tried to first integrate with repect to y
#help-10
lament atlas
then i thought of using polar coordinates
i find the radius to be sqrt(3)/2
idk if that is correct
obtuse pebbleBOT
@lament atlas Has your question been resolved?
obtuse pebbleBOT
@lament atlas Has your question been resolved?
robust raven
$\left| V \right|=\int_{}^{}\int_{D}^{}\left[ 3-2x^{2}-2z^{2}-2x^{2}-2z^{2} \right]dzdx=\\=\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\sqrt{3}}{2}}\left( 3-4r^{2} \right)rdr=_{\cdots }etc$
warm shaleBOT
Joanna Angel
wicked tree
hello
sweet gulch
Start by posting the question
obtuse pebbleBOT
@wicked tree Has your question been resolved?
woeful cove
Ok
Maybe sine law?
Drop the altitudes, find the angles
Remember, you don't need to calculate the sines
prime yacht
$\sin^{-1}$
warm shaleBOT
OceanBro
prime yacht
Use inverse trig functions
I thought you said the angle is not given here
In order to get the angles, you'll need to use inverse trig functions
No?
woeful cove
Wait
Does the altitude divide the triangle?
Nahh
You can use the sine law if you manage to get one angle some how
Holdon
I think I know
Half angle theorems
Sin(A/2)=โ((s-b)(s-c)/bc) where s is the semiperimeter of the triangle
Surely there's an easier way ๐
Oh yeah
The cosine law
Find cosa
Then use that to find sina
There
You have one of your angle
It's simple now
If that's correct, put it on a calc
To get angle from cos inverse function
Do you know how to get angles from your given cosine
Yo holdon
๐
Cos a=277/210?!!!
That's greater than 1
Cosa lies between 0 and 1 in a triangle
Check your calculations bro
Are you sure this time?
Ok then sina=12/13
Find the rest
Using sine law
Sina/a=sineb/b
Also know sinA means the angle opposite to the side a
woeful cove
Also know sinA means the angle opposite to the side a
.
deft fable
12.9 yw
woeful cove
Do you follow this?
Yes I messed up angles a and c
๐๐
But you get the idea right?
??
Please note I totally messed up the variables
I hope you get the idea tho
halcyon sparrow
is it not just 13 tho
obtuse pebbleBOT
12.9 yw
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.
woeful cove
,rotate
warm shaleBOT
woeful cove
Similarly you can find all the altitudes
This time the figures correct
Have fun lol
A bc?
obtuse pebbleBOT
@glad dawn Has your question been resolved?
hearty terrace
how do you do this?
obtuse pebbleBOT
how do you do this?
hearty terrace
the answer is that the limit doesn't exist
After l'hopital i get $\frac{g(x)}{g'(x)}$ which seems to just be $\frac63=2$ but that's wrong
warm shaleBOT
Vัณrtัx-
trail musk
2 seems right
hearty terrace
the answer is that the limit does not exist though
not sure why
the only way for the limit not to exist is if it's discontinuous right? if the delta epsilon thing doesn't work
hearty terrace
2 seems right
nvm it is right
my friend circled the wrong correct answer ๐ญ
my bad
.close
timber pine
i dont know how to approach g
obtuse pebbleBOT
i dont know how to approach g
brave bramble
You're really concerned about the area of said polygon
timber pine
yeah
timber pine
How to know the height? with trigo?
brave bramble
Yeah sin will jump into this, as given by the expected answer
You know the length of the black dotted lines, and the angle between them
brave bramble
The answer expects radians so let's work with that. The angle between each black dotted line is 2ฯ/n
brave bramble
Yeah we can do that! Then you'd be cutting each angle in half
timber pine
What's ฮธ?
pi/n?
brave bramble
Cool! Looks like we're getting there. What's r? Can we write that in terms of d?
timber pine
d/2?
obtuse pebbleBOT
@timber pine Has your question been resolved?
obtuse pebbleBOT
@timber pine Has your question been resolved?
sharp forum
hi cosmo
timber pine
hi
shy tusk
A rectangular sump has an inner length and breadth of 24m and 20m respectively water flows through an inlet pipe at 180m per minute.the cross sectional area of the pipe is 0.5m^3.
. The tank takes half an hour to get filled. Find the depth of the sump (in
m
)
shy tusk
Hi
shy tusk
Yes please
sharp forum
ok so lets take heigh as h
so the volume of water flowing throught the pipe is 180/0.5
shy tusk
Ok
sharp forum
as the area is 0.5 and the water that can flow iis 1180\
so the total volume whcih can flow is 90 into 30
shy tusk
I get it
sharp forum
so now l into b into h will be 270
so 24 into 20 into h is 270
after solving you get it
5.625
easy
shy tusk
Tqs
sharp forum
welcome thank me and close the channel @shy tusk pls thxx
obtuse pebbleBOT
@timber pine Has your question been resolved?
lament atlas
$\left| V \right|=\int_{}^{}\int_{D}^{}\left[ 3-2x^{2}-2z^{2}-2x^{2}-2z^{2} \rig...
but why pi/2
is it becuase x>0 and z>0?
sharp forum
ok thx
interrsting sum
show that C = n*sin(2ฯ/N), where n is greater than 2 and even, we can use the fact that sin(2ฯ) = 0.
if i am not wrong @timber pine
timber pine
Huh
obtuse pebbleBOT
@timber pine Has your question been resolved?
obtuse pebbleBOT
@timber pine Has your question been resolved?
sharp forum
Huh
bro its a fact that sin into 2pie is 0
sin 2ฯ = sin 0ยฐ = y-coordinate of (1, 0) = 0. Hence, sin 2ฯ = 0.
timber pine
yeah
sharp forum
so we can use this right
timber pine
i guess? idk
sharp forum
yeah wait let me show you
timber pine
yeah wait let me show you
ok
robust raven
but why pi/2
yes, your region of an integration is first quadrant in plane Oxz, hence such an angle
timid silo
my question remains unanswered
limpid cape
I didn't understand how to solve it. I think there is a defect in the form, or I am the one who has a defect
hollow parrot
I didn't understand how to solve it. I think there is a defect in the form, or I...
