So just to be sure. Here I have to verify that if $A,B$ are normal subgroups of $G$ , then $g(A \cap B) = (A \cap B) g \forall g \in G$
#help-33
novel juniper
elfin berryBOT
wai
novel juniper
The intersection of two subgroups is a subgroup..Let $x \in g(A \cap B)$. The $\exists t \in A \cap B: x= gt$. But as $t \in A,B$ which are both normal,\exists t' \in A\cap B : gt=t'g$. So $x \in (A \cap B)g$. So $g(A \cap B) \subseteq (A \cap B)g$. $(A \cap B) g \subseteq g(A \cap B)$ similarly. So $g(A \cap B) = (A \cap B)g$
devout mauve
A,B being normal does not mean gt=tg
elfin berryBOT
wai
The intersection of two subgroups is a subgroup..Let $x \in g(A \cap B)$. The $\exists t \in A \cap B: x= gt$. But as $t \in A,B$ which are both normal,\exists t' \in A\cap B : gt=t'g$. So $x \in (A \cap B)g$. So $g(A \cap B) \subseteq (A \cap B)g$. $(A \cap B) g \subseteq g(A \cap B)$ similarly. So $g(A \cap B) = (A \cap B)g$
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.1422 ...t \in A,B$ which are both normal,\exists
t' \in A\cap B : gt=t'g$....
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.
Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
(./796939110815629322.aux)```
devout mauve
thats also not what it means
devout mauve
yes
novel juniper
so how is what I'm saying wrong
novel juniper
sure
marsh citrusBOT
novel juniper
.reopen
marsh citrusBOT
✅ Original question: #help-33 message
novel juniper
there exists t' in A and t'' in B with gt=t'g=t''g
$t'g=t''g$ with both in $G$. Then $t'(gg^{-1}) = t''(gg^{-1}) \implies t'=t''$
elfin berryBOT
wai
novel juniper
does this work
devout mauve
yes
marsh citrusBOT
marsh citrusBOT
Available help channel!
rustic crypt
what is the difference between the two formats
marsh citrusBOT
what is the difference between the two formats
rustic crypt
i had a different ticket opened before and i thought i was crazy but in that ticket i used 0 as true while looking at the left table
if i wrote our the table on the right in using both ttff format and 0011 format would they be Logically equivalent
wary lichen
what is the difference between the two formats
By using TF's and 01's you mean?
Nothing really
wary lichen
You might get the TF
In formal logic
And 01 in Boolean Algebra
But they are essentially the same
rustic crypt
i got myself mixed between the two and stared interpreting 0011 as ttff
wary lichen
Yeah, no, 0011 would be fftt
rustic crypt
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
subtle flare
Integration of sin inverse x whole square
marsh citrusBOT
Integration of sin inverse x whole square
past thicket
,rcw
elfin berryBOT
past thicket
$\int\arcsin^2(x)\di x$
elfin berryBOT
𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³
torpid tapir
Integration of sin inverse x whole square
if i was your teacher i would ban you from you pens
what kind of typing is this
even hebrew is more understandable than this
past thicket
Dude, if you have nothing to contribute to the problem, don’t say it.
past thicket
Exactly, so don’t even talk about the penmanship.
past thicket
I wrote it in latex above you.
past thicket
i can solve it but i cant read it
still, it’s irrelevant to the problem. No need for the comments.
torpid tapir
still, it’s irrelevant to the problem. No need for the comments.
whatever makes you sleep at night
past thicket
!nosols
marsh citrusBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
winged geode
if i was your teacher i would ban you from you pens
what is not understandable?
torpid tapir
what is not understandable?
the typing
winged geode
bro i think its just you
past thicket
the typing
delete the answer that you just sent
winged geode
it doesn't matter what the argument is because it's still in respect to the argument inside the sine
and everything else is understandable perfectly
subtle flare
i can solve it but i cant read it
I also typed the prob bro 😭
And my handwriting isn't that bad 😭
subtle flare
**𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³**
Arcsin ?
winged geode
i think its a combination of IBP and trig sub
subtle flare
means the same thing as inverse sine
Ohh ok
But it's wrong
winged geode
try integrating by parts and then using sine substitution
past thicket
**𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³**
$t=\arcsin(x)\Rightarrow x=\sin(t),dx=\cos(t)\di t$
winged geode
^
past thicket
Ohh ok
But it's wrong
wdym it’s wrong
elfin berryBOT
𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³
subtle flare
Uhh acc to the awnser key
past thicket
Uhh acc to the awnser key
arcsin = sin⁻¹
torpid tapir
arcsin = sin⁻¹
sin^-1 is csc
torpid tapir
it is
winged geode
sin^(-1)(x) ≠ (sin(x))^(-1)
subtle flare
winged geode
no it doesnt
past thicket
well it is maybe some diff use of it between us
it is. Read alex’s answer.
torpid tapir
am out
subtle flare
This is the ans
winged geode
Yes.
Try applying IBP first
And then you'll probably get some trig substition there
u = arcsin²(x) and dv = dx
try from there
winged geode
you can also go from there right away if you want
subtle flare
Ok just a sec
winged geode
**𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³**
but i'd go like this
past thicket
**𝙸𝚗𝚏𝚒𝚗𝚒𝚞𝚖³**
after doing all this you would do int by parts
oak oxide
after doing all this you would do int by parts
-# talk more infinium
-# the more messages the faster you become green 
past thicket
-# the more messages the faster you become green <:emojigg_Huh:13987832154053674...
-# im still pissed after that sekiro came in here to talk about penmanship instead of actually helping 
subtle flare
Wait I think I solved it
past thicket
Wait I think I solved it
Show your steps!
subtle flare
Few more mins
past thicket
okie dokie.
past thicket
Show the work.
winged geode
good answer
subtle flare
👍😃
supple junco
👍😃
Yoo no way
U actually read the manga
"We are gonna study ethics" or whatever
I forgot the name
It's very underrated
past thicket
!done
marsh citrusBOT
If you are done with this channel, please mark your problem as solved by typing .close
subtle flare
.close
marsh citrusBOT
odd agate
do you have a question?
marsh citrusBOT
odd agate
cool
supple junco
do you have a question?
