help-4

[ruby ingot]

calm down

[umbral elbow]

sum of roots

= -b/a

[ruby ingot]

i have an idea i want to try then ill get back

[umbral elbow]

c/a = sum of roots taken two at a time

u dont even need

those equations

rip

so

[distant galleon]

And how do you plan to find the leading coefficient

[umbral elbow]

its 1

[distant galleon]

why

[umbral elbow]

x^3 -x =f(x)

is this correct

[distant galleon]

no

[umbral elbow]

lemme plot in desmos

[distant galleon]

it does not pass through (1/2, -1)

Left hand side is not in Z for 1/2

[umbral elbow]

???

what

[distant galleon]

We mentioned earlier that stationary points of cubics are always halfway between the roots

[umbral elbow]

i dont understand

[distant galleon]

There’s roots at 0,1

So there’s a local minimum at x=1/2

And from the description, we know the value of f at x=1/2 is -1

Hence the graph should pass through (1/2. -1)

[umbral elbow]

my answer is correct i think

plot it in desmos

[fossil mulch]

Can I propose another method

[distant galleon]

Are you gonna sine triple angle it lol

[umbral elbow]

@tame horizon check if x^3 -x is the cubic

@ruby ingot

[distant galleon]

The description implies the local minimum has a value of -1

[ruby ingot]

I HAVE AN ANS

[distant galleon]

Is that true for your function

[ruby ingot]

i got f(x) = 4x^3 -3x

[umbral elbow]

no

[distant galleon]

That’s good

[ruby ingot]

i see

[distant galleon]

It says it “just fits”

And that is also suggested by the picture

[ruby ingot]

i had to just use drivatives

[distant galleon]

If you wanted to be fancy and get a 10 second solve, you could have also noticed that x=sin(theta) results in a connection to sine triple angle

[silver birch]

btw you could've also done this using a system of equations but it's not that fancy

[distant galleon]

Yeah that’s the very

Bashy way

[ruby ingot]

never heard of sine triple angle

[ruby ingot]

i did that

[distant galleon]

Expansion of sin(3x) basically

[ruby ingot]

ah i see

[distant galleon]

bruh I literally gave you a shortcut

[ruby ingot]

i did notice it was an odd function

[distant galleon]

,w k*(x-1)(x)(x+1)+x=-1, x=1/2

[rocky lotus]

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[distant galleon]

,w expand 4(x-1)(x)(x+1)+x

[rocky lotus]

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[ruby ingot]

yh i dont quite understand that

what was the logic behind it

since u got f(x) - x from somewhere

[distant galleon]

zero is nice to work with cause you get factors for free and f(x)-x happens to be zero for three values

[ruby ingot]

yea but surely if it only works for 3 values howe can u be convinced?

[distant galleon]

f(x)-x is cubic

So it can only have three roots?

Fundamental theorem of algebra

polynomial of degree n has n roots (counting multiplicity)

[ruby ingot]

ohhh i see why cause u implemented a k in there to scale after

[distant galleon]

correct

[ruby ingot]

damn thats lowkey goated

[distant galleon]

Yeah standard trick

Anyway 4x^3-3x was the answer right

[ruby ingot]

yea

[distant galleon]

@umbral elbow ^

Read this over

I gave three different ways (system bash, f(x)-x, triple angle), pick one

But I was right about the local minimum being -1 (and also you can’t assume the leading coefficient is 1)

[vale dock]

If you are done with this channel, please mark your problem as solved by typing .close

[ruby ingot]

wait so if i know 3 values for this and the fact taht f(x) - x = 0 for those 3 values, i can then use those to form the factors like in this case x(x-1)(x+1) -x then i can scale the f(x) after?

[distant galleon]

f(x)-x=k(x-1)x(x+1) -> f(x)=x+k(x-1)(x)(x+1)

If that’s what you’re trying to say

[ruby ingot]

yes

solid thanks bro

[distant galleon]

np

[ruby ingot]

.close

[vale dock]

Channel closed

Closed by @ruby ingot

Use .reopen if this was a mistake.

[ruby ingot]

.reopen

[vale dock]

✅

[ruby ingot]

i wanna ask a follow rhrough to this

wait ill send it here and let u try if u want @umbral elbow

try the same thing for a 4 degree polynomail

so for

so i know that f(0) = 1 f(1) = 1 f(-1) = 1

[silver birch]

you can also consider the fact that it's an even function

[ruby ingot]

so f(x) = f(-x) asw

[silver birch]

yep

[ruby ingot]

i mean lowkey, i can do the same substitution bash

but i wanna try smth new

[silver birch]

you said no derivatives?

[distant galleon]

my 30 second solution rip

[ruby ingot]

nah i got u, ima try tat

[distant galleon]

Wait my dumbass thought my solution was the substitution bash

brain gone

This is what I get for working on 2 brain cells lol

[silver birch]

Substitution bash is indeed pretty funny if I might say so

[distant galleon]

oh yeah can confirm you don’t need derivatives

If my mental math is right

Cause I am still walking

💀

[silver birch]

derivatives aren't needed but they can help

though

you might need integration

[distant galleon]

eh

I did it with just pure algebra

[ruby ingot]

bruh

[silver birch]

yeah no you should prob use only substitution bash

[ruby ingot]

give me a hint @distant galleon

[distant galleon]

no

[ruby ingot]

alr sn

[distant galleon]

ok I let you guys cook

https://tenor.com/view/fat-pigeon-pigeon-fat-pigeon-leaves-gif-7805915763287027852

[silver birch]

i would stay here but I gotta get out in like one minute

life calls unfortunately 😔

[ruby ingot]

its fine, ill try solvig

[silver birch]

you can do it I believe in you

[ruby ingot]

nah im confused

i tried f(x) = (x^2-1)(ax^2 + b)

idk how i can get a fourth point on this

waitt

yh no

ima le this marinate, ill eat then come bac to it

[vale dock]

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[tardy pollen]

I don't seem to reach anything solid

[opal pendant]

💀 what level of math is this

[tardy pollen]

<@&268886789983436800>

Okk no worries

<@&286206848099549185>

[sullen summit]

did u try writing sec(2(b-a)pi/5) in terms of w

[tardy pollen]

Yes

[ashen prawn]

hi sir

you remember me?

