help-39

[vestal tapir]

there's probably a concrete reason

[urban galleon]

It depends on the context

[mint spruce]

Theres no context 😭

I feel like I gave a good example

[urban galleon]

You did lol

[vestal tapir]

that's just explaining the difference

[urban galleon]

They clicked ❌ but didn't answer any questions abt the context 😔

[vestal tapir]

the question is more like, why is it better, when it is

one more level of "why"

[urban galleon]

It is more accurate for describing the central tendency of a data set because it is not affected by outliers

[vestal tapir]

what's central tendency

[urban galleon]

Tendencies in the center 😎

[mint spruce]

It describes the middle point of a data set better because it's not influenced by bigger/smaller numbers at the ends

Which is basically what I said before

[vestal tapir]

yes this is all the same answer

why is the thing you;re trying to measure affected negatively by outliers? is it tautological?

i hope not

[urban galleon]

This is why context is important

[vestal tapir]

what

[urban galleon]

🤦‍♂️

[glacial tangle]

What??

[mint spruce]

@glacial tangle What do you mean by "better"

[glacial tangle]

Which is more useful

[mint spruce]

Well in what context are you looking

[vestal tapir]

if the question was like "why is milk better than water", you wouldn't be like this

[mint spruce]

?

[urban galleon]

Am I going crazy

[mint spruce]

Average is good when there's not a bunch of bigger numbers dragging it up

why did he open another help channel 😭

[light helm]

ideally you'd get them all,
arithmetic mean, median, quartiles etc

SD

to analyse your data

just one aspect by itself isn't that helpdul

[mint spruce]

agreed

[vestal tapir]

thanks

[mint spruce]

each of them only offers up one piece of the pie

and not in totality

[light helm]

whether the mean or median better represents your data depends on your data

[vestal tapir]

but when median is clearly better

how does it work

[light helm]

no point if op isn't here

[glacial tangle]

.close

[pearl pond]

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[glacial tangle]

You don’t have the attention.

[mint spruce]

You're the one who wasn't on

You didnt actively explain your shit

[pearl pond]

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[midnight haven]

26 − 7(3 + 5) ÷ 4 + 2

[midnight haven]

Do you distribute the 7?

Why or why not

[regal herald]

you can so long as you maintain the bracket

[midnight haven]

Do you only distribute with equations?

[regal herald]

i wouldnt say theres a single benefit to doing so here though

[midnight haven]

Ohh yeah you right. I forgot to keep the brackets

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[jagged oxide]

Hello 👋 I'm working through some integration problems and I ran across this problem. The answer key says what u-sub we're expected to use, but does not show the steps.

[jagged oxide]

The u-substitution being x * ln(x), du = (1 + ln(x))dx, dx = du/(1 + ln(x))

Not really sure what I can eliminate with that.

[sharp vigil]

to be honest that doesn't seem like the most straightforward substitution you could do

[jagged oxide]

I'm at a loss at what else I could do.

lnx perhaps?

[sharp vigil]

i would try that first

[jagged oxide]

u = ln x
du = dx/x
dx = xdu

integral of 1/lnx

uh

[glacial sequoia]

ln x is u

[jagged oxide]

oh right

that clears it up, thanks a lot!

[worthy lance]

!done

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[thick trench]

need help understanding what the natural pairing is. The dual space is just the set of homomorphisms from V to the complex numbers right?

wikipedia just says this, im not sure hwo the pairing defines a function

[solid pier]

is this from fulton

[thick trench]

oh yes it is

[solid pier]

i think i recognize the font

yeah

i remember getting frustrated at this section actually

i think the natural pairing is gonna be <v*,w>= v*(w) because v* is a map V \to C

or whatever order you want the map <,> to take arguments in lol

here fulton decides that v* comes first

seems pretty natural to me

[thick trench]

oh wait

I have seen this before somewhere in linear algebra it was called something else, yes it makes sense now

thank you for the help

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[heady radish]

200 * cost = x

dx/dt = 12

i used sint = 100/200 to find t = 1/2

differentiated the first equation

-200 * sintheta * dtheta/dt = dx/dt

i solved for dtheta/dt and got -12/100

but the right answer is -3/100

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[heady radish]

e

[pearl pond]

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[heady radish]

e

[heady radish]

is this incorrect because the length of string also changes as x changes

but using tantheta = 100 / x, the height remains constant irrespective of x?

[glacial sequoia]

yup

the amount of string let out is sqrt(100^2 + x^2)

[heady radish]

ah i see

ty

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[mental surge]

Can someone help me with this?

[jolly parrot]

neon

[glacial sequoia]

Then write the same thing in polar form and you'll find the requisite values

[mental surge]

@glacial sequoia I did sth like this

Am I on the right path?

But then I have no idea what to do next for the deduce part

[solemn seal]

Can anyone help me

[neat harbor]

#help-2

ask there

[solemn seal]

Ok thanks

[glacial sequoia]

yeah

let sqrt(A) = a + bi

then square both sides and equate the real and imaginary parts to find a and b

[mental surge]

By the way tho

[mental surge]

I wanna know the reason why we are doing this

Or is it just a rule that we follow?

[glacial sequoia]

well

the square root of a complex number is also a complex number

we don't know what it is but what we do know is that it can be written as a + bi

so we do, and then find a and b using algebra

[mental surge]

Mmmm

Ok ok

Thanks a lot

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[sleek junco]

is this correct

or shud the matrix be
1 0
0 1

y

[sharp vigil]

that would just be a different system entirely

[sleek junco]

whats d/dt x

this the quetion

[sharp vigil]

what do you get if you multiply out the matrix and vector on the right side?