No
marsh citrusBOT
Available help channel!
faint pewter
I have this question with my math work that I've been struggling with-- I understand how to do part a, I just don't understand how to translate the answers from that into the table in part b
marsh citrusBOT
I have this question with my math work that I've been struggling with-- I unders...
fervent tapir
maybe it would help if you can share the answers from part a.
wispy stump
I have this question with my math work that I've been struggling with-- I unders...
You need to find interest, principal and new balance for 3 months
faint pewter
maybe it would help if you can share the answers from part a.
here’s what I put in the calculator
fervent tapir
you have three screenshots here for two blanks in part a though.
faint pewter
the last 2 screenshots was me calculating the interest the way my professor told me to
wispy stump
here’s what I put in the calculator
Why am i getting 582. 35
fervent tapir
so screenshot 1 is the monthly payment and ss2-3 are the total interest
wispy stump
so screenshot 1 is the monthly payment and ss2-3 are the total interest
Yeah but ig she’s done some calculation mistake if im not wrong
faint pewter
Yeah but ig she’s done some calculation mistake if im not wrong
I can't tell what I typed wrong honestly i keep retyping it and i dont know if my calculator is just a bad one or
wispy stump
I can't tell what I typed wrong honestly i keep retyping it and i dont know if m...
No ig, cuz if you did it by calculator then you might he right
marsh citrusBOT
@faint pewter Has your question been resolved?
wispy stump
You need to find interest, principal and new balance for 3 months
@faint pewter you gotta find this for b part
blissful summit
can anyone help with me question a) and question b)
marsh citrusBOT
can anyone help with me question a) and question b)
blissful summit
the marking key for question a) says indepdent events
but if its a bernouli, why are we concerned with independent events
isnt that concern if we are considering if it can be binomial
heres the markig key
its very ambiguous
cunning fiber
but if its a bernouli, why are we concerned with independent events
the probability of success has to be the same every time the experiment is conducted
blissful summit
the probability of success has to be the same every time the experiment is condu...
wait a second i dont understand this
isnt bernouli concerned with only 1 trial
blissful summit
the probability of success has to be the same every time the experiment is condu...
why is it important for probability of success to be same for every time the experiment is conducted
if it changes, cant we define new bernouli for each for those indiviidual trials
blazing pulsar
I'm not sure what is confusing you here. A binomial distribution is a sequence of bernoulli trials with the same probability of success
It sounds like you are overthinking the meaning of part (a)
blissful summit
I'm not sure what is confusing you here. A binomial distribution is a sequence o...
but the question is asking why it can be considered a bernouli trial
and saying that the events are independent sounds like its justifying why it can be considered as binomial instead of bernouli
blissful summit
It sounds like you are overthinking the meaning of part (a)
ye i think im overthinking but why must events be independent for it to be able to be considered as a bernouli trial
blazing pulsar
I don't know where you are reading independence in part (a)
blissful summit
I don't know where you are reading independence in part (a)
the answer key says that "indepedent event" is a reason why it can be considered a bernouli trial
but i dont understand this marking key because it seems to be justifying why it can be considered as binmial instead
blazing pulsar
If you have two independent coinflips, you can think of it as two bernoulli trials or one binomial depending on your perspective
blissful summit
this is the marking key
blissful summit
If you have two independent coinflips, you can think of it as two bernoulli tria...
oo yea thats right
but why must it be indepedent events, like non constant probability of success
cant we define new bernouli parameters for each trial
blazing pulsar
If you have two dependent coinflips there are 4 outcomes not 2
the marginal distribution is bernoulli but the joint is not
blissful summit
how is there 4 outcomes
isnt there only head or tail outcomes
but the probaiblity are different
blazing pulsar
I said for two dependent coinflips. There are HT TH HH TT
At the end of the day it depends on what your random variable is tracking
blissful summit
I said for two dependent coinflips. There are HT TH HH TT
but this is binomial right
we are no longer tracking 1 trial
blazing pulsar
You asked why can't they be dependent events
blissful summit
ye they cant be depedent for binomial distrbutions, but how about bernouli
bernouli context dont even consider whether there are other trials or not, arent bernouli only focused on 1 trial
wait its okay i think its the question, how thye phrased it
they said why this can be considered a bernouli trial
but i still dont understand onn how i should solve question b i)
and the marking key also dont make sense
the marking key does this and i dont understand why do they this
and why do they not include the P(X=0) when considering P(Y=3) and missing P(Y=0) for P(X=3) case
this is question b i)
cann anyone help
@tight lance
torpid tapir
cann anyone help
what do you need
blissful summit
what do you need
i dont undestand why the question do P(X=0)+......P(Y=0) as a method to solve question b i)
and a big issue is that i dont truly understand what question b i) is actually asking
anyone helpers online?
marsh citrusBOT
@blissful summit Has your question been resolved?
granite dove
Is my proof fine?
marsh citrusBOT
Is my proof fine?
still temple
"only if" part is fine
"if" i am not quite sure what you are saying, what do you mean by smallest positive real number?
also the negation of < is $\geq$, just fyi
elfin berryBOT
Aude
granite dove
"if" i am not quite sure what you are saying, what do you mean by smallest posit...
Epsilon not defined to be the smallest positive number?
still temple
I use proof by contradiction
ah yep
still temple
Epsilon not defined to be the smallest positive number?
what is the smallest positive number
granite dove
So |a-b| > epsilon
still temple
can you find it?
granite dove
what is the smallest positive number
Wdym?
Epsilon
Is the smallest number
Like the definition
proud ice
Nope
still temple
Is the smallest number
there is no smallest number
granite dove
What
still temple
epsilon by itself is not defined to be anything it's just a placeholder for a set of values which are defined in the problem
still temple
What
you will learn about this when you continue with your study
i am pretty sure this exercise is in abbott
granite dove
epsilon by itself is not defined to be anything it's just a placeholder for a se...