!occupied
obtuse pebbleBOT
I didn't understand how to solve it. I think there is a defect in the form, or I...
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #โhow-to-get-help for instructions).
obtuse pebbleBOT
@timber pine Has your question been resolved?
timid silo
anyone??
brazen viper
!occupied
obtuse pebbleBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #โhow-to-get-help for instructions).
frigid venture
๐คฉ
sharp forum
you didn't know, you're new ๐
so are you helping vvarad or should i just dm him
brazen viper
neither
this is c0smo3's channel
you can DM him if you want, but he should make his own channel
wait for him to do it
@timber pine So any regular n-sided polygon can be expressed as the sum of the areas of n isocelese triangles. We know the base of each of these triangles and the we can find the height using some trigonometry.
I encourage you to draw out one and then use the base and the angle between the two equal sides to find the area of this triangle.
timber pine
wait they are isosceles??
brazen viper
nbd
timber pine
okay
so
wait i think its better if i illustrate this
theta here is 2pi/2n?
so pi/n
timber pine
wouldnt the area be 1/2 r^2sin(pi/n)cos(pi/n)
times 2n
so r^2nsin(pi/n)cos(pi/n)
brazen viper
Use the double angle formula
timber pine
Where can it be used here?
timber pine
so 1/2 r^2 n sin(2pi/n)
brazen viper
Yup
brazen viper
โค๏ธ
timber pine
Also
for (h), can i draw a pentagon
and show that the longest distance (d) is not 2r
brazen viper
well, you can figure out the distance by using the same idea as finding the base and height
do you know differential calculus?
you can set the derivative to zero to find the maximum
but no actually, corner to corner is the longest
timber pine
do you know differential calculus?
yeah
brazen viper
ok cool
timber pine
you can set the derivative to zero to find the maximum
not sure how to set up an equation for this
brazen viper
So use the angle theta as the parameter
timber pine
Do i need law of sine/cosine
brazen viper
you're going to be anchored in one corner, so find the distance from the top to the base
then use basic trig to find the hypotenuse using cos(alpha) = leg/hypo
timber pine
do you mind illustrating it
brazen viper
I def would but my cat is sleeping on my pen and paper at the moment, so I'll have to shitty mouse draw it
timber pine
oh thats ok
I set something up but i only proved that the longest side of an equilateral triangle is the side itself
brazen viper
yeah, you'll find that dh/dt = 0 at t = 0, but that's a minimum not a maximum
in fact this doesn't have a maximum, so the maximum is at the edge of the range.
timber pine
.close
timber pine
<@&268886789983436800>
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
hearty pendant
whats argument of number= 0 +0*i in complex plane
obtuse pebbleBOT
whats argument of number= 0 +0*i in complex plane
twin oar
vale linden
ODE homework question, wait for me to post the whole question
obtuse pebbleBOT
ODE homework question, wait for me to post the whole question
vale linden
There is a planar (ode) system
x'=y, y'=-x+(1-x^2-y^2)y
Prove that, for any non-zero solutions x(t), y(t), there exists some constant theta, such that:
$\lim_{t\to+\infty}(x(t)-sin(t+\theta))=\lim_{t\to+\infty}(y(t)-cos(t+\theta))=0$
warm shaleBOT
CollinGao-Original
vale linden
I've already converted it into polar coordinates, and proven that x^2+y^2 converges to 1, and the "angular velocity", derivative of arctan(y/x) converges to -1, but I can't seem to prove the required fact
obtuse pebbleBOT
@vale linden Has your question been resolved?
vale linden
<@&286206848099549185> I need some ODE experts...
vale linden
?
sharp forum
you ased for ODE experts a]man
ordinary diffrential equation
sorry ig i should leave if you not interested see ya
vale linden
I need some immediate help
I can wait for you to prove it
vale linden
There is a planar (ode) system
x'=y, y'=-x+(1-x^2-y^2)y
Prove that, for any non...