[sleek junco]

y
x

oh wtf

ok got it

thx

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[vernal hinge]

can someone help me

[glacial sequoia]

Well do you know the multivariable chain rule

[vernal hinge]

yead i know

but if u look the question carefully u'll its harder than normal

ur wrong at the last

[jolly parrot]

neon

[vernal hinge]

i know what u mean

[glacial sequoia]

hmm my edit didn't go through

[vernal hinge]

but just look again the right hand side

[glacial sequoia]

I'd just differentiate both sides wrt t again and sub in theta for t

and just sub in r here

and see what happens

[vernal hinge]

not thing happened

[glacial sequoia]

substitute anything that can be substituted

i don't know if that's right though

[glacial sequoia]

the partials of x and y wrt r and theta

[vernal hinge]

brother can u write on paper and send me

[glacial sequoia]

nah don't have any on me

[fossil drum]

I think maybe you need to do something like this

and then after differntiating everything and subsituting the formula will probably pop out

[vernal hinge]

yeah probably hahha

definitely

let me try again

.clode

[fossil drum]

make sure to not forget the product rule

[pearl pond]

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[woven vale]

Can you help solve the differential equation
I attempted to solve it using substitutions, such as z=sin(y) and z=cos(y), but they did not lead to a straightforward solution. Is there a better approach to tackle this equation

[sharp smelt]

First expand and divide across by x

it should be a linear ordinary differential equation

[woven vale]

Something like that?

do I just need to integrate?

[pearl pond]

@woven vale Has your question been resolved?

[woven vale]

mb I should replace z=siny?

<@&286206848099549185>

[pearl pond]

@woven vale Has your question been resolved?

[woven vale]

😢

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[midnight haven]

The blue

Ignore my leopard print water bottle

I can't get rid of the density

It's because I'm not too sure if I did the differentiatls correctly

<@&286206848099549185>

[sharp smelt]

Have you studied errors yet

at school

if so , use the formulae you learnt there

[pearl pond]

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[dusty heart]

how do i do this

[dusty heart]

this is what i tried

maybe i could use the substitution with tg x/2

but that didnt seem to help at all

[strong relic]

kings maybe

[dusty heart]

huh?

[strong relic]

kings rule

[dusty heart]

i havent heard of that

[strong relic]

wait no that doesnt do anything

[dusty heart]

now im curious, whats the kings rule?

[glacial sequoia]

you should write sin^4 + cos^4 as 1 - 2sin^2cos^2 and divide the top and bottom by cos^4

but yeah first use kings rule to simplify the bounds

[dusty heart]

you mean by cos^2?

[regal heath]

i would see what happens if you transform cos^4x into something with sine or the other way around

[dusty heart]

oh no i see what you mean

[glacial sequoia]

cos^4

yeah

[dusty heart]

yea, but thats solving it using a trick from trigonometric equations

[regal heath]

well its not rly forbidden to do that

[dusty heart]

i feel like there has to be a more clever way

[jolly parrot]

neon

[dusty heart]

whaaaaaaaaaaaaat

im saving this

thats insane

[jolly parrot]

neon

[dusty heart]

why can you apply it twice

cuz it would be f(pi-x)

which would kinda fuck this up, no?

[glacial sequoia]

f(pi - x) is still f(x)

[dusty heart]

THATS RIGHT BECAUSE ITS TO THE 4TH POWER AND THE NEGATIVE COS IS CANCELED

this is genius

although i dont see how it helps me much

but its really interesting

[glacial sequoia]

you need to do it before doing all this

otherwise you have a bunch of points at which tan x is undefined

since we're going to be substituting that next

[storm hatch]

i dont know what that proves except that the function is symmetrical about x=pi/4

[glacial sequoia]

also working with (0, pi/2) bounds is always good with trig functions

[dusty heart]

wait so

[dusty heart]

is what i did here alr?

like, i should be continuing it from there, right?

[storm hatch]

can you notice that $f(x)=\frac{1}{\sin^{4}x+\cos^{4}x}=\frac{\sec^{4}x}{1+\tan^{4}x}=\frac{(1+\tan^{2}x)\sec^{2}x}{1+\tan^{4}x}$ which is in VERY good form for a substitution $u=\tan x$

[jolly parrot]

如月あやみ　Kisaragi Ayami

[glacial sequoia]

.

[glacial sequoia]

you need to divide by cos^6 like that

[glacial sequoia]

and maybe you should apply this one more time and make it 8 times the integral from 0 to pi/4

[regal heath]

alr this is pretty cool

[storm hatch]

but tan x is defined for all of [0,pi/2]

oh, the bound is 2pi

[glacial sequoia]

yeah

that's why you need to do this stuff first

[storm hatch]

yeah i see what you mean

[glacial sequoia]

and probably one more time because the limits will be from 0'to 1

[storm hatch]

yeah visualising the graph cutting it into 8 pieces would be the first thing i do lol

[dusty heart]

wait, i did something

[storm hatch]

not even considering the fact that it eliminates discontinuities

[dusty heart]

does this help anyhow?

