help-23

[gaunt belfry]

i can try but this is not a maths problem, it's physics so....

other people may not like this😅

[safe radish]

@strong nexus Has your question been resolved?

[strong nexus]

Right. Anyways, do you have an idea?

[gaunt belfry]

yes

[strong nexus]

What is it?

[gaunt belfry]

you know about momentum and kinetic energy i assume?

[strong nexus]

Yes

[gaunt belfry]

so let the momentum of the white before hitting the cushion be p

now p = m*v, i hopw you understand what m and v are, right?

in this scenario atleast

[strong nexus]

Yes of course

[gaunt belfry]

hence we can also say that the kinetic energy before the ball hit the cushion is, E_k okay?

[strong nexus]

Yeah, E_k = 1/2 mv^2 with the same m and v

[gaunt belfry]

yes

now before the ball hit cushion, what's the angle of incidence with respect to surface normal?

[strong nexus]

50°

[gaunt belfry]

right, so now we have to decompose p into two directions

let me draw it

[strong nexus]

[In the mean time, for the magnitude: We know that E_beforehand = E_kin = E_afterwards = E_kin*. So 1/2mv^2 = 1/2mv*^2, i.e. |v| = |v*|. But then |p_before| = m|v| = m|v*| = |p_afterwards|, so the magnitude of the momentum does not change]

[strong nexus]

Right

[gaunt belfry]

now after collision with the cushion, which component of that one do you think will change?

[strong nexus]

Is there anything that says that after hitting the cushion, only the horizontal part gets flipped or something?

[gaunt belfry]

unless there is no loss of energy, it should be like that

[strong nexus]

Yeah

[gaunt belfry]

which means we have to assume that the cushion is perfectly elastic

so that kinetic energy is conserved

[strong nexus]

Yep

That the magnitude stays the same would follow right away too, (pcos)^2 + (psin)^2 = p^2

[strong nexus]

No need to do it like this

Now we just need to think about the angle

[gaunt belfry]

hence we can safely say that the magnitude of p*sin(alpha) remains same

after the collision with the cushion that is

let me draw the next part

[gaunt belfry]

yeah

[strong nexus]

The angle 90° - alpha matches because the sin matches because of what we discussed with the magnitudes

So we get this:

And then we should be done

Thanks a lot!

[strong nexus]

What momentum and energy does the cushion take in?

For the momentum, its just the double of the momentum of the ball, right?

It has to absorb and send back

[gaunt belfry]

no no if we take that into account then calculations will be messy

everything here is perfectly elastic

[strong nexus]

I mean the second part of the exercise asks this, Im not using this for what we did

[gaunt belfry]

anyways if you see only the horizontal part is reversed so let's do the calculations for that

[strong nexus]

Right

[gaunt belfry]

it's kinda difficult to write in paint with just mouse let me see what i can do

[strong nexus]

The cushion takes in the momentum $|\Delta p| = 2p_x = 2\sin(\alpha)p$, no?

[flat frigate]

ILikeMathematics

[gaunt belfry]

wait i'm doing some calculations

[safe radish]

@strong nexus Has your question been resolved?

[gaunt belfry]

okay i am back

are you there?

[gaunt belfry]

just read this picture, everything is shown in the above pic 👆

[safe radish]

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[eager junco]

Hi, may I know how I can prove that f(x, y) = summation of bi - ai*x-y is monotone?

[eager junco]

This function is related to the least squares method, where a and b are data points

[violet storm]

I'm trying to figure out the number and type of roots for a problem " x^5 + 4x^3 = 0". I keep getting 3 roots but i thought the number of roots was supposed to be the highest exponent? so far i factored to x^3 (x^2 + 4) =0 and solved from there to get roots of 0, 2i and -2i. I'm unsure whether this is right and im being stupid, or if theres supposed to be 5 roots

[manic patrol]

you're a little too late in this question, I'm afraid. tinyfold has gotten the channel; you should resend your question in a new channel.

[violet storm]

ohhh okay! sorry

[solar hazel]

what is a monotone function with 2 variables?

[manic patrol]

npnp!

[eager junco]

I’m not to sure on the formal definition but in this case, it would be such that g(x) = min(f(x, i)) and the other way around are both monotone

[solar hazel]

makes no sense to me. do you have the original question and something about monotone?

[eager junco]

I am exploring stuff related to linear regression in a project. I know so far that there exists parabolas when I make cross sections. I am looking to know whether or not the vertex of all these parabolas show a pattern.

[eager junco]

In my observation, it seems like these vertices line up create another parabola. I am wondering if there is a rigorous way to prove this.

[solar hazel]

do you have a picture of what you're trying to explain?

[eager junco]

It looks like a hot dog, or a folded piece of paper

Yes, or a taco

[warm lodge]

is this the plot of the equation you first sent. I’m not convinced this looks more like a f(x,y)= ax^2+by^2+cxy of some kind

[eager junco]

No, I have tested this on different platforms, and they all show this result

[warm lodge]

sure, but what’s the formula you plotted?

that was my question really

[eager junco]

This is the formula I think. I have generated a very small dataset with an obvious line of best fit, and put them through a and b

[warm lodge]

what do you mean you put them through a and b. from what i gather you notate (a_i,b_i) your data points and x and y the slope and intercept of your regression line did i parse correctly?

like what is your fitting procedure

[eager junco]

I come from a cp background so please bear with me if my notation is sacrilegious. a and b are arrays of points where a_i and b_i represents the location of the ith datapoint. My f(x, y) function essentially sums up the squared residue of each data point.

This is pretty much what you said lol

[warm lodge]

ok that’s starting to make sense what you did is like $\sum_{i=1}^n (b_i - a_i x - y)^2$?

[flat frigate]

pola_touche

[eager junco]

Yes, that is exactly it

[warm lodge]

well that’s a ax^2+by^2+cxy+dx+ey+f if you expand each term squared and regroup all term of the same kind

but honestly i think it’s simpler to think of it this way let’s say you fix a slope x and you “scroll” through the intercept, then you get something like this :

[eager junco]

Wow, that is very interesting as it greatly simplifies the problem!

