r u rlly making us check if u substituted the values properly?
#help-49
cedar coral
i would most definetly not accept that. for a start, its very disorganized, and units are nowhere to be found
midnight plankBOT
@steep mirage Has your question been resolved?
steep mirage
r u rlly making us check if u substituted the values properly?
Idk if I used the right formulas
And yeah idk if I did the math correctly
steep mirage
i would most definetly not accept that. for a start, its very disorganized, and ...
I cannot write with a mouse for shit, also units are in the end so it's fine.
I'm just wondering if I got the right answers for the question asked
.
trail wasp
its correct
midnight plankBOT
@steep mirage Has your question been resolved?
midnight plankBOT
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placid dragon
bro
lavish venture
trig sub
lavish venture
x = a sin theta
placid dragon
yes i did that
lavish venture
nope
subtle blaze
1 - sin^2 is smarter
lavish venture
!show
midnight plankBOT
Show your work, and if possible, explain where you are stuck.
rare maple
I believe that is correct
last slate
Gigachad meets math final boss
placid dragon
got sm shitty test tmrw
lowkey cooked asl
prof gon grape me
explain bro
otherwise gotta head to stackexchnge
where those mfs will show attitude
and all
zesty asses
rlly gon ask for a 10 page approach and shi
i aint doing all that bs
they be draggin the simplest shi
fk stackexchange
last slate
It seems the intended question is $\int\frac{1}{a^2-x^2}dx$
grand pondBOT
à뜜
last slate
Whose answer is the one in the answer key
A misprint basically (in the question)
In any case,
placid dragon
hmmmmmmmmmmm
last slate
$\int\frac{1}{a^2-x^2}dx = \frac1{2}\ln\left|\frac{a+x}{a-x}\right| + C$
$\int\frac1{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$
last slate
You're welcome, have a great day
placid dragon
.close
midnight plankBOT
grand pondBOT
à뜜
last slate
@placid dragon slight correction before you go^
midnight plankBOT
Available help channel!
dusty portal
https://cdn.savemyexams.com/uploads/2023/01/edexcel-as-further-maths-2021-p2-ms.pdf Markscheme is linked, my answers are a PDF, if you could please grade it that would be really appreciated 
midnight plankBOT
https://cdn.savemyexams.com/uploads/2023/01/edexcel-as-further-maths-2021-p2-ms....
dusty portal
If you don't like the PDF I'll just screenshot my answers
Though it will be a bit difficult to read
dreamy lichen
Hmm cant you just grade it yourself with the markscheme?
dusty portal
Eh
I've done that before but I feel like doing something different
Like sometimes I do things that are non-standard and I want to see how rigorous it is
chrome vessel
Like sometimes I do things that are non-standard and I want to see how rigorous ...
maybe its best to single those solution out then
also where are the questions?💀
dusty portal
also where are the questions?💀
In the PDF
chrome vessel
Oh should've made the "problems" blue, i didn't know its a link
chrome vessel
Like sometimes I do things that are non-standard and I want to see how rigorous ...
so which of the solutions are non-standard?
dusty portal
so which of the solutions are non-standard?
For me, I want to get a new perspective
I don't necessarily know yet that I have non-standard solutions but I want to see how rigorous it is
chrome vessel
I don't necessarily know yet that I have non-standard solutions but I want to se...
u should check the solution guide once to pick out those u r not confident in
dusty portal
<:KEK:586240877358350341> u should check the solution guide once to pick out tho...
Hmm, okay
Thanks
I'll close it
.close
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craggy yew
Hello guys, how do I integrate sin^2x/cosx? I tried to split it by rewriting sin^2x into 1-cos^2x but now I dont know how to integrate 1/cosx
midnight plankBOT
Hello guys, how do I integrate sin^2x/cosx? I tried to split it by rewriting sin...
fallow scarab
Hello guys, how do I integrate sin^2x/cosx? I tried to split it by rewriting sin...
everyone should see this once
https://math.stackexchange.com/questions/6695/ways-to-evaluate-int-sec-theta-mathrm-d-theta
craggy yew
I have no clue what is sec
grand pondBOT
riemann
craggy yew
Thanks 😭🙏🏻
craggy yew
We just dont use that term in our country
fallow scarab
yea i've seen sen used for sine
lavish venture
how do we know he’s spanish?
fallow scarab
tirikis is spanish for gullible
craggy yew
What 😭
lavish venture
it is
lavish venture
lmao
subtle zinc
welcome
craggy yew
20 more diff equations to go
fallow scarab
discover trig identities by doing diffy qs
lavish venture
he won’t know that trig identity
woven quarry
midnight plankBOT
woven quarry
Part a, im confused how the graph of that curve even looks like that if tje domain is 0<theta<rootpi
Would it not just be like a graph which is rotating about the origin anticlockwise at around root(pi) radians
Whilst the graph above is a graph rotated pi degrees from pole
midnight plankBOT
@woven quarry Has your question been resolved?
woven quarry
<@&286206848099549185>
warm palm
Would it not just be like a graph which is rotating about the origin anticlockwi...
