALLY limited
open wedge
as $n \to +\infty$ then $\frac{1}{n} \to 0$
flat frigateBOT
1 divided by 0 equals Infinity
high cliff
sorry no offense im not sure you understood the request
of course i can do this but this isnt calculus, this is analysis
high cliff
theres a giant chunk of text right above that photo
open wedge
theres a giant chunk of text right above that photo
Im out if it's analysis
high cliff
fathom jewel
**1 divided by 0 equals Infinity**
analysis wants to know how
blazing swallow
My impression is that ni can literally be anything as long as it makes the succe...
usually, we will pick n > (something), which means infinitely many options for a given epsilon.
generally, though, the key is to get an expression for n in terms of epsilon (e.g. n > 1/eps), so that when you are given a choice of epsilon, you can find an n that corresponds to it
high cliff
usually, we will pick n > (something), which means infinitely many options for a...
that something is the ni(epsilon)
but my question is Whats the direction of the flow
are we making up a ni(epsilon) and THEN proving the original limit inequality, or are we starting with the limit inequality, THEN finding a ni(epsilon)?
Step by step, what is the process?
worldly lantern
of course i can do this but this isnt calculus, this is analysis
binomial expansion
high cliff
binomial expansion
fathom jewel
binomial expansion
me trying to fit in:
rich wasp
I guess a formal proof would be of the form
Let ϵ>0 be arbitrary. Then for all n ≥ god-given lower bound we have via algebraic manipulation |a_n - L| < ϵ and hence a_n -> L.
But of course in practice you need to find the necessary lower bound for n yourself and hence you'd first manipulate the expression |a_n - L| < ϵ until you get n≥lower bound
high cliff
is it Really a lower bound?
there is no difference from ni(epsilon) and ni(epsilon) + 1
Because both verify the "eventually limited" clause
rich wasp
Sorry sufficient bound
blazing swallow
this is a little complicated. the order in which you do stuff is: \
Let $\epsilon > 0.$ Pick $n_i$ such that $\underline{\qquad}$. Then for $n > n_i,$ $$\left| \frac{1}{n} - 0 \right| = \frac{1}{n} < \frac{1}{n_i} < \epsilon.$$ \
This is the order of the proof. Now, when you're actually doing it, you won't magically know what $n_i$ is. So once you've subbed $n > n_i,$ you do some rough work: \
If $\frac{1}{n_i} < \epsilon,$ solve for $n_i$ to get $\frac{1}{\epsilon} < n_i.$ \
Now, go back and fill in the blank to read $n_i > \frac{1}{\epsilon},$ and the proof is done. So, logically speaking the flow of the proof is to assume $\epsilon > 0,$ find a corresponding $n_i,$ then deduce $|a_n - L| < \epsilon$
delicate shore
I guess a formal proof would be of the form
Let ϵ>0 be arbitrary. Then for all ...
I think this style of writing the proof is a little obscure for no good reason
It is just as good to write out the algebraic manipulation then conclude by saying "hence, the choice of ... will ensure ... < ϵ"
high cliff
It is just as good to write out the algebraic manipulation then conclude by sayi...
so this means that its not ni(epsilon) thats the end result
flat frigateBOT
حسیب ♥
high cliff
its the initial inequality
high cliff
https://discord.com/channels/268882317391429632/903486478049488936/1475480931019...
the addition of max/min/floor/ceil etc, this is what made me confused, because algebrically you cant just like, do that
so my assumption was hey maybe this isnt just algebric manipulation
and there is no real rigor in creatign this ni(epsilon) itself
the REAL rigor is in going backwards
rich wasp
It is just as good to write out the algebraic manipulation then conclude by sayi...