[tardy pollen]

Yes yes jeeneettards

[ashen prawn]

yeah

[sullen summit]

wait yall doing jee asw

[tardy pollen]

Yes 🙏😭

[sullen summit]

Dis frm isi?

[tardy pollen]

I'm not sure abt the source

Just help me with this shit 🥹

[sullen summit]

Wait

I got it i think

one sec

no i didnt

[tardy pollen]

Exactly the same situation I think I got the approaches but when I proceed I get stuck

[sullen summit]

You can try it like rhis i guess

[prime violet]

Is 101 a prime number or I'm tripping? If you could reduce everything modulo 101 in that field and uh

nvm I'm lost

[sullen summit]

@tardy pollen i get this

[weary pilot]

,w is 101 prime

[rocky lotus]

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[sullen summit]

This is very weird though

last time i did complex was in 11th so im very trash at it at the moment

[tardy pollen]

I did this too

But didn't work

And I think the way u put k in Nr is wrong it's 2b-a not 2(b-a)

[sullen summit]

Divide and multiply by w^a

[weary pilot]

ah yes

jee

write down 5 functions together, make it hideous, call it math

[tardy pollen]

Fr 🥹

[tardy pollen]

I see what u did but how will u calculate values of tan

[sullen summit]

shi i suck

yeah thats

what i was thinking

and its to the power 5 asw

so coverting to c and s wont help

Imma cope by saying they wont ask this in 26

[tardy pollen]

Fr let's just pray 🙏

[sullen summit]

yooo

nun

[ruby ingot]

this looks like A level fm roots of unity

[sullen summit]

if u write itan(b-a) in eulers form

U nay get somewhere

but power 5 again ruins it

[sullen summit]

yo can you find the error in this

This was the q

[tardy pollen]

Nah bruh not in mood rn
Going to sleep now
I got absolutely smoked by the questions today

[tardy pollen]

Bro why u used solid angles and stuff it's not that deep 😭

[sullen summit]

This q is pretty ambigous

[tardy pollen]

Just use proportionality energy is proportional to area and solve for it

[ruby ingot]

u can do it with asw

[tardy pollen]

I mean yea its area anyways but why go the long way

[ruby ingot]

true

[sullen summit]

howd you find the area

[tardy pollen]

Not area exactly but ratio

[sullen summit]

and howd u calc the ratio

[tardy pollen]

Similarity

[sullen summit]

the triangles wont be similar

theta would change

[ruby ingot]

its some vague q

[sullen summit]

or half of the apex angle

[sullen summit]

thats what i thought

but ans keys werent changed

[tardy pollen]

I see what's the problem here

[sullen summit]

they didnt mention small aperture so we just assumed it

[ruby ingot]

u have the right ans for value of K?

[tardy pollen]

What's the answer btw ?

[sullen summit]

27

[ruby ingot]

oh i see

have u tried using 1/f = 1/u + 1/v ?

[sullen summit]

1/v - 1/u = 1/f

[ruby ingot]

yhyh

[sullen summit]

Thats how i got virtual source's position

[ruby ingot]

so u got u as -10?

[sullen summit]

yes

v

[ruby ingot]

so distance from the virtual image to P is 50 right?

[sullen summit]

Yeah

[ruby ingot]

alr so what my logic is yh,

lemme type all this up

nah im dumb

i thougt i was onto to smth

[sullen summit]

yu giving jee asw?

[ruby ingot]

no lol, im in the UK

starting undergrad

[sullen summit]

Oh

Good luck

[ruby ingot]

havent done maths for 2 months so need to lock in

[tardy pollen]

@sullen summit
Bro I think the answer u gave 27 have some problem
It's not coming 27 no matter what I do

[sullen summit]

what

[tardy pollen]

No wait intensity reduce to 0.27I
Too it's much change to be possible

[tardy pollen]

Bro ∆@/@ should be -1/6 so @2/@1 will be 5/6 I think

[sullen summit]

delL is negative

[tardy pollen]

Ahh yea the length is decreasing

Yo I got a solid approach but it won't yield the result as 27 @sullen summit

[sullen summit]

What is it

[tardy pollen]

Like everything is same
But we were stuck at area right
For initial condition the light is spread in a sphere of radius 60 till it reaches the screen and for final condition it spreads of sphere of 50cm
And area is proportional to (radius)² so we can take ratio and it would give the ratio of area to us

[sullen summit]

hmmm

I' = 3P/4(50)² I = P/60^²

363/425

27I/25

[tardy pollen]

Seems like it

It is 27 but not with 100

[sullen summit]

this sounds right

[tardy pollen]

It's funny how we turned from that complex no. Problem to wave optics 🥲

[sullen summit]

ray

and wave

this becomes confusing

[sullen summit]

i got a q thou

[tardy pollen]

Nice

[sullen summit]

Do yk hindi?