That is what it is. sqrt(pi) radians is about 101 degrees. When theta is 0, r is 0, and when theta is sqrt(pi), r is also 0. So when the curve comes back to the origin at the end, it's only gone 101 degrees.
You can imagine drawing a tangent line at the end of the curve (when theta is almost sqrt(pi)), and that line would make an angle of 101 degrees with the positive x-axis.
midnight plankBOT
@woven quarry Has your question been resolved?
woven quarry
That is what it is. sqrt(pi) radians is about 101 degrees. When theta is 0, r is...
Yh thanks
I thought it looked 180*
To me
lean jay
<@&286206848099549185> can anyone help me with making a multivariate binary logistic regression?
midnight plankBOT
<@&286206848099549185> can anyone help me with making a multivariate binary logi...
deep vine
<@&286206848099549185> can anyone help me with making a multivariate binary logi...
Please don't occupy multiple help channels.
midnight plankBOT
@lean jay Has your question been resolved?
fallow scarab
.close
tidal turret
midnight plankBOT
tidal turret
guys, can I get some help?
grim vector
So the abc plan must have the same normal vector as Pi
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
So the abc plan must have the same normal vector as Pi
can you elaborate
grim vector
They have to be parralel so they must have the same normal vector
tidal turret
They have to be parralel so they must have the same normal vector
or a scalar multiple, yeah I guess
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
They have to be parralel so they must have the same normal vector
bro may i ping
subtle blaze
Could be good if you translated it
midnight plankBOT
@tidal turret Has your question been resolved?
unique tundra
i need help with augmented matrixes and echoleons
midnight plankBOT
i need help with augmented matrixes and echoleons
sharp coral
you would apply the gaussian elimination algorithm
unique tundra
i have no knowledge on this and don't know how to do that LOL
sharp coral
there are two parts to this:
- know the 3 row operations
- know how to apply the 3 row operations to reduce to echelon form
out of 1 or 2, which one are you unsure about?
sharp coral
from the link above, here are the 3 row operations:
so interchanging rows is fairly self explanatory
for multiplying a row by a constant, you multiply each entry in a row by that constant
for adding a multiple of a row to another row, you add them in corresponding columns
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Available help channel!
tired ferry
How would I find all possible ideals in $\mathbb{Q}$?
midnight plankBOT
How would I find all possible ideals in $\mathbb{Q}$?
grand pondBOT
BOSS
nova yoke
Q is a field...
nova yoke
haha indeed
tired ferry
lmfao ty ty
nova yoke
yw
nova yoke
yea?
tired ferry
I had another question but I didnt want to open a new channel and no one is really respoding lmfao
how exactly do you prove something is an integral domain
Like cancelation laws or do you try and prove axioms
For example something like this:
nova yoke
well first show that it's a commutative ring, that's clear enough here
then i guess you could try to show explicitly that there are no zero divisors
tired ferry
Is that the proper way to go about it? Just explicity show there are no zero devisors?
nova yoke
well in some cases you might be able to use a theorem
nova yoke
for example if you can show that it's isomorhpic to a quotient by a prime ideal
tired ferry
for example if you can show that it's isomorhpic to a quotient by a prime ideal
oh hmm
thats likely much more difficult no
nova yoke
in this case probably easiest just to compute directly
show that if the product of two elements is zero then one of the elements has to be zero
nova yoke
sure
tired ferry
I was trying to find all possible homomorphisms from $\mathbb{Z}6 \rightarrow \mathbb{Z}{15}$ and define each kernel + image
grand pondBOT
BOSS
tired ferry
I just basicly use the idenpotents right
so essencily, let $\phi$ be the homomorphism and let $\phi(1)=a$
grand pondBOT
BOSS
tired ferry
then $a = \phi(1) \phi (1) = a^2$
do you need me to discribe my thinking there further
grand pondBOT
BOSS
nova yoke
so you aren't requiring phi(1) = 1 as part of the definition of ring homomorphism?
tired ferry
no, the identity can be anything when finding all possible homomorphisms right
nova yoke
usually phi(1) = 1 is part of the definition but maybe not for some authors
tired ferry
I have the solution to a similar question that i was just following
nova yoke
however there is this generalization:
"rng" is a ring that doesn't necessarily even have an identity
nova yoke
so ok i assume you're talking about the latter type
nova yoke
ok yea they're using the more general version, that's fine
grand pondBOT
BOSS
tired ferry
all such solutionz in Z_15 are
0,1,6,10
and each homomorphism would be defined like
$\phi(n)=an$
grand pondBOT
BOSS
tired ferry
cool?
tired ferry
I just put it in desmos ngl
nova yoke
and then yea, phi(n) = an follows because phi is an additive group homomorphism
tired ferry
Ok cool now for the kernals and images right
nova yoke
yea
tired ferry
so for a = 0 then we just have everything goes to 0 or the trivial homomphism
where the kernal is Z_6 and the image is like
0
:D
for 1 we just have everything maps to itsself, kernal is 0 image is Z_6
nova yoke
does 1 actually work?