Yeah unfortunately so. I've seen professors who insist that this is wrong and that one should write ad hoc bounds in the final proof instead
high cliff
**حسیب ♥**
im trying to understand this but i honestly have no idea where a lot of this stuff came from
ni is the greek letter ni, not n of i
its a function of epsilon from R to N
blazing swallow
as in, $\nu$?
high cliff
https://discord.com/channels/268882317391429632/903486478049488936/1475480931019...
this is amazing
flat frigateBOT
حسیب ♥
blazing swallow
well, my bad too. i use N instead of nu
high cliff
i think we're using nu because then when we transition to functions we're gonna be using delta instead
so i feel its more natural (?)
blazing swallow
i guess? i mean ultimately it doesn't matter
blazing swallow
im trying to understand this but i honestly have no idea where a lot of this stu...
feel free to ask about whatever, but plant makes a good point (as always), that this proof does obscure a few details. if you understand the other style of proof, then you understand it
high cliff
This is an example of a ϵ-δ proof written in the style I described
this proof clarified a lot but is the case for real function limits not a little different? is the implication not double sided? as in |f(x) - l| < epsilon <=> |x - x_0| < delta(epsilon)
oh okay
so its single sided, oopsie
i think i got it mostly
ill retry the limit i was working on
successions might be the most annoying part of calc 1 so far
how is calc 2 easier than this 💔
fathom jewel
multivar limits ah
blazing swallow
this proof clarified a lot but is the case for real function limits not a little...
i'm pretty sure you could cook up an example where f(x) has arbitrarily close points, but the x values fed into it are very far apart. you might also require δ(ε) to have an inverse which is not guaranteed
blazing swallow
successions might be the most annoying part of calc 1 so far
they are! i always thought ε-δ made more sense, but eventually you get to successions (sequences) of functions, so it is important to be familiar with sucessions
high cliff
step 1
blazing swallow
i would recommend starting with the absolute value, because that's what you'll start with in the proof. but this is correct
high cliff
step 2
blazing swallow
this works, well done :) few notes:
- second-to-last inequality should be 0 ≤ sqrt(4 + 1/n) - 2 < ε
- you should write explicitly that |sqrt(4 + 1/n) - 2| < ε, since that is the definition you have to prove
- you might want to add a line that says "Let ε > 0 be arbitrary"
high cliff
oh yea right
thank you sm
thanks to everyone who helped me actually
i was losing my mind over this haha
should i close the channel now? 👀
fathom jewel
if you are cleared out, sure
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Available help channel!
worldly lantern
safe radishBOT
covert rain
x^2-y^2=a , 2xy=b
vague phoenix
No no
vernal portal
x^2-y^2=a , 2xy=b
but seems like b is complex tho..
vague phoenix
Assume square root of a+ib is x+iy
worldly lantern
x^2-y^2=a , 2xy=b
whats x and y
covert rain
sqrt(z)=x+iy
vague phoenix
Then find a+ib=(x+iy)^2
covert rain
(x+iy)^2=a+ib
vernal portal
but seems like b is complex tho..
@worldly lantern ?
worldly lantern
<@1169685245235318837> ?
b is real ofc
covert rain
x^2-y^2=a , 2xy=b
.
worldly lantern
cuz its a coeff of i
vernal portal
b is real ofc
there is a mod b in the formula?
vague phoenix
I don't know what is Mario talking...
covert rain
I don't know what is Mario talking...
comparing real and imaginary parts
x^2-y^2+2ixy=a+ib
from this you get a system
worldly lantern
hey guys
vernal portal
there is a mod b in the formula?
thats messed up lowk b/mod b is just 1 sob if b is supposed to be real
worldly lantern
when i square an img number
i can take that its mod also got squared right?
so if a + ib had mod z , i can take mod of x+iy as z^2 right?
is it correct
im saying this by using euler form
winged flare
$\sqrt{a+ib} = \sqrt{r} e^{i\frac{\theta}{2}} = \sqrt{r}\cos(\frac{\theta}{2}) + i \cdot \sqrt{r}\sin(\frac{\theta}{2})$\\
use half angle formulae to simplify
flat frigateBOT
safe radishBOT
vague phoenix
No problem
safe radishBOT
Available help channel!
soft lava
I’m not sure what to do next
safe radishBOT
I’m not sure what to do next
lean otter
,rccw
flat frigateBOT
gentle edge
you supposed to write this in only sin and cosin?
lean otter
how does this become that
soft lava
Is this right now?
covert rain
sinx-sin^2x=sinx?
soft lava
Isn’t it?
vague phoenix
Looks better
vague phoenix
Wdym
covert rain
why did you put the question mark on me?
