help-17

[queen mountain]

Hold on guys give me one second it’s 12 am let me do something real fast

[fallow gull]

I’ll be munching awn my noodles then 😍😍

[queen mountain]

😂

[fallow gull]

waiting peacefully

@queen mountain

[queen mountain]

Sorry yall js doing something wit my parent real quick shouldn’t take much longer

[fallow gull]

okay back to this

do u see any other like terms that u can add together

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@queen mountain Has your question been resolved?

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[random heath]

Can anyone help me with rational expressions?

[random heath]

I was absent on the day it was taught

now I'm behind my class

😞

[spiral turtle]

The rules are a^x a^y = a^(x+y) and (a^x)^y = a^(xy)

And 1/a^x = a^(-x)

[random heath]

Yes, but how will I use them step by step on the expressions

[tawny sorrel]

It’s the same thing you’re just adding fractions instead of whole numbers

[random heath]

Ohhh

So for example in number one I will just add the exponents?

[brittle cipher]

yup

[random heath]

then it will become 0

[brittle cipher]

yeah exactly

[random heath]

so the answer is 1

[brittle cipher]

perfect!

[random heath]

Thank you so much I understand it now

[brittle cipher]

you're welcome, although I didn't really do most of the work here haha

[random heath]

hehe thank you for those who helped me

[spiral turtle]

!done

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

[spiral turtle]

(if not we're still happy to help)

[vocal sleet]

@random heath Has your question been resolved?

[random heath]

oh sorry

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[random heath]

.done

😌

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[low haven]

is it true that $\abs{\sin{b}} + \abs{\cos{(a+b)}} \geq \abs{\cos{a}}, \forall a,b \in \mathbb{R}$? if so, how can i prove it?

[twin meteor]

mod numero esperto

[bronze osprey]

https://www.desmos.com/calculator/dwejr5dhcj
interesting, the statement appears to be true

now note that the period of the left-hand side is pi

differentiating this will be a real mess in terms of the sign choices

there should be another way

probably reverse triangle inequality on $|\cos a \cos b - \sin a \sin b|$

[twin meteor]

south, just south

[vocal sleet]

@low haven Has your question been resolved?

[loud walrus]

@low haven use trigonometric identities

cos(a) = cos(a+b-b)

Use this hint

[low haven]

nice, thank you, man

.close

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[near hearth]

how do you determine if each curve is an orientation-respecting or -reversing parametrization

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@near hearth Has your question been resolved?

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[outer warren]

you added 208 and 78 instead of subtracting

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[outer warren]

solutions aren't necessarily nice,
so not a real issue if it doesn't divide exactly
just make sure the work leading up to it is correct

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[cobalt ocean]

why would they treat e^x as v when there's no e^x in the expression

only e^-x

[dim pumice]

both works I guess, whether its u or v

[cobalt ocean]

no

e^-x = 1/e^x

so it would be worse

[hard atlas]

you can also do it the other way

it basically changes only a few factors

do it as an exercise

[cobalt ocean]

you can either have e^-x as u and cos(nx) as v, or e^-x as v and cos(nx) as u, but you shouldn't be doing anything with e^x

[hard atlas]

check signs

I fucked up, sry

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[vocal sleet]

@lean merlin Has your question been resolved?

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[tight apex]

does anyone help my probability and stastics question?

i found 0.038

but its not correct

i made C (28,0) * (0.11)^0 * (0.89)^28

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@tight apex Has your question been resolved?

[hexed needle]

probability that none are independent = 1 - probability that all are independent

[peak matrix]

Sus

[peak matrix]

this seems correct I think

[hexed needle]

why is that sus lol

oh wait a sec

[peak matrix]

Why should it be true?

[hexed needle]

lmao I just woke up, you're right

I shouldn't do math this early, thanks

[peak matrix]

How do you know it's not correct

[hexed needle]

I concur that it should be (0.89)^28

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[tight apex]

my homework site says it is incorrect

[tight apex]

.reopen

[peak matrix]

try rounding it less maybe

[tight apex]

to which number

[peak matrix]

or entering it as (0.89)^28 if it allows it

[tight apex]

let me try it

yeah its done

thank you so much!

@peak matrix KING!

.close

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[cunning olive]

Can someone guide me through this question please

[meager shoal]

What’s the formula

For compound interest

[cunning olive]

A=P(1+r)^n OR A=P(1+(r/x)^nx )

I attempted it the best i could

but i think i may have gotten the x and n values mixed up

<@&286206848099549185>

[vocal sleet]

@cunning olive Has your question been resolved?

[cunning olive]

<@&286206848099549185>

[marsh elk]

hey

I can try to help you I believe

[cunning olive]

If you can, please :)

[cunning olive]

I have 10 of each style question but dont want to attempt the other questions till i know i got the method and calculations right the first time

[marsh elk]

to apply an interest rate of $r$, you multiply a value by $1 + r$

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[marsh elk]

so say you have 1 dollar, and you have an interest rate of 100%, then $r = \frac {100}{100} = 1$

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[marsh elk]

so at the end of the year, you will have $1 * (1 + r) = 2$ dollars

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[marsh elk]

now say this interest rate is applied bi-annually, meaning twice a year

[cunning olive]

we would replace it with 2?

r*

[marsh elk]

I have just one question, is this your correction, or is it the book's or teacher's ?