[warm lodge]

if the intercept y is too low or too high the errors are huge (orange line) so the parabolla shape does not seem like nonsense

[eager junco]

Yes I see

[warm lodge]

i think a similar intuition is there for the slope x

[eager junco]

Ok, it’s starting to make intuitive sense now

<@&286206848099549185> (mostly pola touche) thank you so much for helping me so far. I have to sleep now, and the forum will close soon. I am not sure if I am allowed to do this, but if you have any ideas on this problem or have experience in stuff like this in general, please feel free to dm me. Greatly appreciated!

[warm lodge]

what you plotted is called a loss function, for a simple linear model like your’s its shape is called a paraboloid and there is always a unique minimum/optimal x* and y* that is best to fit the model. The formula for the minima can be computed with some (multivariate) calculus.

[safe radish]

@eager junco Has your question been resolved?

[safe radish]

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[hardy grove]

Is it true that for any irrational number a, the set {m+(n^2)a | m, n in Z} is dense in R?

[pallid magnet]

yes

[solar hazel]

no?

[pallid magnet]

read up on Weyl equidistribution theorem

or i'm not sure actuallu

actually

[solar hazel]

maybe it is actually

[winter whale]

damn a lot of people who're unsure

[pallid magnet]

i'm bad with number theory

[winter whale]

same

[hardy grove]

there is one sentence on the wiki page that says that weyl basically proved my statement? but i havent been able to find a ref for that

[manic patrol]

if n in Z, then wouldn't negative n cause problems?

[hardy grove]

oh i suppose my notation was a little ambiguuous

i meant

m+(n^2)*a

[solar hazel]

i think it is true

[manic patrol]

so m + an^2.

[hardy grove]

yeah.

[manic patrol]

so technically we only want to know whether the fractional parts of an^2 are dense in [0, 1), I believe, since m being an arbitrary integer means that all integers are covered.

[hardy grove]

that would work - or you could project onto a circle - but i havent been able to figure out anything from there :(

[manic patrol]

actually, this then uses the theorem someone mentioned earlier (Weyl's equidistribution theorem for polynomials).

or... hm, I'm not sure. sorry.

[hardy grove]

yeaaa now that i know the existence of that theorem i at least have something to work towards (weyl’s criterion)

[solar hazel]

hm annoyingly this set is not closed under addition

[hardy grove]

yup

[solar hazel]

yea that does prove it

i think chloe is wondering how to prove that

[manic patrol]

I believe that is Weyl's equidistribution theorem mentioned.

[tardy mango]

oh wait I can't scroll up and read

ignore me

[safe radish]

@hardy grove Has your question been resolved?

[hardy grove]

actually i’m just trying to prove that my sequence satisfies the criterion at the moment

[solar hazel]

what criterion?

[rugged arch]

weyl probably

[solar hazel]

oh lol

[safe radish]

@hardy grove Has your question been resolved?

[safe radish]

@hardy grove Has your question been resolved?

[manic patrol]

hi, this channel is still occupied, unfortunately.

[hearty fox]

oh

sorry guys

[safe radish]

@hardy grove Has your question been resolved?

[hardy grove]

hm, maybe i'll have to revisit this one in a few months/years, after i've gotten a little better at analysis

.close

[safe radish]

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[winter whale]

dang

well time for character development ig

[rancid rock]

what do i do with the 2 b squared and their multiplicators

?

[lean otter]

!occupied

[safe radish]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[rancid rock]

what do i do

[primal bone]

@rancid rock MASSIVE COUGH

[vague phoenix]

What did you do

[lean otter]

go to an available help channel

like #help-47

[safe radish]

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[west hedge]

okay, what is it trying to convey

[round egret]

i think Hilbert was trying to say you need to be creative for mathematics

and poetry helps with that ig

idk tho

[lean otter]

so is a poet not creative

[winged flare]

mathematics needs unreal amounts of imagination

poetry is known to have layers of understanding and thought

[primal bone]

Yet Hilbert is arguing that you need to be even more imaginative than a poet to be a mathematician

(Contextually, I believe he was referring to a student of his who'd dropped maths to pursue poetry)

[safe radish]

@west hedge Has your question been resolved?

[primal bone]

According to WikiQuote, the quote is

Good, he did not have enough imagination to become a mathematician.

• Upon hearing that one of his students had dropped out to study poetry, as quoted in [http://books.google.com/?id=nnpChqstvg0C&pg=PA151 The Universal Book of Mathematics (2004) by David J. Darling, p. 151
  which the Chinese article translates thusly:
  很好，他沒有足夠的想像力能成為數學家。
• 當他聽到他的一個學生放棄數學去研讀詩。

[safe radish]

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[lofty cipher]

I already know how to do the question, im just confused on why we draw the triangles like the hint says and not some other way, and also how do you notice that in questions like these you have to draw triangles like that

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[echo gazelle]

im kind of struggling with the concept of continuity

can i pls have some help understanding

[karmic merlin]

A function is continuous at b if lim(x->b) f(x) =f(b)

And of course if lim(x->b) doesnt exist then it isnt continous

[honest perch]

you can draw the graph without lifting ur pencil

[worldly lantern]

for piecewise defined function to be continous, value should be same at breakage points for it to be continous

there fore Left hand limit at x = right hand limit at x = value of function at x

[plucky elk]

read this first https://tutorial.math.lamar.edu/Classes/CalcI/OneSidedLimits.aspx

don't troll in help channels

[safe radish]

@echo gazelle Has your question been resolved?

[echo gazelle]

oooo

[echo gazelle]

ahhhh

[echo gazelle]

wait im confused on what f(b) is?

[worldly lantern]

value of fxn at b

[timid shuttle]

b is a point on the domain of the function or in the case the point where you want to evaluate the limit at

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[kindred slate]

hi

[kindred slate]

so this was my answer

[safe radish]

@kindred slate Has your question been resolved?

[kindred slate]

<@&286206848099549185>

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@kindred slate Has your question been resolved?

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@kindred slate Has your question been resolved?