So how to proof it
still temple
So how to proof it
i won't tell you how but i will help guide you
proud ice
For starters, you wrote you very first line incorrectly.
granite dove
For starters, you wrote you very first line incorrectly.
😢
still temple
you are correct to use contradiction though we just need to clarify some concepts
if epsilon is the smallest positive number then what is epsilon/2?
granite dove
you are correct to use contradiction though we just need to clarify some concept...
I think the problem in forall not?
We need to find some value
It make forall wrong
granite dove
if epsilon is the smallest positive number then what is epsilon/2?
Ah that make sense
proud ice
What you wanted to write was
$$\forall (a,b\in\bR)(a=b\Leftrightarrow \forall(\varepsilon\in\bR^+)(\abs{a-b}<\varepsilon))$$
elfin berryBOT
SWR
granite dove
But why i see in books that epsilon is placeholder for ver small value?
still temple
But why i see in books that epsilon is placeholder for ver small value?
that is its typical use
in analysis
but that doesn't mean it's the smallest value
well, because one doesn't exist
proud ice
It can be arbitrarily small, but there is no smallest value
still temple
think of a and b as two points on the real number line and |a - b| as the distance from a to b
what this definition is saying is that if this distance is arbitrarily small, as SWR said, i.e. for any positive distance you could possibly imagine - think of any insanely small number
then these two are actually equal
granite dove
But that mean a-b=0
If there insanely small number
Is there a number smallest than it not equal to 0
Let 0.00000000001
proud ice
Nope
granite dove
There 0.000000000000000001 smallest than it
still temple
it's saying that for any positive distance someone throws at you (which we will define as epsilon), the actual distance is less than that
and we know that |x| is always 0 or more
so if it can't be more
then it must be 0
but |a - b| is zero IF AND ONLY IF a = b
which you can and also should prove
still temple
it's saying that for any *positive* distance someone throws at you (which we wil...
does this make sense to you?
proud ice
Some properties of $\bR$: if $0<x$, then $0<x/2<x$. That is, no matter how small a positive number is, there is always a smaller one
elfin berryBOT
SWR
granite dove
does this make sense to you?
From that i conclude that the problem is forall epsilon
still temple
yes, that is in the problem statement
granite dove
If the statement is
There exist epsilon
granite dove
Then a=b not necessary
still temple
a = b is sufficient (but not necessary!) for "there exists eps in R^+ such that |a - b| < eps"
granite dove
a = b is sufficient (but not necessary!) for "there exists eps in R^+ such that ...
Can you use that bot to organize it
still temple
sure thing
still temple
$a = b$ is sufficient (not necessary) for the statement $\exists \epsilon \in \mathbb{R}^+$ and $|a - b| < \epsilon$. For two statements, $A$ is sufficient for $B$ is logically equivalent to $A \implies B$. So $a = b \implies \exists \epsilon \in \mathbb{R}^+$ such that $|a - b| < \epsilon$
elfin berryBOT
Aude
granite dove
$a = b$ is sufficient (not necessary) for the statement $\exists \epsilon \in \m...
Thank you
still temple
We need to use some counter example?
we use counter-examples to disprove statements
your instinct to use contradiction is actually the most elegant approach
and it is actually the one thing that ties this problem to the "for all" quantifier for positive epsilon
granite dove
What you think? @still temple
proud ice
What you think? <@456226577798135808>
First looks good. Reading second
Looks good. Just one question: what is your proof that $x\ne0\to\abs{x}>0$? (That is, is this something you have proven already?)
elfin berryBOT
SWR
still temple
What you think? <@456226577798135808>
hello! Sorry! Was making tea
you can take SWR’s word though if you are in a rush, I am sure he is more qualified than me
still temple
What you think? <@456226577798135808>
Good!
granite dove
Looks good. Just one question: what is your proof that $x\ne0\to\abs{x}>0$? (Tha...
Ok
still temple
Also make sure you understand why you can let eps equal the distance between a and b
It’s precisely because of what I said before
This statement holds for all real positive numbers
So that’s why we can choose epsilon to be |a-b|
And we don’t even know what it is - just by the virtue of it being non negative and nonzero it is automatically a “candidate” for epsilon
And if you are done, feel free to type ,close @granite dove
granite dove
And if you are done, feel free to type ,close <@607963152344416256>
I went to eat and now I'm back. I'll solve what SWR sent.
granite dove
Also make sure you understand why you can let eps equal the distance between a a...
Cuz the statement said for all epsilon
And |a-b| > 0 so epsilon can be |a-b|
For all a,b>0 ,There exist epsilon>0 such that epsilon=|a-b|
granite dove
For all a,b>0 ,There exist epsilon>0 such that epsilon=|a-b|
Maybe the quantifiers applying is ass but i understand the concept
granite dove
Looks good. Just one question: what is your proof that $x\ne0\to\abs{x}>0$? (Tha...
@proud ice what you think
still temple
And |a-b| > 0 so epsilon can be |a-b|
yes!
granite dove
Can you see that for me? @still temple
proud ice
Looks fine. Proof might be a little long but still fine
fickle shell
marsh citrusBOT
fickle shell
I see what they did but idk why they did it
How did they decide that r = 3 and the number of elements in X = n
and the answer was 56
Normally these questions involve slots and things that go in those slots
I cant tell what are supposed to be the slots and what are supposed to be the things in the slots
if that makes sense
elfin cairn
the number of subsets of size k of a set of size n is (n choose k)
you choose the elements in the set
elfin berryBOT
Julian
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
fickle shell
you choose the elements in the set
So the subsets are the slots?