[glacial sequoia]

I told you you need to divide by cos^6 if you want to do it like that

we need a sec^2 up top

and write it as 8 times the integral from 0 to pi/4

[dusty heart]

fiiiiiiine, let me redo it

ok, i did it

up until the division by cos^6

cuz it just doesnt make sense to me why you wouldnt divide it by cos^4

[storm hatch]

because in order to prepare for a tan x substitution you need to create a sec^2 x to be in the numerator

[pearl pond]

@dusty heart Has your question been resolved?

[frigid cove]

just simplify

[dusty heart]

this exercise will be the death of me

i dont get it no matter how many time the neon guy tells me there need to be a sec^2 up top

[frigid cove]

bruh

lets start from begining

[dusty heart]

goddamnit

sure

[frigid cove]

the denom can be written as 1 - 2sin^2xcos^2x

[dusty heart]

fax

whats next

[frigid cove]

simplify further

you'll get 2int{0,2pi} 1/1+cos^22x

[noble stratus]

is there a way to check for concavity if the second derivative is too complex to do ??

[dusty heart]

can you write that on paper?

matter of fact

solve the whole exercise on paper and show me if its that easy, maybe then i wont get lost in explinations

[frigid cove]

why tf would I

Im just here to give hints

[dusty heart]

ight then hint me what this means

that kinda resembles a for loop in c++

[frigid cove]

that's the integral after few simplificaitons

using simple trig identities

you then multiply by sec(2x))^2

to make a tan sub

[dusty heart]

wasnt the whole problem with that that the integral is from 0 to 2pi?

[frigid cove]

what

[dusty heart]

look back at the messages

thats what they said

[frigid cove]

okay like that

use the periodicity property and then solve

[dusty heart]

whats the periodicity property

educate me

pls

ill show you where i got in one of my attempts

[frigid cove]

the function repearts itself after pi/2

so 2 pi is just going to be 4 times that integral

[dusty heart]

this is where i got after a few attempts

in another i got here:

[dusty heart]

yea right, did it

[dusty heart]

theres a 4 before the first integral

they both got 4 before em

[frigid cove]

uhh

I don't know how you proceeded but did you understand till the integral I sent

[dusty heart]

what integral

[dusty heart]

if you mean this, then write it in such a way that i can understand it

cuz idk how you got the nominator to be 1 + cos ^22 x

[frigid cove]

multiple angle formula

[dusty heart]

right

dude, i got 4 different paths in my head for this exercise, thats why im getting things mixed up at this point

[dusty heart]

keep going

write your solution

idk what path you wanna take to solve this exercise

[frigid cove]

mutiply with sec^22x

[dusty heart]

aha

so then youd have sec^2 of 2x / (cos^2 of 2x+1) sec^2 of 2x

right?

[frigid cove]

ye

[dusty heart]

and then?

[frigid cove]

make the denom in terms on tan2x

[dusty heart]

you mean tanx right?

[frigid cove]

tan2x

[dusty heart]

its more crazy than i thought

wanna see the solution?

its in romanian, but i think you will understand it anyway

[frigid cove]

sure

[dusty heart]

i wasnt expecting infinities in there dude

this shit beats me

[frigid cove]

yeah it is there

the method I suggested was easy to begin with but complex at the end because of tan2x discontinuity

dude to discontinuity there's change of domains and shit while substituting

but the book method is fine

[dusty heart]

is there no way to do it without sec x or integral from 0 to infinity?

i feel like its way to extra

[frigid cove]

there's no problem with infinity coming in your integrals

[dusty heart]

thats some premium knowledge that i do not possess

[dusty heart]

whatever you say buddy

[frigid cove]

because you'll be having arctan

that's just gonna give pi/2

[dusty heart]

oh

damn

that makes sense

damn i just understood it

bro, ngl, whatever sec you were trying to pull

like, i didnt understand it, and it seemed hard af

[frigid cove]

I'll tell you why

when we're dealing with cosine squared or something like that in the denominator

we multiply num and denom with sec^2x

[dusty heart]

why

[frigid cove]

this is because we can write the denom in terms on tan^2x

[dusty heart]

how

[frigid cove]

because tan^2x+1 = sec^2x

simple identity

[dusty heart]

never been told that before

[frigid cove]

so now you've got the denom in terms of tan

and the numerator had sec^2x

so dy/dx of tanx is just sec^2x

makes life simpler

that's called a tan sub

[dusty heart]

i do tan x/2 substition all the time

like with the sin and cos formulas

i just never worked with sec x

[pearl pond]

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[verbal whale]

Well, first of all you have to check whether that denominator can be 0

Then, you can multiply both sides by that denominator and do the calculations

By writing z as x + iy (as they're suggesting you there...)

[sharp ivy]

z^* is conjugate right?

[bronze coyote]

yes

what do i do after i multiply both sides by denominator

[verbal whale]

This...

[bronze coyote]

i did

[cunning comet]

sort all terms. those with i on one side, those without on the other.

[bronze coyote]

i did that already

[cunning comet]

show.

[cunning comet]

if you have done this, you should have a real number on one side and an imaginary number on the other side, right?

[bronze coyote]

oh

hang on

am i supposed to replace z with x+iy from the beginning

[cunning comet]

that was suggested.

[bronze coyote]

i did that for my first step

i don't get where i messed up

[cunning comet]

after writing as x+iy and after multiplying by the denominator you should have:
2x+2iy = i(3-x+iy)

now simplify and sort all terms. those with i on one side, those without on the other.

[bronze coyote]

oh ok

wiat

isn't that what i did

[cunning comet]

you said this, i just want to help you to check it.

[cunning comet]

but anyway, go on with this.

