paper nebula
and prove that the "construction" works
#help-28
somber whale
i’m having difficulty understanding how to solve trig sub integrals if anybody can help
somber whale
like this one for ex
gritty rose
Sometimes you just gotta try different trig functions
sin, cos, tan are the usual first ones to try
But you probably want 3*sin since there's a 9
gritty rose
You don't need to know?
somber whale
wdym
gritty rose
You don't need a triangle to do every trig sub integral
gritty rose
Well that wasn't part of the instructions
gritty rose
i’m saying what purpose does it serve 😭
The triangle serves little to no purpose
somber whale
so it’s js irrelevant then i can js do it without
gritty rose
nah we need it for credit for this midterm
Idk only your teacher would know
gritty rose
You don't need to know?
Yes that's what I meant here
gritty rose
The triangle serves little to no purpose
Leaning towards no purpose
somber whale
js for show type shi
what’s the equation i need for trig sub then cus i thought it was based around the triangle
coral ore
coral ore
👍
somber whale
i’m saying what purpose does it serve 😭
how does x^2 become 9sin^2 theta tho
somber whale
boo
somber whale
kinda i don’t get how the x^2 just gets a sin
ik the bottom because it’s a-x idk how that translates to the numerator tho
sand escarp
well we have that x = 3sin(theta) true?
you understand how the x^2 in the numerator turns into 9sin(theta) ^ 2 right
somber whale
yea but that’s in the answer key idk how it translates off dome
sand escarp
no that makes sense
try to think of x as a function of theta
x(theta) = 3sin(theta)
now take the derivative
dx/dtheta = 3cos(theta)
and multiply both sides by dtheta
gives dx = 3cos(theta)dtheta
and just replace the dx by 3cos(theta) dtheta
somber whale
so x always has like an imaginary theta next to it basically
and that equals three sin
like just as a rule
sand escarp
that's pretty much what substituting is
if we have that x = 3sin(theta), they actually mean x(theta) = 3sin(theta)
kindof like how y = ax + b
is like f(x) = ax + b
or y(x) = ax + b
somber whale
ohhhhh ok
sand escarp
makes sense?
somber whale
so like for the future tho x is gonna equal 3 sin theta
sand escarp
not in every case
somber whale
how come now tho
sand escarp
becuase we have sqrt(x^2 - 9).
we want to get rid of the sqrt in some way
we know that sin^2 + cos^2 = 1
so maybe we can rewrite x^2 - 9 in those terms
we smartly guess x = 3sin(theta)
gives x^2 = 9 sin(theta) ^ 2
and we have 9sin(theta) ^ 2 - 9
we factor out a 9
gives us sin(theta) ^ 2 - 1
which is cos(theta)^2
if we for example had x^2 - 4
we would take x = 2sin(theta)
because then x^2 = 4sin(theta)^2
somber whale
ohh ok so u literally js sub it out with what fits
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woeful wolf
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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
robust slate
should i js transfer 1 term to rhs?
Yeah I'd do both of those
and this counts as progress btw
try to give whatever you're thinking of/done so far
gives us more context and saves time from explaining things unnecessarily
robust slate
how do i factor this?
You can't rlly factor $x+y+xy$ practically
glossy valveBOT
Civil Service Pigeon
robust slate
ig you could say that $(x+1)(y+1)=xy+x+y+1$ but idk how much that would help here
glossy valveBOT
Civil Service Pigeon
robust slate
Remember that you want dy/dx, so ||try isolating y||
robust slate
y= -x/(1+x)
mhm
woeful wolf
so dy/dx= -(1+x)^(-2) ??
robust slate
so dy/dx= -(1+x)^(-2) ??
mhm
robust slate
alr thanks, have a nice day
woeful wolf
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dull flint
I want to study the topic binomial probability distribution from where I can learn it with problems
fossil hound
does Khan Academy not have a chapter on that? this seems like basic-intermediate statistics that they should have covered
dull flint
does it work for JEE
fossil hound
um I would think that math knowledge is global, so perhaps?
if you are looking for JEE-specific material then I can't help you, sorry
dull flint
um I would think that math knowledge is global, so perhaps?