And the elements are the things that go inside?
gritty vault
sure 3 slots
elfin cairn
So the subsets are the slots?
the subset itself is a slot
fickle shell
sure 3 slots
But how can an element be a slot sry
gritty vault
But how can an element be a slot sry
it goes in the slot
if we have n marbles in X
and 3 slots
its saying 56 different ways to fill the slots
but order doesnt matter
fickle shell
Ok just so that I understand, n choose k, n is always the number of slots and k is always the things i need to pick, right?
just generally not specifically for this question
gritty vault
n is the size of X, k is number of slots, and n choose k gives us the amount of unordered ways to fill the slots
for example if we have the set {1,2,3}
n=3
and we want a subset of size 2
so 2 slots
we have {1,2}, {2,3}, {1,3} = 3 total subsets or ways to fill the slots
we dont count {2,1}, {3,2}, {3,1} cuz order doesnt matter
fickle shell
Ohhh
gritty vault
its kinda confusing the first time maybe watch a video with examples
marsh citrusBOT
marsh citrusBOT
Available help channel!
mental pike
why does this limit equal 1/2?
marsh citrusBOT
why does this limit equal 1/2?
mental pike
assuming it actually does
elfin cairn
intuitevly n^2 - n is approximatly n^2 where n is large
and so your expression is approximately n/(sqrt(n^2) + n) = 1/2
to do it more formal, divide both numerator and denominator by n
and then this should simplify nicely to an easy limit
mental pike
but how do i divide the denominator by n when it has that root
mental pike
oh like the root divided by n + n/n?
gritty vault
so
sqrt(n^2-n)/n = sqrt(n^2-n)/sqrt(n^2)
=sqrt((n^2-n)/n^2)=sqrt(1-1/n)
and 1/n tends to 0
so its sqrt (1-0)
=1
plus the other 1 we have 2 in the denominator
1 in numerator
mental pike
oh so you made n sqrt(n^2) to work with the root
and the division by n is like a standard thing when calculating limits
i think i understand it now
thank you
.close
marsh citrusBOT
mental pike
.reopen
marsh citrusBOT
✅ Original question: #help-33 message
earnest belfry
why does this limit equal 1/2?
I can try to help also
mental pike
wait sorry i dont get this step
im on this step rn
earnest belfry
Okay so you have an algebraic\manipulative method
and you have on the other hand L'Hopital's rule
L'Hopital's rule is applied once you learn derivatives
But, if that isn't the case for now, we resort to algebraic manipulation
earnest belfry
why does this limit equal 1/2?
So here, multiply by the conjugate of the denominator
rationalize the denominator
So, it becomes:
$\frac{n\sqrt{n^{2}-n}}{n^2-n+n}$
wait . . .
So, it becomes:
$\frac{n(\sqrt{n^{2}-n}-n)}{n^2-n+n}$
elfin cairn
and all of this leaves you with something ugly
earnest belfry
My math isn't mathing, nvm
elfin cairn
what I suggested at the beginning is good
silk sable
Perhaps factor out n^2 from the radicand.
elfin cairn
or uhhh 1-1/n?
$\sqrt{\frac{n^2 - n}{n^2}} = \sqrt{1-\frac{1}{n}}$
elfin berryBOT
ExpertEsquieESQUIE
elfin cairn
is this what you don't understand?
mental pike
i think i missed the mention of how 1/n is going to 0
so its 1-0 under the root right?
elfin cairn
yes
mental pike
yeah sorry im scatterbrained
elfin cairn
all good
marsh citrusBOT
marsh citrusBOT
Available help channel!
lyric kelp
i did a proof by case here. to show if x is rational, all the following statements are also rational. however when it comes to x being irrational, idk how to prove (ii) and (iii) to be irrational too
marsh citrusBOT
i did a proof by case here. to show if x is rational, all the following statemen...
wary kite
i did a proof by case here. to show if x is rational, all the following statemen...
i mean seems like you've already done it then?
mellow kraken
you'll need to show four things
- if (i) is true, then (ii) is true
- if (ii) is true, then (i) is true
- if (i) is true, then (iii) is true
- if (iii) is true, then (i) is true
void elm
(technically, you don’t need to show four, only three, but four is probably easier here)
mellow kraken
yeah, technically 3
- if (i) is true, then (ii) is true
- if (ii) is true, then (iii) is true
- if (iii) is true, then (i) is true
basically, you need to set up some chain of implications such that any statemeny implies any other statement.
lyric kelp
oh yeah i can just do a contrapositive of a biconditional oh... normally for p <--> q we have to prove p-->q and q-->p, here we just prove neg p-->neg q and reverse?
okay yeah i forgot i can just do that ty
.close
marsh citrusBOT
mellow kraken
¬p -> ¬q is the same statement as q -> p.
marsh citrusBOT
Available help channel!
twilit kindle
marsh citrusBOT
twilit kindle
Okay so how he got a-x after observing
marsh citrusBOT
@twilit kindle Has your question been resolved?
wheat trench
Which version do y'all prefer?
marsh citrusBOT
Which version do y'all prefer?
wheat trench
The underlined one feels a little redundant, but also kinda sounds better.
night relic
"$\in \mathbb{G}$" and "is a member of $\mathbb{G}$" are pretty much the same. both are accurate and widely used \
personally, though, i think you may want to go with "$a,b,c$ be members of $\mathbb{G}$" if you'd like to keep that text
elfin berryBOT
ηασιβ ♥
wheat trench
To understand what you mean: Are you saying, either keep the text or remove it, but don't have both? like, don't have " in G be members of G" . Or are saying that its fine in any combination?
night relic
it is fine in any combination, because it's understood what the elements are, and what the bigger set is. that's really all that matters.
however, "in G be members of G" is technically redundant, so if we're being really specific, i would go for only one of them
my personal process is this: the "natural" thing to put inside of a group is an element, so if you say $g \in G,$ that's going to be unequivocally understood as an element of a group. if, however, i thought there might be ambiguity, i might say "let $(gh)^2j^{-1}$ be an element of $G$", so it's clear what $\in$ is doing here
elfin berryBOT
ηασιβ ♥
wheat trench
I initially had the first one, just "Let 𝔾 be a group and 𝖺, 𝖻, 𝖼 ∈𝔾.", but it almost sounds like the sentence cuts off, when I read it in my mind. But, if it seems ok to have that, then I'm gonna go for this version.
I was just concerned that it might have been "wrong" in some type of way.
Thanks for the feedback. I appreciate it.