[bronze coyote]

sry for the confusion, i thought i did something wrong

[bronze coyote]

so the side with real numbers is x and the side with imaginary numbers are y?

[cunning comet]

if you have this, you did something wrong.

[bronze coyote]

😭

[cunning comet]

iz should by something like 2x+y on one side and i(3-x-2y) on the other.

[bronze coyote]

i have that

[cunning comet]

so your real number is?

[bronze coyote]

but how can i find value of x and y

[cunning comet]

we come to this soon. please answer: your real number is?

[bronze coyote]

2x+y

[cunning comet]

and your imaginary number is?

[bronze coyote]

3-x-2y

[cunning comet]

exacty it is i(3-x-2y), not 3-x-2y

[bronze coyote]

ok

[cunning comet]

now the most important question: how can a real number equals an imaginary number?

[bronze coyote]

multiply by i

wait no

it can't

[cunning comet]

there is one case, where it can.

[bronze coyote]

when its 0?

[cunning comet]

exactly.

[bronze coyote]

so i get a simultaneous equation of x and y

[cunning comet]

yes

[bronze coyote]

2x+y =0 and 3-x-2y=0

[mental vine]

um what does z* mean? y+xi?

[bronze coyote]

x-iy

[spare lark]

Conjugate complex

[mental vine]

oh ok

is z* means x-iy

nvm yall fixed that alr

oh wait it is alr solved

[bronze coyote]

ye

[mental vine]

guess I was late XD

[bronze coyote]

@cunning comet thank u so much for ur help

.close

[pearl pond]

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[deep socket]

Hi,
PI_1 : 5x-2y-z=-2

[deep socket]

I wanted to understand why i cannot find the normal to pi_2 by doing the cross product of
(1 4 -3) x (5 -2 -1)

i thought that the line of intersection is paralllel to the normal of the two normals of the planes

[pearl pond]

@deep socket Has your question been resolved?

[pearl pond]

@deep socket Has your question been resolved?

[pearl pond]

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[void gyro]

i wanna solve both of these equations separately. i tried doing the first one and i replaced the root of x^2-x-42 with t and got -11t = t^2 but idk what to do from there

[void gyro]

i think the second one must be solved in a similar way

[unkempt yacht]

!xy

[pearl pond]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[errant fable]

t^2+11t=0
t(t+11)=0
t=0 or t=-11

matter of factorizing

[void gyro]

but it cant be -11 since its a square root right

so after that i should use the quadratic formula and find the x1 and x2

[errant fable]

yup

[errant fable]

yup

[void gyro]

x1 = -6 , x2 = 7

so those are the 2 solutions to the first problem?

what about the second one, how can i factorize it?

[errant fable]

try the same way

but first add 20 to both sides

[void gyro]

but under the root its +20 and on the other side its +10

hmm

i got 3t = t^2-10 by adding 10 to both sides

so i make a quadratic equation out of that?

[errant fable]

so you got t^2-3t-10=0

yeah

first solve for t

then substitute back in

and solve again

[void gyro]

these are the ts it gives me

oh no wait

i messed it up

its -2 and 5

my bad

so only 5 is valid

and then solve for t = 5

[errant fable]

ye

[void gyro]

alright thank you so much

happy holidays

.close

[pearl pond]

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[void gyro]

.reopen

[pearl pond]

✅

[void gyro]

im sorry for reopening but i got another question

after finding that t=5

and we replace t with sqrt of x^2 - 4x + 20

[errant fable]

yes

[void gyro]

the equation is sqrt of x^2 - 4x + 20 = 5 right

[errant fable]

yes

[void gyro]

so after that we should do both sides by ^2 and it equals 25

[errant fable]

yes

[void gyro]

so then its x^2 - 4x - 5

not +20

[errant fable]

x^2-4x+20=25

x^2-4x-5=0

[void gyro]

yea

[errant fable]

(x-5)(x+1)=0

[void gyro]

alright thank you once again

.close

[pearl pond]

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[errant fable]

👍

[pearl pond]

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[midnight haven]

एक घन को एक गोलार्ध में इस प्रकार अंकित किया जाता है कि घन के ऊपरी चार शीर्ष गोलार्ध की समतल सतह को छूते हैं और अन्य चार शीर्ष गोलार्ध की घुमावदार सतह को छूते हैं। यदि गोलार्ध की त्रिज्या 1 है, तो अंकित घन का आयतन ज्ञात कीजिये।

[midnight haven]

मुझे नहीं पता कि कहां से शुरू करूं.

!status

[pearl pond]

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

[ember spear]

A cube is inscribed in a hemisphere such that the upper four vertices of the cube touch the flat surface of the hemisphere, and the other four vertices touch the curved surface of the hemisphere. If the radius of the hemisphere is 1, find the volume of the inscribed cube. This is the correct translation right?

[midnight haven]

हां मुझे ऐसा लगता है।

[ember spear]

So, cube is inscribed in hemisphere and we have to find its volume

[midnight haven]

बिलकुल सही

मुझे नहीं पता कि वृत्त की त्रिज्या को घन की भुजा की लंबाई से कैसे जोड़ा जाए।

[ember spear]

Thanks to symmetry we shall only consider a cross section along the diagonal

[midnight haven]

बहुत अच्छा

ये तो मुझे अब समझ आ गया है. आपकी सहायता के लिए धन्यवाद.

[ember spear]

agar bhuja ki lambai x ho to vikran ki lambai kya hogi?

[midnight haven]

?