[cunning olive]

i dont understand your question

[marsh elk]

the picture you sent is a question (in red), followed by what seems to be answers

[cunning olive]

highlighted in red is the question, black is my attempt

[marsh elk]

it's your attempt

ok

[cunning olive]

i have difficulty differentiating the question lines from my lines

[marsh elk]

do you have a clear definition of what "compound" means ?

I am french so I'm not sure it means the same thing for me and you

[cunning olive]

somewhere in my written book yeah

let me quickly find it

interest calculated on both the initial amount and all previously generated interests

[marsh elk]

ok

If my understanding is correct, then you are right

https://en.wikipedia.org/wiki/Compound_interest

this means the interest rate is applied nth times

[cunning olive]

ahh ok, i see how we get the formula now

[marsh elk]

in this context, since there is one dollar, and the duration is a year, the formula is indeed :
$$(1 + \frac 1 n)^n$$

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[marsh elk]

which is a sequence that famously converges to $e$

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[cunning olive]

eulers number

[marsh elk]

yes

https://en.wikipedia.org/wiki/E_(mathematical_constant)

[cunning olive]

If we increase the compounding indefinitely, the value will stop at 2.718281828?

[marsh elk]

It will converge towards $e$, which is approximately that, yes

[twin meteor]

Grui ʕ•͈ⱉ•͈ʔ

[marsh elk]

there is a reason for that, but we would have to dive into a little bit of calculus

(it is because e^x is its own derivative)

[cunning olive]

i cant thank you enough Grui

[marsh elk]

aw

np

[cunning olive]

i have been procrastinating and thinking im stupid for hours, i did the original question what mustve been 4-6 times, got a different answer each time

:)

[marsh elk]

I'm very much familiar with that feeling haha

[cunning olive]

Merci? if i spelt that right

[marsh elk]

you did haha

De rien 😉 (you're welcome)

[cunning olive]

my french is only Bonjour, and merci, thats all i know now :( 😢

[marsh elk]

tha'ts alright haha

[cunning olive]

.close

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[hidden gyro]

How do you sythetically prove that the isogonal conjugate of a point exists? (so no trig ceva)

[vocal sleet]

@hidden gyro Has your question been resolved?

[hidden gyro]

no

<@&286206848099549185>

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[wispy harness]

i need help with this

[flat whale]

!show

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Show your work, and if possible, explain where you are stuck.

[wispy harness]

nevermind got it

.close

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[vocal sleet]

Please don't occupy multiple help channels.

[flat whale]

you're better off translating

[tropic dove]

yeah I was about to do so

[maiden iron]

t'es à quel question

[tropic dove]

o lala

2 b

[flat whale]

oh a french herels has appeared

[tropic dove]

j'utilise le TAF pour montrer f'(a) = 0

mais je sais pas comment montrer la deuxieme

[maiden iron]

TAF c'est theoreme des accroissements finis c'est ça ?

[tropic dove]

oui

sais pas comment faire la deuxieme

f'(1/a) = 0

meme si on a deja montre que f(x) = -f(1/x)

[maiden iron]

tu peux me dire comment tu as fait pour f'(alpha) ?

[tropic dove]

on a f est derivable sure ]0,+inf[ donc derivable sur ]0,1[

on a f est est continue sur [0, +inf[ donc continue sur [0, 1]

et f(1) = f(0) = 0

[maiden iron]

yep, et par le TAF il existe un reel alpha dans ]0,1[ tel que f(1) - f(0) = f'(alpha)

[tropic dove]

donc d'apres taf ou bien precisament le theoreme do rolle

il exist a tel que ....

[tropic dove]

si

et pour f'(1/alpha)

je sais pas

[maiden iron]

On a $0 \leq \alpha \leq 1$

donc on peut retrouver $\frac{1}{\alpha}$

[twin meteor]

Herels

[tropic dove]

comment peut t on projeter l'egalite precedente a cela

[maiden iron]

si alpha est dans ]0,1[, où se trouve 1/alpha a ton avis

[tropic dove]

]1, +inf[ bien sure

[tropic dove]

l'encadrement doit etre stricte btw

[maiden iron]

et sur cet intervalle, est ce que f est derivable et continue ?

[tropic dove]

oui

[maiden iron]

🤔

[tropic dove]

mais attendddddddd

le taf n'est valide que dans un intervalle fermee

c'est une condition primordialee

[maiden iron]

oui je viens de me rappeler

[tropic dove]

en plus je suis 100000% sure qu'on doit utilliser 2)a)

one way or another

[maiden iron]

f(x) = -f(1/x)

cest ca ?