Yes
dull flint
if you are looking for JEE-specific material then I can't help you, sorry
Just asking it
fossil hound
if you don't have the requirement that the resourve be JEE-specific then Khan Academy and Math LibreTexts (written) might be good
dull flint
Ok thx
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cedar marsh
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cedar marsh
stats question i dont undertsand pls someone help me
tall onyx
stats question i dont undertsand pls someone help me
find the probability that Y is even, there are 2 cases x odd or x even, it does not matter what exactly x is
cedar marsh
one part i dont get is the inequaltiy you have to write
2x 4x and 5x greater than or equal to 20
cedar marsh
find the probability that Y is even, there are 2 cases x odd or x even, it does ...
btw the final answres are 5 7 and 9 i think
but im so confused
cedar marsh
2x 4x and 5x greater than or equal to 20
but thta means there will be a ttal of 5 when you want total to be equal to probablity of being even
tall onyx
btw the final answres are 5 7 and 9 i think
there are ,in total, 6 pairs (a,b), and 3 of them are trivial
we don't know what other 3 are
let say y pair in that 3 pairs have even sum
Then find P(Y>=20) in term of y
tall onyx
2x 4x and 5x greater than or equal to 20
working with x is pretty hard and annoying
If (a,b) is not ordered pair then well, you would have to consider x=4 as it own case
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lofty ingot
Hlo
delicate torrent
what question?
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Can anyone help me with this question
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
lofty ingot
what question?
9th question in the page i sended here
delicate torrent
a hemisphere is just 1/2 of the sphere right?
lofty ingot
Yes
delicate torrent
so the surface area of a hemisphere is just 1/2 the original sphere (without counting the bases)
notice that the cylinder's surface area's bases is removed because of the hemisphere
or in other words, the area is surface area of cylinder - 2*base + 2*hemisphere
do you get what im saying?
lofty ingot
Um
Kinda i mean the height of the cylinder is given and the base radius is 3.5cm so we wanna find the surface area of the total solid
Height of the cylinder is 10 cm
delicate torrent
yeah
you can find the area of the hemisphere given the radius right?
1/2 the sphere
lofty ingot
The surface area of the solid will be surface area of cylinder+2*surface area of hemisphere right?
delicate torrent
oh hold up
does the "surface area of hemisphere" you're saying include the circle as the base?
lofty ingot
I mean the hemisphere and cylinder is joined together so the base of the cylinder and hemisphere is the same in the question they said they scooped out the hemisphere
delicate torrent
so here's the thing
im just assuming the surface area of the hemisphere does not include the circle base
it's just the area of the curvy thing
delicate torrent
so if they scoop out one hemisphere
delicate torrent
Can I ask a doubt btw
aight
lofty ingot
You said you ain't including the base of hemisphere right?
delicate torrent
yea
lofty ingot
So basically your telling surface area of cylinder+surface area of hemisphere*2-Base of the cylinder Right?
delicate torrent
yes
delicate torrent
So basically your telling surface area of cylinder+surface area of hemisphere*2...
2 times the base
lofty ingot
2 times the base
Oh yeah there are 2 hemisphere
delicate torrent
Oh yeah there are 2 hemisphere
aight
the reason here is that if you scoop one hemisphere
the surface area changes by adding the surface area of hemisphere and effectively remove the base
delicate torrent
that is with the assumption i made
lofty ingot
👍
Indian question maybe tricky like this
I got one more very interesting question
Btw cya and ty for the help
Lemme solve ig
delicate torrent
!done
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lofty ingot
.close
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lofty ingot
. reopen
delicate torrent
don't include spaces
delicate torrent
dw\
fossil hound
.reopen
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✅ Original question: #help-28 message
delicate torrent
The height of the hemisphere is not given
with the assumption i made, the surface area of the hemisphere is 1/2 the sphere with the same radius
lofty ingot
no spaces for next time
K
lofty ingot
with the assumption i made, the surface area of the hemisphere is 1/2 the sphere...
I mean in the question the height of the cylinder and the total figure is only given
So if we minus the total height by base radius of hemisphere we will get the height of the hemisphere right?
delicate torrent
So if we minus the total height by base radius of hemisphere we will get the hei...
i don't think we even need to find the height of the hemisphere
lofty ingot
Why
delicate torrent
with the assumption i made, the surface area of the hemisphere is 1/2 the sphere...
.
lofty ingot
But there is no sphere in the question
delicate torrent
But there is no sphere in the question
the surface area of the hemisphere with a radius of 3.5cm is 1/2 the sruface area of the sphere with a radius of 3.5cm
that's what i meant here
that is
with the assumption i made again
lofty ingot
Do you think it's true btw
Um in the method i use I dont think i can make an assumption like that
delicate torrent
Um in the method i use I dont think i can make an assumption like that
what's the formula for the surface area of the hemisphere (with the base included)
ofc you can't do it with the assumptions provided
lofty ingot
3pir square
delicate torrent
i think that's called the lateral area
lofty ingot
i think that's called the lateral area
Lateral surface area
lofty ingot
Oh that
So you think we can possibly get height of the hemisphere by minusing base area of the cylinder from height of total solid?
delicate torrent
So you think we can possibly get height of the hemisphere by minusing base area ...
height of the hemisphere
lofty ingot
height of the hemisphere
Ye
delicate torrent
oh it's the radius 
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raven sail
Does someone know geogebra?