[safe radish]

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[terse fulcrum]

I don't entirely understand Sigma Notation, I have looked at how it is done and have a fair understanding. But, I lack the ability to actually 'know' what is happening when doing it. I find it easier to do math knowing what is happening in reality when an event occurs, so this has been a difficulty. 🥀

[dull elk]

You mean summation?

You basically iterate over a set as specified in the notation and add the value of the expression at each iteration to the total

And return the total

The set has to be countable

[hidden gyro]

You got specific examples?

Also, sigma notation is a convention it is entirely up to you to not use it but it is not worth it in the long run.

Sigma notation is basically adding terms of a sequence when you have some kind of closed form to said sequence

[terse fulcrum]

Yes, however, I lack to understand what is actually happening. Like, 2+2 obviously represents how you are adding two of a given object. I do not understand what is happening in reality as I apply it.

[dull elk]

If sum over say n= 5 to 15 and the expression is n²

What do you think happens

[terse fulcrum]

We repeat 5²+6²+7²+8²...+14²+15².

[dull elk]

Yep

[terse fulcrum]

And that equals 1210.

[dull elk]

That notation compresses this long series of additions

In a clean compact expression

[terse fulcrum]

So, similar to a much more compact multiplication?

[dull elk]

What?

This is not a multiplication

[terse fulcrum]

Well, multiplication is simply faster addition, right?

[dull elk]

This is iterated addition

[dull elk]

Multiplication has a stronger condition though

In that the expression is a scalar

So sum over n = 1 to 10 of 3 would be equal to 3 times 10

[pallid magnet]

no, a more "universal" and powerful multiplication kinda

[dull elk]

But if we have n or n² it is different from a multiplication altogether

[terse fulcrum]

Ok, I see. So,

[dull elk]

Multiplication is a special case of iterated addition yeah, but it's silly to say iterated addition is a general case of multiplication imo

[terse fulcrum]

Ok.

What happens when we add additional exponents to "i"?

[dull elk]

So sum over only a subset of Natural numbers ?

The subset being perfect squares?

[terse fulcrum]

Instead of having "i" be ¹ (1), then to have I squared or cubed, or to the fourth.

[dull elk]

Oh

Yeah I mean

If we have like n² = 1 to 10 f(n)

It basically means

f(-3)+f(-2)+...+f(2)+f(3)

You can have any condition on n to iterate over

As long as the set is countable

You cant iterate over real numbers between 0 and 1 for example

The sum would be undefined

[terse fulcrum]

Would n²=-n, then?

[dull elk]

What

[dull elk]

n has to be an integer mind you

[terse fulcrum]

I mean n³

[pallid magnet]

@terse fulcrum for these kind of question involving definitions and such you could try asking the AI models, do verify your info but most of the time they're pretty accurite

[terse fulcrum]

So, what is "i"?

[dull elk]

Don't lol

[pallid magnet]

a variable

[dull elk]

Why would you ever trust ai here

[dull elk]

i represents an element of the set you are iterating over

[terse fulcrum]

Well, I was tough "i" in this case was an imaginary number. Does that still apply to what you are talking about?

[dull elk]

Oh wait you can.

Just define your domain properly though

And make sure the set is countable

[pallid magnet]

most of the time the major AI models are very accurate when it comes to most math, and depend on the setting, i could be the unit or just a variable

[terse fulcrum]

So, what happen when we put "i³=-i"?

[pallid magnet]

could you explain?

[dull elk]

That's a statement which is always true...

So you will iterate over all natural numbers if n is natural

Since for any n, the statement holds

[terse fulcrum]

Well, i³=-i, so, what changes in the equasion if we change i to i³?

[terse fulcrum]

Right, but does the summation look differently?

[dull elk]

Why are you taking the imaginary number i here

[left gyro]

i can mean two different hings

[dull elk]

Where did it even come from

[left gyro]

usually when they do $\sum_{i=1}$ or whatever, the i here just means index

[flat frigate]

mtt

[left gyro]

it wont have any relation to the imaginary number i while its used this way

so for example if you have $i+\sum_{i=1}^32i$

[flat frigate]

mtt

[left gyro]

first off people will tell you this isnt very clear, so theyll get you to change the "dummy variable" or "index variable" from i to a different letter like k

second, itd be calculated as "i + 2 * 1 + 2 * 2 + 2 * 3"

[terse fulcrum]

Was that what Fate_ was using as "n"?

[left gyro]

yep

[terse fulcrum]

Pk

[left gyro]

usually its called dummy variable

think like test dummy

its only there for the purpose of knowing what changes in the sum

[terse fulcrum]

But imaginary u it is still acceptable?

*unit

[left gyro]

you should make sure that different purposes get assigned to different letters

so if youre using i, avoid using i as a dummy variable

other than that youre good

[dull elk]

The summation notation has two parts

1. The condition which restricts the set (countable to iterate over)
2. The expression to be evaluated at each iteration and added to the sum

If that condition is something which is always true, then the set basically becomes the domain(usually the natural numbers)

[pallid magnet]

in this case, only if it's said to be the unit, otherwise it's a dummy

[dull elk]

Usually n = 1 to 15 or whatever restricts the set to natural numbers between and including 1 and 15

[terse fulcrum]

So, if we have the variable "n" set to an assigned value. Then words the bottom of the Sigma it would look "n=2" (2 for easy math), and then towards the right of the Sigma "2"?

[pallid magnet]

yes

[terse fulcrum]

Ok!

I am unsure if I should be prepared to see "!" in these summation, should I practice that?

[pallid magnet]

! as in the factorial?

[terse fulcrum]

Yes, from what I understand it looks similar to Sigma notation, and I see it commonly in equasions that appear similar.

[pallid magnet]

a factorial of n is the product from 1 to n, it will appear in these summations depending on what level of math your in

if you could provide a bit more background on what your studying it could hepl a lot

help

[terse fulcrum]

Ok, like here: (hold on a sec)

I see it pared with this commonly (with e)

[pallid magnet]

yes, it's really common in uni math

and combination/statistics

[terse fulcrum]

And, we use it in dependant probability, yes?

[terse fulcrum]

Ok

I think my answer has been satisfied! Thank you all for assisting me.