the question is above
sharp forum
hmmm wait
x(t) = r(t) * cos(theta(t))
y(t) = r(t) * sin(theta(t))
r(t) represents the radial distance from the origin and theta(t) represents the angle that the line connecting the origin to the point (x(t), y(t)) makes with the positive x-axis for you
Taking the derivatives of x(t) and y(t) with respect to t, we have
x'(t) = r'(t) * cos(theta(t)) - r(t) * sin(theta(t)) * theta'(t)
y'(t) = r'(t) * sin(theta(t)) + r(t) * cos(theta(t)) * theta'(t)
vale linden
For convenience you can actually use rho(t)=x(t)^2+y(t)^2
sharp forum
nah i like it this way
now lets subsitute
r'(t) * cos(theta(t)) - r(t) * sin(theta(t)) * theta'(t) = r(t) * sin(theta(t))
r'(t) * sin(theta(t)) + r(t) * cos(theta(t)) * theta'(t) = -r(t) * cos(theta(t)) + (1 - r(t)^2) * sin(theta(t)) * r(t) * sin(theta(t))
vale linden
anyways I've derived the equations of r and theta, no need for the details, just jump to the result
sharp forum
r'(t) = r(t) * sin(theta(t))
theta'(t) = 1 - r(t)^2
Now, let's consider the magnitude of the radial distance r(t). Since we are given that x(t) and y(t) are non-zero solutions, this implies that r(t) is also non-zero. Therefore, we can divide the first equation by r(t) to obtain:
r'(t)/r(t)
vale linden
wait a sec that isn't the equation i got
vale linden
Well we can actually
sharp forum
hmmm wait then
vale linden
ODE class teacher allows it once it was taught
sharp forum
First, let's define a Lyapunov function V(x, y) as follows:
V(x, y) = (x - sin(t + theta))^2 + (y - cos(t + theta))^2
Taking the derivative of V(x, y) with respect to t, we have:
dV/dt = 2(x - sin(t + theta))(x' - cos(t + theta)) + 2(y - cos(t + theta))(y' + sin(t + theta))
= 2(x - sin(t + theta))(y + sin(t + theta)) + 2(y - cos(t + theta))(-x + (1 - x^2 - y^2)y)
2(x - sin(t + theta))y + 2(x - sin(t + theta))sin(t + theta) - 2(y - cos(t + theta))x + 2(1 - x^2 - y^2)(y - cos(t + theta))y
= 2xy + 2(x - sin(t + theta))sin(t + theta) - 2yx + 2(1 - x^2 - y^2)y - 2xy + 2(x - sin(t + theta))x - 2y(y - cos(t + theta))
= 2(x - sin(t + theta))sin(t + theta) + 2(x - sin(t + theta))x - 2y(y - cos(t + theta))
= 2(x - sin(t + theta))(sin(t + theta) + x
vale linden
You must have done something I've never seen before
the Lyapunov function shouldn't include t
sharp forum
we had to do it thay way
i solved with a diffrent system and somehow ended up with a quadrartic equation with x^@
2
its 5:13 in the morning here sorry my brain aint working
obtuse pebbleBOT
@vale linden Has your question been resolved?
strange fiber
Find a partially ordered set (POS) in which there exists a two-element set that does not have a supremum.
obtuse pebbleBOT
Find a partially ordered set (POS) in which there exists a two-element set that ...
strange fiber
let A=[a,b,c] and define POS on A as a<=a b<=b c<=c
if we take subset [a,b] it doesnt have a supremum right?
vale linden
Right
strange fiber
si it completely fullfills the task? @vale linden
vale linden
true
strange fiber
.close
lilac sun
I am very bad when it comes to "show that" questions. Can anyone give me advice to make my attempt more comprehensive and rigorous. Any advice too? And I don't even know if my proof makes any sense
obtuse pebbleBOT
I am very bad when it comes to "show that" questions. Can anyone give me advice ...
glass dagger
It might help to know that since the sum is convergent, the partial sums form a cauchy sequence
So you can easily bound the difference of two partial sums from above by epsilon
but for a suitable choice of partial sums the difference will be any almost arbitrary term a_k
meaning you can conclude that |a_k| < epsilon
and that should be helpful
Maybe that's what you did, I don't know because I can't really make sense of your proof lol sorry
warm canopy
Looks like they're trying to contrapositive
plucky rivet
Maybe that's what you did, I don't know because I can't really make sense of you...
They trying to prove by contrapositive
past sand
This says nothing
warm canopy
Issue is the negation of lim an = 0 is not that lim an = ฮต > 0
plucky rivet
Yet they dont have the case where limit of an is lesser than 0
warm canopy
Or where it doesn't exist
obtuse pebbleBOT
@lilac sun Has your question been resolved?
lilac sun
This says nothing
I think what I was trying to put into words is that if the limit is not 0 could be negative or positive, then ur always adding something to the limit, and since the limit is not 0 its like recursive and so it diverges since the sum before the limit is larger than the limit in the case that the limit is positive. If the limit is negative then its the other way around
lilac sun
It might help to know that since the sum is convergent, the partial sums form a ...
Is telescoping sum what ur trying to say? I'm kind of confused. I'm really bad with analysis (Idk why we're doing analysis in CS)
I think its bad to do math questions in a computer science perspective lmao
but can't help it im a programmer by nature
lilac sun
Issue is the negation of lim an = 0 is not that lim an = ฮต > 0
Oh I forgot to add the other cases where its negative or doesnt exist. But surely they follow the same principle once I proved it is the case for positive?
past sand
I think its bad to do math questions in a computer science perspective lmao
To each their own 
warm canopy
Well I still can't see what exactly you're doing in this case
past sand
Oh I forgot to add the other cases where its negative or doesnt exist. But surel...
It would if the rest of the proof was coherent
warm canopy
This says nothing
One reason being this
lilac sun
man I just suck at proofs 
past sand
I believe you're trying to say that if the limit of the sequence doesn't converge, then there exists a point where either all the terms are positive or all the terms are negative
Then, since there is still an infinite number of subsequent terms, the sum diverges
lilac sun
yes something like that
but its not a rigorous way of thinking i guess
how do I get better at proofs? Do I just read more proofs to get better at them?
past sand
Study basic logic I suppose
And yes of course, reading and trying to write more proofs is like any other kind of practice, so it will make you better
glass dagger
Proofs related to limits are notoriously difficult when you're first starting out because the you have to be very careful with what the limits are actually saying
so I wouldn't feel bad if I were you
glass dagger
I am very bad when it comes to "show that" questions. Can anyone give me advice ...