I wanted to show myself, what happens if a I createa circle set the circumreference to 6 and its diameter to 2 and then do 6/2 c / d
raven sail
but how do I use the properties
heady heron
setting the diameter of a circle to two distinct values? but also setting its circumference? I don't get what you mean.
what, exactly, are you trying to do?
raven sail
oh, then do 6/2
so circumference / diameter
since i wanna see that the programm tells me its yeah pi
heady heron
so you want a circle of circumference 6 but diameter 2, exactly. that isn't happening though, as I'm sure you're aware of the true value of pi?
I'm very sure you're only allowed to specify either the circumference, or the diameter/radius, but not both, of the circle.
raven sail
oh
hot herald
changing one value, the system would automatically adjust the other values
so if you're trying break it and make it say pi=3, you're outta luck
raven sail
so, I wanna see how a circle looks with 3 instead of pi
would it be more ugly?
is it becoming an ellipse?
heady heron
Geogebra does not roll that way, unfortunately. I don't think you can change the value of pi.
raven sail
but I could make c = 6 and d = 2
heady heron
but you cannot assign it to the same circle!
heady heron
if, by a shape you meant a circle, then no.
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vagrant badger
Hi! need help with converting base-4 number to a decimal equivalent using a calculator. I really can't seem to figure it out. thanks!
wild sleet
you just press ×4 and add a digit, left to right
e.g. 332
start with 0
× 4 + 3
× 4 + 3
× 4 + 2
neon basin
Hi! need help with converting base-4 number to a decimal equivalent using a calc...
Let me ask you this
Do you understand decimal
For example
469
Can you write it in powers of 10
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raw ivy
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https://dm2ec218nt2z5.cloudfront.net/2022-02-07/v6aSdP5z2oRrc9xNY/620076adfdd76b...
raw ivy
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buoyant ravine
<@&268886789983436800>
severe igloo
<@&268886789983436800>
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slender plover
.reopen
Express the given expression as an integer or as a fraction in simplest form.
(e^1/2ln9)
narrow ermine
Is that $e^{\frac{1}{2}\ln(9)}$?
glossy valveBOT
Azyrashacorki
narrow ermine
Yeah ok that's the same
narrow ermine
Okok
Well there's a law of exponents for multiplication
Do you think you can use that?
slender plover
like undoing it?
narrow ermine
Yeah
slender plover
😭 😭 😭
glossy valveBOT
Azyrashacorki
slender plover
okay so then e^(-1/2)^9^ln
narrow ermine
Well you have $e^{-\frac{1}{2} \ln(9) } = (e^{\ln(9)})^\frac{-1}{2}$
glossy valveBOT
Azyrashacorki
narrow ermine
Yes
slender plover
so its just (9)(-1/2)?
narrow ermine
yes
slender plover
so -9/2
narrow ermine
Wait wait
slender plover
or is it 9^-1/2
narrow ermine
Yes that
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slender plover
.reopen
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✅ Original question: #help-28 message
slender plover
question rq
the question ius at the top
but if its 4^x=1/3
why can we just assue that we can log both sides
and itll still be equal?
i dont get the concept of doing that
queen crater
Taking the log of both sides of an equation is always fine as long as the quantity is strictly positive
slender plover
so like i can do anything to both sides if theyre positive
as long as i do it to both songs?
queen crater
Not "anything", but applying a log, yes
Just like adding the same quantity to both sides, or multiplying both sides by the same non-zero quantity
queen crater
That comes from the fact that log is a bijection between (0,+inf) and R
slender plover
no clue what that means ngl but thank you LOL
queen crater
Sure
slender plover
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tardy belfry
Hi
tardy belfry
I need help
severe igloo
Why does it say Request NEw Name
probably because your username might have been inappropiate so mods changed it
tardy belfry
probably because your username might have been inappropiate so mods changed it
It was ApplyNow?
How is that bad??
severe igloo
eh?
severe igloo
How do they get points
they subsituted x values in the interval [5, inf) into the function h(x)
you can see from the table
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raven sail
if, by a shape you meant a **circle**, then no.
would a circle with with an c / d with 2 not just be a not full circle?
devout saddle
can i pls have a hint for this problem?