lean otter
you corrected the step but after that:
sinx + 1 - sin^2x is not equal to 1-sinx
vague phoenix
She is fixing
soft lava
wait I’m confused
What would it equal then
Because I thought if you had sinx-sin^2x it would equal sinx
I don’t know what else it can equal
How do I add them together
lean otter
leave it as it is
its like a + 1 - a^2 is not equal to 1-a
soft lava
How do I simplify it more tho
I’m really confused because I know that is not the final answer but I don’t see how else I can simplify it
lean otter
idk what the question asks you had transformed into only sines and cosines
flat frigateBOT
lean otter
you can open the bracket and multiply by sinx but i dont think so it is more simplifiable
lean otter
🥀
lean otter
happens
soft lava
How do I solve this one without getting the sin cubed
Some how in the answer key they just ignored the ones
vague phoenix
,rccw
flat frigateBOT
lean otter
sniped
vague phoenix
Bro rotate twice
lean otter
How do I solve this one without getting the sin cubed
factor tanx
soft lava
Wdym
I don’t know how I can do it
Like I know how to factor but I don’t see anything I can factor
lean otter
Like I know how to factor but I don’t see anything I can factor
do you see something common ? sinx/cosx
vague phoenix
Take common factor👍
soft lava
Yea I don’t undedtahd them
thorn bronze
see the red circle its common in both the terms so you can take them out
(sinx/cosx) ( sin^2x +cos^2x)
lean otter
can you identify common factor for $2\cdot 3 + 2\cdot 4$
flat frigateBOT
mercy
soft lava
No
lean otter
thorn bronze
If it is from middle school I likely don’t cus of covid
can you expand 2(x+y)
covert rain
is there something the same in all terms?
lean otter
if you have an expression
a.b + a.c you can see a is common/same for both terms
so you can factor a out leaving b and c
a(b+c)
soft lava
can you expand 2(x+y)
2x+2y
thorn bronze
similarily you take out (sinx/cosx) out from both the terms
because its common in both
like 2
thorn bronze
😂 its alr take your time
lean otter
lol
soft lava
similarily you take out (sinx/cosx) out from both the terms
Yea I just don’t see a way it evenly can be taken out since we have sin^2x/1
And you can’t take sinx/cosx out of that
thorn bronze
Yea I just don’t see a way it evenly can be taken out since we have sin^2x/1
its like 2 is sinx/cosx
soft lava
Sorry I don’t understand
I just don’t know how I can factor because to me it doesn’t look like like variables even if they are
thorn bronze
okay dont convert tanx to sinx/cosx then can you take it out from both the terms?
soft lava
I dont know how I would
thorn bronze
i converted csc^2x and sec^2x like you did
and took tanx out
imagine tanx is 2 and sin^2x and cos^2x are x and y
safe radishBOT
@soft lava Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
safe radishBOT
open wedge
,rccw
flat frigateBOT
open wedge
!xy
safe radishBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
open wedge
what's the original question please
cyan coyote
Well it's the question if in done inequalities well
open wedge
send the question
cyan coyote
what's the original question please
My question is that not the prblm it self
cyan coyote
send the question
Why?
open wedge
i need some context
if i don't know what your original problem is
how am i supposed to check your work
near sleet
well
near sleet
But u have Nthg to do with prblm
is it asking to prove roots
open wedge
Just check the roots and confirm inequalities
you didn't even give us the quadratic to check
😭
i need the original question
and your full work
cyan coyote
is it asking to prove roots
Naah bro just look at the question is asked
I don't need help with prblm
It's just smthg i analysed out of prblm
near sleet
Naah bro just look at the question is asked
I don't need help with prblm
It's j...