[terse fulcrum]

Do you know how to close the channel? I don't know how.😅

.close

[safe radish]

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[safe radish]

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[wind laurel]

Hey! Could anyone check this please? I've tried everything but I'm pretty certain that the given substitution does not reduce (1) to (2)

[mild mauve]

Let me do it for you

[wind laurel]

thanks!

[mild mauve]

Start with z=xy find derivative in terms of y

Find first and second derivative

[wind laurel]

I've already done the full substitution, the issue is that i cannot reduce (1) to (2) using the substitution y = z/x

so i'm pretty sure the question is just wrong

but maybe my working is just wrong

[mild mauve]

Follow my process please

[fathom jewel]

Maybe you ought to show your working instead

[mild mauve]

I assume you done that now look at your original equation and group the terms

[wind laurel]

yep let me write it all down on pc

[mild mauve]

Note : multiply the whole equation by X to make the substitution varibale

Group the "second derivative" terms and the "first derivative" terms

Factor out an x from the first group and 4x from the second

Now, substitute the expressions we found in Step 1 into the parentheses

[main mural]

hm i tried it myself and i agree it seems like something is off

[mild mauve]

Divide the entire equation by X assuming xnot equal to 0

That will match Equation (2) exactly

hope it helps !!

bye

[open wedge]

damn

[open wedge]

bro cooked

[wind laurel]

yea sorry i dont have the entire solution infront of me right now, its just that i deleted it because i noticed something was off and rewriting it on pc takes a while

[main mural]

okay but what did you get in the end?

[wind laurel]

but i end up with $$xz'' + (2x^{-1} - 6)z' + (x^{-1} -2x^{-2})z = 0$$

something like that

[flat frigate]

woomy

[main mural]

huh damn i got something very different LOL

[open wedge]

and remember not to post solutions and guide the user

[wind laurel]

ah

let me re do everything and send a photo gimme a min

[main mural]

still not (2) tho :/

[mild mauve]

why

[main mural]

maybe i'm wrong too

[wind laurel]

what do you get?

[mild mauve]

I already helped u do it by my process it's tricky but is easy

[main mural]

$$
z'' + 2z' - (2/x) z' - 2 = 0
$$

[wind laurel]

z' in the denominator?

[main mural]

no sorry

[flat frigate]

artemetra

[wind laurel]

what did you find for y' and y''?

[mild mauve]

https://tenor.com/view/lebron-pointing-lebron-pointing-thumb-lebron-thumb-lebron-meme-gif-15467774302322576430

[main mural]

$$
\frac{dy}{dx} = -\frac{1}{x^2} \frac{dz}{dx}
$$

oh my y'' is wrong mb lol

one sec

[wind laurel]

is it not y' = -z'/x^2 ?

why do you have an extra z

[main mural]

yeah tru

[flat frigate]

artemetra

[main mural]

jesus i forgot this stuff

that makes a bit more sense

[wind laurel]

lol fair tbh i never actually felt like learning implicit differentiation

its pretty much just mainly intuition 😭

[fathom jewel]

What rule did you use?

,w d/dx of z(x)/x

[wind laurel]

i get y' = z' x^-1 - z x^-2

[fathom jewel]

yea

[flat frigate]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by 4nzon

[wind laurel]

and for y'' just derive that againn

[fathom jewel]

nasty

[wind laurel]

yea

[mild mauve]

Y'all are still discussing the problem I solve 15 mins ago

https://tenor.com/view/get-a-load-of-this-get-a-load-of-this-guy-look-at-this-guy-come-on-bruh-gif-18066748999193463069

[open wedge]

!redir

[safe radish]

This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

[open wedge]

ik you helped

but please don't send smth unrelated and not helpful in other people's channel

[wind laurel]

anyways pretty sure they messed up because this is the solution according to wa

[mild mauve]

No I mean like what's the point of helping when someone don't even look at it

[wind laurel]

i looked at your solution and it is incorrect

[open wedge]

now that's what you call The problem I solved 15 mins ago

[mild mauve]

no it's not you didn't do by my process

or you did it incorrectly

[open wedge]

blud

OP read your solution

not

doing by your sols

[wind laurel]

immediatly at the first step, we should not be taking the derivative of z with respect to y, but we need to find dy/dx in terms of dz/dx

[mild mauve]

If that's the case then do everything same just do it by dy/dx

so simple

[wind laurel]

thats exactly what I've done

yet (1) does not reduce to (2)

so i've reached the conclusion that the question is simply stated incorrectly

actually looking at it i think it's a typo because the sub y = z/x^2 seems like it would work

[lean otter]

i agree cannot match the (2)

[wind laurel]

anyways thanks everyone, i'll close the channel now unless someone has something to add

[wind laurel]

thanks :)

.close

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[safe radish]

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[primal gazelle]

why is any k-cycle a composition of k-1 transpositions and what do they mean by composition like taking k permutations where there is just one transposition and composing them?

[primal gazelle]

idk I feel they use the word transposition in two ways

I showed 1-3

[delicate shore]

use some toy cases I guess

how do you create the cycle (1 2 3 4) by using transpositions

[primal gazelle]

I move 1 to the end and thats 3 transpositions

okay let me try some more

ok so if we want cycle say 3 2 1 4
swap 3rd element with the 2nd, 2nd with 1st and 1st with the 4th

from identity

that seems to work

and it works because

[delicate shore]

uhh

by 3rd element you mean the "1"?

[primal gazelle]

1 2 3 4
1 3 2 4
3 1 2 4
4 1 2 3

[delicate shore]

yes ok so you are creating (1 2 3 4)

[primal gazelle]

this is cycle [3 2 1 4] or not

[delicate shore]

well ok that works but do you see how to generalise it to longer cycles

[primal gazelle]

i swap elements at consecutive positions

[delicate shore]

you have shown that (1 2 3 4) = (1 4)(1 2)(2 3)

it does work but can you find a more "organised" way of doing it

one that is more straightforward to generalise to longer cycles

[primal gazelle]

okay

[delicate shore]

hint: compute ||(1 4)(1 3)(1 2)||, what do you find?