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can i pls have a hint for this problem?
devout saddle
i plugged in numbers to find that f(x)=p(1)4/15/26/3...(x+2)/(x-1)
this nicely cancels out a lot of terms but it only works with the positive integers and not the negative or rational/irrational ones
oh wait nvm i got it
.cloe
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knotty marsh
Hi i'm only grade 11 but im doing functions and i dont understand how to do this
The question is 6 = 3x^2 + 9x -5 and i have to factorise by completing the square. ive tried to do it but im stuck and i dont get why
i got 3(x+1.5)+1.75 but im just lost and ion really get it if anyone could help me id appreciate that sm
im kinda slow sorry
hot herald
can you show a pic of your work
knotty marsh
can you show a pic of your work
In jus stuck on question D
hot herald
,rcw
glossy valveBOT
hot herald
you messed up a a sign
lime gate
wait give me a sec
hot herald
also forgot a ^2 in that last line
rare dock
decent handwriting
knotty marsh
also forgot a ^2 in that last line
oh yeah i know but other than that i mean
sorry im taking it in french so the (1.5, 1.75) we call the sommet
i think in english its the vertex
according to desmos its supposed to be (1.5, 11.75)
lime gate
-71/12
hot herald
to keep stuff balanced, that should be - 27/4
knotty marsh
wjat
hot herald
lime gate
i got the turning point as (3/2,-71/12) i’m not sure if it’s right tho
multiply
by 3
the y
coordinate
-71/4
knotty marsh
@hot herald got it sorry, i accidentally wrote + instead of -
and -20 over 4 minus - 27 over 4 = -47 over 4
which is 11.75
lime gate
hot herald
original question as mistyped
lime gate
oh..
hot herald
actual question has y =, not 6 =
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knotty marsh
.reopen
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✅ Original question: #help-28 message
knotty marsh
@hot herald sorry i have 1 more question i might be doing smt wrong but why did i get 1.5 and not -1.5
,rcw
glossy valveBOT
hot herald
$$y = a(x-h)^2 + k$$
the vertex will be $(h,k)$ \
$$x - \frac 32 = x - (-\frac 32)$$
which means that your $h$ will be $-\frac 32$
glossy valveBOT
ραμOmeganato5
hot herald
one sec, formatting
hot herald
and you didn't fix that missing ^2
knotty marsh
yeah yeah and also the 2nd step i didnt do it right
but ill fix it thanks sm
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hasty plume
I am tired, and I am struggling to find out what I did wrong. The solution according to my calculator is supposed to be 79/6.
hasty plume
But I can’t find my error in writing.
I don’t know how I am so far off, I triple checked my work.
I can also clean up my work if it is hard for anyone to read.
plush egret
,w graph y=x^2-3x and x=5
glossy valveBOT
hasty plume
lemme graph it on desmos rq
Ok.
I think my integrals are good, I am wondering if I put the wrong sign on something.
plush egret
you might get into trouble with exactly what they mean by area
hasty plume
you might get into trouble with exactly what they mean by area
The area between the graphs and the x-axis, right?
Because y=0 graph.
plush egret
well it does kind of look like you did it
lime ether
the last line is hard to follow
plush egret
i have an idea
lime ether
also, why the mixed fractions
plush egret
lets call it $f(x) = x^2-3x$
hasty plume
you might get into trouble with exactly what they mean by area
The high-lighted area.
glossy valveBOT
jan Niku
hasty plume
the last line is hard to follow
Ok, let me re-write.
plush egret
so then your areas are $- \int _0 ^3 f(x) \dd x + \int _3 ^5 f(x) \dd x$
glossy valveBOT
jan Niku
plush egret
this is $-(F(3) - F(0)) + (F(5)-F(3))$
glossy valveBOT
jan Niku
hasty plume
also, why the mixed fractions
I ended up with a fraction so I just put all of them over the same denominator instead of decimals. Because I had thirds.
plush egret
or $F(5) - 2F(3)$
glossy valveBOT
jan Niku
lime ether
I ended up with a fraction so I just put all of them over the same denominator i...
but like they don’t seem to match up with the line above it
hasty plume
**jan Niku**
I am lost.
plush egret
idk knief has got it 
lime ether
go on jan
lime ether
I am lost.
have you learned the fundamental theorem of calculus yet
hasty plume
but like they don’t seem to match up with the line above it
I simplified as well.
lime ether
if not then this is probably going over your head
lime ether
**jan Niku**
that’s what he did here
hasty plume
just in nicer notation
Okok.
lime ether
**jan Niku**
note that F(0) = 0
lime ether
happens
hasty plume
(54/6) - (81/6) = (-27/6)
But that whole thing was negative.