nova yoke
well what would phi(7) be
nova yoke
yea but you also have phi(7) = phi(1 + 1 + 1 + 1 + 1 + 1 + 1) = 7phi(1) = 7
and 1 does not equal 7 in Z15
tired ferry
but here we are using equivilance classes right
nova yoke
yes
tired ferry
hmm
nova yoke
the above calculation would imply
phi([7]) = [7]
phi([7]) = phi([1]) = [1]
nova yoke
the issue is that it doesn't even work as additive groups
tired ferry
I think our only inputs are
0,1,2,3,4,5,6
and everything else just is an equivilance clase of one of those
and behaves the same
because we are technically defining as such:
$\phi([n]) = a[n]$
grand pondBOT
BOSS
nova yoke
if phi(1) = 1 then the kernel is trivial but then the image has to have order 6
and 6 does not divide 15
tired ferry
no wait wont the image just be
0,1,2,3,4,5,6
those are the only places its being mapped
nova yoke
but that's not a homomorphism then
nova yoke
yea 6phi(1) would just be 6, but it also has to equal phi(6), which is phi(0), which is 0
tired ferry
Ok wait how about i look at it differently
and start with the subgroups of Z_15
and see mappings to them?
So like, <0>,<1>,<3>,<5> are all subgroups of Z_15 right
nova yoke
right
nova yoke
<1> has order 15, so that can't be the image if the "input" has order 6
nova yoke
<3> has order 5, what order would the kernel have to have?
just looking at the additive groups
tired ferry
We can also get possible kernals, so 3Z/15Z is iso to Z/5Z
so has kernal which is the multiples of 5
0, 5
nova yoke
the kernel's order has to be a divisor of 6
nova yoke
you have the 1st isomorphism theorem
nova yoke
yea Z_6 / ker is isomorphic to the image
and Z_6 / ker has to have order 1,2,3 or 6
it can't be 5
tired ferry
it can't be 5
wait it wont be
nova yoke
that's one solution, are there others?
tired ferry
which gets us our image
tired ferry
that's one solution, are there others?
i dont think so
tired ferry
yeah ur right
0,1,6,10
hmm
ok so going back we have to map to multiples of 5
so im thinking its 10
because 20 = 5 mod 15
so a = 0 and a= 5 are our solutions
but we should have 3 im p sure
nova yoke
yea 10 looks like it works
because 10+10 = 20 = 5, and 10+10+10 = 30 = 0
so that has order 3
tired ferry
Yup
nova yoke
but we should have 3 im p sure
hmm well we excluded <1> and <3> right?
and you found that a has to be 0,1,6,10
and 1 and 6 are ruled out because their orders don't work
tired ferry
0 works 10 works
nova yoke
so unless there are other a's that work then that seems to be it
tired ferry
I think it’s a rule that we have gcd(m,n) homomorphisms
nova yoke
i just checked all the other numbers, only 0,1,6,10 satisfy a^2 = a
tired ferry
I think it’s a rule that we have gcd(m,n) homomorphisms
Because of this
So we are definitely missing 1 somewhere
nova yoke
I think it’s a rule that we have gcd(m,n) homomorphisms
are you sure about this formula?
i found this on MSE
here, w(n) would be the number of prime divisors of 15, which is 2
tired ferry
It was shown in class not my book so I can’t find a proof about it
nova yoke
so the formula would give 2^(2 - 1) = 2
nova yoke
which ones
tired ferry
a= 1 and a=6
nova yoke
we can't have a = 1 because then the image would be all of Z15
but the image's order has to be a divisor of 6
since Z6 is the input ring
tired ferry
we can't have a = 1 because then the image would be all of Z15
It would be all of z6 right
Oh sorry
tired ferry
Yeah i said a = 1
So for every n in z_6
We get 1n which gives us back z_6 in 15
I think it’s just not a group
nova yoke
well you don't even have a well defined map
nova yoke
phi(0) = 0
but phi(0) = phi(6) = 6
nova yoke
then you would have:
phi(1) = 6
phi(2) = 12
phi(3) = 3
phi(4) = 9
phi(5) = 0
so the image has order 5
tired ferry
Isn’t this the subgroup generated by 3
nova yoke
but just looking at the additive groups, you have Z6/kernel is isomorphic to the image
so Z6/kernel would have order 5
but 5 is not a divisor of 6 so that can't happen
nova yoke
Isn’t this the subgroup generated by 3
yep
tired ferry
Wait the kernel here is legit only 5
nova yoke
you get the elements of <3> in a scrambled order
nova yoke
well you would also have phi(0) = 0, and phi(10) = 0
but phi(10) = phi(4)
and phi(15) = 0
but phi(15) = phi(9) = phi(3)
tired ferry
Wait wait we are only worried about 0,1,2,3,4,5 here
Its equivalence classes
For input
nova yoke
yea but still, additivity requires these
nova yoke
phi([10]) = phi([2 x 5]) = phi([2][5]) = [2]phi([5]) = [2][0] = [0]
basically the "addition" part of the homomorphism is incompatible with the "multiplication" part
nova yoke
which one
tired ferry
phi([10]) = phi([2 x 5]) = phi([2][5]) = [2]phi([5]) = [2][0] = [0]
Here
nova yoke
because in Z6, [10] = [4]
and we just calculated phi[10] = [0]
but earlier we found phi[4] = 9
tired ferry
One sec
nova yoke
because it's phi(1+1+1+1) = 6+6+6+6 = 24 = 9 (mod 15)
tired ferry
then you would have:
phi(1) = 6
phi(2) = 12
phi(3) = 3
phi(4) = 9
phi(5) = 0
Cool ok so
tired ferry
because it's phi(1+1+1+1) = 6+6+6+6 = 24 = 9 (mod 15)
That’s good right?
