yay
#help-42
wild marten
elder pawn
@wild marten theres two defs of normal subgroup going round and im not sure u know theyre equivalent
calm coralBOT
@wild marten Has your question been resolved?
calm coralBOT
calm coralBOT
Available help channel!
lean folio
Sorry, it's in french but I think I'm missing something
calm coralBOT
Sorry, it's in french but I think I'm missing something
mortal orbit
Sorry, it's in french but I think I'm missing something
On dit de cette fonction qu'elle est "radialement" continue en 0
lean folio
So, I have a function continuous and we want to know if it's continuous in (0,0) too, so we used the polar coordinate system and we found that the lim of f is 0
Oh, un français
mortal orbit
So, I have a function continuous and we want to know if it's continuous in (0,0)...
par contre ça c'est pas exactement vrai
limite de cette quantité à theta fixé
lean folio
par contre ça c'est pas exactement vrai
comment ça ?
mortal orbit
c'est pas pareil que limite de cette quantité quand theta a le droit de bouger comme il veut, tant que r->0
lean folio
oh okay
mais comment on sait que l'on ne peut pas majorer par une fonction indépendante de thêta ?
mortal orbit
bah là c'est un peu compliqué de l'exprimer en fonction de theta
mais en fonction de x,y on a (x,y) = (t,t^2) qui est un chemin continu qui tend vers (0,0)
et pourtant f(x,y) tend pas vers 0
donc je te laisse trouver à quel (r,theta) ça correspond
r = racine(x^2+y^2)
et tan(theta) = y/x
mortal orbit
le chemin?
mortal orbit
lean folio
et quand on applique f à ce chemin on trouve 1/2 ?
mortal orbit
oui, tu peux le voir facilement
lean folio
à cette courbe en fait
mortal orbit
f(t,t^2) = t^4/(t^4+t^4)
lean folio
f(t,t^2) = t^4/(t^4+t^4)
oui = 1/2
primal bear
mortal orbit
oui = 1/2
donc effectivement, même si f est radialement continue
elle n'est pas continue pour autant
le besoin de laisser theta bouger comme il veut est important
subtle kayak
croissant tres bien 👍🏻
mortal orbit
donc voilà finalement tu aurais besoin, si tu voulais prouver que f est continue en (0,0)
de montrer que $|f(x,y)-f(0,0)|\leq g(r) \to 0$
potent lotusBOT
rafilou is not not born in 2003
mortal orbit
avec $g$ une fonction indépendante de $\theta$
potent lotusBOT
rafilou is not not born in 2003
lean folio
et là c'est pas possible parce qu'on ne peut pas majorer par une constante indépendante de thêta
mortal orbit
on peut pas majorer par une fonction indépendante de theta qui tend vers 0
lean folio
ah je vois
mortal orbit
**rafilou is not not born in 2003**
on peut majorer par 1/2
mais bon 1/2 ça tend pas vers 0
lean folio
mais on pourrait quand meme majorer
ouais je vois le truc
okay merci beaucoup pour ces explications !
mortal orbit
**rafilou is not not born in 2003**
faut montrer exactement ça
sinon
tu trouves un chemin continu qui tend vers (0,0)
tel que f(chemin) tend pas vers f(0,0)
L'idée est la même pour montrer qu'une fonction f définie sur R^2 \ {(0,0)} a / n'a pas de limite en (0,0)
mortal orbit
**rafilou is not not born in 2003**
suffit de remplaçer "f(0,0)" par la présumée limite "L" si tu veux prouver qu'il y a une limite
Sinon pour prouver qu'elle n'existe pas, tu trouves deux chemins qui tendent vers (0,0) qui n'ont pas la même limite
lean folio
suffit de remplaçer "f(0,0)" par la présumée limite "L" si tu veux prouver qu'il...
donc dans ce cas là, L = 0 c'est ça ?
mortal orbit
donc dans ce cas là, L = 0 c'est ça ?
dans notre exemple tous les chemins "ligne droite" tendent vers 0
donc si la limite elle existe ça devrait être 0
mortal orbit
Sinon pour prouver qu'elle n'existe pas, tu trouves **deux** chemins qui tendent...
donc dans notre exemple, si tu veux deux chemins qui n'ont pas la même limite
tu prends un chemin ligne droite au pif
lean folio
mais on a trouvé un chemin tel que f(chemin) ne tend pas vers 0
mortal orbit
ça te donne une première limite potentielle "0"
mortal orbit
mais on a trouvé un chemin tel que f(chemin) ne tend pas vers 0
et voilà tu prends comme deuxième chemin (t,t^2)
lean folio
.close
eager tapir
Why cant we have a bernoulli-binomial conjugacy? a bernoulli prior and a binomial likelihood it should give a bernoulli posterior like it works with bernoulli-beta
calm coralBOT
Why cant we have a bernoulli-binomial conjugacy? a bernoulli prior and a binomia...
