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#help-39
pearl pondBOT
pearl pondBOT
Available help channel!
radiant epoch
Is this proof valid
pearl pondBOT
Is this proof valid
radiant epoch
this question is: prove (2n)! > (2^n)(n!)^2
hallow timber
Yes, it is correct in logic.
mental salmon
Can anyone give me a hint for this problem? As in a place to start idk how to start
mental salmon
I tried to make bounds for m, that m must be greater than or equal to 6 but that doesn’t seem to help
hallow timber
can you factor $m^2-n$ since $n$ is positive?
jolly parrotBOT
flynger
mental salmon
Yes but you get a difference of squares in square roots, and I don’t think that would help
hallow timber
Not so quick lol
always worth trying it out
i said that because look at whats under the 5th root?
hallow timber
yea np
mental salmon
.close
quick venture
can someone explain what he is doing for the rectangle and triangke
quick venture
like why is he doing 600*3
sharp vigil
the total force is the area under the curve
since they are rectangles and triangles you can just use standard geometric formulas
pearl pondBOT
Available help channel!
midnight haven
hi
wary siren
is the target vector B or B'?
midnight haven
is the target vector B or B'?
B’
midnight haven
is the target vector B or B'?
B and B’ are basises though
midnight haven
B' = {(1,3),(-1,-1)} = <-1-1, -1-3> = <-2, -4>
i don’t understand
rough stream
Huh, I've never thought about doing this with elimination before
midnight haven
im confused
rough stream
I'm going to guess that you can put the bases into their own matricies, block them together, then reduce one of them, getting the other
Let me check to be sure though
Well, I'm not sure of a way that explicitly uses guess-elimination. We can do this the standard way tho
You want to write the vectors in B, in terms of the vectors in C
You've written them in terms of some u1,u2. That's fine, technically. We can now equate them:
I'll call your coefficents of the left equation b1,b2 in order to avoid confusion
midnight haven
I'll call your coefficents of the left equation b1,b2 in order to avoid confusio...
isnt it supposed to be the same vector tho?
isnt the point of changing from bases to another is to represent the same vector with different basis?
thats what i know
rough stream
B is like a language. You can express vectors in the basis B.
B' is like a language. You can express vectors in the basis B'.
A vector expressed in B may be the same as a vector expressed in B', but they wouldn't look the same.
You can change your language from B to B' by using a change of basis matrix. A vector expressed in B, when multiplied by your change of basis matrix, will be the same vector but expressed in B'
midnight haven
B is like a language. You can express vectors in the basis B.
B' is like a langu...
wait what
Wdym by same vectors but dont look the same
rough stream
Like, for example, (1,1) expressed in B
is (6,1) expressed in the standard basis.
I got that by doing 1*(2,2) + 1*(4,-1)
There exists a vector in B', that will give (6,1) in the standard basis. We can do the math to figure out what that vector is
midnight haven
Like, for example, (1,1) expressed in B
is (6,1) expressed in the standard basi...
isnt the point of changing base to another to:
Express the same vector in different basis??
like to express(1,1) in both B and B’
rough stream
isnt the point of changing base to another to:
Express the same vector in diffe...
Strictly answering this question, yes.
rough stream
isnt it supposed to be the same vector tho?
Strictly answering this question, no
At least, not the same as in both C's should be the same
midnight haven
At least, not the same as in both C's should be the same
no no I mean
same output
Different linear combinations
i should have renamed my scalars for that matter
rough stream
Maybe we miscommunicated! I'll continue.
(2 4) (b1) = (1 -1) (c1)
(2 -1) (b2) (3 -1) (c2)
That make sense?
That is just equating your (u1, u2)
rough stream
We can solve that for (c1, c2) by taking the inverse of that matrix
Yes. If you bring both (b1, b2) and (c1, c2) to the standard basis, you'll get exactly the same vector in each case
midnight haven
you still have another unknown tho right
midnight haven
Yes. If you bring both (b1, b2) and (c1, c2) to the standard basis, you'll get e...
are u sure?
rough stream
You can't solve for (b1, b2) and (c1, c2) together. The trick is that if you give me a (b1, b2), I can give you a (c1, c2). In other words, we changed a vector from B to B'
midnight haven
but b1,b2 are the scalars for thr non standard one
midnight haven
You can't solve for (b1, b2) and (c1, c2) together. The trick is that if you giv...
oh
rough stream
are u sure?
That's what should happen if the change of basis matrix is correct
midnight haven
That's what *should* happen if the change of basis matrix is correct
i still don’t understand this problem tho
like the solving for B’
rough stream
(b1, b2) is a vector expressed in B
(c1, c2) is a vector expressed in B'
This should hold:
(2 4) (b1) = (1 -1) (c1)
(2 -1) (b2) (3 -1) (c2)
This is noting that, if we take each vector to the standard basis, they'll actually be the same vector
We can use the equation to solve for (c1, c2). This gives us a new equation where we can plug in a (b1, b2) and get a (c1, c2) out. This gives us a change of basis formula
midnight haven
(b1, b2) is a vector expressed in B
(c1, c2) is a vector expressed in B'
This s...
those are scalars tho
rough stream
b1 and b2 are each scalars, but (b1, b2) is a vector
Are you able to solve that equation for (c1, c2)?