nova yoke
yes
tired ferry
Means phi(4) goes to 9
nova yoke
that's what the additive part tells you
tired ferry
Ok
nova yoke
phi has to send [4] to [9]
tired ferry
Ok
nova yoke
but the multiplicative calculation says phi([4]) = phi([10]) = [0] not [9]
but [4] and [10] are the same element of Z6
phi can't send them to two different outputs
tired ferry
Wait but
[10] is the same as [4] right, like we are just passing in 4
If we passed in 16 its still going to map to the same place 4 does
phi([10]) = phi([2 x 5]) = phi([2][5]) = [2]phi([5]) = [2][0] = [0]
Did u mean phi[2]
Oh
I see
U end up getting 12*0
Which is 0
Hmm
Sorry for taking so long to get there lmao
@nova yoke tysm
im gonna close the channel but all of this was incredibally helpful
nova yoke
yw!
tired ferry
.close
midnight plankBOT
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chilly cobalt
if i have a point (x,y) and reflect it across y = -x what would the image be? i kinda forgot..
midnight plankBOT
if i have a point (x,y) and reflect it across y = -x what would the image be? i ...
tidal turret
(-y,-x)
just know that the equation y = -x is telling you y coord is -x
and is telling u x = -y
midnight plankBOT
midnight plankBOT
✅
chilly cobalt
oh🥀
runic hamlet
how much linear algebra do you know
tawdry laurel
i just started it today
gl
chilly cobalt
heres the question i was doing btw
'Let f be the reflection across the x axis, g be the reflection across the y axis, h be the reflection across y = x and j be the reflection across y = -x.
Prove that:
j o h = g o f
h o f o h = g"
runic hamlet
you could always just draw a sketch and then bruteforce it with a bit of geometry
chilly cobalt
.close
midnight plankBOT
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slate berry
Can someone guide me how to solve this prob? Thx🤗
midnight plankBOT
Can someone guide me how to solve this prob? Thx🤗
dreamy lichen
idk if it's the intended way to do it, but it shouldnt be too hard to solve for a, b, c
there is a neat way to do that which has to do with binary btw
slate berry
Oh I’m not really good at binary btw😅
But how do u come out with binary when u look at the problem?
dreamy lichen
Oh I’m not really good at binary btw😅
But how do u come out with binary when u ...
It's not necessary to know binary, the way i came up with it is first i tried guessing a, b, c without using binary, and then i realized that what im doing in my head is the same thing as converting certain number to binary
dreamy lichen
Oh I’m not really good at binary btw😅
But how do u come out with binary when u ...
Can you in any way try to guess what a, b, c could be?
Here is a small hint which may or may not help:
||Multiply both sides by 64 = 2^6||
midnight plankBOT
@slate berry Has your question been resolved?
slate berry
😯 I understand now
The point (or the key step) is to turn the fraction into an integer right?
dreamy lichen
😯 I understand now
The point (or the key step) is to turn the fraction into an ...
Yeah, that works
In binary, 37 is 100101 btw
Which is 2^5 + 2^2 + 2^0
So thats how binary is related
slate berry
Wow that’s really smart 😅anyways Thanks for helping
uneven sandal
are those two sides equal?
midnight plankBOT
are those two sides equal?
uneven sandal
\[
\sum_{\sigma \in S_N} (-1)^\sigma \prod_{i=1}^{N} v_i^{(\sigma(i)-1)} \cdot v_i^{* (\sigma(i)-1)}
\stackrel{?}{=}
\sum_{\sigma \in S_N} (-1)^\sigma \prod_{i=1}^{N} \sum_{j=1}^{N} v_j^{(i-1)} v_j^{* (\sigma(j)-1)}
\]
tobi
lament knoll
midnight plankBOT
@uneven sandal Has your question been resolved?
twilit field
Show if : 0<x;0<y, then 0<x+y
midnight plankBOT
Show if : 0<x;0<y, then 0<x+y
midnight plankBOT
cursive swan
gaunt nimbus
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Available help channel!
twilit field
I would like to prove the monotone convergence thoerm
midnight plankBOT
I would like to prove the monotone convergence thoerm
twilit field
Here's my idea
Let ${a_n}$ be a monotone increasing sequence. As it's bounded, there's a value $M$ such that , $\forall n \in \N, a_n≤M$.
\
We then have , $\forall n\in \N$
\
$a_{n}≤M$
\
$\text{ now here I want to use the defn of a limit, but not sure how to, here's my best bet }$
\
$\forall \varepsilon >0$
$a_n-\varepsilon≤M \implies a_n-M< \varepsilon$
grand pondBOT
What a wonderful world !
twilit field
Let ${a_n}$ be a monotone increasing sequence. As it's bounded, there's a value $M$ such that , $\forall n \in \N, a_n≤M$.