@eager tapir Has your question been resolved?
calm coralBOT
@eager tapir Has your question been resolved?
west shard
Why cant we have a bernoulli-binomial conjugacy? a bernoulli prior and a binomia...
Why couldn’t we?
severe bolt
how the solution equated the parametric equation of line CP with the lenght of line CP thats d
calm coralBOT
@severe bolt Has your question been resolved?
spice venture
how the solution equated the parametric equation of line CP with the lenght of l...
can u plot this
calm coralBOT
@severe bolt Has your question been resolved?
calm coralBOT
Available help channel!
lean folio
Hi, I have some problems with I2, I have to find that it's conditionally convergent by using the Abel's criterion but I don't see how I can do this
indigo otter
bread
lean folio
but i cant help , i am sry
don't worry
lean folio
bread
?
I mean, we have to take 1/t² for the decreasing function so for the integral we have to take sin(1/t²)
and this is the problem
warm nimbus
Hi, I have some problems with I2, I have to find that it's conditionally converg...
for I(2) use sub : 1/t = y , -1/t² dt = dy
warm nimbus
yep
lean folio
and I don't know how to do I3 
warm nimbus
I(2)=I(3)
warm nimbus
and I don't know how to do I3 <:cat_happycry:636407813144576010>
i can help you to find the value of I(3) but i cant prove if it converge
lean folio
i can help you to find the value of I(3) but i cant prove if it converge
yeah fine
is there a Fresnel integral too?
edgy leaf
Hi, I have some problems with I2, I have to find that it's conditionally converg...
try the substitution of u=1/t^2, then use abel's criterion
lean folio
try the substitution of u=1/t^2, then use abel's criterion
I'll try
frank jungle
can you just use Dirichlet's test
lean folio
can you just use Dirichlet's test
don't know this
edgy leaf
i believe they are the same
lean folio
i believe they are the same
oh okay
edgy leaf
nevermind
frank jungle
are you already given that I(3) is convergent?
edgy leaf
they are different
lean folio
are you already given that I(3) is convergent?
no
warm nimbus
they are different
who is different ?
edgy leaf
dirichlet suggests one of the products partial integral must be bounded
and the other has to be monotonic tending to 0
lean folio
so I have the integral of 2u*sqrt(u) sin(u)
and now I have to use the Abel's criterion?
lean folio
show your work for the substitution
lean folio
it doesnt seem entirely right
it's possible
edgy leaf
u = 1/t^2
du = -2t^(-3) dt
but since t=1/√u (by substitution)
dt = -(1/2)u^(-3/2)du
so you get
-∫usin(u)*(1/2)u^(-3/2)du from inf to 0
we change variables from 0 to inf by considering the negative sign
lean folio
u = 1/t^2
du = -2t^(-3) dt
but since t=1/√u (by substitution)
dt = -(1/2)u^(-3...
why do you have u^(-3/2) in the expression of dt?
I have u^(-1/2)
maybe I'm missing something
oh sorryyy
I'm stupid
so yes I have this integral
lean folio
u = 1/t^2
du = -2t^(-3) dt
but since t=1/√u (by substitution)
dt = -(1/2)u^(-3...
okay I just finish the exercise thank you for your help!
.close
next yacht
is there anyone to help
yep
wintry canopy
bro so fast
next yacht
wassup
wintry canopy
am i lucky
next yacht
am i lucky
maybe
wintry canopy
can u tell me why antiderivative at a point gives all the area before that point
im working on this but u dont need to read it
edgy leaf
well the answer is on that page
also "can u tell me why antiderivative at a point gives all the area before that point"
is a wrong statement
difference of antiderivative evaluated at two points gives the area bounded by those two points
wintry canopy
yea
i know
but its said that
the antiderivative evaluated at a point gives area of whatever curve lies before that point
velvet osprey
may i recommend 3b1b's essence of calculus
edgy leaf
well can u tell me where
scroll down to "proof of second part"
wintry canopy
may i recommend 3b1b's essence of calculus
bro just give me the jist or write in a paper so i can follow up i am done reading the same shi 5 hrs st
please dont think i cant read
im tired boss
edgy leaf
that is your problem
velvet osprey
do not call me "bro" please.
wintry canopy
sis
wintry canopy
watt
next yacht
do not call me "bro" please.
he gotta call u DAD?
wintry canopy
send send
wintry canopy
big sis
velvet osprey
<@&268886789983436800> transphobia
wintry canopy
huh
wintry canopy
your higness
wintry canopy
https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t...
