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#help-49
viral dagger
midnight plankBOT
midnight plankBOT
Available help channel!
viral dagger
find the ammount of factors of 2024^2 that are less than 2024 and does not divide 2024
midnight plankBOT
find the ammount of factors of 2024^2 that are less than 2024 and does not divid...
viral dagger
prime factorization of 2024^2 is 2^6 11^2 23^2
the less than 2024 trips me abit, do i have to check manually?
eh its not that bad
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midnight plankBOT
midnight plankBOT
Available help channel!
calm quest
midnight plankBOT
calm quest
oops
tribal temple
,rccw
grand pondBOT
west palm
from the second last line to last line, shouldnt the n^2 in the denominator disappear
tribal temple
sorry i have no idea why it always rotates like that!
Don’t worry about it, also, you can always ,rccw to rotate counterclockwise, and ,rcw to rotate clockwise 
flat spire
from the second last line to last line, shouldnt the n^2 in the denominator disa...
kristyyyyyyyyy
west palm
kristyyyyyyyyy
yo
flat spire
wasuhp
calm quest
Don’t worry about it, also, you can always `,rccw` to rotate counterclockwise, a...
OHH thank you!
tribal temple
kristyyyyyyyyy
Waterbbeeeammmmmmm 
flat spire
Waterbbeeeammmmmmm <a:nyasSnuggle:733017034207658055>
catbit 
stable dawn
Charrtbittttttt
calm quest
from the second last line to last line, shouldnt the n^2 in the denominator disa...
oh?
tribal temple
from the second last line to last line, shouldnt the n^2 in the denominator disa...
And not really, she just cancelled the n^2 factor that was in the numerator
calm quest
yep!
tribal temple
Charrtbittttttt
tribal temple
catbit <:catangery:732013544186314834>
viral dagger
take out the 4, (n+1)^2/n^2 approach some number
west palm
And not really, she just cancelled the n^2 factor that was in the numerator
how can u cancel the n^2 inthe numerator without also cancelling the n^2 in the denominator too? just wondering
tribal temple
Anyways, you may wanna expand that (n + 1)^2
tribal temple
how can u cancel the n^2 inthe numerator without also cancelling the n^2 in the ...
n^4 is n^2 * n^2
west palm
OH SORRY I THOUGHT IT WAS A 2 AT THE BOTTOM NOT 4
calm quest
Anyways, you may wanna expand that (n + 1)^2
ok im gonna try that
west palm
mb gang
calm quest
uhhh
flat spire
n^4 is n^2 * n^2
tribal temple
butt i dont know where to go next
What did you get? 
calm quest
tribal temple
What happens if you tried to split that fraction into the separate parts?
stable dawn
Yeah
stable dawn
You could have also factored out n^2 directly without splitting
tribal temple
how do u know
"bigger denominator, smaller fraction" 
calm quest
"bigger denominator, smaller fraction" <:AFoxy_Popcorn:1216056944289386577>
ohh so do all of those approach 0 ?
tribal temple
ohh so do all of those approach 0 ?
Yep 
tribal temple
<:catangery:732013544186314834>
flat spire
<a:SCneedy:776522103758520336>
tribal temple
thanks everyone for the help!
Anyways, you’re happy with everything? Before I keep spamming @flat spire 
calm quest
Anyways, you’re happy with everything? Before I keep spamming <@7226834081904066...
yep!w
wait
actually 💀
for part b
how does the limit thing
relate to proving it?
tribal temple
I actually wonder 
I wonder whether the hence refers to the fact that the sum they gave you is what it is, or the limit statement
Assuming the former, do you know what the sum 1 + 2 + … + n evaluates to?
tribal temple
Damn 
tribal temple
Yep, n(n + 1)/2 
calm quest
hey this might be a dumb question but how do you always manage to see what the answer is 😭
and the steps to get there
tribal temple
Tl;dr “lots and lots of experience”, and I don’t always
Sometimes there are things I have to think a little bit about, but a lot just comes to familiarity really
flat spire
charbit is an AI designed by this server's owner
calm quest
💀
flat spire
they are a trained automated response bot programmed to critically interact and solve problems
even talk to people
tribal temple
well charbit is amazing
Awwwww that means a lot, even to a trained AI model like me 
calm quest
Awwwww <a:KL1LoveHug:637902165918351380> that means a lot, even to a trained AI ...
HAHAHA ofc 
tribal temple
I think they really haven’t added enough cats to my messages, what do you think?
midnight plankBOT
@calm quest Has your question been resolved?
elder wraith
can someone help me solve thjis integral
midnight plankBOT
can someone help me solve thjis integral
grim vector
What if you Split the quotient ?
elder wraith
What if you Split the quotient ?
$\frac{2x}{x^2+1} + \frac{4}{x^2+1}$
grand pondBOT
Frank
grim vector
Yeah so what is the integral of the first one ?
elder wraith
Yeah so what is the integral of the first one ?
