This is used a lot for multi-variable limits because if you can show $\abs{\text{inside of your limit}} \le (\text{limit of something you know goes to 0})$, then you can conclude straight away that your limit goes to 0.
#help-28
narrow ermine
glossy valveBOT
Azyrashacorki
narrow ermine
It's often easier than checking paths and such
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@bitter tree Has your question been resolved?
bitter tree
ohh oke thx so much
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upbeat tiger
\begin{document}
\textbf{Problem 6.}
In the accompanying figure is represented the cube ( ABCDA'B'C'D' ), with ( AB = 8 ) cm. The diagonals ( AC ) and ( BD ) intersect at point ( O ), and the lines ( A'B ) and ( AB' ) intersect at point ( E ). Point ( F ) is the midpoint of segment ( CC' ).
\begin{enumerate}
\item[(2p)] \textbf{a)} Show that the volume of the cube ( ABCDA'B'C'D' ) is equal to $512$ cm( ^3 ).
\begin{center}
\( V = l^3 = \dots = 512 \) cm\( ^3 \)
\end{center}
\item[(3p)] \textbf{b)} Prove that the lines \( FO \) and \( DE \) are perpendicular.
\end{enumerate}
\vspace{1cm}
(Note: The figure shows a cube in perspective with points ( O ), ( E ), ( F ) marked, together with the space diagonals and some face diagonals.)
\end{document}
glossy valveBOT
Allstars
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torn jolt
<@&286206848099549185>
mossy stratus
<@&286206848099549185>
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pearl thicket
This is the figure
What do you need help with
upbeat tiger
**Allstars**
Compile Error! Click the <:errors:504266556054962196> reaction for ...
I can only do this with pure geometry
I need help with B
Soooo?
upbeat tiger
No
balmy dust
@upbeat tiger
upbeat tiger
Pure geometry only
solid anchor
hi
balmy dust
seems pretty elementary with vectors
solid anchor
whats the doubt?
balmy dust
dk about geo tho
solid anchor
wait is geometry?
balmy dust
whats the doubt?
b part
torn jolt
Pure geometry only
what geometry are you allowed to use
upbeat tiger
what geometry are you allowed to use
Like all the things before vectors
Like pure geoemetry
torn jolt
🥀
upbeat tiger
Im in 8th grade
torn jolt
hm..
balmy dust
!status
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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
1
mossy stratus
for question A since its a cube so base line height and width equal
set those equal l, then volume = l^3 = 8^3
balmy dust
he only has doubt in B supposedly
mossy stratus
o let me see
upbeat tiger
he only has doubt in B supposedly
Yes
upbeat tiger
It is the middle line so it is parallel
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torpid perch
which kind of functions have empty image
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which kind of functions have empty image
mossy stratus
Empty function
mossy stratus
Function f: empty symbol -> B
You know a function is when given any x only return exactly one y value for each x right
Alo, care to respond? -.-
torpid perch
you mean a function with a empty domain
mossy stratus
Yep
torpid perch
that is a empty function
mossy stratus
If no domain then no codomain to image
torpid perch
You know a function is when given any x only return exactly one y value for each...
yeah, precisely
any x in domain
mossy stratus
My bad for respond long i was typing translate and do code
torpid perch
what?
mossy stratus
You nailed it
mossy stratus
care to elaborate
I. Was. Using. Google. Translate.
torpid perch
i see
torpid perch
You know a function is when given any x only return exactly one y value for each...
yeah i think that is the only possibility right
mossy stratus
If you have a domain that have no elements, then you cant have an output
torpid perch
that the domain is empty
mossy stratus
Like putting no fruits in blender and expect to have smoothies
mossy stratus
yeah i think that is the only possibility right
Yes
torpid perch
Like putting no fruits in blender and expect to have smoothies
i see
mossy stratus
Function (empty symbol)
torpid perch
.close
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subtle thorn
I have a couple of PDE-related problems i should be able to solve but am too brainrotten to rn, and deadline is today. If someone could nudge me in the right direction, i'd be very grateful.
1
I have proven a result showing for $x\in\mathbb{R}^3$, $R>0$ [
\int\limits_{|y|=R}\frac{dy}{|x-y|}=\int\limits_{|y|=R}\frac{dy}{\max{(|x|,R)}}.
]Using this, i need to show the following: for $0 \leq f \in L^1(\mathbb{R}^3)$ a radial function, prove that there exists $M>0$ such that for $|x|$ large
enough, we have
[\int\limits_{\mathbb{R}^3}
\frac{f (y)}{
|x - y|} dy \leq \frac{M}{
|x|}.]
glossy valveBOT
🇵🇸μ🔆
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subtle thorn
i think the idea of this problem is to split the integral by $|x|>|y|$ and vice versa, so we integrate over a ball (or its complement) for fixed $|x|$, there the radiality of $f$ should prove useful
but just some step isn't clicking
glossy valveBOT
🇵🇸μ🔆
subtle thorn
it is also obvious the constant will be a multiple of the L1 norm of f
<@&286206848099549185>
solid anchor
Yeah?
subtle thorn
here's an idea of what might work as a solution but i don't think it makes sense.
\begin{align*}
\int\limits_{\mathbb{R}^3}
\frac{f (y)}{|x-y|}
\leq \int_{|x|>|y|}f(y)/|x|dy+\int_{|x|<|y|}f(y)/|x|\leq 1/|x| (\int_{|x|>|y|}|f(y)|dy+\int_{|x|<|y|}|f(y)|dy)\leq ||f||_{L^1}/|x|
\end{align*}
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subtle thorn
:/
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subtle thorn
<@&268886789983436800>
subtle thorn
I think there is shortage of helpers, dont worry <:catthumbsup:61454018874756300...
thx, i'll try to solve it in the meantime
neon rivet
inflicting sensitive topics?
sterile jungle
Good for you, but don't spam in random help channels please
austere cove
Why you invading a help channel if you don't have anything useful to say?
cunning sapphire
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@subtle thorn Has your question been resolved?
subtle thorn
no it hasn't and i don't expect it will be.