well
open wedge
i need the original question
in fact, not only me, but helpers want to take a look at your original question
near sleet
we can’t know if your work is right
if we don’t know what you’re proving
if the goal was to prove a,b >= 4 is necessary, then a = b = 4 is right
cyan coyote
,rccw
flat frigateBOT
cyan coyote
if the goal was to prove a,b >= 4 is necessary, then a = b = 4 is right
It wasnt the goal I was just checking and analysing more and more
Question has no relation with that
These 2 quadratic has roots compulsorily
open wedge
goodness
open wedge
just send the original question already
near sleet
what’s the problem
near sleet
thank you
open wedge
finally
open wedge
near sleet
I want to swear
? no need for that
open wedge
so you solved that $a = b = 4$
flat frigateBOT
1 divided by 0 equals Infinity
near sleet
hm
tall totem
yeah you seem to have done something completely different
cyan coyote
it’s asking for a circle equation, not proof of when real roots exist
That's the reason I wasn't giving u guys the questions
I said I was just analysing prblm
near sleet
**1 divided by 0 equals Infinity**
that seems right
safe radishBOT
near sleet
U here to help to learn or solve work stuffs given from school
can’t help if you don’t provide what you need help with.
tall totem
.reopen
cyan coyote
I literally said 2-3 times to just check if I handled those quadratic well but wtv doesn't matter anymore
open wedge
I literally said 2-3 times to just check if I handled those quadratic well but w...
but you took TOO long to send in the problem
open wedge
we helpers expect helpee to send in the original question
tall totem
that both quadratics have two real roots?
open wedge
or else we don't have enough context to see if your solution is correct or not
@cyan coyote
tall totem
or else we don't have enough context to see if your solution is correct or not
i think pipassionflop here isn't willing to do the problem in terms of circles and wants to go all the way in algebraic rigour
tall totem
by the way, here
cyan coyote
i think pipassionflop here isn't willing to do the problem in terms of circles a...
What's there do with prblm all to do is learn
tall totem
saying a^2/4 >= b doesn't mean you can substitute b^2 as a^4/16
should be careful squaring both sides of an inequality because you don't know the sign of a and b
case in point, -2 > -3 but 4 < 9
lean otter
imagine fighting in a closed help channel only to be lockedouut midfight
tall totem
imagine fighting in a closed help channel only to be lockedouut midfight
i know right
open wedge
imagine fighting in a closed help channel only to be lockedouut midfight
blud look at the chat history
cyan coyote
Well i understood all that how uselessly and carelessly I was doing that
Ohk thanks guys
cyan coyote
Cya
tall totem
fair
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Available help channel!
quick wraith
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quick wraith
I am working on d)
I want to start by proving b
first
I have tried induction
can someone check it?
Then let's work on d)
worldly lantern
b is true cuz of a^x being an increasing function when a>1
and being a decreasing function when a<1
what does sup stand for here
maximum value among the set?
like supreme or smthn?
what grade are you in again?
quick wraith
12th grade high school
quick wraith
what does sup stand for here
supremum
is like
the smallest upper bound
for instance
the supremum of (0, 1) is 1
and the sup of (0, 1] is 1
quick wraith
the supremum of (0, 1) is 1
here maximum does not exist
That is it
@worldly lantern, you there?
quick wraith
okay
worldly lantern
they really just defined
what youve gotta do for irrational power
just take the lowest rational power below the irrational power
what is this topic anyway
worldly lantern
real analysis
well then
quick wraith
I have a few days in this
quick wraith
yeah, It is okay
worldly lantern
there should be a method
quick wraith
Yeah
quick wraith
.close
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iron loom
Hi
signal crane
hi abelisa
iron loom
Structural Analysis Request: Weighted Logic System (3:3:4)
I am seeking a peer review of the following mathematical framework. This is a system designed for Structural Optimization using a 3:3:4 weighted ratio across three variables (x, y, z):
The Objective Function:
Let the total system be S.
S = 0.3x + 0.3y + 0.4z Variable Definitions:
x (Reduction Variable): The process of simplifying a complex set into its base axioms.
y (Equilibrium Variable): The identification of isomorphisms and balance between those axioms.
z (Integrity Variable): The stress-testing phase; applying external constraints to verify the stability of the system.
The Logic Question:
Given that z (Integrity) is weighted at 40%, while x and y are weighted at 30% each, how does this affect the Systemic Stability when external pressure is applied? Is this ratio (3:3:4) a mathematically sound distribution for maintaining a "Steel" (rigid/unbreakable) structure, or would a different ratio provide higher harmony?