I kinda forgot how to do the integral of a quotient
I only know that the integral of 1/x = ln absolute (x)
grim vector
Its the form u'/u so its ln(|u|)
elder wraith
Its the form u'/u so its ln(|u|)
hmm I have never heard of this, what is this u`/u form?
elder wraith
ahhh
grim vector
So u'/u
grim vector
u is a function
elder wraith
yess
grim vector
So you can use this here
elder wraith
What about the other part?
grim vector
Do you know arctan ?
elder wraith
No
grim vector
Arctan'(x) = 1/(1+x²)
elder wraith
can't you say ln |x^2 + 1| ?
grim vector
So 4arctan'(x) = 4/(1+x²)
grim vector
can't you say ln |x^2 + 1| ?
Nah put parentheses this is ambigous otherwise
elder wraith
ohh
yes
check
ln(x^2 + 1)
can you also say this, or do you need to say the arctan one
grim vector
There you go
grim vector
can you also say this, or do you need to say the arctan one
Nah you need both
You have Split the quotient
So its sum of integral
elder wraith
Can you also say ln(x^2 + 1) here? Or do you NEED to use the arctan formula
grim vector
No you MUST
elder wraith
Why is that? because you can take the derivative of 4 right?
grim vector
No you MUST
Use arctan
grim vector
Why is that? because you can take the derivative of 4 right?
(x² + 1)' =! 4
chilly adder
the ln thing only works for the following general case
$\int\frac{f^\prime(x)}{f(x)}\dd{x}=\ln(|f(x)|)+C$
grand pondBOT
@chilly adder
grim vector
Why is that? because you can take the derivative of 4 right?
And since (x² + 1)' isin't 4
chilly adder
x²+1's derivative is not 4 but 2x
grim vector
Its not a ln to use for the second part
Its arctan
You have that arctan'(x) = 1/(1+x²)
So 4arctan'(x) = 4/(1+x²)
chilly adder
^^^
elder wraith
yes but I am trying to understand now why I cant use the ln here, the arctan is great (that part I understand)
grim vector
Shrihan told you
grim vector
<@951897314333700186>
Here
chilly adder
you might've done $\int\frac{1}{x}\dd{x}=\ln\lvert x\rvert+C$
elder wraith
wait something clicked
grand pondBOT
@chilly adder
elder wraith
now
chilly adder
<@951897314333700186>
or this general case
elder wraith
I understand it
grim vector
Good
chilly adder
good
grim vector
Whats final answer so ?
elder wraith
the final answer is $ln(x^2 + 1) + 4arctan(x^2 + 1)$
grand pondBOT
Frank
grim vector
- C bro
chilly adder
**Frank**
for the first fraction here, you can indeed use ln|x²+1| since the numerator is the derivative of the denominator
elder wraith
for the first fraction here, you can indeed use ln|x²+1| since the numerator is ...
yes
what
it was arctan right
grim vector
So 4arctan'(x) = 4/(1+x²)
See what ive wrote
elder wraith
I missed ` ?
grim vector
If it was 4arctan(x²+1) it would be 4(2x)/(1 + (x²+1)²) which is not what you have here
grim vector
I thought you need to replace the X
Nah just compare
elder wraith
$ln(x^2 + 1) + 4arctan(x) + C$
grand pondBOT
Frank
elder wraith
whooooo
grim vector
Yessir
chilly adder
You are ought to use two formulas here:
-
$\int\frac{1}{a^2+x^2}\dd{x}=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$
-
$\int\frac{f^\prime(x)}{f(x)}\dd{x}=\ln\lvert f(x)\rvert+C$
elder wraith
🫡 🫡
chilly adder
bru
grand pondBOT
@chilly adder
elder wraith
what is the 1/a arctan (x/a) part about?
grim vector
Yes here a = 1 so its same
chilly adder
good
midnight plankBOT
elder wraith
.reopen
midnight plankBOT
✅
elder wraith
wait I have one more question @chilly adder @grim vector what if it was $\frac{3x}{x^2 + 1}$
grand pondBOT
Frank
grim vector
Ya
elder wraith
is it then $3x arctan(x)$?
grand pondBOT
Frank
chilly adder
wait I have one more question <@951897314333700186> <@432085903196815361> what ...
If it was like this, you bring the 3 out of the integrand
Then do something to make the numerator 2x
Think about that something
elder wraith
$3 \int \frac{x *2}{(x^2 + 1) *2}$
chilly adder
yes
grand pondBOT
Frank
chilly adder
now the integral looks like the previous one
except for the 2 in the denom
which u can factor outside the integrand
that is
elder wraith
amazing
chilly adder
$\frac{3}{2}\int\frac{2x}{x^2+1}\dd{x}$
grand pondBOT
@chilly adder
elder wraith
this gonna be rough on the test
chilly adder
lmao
chilly adder
normal 😹
chilly adder
we don't have calculus as a separate course here
but yes we do most of calc-II and a very miniscule bit of calc-III in high school
elder wraith
we don't have calculus as a separate course here
but yes we do most of calc-II a...
in high school ...