So it was supposed to be positive 27 over 6.
I feel stupid.
Ok, thank you for the help!
.close
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nimble crane
could somebody double check my argument for this problem? 
assume for a contradiction that such a function f(x) existed, and consider A = {a_0 + a_2x^2 + a_3x^3 + ... | a_i \in R}. this is an algebra because its closed under scalar multiplication, addition, and multiplication (there's no way to make a term of the form kx with these operations). A separates points because for any p neq q in [0, 1], h(x) = x^2 is an element of A for which h(p) neq h(q). A is also nowhere vanishing because g(x) = 1 is nonzero for all x in [0, 1].
the Stone-Weierstrass thm then yields that A is dense in C_0([0, 1]). since the function x lies in that space, we can find a sequence p_n in A that converges uniformly to x. then int_0^1 p_n(x) f(x) dx = 0 for all n, so taking the limit of this as n -> infty (passing it into the integral with uniform convergence), we have int_0^1 x f(x) dx = 0, a contradiction
raven mango
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bronze sand
They tried to bury us. They didn’t know we were seeds.
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bronze sand
could somebody double check my argument for this problem? <:pikathink:8790668725...
it looks sufficient so far, HOWEVER, consider condensing the Stone-Weierstrass thm
if i were to suggest advice, rephrasing the last part is appropriate
nimble crane
it looks sufficient so far, HOWEVER, consider condensing the Stone-Weierstrass t...
ah, checking the hypotheses?
maiden vapor
did you mean C([0,1])
nimble crane
if i were to suggest advice, rephrasing the last part is appropriate
hmm, what about the wording is problematic?
nimble crane
did you mean C([0,1])
is that the space of continuous functions on [0, 1]? 
this textbook uses C_0([0, 1]) for that instead
bronze sand
hmm, what about the wording is problematic? <:pikathink:879066872513970227>
C0[0,1]
maiden vapor
strange notation
bronze sand
it does not mean continous functions vanishing at 0
nimble crane
okay nvm, it's the space of continuous bounded functions on [0, 1] in this book
bronze sand
it means continous funtions that vanish at infinity
nimble crane
but this is compact anyways so same thing
nimble crane
it means continous funtions that vanish at infinity
icic, I was not aware that the notation here conflicts with standard notation
I will keep that in mind 
in that case, I suppose there's nothing else for me to ask here, so I will close the channel 
maiden vapor
yes
nimble crane
thank you @bronze sand and @maiden vapor for the help c:
maiden vapor
obviously A isn't an infinite sum which your current wording kinda suggests it is
nimble crane
ah, right
that's a good point
how should I write it instead then?
span{1, x^2, x^3, ...}?
maiden vapor
that works yes
nimble crane
gotcha
maiden vapor
or something like
diamond$ds A= left lbrace, a_0+sum_{k=2}^N a_k x^k colon N in mbb N,, a_k in mbb R , right rbrace$
glossy valveBOT
Oléagineux Distilliànus VIVII
maiden vapor
span{1, x^2, x^3, ...}?
but this works as well
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maiden vapor
nimble crane
timber veldt
rare dock
maiden vapor
rare dock
blud loves bad ideas
maiden vapor
bideas
rare dock
blud ideas
i need my melody creepy smile emoji again
😌
so many years without you
now you have returned
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plain fox
just started metric spaces and now im stuck on this
Let d be a metric on X and let d:X x X -> R be given by d'(x1,x2) = min[1,d(x1,x2)]. is d' a metric on X
plain fox
cant prove the triangular property / inequality , whatever :c
queen gull
||to show d'(x,y) + d'(y,z) >= d'(x,z), either all of d(x,y),d(x,z),d are at <= 1 and the inequality follows from d being a metric, or one of them is at >1 and the inequality is satisfied immediately. the only case left is when d(x,z) > 1 and d(x,y),d(y,z) <= 1 and in that case
d'(x,y) + d'(y,z) = d(x,y) + d(y,z) >= d(x,z) > 1.||
fast peak
!nosols and all that
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queen gull
delete?
plain fox
||to show d'(x,y) + d'(y,z) >= d'(x,z), either all of d(x,y),d(x,z),d are at <= ...
wait how many cases am i making here?