[tardy pollen]

Ofc

[sullen summit]

Toh bhai

hum isme

Intensity ko

/ area krke thodi likh skte h

Like net power / tsa

The reason being lens se jo rays jaayengi unki intensity alag hogi

And jo jo baaki rays h lens ke baahar waali

unki alg

i mean 1/4 rays absorb hongi lens ke thru jaane waali

Along with that

[tardy pollen]

But energy jo hogi wo uniformly distributed hogi

[sullen summit]

nop

Lens ke thru jitna portion jaayega

usme baaki portion mei distribution ka 3/4 hojayega

aerial distribution ka 3/4

[tardy pollen]

Toh uski 3/4 multiply krke compensate kiya toh

[sullen summit]

Yeah

But 3/4 se multiply krke

Poore sphere ka likha

however woh 3/4 wala sirf lens ke thru jaane wala h

[tardy pollen]

Baki ke saath kya ho raha hume kya krne 3/4 krke humko lens wala part toh mil gaya na jiski requirement thi

[sullen summit]

hmm

I am too dumb for jee dawg💔

imma ask my friend for the ans key again tom

gn

maybe 108 ho fir

[tardy pollen]

Probably
Waise ek logic h jisse maine jo bataya wo galat ho sakta h but whatever going to sleep now to I'm going to get fked tmrow

[vale dock]

@tardy pollen Has your question been resolved?

[vale dock]

@tardy pollen Has your question been resolved?

[distant galleon]

Multiply each factor by $\omega^{-a}$ and consider the distribution of residues modulo $5$

[rocky lotus]

Civil Service Pigeon

[vale dock]

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• Show your work, and if possible, explain where you are stuck.
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[ornate dragon]

Find, if it exists, the limit of an

we can do

an+1/an

if it's less than 1 we'll know it's 0

if it's more than 1 we'll know it's +inf

[mortal temple]

that doesn't sound right

oh did you mean the limit of a_(n+1) / a_n ?

[vale dock]

@ornate dragon Has your question been resolved?

[ornate dragon]

yeah my bad

i've divided everything by 8^n

everything in the root that is

[cobalt crow]

like

if you divided everything by 8^n

then for example you have a sequence (3/8)^n another one (-1/8)^n etc

cant we apply cauchy nth root for any of those terms?, maybe I am tripping

[cobalt crow]

we are taking the limit for a_{n+1} how do you conclude that lim a_n = 0?

[ornate dragon]

we are taking the limit for a_n+1/an

[mortal temple]

ratio test for sequences

[ornate dragon]

this is defining it for series

[cobalt crow]

how is (3/8)^n not approaching 0?

[ornate dragon]

(3/8)^n^1/n

(3/8)^n*1/n

3/8

[mortal temple]

you posted the ratio test for series

[ornate dragon]

yeah

i couldnt find it for sequences

but the idea holds because at any rate a series can not converge if it sequence is greater than 1

[cobalt crow]

like, if you know p series

[ornate dragon]

i do

but this is not a p series

[mortal temple]

the limit calculation looks sketchy

[cobalt crow]

yes

[ornate dragon]

how come

(-1/8)^n is 0

(-1/4)^n is 0

[mortal temple]

you sent (1/8)^n to zero without touching (3/8)^n

[cobalt crow]

its alternating

[ornate dragon]

hm

mind you i usually check these with my calculator in table mode

which is why im confident

(3/8)^x^1/x gave me 3/8

[mortal temple]

in general you can't just take the limit of part of the expression

you can sometimes do it

but it looks sketchy here

[cobalt crow]

like. $\lim_{n \to +\infty} \left(\frac{3}{8}\right)^n = 0$ no?

[rocky lotus]

Renato

[ornate dragon]

yes

[cobalt crow]

we can do auxiliary limit calculations

like, sub calculations so we can figure out what this whole limit approaches to

[mortal temple]

yes but you have to be careful about how you do it

let me give an example

[cobalt crow]

like. we are using dalembert because we are calculating the ratio of the an+1/an

but an+1/an we are going to use cauchy inside the nthroot

[ornate dragon]

it wasn't liking inf for some reason

[cobalt crow]

euler

[mortal temple]

you can define the constant e like this, but you can't just say 1/n goes to 0 because then the limit would end up being 1

[ornate dragon]

everything seems to indicate my work is correct

[cobalt crow]

there is a cauchy and a dalembert test for sequences

[ornate dragon]

yes

this is dalambert

like, starting from their definition

all i’m doing is using dalambert criteria

[cobalt crow]

yes. but do you have the definition from your class notes?

[ornate dragon]

yeah

[cobalt crow]

I am not finding any info online

I speak spanish

[ornate dragon]

oh wait

this is for series

see that last line

[cobalt crow]

thiss

[mortal temple]

it looks right, but you haven't justified it
or i don't understand how you justified it

[ornate dragon]

all i’ve done is divide by 8^n

quotient and numerator

which is allowed

then i’ve basically said

this term goes to 0

this term goes to 0

[mortal temple]

i don't think that's allowed

[ornate dragon]

left with (3/8)^n

[cobalt crow]

we need to find what the limit of an+1/an approaches to, but for that, we need to find what the limit of what's inside the nth root approaches to

[mortal temple]

i don't think that approach is valid either

maybe i'm missing something 🤷

[cobalt crow]

let me explain

I am drawing something axe

check the theorems that ransik sent btw

[mortal temple]

$\lim_{n\to\infty}\sqrt[n]{\frac{2^n+3^n}{8^n}}=\lim_{n\to\infty}\sqrt[n]{\frac{(2/8)^n+(3/8)^n}{1}}=\lim_{n\to\infty}\sqrt[n]{(2/8)^n}=\frac{1}{4}$

[rocky lotus]

Axe

[mortal temple]

what's to stop you from doing this?