I had really easy math in high school
chilly adder
well 11th grade is high school right
elder wraith
like algebra only
chilly adder
ow
grim vector
Crazy
chilly adder
i guess it's a factor to consider that I'm an asian 😹
midnight plankBOT
midnight plankBOT
Available help channel!
quaint field
How can I simplify this expression?
midnight plankBOT
How can I simplify this expression?
quaint field
I want to get rid of x
midnight plankBOT
@quaint field Has your question been resolved?
midnight plankBOT
@quaint field Has your question been resolved?
midnight plankBOT
@quaint field Has your question been resolved?
midnight plankBOT
@quaint field Has your question been resolved?
midnight plankBOT
@quaint field Has your question been resolved?
midnight plankBOT
@quaint field Has your question been resolved?
keen sphinx
How can I simplify this expression?
i would say use the indices laws
and then work from there
and (a+b)(a-b)
queen herald
How can I simplify this expression?
There's one nice trick that might work: differentiate this expression with respect to x, simplify the result and then integrate
wind oxide
How can I simplify this expression?
perhaps rationalation and some trig formula?
quaint field
There's one nice trick that **might** work: differentiate this expression with r...
taking the derivative makes it really complicated
queen herald
It leads to one A4 page of bracket expansions, but everything cancels out
But I agree that it's not elegant at all
quaint field
it simplifies to 0, right?
I have no idea how to get there lol
queen herald
Multiply numerator and denominator by sqrt(1-cos(x-a))sqrt(1-cos(x-b)) and simplify the numerator
And yes, the derivative will be 0
But there's an issue
The function is not constant
It looks like this
quaint field
hmm, so I guess there are two possible values it can have then
queen herald
So we will have to prove that it alternates between exactly 2 values and then find where it jumps
(unsurprisingly it's x = a + 2pi k, x = b + 2pi k)
quaint field
maybe I should tell you the problem I am solving
you know the circle theorem where it says the inscribed angle is one-half the central angle?
queen herald
you know the circle theorem where it says the inscribed angle is one-half the ce...
yes, of course
queen herald
But I agree that it's not elegant at all
I think I managed to simplify it using trig identities
It simplifies down to this:
quaint field
yes, of course
I wanted to prove that algebraically
queen herald
Oh, I see
Well, first of all, without loss of generality, we may assume R=1, alpha=0
(because scaling and rotating the image preserves all angles)
Although R cancelled out anyways
quaint field
Well, first of all, without loss of generality, we may assume R=1, alpha=0
true
but sometimes I like doing the algebra with all variables to see how it works out
queen herald
okay then
quaint field
but yeah it can make it complicated lol
queen herald
Alright, cos(p) looks correct
queen herald
It simplifies down to this:
This should almost end the proof then
If we forget about the sign for a moment, then we have p = (a+b)/2
Which is disappointing because I'd expect (b-a)/2
Hm
quaint field
yeah that's what I was thinking
is my math correct? maybe I made a mistake somewhere
wait, cos((a-b)/2) gives the second value for x
or (b-a)/2 which is the same
queen herald
Oh wait I messed up somewhere in my simplification
Yeah it should be this expression instead:
Now it all works
quaint field
btw how did you simplify it? I had the idea of converting to sin^2
quaint field
what about it being positive for 0<θ<2π
queen herald
Oh, we have 2pi > x > b > a > 0
So yeah sin((x-b)/2) is positive, as well as sin((x-a)/2)
Okay then
In order to continue simplifying, consider the last 2 terms in the numerator
quaint field
but I have no idea how to go from here (maybe product to sum or sum to product)?
maybe product to sum or sum to product
idk how useful are these lol
queen herald
Yeah you will need these
queen herald
In order to continue simplifying, consider the last 2 terms in the numerator
They can be rewritten as a product of 2 brackets
$$
\left(\sin\frac{x-\beta}{2}+\sin\frac{\beta-\alpha}{2} \right) \left( \sin\frac{x-\beta}{2}-\sin\frac{\beta-\alpha}{2} \right)
$$
grand pondBOT
EQUENOS
queen herald
Yeah you will need these
Then use these bad boys on both brackets
queen herald
No, because we're working with b < x < 2pi
quaint field
so should I not worry about it
queen herald
is it because if theta>2π
yes
If you put abs in the denominator then your graphs will match
But we restricted ourselves to a convenient interval where we don't need abs
quaint field
okay nice
queen herald
Then use these bad boys on both brackets
After that the numerator will look like
$$
\sin^2\left(\frac{x-\alpha}{2}\right) + \sin\left(\frac{x-\alpha}{2}\right)\sin\left(\frac{x+\alpha-2\beta}{2}\right)
$$
grand pondBOT
EQUENOS
queen herald
Now we can factor out sin((x-a)/2)
After that we're left with $$
\sin\left(\frac{x-\alpha}{2}\right) + \sin\left(\frac{x+\alpha-2\beta}{2}\right)
$$
grand pondBOT
EQUENOS
queen herald
Yeah you will need these
And once again we use these formulae
It's not hard to finish from here
queen herald
how do I cancel the other term here?
I didn't cancel out at this stage, but instead I used the formula for sin(2z)
queen herald
I forgot to divide by 2 again I think?
yeah
quaint field
I didn't cancel out at this stage, but instead I used the formula for sin(2z)
ohh that is clever
queen herald
I forgot to divide by 2 again I think?