I know this doesn't sound terribly encouraging but I don't think a computer science student should be expected to be able to prove this
lilac sun
Proofs related to limits are notoriously difficult when you're first starting ou...
its not difficult to understand but its difficult to make it rigorous
glass dagger
like it's a genuinely hard problem without a deeper interest in math
warm canopy
To be fair it's good that you tried something. You looked at the statement and identified that contrapositive could be a way to do it
lilac sun
I know this doesn't sound terribly encouraging but I don't think a computer scie...
this is why idk why we're even doing analysis in the first place
warm canopy
That's already better than most
lilac sun
intuitively it makes sense but i struggle to write it in terms of logic
lilac sun
Study basic logic I suppose
I'll try this out
cerulean cedar
obtuse pebbleBOT
gloomy vector
uh do you know what the standard equation for a circle is?
cerulean cedar
i can't remember
past sand
!15m
obtuse pebbleBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
gloomy vector
anyways do you think you can get an idea on how to move the center of it?
past sand
!volunteers
obtuse pebbleBOT
Helpers are just people volunteering their time to help you. Be polite and patient.
past sand
Not our problem
cerulean cedar
Not our problem
just saying
past sand
Please only use the <@&286206848099549185> ping **once** if your question has no...
Just saying
cerulean cedar
Just saying
bruh
cerulean cedar
anyways do you think you can get an idea on how to move the center of it?
can u give me a clue
gloomy vector
uhh
(x)^2+(y)^2=r^2
try to mqke something happen in the brackets
also sorry if i am explaining this horribly i just happen know from tinkering with graphs
placid lynx
imagine you shift origin to (-3,-4) can you calculate coordinate of (x,y) in new coordinate system ?
gloomy vector
np
warm flame
How do I solve this exponential problem?
I can't even make them have the same base, do I have to use logarithm or ln?
obtuse pebbleBOT
How do I solve this exponential problem?
I can't even make them have the same b...
versed stratus
I think you're expected to solve this by inspection
warm flame
how though
versed stratus
ok, notice how 243 can be expressed as the difference of two squares
18 and something else
try it out for yourself!
@warm flame
wispy silo
Or factor out 9^x from the LHS, it will give you
warm shaleBOT
OneTrackPony
wispy silo
Ah, not sure.
warm flame
It's 9
placid lynx
Since the lhs is increasing graph it can only have one solution
by hit and trial x=2
is the only solution
warm flame
log base 10 right
left vault
You can factor it this way $9^{x\cdot\log_{18}(9)}+9^{x}=243$
warm shaleBOT
The Great D
bright junco
yeah the answer should be 2
bright junco
You can factor it this way $9^{x\cdot\log_{18}(9)}+9^{x}=243$
Nah just graph it
@warm flame
Itโs W
2
left vault
You can't always graph it.
marsh geyser
Guys, you're doing a mess
bright junco
Straight line through the y axis on 243
tardy tusk
Guys, you're doing a mess
Teach us
left vault
and then graph 9^x (2^x - 1)
This is not correct factoring of it though.
marsh geyser
18^x-9^x = 243
Factor out 9^x
9^x(2^x-1) = 243
243 = 9^2 * 3
Now look at the equation you have
Compare and solve
bright junco
This is not correct factoring of it though.
Then ur just rlly shit at factoring
Cause itโs right
๐
left vault
Oh my bad yes
bright junco
Well what if it isnโt 2
Trial and error wouldnโt work
Thatโs so stupid
If ur stuck always graph it
Or put it into desmos if u donโt know how to graph it
but I mean if u canโt graph exponentials then ๐คทโโ๏ธ๐คทโโ๏ธ
marsh geyser
Trial and error wouldnโt work
?
warm flame
.close
obtuse pebbleBOT
bright junco
Yes bro
marsh geyser
Don't call me bro
bright junco
Don't call me bro
Why TF r u so pressed
marsh geyser
Whatever, don't talk to me.
rough ember
hi
rough ember
im confused on this
are you allowed to just
randomly multiply by 2
or is there a diff reason why theyve put a 2 infront
like idk it pisses me off just multiplying by 2
this is the full thing
im not confused on anything its just the multiplying by 2 always throws me off
brazen gorge
are you allowed to just
it's in the numerator and denominator
so it cancels out
so we are allowed to multiply by it
brazen gorge
they are doing that to use $\sin(2x) = 2\sin(x)\cos(x)$
warm shaleBOT
artemetra
brazen gorge
no problem
rough ember
.close
severe marsh
why the +1
obtuse pebbleBOT
why the +1
sage geode
Well, would it make sense for s_n to be equal to 2^n - 1?