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fringe siren
In the first equation the right part is supposed to be infinity, so there is a mistake, i think
subtle thorn
In the first equation the right part is supposed to be infinity, so there is a m...
what do you mean by that
fringe siren
you integrate over R^3 the constant, it is infinity.
subtle thorn
In the first equation the right part is supposed to be infinity, so there is a m...
oh you're right, the integral is supposed to be over |y|=R
sorry about that
<@&268886789983436800>
fringe siren
in the second equation also the same?
subtle thorn
in the second equation also the same?
no, it's supposed to be over R^3 in the inequality.
fringe siren
what you mean by f- is a radial function, it depends only on radius?
subtle thorn
yes, ie it's rotationally invariant
fringe siren
the conditions still seem strange to me because for equations shouldn't matter how big x is
i suppose you should rewrite equation in radial coordinates to better understand behavior of integral
$dx dy dz =r^2 \sin(\theta) dr d\theta d\phi$
glossy valveBOT
mikhail_kolesnikov
fringe siren
if f is a radial function it will be actually $L^1(\mathbb{R})$ and and the analysis will be reduced to integration over one variable am not sure how to write |x-y| but i suppose y shoulduse formula |x|^2=(x,x) where (x,x) is a scalar product
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mikhail_kolesnikov
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subtle thorn
if f is a radial function it will be actually $L^1(\mathbb{R})$ and and the anal...
yeah yeah i was just hoping there would be an easier way while incorporating the previous result
like the computation is very much analogous
like i can WLoG rewrite f(y_1,y_2,y_3) =g(|y|)
fringe siren
the problem that f(r) can have pole in r=x
subtle thorn
but it's a removable singularity so do we mind
that's why we split the integral right
and since f is L1 then the integral is finite
fringe siren
honestly idk
subtle thorn
now i'm just not 100% sure what to do on the ball's complement
subtle thorn
honestly idk
to elaborate, by previous result, i already know integral of 1/|x-y| is finite
(on the ball)
so multiple by L1 function is no problemm
fringe siren
to elaborate, by previous result, i already know integral of 1/|x-y| is finite
its finite on sphere by previous result
subtle thorn
exactly
fringe siren
so multiple by L1 function is no problemm
It's not obvious to me
subtle thorn
well, i'm just tryna find the upper bound, right?
so, [
\int_{|x-y|<R}f(y)/|x-y| \leq \int_{|x-y|<R}|f(y)|/|x-y|\leq ||f||{L_1}\int{|x-y|<R}\frac{1}{|x-y|}]
glossy valveBOT
🇵🇸μ🔆
subtle thorn
am i making a logical leap
fringe siren
f(y) is a vector function?
subtle thorn
well $y\in\mathbb{R}^3$
glossy valveBOT
🇵🇸μ🔆
subtle thorn
the whole time $x,y\in\mathbb{R}^3$. Idk if that wasn't super obvious, sorry
glossy valveBOT
🇵🇸μ🔆
fringe siren
no am asking if $ f:R^3\to R$ or $f:R^3\to R^3$
subtle thorn
right dude im so fucking spent lol
glossy valveBOT
mikhail_kolesnikov
fringe siren
**🇵🇸μ🔆**
I don't think the last inequality is correct, it needs to be integrated by parts and carefully
fringe siren
sr gotta go
subtle thorn
ofc, thanks for your time anyways
fringe siren
subtle thorn
you're the only one on here who actually gave big effort it's v appreciated :3
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robust knoll
noble plover
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fresh dune
I need help with proving this problem
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I need help with proving this problem
fresh dune
I tried doing some calculations but got stuck
What am I supposed to do in that problem?
spiral vigil
s^2 + c^2 = 1
fresh dune
s^2 + c^2 = 1
Okay, what am I supposed to do after that one?
spiral vigil
what's your goal again?
fresh dune
I need help with proving this problem
To prove this problem
spiral vigil
with all of these problems the correct thing to do is to immediately convert both sides into sin and cos
using s^2 + c^2 = 1 and t = s/c
fresh dune
Alright, I'll try right now
fresh dune
with all of these problems the correct thing to do is to immediately convert bot...
Oh alright i got it
Thank you
.close
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empty tangle
am I the only one who think this is worded a bit ambiguously ?
empty tangle
to me it sound like shes earning 300 per trip
spiral vigil
that does appear to be what it says
lost python
I think it would be 300 + earnings for apples each trip?
empty tangle
I think it would be 300 + earnings for apples each trip?
so you are saying that a sentence "for each trip lennox earns 300 dollars" does not mean 300 dollars TOTAL ?
empty tangle
ye its so annoying
lost python
but it could be 5 trucks going on 15 trips per week getting $4500?
empty tangle
to me it sound like shes earning 300 per trip
it COULD, but I would say that its a lot more reasonable to interpret it this way
this happens sometimes, not a big deal
noble spade
am I the only one who think this is worded a bit ambiguously ?
Each truck goes on 3 trips per week, 1 trip is 300$ so 3 trips would be 900$ meaning 1 truck earns 900$ per week so there are 5 trucks then right
empty tangle
saying "for each trip lennox earns 300 dollars" would not be accurate then
empty tangle
Does it give any other meaning?
why is this not correct then ?
noble spade
why is this not correct then ?
Because you didnt consider the fact that there are multiple trucks going
Its not ONLY 3t/W its 3t/W x no.of trucks
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foggy prairie
Can someone explain to me why velocity is going downwards
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Can someone explain to me why velocity is going downwards
foggy prairie
I’m confused
foggy vapor
as the acceleration is pointing downwards, it forces the velocity to (eventually) also start pointing down
or were you looking for a more mathematical explanation?
foggy prairie
But I don’t even understand why acceleration is pointing downwards
foggy vapor
gravity
foggy prairie
Acceleration is supposed to be the instantaneous rate of change of velocity right ?