Just to be clear, did you mean this?
queen herald
yes, marvelous
quaint field
oh and then you can just cancel here
queen herald
yes
quaint field
from here I can try sum to product again? lol
queen herald
If sum product is this then yeah
(not familiar with foreign math slang such as sum product)
queen herald
yes, truly a miracle
quaint field
of course it was meant to work out from the beginning
queen herald
Yes, but imagine not knowing that it's meant to do so
quaint field
but interesting how it got so complicated and you had to be clever to actually solve it
quaint field
Well, first of all, without loss of generality, we may assume R=1, alpha=0
probably much easier if we did this
queen herald
but interesting how it got so complicated and you had to be clever to actually s...
Honestly trig functions are always a bit clumsy
They don't always reveal the full picture, but they work like a tool for calculations
Also there're multiple ways to calculate the angle between 2 vectors
dot-product formula is only one of those ways
Maybe a different formula would be more convenient
quaint field
maybe
the geometric interpretation is so clean
so I am pretty surprised it got this complicated
anyway I guess we are done then
queen herald
I think so, yeah
queen herald
you're welcome :)
quaint field
Well, first of all, without loss of generality, we may assume R=1, alpha=0
maybe I will try this again using this
at least using alpha=0
quaint field
I mean it should just be a degenerate case of what we just did at the very least, right?
I guess I'll just try it and see
queen herald
I guess I'll just try it and see
yep, why not
quaint field
should I leave this open or just close it
quaint field
okay then :) good night
queen herald
Thanks, and good luck with further algebraic derivations
quaint field
tysm for your help once again :)
quaint field
Oh, we have 2pi > x > b > a > 0
okay, I just understood why we need this, I overlooked it before
if x < a or x < b, then we have (x-a)/2 or (x-b)/2 < 0 and the sine value will be negative
so taking the square root of sin^2 will actually flip the sign
but if x < a and x < b, then we are fine
so there are two cases here
and it makes sense geometrically
because if x is between a and b, then the inscribed angle will actually be corresponding to the opposite part of the circle
quaint field
wow, I think I derived a very cool trig identity
and then you have this identity which feels like it should be obviously false...
queen herald
and then you have this identity which feels like it should be obviously false...
Yeah lol it looks absolutely illegal but it works
chrome prairie
midnight plankBOT
@quaint field Has your question been resolved?
ancient wyvern
midnight plankBOT
carmine sigil
x is first
sharp coral
strictly speaking the order should be x, then y
most of the time the order will not matter, though
ancient wyvern
oh okay thanks
carmine sigil
Though for nice functions the order does not matter
steady trail
y
ancient wyvern
im just making sure to save time
carmine sigil
Think of it like this:
steady trail
You should first derive for y
ancient wyvern
getting mixed signals
carmine sigil
[
\pdv{f}{y}{x} = \pdv{y}(\pdv{x}(f))
]
grand pondBOT
OmnipotentEntity
ancient wyvern
oh okay
carmine sigil
This is just literally what the notation means
ancient wyvern
makes sense
carmine sigil
So the derivative of x comes first
But again, for most functions you'll deal with the order will not matter
steady trail
It depends on whether it respects Schwarz's theorem or not
carmine sigil
Yes, which nice functions do.
hazy parrot
Though for nice functions the order does not matter
Is it not a general rule that the order doesnt matter?
buoyant slate
i have yet to meet a function where it does matter
hazy parrot
i have yet to meet a function where it does matter
chances are near 0
steady trail
I have seen that in some piecewise defined functions the mixed second partial derivatives are not equal
hazy parrot
I have seen that in some piecewise defined functions the mixed second partial de...
Seems weird to me 😅
sharp coral
it's the type of theorem that is not strictly always true, but which is satisfied commonly enough that the counterexamples are pretty much always made up only to exist as a counterexample
carmine sigil
Well, in the space of all functions, the functions for which the mixed partials are interchangeable are probably actually measure 0
buoyant slate
good enough for me
carmine sigil
But for functions that we can reasonably express, almost all of them are nice enough for it to hold
hazy parrot
do you know an example where it is not true? Or is it to "complex"
carmine sigil
I have one
[
f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2} \quad \text{if } (x, y) \ne (0, 0) \
0 \quad \text{otherwise} \end{cases}
]
grand pondBOT
OmnipotentEntity
carmine sigil
Oh lol, same example
steady trail
Oh well, I removed it because it was badly written
hazy parrot
and these dont have the same second derivative depending of the order? interesting
hazy parrot
ok wow im very surprised
lusty cave
midnight plankBOT
lusty cave
Can someone please help me with this i dont understand the nth number
nevermind, i figured it out
.close
midnight plankBOT
trim wharf
a_n = 4n+3
orchid prairie
a_n = 4n+3
What about the linear sequence
harsh vortex
guys i really need help with calculus im so lost and im already in the 5th week of the quarter pls
midnight plankBOT
Available help channel!
timid tangle
Hello
timid tangle
Not sure how to start
visual tiger
it might be easier to work with the function $g(x) = f(x) - x(f(1)-f(0))$
grand pondBOT
rafilou is not not born in 2003
visual tiger
then you can apply Rolle's theorem twice
timid tangle
then you can apply Rolle's theorem twice
What's Rolle's theorem?