Given 2^n is already in the sum of positive terms that s_n is
Basically, the idea is that when you add 1 to s_n, then you get 1 + 1 + 2 + 4 + 8 + ... + 2^n. Adding first two terms yields 2 + 2 + 4 + 8 + ... + 2^n; Doing it again yields 4 + 4 + 8 + ... + 2^n
Then 8 + 8 + ... + 2^n
Eventually you get to 2^{n+1}
Meaning 1 + s_n = 2^{n+1}
severe marsh
Eventually you get to 2^{n+1}
from 2^n+2^n
i see
thanks
so what's the generic formula for finite sums of geometric series?
sage geode
It's $a + ar + ar^2 + \dots + ar^n = a\frac{r^{n+1}}{r - 1}$, but the same trick doesn't work in the general case
warm shaleBOT
A Lonely Bean
severe marsh
ok yeah that's what i have on my notebook
wait it's slightly different
i have a((r^n) - 1) in the numerator instead
severe marsh
yeah it is
sage geode
Right, that's the same formula but with n replaced by n-1
severe marsh
i think i just got confused when i applied n instead of n+1, because from 2 to 2^n would be n terms
sage geode
It's $a + ar + ar^2 + \dots + ar^n = a\frac{r^{n+1}}{r - 1}$, but the same trick...
Ah wait I meant to type (r^{n+1} - 1)
sage geode
i think i just got confused when i applied n instead of n+1, because from 2 to ...
But it doesn't start at 2
snow dawn
.close oh dear god what have i done
forest sinew
.close oh dear god what have i done
just open a new channel 
obtuse pebbleBOT
Available help channel!
strong pebble
Hello
strong pebble
Lol
Dont worry you'r efine
I'm currently preparing for my SAT, and I'm trying my best to pick up the basics.
If any kind person could explain this question to me with steps I'd be forever grateful
I can't find the exact question online, and I wanna understand how this is solved
I suck at math.
yessss
That's what google said but it doesn't have steps
inland matrix
I'm currently preparing for my SAT, and I'm trying my best to pick up the basics...
Just sub 2x in place of x in the original function
strong pebble
so i have no clue how the answer came
strong pebble
Just sub 2x in place of x in the original function
waitttt reallyy
I'm so dumb haha thanks
ember monolith
Hi
ember monolith
1+2~log_{x}2\cdot log_{4}(10-x)=\frac{2}{log_{4}x}
Can anyone help to solve this?
unreal musk
$ around the equation please 
ember monolith
unreal musk
Any step that doesnโt seem clear in that?
ember monolith
Yup actually I just got the first step rest idek how to do it๐ฅฒ
Is there any other way to do it?
unreal musk
Second one t after they multiplied by that $\log_4(x)$, they used the change of base rule $\log_b(a) = \frac1{\log_a(b)}$
warm shaleBOT
@unreal musk
unreal musk
And well you could but for the most part youโd be using the same log rules at some point or another, just in different orders and stuff
tardy tusk
โ in logs means : in equation
No way
I'm stupid as hell
It's reverse
: means -
Yeah
It won't help there
ember monolith
oh
tardy tusk
+ between logs means ร in logs
Same about โ between logs means : in logs
tardy tusk
ember monolith
Thank you so much!! โฅ๏นโฅ
tardy tusk
ok
You can use Google for more properties
Or YouTube
For more knowledge
You can use photomath or wolphram to solve them if you're really stuck
willow acorn
Hey Iโm really confused how to do q15 any help would be really appreciated
willow acorn
Iโve looked at the mark scheme but I donโt get it
humble wind
hi
u have to complete square the expression in the root
and then use standard integrals
humble wind
yeah
it should be 4(x-1/2)^2-1/2
then do the definite integral bit and it should work out i think
obtuse pebbleBOT
@willow acorn Has your question been resolved?
willow acorn
Omg I actually got there thanks a lot my friend
humble wind
Omg I actually got there thanks a lot my friend
u did all the work bro no worries
u edexcel or cambridge @willow acorn ?
willow acorn
OCR mei, I think itโs Cambridge
willow acorn
Iโve got mocks in Jan
humble wind
Iโve got mocks in Jan
good luck!!
willow acorn
Thanks!
You too
Iโll be back with more questions soon my friend
And Iโm back
That was short lived
.close
obtuse pebbleBOT
willow acorn
Iโll make a new one
fickle turret
Ik it's not very helpful, but because everything cancels out nicely? Is there a particular step, that you need help with understanding?
robust raven
I proved the formula for the "chain rule" for you. In this version, so that you could understand why this formula involves multiplication and not another operation, I provided the proof in a less rigorous version due to the formal details, to make it easier to read the notation.
I repeat to you that, as previously, understanding the formula depends solely on the analysis of the proof and its acceptance, this is not an example illustrating the method, only proof justifying the form of the formula.
fickle turret
I proved the formula for the "chain rule" for you. In this version, so that you ...
Is that your proof? It's beautiful
robust raven
Is that your proof? It's beautiful
yes it is. Apertyx wanted to find out, why formula contains "multiplication" ๐
since there is an outer function f and an inner function g, I decided to use two different notations for the increments of both functions to distinguish them, and finally to use the definition of the derivative at the point for both f and g.
the choice of delta h was not accidental, the idea was that it tended to zero when h also tends to zero and that it related the increase of the function f with the function g,
this is simply "technical" proof, it is enough to understand the definition of derivative in point to agree with this evidence
obtuse pebbleBOT
@timid silo Has your question been resolved?
calm vale
HOW
obtuse pebbleBOT
@calm vale Has your question been resolved?
spark arrow
Yeah
Polynomial division/synthetic division should be best for this problem
You should start with dividing the Polynomial by (x-1)
timid silo
then divide both sides equation you got by x^3
spark arrow
Wdym both sides
There won't be a 2nd side to a Polynomial until it's completely factored
spark arrow
There wouldn't be a 0/xยณ
timid silo
after you resolved root x = 1 you get polynomial of 6th degree
obtuse pebbleBOT
Available help channel!
frozen jasper
hello
frozen jasper
tanphi= -b/a
but here tanc= roots3/1 positive, not negative
the answers are sometimes in positive so sink(x-c) and sometimes negative i cant find the pattern
<@&286206848099549185>
obtuse pebbleBOT
@frozen jasper Has your question been resolved?
obtuse pebbleBOT
@frozen jasper Has your question been resolved?
tardy epoch
,rotate
warm shaleBOT
tardy epoch
the answers are sometimes in positive so sink(x-c) and sometimes negative i cant...