\
We then have , $\forall n\in \N$
\
$a_{n}≤M$
\
$\text{ now here I want to use the defn of a limit, but not sure how to, here's my best bet }$
\
$\forall \varepsilon >0$
$a_n-\varepsilon≤M \implies a_n-M< \varepsilon $
\
$\textbf{ does this imply the following? }$
\
$\abs{a_n-M}< \varepsilon$
grand pondBOT
What a wonderful world !
twilit field
not sure what to do now
because this doesn't allow me to conclude $\abs{a_n-M}< \varepsilon$
grand pondBOT
What a wonderful world !
twilit field
Let ${a_n}$ be a monotone increasing sequence. As it's bounded, there's a value $M$ such that , $\forall n \in \N, a_n≤M$.
\
We then have , $\forall n\in \N$
\
$a_{n}≤M$
\
$\text{ now here I want to use the defn of a limit, but not sure how to, here's my best bet }$
\
$\forall \varepsilon >0$
$a_n-\varepsilon≤M \implies a_n-M< \varepsilon $
\
$\textbf{ does this imply the following? }$
\
$\abs{a_n-M}< \varepsilon$
grand pondBOT
What a wonderful world !
lyric charm
because this doesn't allow me to conclude $\abs{a_n-M}< \varepsilon$
yeah cause you don't know that M equals the limit lol
it may well not!
twilit field
I could take M to be the infimum of the set of upper bounds of the seqeunce
lyric charm
or you could make that not so roundabout and say M = sup a_n.
twilit field
now what
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
<@&286206848099549185>
runic hamlet
now play around with classic supremum properties
in particular, if the a_n do not get close to M, then they would have a smaller supremum
twilit field
in particular, if the a_n do not get close to M, then they would have a smaller ...
okay, so that tells me that M is the supremum, right
heavy falcon
so until now, you said that {a_n} is bounded and denoted by M to be the supremum of the set of elements of this sequence
now what does it mean for M to be the supremum
twilit field
M is the limit of the seqeunce
twilit field
you are trying to prove this no ?
yes
heavy falcon
alright
so what is the definition of supremum
what does "M is the supremum of A={a_n| n in N}" mean by def
twilit field
$M$ is said to be the supremum of a set $S$, if $M≥a \forall x \in S$ and $M$ is the smallest such number
runic hamlet
thats only half of the def
heavy falcon
did you mean to say M>=x?
also is that it ?
or is this the definition of any upper bound
grand pondBOT
What a wonderful world !
heavy falcon
alright, how can you write the second part of your statement in symbols
twilit field
M is said to be the supremum of a set $S$ if $M≥x \forall x \in S$ and $M≤K. \froall K, st k≥x \forall x \in S$
grand pondBOT
What a wonderful world !
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
heavy falcon
alright this is a correct way to write it, but does it give you what you are looking for
heavy falcon
**What a wonderful world !**
you are looking to reach something like the last statement here right ?
|a_n-M|<ε
twilit field
yes
heavy falcon
**What a wonderful world !**
Compile Error! Click the <:errors:50426655605496219...
so try rewriting this by beginning as follows : Given ε>0, M is said to be the supremum of S if ...
twilit field
I don't recall that defn
heavy falcon
so if M is a supremum
twilit field
I think I'll revise these and get back to this soon
heavy falcon
then any real number less than M is not an upper bound
heavy falcon
I think I'll revise these and get back to this soon
alright as you wish
although i think you are pretty close
twilit field
Sorry and thanks
heavy falcon
no need to be sorry because you didnt do anything wrong xD
midnight plankBOT
heavy falcon
just really tired, sem just got over, don't know why I'm doing this
ohhh , yea then go rest for a while
runic hamlet
wai doesnt know the word rest
heavy falcon
its still not too late to look it up 
midnight plankBOT
true lark
not sure where to start
midnight plankBOT
@true lark Has your question been resolved?
true lark
<@&286206848099549185>
spring wave
look at the numbers written on the board + 1
for example if alice writes
3 and 5 (= (3*3 + 1) /2)
then look at
4 and 6
midnight plankBOT
@true lark Has your question been resolved?
true lark
for example if alice writes
3 and 5 (= (3*3 + 1) /2)
then look at
4 and 6
what about them?
spring wave
that is a nice place to start, try find a formula to generate 6 interms of 4 of to generate the next number +1 in terms of the previous number +1
midnight plankBOT
@true lark Has your question been resolved?
odd lagoon
Hello
odd lagoon
If I wanted to find the Volume for the function cos(x) while rotating it around the y-axis on an interval of 0<=x<=pi
what the hell would that mean
should i split it up
into multiple integrals
vital sinew
what the hell would that mean
Do you actually want to know what a solid of revolution is or are you asking how to do it
odd lagoon
i think i know what a solid of revolution is already, like i can do it for sin(x) from 0 to pi
but it just struck me that cos(x) would be strange because one part dips into the negative
idk how that would translate at the end
i can imagine the shape it would make too
cos(x) being rotated around the y-axis with 0<=x<=pi
i want to do this myself but just wondering if my thoughts are correct:
could i solve the problem by
first doing cos(x) from 0<=x<=pi/2 first
get that volume
then cos(x) from pi/2<=x<=pi
odd lagoon
then cos(x) from pi/2<=x<=pi
but for this part
i would have to find the area for a circle with a hole in it
and the hole would be the area i used with the disk method from the first part i already got
or can i do everything using one integral?