.close
chilly fable
i mean seems like you've already done it then?
marsh citrusBOT
@deep spindle Has your question been resolved?
hazy fox
Q3
hazy fox
This is under chapter differentiability so ig its gonna use that topic
marsh citrusBOT
@hazy fox Has your question been resolved?
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
Let $f(x)=x 2^y + y2^{-x}-x-y$ and note that $f(0)=0$ (which means the two sides are equal when $x=0$)
elfin berryBOT
Civil Service Pigeon
cunning fiber
you should see from this that ||it suffices to show f is never zero again||
Further hint: ||is f decreasing? increasing?||
marsh citrusBOT
@hazy fox Has your question been resolved?
hazy fox
Further hint: ||is f decreasing? increasing?||
Yes so how do i show this part
sage nova
Yes so how do i show this part
Couldn't you take the derivative to show if it is decreaing or increasing, and then from that show that f will never equal zero again, and hence the two sides will not be equal?
restive coral
Hello
sage nova
I am taking Differential Equations next semester, but that is just my guess
restive coral
Est ce que une application peut etre non injective ni surjective ou bijective
?
Donner moi un reponse
sage nova
!occupied
marsh citrusBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
restive coral
@marsh citrus est ce que une application peut etre non injective ni sujective ou bijective
hazy fox
Couldn't you take the derivative to show if it is decreaing or increasing, and t...
Its a multivariable function
sage nova
Its a multivariable function
Well yeah but take the partial wrt to y and divide it by the partial wrt to x
I thought that works
hazy fox
Well yeah but take the partial wrt to y and divide it by the partial wrt to x
Idk how to find increasing decreasing nature of a multivariable function 😭
Its not in our syllabus either
sage nova
Idk how to find increasing decreasing nature of a multivariable function 😭
Yeah I am pretty sure that derivative dy/dx is just partial of y divided by the partial with respect to x, then you can check if it’s increasing
Try that
That’s just my guess
hazy fox
Oooh is that so
marsh citrusBOT
@hazy fox Has your question been resolved?
elfin onyx
how to solve (X+3)^3 + 64?
marsh citrusBOT
how to solve (X+3)^3 + 64?
elfin onyx
i mean how to factorize it
weak surge
i mean how to factorize it
Look at it. Is it in the form a^3 + b^3?
weak surge
what do i do after it
Do you know how to factorize a^3 + b^3?
elfin onyx
Do you know how to factorize a^3 + b^3?
yea
elfin onyx
(a+b)(a^2-ab+b^2)
elfin onyx
.close
marsh citrusBOT
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steel plinth
Need help understand the expression I marked up in the red box
marsh citrusBOT
Need help understand the expression I marked up in the red box
steel plinth
How does subtracting two points and dividing them by their distance=x²? (Aka 3²)
devout mauve
A(3.001) is the area from 0 all the way to the second of those two red lines
and A(3) is the area up to the first of them
so A(3.001)-A(3) is the small area between them
the one in yellow
this area is approximately a rectangle
with height 3^2 and base 0.001
so if you divide its area by 0.001 you get 3^2
marsh citrusBOT
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faint grove
The data A has 4 elements and the average is 5,the standard 편차 (표준편차) is 2
And the data B has 6 elements and the average is 5, the standard편차(표준편차) is 3
Then what is the standard편차 of both sets of data A and B?
marsh citrusBOT
The data A has 4 elements and the average is 5,the standard 편차 (표준편차) is 2
And ...
wispy stump
The data A has 4 elements and the average is 5,the standard 편차 (표준편차) is 2
And ...
Since both their mean is 5, so combined mean will be same right?
faint grove
What the heck?
desert dirge
<@&268886789983436800>
wispy stump
Irrelevant bro delete
faint grove
Since both their mean is 5, so combined mean will be same right?
Maybe not if it has something lower than 0?
wispy stump
Since both their mean is 5, so combined mean will be same right?
You can apply formula for combined variance when means are equal
wispy stump
Maybe not if it has something lower than 0?
No they said average is 5 for both right so mean will be 5 for both
faint grove
I have no idea whether the average of combination of both datas will still be 5 or not if it is a variable
wispy stump
We compare overall mean no matter what their individual values
wispy stump
I have no idea whether the average of combination of both datas will still be 5 ...
Its in the question right?
faint grove
The average of combination of data A and B
wispy stump
The average of combination of data A and B
It will be same
smoky parrot
average of both datas will be $$\frac{\sum n_i x_i}{\sum n_i}$$ , x_i being the averages and n_i being the number of averages of each dataset
elfin berryBOT
RadMeerkat62445
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
smoky parrot
so 5
wispy stump
The average of combination of data A and B
Take a data set of 4 numbers (2,4,5,9) and another data set of 5 numbers (3,4,5,6,7) now take their averages individually and take their combined average
It’ll be the same
faint grove
There is a chance of either data A or B containing something lower than. 0
wispy stump
There is a chance of either data A or B containing something lower than. 0
Do it with (-2,6,7,9) and (-5,0,8,10,12)
It’ll be the same mate
When it says average is 5, then it is 5 no matter what numbers you use
wispy stump
Do it with (-2,6,7,9) and (-5,0,8,10,12)
Calculate individual and combined average for both
faint grove
I think it was true
faint grove
Maybe the thing that you said
wispy stump
Maybe the thing that you said
Yeah, remember when it says average is 5, then it is, no matter what number combinations you use
So now we have combined mean 5 right
So use combined variance formula for combined mean
smoky parrot
The average of combination of data A and B
Let us say we have a dataset of m elements $(a_1, a_2, .. a_m)$ and of n elements $(b_1, b_2, .., b_n)$.
Then, the average of the dataset of m elements can be denoted A and is given by $\frac{a_1+a_2+...+a_m}{m} $ and that of the dataset with n elements can be denoted B and is given by $\frac{b_1+b_2+...+b_n}{n}$. But now $mA = a_1+a_2+...+a_m$ and $nB = b_1+b_2+...+b_n$.
We want to find the average of the combined dataset $(a_1, b_1, a_2, b_2, ... a_m, ... b_n)$.