fast peak
well at this point you already went through the trouble of typing it up ¯_(ツ)_/¯
plain fox
if all are less than or equal 1, if any 1 is > 1
fast peak
probably doesnt hurt to just write down all 8 cases. then you can really quickly see which ones are trivial
plain fox
wait lemme reread again
plain fox
probably doesnt hurt to just write down all 8 cases. then you can really quickly...
all 8?
i can only think of 6
fast peak
three numbers, all <1 or >=1
plain fox
ah ok , i forgot for different equalities
bronze sand
!nosols
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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.
plain fox
three numbers, all <1 or >=1
wait , so if i wanna prove that
should i write all 8 cases?
or just write the ones that are gonna be hard to spot
like if all are > 1 then its ez af
same with < 1
as ExpertEsquie said so
fast peak
its just 8
just write them out
then its easier to see which ones are easy and which ones are not
and less risk to accidentally miss one
plain fox
just 8 D:
anyway , thnx to both (ig time to spend like 10 mins on this question now )
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fast peak
you cant get better without investing time
plain fox
thats true
oh wait btw , why are we doing < 1 and > = 1
why not < = 1 or > 1 
or would that not make a difference?
fast peak
try it and see
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fervent briar
Exercise says I gotta evaluate, am I multiplying wrong?
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Exercise says I gotta evaluate, am I multiplying wrong?
fervent briar
I checked the answers and it says it's wrong
nimble ridge
You're applying cross-multiplying to a concept that doesn't use cross-multiplying
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torn jolt
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proven lynx
is this correct
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is this correct
umbral dome
where did that last a go in the simplified form
umbral dome
you should have an a^3 in there, yes
ivory pilot
sure, take out a^2 common
ivory pilot
yeah actual answer is $e^{-ax}(a^2-a^3x)=e^{-ax}a^2(1-ax)$
glossy valveBOT
donkey
proven lynx
yeah actual answer is $e^{-ax}(a^2-a^3x)=e^{-ax}a^2(1-ax)$
but
shouldnt i just leave it like this
in the bracket
ivory pilot
your wish, it doesn't really make a difference
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@proven lynx Has your question been resolved?
ivory pilot
there shd be a negative sign when you move a^2 to RHS
proven lynx
omg 🤦♂️
proven lynx
a/a^2 = 1/a
oh
ohhh
okay got iit
also i dont need to do a case thing here right
for extreme points
ivory pilot
nah idts
proven lynx
okay tysm
ivory pilot
yo did i do something wrong here
you're good
proven lynx
you're good
okay so
i cant do anything further right
thats my final y value
a * e^-1
ivory pilot
bingo
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dull flint
$\text{From a pack of }52\text{ cards, one card is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability that the lost card is a diamond.}$
glossy valveBOT
BlackidoZΣ
fast peak
no reason at all to use \text for this
dull flint
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void jay
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void jay
i've reached a point where i dont know how to continue
neon surge
i've reached a point where i dont know how to continue
what is 2^4 equal to?
void jay
16
neon surge
so rewrite the last line so it looks like one of the answers
void jay
i can split 2^4a ?
neon surge
you can write $2^{4a} = (2^4)^a$
glossy valveBOT
k
brittle sun
,tex .exp rules
glossy valveBOT
Xavier 🌺
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nocturne barn
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grave elm
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grave elm
its with lowercase
nocturne barn
Ok
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civic dust
How to find asymptotes vertical and horizontal ones in those functions
fossil hound
drawing the graphs is the best option if you cannot reason it out
civic dust
I need to show the working I think
fossil hound
is graphing not considered working nowadays
civic dust
I mean I don't know I just have to fill out the table
And it appear that all of e functions have similar horizontal asymptotes
But do they have vertical ones?
fossil hound
have you seen a graph of the function before?
fossil hound
I presume you have drawn it correctly if you're not showing it
if you are, then can you see any obvious vertical asymptotes?
civic dust
There is no vertical asymptote, the function goes to infinite y and x positive cordinates
fossil hound
congratulations, you've answered part of your own question correctly
fossil hound
and I am validating your answers
civic dust
❤️
How does the minus sign in the exponent in equation $2^{-x}$ reflect it over y axis?
glossy valveBOT
JPuXIUim6x
fossil hound
f(-x) is a horizontal reflection of f(x).
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marsh trellis
f(-x) is a horizontal reflection of f(x).
You mean vertical, about y axis
fossil hound
You mean vertical, about y axis
nimble ridge
You mean vertical, about y axis
y-axis is vertical, and the reflection takes every point in the orthogonal direction across it
fossil hound
fossil hound
the direction of reflection gives the reflection its name, not the orientation of the axis of reflection
marsh trellis
the direction of reflection gives the reflection its name, not the orientation o...