So how is it that
foggy vapor
yeah?
gravitational acceleration is just taken as a constant system condition
you are always going to have -9.81 m/s^2
foggy prairie
Again this isn’t explaining how that’s acceleration comes about from what we understand the derivative is
That’s gravity but I don’t get why that’s acceleration which is the instantaneous rate of change of velocity
foggy vapor
i am saying that you should be looking at it from the opposite direction
given the acceleration, what is the velocity at any time t?
foggy prairie
Integrate it
foggy vapor
yes
foggy vapor
Since there is no horizontal acceleration, the horizontal component remains constant
ok i mean
foggy prairie
Yeah Idk man
foggy vapor
we can be proper about it too
foggy prairie
Can you explain it in terms of how acceleration is the rate of change of velocity
dusty scaffold
well we know that the acceleration is the rate of change of the velocity, and we also know that the acceleration is -9.81 m/s^2 (because that's how gravity works), so we compute the velocity from that
more explicitly the thing we end up with is v = (initial velocity) - 9.81t
if you differentiate that then you get -9.81 m/s^2
(which is the same as how gravity should work because we specifically chose this formula to make it work out that way)
foggy vapor
your position at any time $t$ is going to be [
\vv r(t) = \vc xx(t)+\vc yy(t)
]
if you launch the projectile from an initial height $h$ with an initial velocity $v_0$ at an angle $\theta$, the components are going to be
\begin{align*}
x(t) &= (v_0\6\cos\theta)t \
y(t) &= h + (v_0 \sin \theta)t - \412gt^2
\end{align*}
due to the standard kinematics equations. Differentiating those, you get:
\begin{align*}
v_x(t) &= v_0\6\cos\theta \q(\t{note it being constant}) \
v_y(t) &= v_0 \sin \theta - gt
\end{align*}
Take the derivative once more, you get:
\begin{align*}
a_x(t) &= 0 \
a_y(t) &= -g
\end{align*}
Although, truth be told, this is like circular reasoning as the kinematics equations are derived from the above procedure done in reverse, so...
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foggy prairie
well we know that the acceleration is the rate of change of the velocity, and we...
Sorry you lost me at “how velocity works”
foggy vapor
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foggy vapor
DAG: Directed Acyclic Graph
I don't understand the fourth bulletpoint of this proof. What was taken to be the inductive hyptohesis?
split hatch
I guess all DAGs of order <n have a topological ordering
foggy vapor
strong induction?
split hatch
well it's weak induction since we only use the fact for n-1
but it doesn't really matter
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tame sundial
im so confused rn, i should be getting x+4-4 but im getting x-4-4
delicate torrent
,rccw
glossy valveBOT
tame sundial
and i’ve tried using ai to explain it to me but i lwk need human help
noble spade
and i’ve tried using ai to explain it to me but i lwk need human help
can you show your whole process
delicate torrent
and the original question too
delicate torrent
,rccw
glossy valveBOT
delicate torrent
i think you wrote your $x$ as $y$
glossy valveBOT
1 divided by 0 equals Infinity
tame sundial
forget the <-4 that’s me bent a dumbass
delicate torrent
i kinda notice that for a long time
tame sundial
like wdym i switched x and y to find the inverse
im stuck on trying to prove it’s an inverse
delicate torrent
huh?
noble spade
which one are you doing gang
tame sundial
im doing a
glossy valveBOT
1 divided by 0 equals Infinity
glossy valveBOT
1 divided by 0 equals Infinity
tame sundial
well yeah and?
noble spade
im getting f inverse x =2x^2+16x+58 lowk ;-;
tame sundial
im getting f inverse x =2x^2+16x+58 lowk ;-;
that’s js u
noble spade
im so confused rn, i should be getting x+4-4 but im getting x-4-4
so what are you getting as answer?
i mean what is the answer supposed to be
tame sundial
so what are you getting as answer?
well it’s supposed to be x+4-4 which is just x
ik im not crazy
and i asked ai and it says that too but it’s talking abt some absolute value and changing the sign to negative for some reason and idek
glossy valveBOT
delicate torrent
bruh
noble spade
well it’s supposed to be x+4-4 which is just x
im asking the uh f inverse function
tame sundial
**1 divided by 0 equals Infinity**
it’s like saying this but idek what the absolute value has to do with it
noble spade
WOW
noble spade
that’s js u
..??
sacred cloak
there we go
noble spade
mb forgot to divide by 2
tame sundial
i already know that part i just wanna know how to verify that it’s an inverse
sacred cloak
i already know that part i just wanna know how to verify that it’s an inverse
if g(x) = f-1(x) then g(f(x)) = x
tame sundial
and im genuinely right there but like im stuck right here
noble spade
i already know that part i just wanna know how to verify that it’s an inverse
there is a mathematical process rather than just guessing lowk
tame sundial
if g(x) = f-1(x) then g(f(x)) = x
yeah but what im not getting is x im getting -x-4-4
noble spade
noble spade
its not its the f inverse function lol
tame sundial
it is tho
noble spade
its better than swapping x,y
tame sundial
lwk man i give up idek
noble spade
.close if ure done
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timid shore
Hellooo
Does anyone understand why this is?
X² becomes 2x
& -2x becomes (2-1) since x is 1 & it basically is the same thing when you multiply it??
But how did they get -1×2x^0
gritty rose
Do you know power rule for derivatives
timid shore
Do you know power rule for derivatives
No 😅
gritty rose
,tex .diff rules
glossy valveBOT
riemann
gritty rose
More or less you'll need to memorize those. Certain the second and third for this problem
timid shore
More or less you'll need to memorize those. Certain the second and third for thi...
Is this the same?
gritty rose
yes
timid shore
What's the difference between () & []?
gritty rose
nothing
gritty rose
no idea what you just said
gritty rose
?
sonic stratus
For a certain function f which has a derivative, if f ' is positive, then f is increasing (upwards slope), and viceversa for negative values.
if f ' = 0, then the original function f is constant.
timid shore
For a certain function _f_ which has a derivative, if _f '_ is positive, then _f...