[primal gazelle]

what do you mean by (1 4)(1 2)(2 3)

transpositions?

[delicate shore]

that was the transpositions you performed

[delicate shore]

here

(remember, it's read right to left, like function composition)

[primal gazelle]

ooh alright makse sense now

[safe radish]

@primal gazelle Has your question been resolved?

[primal gazelle]

In each step at a1 we have a_k and we exchange it with a_{k+1}, so after all steps at a_1 we will have a2 at a2 we will have a3 and so on

ok this isnt the best explanation

but it works

thank you very much!

.close

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[narrow rapids]

hi

[upbeat swan]

omg it's the king of curses

[lean otter]

hello what brings you here

[winter whale]

not the math question from the heian era

[upbeat swan]

[narrow rapids]

i love MATH

[winter whale]

if you have a question, please ask

[narrow rapids]

me

[fierce kernel]

@narrow rapids Stop trolling and overusing the help channels

[winter whale]

#discussion is for the good ol discussion

[fierce kernel]

<@&268886789983436800> dude is using multiple help channels to ask 1+1

double ping damn sorry cloud

[narrow rapids]

not your bisness

[winter whale]

dang he really got on your nerves.

[fierce kernel]

tutors and undergrads take time out of their day to help people. wasting channels and time is POS behaviour

[median vigil]

@narrow rapids Come back when you are ready to follow the rules.

.close

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[wind lodge]

I'm trying to solve the SLp $f''+\lambda f=0, \ \ f(0)=0, f'(l)=0$. I get that $\lambda=0$ gives $f\equiv0$, and $\lambda\neq0$ gives $f_n(x)=\sin(\pi n x/l)$ with $\lambda_n=\pi^2n^2/l^2$ for $n\in\mathbb{N}$. The answer should be $f_n(x)=\sin((n-1/2)\pi x /l)$, with $\lambda_n=\frac{(n-1/2)^2\pi ^2}{l^2}$, $n\in\mathbb{N}$. I can't figure out where i go wrong

[flat frigate]

Alrik

[plucky elk]

,rotate

[plucky elk]

what does SLp mean

[wind lodge]

Sturm–Liouville problem

I want to find the eigenfunctions and their eigenvalues

[plucky elk]

can you take a picture of the original question. doesn't seem like your f_n(x) satisfies f'(l) = 0. cos(pi n) = + or -1

[plucky elk]

how did you get this eqn

[wind lodge]

That's a good point

[raven heart]

Your solution is exactly what you'd get with f(l) = 0, not f'(l) = 0, looks like you screwed up your computation of f'(l)

[wind lodge]

Yes

I did

because i say i get positive mu for the derivative

$f'(x)=A \mu (e^{\mu x}-e^{-\mu x})$

[flat frigate]

Alrik

[wind lodge]

which is likely wrong, it should be positive here

i should get $f'(x)=A \mu (e^{\mu x}+e^{-\mu x})$

[flat frigate]

Alrik

[raven heart]

indeed

[wind lodge]

Thanks very much!

.close

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[half vortex]

“We choose a number. The difference between the square of the number that follows it and the square of the number that precedes it will always be equal to four times the original number.” I need to write this expression using x as a variable and prove this equality. I dont understand how to prove it and I think idk the number either

[simple gazelle]

try writing it down as an expression of x

[half vortex]

Yeah I did but then idk how to prove it

[simple gazelle]

what did you get

[half vortex]

(x+1)² - (x-1)² = 4x

[simple gazelle]

good

try expanding the squares

there's also that

[half vortex]

So if its a difference, I have to like do a² + b² - 2ab but with my numbers right?

[simple gazelle]

you could do it as said above

notice that it's written in the form a² - b²

then you use the formula

[half vortex]

uhm im kinda lost sorry

[simple gazelle]

always remember that $a^2 - b^2 = (a+b)(a-b)$

[flat frigate]

MarcoMa210

[simple gazelle]

you can substitute (x-1) and (x+1) to get $[(x+1)+(x-1)][(x+1)-(x-1)]$

[flat frigate]

MarcoMa210

[solar hazel]

mommy tag

[half vortex]

doesnt that equals to 0 tho?

[simple gazelle]

no

what's (x+1) + (x-1)?

[half vortex]

0??

[simple gazelle]

why would it be 0

x + 1 + x - 1

[half vortex]

oh wait

2x??

or x²??

[simple gazelle]

yes 2x

now what's (x+1) - (x-1)?

[half vortex]

2

[simple gazelle]

great

you're basically there

what's the expression now?

[half vortex]

yeah then 2x*2 = 4x

[simple gazelle]

which is true

there you have it

[half vortex]

tysm😭

.close

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[bitter breach]

@solar hazel do you know the theory of computation(TOC)?

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[late bobcat]

I m a confused soul trying to plot <3,8,1>

[edgy breach]

you should dot lines from z=1 so that you get a line passing over the x and y coordinates

[late bobcat]

didn’t I dot a line from z = 1

[edgy breach]

thats the height in the yz plane

actually not even

you do it like this

the 3D was throwing you off

[late bobcat]

so what’s the best way

to plot this

dotted gets so confusing ugh

<@&286206848099549185>

[plucky elk]

use https://www.desmos.com/3d

[simple gazelle]

desmos

actually goated

[visual kraken]

Unfortunately any method of plotting 3d points on a 2d piece of paper is gonna get ugly fast

Easiest solution would probably be to use a 3d grapher like desmos

[late bobcat]

I am not allowed to use

graphing software

on exam

[visual kraken]

Then you'll just have to suffer with those dotted lines

[late bobcat]

Can u teach me how to go with the dotted lines

Is it just intuition or is there a method

I mean

[visual kraken]

I recommend separating your x,y,z lines into a more hexagonal shape than square

[late bobcat]

x is parallel to y?

y will be to parallel to x?

like

I m confused how do I go about drawing the dotted lines

[visual kraken]

It's basically what you did up here, but you might notice that the topmost line you drew wouldn't actually be parallel to any of your axis

[late bobcat]

I just watched a

organic chem tutor video

I guess that wasn’t rlly that consistent

[visual kraken]

Like, visually it's parallel, but if you were to "rotate" the system you might notice that that line isn't parallel to the others

If this is something for a test, have you seen examples in your classes?