[tropic dove]

oui

mais comment transformer cela en f'(a) = -f'(1/a)

[maiden iron]

tu derives f(1/x) et on voit
🤔

[tropic dove]

ok i'll try

[maiden iron]

puisque ce quil faut appliquer cest f(b)-f(a) = f'(alpha) (b-a)

[tropic dove]

bro yk how hard it is to be quadrolingual

[maiden iron]

donc faut retrouver lexpression mais avec -f(1/x)

[tropic dove]

xd

my brain is steaming

[maiden iron]

quadrolingual ? 💀

[tropic dove]

talking in french

[maiden iron]

ah

tu parles plusieurs langues

[tropic dove]

ik that word doesn't exist

but fr man

[tropic dove]

oui

[maiden iron]

génie

[tropic dove]

mais quand t'as pa vraiment maitriser une langue

et tu dois faire le processus de traduire les phrases a ton langage dans le cervau puis retraduire la reponse au francais

ca casse la tete

[maiden iron]

Mdrr oui je vois
ca me faisait la meme qd je galerais a l'anglais à l'epoque

[tropic dove]

,rccw

[tropic dove]

voila f’

il manque un ^2

attends

,rccw

[maiden iron]

hmm pas comme ca 😂

jaurais du etre precis

att

[tropic dove]

;-;

precis?

precise c’est quoi precis?

precis comment?

[maiden iron]

j'aurais du préciser ce que je voulais dire quoi

$(f(\frac{1}{x}))' = -\frac{1}{x²} f'(\frac{1}{x})$

[twin meteor]

Herels

[tropic dove]

on doit montrer cela?

[maiden iron]

hmm pas vrmt mais je veux juste voir si mon idée est bonne

[tropic dove]

autre question, cela est vrai?

a oui xd

la derivée de la composée

[maiden iron]

en gros en utilisant le TAF, on a :

$$f(1) - f(0) = f'(\alpha)$$
$$f(1) - f(0) = \frac{1}{\alpha²} f'\left(\frac{1}{\alpha}\right)$$

[twin meteor]

Herels

[maiden iron]

J'explique

$$f(x) = -f\left(\frac{1}{x}\right)$$
$$f'(x) = -\left(-\frac{1}{x²} f'\left(\frac{1}{x}\right)\right)$$
$$f'(x) = \frac{1}{x²} f'\left(\frac{1}{x}\right)$$

[tropic dove]

o

[twin meteor]

Herels

[tropic dove]

j'ai compris l'idee

[maiden iron]

et donc dans l'expression du TAF, j'ai juste remplacer x par alpha

on arrange et on tombe bien sur le résultat recherché

[tropic dove]

et cela devient ?

[maiden iron]

ca

[tropic dove]

em

[maiden iron]

f(1) = f(0) = 0

[tropic dove]

aha?

[maiden iron]

donc tu balance 1/alpha² a gauche et voila

[maiden iron]

bah tu las dit toi meme

et cest vrai en plus

[tropic dove]

je sais mais

…

lemmme get this from the beginning

we have this

f’(alpha) = f(0) - f(1) = 0

puis

f’(alpha) = (1/a^2)*f’(1/alpha)

donc

[maiden iron]

non, f'(a) = 1/a² f'(1/a)

[tropic dove]

donc

(1/alpha^2)*f’(1/alpha)=0

[maiden iron]

oui

[tropic dove]

donc f’(1/alpha)=0

ok ok I get it

damn

[maiden iron]

[tropic dove]

I got stupid there for a second

xd

merci!

[maiden iron]

de rien 🫡

[tropic dove]

des idées pour la question suivante?

[tropic dove]

attend je vais dire ce que je pense

j’ai pensé a 2 methodes

la premiere c’est on ajoute -xln(x) au deux membres

puis une pose une nouvelle fonction

et on calcule l’image de ]0,+inf[ par cette fonction

et on dois trouvere [0,+inf[

la deuxieme

c’est de voir l’inegalité comme ça

ln(1/x)/(1/x) <= ln(e)/e

et d’utiliser taf encore une fois pour la fonction u(x) = ln(x)/x

sur l’intervalle ]1/x,e[ puis ]e,1/x[ selon x

y’a t il d’autres methodes?

[maiden iron]

ah oe ca fait bcp
jai rien compris mais bon

[tropic dove]

xd

[maiden iron]

[tropic dove]

xD

t as pas compris quoi

[maiden iron]

pratiquement tout

[tropic dove]

xd

just tell me how you’d go over that inequality

[maiden iron]

hélas, j'avais pas vraiment d'idée
j'aurais réfléchi à un moyen de brute force, mais je vois pas trop comment

je crois y a un truc qui s'appelle Inégalité des accroissements finis

[tropic dove]

ca va devenir exponentiellement plus compliqué

car l’inegalité introduit des valeurs absolues et bla bla

[maiden iron]

oui malheureusement

après, le truc débile à faire c'est de dériver -x ln(x) et déterminer son maximum

[tropic dove]

o

[maiden iron]

ca marche tres bien meme si c'est un peu long

[tropic dove]

c’est plus simple

que mes methodes

je crois

[maiden iron]

oep

tu auras :
-ln(x) - 1/x

[tropic dove]

I TOTALLY FRICKING IGNORED THAT 1/e IS A CONSTANX

CONSTANT

[maiden iron]

lmao

[tropic dove]

i wanna kms

[maiden iron]

[vocal sleet]

@tropic dove Has your question been resolved?

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[half jetty]

I expanded the e and log(1+x) values, and simlified the denominator into x^3 (x^3 * sinx/x = x^3) what next?

[vocal sleet]

@half jetty Has your question been resolved?

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[atomic osprey]

Finding the radius of convergence and the interval of converge

[atomic osprey]

this is where I'm at with this

I was wondering for the (-5) endpoint test, can i just go straight to the divergence test because the AST wouldn't apply?