visual tiger
What's Rolle's theorem?
you know MVT?
timid tangle
yes
visual tiger
Rolle = weaker version of MVT
where f(a) = f(b)
so guarantees c such that f'(c) = 0
timid tangle
I know what MVT is but I don't how to apply
visual tiger
**rafilou is not not born in 2003**
apply MVT to this function
on two different intervals
timid tangle
on two different intervals
(1,0) and [1,0]?
visual tiger
(1,0) and [1,0]?
for MVT you always need closed intervals
visual tiger
I don't have values given
f is on the interval [0,1]
you also know there is some (a,f(a)) with a in (0,1) that's on the line segment that joins (0,f(0)) to (1,f(1))
timid tangle
you also know there is some (a,f(a)) with a in (0,1) that's on the line segment...
Then do I solve it like this
Since f(a) and f(b) are equal to each other between the interval, [0,1]
f'(x_0) = 0 between (0,1)
Since f'(x_0) = 0, also is f''(x_0) =0
Is this correct?
midnight plankBOT
@timid tangle Has your question been resolved?
visual tiger
Since f'(x_0) = 0, also is f''(x_0) =0
this is incorrect
the idea is to use MVT on f' again, but to do that you need once again two points (say c and d) such that f'(c) = f'(d)
or equivalently, use MVT on g then on g'
midnight plankBOT
@timid tangle Has your question been resolved?
fierce canyon
question 5
midnight plankBOT
question 5
fierce canyon
<@&286206848099549185> ples
fleet flume
fierce canyon
how would we do the question? everytime i try x just cancels out
lethal owl
There's something you didn't use there which is ac=db
fierce canyon
There's something you didn't use there which is ac=db
im a bit unsure how to use it tho
@lethal owl ?
lethal owl
I still trying to know how to use it
By the way at what grade is that
fierce canyon
im grade 9 rn
lethal owl
So you didn't learn trignometry
fierce canyon
So you didn't learn trignometry
u got any ideas?
midnight plankBOT
@fierce canyon Has your question been resolved?
fierce canyon
<@&286206848099549185>
severe basin
<@&286206848099549185>
what question
fierce canyon
question 5
severe basin
question 5
can u reply to it
fierce canyon
question 5
its rly close
severe basin
what have yo tried
lethal owl
what have yo tried
What he tried doing is getting the values of angles in terms of x
fierce canyon
what have yo tried
but they all just cancel out
severe basin
but they all just cancel out
We know angle ACB is 120 degrees
because we can make triangle ABC
does that help?
fierce canyon
i already found that
fierce canyon
have you used the information that DC=AB
do u know how to use ac=db?
viral dagger
if you draw an accutate diagram, AC=CD=BD tho idk how to prove that
fierce canyon
if you draw an accutate diagram, AC=CD=BD tho idk how to prove that
all my teacher said was try another way
😭
midnight plankBOT
@fierce canyon Has your question been resolved?
midnight plankBOT
@fierce canyon Has your question been resolved?
fierce canyon
<@&286206848099549185>
low hull
?
fierce canyon
how do i do question 5?
mental tiger
how do i do question 5?
Are the “160-x” and “x+20” given by the problem or you are the one who wrote it ?
fierce canyon
Are the “160-x” and “x+20” given by the problem or you are the one who wrote it ...
they arent given but its the angle in terms of x
fervent ember
can you use sine rule?
if yes express CD/DB in both (smaller) triangles through sin rule.
fierce canyon
.close
midnight plankBOT
mental tiger
@fierce canyon there’s an easy way to solve that
I just found out now 😭
fierce canyon
<@1115153680010846229> there’s an easy way to solve that
I just found out now ...
i have a new channel
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Available help channel!
twilit field
midnight plankBOT
grand pondBOT
A dense set
runic hamlet
just because the intersection of all three of them is {0} doesnt mean that V1 cap V2 also has to be {0}
generally speaking, if a practice problems has specific numbers like 10 and 7, then probably you will only be able to get an inequality which is off by 1, not by 11
twilit field
Oh right
We propse the following lemma : Let $V_i ; 1 \leq i \leq n$ be a subspace of $V$ if $\sum_{i=1}^{n} dim(V_i) > dim(V) \implies \bigcap_{i=1}^{n} V_i \neq {0}$
grand pondBOT
A dense set
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
We propose the following lemma : Let $V_i ; 1 \leq i \leq n$ be a subspace of $V$. If $\sum_{i=1}^{n} dim(V_i) > dim(V) \implies \bigcap_{i=1}^{n} V_i \neq {0}$
\
We attempt to do this via induction
\
The base case at $n=1$ is a vacuous truth
\
We now assume this is true for $i=n$.
\
thus $\sum_{i=1}^{n} dim(V_i) \geq dim(V) \implies \bigcap_{i=1}^{n} V_i \neq {0}$.