man i didnt understand shiiit
velvet osprey
@next yacht do you wanna apologize to me there at least
wintry canopy
i think i saw that
next yacht
<@957552694451666995> do you wanna apologize to me there at least
okay okay, im sorry sister
short salmon
something something integrals are actually real-valued linear operators on a function space
BRO WHY AM I BLUE AGAIN
but a different blue
wtf is up with the colors today
fickle musk
,, g(x) = \int f(x),
\int ^a _0 f(x)
edgy leaf
revolutionary latex
potent lotusBOT
David (דוד)
fickle musk
revolutionary latex
I know I am bad 😭
short salmon
man i didnt understand shiiit
bro watched it for 2 mins 💀💀
fickle musk
Then I think g(a) equals the thing on the right
potent smelt
**David (דוד)**
These two things are subtly different
The indefinite integral of f(x) will include a +C. The definite integral of f(a) da from 0 to x will not
next yacht
bro watched it for 2 mins 💀💀
LOL
next yacht
can u tell me why antiderivative at a point gives all the area before that point
u know?
antiderivative works like area tracker.
at any point on the graph
wintry canopy
tell tell
next yacht
it tells you how much total area has been accumulated under the curve
wintry canopy
yea
next yacht
starting from some fixed point
next yacht
This thing works becoz
wintry canopy
yes yes i wanna know that
next yacht
differentiation and integration are inverse processes
wintry canopy
ok?!
next yacht
ok?!
and the
fundamental theorem of calculus guarantees that
if you take an antiderivative
its derivative will match the original function
meaning that
it perfectly tracks how much area grows
u understood?
wintry canopy
it perfectly tracks how much area grows
well i didnt digest that
next yacht
well i didnt digest that
okay lemme explain
imagine that
you are filling a tank
with water using something
where the flow rate changes over time
like f(x)
The antiderivative is like a gauge that shows how much water has been added up to any moment
meaning
it accumulates all previous flow rates
into a single number
understood now?
where did u go?
wintry canopy
ill try to understand mathematically
short salmon
because it's really the definite integral that tells you that
wintry canopy
can i tell what i know 1st
short salmon
go ahead
wintry canopy
well can we start from why definite integral gives the area under the curve
between two points
i had managed to mathematically find that the derivative of area function is f(x)
so its antiderivative is A(x)
and i dont know how the limits are arranged in that integral sign and why the definite integral evaluated at b gives all the area before b and definite integral evaluated at a gives all the area before a
and when we minus that we get the area of the curve
short salmon
what?!?
ok, your writing is very hard to digest
and also wrong, because that's not what the definite integral is doing
I would say not to look into it too much until you take analysis. just think that the limit of riemann sums is conveniently defined such that for your integrable function on the x-y plane, a definite integral will line up correctly with the area under the curve
silver rover
i had managed to mathematically find that the derivative of area function is f(x...
Are you familiar with summation?
wintry canopy
yep
wintry canopy
first u can explain why definite integrals give the area between two points and also why the definite integral evaluated at a certain point gives area all teh way before that point
silver rover
Ok do you know what the bounderys on the integral sign mean?
What the definite integral does is summate all the area in infinitly small pieces through the boundery a to b
While the indefinite integral finds a general equation to the area of the function
Ill try to graph something for you
wintry canopy
ok
silver rover
in this graph, the function is f(x) so the indefinite integral or antiderivative of the function would be x^2
Meaning, if you would want to find an area between say 1 - 5, you would use the fundemental theorem of calculus
wintry canopy
in this graph, the function is f(x) so the indefinite integral or antiderivative...
yes understood
wintry canopy
well how ??
silver rover
If you would take the definite integral between 1 - 5 you would get an area of 24
silver rover
Sure
wintry canopy
ok then wait
man
phone dead
wait
fuc
is there a way to do it from laptop
???
until then lemme tell what i know
silver rover
Yes
wintry canopy
few mins and it will be open
wintry canopy
i took y=f(x) plotted on the x y plane
i took point x , x distance away from the centre and assumed the area between x and o is given by A(x) similarly i took delta x (h) , the area between h and 0 given by a(x+h)
the area of teh thin strip is given by A(x+h)-A(x) equivalent to h*f(x)
then i divided both sides by h and took limit as h approaches zero
this gives derivative of area and as the change in area for h approaches zero the derivative of A(x) =f(x)
which imply its antiderivative
antiderivative of fx is A(x) area function ( well im still not familiar with area function i just learned or rather have seen it today)
leaden marsh
“distance away from the centre”?
wintry canopy
um
leaden marsh
by “area between x and o”, do you mean “area under f(x) between x and o”?
wintry canopy
yea area between f(x) and 0
leaden marsh
by “area between h and o”, do you mean “area under f(x) between x+h and 0”?
leaden marsh
f(h) is very very near the origin
wintry canopy
A(a+H)
wintry canopy
f(h) is very very near the origin
no
leaden marsh
I think you mean x+h because you say A(x+h)
h is the small difference that is close to 0
wintry canopy
nooo to x
tf
x is x distance away from centre
and h is h distance away from the centre
leaden marsh
and you’re taking $\lim_{h\rightarrow0}$?
potent lotusBOT
blahaquil
leaden marsh
so if x=like 100 surely the difference between x and h is massive
wintry canopy
no
leaden marsh
sir I think you mean x+h
leaden marsh
well exactly 100 as h approaches zero which actually isn’t that massive
you are thinking of taking a very thin strip right
leaden marsh
thats only possible when you take the difference between x and x+h
h is the tiny tiny difference
wintry canopy
yep
leaden marsh
but the actual second value is x+h
leaden marsh
yes
wintry canopy
yea
leaden marsh
ok so we’re on the same page now?
leaden marsh
then assuming u mean those things yea thats accurate
leaden marsh
exactly wow yes genius of our time 🤩
wintry canopy
no
leaden marsh
wait lemme scroll up
leaden marsh
i took y=f(x) plotted on the x y plane
i took point x , x distance away from th...