midnight haven
but now i have unknowns
rough stream
Don't worry, this will end up okay
Inverse of
(1 -1)
(3 -1)
Inverses of 2x2s are pretty easy
midnight haven
Inverses of 2x2s are pretty easy
no not this
its just confusing like
Now i have 3 matrices on the lhs
rough stream
Still there
Remember we want to solve for (c1,c2)
I think that's the third matrix you are referring to. I misunderstood. I don't typically call a 1x2 a "matrix" but I see where your head is at
Like, let's say I wanted you to solve for y in:
2x = 3y
You wouldn't be confused that x or y are present
midnight haven
Remember we want to solve for (c1,c2)
but
i thought we are looking at B’
alright bro im just lost at this point
heres the q again
its so weird bro
i dont need to do no freakin elimination
okay actually thats for the inverse
rough stream
So I think I see a way to do this with elimination
midnight haven
nvm
midnight haven
this is frustrating
rough stream
,w {(1, -1),(3, -1)}^-1 * {(2,4),(2, -1)}
jolly parrotBOT
rough stream
So rearranged you get:
(0 -5/2) (b1) = (c1)
(-2 -13/2) (b2) (c2)
That gives us a way to take a vector (b1,b2) and compute a vector (c1,c2)
In other words, this is a change of basis matrix
(0 -5/2)
(-2 -13/2)
Is our answer
Now, I wanna check my original strategy
midnight haven
heres the q again
they’re saying from B to B’
shouldnt that be solving for the scalars on rhs???
rough stream
Are c1 and c2 not those scalars?
Can you be more specific
,w row reduce {(1,-1,2,4),(3,-1,2,-1)}
jolly parrotBOT
midnight haven
Are c1 and c2 not those scalars?
no?
rough stream
Yes! That's a way to do this by a single elimination. Maybe that's what they want. Put your matricies together and row reduce
rough stream
You've got to say what's confusing you.
midnight haven
I changed the scalars on rhs to d1 d2
rough stream
"The scalars on the rhs" is not clear here
rough stream
Note I was writing
(2 4) (b1) = (1 -1) (c1)
(2 -1) (b2) (3 -1) (c2)
So we swapped different variable names. Maybe that's our confusion!
midnight haven
Note I was writing
(2 4) (b1) = (1 -1) (c1)
(2 -1) (b2) (3 -1) (c2)
So we swa...
oh God 😭😭😭😭😭😭
rough stream
In this case, you've got (c1,c2) as the vector in B, and (b1,b2) as the vector in B'
midnight haven
bruh omg
midnight haven
In this case, you've got (c1,c2) as the vector in B, and (b1,b2) as the vector i...
yes
rough stream
So yes, you'd want to solve for (b1,b2)
midnight haven
so when they say they want the transition matrix from B to B’
what does this whole statement even mean
rough stream
A matrix that, when multiplied by a vector expressed in B, can give "the same" vector, but expressed in B'
"The same", meaning that if you took both vectors back to the standard basis, they would be exactly the same vector
You will get such a matrix by solving for (b1,b2) here
midnight haven
A matrix that, when multiplied by a vector expressed in B, can give "the same" v...
can u vc?
i really cant express my thoughts rn properly
hallow timber
leon you already asked this question before and already got the answer lmao
remember
$P_B[x]B=[x]S=P{B'}[x]{B'} \implies P_{B'}^{-1} P_B[x]B=[x]{B'}$
jolly parrotBOT
flynger
midnight haven
leon you already asked this question before and already got the answer lmao
I still didn’t understand tbh
I actually dont get it
hallow timber
Does it make sense that for $P_B=\begin{bmatrix}2 & 4\ 2 & -1\end{bmatrix}$,
$P_B[x]_B=[x]_S$
Because $[x]_B$ is the scalar coefficients of the basis vectors of $B$
hallow timber
By the same logic does it make sense that for $P_{B'}=\begin{bmatrix}1 & -1\ 3 & -1\end{bmatrix}$,
$P_{B'}[x]_{B'}=[x]_S$
midnight haven
its just a representation of the system in Ax=b
jolly parrotBOT
flynger
hallow timber
its just a representation of the system in Ax=b
That's like asking this:
Problem: If double the amount of apples I have 6, then how many apples do I have?
Solution: You find this by doing 2a=6.
But you're asking whats the point? Aren't you just doing algebra/solving a linear equation? Yes, but the problem is asking something that you're able to solving using the linear framework, doesn't mean you shouldn't see it as a problem.
Do you see what's weird about asking why everything is Ax=b?
midnight haven
That's like asking this:
Problem: If double the amount of apples I have 6, then...
no no i get why its Ax=b its a representation but like
Why is A called the transition matrix
like
why
hallow timber
its transitioning the vector from basis B to B'
They just gave whatever name they saw fit
You're worrying too much about semantics
There's no irrefutable reason to call it that; all names are given
midnight haven
its transitioning the vector from basis B to B'
no im asking why
or how is it transforming
wheres the transformation
wait
hallow timber
The matrix is an operator.... since the input is x in basis B and output is x in basis B'
... thats the transition
midnight haven
never mind bro the frustration got me forgetting stuff
midnight haven
**flynger**
yes
hallow timber
Those two give you $P_B[x]B=P{B'}[x]_{B'}$ right?
jolly parrotBOT
flynger
hallow timber
And you should be able to proceed from here
midnight haven
**flynger**
i get this part but its what theyre asking for that i dont get
how do I continue from here
@hallow timber this is what ur talkin about right
hallow timber
Yes basically
But you need both matrices on the left side
do you know matrix inverses?
midnight haven
yes
waaait
i think they just want the inverse
we cant really solve for the coordinate vectors unless we are given the spanned vector right
like in part c
midnight haven
that you can apply
i see
hallow timber
**flynger**
Basically this gives you $P_{B'}^{-1}P_B[x]B=[x]{B'}$ right?