Got it
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twin wolf
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twin wolf
Since C is just a union of the edges of the polygon can i write
glossy valveBOT
water beam
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@twin wolf Has your question been resolved?
brittle sun
Not enough reacts on that bot message
bright bronze
doesnt the individual term of the summation just correspond to the corresponding side just rotated 90deg?
bright bronze
twin wolf
https://tenor.com/view/the-simpsons-homer-hiding-one-with-the-bush-back-away-gif...
y u leave
come back
twin wolf
doesnt the individual term of the summation just correspond to the corresponding...
um im not sure but. we are trying to show that the outward normals of a closed polgyon sum to 0 e.g greens theorem
bright bronze
well, normal is just a unit vector rotated 90deg, and ||p_i - p_(i-1)|| is just the side length
so that term becomes same as the side vector rotated
and closed polygon means sum of side vecs is 0
twin wolf
i already proved the previous parts
twin wolf
**water beam**
but if what i wrote here is valid then it is just easy substitution i can sub what i found into the summation of the integral
but am i allowed to do that
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topaz cargo
9+10
pearl trout
<@&268886789983436800>
austere cove
9+10 = 21 hours
cunning sapphire
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wraith cargo
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wraith cargo
im meant to be finding the eigenvalues of this matrix or something and i got a bunch of decimals :(
im also not completely sure if my part a is correct which means part b might be fully incorrect !
i tried setting using the k value i got in part i and putting that into the matrix, and then solving for det(M-(lamda)I) by setting it equal to zero but i messed up somewhere in the expanding or adding or something which is not super uncommon for me but i cant seem to see where i went wrong
im going to try it again whilst i wait for help ^_^
oh i mightve seen where i went wrong give me a sec
aaahh all good ! just accidentally did bc-ad on one of the determinants instead of ad-bc
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inner mauve
question: Check whether the following calculation of the expected value of a random variable X, which can take on the three values 1, 2, or 3, can be correct.
$E(X)=1 * 0.2 + 2 * 0.6 + 3 * 0.1 = 1.7$
Reason: No, because the sum of all probabilities must be 1.
just want to clarify. is my answer to this correct?
glossy valveBOT
J. Crestwood
keen vector
yes
inner mauve
is there a way I could word this more 'professionally'?
keen vector
no
inner mauve
okay! thank you
robust slate
**J. Crestwood**
inner mauve
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thorn tartan
Someone had solved it here
fast peak
$\int_0^\infty \frac{\sin^2(\ln(x))}{x\ln^2(x)(x+1)} dx$ ?
glossy valveBOT
Denascite
thorn tartan
Prolly ln(x) = t first?
You can simplify this to
1/2 of integral of sin^2(t)/t^2 ig
Ig it's trivial after that?
Ye just need a lil manipulation
Remember kings
Not exactly kings put apply it
It's an even fxn
Yes
Not sure
Nice
Welcome
fast peak
the sub changes the limits
thorn tartan
Stop after differentiating 1 time
Sin(2ax)/x from 0 to inf
It's a well known integral
You can just let 2ax= t
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oblique tusk
How is this simple proof?
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How is this simple proof?
rugged void
works out
oblique tusk
thanks
delicate torrent
oh it's !done
!done
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If you are done with this channel, please mark your problem as solved by typing .close
delicate torrent
let me do the honors
woeful girder
j- jason bourne?
rugged void
let me do the honors
:c
slate glacier
do you have to ask something else @oblique tusk ?
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delicate torrent
why don't i have my green 🥀
delicate torrent
🥀
slate glacier
bro is emeritus
rugged void
lmao
rugged void
it takes a while lol
slate glacier
i think im in the banned list of green or smth
Well you have very active, I don't
delicate torrent
it takes a while lol
ik someone who's in the server for 7 days and got green
slate glacier
ik someone who's in the server for **7** days and got green
heck even I got it in like 3 months
delicate torrent
heck even I got it in like 3 months
i helped like
i think a year now
like 365 days
not from 2025->2026
slate glacier
😭
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final cedar
Is it true to say that y = sqrt{x} does not have an inverse function since it is not bijective?
queen crater
Depends on the domain / codomain
final cedar
how so?