[plucky elk]

yea you use it to learn how to draw

just copy what desmos gives

[late bobcat]

Our professor

just did

1 example

I want to learn how to manually draw

like is there any set of rules

[visual kraken]

Not really

Like, this sort of thing is very much up to personal preference

[visual kraken]

Like, this is maybe a bit messy, but honestly it's not terrible

[late bobcat]

Can u tell me

how would u prefer

like

step by step

All YouTube videos

have failed

and AI sucks too

[plucky elk]

https://jillwilliams.github.io/Points-in-Three-Dimensions/

do the 10 examples

[visual kraken]

Yeah go through that

That looks like it could explain what you're looking for better than I could

[safe radish]

@late bobcat Has your question been resolved?

[safe radish]

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[formal peak]

Can anyone help me do the f problem ? I can’t calculate the integral

[formal peak]

I tried with (y-y1)/(y2-y1) = (x-x1)/(x2-x1) but it won’t do

[summer coral]

you can just use the formula for area of a triangle

[fathom jewel]

yeah, you just need to identify the geometric shapes, which are mostly triangles/rectangles

[formal peak]

Oh-

[summer coral]

for all of these problems, each integral can be split into some combination of either a triangle or a rectangle

[summer coral]

and remember for problem f, it is under the x-axis. therefore check whether your area should be positive or negative for this problem

[formal peak]

Yeah so it’s negative

Thank u, I will do it now

[summer coral]

ofc!

[safe radish]

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[knotty ledge]

I am confused on where I went wrong, I understand the grouping and getting the GCF's of both groups but am confused after that.

[summer coral]

your work looks fine so far, but x^2+3x+2 can be factored even more

recall either the diamond method or the table method, whichever you hae used to factor

[knotty ledge]

I do not know either of those

ohhh

[summer coral]

where you have to find two numbers that add to 3 and multiply to 2

[knotty ledge]

a x c and adds up to b

[summer coral]

yes

[knotty ledge]

ahhh I forgot

ty I understand now

[summer coral]

ofc!

[knotty ledge]

.close

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[safe radish]

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[vast sequoia]

is this ukmt?

where did you find this problem

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[vast sequoia]

did you solve it?

@livid cloud ?

your idea about symmetry was right

think about the mid point of the length AD

sorry no

the length of A to the midpoint of HG

and compare with the B to the midpoint of EF

what can you say about the positions of thise two midpoints btw

mvm thought you said yes to the ping

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[safe radish]

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

[vague phoenix]

Find the area of 2 triangles

@livid cloud

No need

The area formula you need is 1/2ab*sin

Area of 2 small triangles

Call S1 and S2, then find S1/S2

They have the same height lol

Then a/b =S1/S2

Trust, find areas of 2 small triangles

Using formula $S=\frac{1}{2}ab*sin\theta$

[flat frigate]

Minhh

[vague phoenix]

This. Thats the easiest way to find

Now find 2 areas in term of h where h is the height

[thick tendon]

What's 8.1853527719²

[vague phoenix]

!occupied

[safe radish]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[vague phoenix]

Use other channels

[thick tendon]

Ok

[vague phoenix]

Yeah you should learn it

For example, the small triangle on the right is $\frac{1/2}6h*sin30$

[flat frigate]

Minhh

[vague phoenix]

do the same to other triangle

Finding c wont help lol

Yes why dont you listen

We dont need height

Just find area of small triangles on the right and left interm of h

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[vague phoenix]

Ok ask if need help

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[primal wasp]

Can someone help

[cunning pasture]

What is the question?

Oh, find BD.

[vague phoenix]

Fire drawing on the right. Sorry if not related

[hasty wagon]

those x's are lengths right?

[primal wasp]

Yes

Its supposed to be triangle similaroty related subject idk it in english

[hasty wagon]

then the similar triangles doesn't looks right

ACM~DBM right?

[primal wasp]

Yes

[vague phoenix]

AC=6

[primal wasp]

No, AC is 6

[vague phoenix]

Is that all data given?

[primal wasp]

Yes

[vague phoenix]

You rotate location A and C

[primal wasp]

Ab and cd are chords

Oops

[primal wasp]

2x is given

[hasty wagon]

i know

i mean the given values seems to be wrong

not sure which though

[vague phoenix]

Use chord theorems

AM.MB = CM.MD

[primal wasp]

What does the . Mean

: 3

[hasty wagon]

multiply

[vague phoenix]

I will do it i think i find the solutions

Wait 5 mins i ss

[primal wasp]

Oh wait

[vague phoenix]

i think this might correct

[primal wasp]

Ohh I seee

Thank youuu

[vague phoenix]

No worries!

@primal wasp This might be helpful for your study by the way!👍

[primal wasp]

Thanks 🤤

[vague phoenix]

!done

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[primal wasp]

.close

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[versed wave]

what's the motivation behind chi-squared distribution? this definition imo just seems kinda arbitrary

[peak estuary]

its the sum of independent squared normal distributions

[versed wave]

oh interesting

i assume that v here is the number of squared normal rvs?

[peak estuary]

yes

but thats about as far as my knowledge goes

[versed wave]

hmm

do you know how was it discovered that chi-squared is just a special case of gamma?

[peak estuary]

probability guys are very good at recognising pdfs that look similar

or do you mean how it was discovered that chi squared has this pdf?

presumably you can just calculate that

[versed wave]

yeah, i was asking how did those people even find a correlation between gamma distribution and the sum of squared normal variables, in that the former is just Gamma(v/2, 2)

did the pdf just happen to line up with the gamma dist family, or from this definition you can deduce it's part of the gamma family

[peak estuary]

my guess is that it just happened to line up but I dont know

[versed wave]

fair enough

i'll keep this channel open in case anyone knows the answer

[safe radish]

@versed wave Has your question been resolved?

[safe radish]

@versed wave Has your question been resolved?