[cobalt crow]

@ornate dragon

what do you think about this?

maybe I am tripping hard, but then if you apply dalembert with L = 0 we get that . . .

[cobalt crow]

this limit is 0, because of cauchy theorem for sequences that ransik sent above

this

let me translate it

[mortal temple]

i understand it

oh you meant translate it into english

i thought you meant translate the math

[cobalt crow]

yes, let me translate it to english

[mortal temple]

anyway i think i understand, i found the test described in english

[cobalt crow]

Theorem: Root Criteria (Cauchy for sequences)
Hypothesis: Let (an) be a sequence of positive terms and L = lim n->infty nthroot an exists
then:
L < 1 ==> an -> 0 ,
L > 1 ==> an -> =infty

[mortal temple]

this is not the right test

[cobalt crow]

is similar to root test for series but, this actually says that the sequence converges to 0

[mortal temple]

we were using the ratio test for sequences

not the root test

[cobalt crow]

is confusing but, we need to apply dalembert criterium first, but since the ratio of the sequences is an nth root, we need to apply the cauchy criterium first to then apply the dalembert criterium @ornate dragon

[mortal temple]

no

[ornate dragon]

sorry i was getting dinner

we’re not applying root theorem

the fact that there is an nth root does not mean this is not root theorem

this is quotient theorem

lim an+1/an -> 0 < L < 1 implies
lim an -> 0

[cobalt crow]

ye

this is dalembert

but first we need to apply cauchy to find what the ratio of the sequences converges to

you guys checked the shit I sent above? the auxiliary calculations?

[mortal temple]

not really

honestly i feel you're wasting ransik's time

[cobalt crow]

ok, I will leave then

[ornate dragon]

so you’re like

trying to link the two theorems?

solving through root then quotient then getting an?

[ornate dragon]

nah its fine i’m resting for a bit

[mortal temple]

even if you're trying to use cauchy, that theorem is for sequences, so i don't see how you could apply it here

[mortal temple]

anyway this demonstrates why your work is not properly justified, ransik

[cobalt crow]

what is inside the nth root is sequence himself

[mortal temple]

it can be seen as a series

but a sequence?

[cobalt crow]

what?

[mortal temple]

oh

no you're right

i expected you to say something different

yes it is a sequence itself

[cobalt crow]

if you guys agree that what is inside the nth root is a sequence. . . then you can apply the criterion for sequences, like cauchy

[mortal temple]

and what would you conclude from cauchy?

[ornate dragon]

it’s a subsequence

call it b_n if you’d like

[mortal temple]

wait this makes no sense, you would still need to find the limit of the nth root

[ornate dragon]

(b_n)^1/n

[cobalt crow]

this is what I mean, this sequence is all composed of little sequences

[mortal temple]

so what?

[cobalt crow]

then apply the sequence criteriums for cn, dn, en, fn

[mortal temple]

to what effect?

[cobalt crow]

to find what bn approaches to

[mortal temple]

and what is that?

[cobalt crow]

thats what the sequence bn approaches to as n -> +infty

[mortal temple]

and what is that value?

[cobalt crow]

I found that bn approaches zero

[mortal temple]

this leads nowhere

[mortal temple]

you want to apply this to what, a_n or b_n?

[cobalt crow]

I mean you are right, we would need to find what $\lim_{n \to +\infty} \sqrt[n]{0}$ approaches to

[rocky lotus]

Renato

[cobalt crow]

idk if we can conclude that this limit is 0

,w lim n to +infinity of (0)^(1/n)

[rocky lotus]

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[mortal temple]

we would need to find $\lim_{n\to\infty}\sqrt[n]{b_n}$

[rocky lotus]

Axe

[mortal temple]

this is not the same thing

[cobalt crow]

bn approaches 0 as n->+infty

[mortal temple]

actually this can be justified with one of the composition rules for limits

[cobalt crow]

@ornate dragon are you even here ma dude?

[mortal temple]

hmm

wait

no it can't

[ornate dragon]

yeah

i was washing

[cobalt crow]

can you read what I sent ?

[ornate dragon]

i saw it

i think it makes sense

[cobalt crow]

axe disagrees that what I am saying holds, but just wanted you to check this before I dip out, most likely you have seen something like this in your classes, no?

[ornate dragon]

you are redefining a new subsequence b_n

not really no

this is off an exam

my exam is tomorrow

[cobalt crow]

analisis matematico?

[ornate dragon]

II

[cobalt crow]

anyways, I already passed this class, but when I took it in Buenos aires, we justified it like this, maybe I am saying nonsense, but just wanted you to give you the idea, after this you would need to take the nth root of 0 and conclude that this is zero, there is a theorem for this, that any limit n to +infinity of an nth root of a constant is 0, then conclude the ratio of the limits approaches 0 and you are finished

[ornate dragon]

where'd you take it

[cobalt crow]

UBA

[ornate dragon]

that makes sense to me

ah

i'm in ITBA

alright thanks for the help

[ornate dragon]

i think this makes sense, axe

[mortal temple]

it doesn't

otherwise you could say this

$\lim_{n\to\infty}\sqrt[n]{(1/2)^n}=\lim_{n\to\infty}\sqrt[n]{0}=0$

[rocky lotus]

Axe

[mortal temple]

but the correct answer is 1/2

[ornate dragon]

right that's what i'm basically doing

i'm left with (3/8)^n^1/n

= 3/8

[cobalt crow]

true, but there was no need to use cauchy here if you already had the nth root of a power of n, but I see your point, this is why It makes sense aswell that you disagree

would be useful if you contact a friend of yours and ask him what he says, because we are running in circles here

[ornate dragon]

hold on

let me look it up

on the google drive

[cobalt crow]

yes please

[mortal temple]

you have the answer sheet?