Ah okay tysm
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pulsar kiln
\textbf{Problem (English).} Let $f=f(x,y)$ satisfy the partial differential equation
[
\frac{\partial f}{\partial y}=\frac{\partial^2 f}{\partial x^2}-2\frac{\partial f}{\partial x}.
]
Show that one can choose real numbers $a,b\in\mathbb{R}$ such that the function
[
u(x,y)=e^{ax+by},f(x,y)
]
satisfies the simpler equation
[
\frac{\partial u}{\partial y}=\frac{\partial^2 u}{\partial x^2}.
]
Also determine the values of $a$ and $b$.
glossy valveBOT
Quantie
pulsar kiln
so idk what im supposed to do
do i first solve the last equation du/dy = d^2u/dx^2
then show that that also solved the first equation
gritty rose
you plug u(x, y) into the last equation and show it's true
pulsar kiln
you plug u(x, y) into the last equation and show it's true
so the first equation is just a factiod
well it could be, the dudes that wrote my literature love to do that kind of stuff
gritty rose
i don't know what you mean by factiod
pulsar kiln
i don't know what you mean by factiod
just fun thing to know
i could try to solve it too
pulsar kiln
ok but imma solve the last equation
after deriving and simplifying i get
bf + df/dy = a^2f + 2adf/dx + d^2f/dx^2
which i think is wrong
do you know how id go with starting to deriving
this is just deriving u(x,y)
gritty rose
the word is differentiating
pulsar kiln
oh right yeah, my prof talked about that
in swedish, differentiate and deriving are two different things lol
pulsar kiln
ok but imma solve the last equation
after deriving and simplifying i get
bf ...
ohh ok i plug this into the first equation then i get an easier equation to solve
dope
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iron pond
I asked chatgpt and looked online but I cant understand how to integrate two unrelated trignometric functions. Can anyone explain how I'm supposed to do this?
gritty rose
integrate by parts 
iron pond
integrate by parts <:thumbsupanimegirl:1203420374243610665>
could you elaborate?
gritty rose
noble spade
iron pond
I'm sorry but I dont understand this formula
noble spade
v is generally the one which is easy to integrate and u is the one which gives simple things on differentiation
gritty rose
read the beginning of https://tutorial.math.lamar.edu/classes/calcII/IntegrationByParts.aspx
noble spade
in this case, better to take u=x and v=sinx
gritty rose
it follows from product rule for derivatives
iron pond
v is generally the one which is easy to integrate and u is the one which gives s...
I see so I have to integrate and differentiate at the same time
iron pond
read the beginning of https://tutorial.math.lamar.edu/classes/calcII/Integration...
thank you I will
gritty rose
f' = derivative of f yes
noble spade
I see so I have to integrate and differentiate at the same time
yes but its very simple just try it
rugged void
!done
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iron pond
!done
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rugged void
(if you think you'll need help keep it open)
iron pond
.close
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hoary ember
theres alot of useless info here
number of bathrooms and time in the shower
well i mean, from what i sse in your writing, you used those informations? honestly im not even sure what im seeing right now as its incredibly messy. also making sure again, your only using divisions right as a challange?
honestly i reccomend trying to do it normally, then trying to fit the challange for division after you solved it regularly
cause honestly trying to do it directly is just an extra headache
basically i would try and just spam a×b=a/(1/b)
as doing it regularly should be much simpler
knotty grail
were you able to find a formula for the total money spent first?
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@empty tangle Has your question been resolved?
noble spade
the 3 bathrooms and 480 minutes information is practically not used at all
i mean not needed
72 litres per person, meaning 432 litres for 6 people from there u can calculate
i dont understand it either how did you get here?
noble spade
72 litres per person, meaning 432 litres for 6 people from there u can calculate
how about you try this method to get to the answer tho ?
x6p
knotty grail
works i guess
people normally find multiplication more intuitive though
in case i had 8 boxes with 9 pencils in each i would do $8 box \times \frac{9 pencil}{box}$ rather than $\frac{8 box}{\frac{box}{9 pencil}}$ due to how we describe things in english
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sly anchor
this is a Magnetic flux density-time graph the figure related to it is on top.can some one explain me plz why E2 is reduced? why it is not being added ,as area increase?
neon silo
Physics server might be slightly more helpful
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cyan trellis
Can I get help at b? Im quite stuck. It says to bring the expression to its simplest form (for context tg is tan and ctg is cotangent)
indigo drift
Are you trying to prove both sides are equal to each other?
cyan trellis
No, there's an expression and i need to bring it to the simplest form
balmy dust
do you know the identity $a^2 - b^2 = (a+b)(a-b)$
glossy valveBOT
parthisjoking
flint hull
Can I get help at b? Im quite stuck. It says to bring the expression to its simp...
is it cos(x^4) or cos^2(x)? if the latter, then you can use the identity the person gave above
remember that cos^4(x) = cos^2(x) * cos^2(x)
cyan trellis
Hmm let me see
So uh at the part with (cos²x-sin²x)²/(sinx*cosx) do i apply that for (cos²x-sin²x)²? Or at the last thing i wrote
balmy dust
So uh at the part with (cos²x-sin²x)²/(sinx*cosx) do i apply that for (cos²x-sin...
just apply the identity i told u in the question
doing cos and sin is a waste of time
solve it in tan and cotangent
cyan trellis
Oh, indeed I see it now
My teacher usually told us to do sin cos at the beginning bruh
It's easier working with tan and cotan, thanks for the idea, ill see what I do
Is this alr?
balmy dust
looks perfectly fine to me
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remote bronze
What is this formula and how can I use it?