From taking inverse on both sides
jolly parrotBOT
flynger
hallow timber
So the question is to solve for $P^{-1} = P_{B'}^{-1}P_B$ using Gauss-Jordan
jolly parrotBOT
flynger
pearl pondBOT
@midnight haven Has your question been resolved?
shadow reef
What did I do wrong?
tall flint
how'd you get between these two lines?
tall flint
and what's h'?
tall flint
next question, where's this coming from?
shadow reef
Oh I see
I accidentally differentiated again
Should I put my answer as negative or positive because the question only asks how fast it's changing
stoic imp
pearl pondBOT
jolly parrotBOT
Renato
cursive wraith
**Renato**
what have you tried?
stoic imp
i don't know how to start this
cursive wraith
i don't know how to start this
what does "primitive sixth root of unity" mean first?
And then we can build up from that definition
stoic imp
w^6 = 1
cursive wraith
primitive means that w is not a k-th root of unity for values of k less than 6
so w^k is not 1 if 0 < k < 6
cursive wraith
?
stoic imp
i don't think i follow
cursive wraith
Imagine the sequence 1, w, w^2, ...
If w is a k-th root of unity, that means we come back to 1 after k steps
so w^k = 1
w is a primitive k-th root of unity if w^k = 1 is the FIRST time you come back to 1 in that sequence
which means you didn't get 1 previously
meaning w, w^2, ..., w^(k-1) are all different from 1
stoic imp
okaay
cursive wraith
so
stoic imp
so w^6=1
cursive wraith
w is a *primitive* k-th root of unity if w^k = 1 is the FIRST time you come back...
based on what we said
cursive wraith
well not necessarily garbage, but different from 1
feral sedge
mfw -1 is garbage
cursive wraith
mfw -1 is garbage
What is up with people spoiling
anyways, w^3 for example is not garbage, even if it's different from 1
stoic imp
I'm new to roots of unity, no spoilers please
cursive wraith
and since w^6 = 1, you can deduce what w^3 is...
feral sedge
eh i mean there wasn't even a problem that i spoiled
i think i "spoiled" a question that he wanted to ask but hadn't yet?
thats the only explanation i have
stoic imp
and since w^6 = 1, you can deduce what w^3 is...
-1^2=1
cursive wraith
"mfw any number different from 1 is garbage" would have been okay for me
Idk maybe I overreacted
sry if I did
stoic imp
well is a sum of cubes
or we can use polynomial division
-1 is a root
$\polylongdiv{x^3+1}{x+1}$
jolly parrotBOT
Renato
cursive wraith
you want me to use polynomial division or what
well sum of cubes was good
but the point is (w+1)(w^2-w+1) = 0
stoic imp
yea so
i am missing the big picture
i don't see the correlation
i already forgot what we are doing
cursive wraith
i already forgot what we are doing
we are trying to come up with the smallest 'easy' polynomial that w is a root of
that way, see if it divides f
if it does, we win
stoic imp
okay so
cursive wraith
but the point is (w+1)(w^2-w+1) = 0
from here, we have two options
can you find out if both are valid? only one is valid?
stoic imp
i would need to expand the second factor
cursive wraith
i would need to expand the second factor
factor*
but not really needed
think about if w+1 = 0 is possible
stoic imp
stoic imp
and w^2 - w + 1 = 0
who knows
cursive wraith
from there, no real need to factor this
stoic imp
i need to expand dat
cursive wraith
factor*
and we don't need to
if we come back to the definition of G_6, we can realize that there are exactly two primitive roots
stoic imp
w^0
cursive wraith
$G_6 = {1,\zeta,\zeta^2,\zeta^3,\zeta^4,\zeta^5}$
jolly parrotBOT
Raphaelisius Maximus MMIII
cursive wraith
where $\zeta = e^{\frac{i\pi}{3}}$
jolly parrotBOT
Raphaelisius Maximus MMIII
stoic imp
**Raphaelisius Maximus MMIII**
what's with this zeta notation
cursive wraith
what's with this zeta notation
well usually I use omega
but it's already used
give a name to the smallest (in argument) root of unity
stoic imp
give a name to the smallest (in argument) root of unity
in argument?
cursive wraith
in argument?
the root with the smallest arg
stoic imp
w^0
cursive wraith
e^(i * Arg(...))
stoic imp
w^6
stoic imp
give a name to the smallest (in argument) root of unity
ahh not primitive
cursive wraith
w^0 = w^6, and no
stoic imp
you mean w^2
cursive wraith
**Raphaelisius Maximus MMIII**
e^(ipi/3) is the one
cursive wraith
you mean w^2
neither?