[boreal remnant]

alternating series only applies if the magnitude decreases

[vocal sleet]

@atomic osprey Has your question been resolved?

[atomic osprey]

Ok so then i would just go straight for the divergence test

and that'd be valid?

[vocal sleet]

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[olive igloo]

bro what

[flat whale]

<@&268886789983436800>

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[strange charm]

just generally how do i solve this integral

[flat whale]

why do you think that converges at all

gotta give lambdas and a ranges

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[crystal sail]

Okay so I have an integral problem using average integrals and I'm a little lost on the final part of the problem where the function is set equal to the average of the function and solved for that. I think I should have two answers but I'm not sure how to get the second because of a lack of familiar with inverse trig functions and trig functions in general. If I could get some help, that would be really appreciated. Thanks!

[olive igloo]

you have to calculate the average value first

[olive igloo]

do you know what the average value would be?

[crystal sail]

Yeah I got that part

[olive igloo]

,w integrate pi/8 * sin(x) from x = 0 to pi

[twin meteor]

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[olive igloo]

you made a mistake

you multiply the integral by $\pi/8$

[twin meteor]

rain

[crystal sail]

As in you're supposed to multiply it, or as in I multiplied it?

[olive igloo]

you multiplied it

wait

why did you put it twice

you have to divide by the length of the interval

[crystal sail]

At which step?

[olive igloo]

when you set up the integral

[crystal sail]

Oh wait at the start of fbar?

[olive igloo]

you have pi/8 on the front for no reason

you put it twice

[crystal sail]

Oh that should be 1/pi

[olive igloo]

yes

[crystal sail]

So the average is 1/8

[olive igloo]

i think its 1/4

[crystal sail]

Isn't the antiderivative of sinx -cosx?

[olive igloo]

ok

[crystal sail]

And then just constant multiple makes the pi/8 stay

[olive igloo]

$\cos\pi = -1, \cos 0 = 1$

[twin meteor]

rain

[crystal sail]

So antiderivative of pi/8 * sinx is -pi/8 * cosx

[olive igloo]

ok

[crystal sail]

-pi/8 * -1 = pi/8

pi/8 * 1/pi = 1/8

[olive igloo]

bro

you have

-pi/8(-1 - 1) = pi/4

divide by pi its 1/4

you forgot to include the 0 term

aka you cant assume cos0 = 0

[crystal sail]

Oh that's what I'm missing okay gotcha

Okay so that's one of the two answers I think I'm supposed to have

Now, am I missing something about how arcsin works?

Because I have x = 0.7 (rounded)

But I think I should have a second answer and I don't know how to get it

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[teal mauve]

This assignment was due a while ago, and I'm not quite sure where to start

[potent anchor]

what are the roots?

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[normal haven]

I need help continuing the solution. I'm not sure where to go from here.

[vocal sleet]

Please don't occupy multiple help channels.

[prisma rain]

well you have it

[normal haven]

But I don't think it's in the right form specified in the problem.

[prisma rain]

it is in that form, this is the same as if $c_n = \frac{1}{n}$, $x = -1$ and $a = 0$ for example

[twin meteor]

kaue

[normal haven]

I think I kind of understand, but how would you formulate an answer for the question?

[prisma rain]

just do what you did in paper and then give the examples for c_n, x and a if you want to prove it's in that form, but it's pretty explicit that it is

just fix the mix of u and x

[normal haven]

Got it, thank you so much again!

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[arctic wave]

Hey guys, i got lost in my word problem, can you help me out

[arctic wave]

is it more like this?

oops it's 350

i figured it out, it's been a few months since i've done these 😭

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[strong cloud]

having trouble with this question can somebody help me?

[hushed pewter]

The rotation is basically going to create two cones joined at their bases

Find the radius of the shared base and the height of each cone and you are golden

[strong cloud]

67.86?

[hushed pewter]

I don't have a calculator

So i can't say

[tawny ginkgo]

Did it in my head real quick, simple math really. Answer is 67.86

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[scenic ravine]

could someone walk me though the last part

[scenic ravine]

,w eigenvalues of [1,2][3,2]

[twin meteor]

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[scenic ravine]

so the diagonal matrix would be $\begin{bmatrix} 4&0\0&-1\end{bmatrix}$

[twin meteor]

A dense set(Ping when reply)

[scenic ravine]

but how do I find P?

by normalising these vectors using gram scmidt ?

[gaunt sparrow]

P's columns are the eigenvectors

[scenic ravine]

Yea

wait

what

[drifting jackal]

Find the eigenvectors with the corresponding eigenvalues

And your P matrix are those eigenvectors

[heavy yoke]

note that A and D are both matrices which represent the same linear transformation, with respect to different bases (A wrt the standard basis, D wrt an eigenbasis). so all you need to relate them is the change of basis matrix P (and in general the change of basis matrix is formed by putting your basis vectors as columns)

note that any basis of eigenvectors will suffice as long as the order corresponds to the order of eigenvalues in your diagonal matrix

[scenic ravine]

so $P = \begin{bmatrix} 4&-2\12&-2\end{bmatrix}$

[twin meteor]

A dense set(Ping when reply)

[scenic ravine]

I now normalise this

[heavy yoke]

if those are the correct eigenvectors corresponding to 4 and -1 respectively (double check this by multiplying them with your matrix)

[heavy yoke]

that's doable but not strictly necessary

[scenic ravine]

hmm?