so i do this part first
and then i do the outer circle, but since i already did the inner first, i would subtract the area of the inner circle from the larger outer circle
or have i misunderstood, can i just use area of the whole thing off the get go?
midnight plankBOT
@odd lagoon Has your question been resolved?
rancid vigil
$\int\frac{1}{4x(\ln(x^2)-1)}dx$
midnight plankBOT
$\int\frac{1}{4x(\ln(x^2)-1)}dx$
grand pondBOT
<rajel />
grand pondBOT
;(
rancid vigil
As I mentioned, just use $\ln(x^2)=2\ln|x|$
So that's what I got
Oh I forgot to change dx with its value
dusty portal
Huh?
fallow scarab
simplify by cancelling
empty ivy
Hi I don't really get part i
midnight plankBOT
Hi I don't really get part i
night hawk
Hi I don't really get part i
The particle is not always travelling in a single straight direction. Instead, its vertical acceleration is -10 because of gravity.
empty ivy
The particle is not always travelling in a single straight direction. Instead, i...
Yea.. but how does that help me get the angle of projection?
night hawk
Yea.. but how does that help me get the angle of projection?
It seems like you understand parametric functions, so you need to use:
$$v|_{t = 0} = \left<50 \cos(a), 50 \sin(t) \right>$$
$$a = \left<0, -10 \right>$$
grand pondBOT
@night hawk
empty ivy
It seems like you understand parametric functions, so you need to use:
$$v|_{t =...
am i on the right track?
night hawk
am i on the right track?
Something messed up here
empty ivy
Something messed up here
Uhh I don't see it
night hawk
Uhh I don't see it
$$y = -5t^2 - 50 \textcolor{red}{t} \cos(a) + \mathrm{C}$$
grand pondBOT
@night hawk
midnight plankBOT
midnight plankBOT
Available help channel!
storm remnant
midnight plankBOT
storm remnant
was watching a tutorial on how to solve this
steps are kinda messy but I was confused at a point
at the ln(y) + c = kt + c part
he substracts c from both sides as he's trying to isolate the y
so how come on the left side there's no more c but right side the c still remains even though he subtracted from BOTH sides?
polar star
c1 and c2 can be different
he subtracted c1 from both sides
and defined a new constant C = c2-c1
floral apex
what a mess this persons work
storm remnant
what a mess this persons work
LMAO he explains really well though which is nice
takes his time
storm remnant
c1 and c2 can be different
it still kinda doesn't make sense to me but idk I'm thinking of it like idk y + x = 2x so you subtract x from both sides which leaves y = x
polar star
if c2 = 3 and c1 = 1 then c2-c1 = 2
does that look like 0
different subscripts -> different variables
so instead of thinking like
y+x = 2x
think
y+z = 2x
storm remnant
does that look like 0
ight bro
but yeah I get it now
thanks for the help
.close
midnight plankBOT
midnight plankBOT
Available help channel!
storm remnant
midnight plankBOT
storm remnant
unsure of how to start
I'm also running into an issue factoring for this problem
damn
midnight plankBOT
@storm remnant Has your question been resolved?
tidal turret
midnight plankBOT
floral apex
tidal turret
if there is infinite many solutions then most likely the kernel is non empty
A^2x - Ax = 0
floral apex
yea
tidal turret
A(Ax - x) = 0
floral apex
maybe (A^2 - A)x = 0 is more helpful?
tidal turret
true dat, im so blind
twilit field
If there are infinite solutions, the dimension of the kernel is atleast 1
floral apex
ur good
tidal turret
that is indeed way more helpful
tidal turret
If there are infinite solutions, the dimension of the kernel is atleast 1
yes, exactly!!!!
aka, det is not zero
or something...
twilit field
Try re arranging the rows of the matrix so that you can find the rank and nullity of the linear transformation
floral apex
seems like you have it sorted tho
twilit field
Try re arranging the rows of the matrix so that you can find the rank and nulli...
and then find values of alpha that allow for a nullity of atleast 1
tidal turret
Try re arranging the rows of the matrix so that you can find the rank and nulli...
we havent covered linear transformations yet, thoo
twilit field
Nevermind then
floral apex
how about REF
its annoying
wait you have det but not linear transform?
det is another choice
theyre both a little annoying
twilit field
very
floral apex
yea
tidal turret
ok
floral apex
how do we know that 
terse lodge
maybe theres some way without it
you can compute det(A) and det(A-I) and multiply them?
tidal turret
so we know Ax=x
that is not true
floral apex
IDK i dont know anything about linear algebra tbh
tidal turret
XD
floral apex
to me i would do whatever is easy
twilit field
?