it's the one with k = 1 in your formula
the 6th roots of unity are $e^{\frac{2ik\pi}{6}}$ where $k = 0,1,...,5$
jolly parrotBOT
Raphaelisius Maximus MMIII
cursive wraith
**Raphaelisius Maximus MMIII**
so k = 1 gives this root
cursive wraith
k=1 always gives a primitive root
if you do a little bit of digging, you realize k = 5 is the other primitive root
cursive wraith
**Raphaelisius Maximus MMIII**
if you take z = (this k = 1 root)
so z = e^(ipi/3)
you know 1,z,z^2,...,z^5 are all the 6th roots of unity
so z,...,z^5 are not equal to 1
so z is primitive
does this at least make sense?
stoic imp
so z is primitive
how is that
cursive wraith
how is that
z^6 = 1
cursive wraith
so z,...,z^5 are not equal to 1
and z,...,z^5 are not 1
because 1,z,...,z^5 are the 6 different 6th roots of unity
stoic imp
so z is primitive
primitive means that z^6 = 1 and it's the first occurrence?
pearl pondBOT
@stoic imp Has your question been resolved?
humble axle
equation f(X) = x-2 can be written as x^3 +px^2+qx+r find p q r
humble axle
how do i do this
quick star
equation f(X) = x-2 can be written as x^3 +px^2+qx+r find p q r
what
pearl pondBOT
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.
quick star
how can you write a linear function as a cubic
unkempt yacht
equation f(X) = x-2 can be written as x^3 +px^2+qx+r find p q r
do you have a screenshot of where the problem came from?
humble axle
my exam
languid plover
u sure its not a dual function problem... like there's f(x) and g(x) or h(x)
quick star
i tried (x-2)^3 and expanding
why did you cube it?
quick star
i think the actual question is probably f(x) = (x - 2)^3
which can be x^3 +px^2+qx+r for some p,q, and r
find the values of p,q, and r
languid plover
then expand and compare coefficient
humble axle
i did but my teacher said its wrong
wait
yeah nah
idh the question
its like a multi section question like
theres 5A 5B 5C 5D
im pretty sure there was +qx + r
languid plover
I got it
languid plover
what r the three sols
languid plover
You need to solve the interection (there will be three points i think) between the graph provided of f(x) and the line y=x-2
humble axle
then what is p q r
humble axle
what
fluid crystal
Is anyone free to help with homework i have exam tmr and im so cooked
humble axle
so i draw line x-2?
quick star
show the very top of this
i found a similar question
as you can see
they've defined a function f(x) = 20/x
languid plover
actually I might not be seeing anything
languid plover
show the very top of this
that would help
quick star
humble axle
thats the whole question
quick star
then they ask you to do this
humble axle
quick star
in your question we don't know what is f(x) because you never gave us anything
there
humble axle
?
quick star
,, f(x) = x^2 - \frac 2x = x-2
jolly parrotBOT
mmmm7
quick star
multiplying through by x gets you to
x^3 - 2 = x^2 - 2x
move everything to the left and you'll have your answer
pearl pondBOT
@humble axle Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
shut flicker
i was wondering how i would show this, im not sure if its useful, but i thought fibonnaci numbers could have helped, since i have found 10 numbers, such that there are no 4 numbers such that a+b=c+d, i suppose adding 100 to the list can make it 11. should i be approaching this differently?
proper nova
i was wondering how i would show this, im not sure if its useful, but i thought ...
sounds like a dirichlet problem
compact ridge
hmm, (16 choose 2) = 120, so you need 20 of those pairwise sums to not be unique
it actually might be using binary numbers
proper nova
i think instead we think about $a + b = c + d$, we think about $a + b = a - m + b + m$, where $c = a - m$ and $d = b + m$
jolly parrotBOT
1 divided by 0 equals Infinity
shut flicker
hmm feel like pidgeon hole principle may be used somewhere
il further think
.close
void estuary
$find the local maximum of f(x)= (\frac{\sqrt{3e}}{2sinx})^{\sin^2x} , x\in(0,\frac{\pi}{2})$
jolly parrotBOT
Prathmesh
void estuary
is there any alternate method to this except differentiation?
cuz won't differentiation look ugly?
warm tiger
cuz won't differentiation look ugly?
Maybe try taking log both sides
Have you tried that?
void estuary
no not yet
I'll try that immediately
$\frac{y'}{y} = \sin2x((\log{(\sqrt{3}))+1}-\frac{1}{4} - \log{(2 \sin{x})})$
jolly parrotBOT
Prathmesh
pearl pondBOT
@void estuary Has your question been resolved?
inland laurel
The number of solutions of the equation
[
\sin x + \cos x - 1 = \sqrt{ \ln\left( e^{3 - 2\sqrt{2}} \right) }
]
in ([0,2\pi]) is:
A) 1 \quad B) 5 \quad C) 6 \quad D) 7
inland laurel
!status
pearl pondBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
toxic lichen
simplify right hand side root first
inland laurel
i wrote latex of question wrong
jolly parrotBOT
BlackidoZΣ
inland laurel
okay now its correct
toxic lichen
my instruction stands
inland laurel
okay so rhs = sqrt (2) -1
inland laurel
basically i attempted this question before and did this same step there also
inland laurel
if i follow your instruction ill know where i was doing mistake rather than sending my solution and saying to check my mistake
inland laurel
so your equation becomes sin(x)+cos(x)=sqrt(2)
yes
toxic lichen
if i follow your instruction ill know where i was doing mistake rather than send...