[ornate dragon]

not for this one no

but probably someone else's solution

jesus

okay they used sandwich

[cobalt crow]

,w limit n to +infinity of ((-1/8)^n)^(1/n)

[rocky lotus]

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[cobalt crow]

I guess they didnt saw that you could divide by 8^n

nth root of (-1)^n = 1 > 0 that is true

but nth root of (-1/8)^n is always strictly lesser than 1

regardless if n is odd or even

but this class is not about getting to the correct answer, but justifying your way through, correctly

if you are more convinced with the solution that was provided by another classmate then take that

[cobalt crow]

also the exercise was this, idk why they used m at the start to denote the nth root?

[ornate dragon]

that’s just cursive

or they’re used to the cursive n

which looks like m

cursive m looks like two ns together

[cobalt crow]

m = 1/n?

[ornate dragon]

ah

i see

yeah no thats n still

i’ve seen some profs do that

[cobalt crow]

yet still is a different limit I suppose?

looks very similar yet, is not exactly the same, but we both concluded bn -> 0

like the inner sequence

is this the same exercise or not?

let me ask a friend, about this limit

dont close this channel

[mortal temple]

i think squeeze works

[cobalt crow]

lets use it

[mortal temple]

you don't even have to use the squeeze theorem to find the exact value

you can just bound it

between 0 and 1

[cobalt crow]

ok, whats your idea?

and what would be the bounds?

[mortal temple]

idk nevermind

[vale dock]

@ornate dragon Has your question been resolved?

[cobalt crow]

,align \lim_{n \to +\infty} \frac{a_{n+1}}{a_n} &= \lim_{n \to +\infty} \sqrt[n]{\frac{(-1)^n + 3^n}{(-2)^n + 8^n}} \ &= \lim_{n \to +\infty} \sqrt[n]{\frac{3^n\left( \frac{(-1)^n}{3^n} + 1\right)}{8^n\left(\frac{(-2)^n}{8^n} + 1\right)}} \ &= \lim_{n \to +\infty} \sqrt[n]{\frac{3^n\left( \left(\frac{-1}{3}\right)^n + 1\right)}{8^n\left(\left(\frac{-1}{4}\right)^n + 1\right)}} \ &= \lim_{n \to +\infty} \frac{3}{8} \cdot \sqrt[n]{\frac{\left(\frac{-1}{3}\right)^n + 1}{\left(\frac{-1}{4}\right)^n + 1}} \ &= \frac{3}{8} \cdot \lim_{n \to +\infty} \sqrt[n]{\frac{(-1)^n \cdot \frac{1}{3^n} + 1}{(-1)^n \cdot \frac{1}{4^n} + 1}}

here you would have to agree that,
0 * bounded = 0
like that, the sequence (-1)^n is bounded between 1 and -1

[rocky lotus]

Renato

[cobalt crow]

no, I think this is the best argument I could make, just that 0 * bounded = 0, where (-1)^n bounded between 1 and -1

then you get the nth root of something approaching 1/1 and that is equal to 1

so the whole limit of the ratio of an+1/an is equal to 3/8

and because 3/8 is less than 1, by dalembert we get that this sequence is approaching 0

this is what my friend did, hope this is good enough tbh

,w limit n to +infinity of (((-1/3)^n + 1)/((-1/4)^n + 1))^(1/n)

[rocky lotus]

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[cobalt crow]

@ornate dragon you here? my guy

we showed the ratio of sequences converges to 3/8

[ornate dragon]

@cobalt crow yeah

sorry

like i said, exam is tomorrow

i'm a bit sick

lots of stuff going on

i appreciate the work

[ornate dragon]

yeah, this makes sense

[cobalt crow]

good luck with the exam dude 👍 you got this

[ornate dragon]

thanks

appreciate all the help from you two

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[drowsy atlas]

how do i solve these equations without using matrices???

[drowsy atlas]

i have 3 equations and 3 variables, whaty do i do

[gaunt flint]

You can expand the left hand side to look like:
$$
\begin{bmatrix}0.390\-0.488\0.780\end{bmatrix}T_A+\begin{bmatrix}-0.518\-0.605\0.605\end{bmatrix}T_B+\begin{bmatrix}-0.512\0.768\0.384\end{bmatrix}T_c=\begin{bmatrix}0\0\500\end{bmatrix}
$$

[rocky lotus]

za

[gaunt flint]

Then you can write 3 equations separately, for example the first row would be,
$$ 0.390T_A-0.518T_B-0.512T_C=0$$

and you keep going along the rows to get the 3 equations of 3 unknowns

[rocky lotus]

za

[vale dock]

@drowsy atlas Has your question been resolved?

[drowsy atlas]

i got the 3 rows

whats next??

[gaunt flint]

then you solve the system of equation normally. Solve a variable (like T_A) and then substitute it back in to find the variables. If you can rewrite T_A in terms of T_B and T_C using equation 1, then you can plug that in the second and third equations to find the solution for T_B and T_C.