I’ve only been reading this
wintry canopy
yea
yea
read
then from that it came as
that
the derivative of the area is f(x)
leaden marsh
yea
leaden marsh
yea
wintry canopy
so from that point can u tell me how to get the area between two points
from all that
and what i am missing
leaden marsh
A(b)-A(a)
leaden marsh
ya
wintry canopy
A(b)-A(a)
that definite integral with a and b in the integral symbol will calculate the area between two points via this right
leaden marsh
yes
wintry canopy
in our case integral of f(x) came A(x)
can we say it as the A(x)-A(0) similarly like that
??
am i wrong or correnct in this one
leaden marsh
umm
ok see
the indefinite integral
is actually like a family of functions
a family of antiderivatives
leaden marsh
bc of the PLUS C
wintry canopy
thats indefinite integrals
we still dont know
the c
that corresponds to our
function
leaden marsh
like since we have no bounds on an indefinite integrals we adjust for the lower bound with the C
and we plug the upper bound in
thats the thought behind why it’s useful in the first place
leaden marsh
that also makes mathematical sense since differentiating constants =0 and so they’re all the same in their derivative
idk man
wintry canopy
yea
leaden marsh
the A(x) you’re talking about is probably an indefinite integral with C=0
wintry canopy
no its above the centre wait if my mobiles charged ill pinge u
@leaden marsh
lmao
leaden marsh
camera bro
leaden marsh
wait what’s ur question????
wintry canopy
find the are aunder curve yea do that
leaden marsh
what 😭
leaden marsh
the area under curve from where to where??
wintry canopy
from any point a to b
leaden marsh
in general it’s just A(b)-A(a)
wintry canopy
like proof
leaden marsh
what
short salmon
dude
leaden marsh
like
short salmon
if you truly want to understand a riemann integral
you need to read an analysis book
leaden marsh
the area from 0 to b minus the area from 0 to a is the area from a to b
wintry canopy
is that reimman integral?
short salmon
is that reimman integral?
yes
wintry canopy
the area from 0 to b minus the area from 0 to a is the area from a to b
you mean like 0 to h and o to x
leaden marsh
although we just said “let A(x) be the area” and thats the tricky part
0 to x+h and 0 to x
leaden marsh
yea
wintry canopy
and
wintry canopy
yea
leaden marsh
what are u trying to do rn 😭
leaden marsh
what are you trying to prove
wintry canopy
trying to prove that
the area between two points a and b is A(b)-A(a)
do u get me
leaden marsh
you just have to say “observe that the area from 0 to some point b minus the area from 0 to some point a must equal the area from a to b, therefore the area from a to b is A(b)-A(a)”
which is what I’ve been telling you 😭
wintry canopy
wait wait wait and i dont get putting the bounds in the integral sybol can i put bounds after saying that
???
leaden marsh
like if you want a more accurate proof
short salmon
you would need to use some more rigorous tools to do this than what you're doing right now
leaden marsh
you need to like really REALLY study analysis
short salmon
yes. as I said, this is an analysis thing
short salmon
I can point you to some good reading on the construction of the riemann integral if you'd like
leaden marsh
thats a Riemann sum
wintry canopy
wait wait wait and i dont get putting the bounds in the integral sybol can i put...
but bro ??
leaden marsh
do kids not get the accumulation function seminar before studying integration anymore or like 😭
short salmon
chapter 4.2
wintry canopy
do kids not get the accumulation function seminar before studying integration an...
the fact is im 16
and our collage do not teach
this
or
neither any collage in our country
i jus wanted to know
short salmon
so you have 2 options
- give up because elementary calculus doesn't have the tools for this
- study analysis
(3. be a genius and develop the rigorous theory independently)
wintry canopy
studying analysis is what is even that all about
leaden marsh
the fact is im 16
im literally in middle school jus u need RIGOR man
(>13 don’t shoot me)
wintry canopy
hum
leaden marsh
it’s like the formal study of functions
wintry canopy
whats ur age
leaden marsh
i am not telling u that information 👍
leaden marsh
i would think smaller your age
leaden marsh
that doesnt matter tho
leaden marsh
wait wait wait and i dont get putting the bounds in the integral sybol can i put...