oh so you purposely hide your solution bc you want to make us give you instruction
versed mica
did you rewrite the equation after the simplification?
inland laurel
sinx + cosx = root2
versed mica
ok then divide both sides by root 2 and use the angle sum formula for sine
inland laurel
sin(x + pi/4) = 1
mighty epoch
hi
inland laurel
i am forgetting formula
versed mica
sin(x) = 1 when x = pi/2 + 2npi
inland laurel
yes
versed mica
yep
inland laurel
i was getting an weird equation which collpased to sin2x = 2 which is false when i shifted 1 to rhs in question and squared both sides
versed mica
ambigous?
pearl pondBOT
versed mica
its 3 am so i might get high sometimes
word
toxic lichen
its 3 am so i might get high sometimes
WHY ARE YOU DOING MATH AT 3 IN THE MORNING
versed mica
why not
inland laurel
WHY ARE YOU DOING MATH AT 3 IN THE MORNING
i am doing it from staright 8 pm to till now
versed mica
🔥
inland laurel
i took brakes for dinner obv
toxic lichen
bro fucks his own sleep schedule
inland laurel
bro fucks his own sleep schedule
your true i sleep one day 6 hours and other day 3 hours only
versed mica
been there
toxic lichen
or idk maybe all of yall think its hip and cool to sleep 0 hours a day and AAAAAAAAAAAA SIGMA SIGMA TRIPLE SIGMA GRINDSET HARDER HARDER HUSTLE NO SLEEP AAAAAA
versed mica
do you have a test
toxic lichen
your true i sleep one day 6 hours and other day 3 hours only
yeah that sleep schedule is mega fucked
you should maybe
versed mica
or idk maybe all of yall think its hip and cool to sleep 0 hours a day and AAAAA...
weird
inland laurel
i am analaysing test problems
warped violet
8+
toxic lichen
crore times more important than +20% math problem grind
versed mica
or idk maybe all of yall think its hip and cool to sleep 0 hours a day and AAAAA...
have you had your coffee yet today?
toxic lichen
it's 23:36 and i am winding down
toxic lichen
crore times more important than +20% math problem grind
this is 100% serious btw
inland laurel
crore times more important than +20% math problem grind
but everythings happneing really bad with me in exams
toxic lichen
bad exam scores CANNOT be fixed by sacrificing sleep schedule
versed mica
depends
toxic lichen
.reopen btw
pearl pondBOT
✅ Original question: #help-39 message
versed mica
if you haven't learned the material then obviously staying up is better
but if you're just reviewing then yea get sleep
warped violet
Man, you will just perform worse if you don't sleep.
toxic lichen
but everythings happneing really bad with me in exams
it'll keep getting worse and worse and worse and worse and worse if you keep your "sometimes 6 hrs sometimes 3 hrs" sleep cycle
completely worse worse
bad worse
you have to sleep
inland laurel
but then in mornings how will i manage whole time stuff
toxic lichen
you get consistent sleep schedule
warm tiger
get 7 or 8 hrs a day
reading this at 3 am when i have to wake up at 8 am
toxic lichen
something like 23:00 sleep 07:00 wake
doesn't need to be precise to the minute, but it does need to be consistently like that
versed mica
but then in mornings how will i manage whole time stuff
wake up earlier
toxic lichen
wake up earlier
devil talk. if wake early then also bed early!
@inland laurel when is your nearest test in the future
toxic lichen
or rather high stakes exam
versed mica
4 days?
inland laurel
i have to cover all the chapters
toxic lichen
fucking 21
21? like the 21st? 4 days from now?
inland laurel
its totally complete syllbaus
toxic lichen
i have to cover all the chapters
and if you score <100% then?
inland laurel
21? like the 21st? **4 days from now**?
21st november
toxic lichen
life over?
versed mica
why are you staying up this late if your exam isn't even until the 21st
😭
is this your normal schedule
inland laurel
you know i took a mock between 8pm to 11pm today and i just scored TM 3 out of 300
in math i scored only 2 /100 and P 7 and chem -6 everything is happening with me so much weird and still i am analaysing that test
versed mica
you got a 2%?
inland laurel
I feel i forgot all my practice i couldn't solve problems anyhow
toxic lichen
I feel i forgot all my practice i couldn't solve problems anyhow
yeah that sounds like sleep deprivation dude
YOU'RE GETTING NO SLEEP THAT'S WHY YOUR SCORES ARE SO SHIT
inland laurel
like in trigonometry and series
toxic lichen
you can cram for lakh crore hours but your brain is like a strainer
it is so tired and burned out it holds nothing
and this WON'T be fixed by cramming for another lakh crore hours
YOU'RE GETTING NOT ENOUGH SLEEP, THAT'S WHY YOUR SCORES ARE SO SHIT
versed mica
so lively tonight ann
toxic lichen
100% serious
toxic lichen
so lively tonight ann
(derogatory)?
versed mica
what?
inland laurel
i was getting much better score 6 months ago
toxic lichen
did you mean that in a sarcastic and/or derogatory way
warped violet
😭
versed mica
how is that derogatory
toxic lichen
off to #「helpers-lounge」 we go
versed mica
no im ok
warped violet
i was getting much better score 6 months ago
Will you take Ann's well placed advice
inland laurel
after that i got hugedecrement in my marks in math and other 2 subjects, i just score only 2 or 4 continously there
inland laurel
Will you take Ann's well placed advice
yes
toxic lichen
Will you take Ann's well placed advice
nah obv not
why would anyone ever take advice from someone like me?