[mortal temple]

at that point you'll have a system of 2 equations. so you'll have to solve for one of the remaining variables and then do another substitution

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[graceful galleon]

I’m aware that the centroid of an equilateral triangle divides each median in a 2:1 ratio however from there I’m not sure where to go with this question

I'm also aware that i'll need to find a side then rotate it by cis(pi/3) however i need to find how to express that first

[white adder]

using complex numbers and vector geometry? lol what

[craggy rock]

........it's irritating but it sure can be done

[flint phoenix]

Have you tried giving them coordinates? It might be helpful for you to introduce algebric arppach

[white adder]

maybe set A as the origin?

[flint phoenix]

I'd set Z myself

[white adder]

that also works

does it matter tho lol

[flint phoenix]

nah 😂

[vale dock]

@graceful galleon Has your question been resolved?

[vale dock]

@graceful galleon Has your question been resolved?

[flint phoenix]

@graceful galleon Have you tried setting coordinates?

[graceful galleon]

I tried doing it

however it gets quite messy

i mean of course thats how its mean to go but it seems its convoluted to go anyuwhere

[flint phoenix]

it's alright, post everything you've done so far

We'll figure it out together

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@graceful galleon Has your question been resolved?

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[errant scarab]

I need help with proving continous function multivariable function

[errant scarab]

And im trying to prove its continouty with epsilon delta

So im trying to prove its continouty at the point P_0 =(0,0)

So let ε>0

[ivory valley]

Have you tried the squeeze theorem?

[errant scarab]

Nope

[polar sphinx]

Let me check

[errant scarab]

Im not sure if i have that theorem in my book

So i was thinking that then

I know that if i choose P_0 =(0,0)

Then d(P,P_0) < δ

[ivory valley]

Actually nvm, try some paths

What happens if ||x=0 and y->0 and vice versa||

[errant scarab]

d(P, P₀) = √(x² + y²) < δ

Oh so by limits_

[lyric sundial]

Yeah just compute the limit

There's no need of ε - δ

(luckily hahaha)

[errant scarab]

Oh the task i have is just to make sure if the function is continous at (0,0)

So i dont have to prove it

[lyric sundial]

Mmh it is not continuous though

[errant scarab]

But like if im using limit, is it just elementary maths or wha

[lyric sundial]

Remember that in multivar limits, if the limit exists then it's the same no matter what direction you approach from

[errant scarab]

By that i mean that if i plug just numbers in to the function and if the limits are the same, then its continous and if not then not, so i check if x,y -> 0,0 (from left) and x,y-> (0,0) from right and if i get its 1 then its continous at (0,0)

[errant scarab]

Oh oki

[lyric sundial]

The concept of "left and right" has no meaning in two variables limits

[polar sphinx]

I found answer

[errant scarab]

Oh so thats the difference between multivariable functions and basic functions

in limits

[lyric sundial]

Sure

[polar sphinx]

The function f(x,y) is not continuous at (0,0)

[lyric sundial]

Haven't you been taught it?

[errant scarab]

No not yet

I was given homework on continous functions but at the moment we have studied closed and open sets

[lyric sundial]

Mmmh so i wonder how you can do this exercise 😅

[polar sphinx]

I tried to prove the continuity of the function at (0,0) using the epsilon-delta definition. However, by examining the limit along the line y=mx , I found that the limit depends on m, which means the limit as (x,y) → (0,0) does not exist. Therefore, the function is not continuous at (0,0)

[errant scarab]

Oh i see

Got it

Thank you very much!

I really appreciate the help!

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[signal chasm]

quick question

how would sin45 have a pair

wouldnt sin 45 just be sin 45

oh wait nvm guys

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[velvet bramble]

Hi!

[velvet bramble]

I have a problem here

"f(x) is a function defined in all of R. If x=c is a vertical asymptote of f(x), then the function f(x) does not have an absolute maximum nor absolute minimum"

True or false?

At first, I thought that if f(x) is defined in all of R, there cant be a vertical asymptote, but chatgpt told me a case in piecewise functions where the function can be defined in the point x=c and still having a vertical asymptote, so idk

[solemn salmon]

Is this year 11?

[velvet bramble]

what do you mean

[solemn salmon]

Math 11

[velvet bramble]

I'm from Argentina this is from calculus I in college

first year

[sullen summit]

imma evaporate

[velvet bramble]

hahah

[sullen summit]

or

if u take f(x) = mod tanx

for x > 0

the x = pi/2 is an asymtote

[velvet bramble]

i mean if i have a vertical asymptote going minus infinite of course i could have a maximum absolute

[sullen summit]

and x = -1 remains a minima

[velvet bramble]

but i still dont understand how can a vertical asymptote exist if all of R is defined

[velvet bramble]

I think in the exam we are supposed to justify it with words

since we probably wont be able to test specific functions or even be able to remember them

its more a theorical question

[sullen summit]

hmm

[solemn salmon]

Huh

I think its false if theres no va to begin with

Am i tweaking

[velvet bramble]

this is

what chatgpt showed me

[light saddle]

f is not necessarily continuous

[velvet bramble]

it says that here the function is in fact defined in all of R

and there is a VA in 2

since limits tend to infinite

[light saddle]

yes but you can still define your function at x = c

like let f be the function such that f(x)=1/x^2 if x is not 0, and f(0)=0

[velvet bramble]

oh i see

so the cases where this can be real is in piecewise functions right? where we can patch that point

[light saddle]

f cannot continuous yeah

[velvet bramble]

Okay. since there is a VA, can absolute maximums and minimums exist at the same time?

I think one of them can obviously exist, since, if you have 1/x^2 if x is not 0, and f(0)=-1, x=0 could be a minima

[light saddle]

f is not bounded in a neighbourhood of c, so it can't have both

[velvet bramble]

so in this case that red point could be a local maximum but not an absolute

right?