[ember spear]

sorry my hindi is bad

Did you get the problem?

[midnight haven]

यदि घन की भुजा की लंबाई x है, तो त्रिज्या होगी (जैसा कि मैंने गणना की) x * वर्गमूल(3/2)

Basically the volume of the cube will be (2/3)(root(2/3))

No wait

Is this correct?

[glacial sequoia]

yeah

[midnight haven]

thanks

🙂

.clsoe

.clsoe

.clsoe

.close

[pearl pond]

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[ember spear]

How do you type in Hindi???

[pearl pond]

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[midnight haven]

.by the keyboard

.it isn't hard

[north talon]

Can anyone guide me through the solution of this question?

[north talon]

Consider a 25x25 rectangular grid where each side is parallel to the x-axis and y-axis. In each row and each column, there is exactly one cell colored red. Find the largest positive integer 𝑛 such that there exists a polynomial 𝑃 with real coefficients whose graph 𝑦 = 𝑃(𝑥) passes through 𝑛 points on the grid but does not pass through any red cells. (We define that the graph passes through a cell if there exists a point on the graph that lies on the edges or the corner of the cell.)

the position of the grid is not given btw

[pearl pond]

@north talon Has your question been resolved?

[north talon]

<@&286206848099549185>

[obtuse tangle]

Where is your attempt

[north talon]

i answered 600

[obtuse tangle]

Show your working so that we can help ur

[north talon]

where all the red squares are on the top

wait that doesnt work

[north talon]

im not sure where to start

but the answer is probably 600 cause sin(x) expansion right?

[obtuse tangle]

the is 600 but what is sinx doing here

ans is 600

[north talon]

taylor expansion of sinx has real coefficients

[obtuse tangle]

So what

[obtuse tangle]

Just go and evaluate your life if you do not need help and stop waiting our time

[north talon]

what?? are youe evn helping

[north talon]

we can make a polynomial that looks kinda like this

hold on

why is that so gpt looking (sorry if it isnt)

[obtuse tangle]

They did not specify where the red are located

[north talon]

they didnt

but its the maximum

so this is possible (i think)

[obtuse tangle]

they said in each row and column the exist only 1 red read the question carefully

[north talon]

oh dang

[obtuse tangle]

so the reds form a diagonal

[north talon]

not neccesarily

[north talon]

mb

[obtuse tangle]

?

Bye i will see ur later

[north talon]

oh do you have to go

bye

[obtuse tangle]

No

[obtuse tangle]

No

[north talon]

?

i think im starting to find something?

[obtuse tangle]

do you understand now

[north talon]

in every column P(x) can either go over or under the red square

[north talon]

this is also possible btw

[north talon]

i dont get what youre saying

[obtuse tangle]

Let me draw the 25x25

are you there

[north talon]

yes

i was thinking if the reds were in this shape

then P(x) could pass through most of the squares

maybe im over complicating

[obtuse tangle]

i will show you something

give me a second

Note: The must only be 1 red in any column and any row

[obtuse tangle]

Note: The must only be 1 red in any column and any row

[north talon]

yeah?

that satisfies

[obtuse tangle]

You still get 600

you can use any method you want

i am. Going bye for real

.close

[north talon]

how do you get 600?

excuse me😭

[obtuse tangle]

This

[north talon]

okay then if n is 600 there then it would have to hit every non red tile which isnt possible in that img

if it crosses corners by definition it hits the red tile

[obtuse tangle]

It must form a diagonal

[north talon]

it doesnt have to

[obtuse tangle]

Why

[north talon]

this works

[obtuse tangle]

We wont have a maximum

then

[north talon]

how do you know its a maximum?

im confused

[obtuse tangle]

The way you drawn it where will a line pasd

pass

[brave sluice]

the polynomial can't pass both above and below a red square

[north talon]

yeah thats what im saying bro😭

[brave sluice]

did you get an answer?

[north talon]

no

well i have one answer but idk if its correct

[north talon]

if i draw on mine itll be super messy

[north talon]

my ans is from this

[brave sluice]

456?

i got 456

[north talon]

wait gimme a sec

i got 481

[obtuse tangle]

I got 600

[north talon]

how?

[obtuse tangle]

let me run it on python

[brave sluice]

it should achieve a max by passing through all the blue cells

[north talon]

black lines are the ones that P(x) doesnt pass

[north talon]

interesting

[brave sluice]

it should give the same answer as yours

but i counted 456 not 481

[north talon]

wait youre right

chat i think i did addition wrong😓

[brave sluice]

$\sum_{n=1}^{25} \max(n-1,25-n)$

[jolly parrot]

Axe

[north talon]

whys that equivalent?

or is it because we're choosing either above or below the red tile

[brave sluice]

yeah exactly

choose whichever is bigger

[north talon]

ahh

thats so cool

thanks! :3

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[midnight haven]

Sorry in advance if this question is a bit stupid

[midnight haven]

$$\gamma =\frac{L_{(A, A)}}{L_{(A, B)}} = \frac{L_{(B, B)}}{L_{(B, A)}}$$

[jolly parrot]

Edmund Cloudsley

[midnight haven]

how do I find an expression for gamm in terms of L_(A, A) and L_(B, A)

nvm I see i thanks

.close

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[hoary nacelle]

Help

[hoary nacelle]

I have a paradox

Wouldn’t this break sequential limit test?

@rough forge @prime bramble

Mighty mathematician help me

[dawn dirge]

The limit doesn't exist, so I don't think you can do algebra with it

[rough forge]

How do you get there?