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What is this formula and how can I use it?
gritty rose
show the full context
remote bronze
gritty rose
and also understand what alpha_i represents
prime forum
Alpha I is basically the power of (in 2^7, 7 is alpha I
grave elm
product, you multiply all the numbers from i = 1 to i=k
grave elm
It looks like pi?
and yeah, it is a capital pi
remote bronze
ai+1 ways to choose how times it appears in the factor why?
gritty rose
Alpha I is basically the power of (in 2^7, 7 is alpha I
did you understand this
remote bronze
Yeah
glossy valveBOT
Result:
128
try listing all the factors of 128
remote bronze
I'd be here all day lol
rugged void
,calc 2^7
i don't know why but i expected you to just have memorized all the powers of 2 upto like 32 ngl
remote bronze
1,2,4,8,64,128?
gritty rose
amazing
rugged void
missed a few
gritty rose
jsut missed two
remote bronze
16
rugged void
1,2,4,8,64,128?
usually a good idea is to pair them up like d, and n/d, so they multiply to n
and
one more
remote bronze
18
rugged void
... no
rugged void
usually a good idea is to pair them up like d, and n/d, so they multiply to n
again
try it like this
rugged void
i don't know why but i expected you to just have memorized all the powers of 2 u...
shush okay
remote bronze
32
rugged void
correct\
rare dock
the reason why the formula works is just basic combinatorics. which nobody has explained for some reason
rugged void
ngl, just following riemann's lead
rare dock
to make divisors of n you can pick some number between 0 and alpha_1 to be the exponent on p_1. similar with all the other primes. there are 1 + alpha_1 choices for the exponent on p_1
elfin crescent
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
Hey 2048 taught me something
rare dock
to make divisors of n you can pick some number between 0 and alpha_1 to be the e...
following multiplication principle, the number of ways to pick an exponent on each prime is the product in question
grave elm
to make divisors of n you can pick some number between 0 and alpha_1 to be the e...
the question is whether they even understand what prime factorization has to do with sth being a divisor, thats where riemann was heading i think
remote bronze
to make divisors of n you can pick some number between 0 and alpha_1 to be the e...
1+alpha b/c of the zero?
rare dock
yes
remote bronze
i just don't understand why this is even a part of this book
like anyways youd have to know the prime factorization of the nubmer
rare dock
yea
remote bronze
why would we even select a number less than alpha?
rare dock
because doing that makes more divisors
rare dock
to make divisors of n you can pick some number between 0 and alpha_1 to be the e...
doing this makes all of them
remote bronze
So 0 | n always right, but if you have a choice between 0-alpha every single time, aren't you counting 0 multiple times?
for each prime
rare dock
0 does not divide n
also when you pick 0 on all the exponents, the corresponding number is 1
not 0
when you don’t pick 0 on all the exponents, the corresponding number will not be 1
there is no double counting going on but maybe that takes a little bit of convincing
it’s basically just that prime factorization is unique and so each distinct choice of exponents gives a distinct product
remote bronze
Oh mb x | 0, irrelevant here though mb
remote bronze
it’s basically just that prime factorization is unique and so each distinct choi...
but can't you multiply them in any order and that also produces a factor?
rare dock
are you suggesting that like… picking (1,2,5) as exponents will give the same number as when we pick (2,1,5) for exponents
remote bronze
Like for the example 84, the prime factorization is 2^23 * 7. We can do 2^03*7 to produce 21 as a factor? Is that the purpose of the formula?
rare dock
yes?
remote bronze
That typed out really weird mb
rare dock
That typed out really weird mb
you can type \* to avoid that
every triple (a,b,c) with 0 <= a <= 2, and 0 <= b, c <= 1 corresponds to a unique (positive) divisor of 84
there are 3 * 2 * 2 such triples
rare dock
remote bronze
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lyric field
problem in probability
I dont understand why this isn't a "stars and bars" problem where we put the green tomatos first which is x_1+...+x_n=n then complete the rest with red tomatos. I might be making this too combinatoric..
rare dock
what exactly are you going to count with stars and bars?
you can do it with stars and bars but kinda complicated for no reason
lyric field
number of ways to distribute green tomatoes then complete the rest with reds
lyric field
you can do it with stars and bars but kinda complicated for no reason
it appears to be wrong though..
rare dock
the thing is that a constraint like 0 <= x_i <= 2 is maybe more difficult to deal with than you’re imagining
lyric field
i forgot to bound them by 2
rare dock
it can be done. just not as easy
rare dock
honestly it’s pretty irrelevant
lyric field
im pretty sure i was wrong by assuming the tomatos of same color are indistinguishable
rare dock
the answer will be the same whether you treat them as distinguishable or not
but if you get into some messy combinatorics instead of taking a probabilistic approach, you do need to decide whether to make them distinguishable or not
lyric field
got it
though just to be clear, if we treat them as indistinguishable then there's only 1 configuration for this event, and thus the probability is 1 over the size of the sample space right?
which is the answer to the stars and bars calculation
rare dock
though just to be clear, if we treat them as indistinguishable then there's only...
hmm i have an objection to this
the “total configurations” are not all equiprobable i think
when you look at it like that
can i give you a suggestion on how to do the problem
rare dock
start with making 1 basket. what’s the probability there is one red and one green tomato?
you pick one tomato at random, then another from amongst the rest
lyric field
n choose 1 times n choose 1?
rare dock
it should be depend on n
we pick a tomato. then the probability we pick a different color tomato to go with it is n/(2n -1)
there are 2n-1 total tomatoes left after after we pick one, and n are the other color tomatoes
lyric field
yes i understand
you set one then figure the probability of getting the other
so we start with 1 basket and then?
rare dock
go on to the next basket
lyric field
(n-1)/(2n-3)
then the probability of this event is (n!)/(2n-1)!!
alright i think i got enough to go on from here, thanks
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dusky sand
Question
2 circles of radii 2cm have their centres 2√2 cm apart. Find the perimeter and the area of the shaded region (as shown in diagram)
dusky sand
My solution, what went wrong and where and why?