[light saddle]

yeh

[velvet bramble]

thank you so much!!

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[floral shell]

Four cards are to be arranged to form a proper fraction such that even the number cards are arranged side by side.
Find the number of different proper fractions that can be formed.

[floral shell]

hello, im very new to this topic, so idk how i should approach this question

[vale dock]

@floral shell Has your question been resolved?

[floral shell]

<@&286206848099549185>

[dire cloud]

i dont understand the question, what does it mean that "form a proper fraction such that even the number cards are arranged side by side."?

[floral shell]

im confused as well

i think its like 26 over 79

theres only 2 even numbers

so 2 and 6 has to be besides each other in the permutation

i think thats what it meant

[flint phoenix]

Like $\frac{26}{79}$ and find the way how many fractions like this do exist?

[rocky lotus]

This is sad 😢

[floral shell]

yes i think sooo??

it has 2 conditions, the fraction has to be a proper fraction and the even numbers has to be side by side

[flint phoenix]

Is there any other restriction? Like the denominator should be greater the the numerator or something

[floral shell]

but i dont know how too

[floral shell]

yes yes

[flint phoenix]

than*

[flint phoenix]

Alright, could you rewrite the question and lay down the current information we have in hand?

[floral shell]

sure, like the conditions?

[flint phoenix]

yes

Because your original post is very confusing. It's better that we know everything well in advance so that there wouldn't be some drama when we are solving it in the midway

🧡

[floral shell]

1. even numbers from the cards has to be side by side in the fraction
2. the denominator has to be bigger than the numerator
3. the question is : four cards are to be arranged with these conditions, find the number of fractions that can be formed

the cards consists of : 1 , 2 , 5 , 6 , 7 , 9

so i know that we got 6 cards, and we have to use 4 cards, right

so i can just use 6P4

but

thats the number of total ways that u can form a fraction without considering those 2 other conditions

so thats where im stuck

[flint phoenix]

Alright, I'll pin it just in case

[floral shell]

alright

[flint phoenix]

Let's say $\frac{XY}{ZW}$

[rocky lotus]

This is sad 😢

[flint phoenix]

What do you notice in this fraction?

[floral shell]

hmmm theres 2 numbers in numerator and denominator

?

[flint phoenix]

yep, that's one of them

what else have you noticed?

[floral shell]

thats all

OH

it consists of 4 numbers??

[flint phoenix]

yes, anything else?

[floral shell]

noo

[flint phoenix]

Hint: About orders

[floral shell]

ohhh

Z has to be grater than W?

[flint phoenix]

Make another guess then we'll begin :))

[floral shell]

okeyy

the numerator has to be in increasing order?

😔

im sorry

[flint phoenix]

Nope :((

It's alright

There are two crucial points
Firstly, $X \ne Y \ne Z \ne W$
Second, Z > X

[rocky lotus]

This is sad 😢

[floral shell]

oh okey

[flint phoenix]

Do you agree?

[floral shell]

ill wirte that down

[floral shell]

it should be like that right, cus each cards consists of diff numbers,, righhhhttt

[flint phoenix]

Yes, you got it

!!

[floral shell]

YAY

[flint phoenix]

Do you know why Z should be greater than X?

[floral shell]

becauseee then it wouldnt be a proper fraction

the numerator would be greater

right

[flint phoenix]

yep

Do you have any idea how to link this with permutation from here?

[floral shell]

uhh no not really

i actually just learnt permutations a couple days ago

[hazy thorn]

dude, just show the full problem

[floral shell]

i did

[hazy thorn]

you showed the diagram but not the question

[floral shell]

oh sorry, i copied and pasted the question

[hazy thorn]

maybe it's a bad translation. That question is gibberish

[flint phoenix]

Alright

Look, so you have
$1, 2, 5, 6, 7, 9$ in the sequence

[rocky lotus]

This is sad 😢

[flint phoenix]

We have the information that Z must be greater than X, right?

@floral shell

[floral shell]

uhuh

[flint phoenix]

So in this case, Z will be on the right side of X

Is it okay so far?

[floral shell]

yess

[flint phoenix]

How many ways can Z and X be in this case?

Calcualte it for me before we move on

[floral shell]

oh wait i got confused

what do you mean right side of x

oh in the sequence you mean?

[flint phoenix]

Z will be on teh right side of x means A, B, C, D, X, Z
A, X, B, Z, C, D etc

just on its right side

[flint phoenix]

yes

[floral shell]

ohh

theres 720 ways?? is it cus 6P6

im so sorry i have to leave this question, i have to go somewhere

thank you so much @flint phoenix for helping meee

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[flint phoenix]

no worries

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[ruby ingot]

could i have a hint on how i can prove the former?

[ruby ingot]

cause i know that fir a linear map

M.(v+v) = M.u + M.v

M.(au) = aM.u

but

how can i use that in relation to a derivate

[vale dock]

@ruby ingot Has your question been resolved?

[flint umbra]

I guess P_n(the space of all polynomials of degree at most n) is in a sense a hyperplane in the subspace of P_(n+1) (the space of n+1 degree polynomials) in that it can be represented by a basis of dimension one smaller than some other vector space and the derivative maps P_n to P_n.

[ruby ingot]

whats a hyperplane?