We remember that the average is the sum of all observations (i.e., the sum of all elements $(a_1, b_1, a_2, b_2, ... a_m, ... b_n)$ divided by the number of observations. In our case, we have m observations from the first dataset and n observations from the second dataset, so we have m+n observations.
Then, the combined average C is $\frac{a_1+b+1+a_2+b_2+...+a_m+...+b_n}{m+n}$. But from above, $mA = a_1+a_2+...+a_m$ and $nB = b_1+b_2+...+b_n$. Hence C = $\frac{mA+nB}{m+n}$.
elfin berryBOT
RadMeerkat62445
marsh citrusBOT
@faint grove Has your question been resolved?
wispy stump
@faint grove You dont know the formula for combined variance?
faint grove
I don't know
wispy stump
I don't know
Alright, write it down then
wispy stump
I don't know
@faint grove You get it now?
Lmk if you want me to explain it with our question
faint grove
Yes
wispy stump
Yes
We have a data set of 4 elements
So n1 = 4
With standard deviation 2
So square of standard deviation is variance
So sigma1^2 =2^2
Can you try for n2 and sigma2^2? @faint grove
faint grove
( 4 * 2 * 2 + 6 * 3 * 3 ) / 10 then?
wispy stump
( 4 * 2 * 2 + 6 * 3 * 3 ) / 10 then?
Yeah
Then what you’ll get is variance
And sqrt of variance is standard deviation
Its done
@faint grove Got it?
faint grove
Yes
wispy stump
Lmk your answer so can verify
faint grove
7
marsh citrusBOT
@faint grove Has your question been resolved?
hazy fox
Question 9
marsh citrusBOT
Question 9
hazy fox
My thought process yet
Use rolles theorem so there exists a c for which f'(c) and f''(c)=0 for some c€(a,b)
Trying to define a function on which i can use imvt but i cant put my hands on that part
elfin cairn
you are close
the two c's that you get from rolle's theorem are not the same necessarily
and yes, try to think what can you use IVT on
also I think your function needs to be continuously second differentiable
hazy fox
and yes, try to think what can you use IVT on
Yes i cant think of it 😭
elfin cairn
the function in the equation 
hazy fox
the function in the equation <:catcothink:827248531730726932>
if i consider that as a function i would get f'''
I dont have any info about f'''
elfin cairn
the function in the equation is
f'' - lambda (f')^2
hazy fox
Yes but rolles theorem is f'(c)= 0
elfin cairn
yes
hazy fox
The derivstive of that equation will give me f'''
marsh citrusBOT
@hazy fox Has your question been resolved?
neat grove
lol
hey
bro
this is NOT highschool math
@hazy fox
dont we have to use
rolles theorem
marsh citrusBOT
@hazy fox Has your question been resolved?
hazy fox
this is NOT highschool math
Its entrance exam prep for college
So technically pre uni :😭
hazy fox
rolles theorem
Yes
vagrant raft
The derivstive of that equation will give me f'''
You're not supposed to take the derivative
vagrant raft
We don't know is f''' even exists
hazy fox
We don't know is f''' even exists
What is the function i should assume then
/what to do
neat grove
@cunning fiber help him out bro😪
vagrant raft
What is the function i should assume then
you'll want to use IVT here
hazy fox
you'll want to use IVT here
Yes i figured that much out
vagrant raft
so you need to find a point where f''+ alpha f'^2 >=0 and another where it is <= 0
for that you can look at points where f' = 0
marsh citrusBOT
@hazy fox Has your question been resolved?
foggy sphinx
Need help with quadratic equations
marsh citrusBOT
Need help with quadratic equations
stoic saddle
send your question(s) here
stoic saddle
I just don’t understand how this works
the vertex of a parabola is its "turning point" i.e. where it switches from going up to going down (or vice versa)
looking at this graph can you scribble over it to pinpoint where the vertex is
(or if you feel up for it, name the coords right away -- but a visual will do fine)
foggy sphinx
umbral oasis
What are its coordinates?
foggy sphinx
I honestly do not know I don’t know what I’m supposed to be looking at
stoic saddle
right, would've liked a more precise reading
tho you are in the right ish place
gimme a sec
umbral oasis
I honestly do not know I don’t know what I’m supposed to be looking at
This point
You see it's derivative there is 0
stoic saddle
You see it's derivative there is 0
this is way way too early to be talking about derivatives
fervent tapir
OP, maybe it helps to think of the vertex of a parabola as the "tip" of the parabola.
stoic saddle
foggy sphinx
I see that’s 8 right
umbral oasis
That's not the point of what I'm talking about
umbral oasis
I see that’s 8 right
Yes
stoic saddle
8 is its y-coordinate
umbral oasis
So what is the tuple
stoic saddle
That's not the point of what I'm talking about
i would bet €20 op does not yet know what a derivative is
given he is just studying quadratics atp
fervent tapir
not sure if we should also use the word "tuple" here too, given the same thing.
foggy sphinx
Wouldn’t X be -3
stoic saddle
also yes i would ask for coords
stoic saddle
Wouldn’t X be -3
the point is sitting directly on the y-axis
what should its x coord be
(also dont confuse this point with other pts of interest on the parabola)
foggy sphinx
So what should I be looking at for the X cord
stoic saddle
@umbral oasis i think my point is quite clear, your explanation is overengineered to the point of not connecting with op's knowledge. simple as that.
stoic saddle
So what should I be looking at for the X cord
what is the x-coord of a point that's sitting directly on the y-axis?
this doesnt require you to know anything about parabolas specifically
just reading coordinates from an xy grid
umbral oasis
It's not overengineered. I literally used the word derivative once. If he doesn't understand it he can choose to completely ignore that part of my "explanation"
Or ignore my explanation entirely
foggy sphinx
10?
foggy sphinx
Idk what I’m supposed to be looking at to find the x cord
stoic saddle
you're not supposed to be like, "finding" the x-coord by means of a formula
are you OK with us going over some basics of coordinates real quick
foggy sphinx
Yes
stoic saddle
ok right
stoic saddle
the graph is not fully marked
we're trying to establish what the x coord is.
sacred idol
oh okk
foggy sphinx
the graph is not fully marked
We already have y we need X
stoic saddle
you know the point where the x and y axes meet?
this point has a special name
do you knwow this na-
augh.
well, i can't really continue my explanation if other people are going to talk over me, i think.