[safe radish]

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[violet fossil]

Hi guys
I need help with this sales report thing which has accounting too and I have never done accounting before.

[broken forum]

Ask away, there's all kinds of people who may be able to answer

[vague phoenix]

Ok any specific questions?

[violet fossil]

The excel formulas are not working out like.
I did everything manually and when I'm doing the formula stuff the numbers are just different
It's like a whole excel data idk how am I supposed to show you guys

[vague phoenix]

For example?

[honest perch]

profit = revenue - cost

[violet fossil]

IKKKK😭

[honest perch]

good

[violet fossil]

But like manual work I did for like a month and the formula thing I did are totally different

I literally would have to share my screen to make you guys understand

[broken forum]

Screenshots work

[violet fossil]

Okay wait

[broken forum]

np

[violet fossil]

My boss literally said "stop working on it"
Because I think I messed up 😭

[vague phoenix]

Chill and find a way to fix it👍

[violet fossil]

why is it $7,521? 😭

[broken forum]

You need to keep commas and dots consistent generally

[primal bone]

Also, I believe you SHOULDN'T be explicitly writing thousands-separators in Excel

[broken forum]

,w 56819-49242.12

[flat frigate]

Results provided by WolframAlpha

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Requested by nyxzore

[primal bone]

Excel freaks out because it doesn't recognise 4,329 as a NUMBER, but as a STRING

[broken forum]

Exakery

[primal bone]

So it won't do a mathematical calculation on that sort of data type

[broken forum]

4.329 is a number 4,329 is not

If I want to write out four thousand it's just 4000 not 4,000

[violet fossil]

ohhhhh

[primal bone]

If it's any consolidation, my dad asked me a similar question the other day, whose solution was the same as this one

It's quite easy to forget if you're not used to data types

[broken forum]

This also looks like a painfully old version of sheets

[primal bone]

I think the file is old, not the software

I mean it still has a dark mode

[violet fossil]

do i have to manually change each number now?

[primal bone]

Only the ones with commas, tbh

[violet fossil]

ok

[broken forum]

Well you don't need to change the 3rd column cuz they are results of formulas

Just everything in 1 and 2

[violet fossil]

hey but some are not even in commas but then why is it showing in commas

look

[primal bone]

Because the raw value is not what's in the cell

The raw value is in where that middle arrow is

The top arrow is what's converting what that raw value looks like, i.e. how it appears in the cell at the bottom

[violet fossil]

oh

[primal bone]

Same here - the raw value has no comma

[violet fossil]

so i should just choose it as a number not currency right?

[primal bone]

You can format it as a currency, that's fine

But the raw value is the important thing to note

Note how the raw value here isn't even a number - it's just a pointer to the cell to the left

All the formula stuff will still work, as long as you're a little more conscious that that's what's going on

[violet fossil]

ok

thank you guys for helping
really appreciate it!

i need to learn how to use excel first

[safe radish]

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[strong nexus]

A stone falls vertically downwards from a height of h = 100 m from a tower in 51° north latitude. The stone is undisturbed when released. How far from the tower's plumb point does the stone hit the ground? State the magnitude and direction of the deviations. Note: Consider the influence of centrifugal force and Coriolis force (here a double effect)!

[strong nexus]

The magnitude can be calculated with 2mvw sin(alpha)
Where alpha is the angle between v and w
Is that 51°?

[charred summit]

Make a sketch

[strong nexus]

Oh wait its 90° - 51° by that

[strong nexus]

So with this, we can calculate how far off the Coriolis force will get it away from the tower

Now how can we be sure it doesnt overlap with centrifugal at all?

In the sense that nothing cancels

[charred summit]

In this problem they're perpendicular so they don't cancel

[vernal portal]

if two forces are perpendicular to each other, they dont cancel so try finding the direction of the forces ig

[charred summit]

You can use the right hand rule to verify

[strong nexus]

For Coriolis my thumb points at v, index at w and middle finger goes to the back

How do you do centrifugal?

[charred summit]

By definition it always points outwards from the axis of rotation like he said

[strong nexus]

What does that mean, how do I position my fingers?

[charred summit]

You just don't use the rule for the centrifugal force since its direction is defined

[strong nexus]

Oh wait, its not RHR at all

[strong nexus]

Alright

[strong nexus]

What does that mean? In parallel to the axis of rotation?

[vernal portal]

like perpendicular to earths surface

[strong nexus]

Like this?

[charred summit]

Yes

[strong nexus]

Why not in the other direction

[charred summit]

Perpendicular to the rotation axis

[strong nexus]

uh

How can it point outwards from the axis of rotation

How can something point outwards from a vector

[charred summit]

The axis is not a vector, it's just a line

[strong nexus]

Ok, yeah

But still

Ok, F_z = mw^2 * r

So it points in the same direction as r

Isnt r the falling stone?

Then it should point from the center to v

[charred summit]

r is measured from the rotation axis, not the center of earth

Fz points in the same direction as r and away from the rotation axis

Not from earth's center

[strong nexus]

But arent there infinitely many possible ways to "point away from a line", what does this mean?

Arent these all "pointing outwards of the axis of rotation"

[charred summit]

There are an infinite amount of ways the point away from a line, although they should be perpendicular though for centrifugal force

That doesn't mean you have to draw them all

Centrifugal force works on all sides of earth, doesn't it?

[strong nexus]

Oh, right

[charred summit]

You just need to draw the one that works on the stone

[charred summit]

Except the north and south pole of course

[safe radish]

@strong nexus Has your question been resolved?

[strong nexus]

Wait so it applies outwards?

Something like this?

[charred summit]

Yes it does, it's the same force that causes a ball to fly outwards when spinning it while it's attached to a string

[charred summit]

Your drawing isn't very clear anymore tbf

[strong nexus]

True haha, anyways basically it wants to send the stone away from the earth

[charred summit]

Correct

[strong nexus]

Thanks! And F_C applies "horizontally to the surface" so its perpendicular to that

Thanks!

[charred summit]

You're welcome

[strong nexus]

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[strong nexus]

Two stars with masses M and m move in circular orbits (with radii r1 and r2) around their common center of mass. The distance between the stars is d. Determine the orbital period T for the two stars as a function of the two masses m and M and the distance d.