like abt this you just need to define notation from there if we’re working at such a coarse level
wintry canopy
like abt this you just need to define notation from there if we’re working at su...
huh
continue
leaden marsh
like literally just define $A(b)-A(a)=\int_a^b f(x) dx$
potent lotusBOT
blahaquil
wintry canopy
yea
leaden marsh
or better yet define indefinite notation when you’re defining A the first time
wintry canopy
thats what im trying to do
leaden marsh
ok so then you have it
wintry canopy
have what
leaden marsh
thats what im trying to do
wait what are you saying here ?
by define i literally just mean “let this symbol mean this”
wintry canopy
**blahaquil**
i wanna know why the definite integral gives the area between two points in a curve which this picture shows isnt it ??
leaden marsh
or if you define the indefinite integral beforehand you have more justification
wintry canopy
wait what are you saying here ?
'""
west shard
what's the question?
west shard
i wanna know why the definite integral gives the area between two points in a cu...
this?
leaden marsh
i wanna know why the definite integral gives the area between two points in a cu...
this is not strictly true in the way you think it is
wintry canopy
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwat
leaden marsh
the C accounts for the lower bound
wintry canopy
the C accounts for the lower bound
tf
leaden marsh
we plug in the upper bound as the x-value after we evaluate the integral
look think of it as an accumulation function
west shard
you're undergrad or high school?
wintry canopy
like how can i tell this education system im in the 11th now
lmao
what is evn that
highschool in america is like collage no before uni
short salmon
he's a high schooler
leaden marsh
wait is 11th grade college for you?
wintry canopy
tyapa
leaden marsh
what
wintry canopy
um
west shard
Depends on where you're from - in the UK we call the two years before university "sixth form" or college
leaden marsh
i see
west shard
but that's just a matter of naming
west shard
In that case, do you know how to approximate the area underneath a graph over an interval?
west shard
If I asked you to use four rectangles to approximate the area of x^2 between 0 and 1 do you know what to do?
also where are you from?
west shard
If I asked you to use four rectangles to approximate the area of x^2 between 0 a...
just answer yes or no
wintry canopy
yea base times height for 4 rectanglesand sum
west shard
how do you get the height?
wintry canopy
f(x)
west shard
where on f(x)
west shard
the first rectangle represents [0,0.25) right?
wintry canopy
huh
leaden marsh
the base
west shard
The first rectangle you draw represent 0 to 1/4 on the domain
wintry canopy
i just drew it without considering any points and all
west shard
can you vc?
wintry canopy
vc? what is that
west shard
voice
west shard
are there no voice channels anymore
wintry canopy
lmao what are these voice channels and where are they
calm coralBOT
@wintry canopy Has your question been resolved?
west shard
this?
wild marten
Is this a proper generalization, normal subgroups are just elements that do not commute with all elements in the main group
twin raptor
im not sure i understand what you mean
wild marten
like
if there is a group H which is a subgroup of G
and there is an $h \in H$ which does not commute $\forall g \in G$
can i stop here and just say H is not normal?
twin raptor
h does not commute with what?
wild marten
@potent lotus sorry fixed it
potent lotusBOT
BOSS
wild marten
so, for each g I would need to get the left and right cosets until one is not equal?
twin raptor
for all $g\in G$, we require $gH = Hg$
potent lotusBOT
flying_fly
twin raptor
but this doesn't mean $gh = hg$ individually for every $h\in H$
potent lotusBOT
flying_fly
wild marten
ah
hmm that makes sense
but if it does not commute, we can easily prove its not equal right
or do we have to write out the entire left and right coset each time
twin raptor
or do we have to write out the entire left and right coset each time
yeah, you want to work with cosets as opposed to individual elements
i am hesitant to say you "have to" do this because you will learn other ways to determine whether a subgroup is normal
twin raptor
but if it does not commute, we can easily prove its not equal right
if a single element does not commute, it's still possible for $H$ to be normal. for example in the dihedral group $D_n$, the subgroup of rotations is normal, even though most rotations don't commute with reflections
potent lotusBOT
flying_fly
wild marten
the stubgroup of reflections is not tho for the reason that they do not commute
hmm
but yeah i get what ur saying
you do have to kinda check and look at the entire coset
twin raptor
the stubgroup of reflections is not tho for the reason that they do not commute
the set of reflections isn't a subgroup of Dn
wild marten
if a single element does not commute, it's still possible for $H$ to be normal. ...
because this in my mind is the group generated by r
wild marten
tbh im just revising factor/normal subgroups so i can understand the first isomorphism theorem better
@twin raptor if ur still here
lets say im trying to prove that $H = eH = hH \quad \forall h \in H$
potent lotusBOT
BOSS
calm coralBOT
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cold basin
calm coralBOT
west shard
u=1/x
cold basin
Can i show u what i did
west shard
Go
cold basin
They dont teach us substitution here they instead tell us to modify the differential
Which ig is the same thing
cold basin
They dont teach us substitution here they instead tell us to modify the differen...