[heavy yoke]

the eigenvectors don't have to be unit vectors to work for this

[scenic ravine]

okay

so I'm done now, right

Or do I compute PAP^T

[heavy yoke]

i would double check your eigenvectors just to be sure, but if they're correct you're done

[scenic ravine]

,w [4,-2][12,-2] \times [4,0][0,-1]

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[heavy yoke]

you can compute the right side of either D = P^(-1) A P or A = P D P^(-1) and compare it to the left. neither of which is necessary but can serve to check your arithmetic

[scenic ravine]

yea, messed up

The first entry I'm getting is 544

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[timber kite]

Summation for
0•1,1•2,2•3…..

It is
(n-1)(n+1)n/3
But thats when i make n = n-1
After I’ve gotten
n(n+2)(n+1)/3 using the summation for n and n^2 formula.

Now lets say the sequence starts from
5•6,6•7,7•8…..
do just simply just make n=n+4 and get (n+5)(n+7)/2 ?

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[velvet karma]

..

extremely

uhhh

what power

does the base need to be raised

to make the given number

thoughts on that?

ok so

u have lik

log_2(8) = 3

this means

that 2 needs to be raised to ^3 to create 8

and if theres no base

its base 10

thats like the default

[vast shale]

take log(2^ 2012)

[dark kiln]

you need to convert the base

[vast shale]

which is 2012 * log 2

Now a number of the form 10^n has n digits

[dark kiln]

oh yeah that's shorter

[vast shale]

For a number like 1050, log(1050) = 3.02118929907

Now, observe the value of log(1050) and the number of digits in 1050

Ok lemme just tell you

If log to the base 10 of a number X is n, then X has ⌈n⌉ digits

Sorry, I accidentally used the floor function

It's the ceiling function

[prisma rain]

actually, floor(n) + 1

10 has 2 digits

[vast shale]

well that's the same thing

i used the ceiling function

[prisma rain]

not really

[vast shale]

that gives the next greatest integer

How?

Oh

[prisma rain]

try to find how many digits 10 has

[vast shale]

yes

so it's floor(n) + 1

[prisma rain]

find floor(2012*log(2)) + 1

[vast shale]

,calc 2012 * log 2

[twin meteor]

The following error occured while calculating:
Error: Unexpected type of argument in function multiplyScalar (expected: number or Complex or BigNumber or bigint or Fraction or Unit or string or boolean, actual: function, index: 0)

[prisma rain]

floor just means round down btw

[vast shale]

,w 2012 * ln2

it has 1395 digits

[prisma rain]

wait

wolfram might reas log as natural log

[twin meteor]

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[vast shale]

ah

[prisma rain]

,w 2012 * log10(2)

[twin meteor]

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[prisma rain]

so its 606

[vast shale]

dang it

605.672351276

[vast shale]

yeah

[prisma rain]

you would need to aproximate log10(2) somehow

many ways

nice

yay

yowpey

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[vast shale]

gg

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[long jackal]

where is the function f(x)=1-$\sqrt{1-x^2}$ continuous ?

[twin meteor]

Hamdy Hisham

[long jackal]

Should it be (-1,1) or [-1,1]

[prisma rain]

do you mean where it is defined?

[cursive turret]

everywhere, where its defined.

[prisma rain]

but you can't take the limit from both directions on the boundaries

[long jackal]

so it's (-1,1) right ?

[prisma rain]

yeah

[cursive turret]

i am not sure, whats your definition of a continuous function is. the definition i know refers to the functions domain.

[prisma rain]

its continuous at a point a if lim x -> a f(x) = f(a)

[cursive turret]

so there s no need to take a limit from a side where the function is not defined.

[prisma rain]

lim only exists if it exists on both sides

[long jackal]

isn't this type of function continuous on it's domain ?

[cursive turret]

... x approaches through the domain of f ... no need for both sides.

[long jackal]

oh through the domain so that means it can be 1 sided

[prisma rain]

then i learned the wrong definition

[cursive turret]

what i said: the definition i know refers to the functions domain. maybe you have other definitions.

[bitter pilot]

no i agree with kaue

the most you can tell for the end points is that it is left/right-side continuous

[hard atlas]

it makes no sense to talk about limits from a side where the function doesnt even exist

if you check the epsilon def of limits then only points in the domain appear

[long jackal]

so that means it'll be (-1,1) ?

[cursive turret]

the function is defined at x = 1.

[long jackal]

right

is it continuous at x=1 ?

[cursive turret]

yes.

[bitter pilot]

but how can you say the limit exists if you approach it from only one side

you are ignoring that

[long jackal]

here @bitter pilot

[cursive turret]

the definition of a continous function refers to the domain of the function. so if the function is only defined on the left of 1 i only need to check limits from the lef.t.

[cursive turret]

you are claiming a both sides limit. show the definition of a continous function where a both sides limit is required.

anyway. I have shown the definition of a continous function I know. I have argued based on this definition and i have shown my result based on this. If someone agrees its fine, if not, its also fine for me. If someone has another definition the result can differ, of course.