[hoary nacelle]

It’s just to lead contradiction

[hoary nacelle]

From the other limit manipulation

[rough forge]

I mean how do you conclude the limit is 0

[dawn dirge]

x = -x is the idea there

[dawn dirge]

But I don't think you can just do DNE = DNE and proceed

[hoary nacelle]

I just thought a proof so I wrote it

[dawn dirge]

And what are you trying to contradict? What are the assumptions?

[hoary nacelle]

The contradiction is that the limit exists

[rough forge]

Just because you made a change of variables doesn't mean the limit is 0

[hoary nacelle]

But since sin^2x + cos^2x =1 it contradicts

But the function defined if we let x in N

Then it has a limit of 0

[hoary nacelle]

I didn’t change variable though

[rough forge]

oh

[hoary nacelle]

It’s just manipulation I can do it with x but less neat t

[rough forge]

You wrote it in the end x in N

[hoary nacelle]

Yes

If x is any real number it doesn’t exist but if it’s a natural number it does

[rough forge]

but why are you saying then DNE in the beginning if x is natural?

oh I get what you did lool

[hoary nacelle]

Shorthand version claim for the sake of contradiction

Ya mathematician has a thing called conciseness right?

[rough forge]

I am still confused by your contradiction lol

[hoary nacelle]

That proof is rigorous btw

I used it for a math competition when I was at high school

[dawn dirge]

For $n \in \mathbb{N}$, the first manipulation line is saying $\operatorname{lim}_{n \to \infty} \sin(\pi n + \pi) = - \operatorname{lim}_{n \to \infty} \sin(\pi n),$ which is just 0 = 0, right? How exactly are we saying $\sin^2(x) + \cos^2(x) = 1$ is contradicted?

[jolly parrot]

elliot

[hoary nacelle]

The point is, when x is natural number in this case the function actually has a limit whereas x is a real number it doesn’t have limit

I will refine it then

[rough forge]

Here x is real, so is y. But how do you jump to the conclusion the limits are 0?

[dawn dirge]

For f(x) = sin(\pi x) that is true

[hoary nacelle]

Just wait

[dawn dirge]

But consider g(x) = cos(\pi x), this does not have limit 0, the arguments need to be the same to apply the trig identity. I think it is weird to do an identity like this as a limit, particularly when the limits in R do not exist

[hoary nacelle]

It’s for the sake of contradiction though

And one assumption without relying on any existing theorem lemma or corollaries

[hoary nacelle]

Here

[rough forge]

ok i was really dumb

[hoary nacelle]

But that step doesn’t matter

What I care is

X is in R

[rough forge]

but hmm

[hoary nacelle]

F has no limit

But if x in N

The f has a limit

[rough forge]

The thing I am concerned with is you are assuming both limits are equal

[hoary nacelle]

Isn’t that very weird

[hoary nacelle]

It was for contradiction though because saying sinx s limit doesn’t exist is quite hard

[dawn dirge]

That's very possible, especially when you have periodic oscillations like this

[hoary nacelle]

x=-X what’s x?

[hoary nacelle]

It’s not contradicting to sequential test right?

So sequential test is like whenever the yielded value differs it proves that the limit doesn’t exist

[dawn dirge]

A function f:R \to R is not a sequence, a function g:N \to R is

[hoary nacelle]

True still I just feel a bit weird

But it can the case that a function has no limit but using a sequence can lead it to a limit?

[dawn dirge]

Yes, maybe its like including all of R can add too much info that breaks the nice countable and discrete behaviour you get with N

There's a whole study of pointwise convergence of functions, which is when you have a sequence f_n: R \to R and look at the values for some x \in R

[hoary nacelle]

That’s how it is… I was always confused with this example though

[rough forge]

Now thinking about it I accept it

[hoary nacelle]

I genuinely get a excellent mark for competition at high school with that proof which isn’t my original I think but I don’t recall where I got it

[rough forge]

yea no doubt it lol

i just had to understand

[dawn dirge]

What does the question ask to prove?

That limit DNE?

Ah okay I think I see what you're trying to do now

[hoary nacelle]

There’s no question it’s just I am exploring reasons why x in N limit exist and x in R it doesn’t

And to make it concrete I prove that X in r it doesn’t have limit

[dawn dirge]

But when you say "that proof" what are you referring to?

proof of what?

[hoary nacelle]

The limit of sinx doesn’t exist

X in R

[dawn dirge]

Yeah okay and then I accept your argument, I just didn't realise at the beginning we assumed it had a limit

[hoary nacelle]

I like mathematician like Sheldon so I learned to be super concise but I think I have done it in an awful way 😭

[dawn dirge]

You have exploited periodicity to force the limit to be 0, but you could have very well forced it to be any value in [-1, 1] by the same logic

[hoary nacelle]

Yes

Indeed but it doesn’t stop me forming valid contractions

Lim (sin^2x + cos^2x)=0 which is clearly contradiction

[dawn dirge]

Well what does it mean to have limit 0 rigorously, could you contradict this from knowledge about sin(x)?