floral apex
youre good its your channel
tidal turret
I am clueless ngl
twilit field
We have two cases actually, det A can be zero ( because then diemsnion is mapped to 0)
tidal turret
det A being 0 or not depends on alpha
floral apex
yea
twilit field
I'm thinking about this in terms of Linear transformations
sorry
so if det A ≠0, you just want the eigenvector(s)
if det (A) is zero, well, that's much easier
tidal turret
dude there is no need to find the eigenspaces wtf
twilit field
dude there is no need to find the eigenspaces wtf
why not
tidal turret
also, not covered until second midterm
floral apex
lets just find A^2 - A
fallen sparrow
lets just find A^2 - A
Yeah lmao
twilit field
okay, then do what Jan Niiku says
fallen sparrow
No need for cases
floral apex
i mean do whatever you want
im sure what wai is suggesting is probably less steps
what im suggesting is a massive headache
tushar suggested something
but IDK how much det properties you have access to
tidal turret
,w {{0,-1,1},{5,k,1},{0,0,k}}^2 - {{0,-1,1},{5,k,1},{0,0,k}}
grand pondBOT
floral apex
strange it shows it twice
this is actually kind of nice
this det will be not so bad
tidal turret
what im suggesting is a massive headache
yes, idk , maybe wai is right we should find the eigenspaces but we havent covered
tidal turret
this det will be not so bad
?
twilit field
yes, idk , maybe wai is right we should find the eigenspaces but we havent cover...
eigenspaces is kind of wrong, we want vectors that map to 0
my bad
tidal turret
eigenvectors, then, yeah srry
floral apex
,w determinant {{0,-1,1},{5,k,1},{0,0,k}}^2 - {{0,-1,1},{5,k,1},{0,0,k}}
grand pondBOT
tidal turret
You can solve the det from last row, which has 2 0's which makes it easy
ok true, I see it now
floral apex
woo
tidal turret
this is a cubic poly
floral apex
even better
tidal turret
no
floral apex
its det(A^2 - A)
tidal turret
oh yeah
floral apex
so what should we do
tidal turret
but, why depressed
floral apex
its just fancy words
we mean that k is a factor
its really a quadratic
$-5k^3 + 35k^2 - 30k = k(-5k^2+35k-30)$
tidal turret
we should look for the roots of this?
grand pondBOT
jan Niku
floral apex
we should look for the roots of this?
yea
what does it mean if we find some k for which det(A^2-A) = 0?
tidal turret
- k = 0
- something
by zero product property
tidal turret
what does it mean if we find some k for which det(A^2-A) = 0?
idk
floral apex
well say we have Ax = 0 as an equation were trying to solve
tidal turret
we have infinitely many solutions
floral apex
we dont care about x=0
floral apex
we have infinitely many solutions
yea, infinitely many solutions to (A^2 - A)x = 0, if det(A^2-A)=0
tidal turret
-5k^2 + 35k - 30 = 0
we can use bhaskhara
let me just cheat with wolfram for a sec
,w -5k^2 + 35k - 30 = 0
grand pondBOT
floral apex
whats a bhaskhara
floral apex
tidal turret
is in spanish, srry
floral apex
oh
oh is that quadratic
sorry im a dumb american
anyways, yea, so youre done i guess 
unless you are still confused about anything
modern sapphire
interesting, I didnt know they called it that even outside india
floral apex
yea that does not look like a spanish word at all
tidal turret
is probably the name of the creator
modern sapphire
yea, that was old old indian dude who came up with it
floral apex
i thought this was from antiquity
tidal turret
unless you are still confused about anything
well, doing A^2 is it hard by hand?
nah I think its fine
floral apex
no, it just has the same problem as row reducing
its tedious and error prone
the natural impulse is to skip steps and not write everything out
then you make some small mistake and its just a nightmare sometimes
depends on how careful you can force yourself to be
tidal turret
no, it just has the same problem as row reducing
I usually dont do matrix multiplication just do power of two for all the elements in the matrix, is that right?
for A^2 I mean
tidal turret
wdym?
floral apex
its not true that to find A^2, you just square each element
its a bunch of dot products
tidal turret
,w {{0,-1,1},{5,k,1},{0,0,k}}^2
grand pondBOT
tidal turret
omg you are right haha
floral apex
its a good exercise
tidal turret
I feel so ashamed
floral apex
or maybe you pick an element or two and verify it
floral apex
I feel so ashamed
i wouldnt worry too much about it
think freshmans dream is so common it has a name
these are common misconceptions 
tidal turret
wdym?
fallen sparrow
I usually dont do matrix multiplication just do power of two for all the element...
tidal turret
XD
tidal turret
ok I appreciate the help, I would need to knkw how to do the determinant and matrix multiplication but other than that I think its clear
floral apex
you should def practice the matrix multiplication
tidal turret
you can compute det(A) and det(A-I) and multiply them?
that is crazy smart aswell using det(AB) = det(A).det(B)
floral apex
yea tushar is really good at math
tidal turret
srry I didnt took the time to read all the approaches
floral apex
i mostly just fumble through it
tidal turret
interesting, I didnt know they called it that even outside india
is the popular name in argentina I guess, idk how it is in america
fallen sparrow
Btw for determinants you can follow any rule i.r R - R, R -C, C - C
But for matrices only R C rule is correct
tidal turret
Btw for determinants you can follow any rule i.r R - R, R -C, C - C
wdym?
fallen sparrow
wdym?