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@iron loom Has your question been resolved?
dreamy mirage
I have questions on the sieve of erastosthenes and euclidean algorithm. For sieve, when we arrive at that number, how do we know if it is a prime or not?
safe radishBOT
I have questions on the sieve of erastosthenes and euclidean algorithm. For siev...
quiet plume
I have questions on the sieve of erastosthenes and euclidean algorithm. For siev...
If it's not been crossed so far it's prime.
dreamy mirage
Why though?
quiet plume
Because you've already crossed all numbers that are multiples of numbers less than it
So it can't be a multiple of those
Otherwise it would be crossed
dreamy mirage
So if there was factor, we would have found it before reaching that number?
quiet plume
Yes
dreamy mirage
Okay. And for the euclidean algorithm, how can i understand it intuitively? Like I see the formula but I don't understand how that works
timid ridge
gcd?
dreamy mirage
Yeah it computes the gcd, I think its the fastest algorithm to do that
I've memorized the formula before, and it helped but I keep forgetting it because I don't even understand it
quiet plume
Well you should convince yourself that gcd(a,b) = gcd(a, a-b) where a>b
Then it's just doing this repeatedly until you get 0 somewhere
dreamy mirage
Well you should convince yourself that gcd(a,b) = gcd(a, a-b) where a>b
Is this because since a > b, then the gcd has to be <= a? How can I be sure that a-b doesn't get rid of the gcd of the b part?
timid ridge
Is this because since a > b, then the gcd has to be <= a? How can I be sure that...
the gcd would be a divisor of a - b
so you cut a perfect multiple of the gcd away
you still have a multiple of the gcd left
dreamy mirage
the gcd would be a divisor of a - b
so if gcd | a and the gcd | b, gcd | a-b?
timid ridge
yes
because a and b are both multiples of the gcd
if you subtract one from the other you are still left with a multiple of the gcd
just imagine dividing everything by the gcd
dreamy mirage
if you subtract one from the other you are still left with a multiple of the gcd
why does a multiple minus a multiple equal a multiple?
timid ridge
$kp - kq = k(p - q)$
flat frigateBOT
enj ʚɞ
timid ridge
multiples of k, result is a multiple of k
dreamy mirage
Actually I think I figured it out. A multiple is a constant times the gcd, if you subtract two of these, then you still left with c*gcd-d*gcd=gcd(c-d)
Yeah mb im new to divisibility and its really confusing since you really have to think about everything yourself
So you just subtract since everything stays as a multiple of the gcd. Until you are eventually left by each side being the gcd?
strange flint
You could do gcd(a, b) = gcd (b, a-bq) to make it faster
dreamy mirage
You could do gcd(a, b) = gcd (b, a-bq) to make it faster
What is q?
strange flint
It's the result of the subtraction
a = bq + r
gcd (a, b) = gcd (b, r)
So for example if you have gcd(60, 16)
60 = 16×3 + 12
16 = 12×1 + 4
12 = 4×3 + 0
dreamy mirage
Then we have to solve for q right?
strange flint
So each time you hold onto your b value, and divide it by the remainder
Until you get a remainder of 0
So in that example:
gcd (60, 16) = gcd (16, 12) = gcd (12, 4) = gcd (4, 0)
And the gcd is that last remainder that's not equal to zero
If that makes sense
iron loom
Did I do something wrong? 🙁
dreamy mirage
.close
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broken finch
I need help
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I need help
covert rain
i think he need to show that from [ tanA+sinA=a and tanA-sinA=b ] ->a^2-b^2=4sqrt(ab)
frigid spruce
if allat = a and allat = b then prove allat
worldly lantern
a + b = 2tanA by adding
b-a = 2 sinA
multiply
and use values of tan and sin from given equations
safe radishBOT
@broken finch Has your question been resolved?
sleek hornet
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sleek hornet
Is there a way to continue
vague phoenix
Why 4th line you write $b(\not b)$
flat frigateBOT
Minhh
vague phoenix
Fuh
vague phoenix
Why you wrote b(not b) and p (not b)
sleek hornet
vague phoenix
You can remove brackets to make it clearer
sleek hornet
Oh forgot the write the up arrow
vague phoenix
No, it should be not b and b
sleek hornet
But after that is there any way to continue
vague phoenix
But after that is there any way to continue
Yes
sleek hornet
Corrected
flat frigateBOT
vague phoenix
Since b and not b always false
sleek hornet
It's like saying F OR (P and not B)
vague phoenix
F is false
sleek hornet
That last line is the final ans?
vague phoenix
You can still use distributive law here
I think this can be simplified to False
You still there?