sturdy valve
if domain is x>=0 it does have an inverse
stiff musk
it's bijective on [0,infty)
queen crater
If f(x) = sqrt(x) with domain and codomain [0, +inf), then f^-1(x) = x^2, with the same domain and codomain
stiff musk
then it's injective but not surjective
queen crater
A function is bijective iff it's both injective and surjective
Surjectivity is really just codomain = range
final cedar
right
don't we normally set the codomain to all real numbers and only restrict the domain when finding inverses?
queen crater
Up to you really
stiff musk
if you have a function that is injective but not surjective, it has a left inverse, perhaps that's what you mean?
final cedar
Like intro to functions, find the inverse, do they just assume there are no restrictions on the domain?
https://www.mathway.com/popular-problems/Precalculus/939645 stuff like this
doesn't really make sense to me
why would it have an inverse in this case? Is it just assumed to have one in the least restrictive sense?
queen crater
https://www.mathway.com/popular-problems/Precalculus/939645 stuff like this
Well that's not very rigorous
As you can see the answer in blue can't really be the actual inverse
If you restrict it to x>=0, then it's correct
stiff musk
you don't really, the argument made at that website implicitly assumes that x and y are suitably restricted
queen crater
From your link:
stiff musk
otherwise the argument is bogus
queen crater
Except f(f^-1(-1)) is undefined
final cedar
you don't really, the argument made at that website implicitly assumes that x an...
If it's implicit, what is the standard implication to follow in academia?
or is it arbitrary
queen crater
The standard kind of depends on the level of rigor
stiff musk
and f(f^-1(-3)) is defined but doesn't equal -3
queen crater
Like here they are teaching you how to manipulate the expression of a function to find the expression of the inverse
They aren't teaching you about domains and codomains
stiff musk
If it's implicit, what is the standard implication to follow in academia?
even at the elementary level i'd expect at least some attention to be paid to domains and codomains when finding inverse functions, otherwise you can get conclusions that are simply false
false as in claiming things like sqrt((-1)^2) = -1
queen crater
Except f(f^-1(-1)) is undefined
Oh sorry I meant this equals 1, not -1
final cedar
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atomic venture
heya im trying to use this book as a reference, but i dont really understand this section; what would 'conjugate' mean in this context?
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heya im trying to use this book as a reference, but i dont really understand thi...
@atomic venture Has your question been resolved?
atomic venture
<@&286206848099549185> 🥺
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@atomic venture Has your question been resolved?
minor crater
heya im trying to use this book as a reference, but i dont really understand thi...
pretty sure it means "equivalent" in some sense
any p(x) can be written in the form of p_c(z)
glossy valveBOT
woeful girder
good handwriting
atomic venture
thanks :>
woeful girder
🫂
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dull flint
The value of
[
\cot x + \cot(60^\circ + x) + \cot(120^\circ + x)
]
is equal to:
(A)\ \cos 3x
(B)\ \tan 3x
(C)\ 3\tan 3x
(D)\ 3\cot 3x
glossy valveBOT
BlackidoZΣ
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
marsh trellis
Don’t brute force like that
dull flint
Yes I stopped that
marsh trellis
**BlackidoZΣ**
Compile Error! Click the <:errors:504266556054962196> reaction fo...
PS - if this came in an exam I’d guess D
marsh trellis
Cuz put x=0
You get infinity
In the options only D gives infinity
Anyways how you solve it is take any two terms, write it as cos and sin, and then use the cos(A+B) formula
dull flint
Anyways how you solve it is take any two terms, write it as cos and sin, and the...
okay I'll not try
dull flint
You get infinity
I try this
marsh trellis
I try this
Lmao is your exam MCQ? Or proofs
dull flint
Mcq
but idk how did yoy guess that trick to put zero and check infinity
Maybe practice practice
sturdy valve
its a standard approach..putting values
woeful girder
good handwriting
marsh trellis
but idk how did yoy guess that trick to put zero and check infinity
Meh it’s normal
I’m a lazy guy
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long helm
How do i integrate cos^2(x)?
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How do i integrate cos^2(x)?
long helm
$\int\cos^2(x) dx$
glossy valveBOT
dream
sick karma
use a trig identity that reduces the power of cosine
long helm
So i always need to remove the exponent?
long helm
use a trig identity that reduces the power of cosine
Well $\cos^2(x) = 1-\sin^2(x)$
glossy valveBOT
dream
sick karma
in general for trig functions, yes
long helm
in general for trig functions, yes
Okaiokai
sick karma
it is better to either look up a trig identity that relates cos(x)^2 to cosine, or try deriving the identity yourself
queen crater
Either use another identity or do IBP
vital citrus
Or should i do something else
whats cos(2x)