[hoary nacelle]

This is assumed for the same of contradiction

[hoary nacelle]

Whenever x-c is within delta neighborhood then it follows sinx - sinc is within epsilon neighborhood

No I said x in N it had limit 0

Not when x in R I think you misunderstood

[dawn dirge]

I thought the goal was to show $\operatorname{lim}_{x \to \infty} \sin(x)$ does not exist

[jolly parrot]

elliot

[hoary nacelle]

I assumed the limit for x in R exist then I started the manipulation, with which I showed that sin^2x + cos^2x =0 contradiction

[hoary nacelle]

No I was just asking why when x in N it has 0 as limit but when x in R it doesn’t

That’s it

I never said sinx x to infinity sinx=0

You

You see here lim at the end of the proof

It doesn’t have lim x to infinity

It’s lim

So it’s for natural number

I just wanted to discuss why x in N f has limit and x in R it doesn’t

[dawn dirge]

Well it's essentially because sequences are discrete objects which behave very differently to continuous functions on R. There's a sort of inclusion going on also from N to R

[hoary nacelle]

Now I kinda understand

Bcs I am not studying mathematics as my major so these logics are hard

Btw that proof for nonexistence of limit is super elegant, it relies no existing theorem, lemma or corollaries. It made only one assumption and led to the contradiction

[dawn dirge]

What is that assumption?

[hoary nacelle]

The limit x to infinity sinx exist and let it be L

Didn’t write it there I have bad habit of omitting great deal of detail for copying mathematicians wrong

[dawn dirge]

Yeah I agree it's nice. My instinct after saying L=0 necessarily would be to essentially argue that sin(x) = 1 for arbitrarily large x, which would contradict that convergence

[hoary nacelle]

I got it when I was at middle school

From one senior I think

And most importantly this proof is super short, in a way that it relies nothing than basic calculation

.solved

[pearl pond]

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[rough forge]

https://cdn.discordapp.com/attachments/1080555115066437686/1275095275140026459/1112374071854891058.gif

[pearl pond]

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[hoary nacelle]

Help

[hoary nacelle]

I need someone to verify if I find the correct pointwise limit for the sequence of function

Here’s my solution

[midnight haven]

What grade lvl is that??

Srry im just curious

[hoary nacelle]

The question grade level? It’s an elementary analysis proof

[midnight haven]

Ah

[hoary nacelle]

Help me

As a Econ student I am sure I had made a lot of formatting mistakes so please help me to point out them and also help me with the refinement

[hoary nacelle]

I wouldn’t say its elementary though but it’s optional for economic, engineering, and humanities majors so it’s not very hard either

[rough forge]

Hav you shown the critical point is indeed a max?

[midnight haven]

So its like uni level

[hoary nacelle]

Hell I made that mistake again😭😭😭

[rough forge]

haha

It's not a mistake if you haven't taken action yet

[hoary nacelle]

This time look at pink part

[rough forge]

O

I can hardly read it

[hoary nacelle]

Here you go

It shows for all x in the interval it converges to 0

And even this fixes the issues I must really remember not to make that mistake again

a=x^2

[hoary nacelle]

Yes but for most people it’s freshmen course

[rough forge]

btw if |g| < 1 ln(g) is not defined for g <= 0

[hoary nacelle]

q is defined on (0,1)

I think you mean q right?

[rough forge]

g or q what that is

|q| < 1 means q is on (-1,1)

[hoary nacelle]

Yes that’s a point

I should have fixed it I am fixing it actually

Done

I mean I am bad at math so those stupid mistakes almost seem inevitable for me which made me sad

[rough forge]

yap yap yap

[midnight haven]

Oh

I dont think im ready for it lol

[hoary nacelle]

I really am… especially when I was at middle school

[rough forge]

Why did you substitute a = x²?

[hoary nacelle]

Only to make algebra neat

And since it’s in the interval it doesn’t matter

X in (0,1) so lnx is always negative

[rough forge]

I don't think it's true that the limit is 0 for [0,1]

[hoary nacelle]

Only for a sequence x_n = that long expression

[hoary nacelle]

I think it’s called moving sequence or something?

so it’s always 0 for any fixed but it diverges when x takes the diagonal sequence

x_n should be called a diagonal sequence if my memory serves

[rough forge]

wait i think you are right

just looked at my notes too

[hoary nacelle]

There’s some introduction of this on rudins

And since my school incorporates rudins principle of mathematical analysis as text book for RA it’ll be my textbook too soon

[rough forge]

ok

[hoary nacelle]

How about other parts

[rough forge]

e^(-1/2) i think instead of e^(1/2)

[hoary nacelle]

Or it should be called limit superior

[hoary nacelle]

Yes you’re correct

I will fix it now

[rough forge]

Also what is that next part? Do you wanna show that it's not uniformly convergent?

[hoary nacelle]

Find the interval where it converge uniformly

I haven’t had any clue

Yet

Fixed

[rough forge]

Well you shown that it doesn't converge uniformly on [0,1]

[hoary nacelle]

I need to determine sub interval of [0,1] if there can be any that uniform convergence holds

It’s on rudins book

[rough forge]

For x > 1 you could show that it doesn't converge point.wise

[hoary nacelle]

Yes you’re

Right

Huh this question is weird then

[rough forge]

You would basically have a product of numbers greater than 1 which would go infinity

[hoary nacelle]

It asks me the pointwise limit if I were to find all then I solved the third question already

And maybe that diagonal sequence is to be found at last but I found it before

On rudin s book, I don’t think this is hardest

I have one involving factorial

Which is making me so clueless

And also arctan(nx)

[rough forge]

(uniform convergent => pointwise convergent) <=> (not pointwise convergent => not uniform convergent)

[hoary nacelle]

I have Screenshot it

[rough forge]

cool

[hoary nacelle]

I need to master all exercise on rudins so that I won’t drop out from RA

[rough forge]

That was my very first assignment exercise last semester

on D = R

[hoary nacelle]

Will final for freshmen level analysis course include these question

D=R?