dusky sand
but howd u know
glossy valveBOT
parthisjoking
balmy dust
while calculating
dusky sand
Oh that
balmy dust
area of sector
balmy dust
i love people who use pythagorean theorem here
dusky sand
area of sector
Yea, as both triangles are congruent, so x = 45, and theta is formed by 2x so 90
dusky sand
i love people who use pythagorean theorem here
Can't tel if this is sarcasm or not-
balmy dust
Can't tel if this is sarcasm or not-
it is
balmy dust
Yea, as both triangles are congruent, so x = 45, and theta is formed by 2x so 90
but u took theta as pi while calculating area of sector
dusky sand
but u took theta as pi while calculating area of sector
Oh shi- yes it's supposed to be π/2
Lemme recalculate
balmy dust
ye
dusky sand
i love people who use pythagorean theorem here
What method would you consider using here?
dusky sand
vector component
Could you explain me a bit?
balmy dust
are you familiar with vectors
dusky sand
are you familiar with vectors
I am a bit
balmy dust
atleast the projection of a vector on another
dusky sand
Yea
dusky sand
dot product to be specifc
So how does it play a role here?
dusky sand
you agree this is true?
Yea, A can be split into Acostheta and Asintheta right?
balmy dust
yes
similarly
in the question
try splitting the length of 2sqrt2
into two parts
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quick aspen
How do I find the vertex with a standard quadratic form?
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How do I find the vertex with a standard quadratic form?
sudden condor
How do I find the vertex with a standard quadratic form?
You know how to convert from standard form to vertex form?
minor sail
How do I find the vertex with a standard quadratic form?
h=-b/2a
glad plover
How do I find the vertex with a standard quadratic form?
Using the vertex form: since the squared term has a minus next to it, then the maximum of f is when that term is 0. Meaning for x=0.5, f is max, which makes f(x)=6.25
The vertex form is simply a clear way to see the max/min of the function
Without the vertex form:
The vertex is
(-b/2a , f(-b/2a))
That is just always truee
minor sail
because if vertex form is a(x-h)^2 + k, expand:
ax^2-2axh+ah^2+k,
ax^2 = ax^2, bx must equal -2axh, c = ah^2 + k,
bx = -2axh,
b=-2ah,
h=b/-2a
quick aspen
because if vertex form is a(x-h)^2 + k, expand:
ax^2-2axh+ah^2+k,
ax^2 = ax^2, b...
I thought it was -b/2a?
sand bone
because if vertex form is a(x-h)^2 + k, expand:
ax^2-2axh+ah^2+k,
ax^2 = ax^2, b...
This is likely not an appropriate question to ask... but, is there a channel I can go for math at a lower level, or are all the channels similar to this?😅
quick aspen
I was able to find 2 points (0, 6) and (1/2, 0)
minor sail
This is likely not an appropriate question to ask... but, is there a channel I c...
idk bruh i js joined this server 20 minutes ago
wrong guy to ask
quick aspen
I was able to find 2 points (0, 6) and (1/2, 0)
Would I plug these points in
Or is that just for finding A
minor sail
im even a bit intimidated
sand bone
<#488120190538743810>
Yes, I would appreciate that.
sudden condor
I was able to find 2 points (0, 6) and (1/2, 0)
Let’s take it slow
First what does x-intercept mean?
quick aspen
Let’s take it slow
First what does x-intercept mean?
The point thats on the X axis
sudden condor
The point thats on the X axis
And the x-axis is where y=0
Do you agree?
quick aspen
No
sudden condor
Why not?
sudden condor
It’s the reverse
If a point lies entirely on the x-axis
The y value of said point is 0
sudden condor
You took linear equations before yeah?
quick aspen
Yeah
sudden condor
y=0 is the same as the x-axis entirely
Moreover, x=0 is the same as the y-axis entirely
quick aspen
Huh
sudden condor
Huh
When you plot the line y=0
this line lies entirely on the x-axis
Since you vary x, but y is just 0
quick aspen
Yea
sudden condor
same with the x=0
y varies but x is always 0
So the line lies on the y-axis
Is that clear?
quick aspen
Yes
sudden condor
So from this conclusion we can have two rules
x-intercept is where the y value is 0
y-intercept is where the x value is 0
clear?
quick aspen
Yes
I substituted those
But for some reason i could not find 6.25
Even though it says it on the vertex form
sudden condor
But for some reason i could not find 6.25
That is the last part
Are you familiar with completing the square ?
quick aspen
Yeah
sudden condor
Add on subtract to in the same side
+6-6 (is still 0 so the expression is equivalent)
- there was a mistake on my part
It was supposed to be -x^2+1x+6
sudden condor
When completing the square think of a scale where the weights are equivalent.
Suppose I took away or added extra weight.
The scale side won’t be equal anymore so we try to make up for the lost weight to preserve the equality by adding what we took away, or taking away what we added.
In completing the square the idea still occurs
We add the square of the middle term divided by 2 but we also subtract to keep both sides equivalent
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past path
I don’t understand how this integral is divergent. Wouldn’t the limits make those terms go to 0?
sudden condor
I don’t understand how this integral is divergent. Wouldn’t the limits make thos...
Check again
You got zero inside your interval
past path
Check again
You got zero inside your interval
Doesn’t breaking the integrals up allow you to avoid that.
sudden condor
Doesn’t breaking the integrals up allow you to avoid that.
Not necessarily
At x=0 the limits shoot up to infinity
lim x->0 1/x^p doesn’t even exist for odd powers and is infinity for even powers
past path
Not necessarily
At x=0 the limits shoot up to infinity
lim x->0 1/x^p doesn’t ...
So the first limit would go to - infinity which makes it div?
sudden condor
So the first limit would go to - infinity which makes it div?
Yes
And also the second term
In anyway
If one of those separated integrals diverges the whole integral diverges
past path
Yes
And also the second term
In anyway
If one of those separated integrals div...
Ah, I see, ty!