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[tropic dove]

Let $f(x) = \frac{-xlnx}{1+x^2}$ if $x \ne 0$ and $f(0) = 0 \ \$ 1) prove that f is continuous in $]0, +\infty[ \ \$ 2)a) prove that $\forall x \in ]0, +\infty[ , f(\frac{1}{x}) = -f(x) \$ 2)b) prove that $\exists \alpha \in ]0,1[ , f'(\alpha) = f'(\frac{1}{\alpha}) = 0 \ \$ 3)a prove that $\forall x \in ]0, +\infty[, -xln(x) \le \frac{1}{e} \$ 3)b) deduct that $\forall x \in ]0, +\infty[, f(x) \le \frac{1}{e} \$ 4) We admit that $\alpha$ and $\frac{1}{\alpha}$ are the only solutions to the equation $f'(x) = 0 \$ a) prove that $f(\alpha) > 0$ and $f(\frac{1}{\alpha}) < 0 \ \$ b) Prove that $f'$ is continuous in $]0,+\infty[\ \$ c) Prove that $f'$ doesn't change its sign in every one of the following intervals: $[0, \alpha]$, $]\alpha, \frac{1}{\alpha}]$ and $[\frac{1}{\alpha}, +\infty[$

[tropic dove]

any ideas on how we can use all the information from 4, 4)a, 4)b to prove 4)c?

[twin meteor]

Froggy

[tropic dove]

can you help guys?

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[tropic dove]

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[wary fiber]

HOW DO I APPLY EULERS FORMULA HERE

e^ix= cos x+I sin x

HOW DID ROOT 3 THITN BECOME E

[wary fiber]

How

Bdueuhdhduf

[lone linden]

Draw a sketch of the two cases and consider drawing a perpendicular down to the real axis to find the argument

[bronze osprey]

you should draw the points (-1/sqrt2, 1/sqrt2) and (-1/sqrt2, -1/sqrt2)

[wary fiber]

Wha

[bronze osprey]

then you will see the angles are pi - pi/4 and pi + pi/4

[lone linden]

And then for the modulus, take the distance of those points from the origin

[wary fiber]

Can't I do without that

I don't understand a thing

[bronze osprey]

drawing will help you understand

or plotting those points on desmos

[wary fiber]

Draw what

I'm ready to draw

[bronze osprey]

these points

[wary fiber]

Let me

Wait

[wary fiber]

It's in root

And why specifically this

Pls pls

@lone linden

@bronze osprey

[bronze osprey]

cause you know how the complex number $a + bi$ on the complex plane

is literally just $(a, b)$ on the xy-plane

[twin meteor]

southlander!

[wary fiber]

What

Complex plane?

I'm very confused with that

[bronze osprey]

yeah like you just swap the axes names

[wary fiber]

Argand plane

Re and imaginary

[bronze osprey]

the complex plane has real and imaginary axes

yes

[wary fiber]

Pls explain sir

Teacher did not teach properly

I'll pay u

[bronze osprey]

I mean, it's not a deep concept

[wary fiber]

Ok here what do I plot

If it's 4+3i how to plot?

[bronze osprey]

these

well like it's the same as (4, 3) then

[wary fiber]

And I read something sq a^2+b^2

[wary fiber]

Oh

[bronze osprey]

yes that's the modulus squared

[wary fiber]

1 more question

[wary fiber]

???

Isn't modulus the absolute value

Like mod -6 is 6

[bronze osprey]

correct, but there's a generalisation to complex numbers

the modulus is always the distance to the origin

[wary fiber]

Sq root a^2+b^2 is modulus?

[bronze osprey]

yep!

[wary fiber]

By definition?

[bronze osprey]

yeah

[wary fiber]

Wait a second.. is that using distance formula?

[wary fiber]

Ok so here

[bronze osprey]

exactly

now you're getting it

[wary fiber]

So this is the expression

After using quadratic formula

[wary fiber]

Why is that converted to this weird form with sq roots?

What

What box

Nooo pls don't go

[lone linden]

$\frac{-1 \pm i}{\sqrt 2}=e^{\pm \frac{3\pi}{4} i} \implies \sqrt 3 \left(\frac{-1 \pm i}{\sqrt 2} \right)=\sqrt{3} e^{\pm \frac{3\pi}{4} i}$

[twin meteor]

Civil Service Pigeon

[wary fiber]

😭

[lone linden]

The “coefficient” of the exponential factor is the modulus

[wary fiber]

How 3/4

[lone linden]

Didn’t south say that already

[lone linden]

.

[wary fiber]

Still in the middle of it

And I don't understand

[lone linden]

It would help if you sent the diagram you have rn

[wary fiber]

Why do I plot specifically that?
Why can't I just use -1+i
Sorry for too many questions

Why should I find an angle?

This is all I got rn

[lone linden]

You can, but then there’s a factor of sqrt(3)/sqrt(2) you have to deal with instead

It doesn’t really matter

Just pick one and do it

[wary fiber]

Ok how to do"it"

What is it

[lone linden]

Plot either $\frac{-1 \pm i}{\sqrt 2}$ or $-1 \pm i$?

[twin meteor]

Civil Service Pigeon

[lone linden]

.

[wary fiber]

HOW

I WILL DO 2ND ONE.

ITS -1, 1 RIGHT

[lone linden]

For the $-1+i$ case, yeah.