inland laurel
why would anyone ever take advice from ***someone like me***?
no your saying correct
warped violet
no your saying correct
Then what are you still doing here
Literally press one button
inland laurel
how much sleep do you take @toxic lichen
hallow timber
Whatever doesn't kill you can also cripple you
pastel umbra
Lie down* 
toxic lichen
how much sleep do you take <@183668144404037632>
about 7 hours
warped violet
Lie down* <:KEK:586240877358350341>
SORRY
inland laurel
Whatever doesn't kill you can also cripple you
enough
warped violet
😂 what did I just say
inland laurel
.close
pearl pondBOT
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naive marlin
yo
naive marlin
hm
Can someone explain me to the intuition of riemann stiltjes integral
like i understand definition somewhat
but like what is the visual of it
versed mica
did you ping me?
pearl pondBOT
@naive marlin Has your question been resolved?
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fair echo
can anyone help explain this?
pearl pondBOT
can anyone help explain this?
fair echo
.close
pearl pondBOT
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hoary meadow
Let $(x_n^{(k)}){k \in \mathbb{N}}$ be a Cauchy sequence in $X$. Then there exist sequences $(x_n^{(1)}), (x_n^{(2)}), \dots$ in $X$ such that $|x_n^{(k)}| \le 1$ for all $k \in \mathbb{N}$ and all $n \in \mathbb{N}$. Since $(x_n^{(k)}){k \in \mathbb{N}}$ is Cauchy in $X$, we have
[
\lim_{k,m \to \infty} d\big((x_n^{(k)}), (x_n^{(m)})\big)
= \lim_{k,m \to \infty} \sup_{n \in \mathbb{N}} |x_n^{(k)} - x_n^{(m)}| = 0.
]
By the definition of the supremum, for all $k,m \in \mathbb{N}$ and for each $n \in \mathbb{N}$,
[
0 \le |x_n^{(k)} - x_n^{(m)}| \le \sup_{n \in \mathbb{N}} |x_n^{(k)} - x_n^{(m)}|.
]
Hence, by the squeeze principle,
[
\lim_{k,m \to \infty} |x_n^{(k)} - x_n^{(m)}| = 0 \quad \text{for each } n \in \mathbb{N}.
]
Thus, for each fixed $n$, $(x_n^{(k)}){k \in \mathbb{N}}$ is Cauchy in $\mathbb{R}$. Since $\mathbb{R}$ is complete, it converges, so there exists a sequence $(L_n){n \in \mathbb{N}}$ with
[
L_n = \lim_{k \to \infty} x_n^{(k)} \quad \text{and} \quad |L_n| \le 1 \text{ for all } n \in \mathbb{N}.
]
jolly parrotBOT
ashyboi
hoary meadow
im struggling to show it converges with the sup metric
i got as far as showing each sequence component converges under the standard metric and showed such a limit sequence exists
grim fractal
what exactly are you trying to show here
hoary meadow
what exactly are you trying to show here
completness
for 3c
im trying to show the cauchy sequence is convergent with the sup metric
hoary meadow
What can you say about those sequences?
umm
idk
they have convergent subsequences?
formal kernel
Yes they have
I'm pretty sure you have to use that. Because if you take a Cauchy sequence there is something to say about how close the limit points are and can help you conclude
hoary meadow
I'm pretty sure you have to use that. Because if you take a Cauchy sequence ther...
but isnt this normally under the standard metric
idk
formal kernel
Sure. I'm talking about elements of X. Take a Cauchy sequence of X call it (y_n). For all n there is a limit point u_n in [0,1]. Show that u_n is a Cauchy sequence
limber nimbus
it might be easiest to start with the limit and show it works
we have a sequence of sequences (x_n)_m, and we know that if we iterate over m (an entrywise sequence), this is bounded and has a convergent subsequence that converges to some (x*_n)
define x* = (x*_1, x*_2, …)
try to show that x* is the limit of the sequence of sequences, so for any epsilon you can find a large enough index such that all following sequences are within the epsilon neighborhood of x*
pearl pondBOT
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hoary meadow
we have a sequence of sequences (x_n)_m, and we know that if we iterate over m (...
but isnt this still with the standard metric?
pearl pondBOT
but isnt this still with the standard metric?
limber nimbus
but isnt this still with the standard metric?
yeah, what i mean is that
you should first show that a pointwise limit exists (which would be a sequence) and is in the space
then you should show that that sequence is also the limit under the broader sup metrix
the existence of the pointwise limit comes from the sequence of sequences being cauchy in any given index wrt the standard metric on R
then, you need to show that each index converges to its target "uniformly"
i didnt give you any insight on that last bit, but constructing the limit is an important first step
to do the last part, you basically want to show that
$\forall \epsilon > 0, \exists N ; \forall n > N, \forall k, |x^n_k - x^L_k| < \epsilon/2$ where $(x^L_k)$ is the pointwise limit of $(x^n_k)$ as $n \to \infty$
jolly parrotBOT
snowflake
limber nimbus
(the k is moving along a given sequence, the n is moving across the sequences)
if you can show that epsilon/2 inequality, then the sup norm $d((x^n)_k, (x^L)_k)$ will be $\leq \epsilon / 2 < \epsilon$ which will prove your result
jolly parrotBOT
snowflake
pearl pondBOT
@hoary meadow Has your question been resolved?