What's your question
verbal niche
does this 9 mean 9 terms?
obtuse pebbleBOT
does this 9 mean 9 terms?
warm flame
do I have to square 46 or square root 46?
obtuse pebbleBOT
do I have to square 46 or square root 46?
warm flame
how do u rotate
๐
I'm a bit confused because I'm following the rule (x-h)ยฒ+(y-k)ยฒ=rยฒ
.stop
.close
latent bobcat
hello I'm stuck on a problem
obtuse pebbleBOT
hello I'm stuck on a problem
latent bobcat
z^2= -5
forest sinew
what'd you try?
obtuse pebbleBOT
@latent bobcat Has your question been resolved?
latent bobcat
what'd you try?
i couldn try much
didnt understand
i mean i got to like z= square root of -5
idk how to represent that
whats next
sorry i i took like 13 minutes to respond
tardy epoch
idk how to represent that
Use i=โ-1
latent bobcat
really?
idk what that even
how does it work
like how do you solve the question
quick blaze
have you learned imaginary numbers?
quick blaze
okay, just a quick check - how would you solve x^2 = +5?
quick blaze
remember that quadratics have TWO solutions
latent bobcat
how do i even makie the symbal
ohh yeah
yeah
its
like
x=+-โ5 idk how to do the plus on top of the minus
i had to copy and paste this
quick blaze
yeah, thats good
latent bobcat
โ
quick blaze
so now if you know imaginary numbers i = sqrt(-1)
so if we have x^2 = -5
x = +- sqrt(-5)
properties of square roots... rewrite as sqrt(-5 * 1) and split it apart
can you try that and see what you get?
quick blaze
sqrt(ab) = sqrt(a)sqrt(b)
(as long as both a and b aren't negative - which isnt the case here so don't worry about that)
latent bobcat
ohh wait
is it z= sqrt(-5)sqrt(1)
is it that
cuz you siad split
would that be the answer?
quick blaze
is it z= sqrt(-5)sqrt(1)
can you retry it now lol?
latent bobcat
ok
wait would that just be the answer?
the book straight up just wants me to solve it
quick blaze
z = +-sqrt(-1 * 5)
z = +-sqrt(-1) * sqrt(5)
z = +- isqrt(5)
quick blaze
well, i is just by definition sqrt(-1)
quick blaze
yeah... or sometimes defined i^2 = -1, both are valid
latent bobcat
oh ok
quick blaze
if the book taught you imaginary numbers that's the solution it wants. if it hasn't taught it... then you just say "no real solutions"
latent bobcat
yeahhh
yeah
yeah i mean school hasnt really talkied about imaginary numbers
but most of my class probably does
we learnign 2 grades ahead in math i used to be really good at math but i fell off
quick blaze
yeah i mean school hasnt really talkied about imaginary numbers
eh, if they brought it up at all I would write the imaginary solution
quick blaze
we learnign 2 grades ahead in math i used to be really good at math but i fell o...
it happens man... keep going, you'll catch up. try to understand whats happening and not just memorize stuff lol
latent bobcat
its slightly trippy cuz the on level kids are learning the stuff we learned 2-3 years ago
latent bobcat
it happens man... keep going, you'll catch up. try to understand whats happening...
idk i remeber and understand more things with like a friend\
i ask my friend and i actually remeber
remember*
but if i listen to the techer in clsss
class
idk
quick blaze
but if i listen to the techer in clsss
eh, if what the teacher is doing isn't helping try taking notes that you would understand
latent bobcat
i mean she makes us take notes mostly but idk everytime it feels like my classmates have all learned the concept outside of school
chich they have
which*
but i just cant remeber
i just cant remember how to do things
tardy epoch
i just cant remember how to do things
obtuse pebbleBOT
@latent bobcat Has your question been resolved?
latent bobcat
sure man
burnt vector
there are a 2 intended solutions for the problem statement below which uses funny algorithms... but there's one which uses math. And it's... insane.
obtuse pebbleBOT
there are a 2 intended solutions for the problem statement below which uses funn...
burnt vector
the answer for a given n simply boils down to simply evaluating this polynomial (x = n).
I'm trying to prove why this is the case, but my efforts have been quite futile.
I'm almost entirely certain to say there is no "easy" way to do this via induction of any sort. If there were, it would have resulted in a recurrence formula, which would have been made into an intended solution...
so instead, I'm trying to somehow bound the answer above and below by 10 degree polynomials, so as to assert that if the answer can be modelled by a polynomial's growth, then it has to be 10 degree.
Aside from showing that this IS actually modelled by a polynomial, forming the upper and lower bounds is not easy.
(number of tuples which satisfy the above conditions for <= n) - (number of tuples which satisfy it for <= n-1) = (number of tuples which satisfy the above conditions for exactly = n)
this is the "discrete derivative" of our answer (where the "function" is basically the number of tuples which satisfy the conditions for <= n). And that has an upper bound of n+9 Choose 9 (stars and bars. We overcount a bit since some of the selections won't satisfy the required inequalities)
that's basically- (n+1)(n+2)...(n+9), which is a 9 degree polynomial.