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glossy tapir
help me
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help me
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
3
mossy stratus
Tính delta
glossy tapir
bài 3.1 anh ơi đ phải 3.3
mossy stratus
Đúng rồi
glossy tapir
3.1 delta j hả a
mossy stratus
tiền lãi 1 xe là bn
=giá bán - giá sản xuất
Tính ra rồi thì đó là lãi 1 xe, rồi lấy tổng tiền vốn chia tiền lãi 1 xe là ra số xe cần
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left moon
Why isn’t tan2x equal to 2sinxcosx
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Why isn’t tan2x equal to 2sinxcosx
left moon
rugged void
tan^2 u + 1 is sec^2 u
left moon
Yes
rugged void
not 1 - tan^2 u
rugged void
lmao
left moon
💀
rugged void
Hopefully not on tests
luckily not
rugged void
anyway .close if ur done
left moon
Yuh
left moon
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rugged void
alr
torn jolt
<@&268886789983436800>
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fair ocean
hi!!! just checking if my work is correct
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hi!!! just checking if my work is correct
fair ocean
sturdy valve
,w integrate sqrt(y+1) from 0 to 3
glossy valveBOT
woeful girder
Dang man good handwriting
sturdy valve
,w integrate t^3(1+t^4)^3 from 0 to 1
glossy valveBOT
visual kiln
do u guys know anything abt binomials and inverse and norm distributions and all
fair ocean
Dang man good handwriting
thank you i try!!!
sturdy valve
hi!!! just checking if my work is correct
both r correct
woeful girder
Cooked
sturdy valve
do u guys know anything abt binomials and inverse and norm distributions and all
!occupied
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do u guys know anything abt binomials and inverse and norm distributions and all
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).
woeful girder
Good thing i misspelled
fair ocean
fhank you sonmuch !
woeful girder
Ahem no one saw that
sturdy valve
👍
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visual kiln
y do u guys use ! this sm
woeful girder
y do u guys use ! this sm
Cuz we're jerks
sturdy valve
y do u guys use ! this sm
!redir
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y do u guys use ! this sm
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
visual kiln
oowww
warped frost
either way, if you want to ask your question, please use an available channel
visual kiln
thats acc cool ngl
sturdy valve
Don't be that guy now yoda
it was a joke bruh cz they were asking y we use it so much
woeful girder
Are you still upset at me about the pre uni remarks 😭
sturdy valve
YES
woeful girder
Oh
sturdy valve
im a valued member of the preuni community
woeful girder
You are
sturdy valve
:)
woeful girder
I value pre uni community too cough
torn jolt
tuff members of pre uni gng
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lethal trout
I need help i have a division test tomorrow on math from 1st quarter to 4th quarter lessons, I'm grade 10
torn jolt
I need help i have a division test tomorrow on math from 1st quarter to 4th quar...
what kind of division
lethal trout
Well mostly i need help with circle, tangent and secant
lethal trout
what kind of division
It is like from the 1st quarter lesson to the 4th quarter lessons, we'll do a test on that
sterile sinew
Wow looks hard
lethal trout
I'm a fast learner but I want to ace this lol
sterile sinew
Dangg, i believe in u, keep going
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drifting summit
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drifting summit
Hi
can someone view my proof?
i'm not sure if i'm doing transfinite induction correctly
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trail barn
This looks fine to me (although it's been a while since I've done set theory)
I assume "$V_\beta^$ is transitive by $()\beta$" is meant to be typeset more like "$V{\beta^}$ is transitive by $()_{\beta^*}$"?
glossy valveBOT
Edward II
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misty garden
Hi, given that a general quadratic polynomial around two variables to be f(x, y) = ax^2+bxy+cy^2+dx+ey+f. I know that cross sections of this polynomial form a parabola. How can I prove that the minimal point of all the cross sections also form a parabola?
gritty rose
kinda vague but you can parametrize the "minimal points" and show that the resulting eqn is a parabola
misty garden
This is what I am working with
wise tartan
Hi, given that a general quadratic polynomial around two variables to be f(x, y)...
cross sections with x constant or y constant or something else
misty garden
kinda vague but you can parametrize the "minimal points" and show that the resul...
Sorry I am not sure what you mean by parameterize. The process goes like this: for all the cross section parabola, extract the vertex(or the minimum point) and put it in a set. May I know if this infinite set forms a parabola as well?
misty garden
cross sections with x constant or y constant or something else
I think constant y is what I am looking for
wise tartan
Sorry I am not sure what you mean by parameterize. The process goes like this: f...
i think hes saying that you pick the constant y, find the minimum point based on that y, thus parameterising them
misty garden
Sorry, I still don’t understand. May I know how we pick the constant y? I thought we were using all cross sections in the form of y=#
gritty rose
maybe find two or three non overlapping parallel parabolas first and write them here along with the conic with actual constants
misty garden
Sure, I can do that! Just making sure, do you want me to find the vertex of three cross sections parabolas to find the parabola containing all the vertices?
<@&286206848099549185>
wise tartan
Sorry, I still don’t understand. May I know how we pick the constant y? I though...
say if i look at $f(x,y_0) = ...$, i take the derivative with respect to x $\frac{d}{dx} f(x,y_0)$ and i equate it to zero, ill find the x, which for that $y_0$ gives the minimum of that cross section, in this case $x=\frac{-by+d}{2a}$
glossy valveBOT
CherryMan
wise tartan
If I let g be the minimum value of f(x,y) for a specific y, then $g(y) = a(\frac{-by+d}{2a})^2 + b(\frac{-by+d}{2a})y + cy^2 + d(\frac{-by+d}{2a}) + ey + f$
glossy valveBOT
CherryMan
wise tartan
that means, the minimums do form a parabola
and the plane the parabola is formed in is defined by x = -by+d /2a
(consequently the same holds for cross sections with x constant)
idk i might have made a mistake
misty garden
Ok, that makes sense to me! So you first took the partial derivative to find a general equation of a cross section vertex. Then you plugged it back to the original function and it was still a general quadratic right?
misty garden
Got it, thank you so much! May I know if the same logic applies when determining if a general cubic equation around three variables have cross sections that satisfies the property of a general quadratic?
misty garden
Oh ok, I think I got it for quadratics, but have not understood how to generalize it for high degree polynomial
<@&286206848099549185> can anyone help me prove or disprove the generalization of this problem? Greatly appreciated!