limber nimbus
a nice trick we can use here is that
$|x_k^n - x^L_k| = \lim_{m \to \infty} |x_k^n - x^m_k|$
for all $n$. If you pick a large enough n such that the Cauchy condition kicks in, then you can bound this limit
jolly parrotBOT
snowflake
hoary meadow
yeah, what i mean is that
you should first show that a pointwise limit exists (...
yes i did do this already
hoary meadow
**ashyboi**
using a bunch of squeeze theorems lol
limber nimbus
im not sure if you can fully avoid epsilons here but maybe
hoary meadow
my problem is now showing the uniform convergence part
limber nimbus
**snowflake**
this method isnt too bad though
hoary meadow
**snowflake**
the cauchy condition?
limber nimbus
like, for all epsilon there exists N s.t. all pairs have a difference < epsilon
so if you pick N for epsilon/2, and just look at n > N, then that limit will be bounded
pearl pondBOT
hoary meadow
oh ye
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hoary meadow
d
limber nimbus
and then you've bounded the LHS, essentially for all k, which means you can bound the sup norm of x^n - x^L for n > N
meaning the sequence of sequences converges in the metric
pearl pondBOT
@hoary meadow Has your question been resolved?
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minor dock
i have trouble finding a double integration questions where we need to change the order and find the region, i found the limits correctly and changed the order but it seems like i am making a mistake in the integration part can you help me figure where i am going wrong
pearl pondBOT
i have trouble finding a double integration questions where we need to change th...
@minor dock Has your question been resolved?
minor dock
<@&286206848099549185>
compact ridge
,w (integrate from 0 to 3 (integrate from 1 to sqrt(4 - y) (x + y) dx) dy)
jolly parrotBOT
minor dock
exactly the answer is 241/60
jolly parrotBOT
south
minor dock
?
compact ridge
it's $\frac{16 - 8x^2 + x^4}{2} = 8 - 4x^2 + \frac{x^4}{2}$
jolly parrotBOT
south
compact ridge
I can't read that step you did
minor dock
I can't read that step you did
compact ridge
no, like your numbers are squiggles and when you blot them out, it's hard to read
minor dock
Hmm
compact ridge
like I could not see that was a 2x^2
but anyway after you integrated, but before you substituted is correct
it's an arithmetic error
minor dock
Yea sry i was doing it roughly
minor dock
it's an arithmetic error
Yea i gues i wil. just redo it again
compact ridge
but anyway after you integrated, but before you substituted is correct
,w f(x) = 2x^2 - x^4 /4 + 8x + x^5/10 - 4x^3 / 3, find f(2)-f(1)
jolly parrotBOT
minor dock
What
compact ridge
this is how you can check your work yourself
pearl pondBOT
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Available help channel!
stoic imp
pearl pondBOT
jolly parrotBOT
Renato
tropic saddle
what have you tried
pearl pondBOT
@stoic imp Has your question been resolved?
stoic imp
what have you tried
tropic saddle
p | 95, not the other way around
and from there you have another finite list of primes
stoic imp
and from there you have another finite list of primes
care to elaborate
stoic imp
p | 95, not the other way around
what would be the definition of divisibility?
stoic imp
and from there you have another finite list of primes
gcd(36,p) = 1 ==> p | 95
gcd(36,p) =/= 1 => p = 2
tropic saddle
well how many primes divide 95
stoic imp
and from there you have another finite list of primes
in general if p divides 95 then what can i same about 95
stoic imp
well how many primes divide 95
i would need to know the primes that divide this
tropic saddle
95 is a known small number
you can say everything about it
a 5th grader can tell you those prime numbers that divide it
stoic imp
help please
stoic imp
95 = 5x19
stoic imp
95 is very obviously divisible by 5
gcd(36,p) = 1 ==> p | 95 => p = 5 or p = 19
gcd(36,p) =/= 1 => p = 2
is that it
stoic imp
p | 95, not the other way around
i almost got it by myself, i just needed to push a little harder and remember my defs correctly
no?
tropic saddle
yes
stoic imp
help with 2
tropic saddle
what do you not understand about it
tropic saddle
how do divisors of 3^200 look like
stoic imp
who knows
tropic saddle
can they have a 7 in their prime factorization
stoic imp
how do divisors of 3^200 look like
prime divisors?
prime divisors is only 3 by euclids lemma
i might be tripping
stoic imp
how do divisors of 3^200 look like
multiples of 3
stoic imp
so only 3^k
how do you know that
tropic saddle
so you are asked to solve 3^k = 82 mod 119
stoic imp
euclids?
tropic saddle
what else should it be
any other thing would have another prime divisor
any multiple of 3 that is not a power of 3 has some other prime divisor other than 3
stoic imp
for me sounds like euclids lemma applied 200 times
tropic saddle
so?