[twin meteor]

Civil Service Pigeon

[wary fiber]

AND FOR - IT IS -1 -1 RIGHT

[lone linden]

Yeah

[wary fiber]

NOW WHAT

WAIT IM GETTING S REFLECTION

[lone linden]

Yup

They’re conjugates, so they’re reflections of each other over the real axis

You can actually use that to find one of the arguments given the earlier, but it’s not necessary

[wary fiber]

Ok now next

[lone linden]

.

[bronze osprey]

btw thanks for taking over

[lone linden]

Eh I’m on the train so not much else to do

[wary fiber]

PLS CONTINUE HELPING ME

DO NOT LEAVE

[wary fiber]

MY EYE DOESNT SEE ANY ANGLE SIRR

@lone linden

I see two points

[bronze osprey]

you want the angle with the positive x-axis

say the vector (1, 0)

[wary fiber]

With origin?

angle with origin?

How to get angle now.....

@lone linden pls I beg u

@bronze osprey

[bronze osprey]

great, so do you see how it's pi - pi/4 and pi + pi/4 radians now

[wary fiber]

I don't see it

[bronze osprey]

think of an isosceles right triangle with 45-45-90 degree angles

[wary fiber]

How can I just estimate it right away

Is there a formal way

Using dot product or something

[wary fiber]

How can I say one of the angles are 90

[bronze osprey]

the x-length has the same length as the y-length

[wary fiber]

I know this.. how can I say the angle is 90

[bronze osprey]

the y-axis is perpendicular to the x-axis

[wary fiber]

Wait which triangle are u saying is isosceles?

OH

I GOT IT

I SEE WHICH

SO THE ANGLES ARE PI/4 AND -PI/4

NOW

@bronze osprey

@lone linden

[bronze osprey]

yeahhhhh

[wary fiber]

Now what

Pls

@bronze osprey

@lone linden

<@&268886789983436800>

@everyone

@trump

[mild flower]

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[bronze osprey]

use Pythagoras to find the modulus

[bronze osprey]

you should get the modulus of both of them is 1

so you have $r = 1$ in $re^{i \theta}$, that's polar form of a complex number btw

[twin meteor]

southlander!

[bronze osprey]

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[hazy coyote]

Is this correct for counter clockwise unit inverse unit circle

[hazy coyote]

<@&286206848099549185>

😭😭

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[hazy coyote]

Ok

<@&286206848099549185> plz help me

[hidden gyro]

wdym by inverse unit circle?

wont 90* be (0,1)

counter clockwise

[hazy coyote]

I give up

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[glass forge]

I got a triangle, one side is 5cm other is 9cm. Area is 20cm^2. I need to claculate the perimeter, how'd I do it?

[boreal remnant]

i don't think that's enough info mr president

[glass forge]

I just took extended trigonometry class

That's the entire task

I know you need a*b*sin(alpha)/2 to get one of the angles but then I'm not sure how to go further from that

[strong grove]

probably to do that and then use cosine law to find the 3rd side

[wispy kindle]

do you know sine rule and cosine rule

[strong grove]

and add them together

[glass forge]

No, I don't

[strong grove]

u dont know
c^2 = a^2 + b^2 - 2abcos(C)?

might be tough to find it then without that

[glass forge]

No, don't think I've learnt that yet

[wispy kindle]

do you know how to add vectors?

[glass forge]

Yes

[wispy kindle]

try to use that

by thinking of the sides as vectors

and parellely shifting them to get the third side

[glass forge]

Oh I see what you mean, thanks guys

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[open sage]

Hey folks

I'm solving b), I know the solution I'm just not sure how to express it

b) Indicate whether or not the subspace S is a subspace of Rnxn. Justify your answer.

So I'm pretty sure it is. indeed a subspace.

Any matrix with a column or row of 0 will have its det = 0

so say we set the third column for a 3x3 matrix to be 0 then the rest of the numbers can be any scalars

and this verifies all three axioms for a subspace

but i'm thinking 1x1 might be a different case? unless it's just 0 which is a subspace

I'm mostly just confused with how to prove that it's a subspace

i can make up a 3x3 matrix as an example but that doesn't prove it for all nxn

[river minnow]

I am confused with your reasoning, why are you setting anything to 0 inside the matrix? There are matrices with all entries nonzero and determinant zero

[open sage]

true

it's just one "obvious" case i thought of

but yes a matrix 2x2 where all its elements are 2 will have det = 0 for instance

[river minnow]

I don't see why that's relevant

[open sage]

well it's just

my thought process

i'm trying to come up with a solution i suppose i was thinking of which matrices may have their det = 0

i assumed it's not entirely relevant since we're not going to solve this by giving examples but

it was just my thought process

Like I could say the following:

1- A zero matrix's determinant will always be 0 for all Rnxn matrices

2- Let A,B be matrices with det=0, det(A+B) = 0

3- Let A be a matrix with det=0, det(kA) = kdet(A) = 0

i suppose I just need to prove the second one

[river minnow]

det(kA) is not necessarily equal to kdet(A)

And yeah proving 2 will be a challenge

[open sage]

sorry, yeah

det(kA)=k^ndet(A)

[river minnow]

Hint: ||If you can't prove something, perhaps it's false, consider looking for a counterexample to 2||

[open sage]

which since det(A) is 0 is just 0

[open sage]

oh

okay

[open sage]

yep i think that works

[river minnow]

That's it yeah

[open sage]

alright perfect

thanks!