Before doing by parts
west shard
Yes
cold basin
1 sec have to write it clearly
west shard
Something went awfully wrong here
cold basin
Thats what they tell us to do
Integrate one of the functions in our head and put it in the differential
Which is just the long formula for by parts
west shard
,w integrate arctan(x)/x^2
potent lotusBOT
cold basin
Im down to the integral of dx/x(x^2+1)
west shard
Sounds good
potent lotusBOT
temp0937249365
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
cold basin
[
\int \frac{dx}{x(x^2 + 1)}
]
potent lotusBOT
temp0937249365
cold basin
How do i solve this
west shard
Something went awfully wrong here
Something went wrong here
west shard
How do i solve this
Parts
cold basin
How exactly
west shard
Partial fractions i mean
cold basin
What do i put in the differential
west shard
Sorry
cold basin
Oh
Lemme try
What did I do wrong
No way I have to use the quadratic formula
Im sorry the second one needs to be BX+C
west shard
calm coralBOT
@cold basin Has your question been resolved?
turbid stratus
Im trying to show that the area of the parallelogram, or v x w, is the determinant after transforming the unit vectors v,w with a general matrix. Where did my math go wrong?
turbid stratus
My end result is exactly the negative of what my textbook says
<@&286206848099549185>
calm coralBOT
@turbid stratus Has your question been resolved?
turbid stratus
If i use determinants, then i get exactly the negstive of my geometry result
calm coralBOT
@turbid stratus Has your question been resolved?
calm coralBOT
@turbid stratus Has your question been resolved?
elder pawn
@turbid stratus det keeps track of how the vectors are oriented. if they follow the right hand rule then det>0, otherwise det<0
if u dont wanna care about orientation then u need to say area=|det|
buoyant ice
Can you tell me what math is this
turbid stratus
Can you tell me what math is this
This is precalc, specifically for competition (from Volume 2 by Aops, chapter 11 to be exact)
turbid stratus
if u dont wanna care about orientation then u need to say area=|det|
Ok, but since my answer is exactly negative of the determinant, does my mistake have to do with the orienation?
opaque yoke
calm coralBOT
opaque yoke
please help i have no idea what to do
so i used quadratic formula
answers are 2+-sqrt5
only 2-sqrt5 can work in inverse sin, so I use that
its negative so I add 2pi
im not sure what else to do
unkempt drift
alright, so given your larger value is correct
first do 6.0449 - 2pi
then use the identity sin(pi - x) = sin(x)
calm coralBOT
@opaque yoke Has your question been resolved?
unkempt drift
np!!
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lilac rock
Let the urn contain 3 white balls and 4 black balls. Two balls are extracted one after the other. Suppose that the first ball drawn out of the urn is returned to it and mixed with the rest of the balls, after which a ball is again drawn out. Find the probability that both these balls will be white.
lilac rock
Can someone confirm if the answer is 9/49?
vagrant oak
Can someone confirm if the answer is 9/49?
Seems right
gaunt hornet
I don't understand how they got to the answer at c)
I started by writing n terms of x > x = 1-z
then i put that into the first equation:
2(1-z) -4y = 10
and i wrote in terms of y
y = .5z -2
and then z = 1-x
But now I'm lost as what to do?
calm coralBOT
I don't understand how they got to the answer at c)
I started by writing n term...
@gaunt hornet Has your question been resolved?
calm coralBOT
@gaunt hornet Has your question been resolved?
calm coralBOT
@gaunt hornet Has your question been resolved?
calm coralBOT
@gaunt hornet Has your question been resolved?
rustic osprey
I don't understand how they got to the answer at c)
I started by writing n term...
ok I think that the point was to solve for x, y, z in terms of w (where they let w be p for some reason)
,w rref{(2,-4,0,10),(0,2,w,2),(1,0,1,1)}
potent lotusBOT
rustic osprey
so basically just rref the coefficient matrix
gaunt hornet
and how do u arrive at that answer
because I thought u had to write in terms of x y z
rustic osprey
and how do u arrive at that answer
this is your coefficient matrix after you rref [A|b]
and this is that vector b, which represents the solution (x,y,z)
you should see that it matches the given answer
rustic osprey
You do elementary row operations?
Consider doing $R_1-2R_3 \to R_1$ and $R_2-wR_3 \to R_2$
potent lotusBOT
Civil Service Pigeon
rustic osprey
**Civil Service Pigeon**
Note that this is not the only valid set of row operations you can do
As an exercise, figure out some other ways yourself
@gaunt hornet Are you there?
rustic osprey
Row 1 minus 2 times row 3?