@foggy sphinx the point where the x and y axes cross each other is special enough to have its own name. do you know what it is called?
foggy sphinx
<@910700208789258300> the point where the x and y axes cross each other is speci...
No
foggy sphinx
A lil
stoic saddle
does it ring enough of a bell that you can tell me what the coordinates of the origin are?
(again, forget about parabolas, we're just going over coordinate grid basics)
foggy sphinx
I feel like it’s zero
stoic saddle
a point on the plane has two coordinates, written in brackets (obligatory, you can't drop them, ever) and with a comma in between.
foggy sphinx
Huh?
stoic saddle
it takes two numbers to tell where a point is
an x and a y
they are written as follows: (x,y)
i was also emphasizing that dropping the brackets and writing x,y is wrong.
go give this a watch tbh
foggy sphinx
K
marsh citrusBOT
@foggy sphinx Has your question been resolved?
fervent rampart
looks like a calligraphic C, $\mathcal{C}$
elfin berryBOT
κλαουντ ☁ (cloud)
fervent rampart
they don't look exactly the same because they are in different fonts (the C in that message appears to be from the ams euler calligraphic font)
marsh citrusBOT
Available help channel!
upper star
please help me solve. i don't even know how to start solving this
marsh citrusBOT
please help me solve. i don't even know how to start solving this
stoic saddle
do you know how to find intersections of straight lines on a coordinate plane?
upper star
no.
upper star
🙁
restive sparrow
What topic are you doing right now?
upper star
this is the bonus problem of my alg homework
stoic saddle
actually wait let's go further back
if they asked for the length of AB, would you know how to find that
stoic saddle
ok
stoic saddle
58?
restive sparrow
this is the bonus problem of my alg homework
What was the homework about in particular?
wise lance
But what if F isn't perfectly on the point
stoic saddle
But what if F isn't perfectly on the point
this is a square grid
- i am tracing a horizontal thru C
upper star
this is a square grid
even i know that
knotty trellis
<@&268886789983436800>
upper star
WHAT
inland frigate
<@&268886789983436800> SPAMMER
charred silo
Yikes. Has been muted and presumably banned
stoic saddle
yuck
lean cedar
did someone @ me?
stoic saddle
did someone @ me?
mistaken modping
lean cedar
lol
stoic saddle
might wanna change your nick
upper star
i'm very afraid rn
stoic saddle
i MENT MODERATORS
calm down, it's been handled already
upper star
i was looking for help with math...
upper star
ok
sleek nimbus
I was thinking about setting axis and finding the equation of the lines ab and dc and then find the point e as their intersection
stoic saddle
I was thinking about setting axis and finding the equation of the lines ab and d...
i tried feeling out that route
but op said he doesn't know how to proceed from that
knotty trellis
this could also work if u know what i mean by that
upper star
i can do that
knotty trellis
sleek nimbus
if you know how to find line equations then finding the intersection is a very simple idea
knotty trellis
can u notice sth with this pic
knotty trellis
the distances between the blue lines is
thats not really true
upper star
also they are all parallel
knotty trellis
what's the proportion of (red+orange) : yellow
upper star
1:1
knotty trellis
sorry, (red+orange) : yellow
upper star
2:1
knotty trellis
yeah, and can you find the total length of the colored segment?
i.e. red+orange+yellow?
upper star
1+sqrt(2)
knotty trellis
hmm not really
how did u get that?
notice that the endpoints lie on the grid
so u can just pythagoras it
upper star
yes
knotty trellis
i think u did ur pythagoras wrongly then
knotty trellis
can u now find the distance from the left endpoint of red to E?
knotty trellis
np
upper star
.close
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slim ginkgo
Linear transforms
marsh citrusBOT
Linear transforms
jagged relic
What's your question / what have you tried?
sleek nimbus
!status
marsh citrusBOT
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
.close
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safe cloud
Sorry for the amount of questions I’m asking but I’m trying to solve for #17 b.
safe cloud
and im at the point where im finding the limits
based on what ive done for the cusp im tryna do the same to see if theres a vertical tan line but
would that mean i would have to find the limits for all 3 critical points
or is there some shortcut i can take/ are there steps that im doing wrong
faint creek
Is 6 a critical point? Is it not just 0 and 12?
faint creek
But anyway, since f'(6) = 0, we can tell that it's going to be flat rather than a cusp or vertical tan
faint creek
Well the other two have f'(c) approaching +- infinity, which implies a cusp or vertical tan
faint creek
A cusp is what you have in (a) right? Where the function makes a sort of spike shape?
safe cloud
yeah i didnt solve it on the paper but its a cusp on a
i was doing it in my notebook, im still a little iffy on it but i do get it
faint creek
Yeah, so if you remember how the way a curve tilts up/down relates to the positiveness/negativeness of the derivative
You can see how the critical point in (a) is a cusp because f' goes to negative infinity from the left (So it's going down), while it goes to positive infinity from the right (so it's going up)
While if you had a vertical tan, it would have to be approaching either negative or positive infinity from both sides
So once you've found either a cusp or vertical tan, you can tell the difference by analyzing the sign of the derivative
Does that make sense at all?
safe cloud
So once you've found either a cusp or vertical tan, you can tell the difference ...
could you explain this part a bit more
faint creek
Sure thing
How about looking at this graph of your work from (a)
The blue line is your original function, while the green one is the derivative
You can see that the derivative is going to -infinity from the left, so the curve is going down really fast
safe cloud
oh my god this visualization actually just simplified it so much
😭 thank you anyways yes continue
faint creek
While it's going to +infinity from the right, so it's going up really fast!