[rough forge]

If you drop out I will lose my mind

[rough forge]

domain all real numbers

[hoary nacelle]

Is that a part of real analysis

[rough forge]

funny thing is i never did this part as rigorous as you

[rough forge]

haha i meant arctan(nx) was defined for x on (-oo,oo) in my assignment

[hoary nacelle]

I want to show that it converges uniformly for all restricted domain (or equivalent a compact set)

[hoary nacelle]

And this one has been troubling me for a while

[rough forge]

haha no

don't assume things to be true yet

[hoary nacelle]

It’s on rudins so it has to be true just how it can be proven

[rough forge]

first always make sure that some conditions are met before you prove something that appears to be true

arctan(nx) is uniform convergent??

[hoary nacelle]

Yes on restricted domain

[rough forge]

ah then yea

lol

[hoary nacelle]

[-M, M]

[rough forge]

and M?

[hoary nacelle]

Arbitrary

[rough forge]

hmm

[hoary nacelle]

Any hint?

I have been on this for a week actually

[rough forge]

What I have in my notes is that the limit function wouldn't be continuous

If you consider the limits on [-M,0), x = 0 and (0, M]

You know arctan(x) right?

[hoary nacelle]

Yes

I am graphing the family

Yes if it’s 0 then it’s weird

Or actually it’s only 0

[rough forge]

if x = 0 yes

trivially

[hoary nacelle]

Yes it’s investigate

I just read it too after graphing it

[rough forge]

so you see it will have a jump in the origin

[hoary nacelle]

Investigate and show uniform convergence of arctan(nx) with respective domain

[rough forge]

limit as n->oo will be pi/2 and as n -> -oo will be -pi/2

[rough forge]

You said on [-M, M]?

[hoary nacelle]

Investigate it for whether it converges uniformly on [-M, M]

[rough forge]

well unless M = 0 it can't

[hoary nacelle]

Can it be

[-M, 0-delta] unions [delta, M]

That it converges uniformly on both sets respectively

[rough forge]

if delta > 0 yes

respectively is the key word

[hoary nacelle]

Then it should be the answer

Still it’s arctanx

I don’t even know Bernoulli number how the fuck can I even expand it

I am definitely going to fail

And this another insane thing sum x^k/k! and prove its uniform convergence interval

[rough forge]

I love how a non-math student does stuff more rigorous than I have done before 🥰

[hoary nacelle]

Not I am not rigorous

[rough forge]

😂

[hoary nacelle]

I made too many mistakes

[rough forge]

🫵🏻 ?

[hoary nacelle]

And I haven’t mastered rudin after 2 months

Like I spend 2 hours more each day

[rough forge]

maybe rudin is no that rudimentary

after all

[hoary nacelle]

Especially those questions with larger index number

And those inequality proof is insane

[rough forge]

You talking about proving or arguing the limit?

[hoary nacelle]

Show that |a-b|^alpha\geq than |a^alpha -b^alpha|

[hoary nacelle]

For that convergence I must do expansion right?

[rough forge]

Show that $|a-b|^{\alpha} \geq |a^{\alpha} -b^{\alpha}|.$

[jolly parrot]

𝔸dωn𝓲²s

[hoary nacelle]

This inequality after the introduction of holder inequality is insanely complicated

Alpha is in (0,1)

Also on that book

Even after expansion I still don’t have clue

Like I don’t know what to argue

Alpha is in (0,1)

I don’t know you expand it and I don’t even have a clue for that direction

[rough forge]

Is that inequality from a book?

[hoary nacelle]

I was doing through construction function

And use concave property to show it

Yes

For rudin s

After the chapter where holder’s inequality is introduced

My thought is that concave function might hint the result though but manipulation is somehow too complicated

Oh maybe it’s not

I actually think of a new way using the same argument

Iam on it now

[rough forge]

,, \abs{x^n-b^n} \leq \abs{x-y}^n

[jolly parrot]

𝔸dωn𝓲²s

[hoary nacelle]

Do you think the concave property is the way?

[rough forge]

I never even have heard of this before

Guess I haven taken real analysis properly yet

[rough forge]

yae

[hoary nacelle]

If I solve this question then I solved 2/3 of all questions related to inequality on the book

From rudins

I am working on it marry Christmas again

[rough forge]

If you apply the concave property are you defining a function?

[hoary nacelle]

Not necessarily but a parameter to formalize the proof I think

In an abstract way yes

A general function I guess

[rough forge]

Since the domain (0,1)

We could take f(x) = x^n

[hoary nacelle]

Last time it was a failure because I was too specific I guess

Yes

That’s my current thought

[rough forge]

f need to be something that satisfies that concave property

[hoary nacelle]

I was originally doing a specific function construction which turned out to be dumb

[rough forge]

I think x^n is the simplest for that matter

[hoary nacelle]

This must be a reason it follows the part of concave and convex function and holders so I will think this direction

[rough forge]

f''(x) = n(n-1)x^(n-2)

[hoary nacelle]

Don’t spoil I must finish my own

Or I will fail real analysis

😭😭😭

[rough forge]

You spoil me

[hoary nacelle]

I can’t even master the stupid textbook for 3 months almost like I was crying almost

[rough forge]

don't worry i wouldnt bee able to prove half this stuff