Yeah that's what I said
and make that the new row 1
Note that I'm using two elementary row operations at once - I'm multiplying row 3 by a nonzero scalar (namely -2), then I'm adding that scalar multiple of row 3 to row 1.
calm coralBOT
@gaunt hornet Has your question been resolved?
prime topaz
Hi so I didn't see this at first and I cross multiplied it and got a quadratic where lambda = 2/3 and 1. Why can't it be 1?
prime topaz
this is part of the solutions to part iii
jaunty snow
what exactly did you do? Multiplying both sides by lambda+1 just gives a linear equation whose only solution is lambda=2/3
prime topaz
what exactly did you do? Multiplying both sides by lambda+1 just gives a linear ...
uhh did i do a silly mistake
jaunty snow
The other solution of 3lambda^2+lambda-2=0 is negative 1, not positive 1, which you can eleminate since that results in a division by 0 in the original equation
prime topaz
The other solution of 3lambda^2+lambda-2=0 is negative 1, not positive 1, which ...
OH thank you!
.close
calm coralBOT
jaunty snow
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wintry canopy
.reopen
lunar iron
what did I do wrong for this question. currenly using completing the square method.
drifting seal
what did I do wrong for this question. currenly using completing the square meth...
it says + 5/2 x
unkempt drift
yeah so it's not (x - 5/2)^2 but... ?
lunar iron
it says + 5/2 x
ohh, so like the - in the brackets is not fixed but rather referring to the bx of the equation?
drifting seal
wat
unkempt drift
ohh, so like the - in the brackets is not fixed but rather referring to the bx o...
yeah so the sign has to match
if you see +5x then it splits as +5/2 x and +5/2 x
hence (x + 5/2)^2
unkempt drift
so you end up with $x + 5/2 = \sqrt{49/4}$ instead
potent lotusBOT
south
unkempt drift
everything else actually doesn't change
lunar iron
👍
lunar iron
.close
calm coralBOT
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covert orchid
Can someone remind me what to do now
calm coralBOT
Can someone remind me what to do now
covert orchid
,rccw
potent lotusBOT
fallow haven
a-2b+c=0
6a+5b-4c=0
how do i find a : b : c from this?
calm coralBOT
a-2b+c=0
6a+5b-4c=0
how do i find a : b : c from this?
fallow haven
i was doing a 3d geometry question and got stuck on this
unreal isle
!status
calm coralBOT
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
polar bison
I don't think it's possible
unreal isle
it is
polar bison
You need 3 equations to solve an equation in 3 variables
polar bison
Oh
unreal isle
there's infinite solutions in that ratio then
fallow haven
You need 3 equations to solve an equation in 3 variables
(a,b,c) is a direction ratio
unreal isle
but i need to know where OP is
polar bison
dread is OP?
fallow haven
idk ive never solved triple ratio before
unreal isle
yes
ooh okay well what comes to your mind
how do you THINK this would be solved
doesnt matter if it's wrong
fallow haven
i tried to add an eqn a^2+b^2+c^2=1 so itll become a direction cosine and i can find it
but solving that was too lengthy
unreal isle
you dont need to do all that, it's pretty simple
so
lets see
the problem at hand
we have 2 equations only
let's for a sec imagine we had 3
what would we do first
fallow haven
eliminate a variable
unreal isle
perfect
fallow haven
wait
unreal isle
use the first eq to write an expression for 1 var in terms of the other 2
unreal isle
let's use a since it's the easiest to work with here
unreal isle
i think i can make quadtratic equations in a/b and b/c
you dont necessarily need to do that
fallow haven
not quadratic
unreal isle
once you form an eq in 2 variables
you can find the ratio bw them
so let's say you find b=x*c
then you use that expression for b in your expression of a
a was originally alpha*b + beta*c for instance
but after substituting b in terms of c you'll get a in terms of only c
then write it as a:b :c
and replace them all in terms of c
and simplify!
if you want i can spoiler tag the answer
||3:10:17|| click this for the answer
i gtg, good luck!
fallow haven
thanks i got it
do you know how they've done it in this solution
calm coralBOT
@fallow haven Has your question been resolved?
polar bison
EXACTLY I never got this cross multiplication stuff
I did it another way don't remember
calm coralBOT
@fallow haven Has your question been resolved?
calm coralBOT
@fallow haven Has your question been resolved?
covert orchid
What to do?
calm coralBOT
What to do?
winter elbow
Is this an ongoing test?
wanton thorn
What to do?
Is it -cot(x/2)
covert orchid
Is this an ongoing test?