[flint umbra]

Well hmm I got the terminology wrong

But it's this https://mathworld.wolfram.com/Hyperplane.html

Actually

[ruby ingot]

im very new to this terminology 😭

never seen derivatives w matrices

[flint umbra]

I'm not sure if I have the right sense of what it meant but I wouldn't worry about the former statement for the exercise

You don't need it for answering the latter question

[ruby ingot]

i did find what the formere answe was

i had the right idea in saying that

a + bx maps to b \

but

ohhh

its essentially saying

whats a matrix that maps (a,b) to (0,b)

thats bs wording icl

[flint umbra]

Oh I thought it tried to speak in a more general sense for all polynomials

[ruby ingot]

yes it does that asw

thata this problem

but

ill have a go first

that took too long for me to even understand the former statemtn 😭

does Uni maths wording acc get easier over time?

[flint umbra]

yeah

[ruby ingot]

lol ty for the reassurance ❤️ and the help

[flint umbra]

np

[ruby ingot]

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[midnight pier]

help

[flint phoenix]

yo yo yo

[midnight pier]

yooo

[flint phoenix]

what do you need help with today ?

[fossil mulch]

Oy oy oy

[midnight pier]

its actually the same problem

@fossil mulch not THAT problem 😂

[fossil mulch]

Thank god

[midnight pier]

when i do the dot product i get 0

0/anything = 0

[midnight pier]

so then its arccos(0)

[fossil mulch]

Wait nvm

When the dot product is 0

That just means the vectors are perpendicular

[stiff lily]

did you try pi/2 instead

[midnight pier]

i think thats the issue yea

its a formatting problem

cause when i put arccos(0) it worked

btw

whenever you solve a trig problem for theta, is the result by default in degrees or radians

when you do it by hand

not calculator

[stiff lily]

depends on what you're doing

[midnight pier]

youre solving for theta in general

[stiff lily]

calculus usually radians

introductory trig usually degrees

[fossil mulch]

If the function is using degrees the usual derivative rules for trig functions don't hold

[midnight pier]

but its the same operations

[stiff lily]

whatever units the question is presented in

or whatever they ask for

[midnight pier]

no like jjust suppose im solving for theta normally

like arccos(something) = theta or arctan(something) = theta

[stiff lily]

consider all of what I just said

[midnight pier]

wait no

[stiff lily]

radians and degrees are units

[midnight pier]

but theyre different

when you solve for an angle what is it

[stiff lily]

each has an equivalent in the other unit

[midnight pier]

it cant be two things at the same time

[fossil mulch]

They refer to the same type of quantity so I mean

Like how 1m is something many inches

Ion exactly how many

But I'm sure it's some many

[midnight pier]

theyre different

[stiff lily]

that's a calculator/ system issue

[fossil mulch]

Also in this case you are specifically asked to give degrees and radians

So unless they specify in exam which unit you should use

It really doesn't matter I guess

[midnight pier]

so i assume its radians by default

[stiff lily]

the calculator pumps out what it's programmed to

[fossil mulch]

Unless you doing calc

[midnight pier]

cause they interpreted the normal operation as radians

[stiff lily]

depends on the calculator

[fossil mulch]

Then just stick with radians

[stiff lily]

ideally you'd use the value you calculate

[midnight pier]

but the value i calculate, is it represented in degrees or radians

[stiff lily]

the calculator pumps out what it's programmed to

[midnight pier]

i mean it has to be one or the other

[midnight pier]

right but when humans do it

the value of arccos(something)

[stiff lily]

either one, up to personal preference

[midnight pier]

oh really?

[stiff lily]

both values have the same value

[midnight pier]

wait i mean they are the same value right

ohhhh

yea

its like 2 and 3-1 i guess

[stiff lily]

like 1m or 100cm

different units same value

[midnight pier]

ohh

i see

why is this wrong

the way i calculated it is like magnitude of v^2 dot u

[stiff lily]

that's not what it asks for though

[midnight pier]

380 - 280 + 5*80

[midnight pier]

wdym

oh

yeah

[gritty sinew]

(u.v)v means you take v (the vector), and you scale it by u.v

[midnight pier]

right

[gritty sinew]

result is a vector then

[midnight pier]

i did dot product on v and v there

💀

[vale dock]

@midnight pier Has your question been resolved?

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Use .reopen if this was a mistake.

[vale dock]

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[proper raft]

quick question

[proper raft]

forgot how to do these

is it you minus all exponents by the 5?

or add all exponents by the 5?

or is it you minus them from eachother

idk

[crimson yoke]

do, divide them by 5

[proper raft]

OHHH

i remember now

x15/5

y30/5

ok

yea

[stark wedge]

use the symbol ^ for exponents.

x^(15/5)

[proper raft]

i know

just asking i dont wanna be insensitive

but is that a personal preference thing?

i usually use ^

i was just speaking informally because i wanted help quickly

i do appreciate it though

[crimson yoke]

it's fine as long as it is understandable

[proper raft]

alright thanks

.close

[vale dock]

Channel closed

Closed by @proper raft

Use .reopen if this was a mistake.

[crimson yoke]

but prefer using ^ cause some people might not understand it

[proper raft]

ok thank you

x^(15/5)y^(30/5)

i believe

[crimson yoke]

yes

[proper raft]

would it come out without the ^5rad?

like uh

radx^(3)y^(6)

i think if im correct

15/5 is 3, 3 and 5. 30/5 is 6. 6 x 5.

ok

alright

.reopen

[vale dock]

✅

[proper raft]

something like this? @crimson yoke

[crimson yoke]

yep

[proper raft]

ohhh

[crimson yoke]

no no no

[proper raft]

cause the 5 on the outside divides by itself?

[crimson yoke]

mb

[proper raft]

oh

ok

[crimson yoke]

remove the sqrt sign

[proper raft]

this good?

[crimson yoke]

yea

[proper raft]

ty

👍