Row to row multiplication , column to column multiplication and Row to column multiplication
terse lodge
computing the determinant of A is easy, since swapping the first two rows makes it upper triangular, so you can read off the eigenvalues from the diagonal
the determinant is the product of the eigenvalues
tidal turret
computing the determinant of A is easy, since swapping the first two rows makes ...
swapping rows affect the det
no?
terse lodge
swapping rows affect the det
yes, by a factor of -1
fallen sparrow
swapping rows affect the det
Just multiply by -1
terse lodge
and computing the determinant of A - I is the same, it just adds a -1 to each term in the diagonal
tidal turret
computing the determinant of A is easy, since swapping the first two rows makes ...
thanks
imma be closing this, thanks
.solved
twilit field
So I was helping someone with a problem yesterday, want to make sure I didn't miss any solutions
midnight plankBOT
So I was helping someone with a problem yesterday, want to make sure I didn't mi...
twilit field
f(x)= 1+ ax, f(x) is its own inverse
find values of a
What I advised was f(f^{-1}(x)) = x, but f=f^{-1}(0)=1
so f(1)=0
we just need to solve this
but I'm concerned it's missing some solutions
subtle blaze
Why would f = f^-1(0) = 1
twilit field
I meant f(0)=f^{-1}(0)=1
subtle blaze
I would’ve just went f² = id
twilit field
I think I'm missing out on a lot of solns, yea
subtle blaze
So 1 + a(1 + ax) = x
twilit field
,w solve 1+a(1+ax)=x in a
grand pondBOT
Requested by math_rocks
lavish venture
I meant f(0)=f^{-1}(0)=1
yea you don’t need to do this
twilit field
,w solve 1+a(1+ax)=x for a
grand pondBOT
subtle blaze
,w solve 1+a(1+ax)=x for a
Now you see that your choice of x has fucked up the solutions
twilit field
yea, got it
subtle blaze
Because perhaps it wasn’t in the image to begin with
An assumption that you made without realising it
It’s good you noticed that you might have lost solutions though
subtle blaze
Also the other one clearly only works when the domain has but 1 x value
twilit field
It’s good you noticed that you might have lost solutions though
tbf (x-1)/x isn't really a "valid" solution as it isn't true for all x
subtle blaze
But I mean, could be the case
midnight plankBOT
midnight plankBOT
Available help channel!
molten bay
{1,-1,i,-i} is not klien four group
midnight plankBOT
{1,-1,i,-i} is not klien four group
molten bay
Because they are not satisfying i×i=-1 and identity is 1
nova yoke
is there a question here?
main current
The reasoning is spotty. Why should I care that i×i = -1? What does the identity have to do with this?
If you don't necessarily care about the reasoning, then yes you are correct that that's not the klien four group
lyric charm
(it's spelled klein)
main current
Dang it
nova yoke
as soon as you observe that $i^2 \neq 1$, you know it's not isomorphic to the klein group (why?)
grand pondBOT
Bungo
molten bay
If you don't necessarily care about the reasoning, then yes you are correct that...
I meant klein group has each element should be self inverse
And it is not satisfying
@nova yoke
nova yoke
which group? you can be more specific
midnight plankBOT
@molten bay Has your question been resolved?
molten bay
which group? you can be more specific
I only say this
midnight plankBOT
midnight plankBOT
Available help channel!
astral talon
what does the notation P(L\P) mean
for context im finding the probability someone bought only a lottery ticket. i assume it means L not P?
midnight plankBOT
what does the notation P(L\P) mean
for context im finding the probability some...
lyric charm
L without P
midnight plankBOT
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Available help channel!
astral talon
to find the modal class of outcomes of a binomial series of events, my teacher told us to find the expected value and see the highest probability around it.
would it be incorrect to instead do (n+1)p and round down? thus far all of them have been correct when i did it like this but it's not in my notes
midnight plankBOT
to find the modal class of outcomes of a binomial series of events, my teacher t...
flat spire
tbnd
astral talon
waterrbeam
midnight plankBOT
@astral talon Has your question been resolved?
astral talon
<@&286206848099549185>
tired ferry
to find the modal class of outcomes of a binomial series of events, my teacher t...
uhh
hmm
this is just optimization right
astral talon
wdym optimization
midnight plankBOT
@astral talon Has your question been resolved?
midnight plankBOT
@astral talon Has your question been resolved?
lyric charm
to find the modal class of outcomes of a binomial series of events, my teacher t...
lemme play around w that
ok so messing around w it on desmos reveals that your formula does look correct
im thinking that this can be proved by establishing that for fixed n, two adjacent probability masses are equal iff p = k/(n+1)
devout mirage
long dagger
to find the modal class of outcomes of a binomial series of events, my teacher t...
ts bozo needs help now
bro got helper role by asking for help 
midnight plankBOT
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tidal turret
in which case might the other equations of the plane come useful? because my course only teach Ax+Bx+Cy+D = 0 like the cartesian one, but In google I am seeing that also theres available point normal, intercept and vector form, is it a matter of how the exercise is designed? or I dont get it
midnight plankBOT
in which case might the other equations of the plane come useful? because my cou...
twilit field
yes