No lol
covert orchid
Is it -cot(x/2)
Idk..
modern briar
i cant make the rest
covert orchid
Its just homework
valid linden
maybe we should write cos(x) as 1-2sin²(x/2)?
modern briar
maybe we should write cos(x) as 1-2sin²(x/2)?
i tried to do that but it becomes 0/0
i might be wrong somewhere
valid linden
Idk im pretty bad with integration
valid linden
i tried to do that but it becomes 0/0
How
modern briar
gimme a sec i will do it again i probably made a mistake somewhere
valid linden
ohk
covert orchid
use those boxed formulas
They used them to get from the top integral to the one I sent, idk if using them again helps..
surreal patio
They used them to get from the top integral to the one I sent, idk if using them...
use those substitution , its a pretty neat sub
valid linden
i found where i made a mistake lol
u got the answer?
covert orchid
Yea
covert orchid
This is where im at idk
covert orchid
Lol
modern briar
IM LOST
wanton thorn
@covert orchid u got this method ???
wanton thorn
Its alr
modern briar
covert orchid
Anyone know how to continue with this method ?
covert orchid
<:bleak:877093280905592842>
Ty for trying too🫡
vague bison
This is where im at idk
!nosols
calm coralBOT
This is where im at idk
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.
vague bison
sorry tagged wrong person
vague bison
!nosols
calm coralBOT
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.
vague bison
Anyone know how to continue with this method ?
in general, you feel stuck in integrals involving trigonometric functions, you may consider Weierstrass's substitution
vague bison
btw i got stuck here so
see my comment above
modern briar
btw i got stuck here so
modern briar
in general, you feel stuck in integrals involving trigonometric functions, you m...
oh
im gonna note that down
thx!
covert orchid
Anyone know how to continue with this method ?
Im using weierstrass substitution here I think (?)
vague bison
Im using weierstrass substitution here I think (?)
weierstrass substitution is "let t = tan(x/2), then dx = 2dt / (1 + t²), ..."
that's tangent half-angle substitution
covert orchid
For clarification
covert orchid
Its just homework
What are the boxed formulas called
modern briar
modern briar
What are the boxed formulas called
whats boxed?
modern briar
ive never seen them in my books or anything
covert orchid
Oh hahah
covert orchid
So I think the right one in the second box weierstrass substituion and the other two are tangent half angle substitution
vague bison
ive never seen them in my books or anything
if you feel that you can do every exercise question easily in your book, you should change to a more challenging book, so that youi can learn better and harder math]
i wish i knew that when i was a high school student
modern briar
i wish i knew that when i was a high school student
🫂
covert orchid
Anyone know how to continue with this method ?
Can someone please explain what to do when I have variable t in my function and dx from integral
If i substitute back tan(x/2) im stuck again
vague bison
Can someone please explain what to do when I have variable t in my function and ...
see my above comment
covert orchid
Something is wrong
vague bison
the purpose of Weierstrass method is to replace everying containing "x" with something involving t, so you don't "substitute back" before integration
vague bison
Something is wrong
,rccw
potent lotusBOT
vague bison
please rotate your image so that it's easier to read for PC users
vague bison
I skipped a few steps
you don't mix t and x in one line containing the integral sign
when you do substitution, you need to do so thoroughly
covert orchid
I will do it again from beginning
vague bison
to check your work in tests/exams, you may differentiate your result and see whether it matches the integrand
that's sth you can do alone in exams
calm coralBOT
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Available help channel!
nimble portal
Does Cos(5n)/n series diverge or converge
Actual series we have to find(if it converges or diverges)
velvet osprey
it is inappropriate to replace the numerator with cos(5n).
$\cos^2(5n)$ is bounded between 0 and 1, therefore the numerator is bounded between $\sqrt{2}$ and $\sqrt{3}$ and we can safely pretend it is constant
potent lotusBOT
Ann
velvet osprey
the denominator behaves like n.
unkempt plume
then it diverges
mortal orbit
<@&268886789983436800>
mortal orbit
then it diverges
to be more "rigorous" (even though what we did is fine), we can now see that a lower bound is more important since we want to show divergence. With what was previously established, we can say the general term is lower bounded by $\frac{\sqrt 2}{n}$
potent lotusBOT
rafilou is not not born in 2003
mortal orbit
with its associated series diverging
velvet osprey
btw op has disappeared
mortal orbit
oh right huh xd
elder pawn
@nimble portal wake up
nimble portal
$\cos^2(5n)$ is bounded between 0 and 1, therefore the numerator is bounded betw...
So that means you can use the test where if a smaller series diverges so does the bigger one, since 1/n diverges so does (sqrt(2) + cos(5n))/n
velvet osprey
how did $\sqrt{2} + \cos(5n)$ happen?
potent lotusBOT
Ann
nimble portal
how did $\sqrt{2} + \cos(5n)$ happen?
Just simplify the numerator of the original series
potent lotusBOT
Ann
velvet osprey
@nimble portal
calm coralBOT
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narrow hamlet
how come is this true
questions is to prove true or false
calm coralBOT
how come is this true
questions is to prove true or false
narrow hamlet
per the reasoning this, i dont rewally understand
i thought false as f(0) = 1 but saying f(c) = c^3
would be f(0) = 0
however im aware my undertsanding of funcitons is flawed here lmao but idk how
so f(c) and f(x) are differnet?
velvet osprey
i thought false as f(0) = 1 but saying f(c) = c^3
it doesn't say that f(c)=c^3 is supposed to hold for ALL c
only that there EXISTS a value of c where these match
narrow hamlet
oh yea