Treating this as a real valued function and integrating to derive an upper bound for the number of tuples which satisfy the above conditions for <= n... we get something of the 10 degree, which shows that the number of tuples which satisfies the conditions for a given n is therefore bounded above by a 10 degree polynomial... (or if you wanted to be more formal about it, one could pull off an argument similar to the proof for the integral test for convergence)
but that's merely the number of tuples, not the sum of all the tuple's products. what's bad is that the maximum value of the product is... it's (n/5)^5. That's no good, that means I can't just multiply the two and say "it's still 10 degree"
as for a lower bound, I'm not quite sure either.... for a given n, it's clearly bounded by the answer for n-1, but that's not a good bound 
obtuse pebbleBOT
@burnt vector Has your question been resolved?
echo heron
The first thing would do is rewrite the condition as
$2(s_1+\dots+e_1)+(s_2-s_1)+\dots+(e_2-e_1) \leq N$
warm shaleBOT
elon mask
burnt vector
oh? Wait... why?
echo heron
Well then each term is completely โunrelatedโ to each other
burnt vector
oh wait, just rewriting
echo heron
Cuz itโs just 2(sum of 5 nonnegative integers)+(sum do 5 positive integers)
echo heron
Hmm, Iโm trying to interpret the answer combinatorially
burnt vector
I got smth related to that if you'd like to see
echo heron
Sure
burnt vector
from the editorial
it gives a rough idea of how to translate it to a combinatorial problem
and then computes it with matrix exponentiation yes, and when they mentioned that line at the bottom, it's referring to the polynomial I mentioned
I honestly kinda struggle to understand how this combinatorial approach works 
but yeah, u got the first part (of the intended solution) down 
that's some sharp intuition
echo heron
The sum of $abcde$ for $a,b,c,d,e$ positive integers where $a+b+c+d+e \leq N$ should be just $\binom{N+5}{10}$ if my thinking is correct
warm shaleBOT
elon mask
burnt vector
hmm
echo heron
**elon mask**
Cuz you can think about putting 5 dividers in 1, โฆ, N and then choosing one number in each of the leftmost five portions
Maybe this can give a purely combinatorial proof that it is a degree10 polynomial
burnt vector
Interesting...
burnt vector
from the editorial
I honestly do not understand this part:
After splitting, the ways of choosing one ball from each delta s, delta n, delta u, delta k, delta e correspond to the answer one by one (has a bijection)
echo heron
Because given a certain splitting, there are delta s * โฆ * delta e ways to pick the balls
So that is exactly what we want to sum up
burnt vector
ahh
echo heron
Oh ok
echo heron
Ok i kind of see how it would be implemented, but how is it related to matrix exponentiation?
burnt vector
that I'm not so sure either... this doesn't even give a linear recurrence relation
burnt vector
Because given a certain splitting, there are delta s * โฆ * delta e ways to pick ...
aanddd you pick the balls within the partition... right-
so like, you "insert" 5 sticks to partition the balls, and then between the start and the first stick you pick delta s, 1st and 2nd u pick delta n and so on...
~~ugh I'm so bad at combi ~~
wait-
... wait that doesn't sound right
ugh
burnt vector
there are a 2 intended solutions for the problem statement below which uses funn...
here
if we just put the polynomial aside for a sec
burnt vector
from the editorial
I still don't fully understand how the bijection works
yes, this problem stems from a competitive programming contest
however, that's not quite the point of my question-
instead, it's about how the combinatorics plays out... and if I can use it to verify the accuracy of the polynomial
burnt vector
aanddd you pick the balls within the partition... right-
so like, you "insert" 5...
hm, maybe I'm overcomplicating it...
It simply is- if we fixed the "positions" of the "sticks" representing s, n, u, k and e (which must have an even number of balls before them) and inserted 5 more sticks representing the deltas with at least 1 ball between the stick before it, then the number of possible ways to pick 1 ball between these partitions corresponds to the value of the product of (s2 - s1)(n2-n1)....(e2-e1)
....
we're using another 5 sticks to count the number of ways we can pick a ball between each partition
but... n-5... eh?
....
burnt vector
**elon mask**
lemme run a bruteforce to check
I sure hope I don't get integer overflow
from 1 to 11
echo heron
How is it going
burnt vector
the code: ```cpp
#include <bits/stdc++.h>
using namespace std;
#define int long long
int generateTuples(int n) {
int ans = 0;
for(int a = 0; a <= n; ++a)
for(int b = 0; b <= n - a; ++b)
for(int c = 0; c <= n - a - b; ++c)
for(int d = 0; d <= n - a - b - c; ++d)
for(int e = 0; e <= n - a - b - c - d; ++e)
ans += abcde;
return ans;
}
signed main() {
ios::sync_with_stdio(0); cin.tie(0);
int ans[12]; fill(ans, ans+12, 0);
ans[1] = ans[2] = ans[3] = ans[4] = 0; ans[5] = 1;
for(int i = 6; i <= 11; i++){
ans[i] += ans[i-1];
ans[i] += generateTuples(i);
}
for(int i = 1; i <= 11; i++) cout << ans[i] << " ";
return 0;
}
11 choose 10 is 11, off by 1
,w 13 choose 10
echo heron
Hmm.