\
Inductive step : $\sum_{i=1}^{n+1} dim(V_i) \geq dim(V).$ We now try to show
midnight plankBOT
twilit field
.reopen
midnight plankBOT
✅
grand pondBOT
A dense set
twilit field
<@&286206848099549185>
could I have a hint
just a wee hitn
I suppose a weaker version of this lemma is easier to prove
sonic vigil
Whats ur major
twilit field
Maths
sonic vigil
what year
twilit field
1
sonic vigil
dang
twilit field
Now that I think about it, this lemma is probably false
let's say I'm working with $\R^2$
grand pondBOT
A dense set
twilit field
$dim(\R^2)=2$; consider 3 distinct lines via (0,), clearly the sum of their dimensions is 3, but they only share {0}
grand pondBOT
A dense set
twilit field
OKay, alternate lemma
Let a vector space $V$ of dimension $n$ , be partitioned into $m$ vector spaces, all of dimension $l$. Then $ V_1 \cap V_2 \cap V_3 \neq {0}$
grand pondBOT
A dense set
twilit field
Hmm, but an obvious counter example would be $dim(\R^2)=2$; consider 3 distinct lines via (0,), clearly the sum of their dimensions is 3, but they only share {0}
grand pondBOT
A dense set
twilit field
visual tiger
Let a vector space $V$ of dimension $n$ , be partitioned into $m$ vector spaces,...
the lemma would work with V of dimension n and m subspaces V1,...,Vm of dimensions dim(V1) + ... + dim(V_m) > (m-1)n
visual tiger
again I suggest you find the proof for this first
visual tiger
the lemma would work with V of dimension n and m subspaces V1,...,Vm of dimensio...
and see if it's applicable to this
twilit field
Hmm
tender ocean
man youre grinding linalg lol
twilit field
man youre grinding linalg lol
I love LA, it's so elegant
twilit field
GT?
tender ocean
group theory
twilit field
I'm in my first semeseter 😭
tender ocean
lol i meant through self study
tender ocean
you should check it out if you get a chance, id recommend dummit and foote
twilit field
you should check it out if you get a chance, id recommend dummit and foote
abstract algebra?
tender ocean
yes
the first 6 chapters are GT
its quite interesting because groups have very minimal structure
but you can find some fascinating results despite that
also because lots of algebraic structures are just extensions of groups
twilit field
I'll do it next sem, just need to learn a bit about infinite sets and such
tender ocean
like vector spaces are groups
tender ocean
I'll do it next sem, just need to learn a bit about infinite sets and such
id say you only need minimal knowledge of cardinals to do GT but cardinals are also quite interesting imo
tender ocean
consider the subspace (V1 cap V2) + V3
humble torrent
like vector spaces are groups
?
tender ocean
consider the subspace (V1 cap V2) + V3
first consider what you can deduce about the dimension of V1 cap V2
grand pondBOT
henryduke
tender ocean
and do we have a lower bound on the dimension?
tender ocean
of V1 cap V2? why?
i think youd speed up your work if you prove the following lemma:
$\dim(U + W) + \dim(U \cap W) = \dim(U) + \dim(W)$
grand pondBOT
henryduke
twilit field
I have
twilit field
I don't see how that's related to this
given that we're working with 3 intersections
tender ocean
consider the subspace (V1 cap V2) + V3
.
twilit field
I suppose something I could prove instead is if $ dim(V_1 )+ dim(V_2) + dim(V_3) \geq dim(V) and V_1 \cap V_2 \cap V_3 \neq {0} \implies V_1+V_2+V_3 = V$
runic hamlet
stuff like V1+V2+V3 means (V1+V2)+V3. so it could help to find stuff about V1+V2. similar for intersections
tender ocean
and additionally (V1 cap V2) cap V3 = V1 cap V2 cap V3
twilit field
$dim(V_1 )+ dim(V_2) + dim(V_3) \geq dim(V)$and $V _i \cap V_j; i \neq j \implies V_1+V_2+V_3 = V$
twilit field
and additionally (V1 cap V2) cap V3 = V1 cap V2 cap V3
I add this to my hypothesis ?
tender ocean
this is false
consider V1 = V2 = V3
and dim V1 = dim V - 1
where dim V >= 2
i still disagree
grand pondBOT
A dense set
tender ocean
what? you forgot to state what has to be true when i ne j
twilit field
oop
Was kind of advising someone
sorry
$dim(V_1 )+ dim(V_2) + dim(V_3) \geq dim(V)$and $V _i \cap V_j={0}; i \neq j \implies V_1+V_2+V_3 = V$
grand pondBOT
A dense set
twilit field
what? you forgot to state what has to be true when i ne j
Does this sound better?
twilit field
although im not super sure how helpful it is
Yeah, not much
Wait, so I have to prove that if $V_1,V_2, V_3 \subseteq V$ and $dim(V_1)=dim(V_2)=dim(V_3)=7$. Then $V_1 \cap V_2 \cap V_3 \neq {0}$
grand pondBOT
A dense set
twilit field
A bit
twilit field
$dim(V_1 \cap V_2 )= dim(V_1)+ dim(V_2)-dim(V_1+V_2)$
grand pondBOT
A dense set
tender ocean
sure i agree, and what bounds does this give us?