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neon surge
<@&286206848099549185> can anyone help me prove or disprove the generalization o...
for n = 3 it is not true the resulting curve neednt be a polynomial take $f(x) = \frac{1}{3}x^3 + xy^2 - x$, if you take set y constant and take the partial derivative wrt x, you get the minimum at $x = +- \sqrt{1-y^2}
glossy valveBOT
k
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
neon surge
so points (+-sqrt(1-y^2), y, some z)
misty garden
I understand, thank you for showing a counter example! 🙏
neon surge
I understand, thank you for showing a counter example! 🙏
yw
misty garden
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azure bridge
Matrix matrix multiplication?
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Matrix matrix multiplication?
nova briar
X(n) = M^n V where M is your matrix, and V is your vector
Do you know how to find the nth power of a square matrix?
azure bridge
Do you know how to find the nth power of a square matrix?
no
i just started this unit
Do I have the right setup?
ive been stuck on this question for an hour 😭
fathom saddle
The numbers are a bit jumbled. Currently, you've got a 20% chance that a juvenile becomes a juvenile, where you want a 20% chance they become an adult
fathom saddle
Knowing how to mutiply a vector by a matrix is needed for this problem, make sure you know how to do that
But, even further, you are going to multiply two matricies together too
Are you studying ahead? This is a bit far from the beginning of a course
azure bridge
im only one lesson ahead sadly
fathom saddle
Jeez they're not taking it easy on you haha
azure bridge
ah okay I got it now, i thought I was doing matrix matrix multiplication
azure bridge
Jeez they're not taking it easy on you haha
i guess thats what happens when the midterm was as light as it was lmaoo
easy probability and easy recurrence solving + bernoulli's equation
and now after the midterm they have us doing expectation, variance, std dev, and now this
in a week
😭
azure bridge
ah okay I got it now, i thought I was doing matrix matrix multiplication
is a vextor just a single column matrix?
fathom saddle
Yes! A vector-matrix multiplication is still just a matrix-matrix multiplication
Or, a matrix-matrix multiplication is where you take the right matrix, break it into column vectors, multiply each by the left matrix, then reassemble the result
azure bridge
Or, a matrix-matrix multiplication is where you take the right matrix, break it ...
ah perfect tysm
i’m done looking ahead for today
fathom saddle
I also like this graphic for visualizing matrix multiplications
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tacit token
How to solve x^{5}+10x^{4}-16x^{3}+7x^{2}-10x+1 = 0?
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How to solve x^{5}+10x^{4}-16x^{3}+7x^{2}-10x+1 = 0?
mossy stratus
$x^5+10x^4-16x^3+7x^2-10x+1=0$
glossy valveBOT
Minhh
cinder thorn
Wolframalpha?
tacit token
No machines allowed.
mossy stratus
He need to understand how to get to that answer
cinder thorn
You won't have rational roots here
mossy stratus
Not to rely on AI or digital tools
tacit token
Wolframalpha?
Not AIs like WA.
cinder thorn
Do you know galois stuff maybe
tacit token
nope
cinder thorn
And how have you solved quintics until now?
Because I'm not even sure this is possible
tacit token
And how have you solved quintics until now?
i have not
cinder thorn
Where did you find this problem
tacit token
self-question
mossy stratus
This one doesnt have beautiful coefficients.
This involves complex solutions for sure.
cinder thorn
Yeah i kinda expected it
mossy stratus
At least 2 conjugate pairs
tacit token
reason?
mossy stratus
Cuz it have real coefficients
mossy stratus
Maybe 1 pair minimum
tacit token
isn't that just a guess?
mossy stratus
You can use conjugate pair root theorem, but it must have complex sols
cinder thorn
isn't that just a guess?
In any case you probably won't find any roots because they are very complicated (there is no rational solution here)
mossy stratus
I just chucked random x values in my calc, i only approximate a real sols in the interval -11, -12, so i assume the rest of 4 are complex, then 2 pairs, could be wrong
tacit token
In any case you probably won't find any roots because they are very complicated ...
shouldn't there be a procedure?
cinder thorn
No
Especially for degree 5 and above
You could be in a special case but then you'd have to use galois and such
atomic hare
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craggy tapir
for part b, how do i calculate P(X = 10) if i am approximating as a normal distribution?
balmy dust
total probability theorem?
craggy tapir
how is it applicable here?
craggy tapir
for part b, how do i calculate P(X = 10) if i am approximating as a normal distr...
i have an idea, that is i calculate P(X <= 10) - P(X <= 9), but i'm not sure if that's correct
balmy dust
is 5% the prob of a board being defective or is it the percentage of the total boards that are expected to be defective
craggy tapir
i don't see the difference
balmy dust
i have an idea, that is i calculate P(X <= 10) - P(X <= 9), but i'm not sure if ...
well i mean that works
balmy dust
i don't see the difference
if it is the former, then you can just apply total prob theorem
if latter then, 5% is the expected value , i.e. the mean observation
wide sundial
for part b, how do i calculate P(X = 10) if i am approximating as a normal distr...
Have you seen continuity correction
craggy tapir
i briefly read about it when the formula when the formula for P(X <= x) to approximate bin dist is introduced
but i'm not sure how to apply it to calculate P(X = x)
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@craggy tapir Has your question been resolved?
neon rivet
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noble spade
What formulaes do you know about vector addition?
neon rivet
what did i do wrong
noble spade
They are not perpendicular
...
neon rivet
what is not perpendicular?
noble spade
Bro its not a right angle triangle u cant use pythagoras here
turbid tinsel
what is not perpendicular?
these two vectors must be in a right angle to use the pythagoren theorem
simple vector addition works
prime pier
a bar is not the hypotenuse
perhaps you meant that the triangle isn't right-angled? $\va$ is still the hypotenuse
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noble spade
They are not in the same direction