warm shaleBOT
burnt vector
it's lower than the actual value
echo heron
Interestingly, 12-1=11 and 78-12=66โฆ
echo heron
Why would the answer for N=6 be 12
The cases are (2,1,1,1,1) with five permutations, so 10, plus (1,1,1,1,1) which is 1
echo heron
Why did you sum up the ans like that
burnt vector
Why did you sum up the ans like that
isn't it the sum of all possible stuff which makes the value n?
echo heron
The first part generateTuples already counts all cases summing to <=N, so you donโt need to sum them again
burnt vector
huh. I thought it's exactly n
echo heron
Nah but youโre choosing e such that a+b+c+d+e is at most N
echo heron
You should get my answer now
burnt vector
sheesh.
echo heron
Yeah?
burnt vector
I wonder if this interpolates to some funny polynomial too
echo heron
Well it does because itโs(polynomial) choose (constant)
You can even write it out
The polynomial
burnt vector
Well it does because itโs(polynomial) choose (constant)
oh...
right...
๐คก
ahem
now to find that polynomial for the funny
echo heron
You can even write it out
(N+5)(N+4)โฆ(N-4)/10!
Which is degree 10 which is a bit sus cuz wouldnโt that mean the partition with 15 numbers chosen would be like close to degree 15 or sth
In the original problem
burnt vector
exactly!!!
burnt vector
but that's merely the number of tuples, not the sum of all the tuple's products....
like here if u looked at worst case u get smth of degree 15 too
found your polynomial
hm
now how can I use this..
burnt vector
the code: ```cpp
#include <bits/stdc++.h>
using namespace std;
#define int long ...
lmfao @wild swallow lurking?
burnt vector
there are a 2 intended solutions for the problem statement below which uses funn...
in this case how my numbers differs is... instead of 5 variables <= n my variables are kinda funni
trying to throw some funny contradiction to show that the result must be represented by some polynomial
cuz if I can prove that, plus the fact that we've bounded the degree by 15...
then it simply boils down to interpolating 15 points and going "there we go!"
... well, assume that there does not exist a polynomial such that P(n) = ans(n) for all n... then for any polynomial, P(n) != ans(n) for some n
Mmm
wait
let's denote this point with n1
and WLOG let it be the smallest value of n such that P(n) != ans (n)
now how do I get my juicy contradiction..
that doesn't sound right, actually... Like, I need to show that there exists a polynomial which actually works... ugh idk
well, it's clear that if I used an arbitary polynomial
I'll get nowhere
burnt vector
the answer for a given n simply boils down to simply evaluating this polynomial ...
but if I just shoved this polynomial
....
okay, it's not gonna be neat but
u gotta do what u gotta do
now what in the actual fck do I do
burnt vector
from 1 to 11
the values are also somewhat similar...
burnt vector
,w (35 - (42983 x)/420 + (773131 x^2)/6300 - (973 x^3)/12 + (429901 x^4)/12960 - (12793 x^5)/1440 + (34441 x^6)/21600 - (193 x^7)/1008 + (451 x^8)/30240 - x^9/1440 + x^10/64800)-(x/1260 + x^2/6300 - (41 x^3)/36288 - (41 x^4)/181440 + (13 x^5)/34560 + (13 x^6)/172800 - x^7/24192 - x^8/120960 + x^9/725760 + x^10/3628800)
warm shaleBOT
Requested by kiameimon
obtuse pebbleBOT
@burnt vector Has your question been resolved?
burnt vector
... maybe I should just drop it
obtuse pebbleBOT
@timid silo Has your question been resolved?
sharp forum
yes
what is it you need
@timid silo
i am help
idiot
soory
i am not supposed to say that @timid silo
what all i did was try to help you
where
i asked you
thankyou
polar fossil
garv if you have the helper role you're going to get pinged for help on questions, if you don't want it you can remove the role in id:customize
sharp forum
no i want it that is why i came
you need the detrimant right
@timid silo
but i need the elemnts of a matrix
polar fossil
garv i don't think you're qualified to answer this question, please leave
sharp forum
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
i am i solved it
bruh i am in undergrad
polar fossil
inversions give me a headache but i think this is correct yes
polar fossil
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
this is for a 3x3 matrix but the original post is talking about a 6x6
tepid sapphire
6, 3 should be negative?
polar fossil
don't think so because this is the minor when we look at 63 -- that should be positive right?
uhh
- and - are the elements we expanded
x are the elemnts we removed
. is the remaining matrix
o is 6,3
tepid eagle
Hi, I need only a quick clarification
obtuse pebbleBOT
Hi, I need only a quick clarification
tepid eagle
alpine bison
polar fossil
<:catthumbsup:614540188747563008>
what do you mean by this 
alpine bison
it goes to zero isn't it ? its basically sin(0)
alpine bison
what do you mean by this <:Sobbingcrying:1010405017385189426>
it's a cuttie cat that is ready to help everyone
tepid eagle
it goes to zero isn't it ? its basically sin(0)
Oh yes, sorry that's a mistake
But it results:
-t/0 so -infinity
alpine bison
no its an indeterminate form
warm shaleBOT
everg
alpine bison
on second thought, if you say $\frac{1-\cos t}{t^2}\to 1/2$ then it s impossible to get $\frac{1-\cos t}{t}\to 1$
warm shaleBOT
everg
tepid eagle
Yes you're right, I made a mistake.
Now it's clear
Thank you very much for the clarification and for the help! ๐
alpine bison
np
tepid eagle
.close
obtuse pebbleBOT