Lmao no problem, graphing a problem is often a great idea if you're having trouble understanding it :)
So a cusp is when the derivative is going to +infinity from one side and -infinity from the other, causing that sort of spike shape
While with a tangent line the derivative will go to one of -infinity or +infinity from both sides, since the curve won't be changing direction as it goes through the critical point
Remember, positive derivative means it's going up, negative derivative means it's going down
safe cloud
that'll be it thank you
faint creek
Cool, can you use .close to free the channel then?
safe cloud
.close
marsh citrusBOT
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dusty bone
How do I solve question (ii)
marsh citrusBOT
How do I solve question (ii)
late geode
use the slope and y-intercept in the y-intercept form and you'll have your equation
late geode
yes
marsh citrusBOT
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chilly fable
😋
indigo nest
Given that they just joined and that username, I feel confident saying they're a bot
ruby mulch
.close I don't think there is any real question here.
hard laurel
an is absolutely convergent and bn is convergent series then their multiplication anbn will be convergent?
marsh citrusBOT
an is absolutely convergent and bn is convergent series then their multiplicatio...
hard laurel
I am looking for examples so that i can be sure
sick charm
What kind of examples are you looking for? Do you have examples for absolutely convergent series?
elfin berryBOT
Ann
stoic saddle
the zero series is in fact convergent, yes
hard laurel
I was looking for such examples where it fails if does
static quarry
you do some manipulations and at some point you say, "it would be nice if b_n were a ____ sequence"
sick charm
at some point, |anbn| would be smaller than |an|.
marsh citrusBOT
calm harbor
hard laurel
Hey?
calm harbor
hello
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Available help channel!
sacred idol
Let ${\delta_1, \delta_2, ..., \delta_k}$ be the set of determinants formed by the set of 9 real and non-zero numbers ${a_1, a_2, a_3, ..., a_9}$.\\
what is the value of k?
marsh citrusBOT
Let ${\delta_1, \delta_2, ..., \delta_k}$ be the set of determinants formed by t...
elfin berryBOT
sick charm
What is a determinate for a set of numbers?
sacred idol
nah its okay
just consider if 9!/18 would be correct
im not sure what the answer is
indigo nest
We can't help you with the question if you don't clarify what the question says
stoic saddle
What is a determinate for a set of numbers?
i think what's meant is like, all the possible values you can get by sticking the a's into a 3×3 determinant in every which way?
sacred idol
We can't help you with the question if you don't clarify what the question says
oh i didnt realize he wanted clarification on the question
mb
determinant with the you know
lavish pelican
so can the question be rephrased like
How many determinants can be formed by the set of 9 real non zero numbers
sacred idol
so can the question be rephrased like
How many determinants can be formed by the...
sure
uhh wait nvm distinct is not given
but how would you do it if it was distinct
lavish pelican
if its distinct it should be 9! right
sacred idol
if its not distinct
lavish pelican
that makes it a little tougher
i dont know
cause there are too many combinations
indigo nest
if its distinct it should be 9! right
No cuz you can rearrange rows and columns to get the same determinant for different matrices
marsh citrusBOT
@sacred idol Has your question been resolved?
marsh citrusBOT
@sacred idol Has your question been resolved?
faint creek
Hmm... what if we look at 2x2 matrices first? So just {a1,a2,a3,a4} ?
Or rather {a,b,c,d}
I think the maximum number of determinants made from 4 values would be 6, from
ab-cd
ab-dc
ac-bd
ac-db
ad-bc
ad-cb
Or... actually 12 because we would need the negatives of all of those
vagrant raft
Do negatives count though
sacred idol
duh??
faint creek
Well 1 =/= -1 so I assume so
vagrant raft
Since we chose a1... a9 and not -a1... -a9
Actually it doesn't matter since we can get the opposite determinant by swapping columns
faint creek
"... by swapping columns" that's what I meant by the negatives
vagrant raft
I think the maximum number of determinants made from 4 values would be 6, from
`...
Aren't some of them duplicate?
faint creek
No! This has all the duplicates taken out
I just also took out the negatives by accident
Or....... No wait I'm dumb????
Too much screwy algebra, I forgot cd=dc lol
vagrant raft
Why isn't ad-cd equal to ad-dc
sacred idol
I think the maximum number of determinants made from 4 values would be 6, from
`...
ab - cd
ab - dc
bruh
sacred idol
i would say
faint creek
Anyway, so take out my doubled half and add in the negatives and we get back to 6
sacred idol
for a given matrix
either a transpose is performed
or/and one of these two choices
- swap rows AND columns
- do nothing
which gives 2 x 2 = 4 matrices in a group with same det
so 24/4 = 6
faint creek
Well there's only 1 way to do a row + column swap isn't there?
I think these 4 matrices should have the same det
[a b]
[c d]
[d c]
[b a]
[a c]
[b d]
[d b]
[c a]
yeah so thats one choice
and the other one is to do nothing
faint creek
Those last 2 matrices can't be gotten from the first 2 by row/column operations
vagrant raft
Those last 2 matrices can't be gotten from the first 2 by row/column operations
Transpose is the other option
vagrant raft
So for the 3x3 case we would have at most 9! possible configuration divided by 433 : even number of row operation and column operation, odd number of row and column operations, and twice that number for transpose?
faint creek
So if it's transpose and double operations... we have 2 * |S_3|^2/2 = 36 redundant permutations?
vagrant raft
I got the same
faint creek
Hell yeah
sacred idol
So if it's transpose and double operations... we have 2 * |S_3|^2/2 = 36 redund...
guess so
faint creek
But that's assuming we've got them all
sacred idol
ill try with an example
faint creek
All the ways of making a redundant permutation I mean
sacred idol
we should get 10080
faint creek
That's a lot of cases to check
vagrant raft
I think we can choose 9 transcendent numbers and that will make sure that we can't get the same det with different formulas
Actually square roots of prime numbers should work
marsh citrusBOT
@sacred idol Has your question been resolved?
marsh citrusBOT
@sacred idol Has your question been resolved?
vagrant raft
Wdym
marsh citrusBOT
@sacred idol Has your question been resolved?
sacred idol
uhh nvm