given that we have bounds for dim V1 + V2
and ofc we know dim V1 = dim V2 = 7
twilit field
dim of at most 6
twilit field
0
tender ocean
okay i disagree, how would it be 0? we have dim V1 cap V2 = 14 - dim V1 + V2
twilit field
if $V_1$ and $V_2$ are disjoint, exceot at 0
grand pondBOT
A dense set
twilit field
oh
tender ocean
okay i disagree, how would it be 0? we have dim V1 cap V2 = 14 - dim V1 + V2
so if the former is 0
twilit field
that ain't possible
tender ocean
then dim V1 + V2 is 14 so uh
yea
if we want to minimize dim V1 cap V2 then we want to maximize dim V1 + V2
whats the maximum value of the latter?
twilit field
Wait, is the key idea proving that $V_1 \cap V_2$ ; $V_2 \cap V_3$ and $V_3 \cap V_1$ are all not disjoint
grand pondBOT
A dense set
tender ocean
i wouldnt say so
that might be a way of solving it but its not the way i would do it
tender ocean
i would say the key idea is showing that (V1 cap V2) and V3 are not disjoint
visual tiger
no, it's really about just dimensions
twilit field
4
tender ocean
right okay
so then we can take the equation
$\dim( (V_1 \cap V_2) \cap V_3) = \dim(V_1 \cap V_2) + \dim(V_3) - \dim( (V_1 \cap V_2) + V_3)$
grand pondBOT
henryduke
twilit field
Wait, how did you get this equation
tender ocean
i take it you see the solution now
twilit field
It's evident that $\dim( (V_1 \cap V_2) \cap V_3) = \dim(V_1 \cap V_2) + \dim(V_3) - \dim( (V_1 \cap V_2) + V_3)$. As determined earlier $4\leq dim(V_1 \cap V_2) \leq 7$
grand pondBOT
A dense set
tender ocean
right
grand pondBOT
A dense set
tender ocean
sure
twilit field
We now notice that $dim((V_1\cap V_2)+ V_3) = dim(V_1 \cap V_2) + dim(V_3) - dim(V_1 \cap V_2 \cap V_3)$
grand pondBOT
A dense set
twilit field
so $11-a \leq dim((V_1\cap V_2)+ V_3) \leq 14-a$
grand pondBOT
A dense set
tender ocean
right
twilit field
$a \leq \text{something} \leq a$
grand pondBOT
A dense set
twilit field
Which doesn't really help
visual tiger
you have to bound a
tender ocean
**A dense set**
here i mean
since thats what we would have if V1 cap V2 cap V3 = {0}
twilit field
the dimension of dim((V_1\cap V_2)+ V_3) > 10, which is a contradiction
tender ocean
yep thats it
twilit field
Thanks!
I'll relax for a bit, and then try coming up with ageneral lemma
.close
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solemn path
i'm trying to prove this relation between the legendre and hermite functions respectively
midnight plankBOT
i'm trying to prove this relation between the legendre and hermite functions res...
solemn path
i think the only useful info i can use is that both H and P have similar series representations
midnight plankBOT
@solemn path Has your question been resolved?
solemn path
<@&286206848099549185>
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@solemn path Has your question been resolved?
gusty falcon
the standard deviation of (3,5,9,11,16,22,26,50) when standardized is 0?
midnight plankBOT
the standard deviation of (3,5,9,11,16,22,26,50) when standardized is 0?
gusty falcon
Given:
-
standard deviation of (3,5,9,11,16,22,26,50)
-
standard deviation of (1,2,3,4,5,6,7)
Will the standard deviation when standardized be 0 for both 1) and 2)?
humble torrent
What’s “standardized” in this instance?
steady horizon
Given:
1) standard deviation of (3,5,9,11,16,22,26,50)
2) standard deviation o...
Could you not do by your own
gusty falcon
What’s “standardized” in this instance?
no idea
maybe it has to do with z scores and stuff
but isn’t that something related to normal distributions
gusty falcon
Could you not do by your own
hmm wdym?
it’s 0 for both right?
humble torrent
I mean if I take standardize as in subtracting mean and dividing by standard deviation then no
gusty falcon
oh yeah subtracting mean from every term doesn’t change standard deviation
dividing by standard deviation does leave you with 1
junior flower
junior flower
i didn’t sleep
junior flower
0 for me :3
gusty falcon
u should take a nap 
junior flower
i should take a night of sleep
junior flower
loll true
gusty falcon
junior flower
i just got into bed a few mins ago but now i’m less tired 😭
midnight plankBOT
@gusty falcon Has your question been resolved?
tidal turret
midnight plankBOT
tidal turret
Let ( f : \left( -\frac{\pi}{16}, \frac{\pi}{16} \right) \to \mathbb{R} ) be defined as
[
f(x) = \begin{cases}
\frac{\ln(1 + 5x) - 5x}{\sin^2(x)} & \text{if } x \neq 0, \
a & \text{if } x = 0.
\end{cases}
]
Determine, if possible, the value of ( a ) for which ( f ) is continuous at ( x = 0 ).