I help a bit more then sleep
safe radishBOT
Available help channel!
mossy lotus
<@&268886789983436800> paycheck is here
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primal onyx
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primal onyx
World Population
The United Nations publish data and forecasts on the development of the world population.
Data source: United Nations, Department of Economic and Social Affairs, Population Division (2019). World Population Prospects 2019, Online Edition. Rev. 1.
a) In 1950, the world population according to this data was 2.536 billion people.
In a model, it is assumed that the world population grew by 1.9% per year from 1950 to 1980, compared to the respective previous year.
The development of the world population starting in 1950 is approximated by the function N₁.
• t … time in years with t = 0 for the year 1950
• N₁(t) … world population at time t in billions of people
- Set up an equation for the function N₁.
N_1(t) = ; ?
(1 point)
- Calculate the time period in which the world population doubles according to this model.
Here is the english translation if someone can help me understand what i need to do here
fathom jewel
Weißt du wie eine Exponentialfunktion aussieht
primal onyx
Ja
flat frigateBOT
primal onyx
A ist der startwert? Also 2,536
fathom jewel
Ja
flat frigateBOT
fathom jewel
Ja
flat frigateBOT
primal onyx
und das war 1) fertig oder?
primal onyx
fathom jewel
Ja das ist die Idee, kann man algebraisch lösen
neon locust
es gibt deutsch in diesem server
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sand bay
I think my integral result is super wrong, I don't think the integral should diverge because that denominator goes to zero really quickly, where did I go wrong?
safe radishBOT
I think my integral result is super wrong, I don't think the integral should div...
vague phoenix
is u here just a substitution?
sand bay
yes
vague phoenix
i think you made a mistake at the partial fractions
first page
if im not wrong, partial fraction is for polynomials (rational functions), and u^3/4 here is radical
perhaps using other methods here
fathom jewel
I think my integral result is super wrong, I don't think the integral should div...
Is the integral 1/(x^4+1)?
sand bay
yes
fathom jewel
yes that doesnt apply pfd
sand bay
oh so that's the mistake I see. thank you
flat frigateBOT
fathom jewel
or go over C
i assume you havent taken complex analysis because that integral could be nicely done with residues
fathom jewel
oh so that's the mistake I see. thank you
but then do you have to find the exact value or only investigate convergence/divergence?
sand bay
I have to find the exact value
I was just curious to solve that integral that had that limit
I see where I went wrong, so what's the correct way to solve it?
slim lion
I see where I went wrong, so what's the correct way to solve it?
wait is the integral
$\int_0^\infty \frac{1}{x^u+1} dx$
flat frigateBOT
Flatus
slim lion
or am i reading it wrobg
fathom jewel
I see where I went wrong, so what's the correct way to solve it?
well i provided 3 ways to solve it, but ig it's a tedious one
slim lion
instead of u is 4
ohhhh
vague phoenix
no its nice
slim lion
yea i would do pfd on this
flat frigateBOT
fathom jewel
depends whethere A and C end up 0, might need to complete the square then
slim lion
$x^4+1 = x^4+2x^2+1-2x^2 = (x^2+1)^2-2x^2 = (x^2+\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$
flat frigateBOT
Flatus
slim lion
it's a nice completing the square trick
fathom jewel
kanena problima
mortal sandal
Why are the denominators the same 
fathom jewel
typo argh
radiant canopy
guys how did newton integrate x^-1 without Euler's number, natural log
vague phoenix
he used power series expansion
pallid magnet
guys how did newton integrate x^-1 without Euler's number, natural log
power series
vague phoenix
with the curve in formula of y = 1/(1+x)
pallid magnet
he didn't intergrate
radiant canopy
guys i got some questions
how did we derive differentiation
how did we derive integration
when to use Du or Dx
round shore
omg riemann got a rebrand td
radiant canopy
https://en.wikipedia.org/wiki/History_of_calculus
when will u become
HONOURABLE from honourable
TY again
.close
safe radishBOT
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Available help channel!