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[crystal surge]

Q. The percentage error in determining sin 30° by assuming that between 0° and 45°, the sine function may be approximated by a straight line is
a) 9.3% b) 7.5% c) 5.7% d)3.6%

[crystal surge]

idk what to do at all

[dark kiln]

,calc sqrt(2)/2

[twin meteor]

Result:

0.70710678118655

[dark kiln]

,calc sin(pi/4)

[twin meteor]

Result:

0.70710678118655

[dark kiln]

so like you would get 30/45 of that value

[crystal surge]

i didn't quite catch what you did there

[dark kiln]

i assume what they mean is that you know sin45°

[crystal surge]

maybe

there is nothing else in the question

[dark kiln]

and you draw a graph and it's a straight line

[crystal surge]

the approximation might work on the middle part but will falter near 0° and 45­° right?

[dark kiln]

not it's the opposite

[crystal surge]

huh

[dark kiln]

it works everywhere pretty much, but especially well near the ends

here's a straight line for comparison

i didn't know this

[crystal surge]

oo

so how do we find the error?

[dark kiln]

,calc sin(pi/4)/45 * 30

[twin meteor]

Result:

0.47140452079103

[dark kiln]

this is our approximated value

it's a linear relationship

30/45 of the argument will give 30/45 of the value

[crystal surge]

but why 30/45?

i'm lost tbh

olympiads be playing with my mind

[dark kiln]

i don't know it's just how lines work

if x = x gives y = 2.6 then x = x/4 gives 2.6/4

probably has to start from 0

but we have that part

[crystal surge]

ooh

so the final answer?

[dark kiln]

idk i'll check if you try

[crystal surge]

idk i'm lost

<@&286206848099549185>

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[crystal surge]

what

i used it just now

[dark kiln]

yeah that was weird

[crystal surge]

it's over 15 minutes already

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[crystal surge]

<@&286206848099549185>

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[hidden wave]

How to prove it wihtout using antng more than Legendre symbol properties and index (discrete log)

[hybrid flicker]

do you know the following result that is correlated to legendre: if $gcd(a,p) = 1$,
$x^2 \equiv a$ mod $p$ has solutions if and only if $a^{\frac{p-1}{2}}\equiv 1$ mod $p$

"$x^2 \equiv a$ mod $p$ has solutions" literally being $\big(\frac ap\big) = 1$ when $gcd(a,p) = 1$

[hybrid flicker]

so can you try to prove that $\big(\frac ap\big) = 1$ if and only if $\big(\frac aq\big) = 1$

[twin meteor]

rafilou is not not born in 2003

rafilou is not not born in 2003

rafilou is not not born in 2003

[hybrid flicker]

(few edits)

[hidden wave]

Yes I know them

[hybrid flicker]

forgot to add that this is true when p is odd prime

[hidden wave]

Yeah, I know these theorems

[hybrid flicker]

alright

[hybrid flicker]

so perhaps this is a way you could go

[hidden wave]

The struggle is mod 4a. I can do it for p ≅ q mod 4 but how to do it for 4a

[hybrid flicker]

you mean you've done it for p = q + 4a already

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@hidden wave Has your question been resolved?

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@late osprey Has your question been resolved?

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[uneven quarry]

if a video tutorial on math skips a topic, but says "see [inserts its own url here] to learn this topic", is it lying?

[sweet flower]

Your language is a bit ambiguous, but if it doesn’t explain the topic then yeah I guess

But also is this a math question or a #discussion question?

[uneven quarry]

on an unrelated note, the topic of the video tutorial is recursion

[open igloo]

what a question its that

[uneven quarry]

i guess i wanted to make it more rigorous but it probably started as a #discussion question

[sweet flower]

If the self referential lesson doesn’t include the base case you’ll reach an infinite loop

[uneven quarry]

this is why i can't decide if it's lying or not

(and in some sense it does teach you about recursion, but that's not very rigorous)

[sweet flower]

It shouldn’t call itself before determining the base case

[uneven quarry]

ok

it's "lying" or "not defined" in that sense then?

sorry, the words i'm using are really too vague to be rigorous

[sweet flower]

Yeah you could argue it’s lying, but you could also argue it does its job of interesting you on the topic and encouraging you to learn more

[uneven quarry]

(thinking out loud - i guess a recursion either must have a base case defined, or can only either never end or run into undefined behavior (if we're talking about some code maybe that is run))

so if it has no base case, that implies it doesn't really have a defined "end result"

[uneven quarry]

but this is true too 🙂

[sweet flower]

I think you should look into search algorithms, backtracking, and dynamic programming.

[uneven quarry]

ok, definitely

those sound interesting

[sweet flower]

They are :)

[uneven quarry]

in my notes. thanks for the help @sweet flower

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[jovial nexus]

I want to learn lebinitz theorem please if someone is willing to call and teach id be happy

[jovial nexus]

<@&286206848099549185>

[hushed pewter]

!15m

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[visual forge]

watch a youtube video and come back here with specific questions

[jovial nexus]

yea like pls reccomend some like im not finding the right one soo

give me a good best vid which will make me understand that

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[queen root]

welp

go to another help channel

[spare smelt]

ok