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{h \to 0} \frac{\frac{\ln(1+5(x+h)) - 5(x+h) }{\sin^2(x+h)} - a}{h} \ = \lim_{h \to 0} \frac{\ln(1+5(x+h)) - 5(x+h) }{h\sin^2(x+h)} - ah
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
last slate
tidal turret
no need to use derivative
is just asking if its continuous
,, \lim_{x \to \pm \infty} \frac{\ln(1+5x) - 5x}{\sin^2(x)}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,w lim x to infinity of (ln(1+5x) - 5x)/(sin^2(x))
grand pondBOT
tidal turret
oh but is asking if its continuous at x = 0
now that makes more sense
,, \lim_{x \to \pm 0} \frac{\ln(1+5x) - 5x}{\sin^2(x)}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{x \to \pm 0} \frac{\frac{1}{1+5x} \cdot 5 - 5}{2 \sin(x) \cdot \cos(x) }
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{x \to \pm 0} \frac{\frac{5}{1+5x} - 5}{2 \sin(x) \cos(x)}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{x \to \pm 0} \frac{5\frac{d}{dx} \left[(1+5x)^{-1} \right]}{2\frac{d}{dx} \left[ \sin(x)\cos(x)\right]}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, \lim_{x \to \pm 0} \frac{5 \cdot -5(1+5x)^{-2}}{2 \cdot \left[\cos^2(x) - \sin^2(x)\right]} = \frac{-25}{5}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
,, a = -\frac{25}{5}
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
.solved
midnight plankBOT
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lone yacht
why is this incorrect
midnight plankBOT
why is this incorrect
steady trail
Why 20?
midnight plankBOT
Show your work, and if possible, explain where you are stuck.
lone yacht
cause 4sin^3(x) is multiplied with 5
steady trail
Can I see how you do it?
steady trail
What is d'=5
steady trail
Okok
steady trail
But in the photo you sent you wrote a different answer
lone yacht
yea i was showing the steps i used to get to thaat answer
i just multiplied the 5 and 4
steady trail
I still haven't figured out where that 20 comes from
lone yacht
steady trail
But you can't do that
lone yacht
this is wrong aswell
steady trail
But
You must remove a 5 from the numerator on the right after the -
Why did you write it?
steady trail
👍
midnight plankBOT
@lone yacht Has your question been resolved?
dense temple
help
last slate
bro
midnight plankBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
low hull
Hm
dense temple
what i did was change the square root (7) - square root (11) into -square root (7) and + square root (11)
and the two negatives on square root 7 turn into positive
then i cross multiply and mutiply the bottom
i got this as answer however they might not be a 2 in the start
floral trench
i got the same thing
untold pollen
same
untold pollen
so what did you need help with exactly? were you just hoping to check your answer?
midnight plankBOT
@dense temple Has your question been resolved?
dense temple
so what did you need help with exactly? were you just hoping to check your answe...
yeah to check
did u get the same as me
?
or without the 2
.close
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dawn blade
so i have this factored form of a polynomial function, y=1.02728468⋅10^-1 (x+12)^2 (x+4)^2 (x-5)^2 (x-13)^2, how do i convert it into standard polynomial function?
midnight plankBOT
so i have this factored form of a polynomial function, y=1.02728468⋅10^-1 (x+12)...
@dawn blade Has your question been resolved?
dawn blade
@stiff heart please help
the only way i know how todo this is to expand foil and cross multiply, but for this amount of foiling and cross multiplying, is there an easier way?
midnight plankBOT
@dawn blade Has your question been resolved?
thick granite
midnight plankBOT
thick granite
I’m just completetly stuck on this question
thick granite
$5?
oak slate
Try writing the function that describes how many toys he will sell in one month given the price of each toy
thick granite
Try writing the function that describes how many toys he will sell in one month ...
How would I do that
oak slate
just try some values and find the general expression
$5 --> 600 toys
$7 --> 600 - 25 toys
$9 --> 600 - 25 - 25 toys
$11 --> 600 - 25 - 25 - 25 toys
paper inlet
funky boy have slope
stoic jolt
I dont understand where the -4(64) comes from
midnight plankBOT
I dont understand where the -4(64) comes from
night lark
I dont understand where the -4(64) comes from
U took out -4 from all the terms including -254, -254/-4 is 64
stoic jolt
.close
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jolly plinth
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spring latch
(i-1)p = 84
i + p = 25
midnight plankBOT
(i-1)p = 84
i + p = 25
spring latch
im genuinely blanking
fallow scarab
Solve for p in the second equation and plug into the first equation
midnight plankBOT
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steep spear
Hi
steep spear
Can you do this please.
errant stirrup
nobody is going to do your homework for you
steep spear
nobody is going to do your homework for you
mf i need help
errant stirrup
refer to my previous message
midnight plankBOT
@steep spear Has your question been resolved?
steep spear
refer to my previous message
stfu
deft frost
formal blade
stfu
People aren't going to do your work here for you. Be civil and follow the rules. You're muted for the day. You can come back tomorrow.
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midnight plankBOT
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ionic magnet
find sum of all answers
midnight plankBOT
find sum of all answers
upper crag
what have you tried so far
slim saddle
!status
midnight plankBOT
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
@ionic magnet Has your question been resolved?
last slate
how many unique triangles can be formed if two sides are fixed and an angle opposite to one of the sides if given?
midnight plankBOT
how many unique triangles can be formed if two sides are fixed and an angle oppo...
@last slate Has your question been resolved?