[strong nexus]

Do we need to determine the common center of mass first?

[plucky elk]

That's probably what CM stands for

[strong nexus]

Yeah

[strong nexus]

Apparently we can say mr1 = Mr2?

[safe radish]

@strong nexus Has your question been resolved?

[fierce kernel]

yes

[strong nexus]

Why

[fierce kernel]

fundamental definition of the centre of mass. we know that if you put it all together it should equal zero m(-r1)+M(r2)=0

[strong nexus]

I define the center as r = sum m_i r_i /sum m_i

[fierce kernel]

astronomy and physics arent my thing i cant help you much, but you seem to be stuck here for a bit so i thought id give you a nudge. try to identify more relationships

[fierce kernel]

thats correct as well, you can reach the same conclusion with this

[strong nexus]

By setting r = 0?

[fierce kernel]

exactly!

and you know how your denominators act, they cant be zero, you can get to the exact same conclusion this way

[strong nexus]

But why can we just set r = 0 like that?

[fierce kernel]

you have basically just set it (the cm) as your origin

do you want me to tex it for you so you get which r I mean?

[strong nexus]

Oh, so we can do that WLOG

[fierce kernel]

sure

@strong nexus got it now?

[strong nexus]

Yep, thanks a lot! From here on we can set F_G = F_Z and we should be able to solve for T

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[deft furnace]

you guys, im struggling at the part where we derivate the 3(f(x)), do i do it in the muliplication type of thingy or just put 3(f'(x))

[simple gazelle]

always remember that $(cf(x))' = cf'(x)$

[flat frigate]

MarcoMa210

[simple gazelle]

you can take the constant out of the derivative

[deft furnace]

ohh ok tysmmm!!!

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[lilac relic]

What's n?

When y = 1,5,5,4

Maybe I need to give more information

OH N = THE SAMPLES

So n -1 would be

3

in this case

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[obtuse drum]

I’m not sure how to continue with finding the zeros

[pallid magnet]

i am not sure what your question is exactly

it's a circle centered on the origin

with a radius of 1

[modest tusk]

It is actually not a circle, but the equation is very similar.

[pallid magnet]

so the answer is 1

[pallid magnet]

wdym?

[light shoal]

not for the stated reason, though

[obtuse drum]

They want me to find the critical point basically

[manic patrol]

I'm pretty sure that's a hyperbola, not a circle.

[modest tusk]

The equation for a circle would be x^2+y^2=1, not -

[obtuse drum]

And using that with a constraint equation which is the x2-y2=1

[modest tusk]

For this problem, I recommend starting by rewriting it in terms of x.

[pallid magnet]

OH mb

[obtuse drum]

To find like where on that equation is the shortest distance but I’m not sure how to start with it 😭

[light shoal]

did you try solving for lambda? (be careful about the cases x=0 or y=0)

[modest tusk]

This gives you y=sqrt(x^2-1)

[pallid magnet]

lemme redo it real quick

[obtuse drum]

Yes I tried with x=0 and plugged it in but y doesn’t exist I think

[light shoal]

what did you get?

[obtuse drum]

Cause I got lambda =+_1 and x=0 and y=0

[pallid magnet]

try the other way around

[modest tusk]

With the equation I mentioned before, notably here you actually will get + or - the square root, giving you the positive and negative ranges of this function.

It is then a bit easier to handle as there are now only single values for y for any given x as opposed to 2 potential ones before.

[light shoal]

can you show your work for this?

[obtuse drum]

I got x is plus negative 1?

[obtuse drum]

Wait lememtryu that

[light shoal]

look at the equation for the curve, x^2 - y^2 = 1

one of the variables can't be 0 (which one?)

[obtuse drum]

X

[light shoal]

right

so use the dF/dx equation and solve for lambda

and plug it into the other equation

[obtuse drum]

Would lambda be -1?

[light shoal]

yep

[obtuse drum]

Plug into df/dy right or the constraint

[light shoal]

into the dF/dy

and solve for y

[pallid magnet]

I solved it from $x^2 + y^2 = (x^2 - y^2) + 2y^2$

[obtuse drum]

I got 4y=0 so the minimum would be at (0,1)?

[flat frigate]

IdelUser404

[light shoal]

i assume the intent here is to do it using lagrange multipliers

[obtuse drum]

Yeah

Idk if I did it right tho

[pallid magnet]

i guess but i feel like it's overkill

[light shoal]

y=0 is correct, there are two possible x values that go with that right?

[pallid magnet]

and...?

[obtuse drum]

Wait this is with partial derivatives right

[light shoal]

the goal is presumably to show how to use lagrange multipliers in a simple setting

[obtuse drum]

OHH

Plus minus 1 mb

[light shoal]

yep

[obtuse drum]

Wait what’s the less overkill way to solve this

Just like do partial and plug it in for one of the variables?

[light shoal]

btw if you look at the graph you can see visually that your answer is correct

[obtuse drum]

Ohh ok tyy

[pallid magnet]

just algebra and factoring tricks

like the one above

[obtuse drum]

Where’d you get the 2y^2 from?

[pallid magnet]

um...

$y^2 - 2y^2 = y^2$

[obtuse drum]

Oh nvm I see what you did

[flat frigate]

IdelUser404

[obtuse drum]

😭😭 it’s the same thing when u simplify you just made it two terms

Ok ty!

.close

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[faint glade]

How do you factor something like $9-(2x+1)^2$?

[flat frigate]

Vortac

[faint glade]

I know for trinomials, you multiply the first and last term, then find m+n and mn

[tardy mango]

difference of squares

[faint glade]

?

[lament violet]

9 is a perfect square and (2x+1)^2 is also

[austere forge]

a^2 - b^2 = (a-b)(a+b)

can you apply that here?

[vague phoenix]

You can apply this formula, note that 9 is a square of 3 @faint glade

[lament violet]

very nicely noticed tbh i didnt think to apply that lol

[faint glade]

Do I just memorize a few of the special identities and then practice enough to spot them?