late bobcat
safe radishBOT
fathom jewel
,rccw
flat frigateBOT
late bobcat
fathom jewel
35.
the line should be contained
instead it's perpendicular to the plane because you used its direction as the normal vector of the plane
late bobcat
Uhh I don’t get it
fathom jewel
the line you were given should be inside the plane
late bobcat
Oh
So the line we r given is contained
and we need a line
that is normal
to the plane
to find
fathom jewel
[ a(x-x_0)+b(y-y_0)+c(z-z_0)=0 ]
$(a,b,c)$ denotes the normal vector of the plane. So when you use a vector, facing into the same direction as the line, as the normal vector of the plane, then the line will be perpendicular to the plane, since it's parallel to the normal.
late bobcat
a direction vector
flat frigateBOT
fathom jewel
Imagine the line you are given
for t=0 you get (4,-1,0) and from there you have the direction vector (-1,2,-3)
Now you want to find a second direction vector from there
that points to (3,5,-1)
fathom jewel
when you found that direction vector, you can take the cross product of both of these, and use that
late bobcat
We need a point and a direction vector
to find the eq of a plane
we have the point
and the vector
I m confused
fathom jewel
let me try to pull off a visual
late bobcat
Okay
fathom jewel
You start your way with the line, pick any starting point, I chose simply t=0 => (4,-1,0) starting point (red)
Now obviously, we are given the direction of the line (-1,2,-3) orange vec
but now we need to find a second direction vec, if we want to produce a plane, containing the blue point (3,5,-1)
you can find that green vec, by simple vector geometry
green vec = (3,5,-1) to (0,0,0) + (0,0,0) to (4,-1,0)
fathom jewel
when you got both vecs (orange and green) you can produce the correct normal vector, so that you can set a,b,c correctly
late bobcat
I wanna cry
😭
This is telling another different story
Can someone explain it to me as if I m a 4 year old baby
late bobcat
Ok where’s (3,5,-1) here
fathom jewel
You start your way with the line, pick any starting point, I chose simply t=0 =>...
i documented everything
it was the blue point
fathom jewel
when you found that direction vector, you can take the cross product of both of ...
which i mentioned here
fathom jewel
when you got both vecs (orange and green) you can produce the correct normal vec...
and here
late bobcat
But why is the cross product
of them
The normal vector
I wasn’t taught this
rule
fathom jewel
the two vectors (green and orange) they span the plane
a normal vector was a vector, that was perpendicular to a plane
late bobcat
Oh I remember
fathom jewel
so if you take the cross product of the vectors (green and orange) spanning the plane, you are producing a new vector that is perpendicular to both
late bobcat
a x b is orthogonal to both a and b
fathom jewel
yes
you need a vector from the line's pov that faces to (3,5,-1) the blue one
so that's why we pick some point on the line an easy choice is always t=0
fathom jewel
that doesn't make sense
you can't have a plane if the second point is not outside the line
because all the information would end up being contained in the line
so all you'd have left is the line again, not a plane
safe radishBOT
@late bobcat Has your question been resolved?
still charm
Quick question
safe radishBOT
Quick question
buoyant shadow
the quick question, the slow lie in wait
still charm
From stirling we have that:
n! ~ sqrt(2pi n) (n/e)^n
But using big O notation we can just say n! = O(n^n), right?
still charm
I've seen both tbh
Like f = O(g)
Even though yeah set inclusion makes way more sense
safe radishBOT
@still charm Has your question been resolved?
still charm
Wdym
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zinc panther
I need help visualizing the situation. I dont have the clearest grasp on how A= (1/2)absin(theta) but I've made it make enough sense to solve the problem. can someone walk me through it?
safe radishBOT
I need help visualizing the situation. I dont have the clearest grasp on how A= ...
timid ridge
I need help visualizing the situation. I dont have the clearest grasp on how A= ...
differentiate the Area w.r.t to time
zinc panther
differentiate the Area w.r.t to time
I know thats what I need to do but im having trouble understanding the actual situation itself
the triangle