stoic imp
i am trying to understand the rationale
tropic saddle
if you are only allowed to use the prime divisor 3 to build a divisor, then how should you end up with anything other than a power of 3
you can only multiply 3s together
stoic imp
ok, then what
tropic saddle
you cannot involve any other prime
stoic imp
chinese theorem
tropic saddle
so you are asked to solve 3^k = 82 mod 119
.
stoic imp
and little fermat
tropic saddle
you said it yourself
what do you want from me
CRT
you have to think about problems for more than 2 seconds before crying for help
stoic imp
how do i solve this non linear diophantine
stoic imp
you have to think about problems for more than 2 seconds before crying for help
i am trying my best dude, I'm just bad
tropic saddle
crying for help without even trying to think about the problem is not "trying your best"
stoic imp
i am trying but am stuck
tropic saddle
CRT says: instead of solving 3^k = 82 mod 119, you can instead solve 3^k=82 mod a_i for smaller numbers a_i
thats easier
you can find k literally just by repeatedly multiplying by 3
stoic imp
in order to use crt i need linear diophantine
tropic saddle
thats not what CRT is saying
CRT says x = something mod 119 is equivalent to several equations of the form x=something mod a_i for smaller numbers a_i
x is 3^k
stoic imp
so i have to apply crt on X = 82 (mod 119)?
i guess i can do that
the problem for me is this exponents give me trouble
stoic imp
CRT says x = something mod 119 is equivalent to several equations of the form x=...
what?
tropic saddle
3^k = 82 mod 119 is equivalent to 3^k = 5 mod 7 and 3^k = 14 mod 17
thats the result you want
everything you did afterwards is irrelevant
solving 3^k = 5 mod 7 can be done by hand
so can 3^k=14 mod 17
stoic imp
stoic imp
3^k = 82 mod 119 is equivalent to 3^k = 5 mod 7 and 3^k = 14 mod 17
i am still in the same spot because i still have a non linear diophantine, just that now i got 2 of them
stoic imp
how do you know the exponent k is in Z/7Z
tropic saddle
I didnt say that
stoic imp
you are implying i try all the possible k? there's like a million possibilities
tropic saddle
literally just compute 3^1, 3^2, 3^3, 3^4, ... mod 7
and do this
stoic imp
ok so the idea is that i try a couple k values until i get either 0,1,2,3,4,5,6 as the remainder under mod 7
then what do i do with those k?
rough forge
-# Hanako lurk
stoic imp
dude
i tried from 3^0 to 3^7 still no remainder 5
there must be a better way of solving this non linear diophantine
can we apply fermat little theorem and separate into cases?
right? since we already separated the congruence in 2 systems with pairwise prime moduli
stoic imp
really?
this is why i hate remainder tables
my approach though is not too far off no?
tropic saddle
oh no, one digit multiplications
stoic imp
like using little fermat
tropic saddle
little fermat will come up in a little bit, yes
stoic imp
oh no, one digit multiplications
there's multiple ways of messing up and getting to false conclusions
tropic saddle
if you cant multiply one digit numbers accurately then thats just a skill issue
stoic imp
there is a remainder 5 in those
which one
tropic saddle
show your work
stoic imp
stoic imp
how are you getting 4*3=9
4×3=4
tropic saddle
dude...
stoic imp
27 = 6 (mod 7)
3^3 is correct
tropic saddle
you had 9 in the k=5 column and 1 in the k=6 column
the only conclusion I can draw from that is that you think 9*3=1
stoic imp
5x3 = 15 = 1 (mod 7)
3x6 = 18 = 4 (mod 7)
ok let's do the table again because i noticed a ton of mistakes
wait, this is not equivalent
3^4 is not 3x4
oh my
let me re do this
@tropic saddle
k = 5 leaves remainder 5
stoic imp
who knows
stoic imp
is that the only k?
you want me to brute force some ks and try to find a correlation between them
stoic imp
yes
3^5 is the first rem 5
3^(6k) is just multiplying by 1 under mod 7
so the formula is 3^{5 + 6k}
@tropic saddle
tropic saddle
you noticed it repeats every 6
that is the same as working mod 6
you used fermat when saying 3^(6k) is just 1
stoic imp
you used fermat when saying 3^(6k) is just 1
me? when
tropic saddle
"assuming"
stoic imp
inverse exists so no problem there but
rough forge
"assuming"
it could be
stoic imp
so 3^k=5 mod 7 means k=5 mod 6
actually this is really clever
i get it now, i appreciate the help, this exercise has been incredibly fruitful for me
pearl pondBOT
@stoic imp Has your question been resolved?
opaque wave
pearl pondBOT
warm tiger
Do you notice something about the second determinant?
opaque wave
ye
swift narwhal
check if first and second are somehow skew symmetric or nah
it gets surprisingly simple
warm tiger
Second is skew symmetric
With odd order
So its determinant is 0
g(theta) should be |sin theta| + |cos theta|
And looks like you haven't posted the full question
warm tiger
p(x) = k(x-1)(x-√2)
You can solve for k pretty easily
Now just plug in every value from the option
@opaque wave
swift narwhal
where did u find this question btw
warm tiger
where did u find this question btw
It's from a competitive exam in India
JEE Advanced
swift narwhal
pyq?
warm tiger
Yep
warm tiger
Asked in 2022 if i remember correctly
swift narwhal
damn
warm tiger
I gave the exam myself this year so i remember all this
warm tiger
Thanks but this rank sucks honestly
swift narwhal
nah