help-49

[glacial nymph]

what did i mess up

[sharp coral]

do you know what the multiplicity of a root is?

[glacial nymph]

i think so

[sharp coral]

can you explain what it is?

[midnight plank]

@glacial nymph Has your question been resolved?

[midnight plank]

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[glacial nymph]

is it smth like this?

[glacial nymph]

.close

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[upbeat plinth]

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[upbeat plinth]

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[glacial elm]

here the power of t is just -(3/2)

u can just use power rule

[upbeat plinth]

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[glacial elm]

for sqrt the power is 1/2

[upbeat plinth]

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[glacial elm]

mutiply by one t which is 1+1/2=3/2

since it is at denominator , the power is neg

so is -3/2

[upbeat plinth]

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[dusty portal]

so, you got it?

[upbeat plinth]

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[glacial elm]

yeah

[grand pond]

Kenzo

[upbeat plinth]

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[glacial elm]

then just use power rule of integration

[upbeat plinth]

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[lavish venture]

you have to use {}

[upbeat plinth]

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[dusty portal]

👍

[midnight plank]

✅

[upbeat plinth]

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[lavish venture]

also don’t use * use \cdot

only one t

[grand pond]

Kenzo

[lavish venture]

but it’s also t^-1

since it’s in the denominator

[upbeat plinth]

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[lavish venture]

$\int t^{-\frac{3}{2}} \dd{t}$

[grand pond]

knief

[upbeat plinth]

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[lavish venture]

why is dt here

and why is there another t

$\int x^m \dd{x} = \frac{x^{m+1}}{m+1} \quad m \neq -1$

[upbeat plinth]

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[grand pond]

knief

[lavish venture]

finally

[lavish venture]

yes

[upbeat plinth]

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[lavish venture]

and who cares about + C

[last slate]

use exponent laws so t^-(1+1/2) and then use the power law for intergration.

[grand pond]

Kenzo

[upbeat plinth]

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[lavish venture]

why is there a t^2

[lavish venture]

^

[upbeat plinth]

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[lavish venture]

no

look at the integral i wrote again

what is -3/2 + 1?

[upbeat plinth]

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[lavish venture]

why is there another t??

{}

[lavish venture]

where do you see an extra x here

[upbeat plinth]

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[lavish venture]

you’re missing a }

for the first \frac

[last slate]

lol latex

[lavish venture]

right but why is there a t

[grand pond]

Kenzo

[lavish venture]

there ya go

[upbeat plinth]

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[lavish venture]

why was it there?

[lavish venture]

that’s because it’s 1 dt

[upbeat plinth]

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[lavish venture]

it’s t^-3/2

not 1

[upbeat plinth]

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[lavish venture]

also just treat the dt as an indicator for what you’re integrating with respect to that’s all you really need to care about

[upbeat plinth]

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[lavish venture]

ok ok

[upbeat plinth]

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[midnight plank]

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[midnight plank]

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[prime swallow]

I need help with 7-6 just in general

[midnight plank]

@prime swallow Has your question been resolved?

[midnight plank]

@prime swallow Has your question been resolved?

[olive yew]

what are the things that you dont understand?

[midnight plank]

@prime swallow Has your question been resolved?

[prime swallow]

The flowchart

[heady plume]

Two lines are parallel if the co-interior angles made by a transversal are supplementary, i.e., in this case, MN || PO if NMP + OPM = 180°, which is indeed the case.

b. Consider the given figure, if MN = PO != MP, you have a rectangle in your hands. If MN = PO = MP, a square, if MN != PO, a trapezoid [note that trapezoid is defined as having **only** one pair of sides parallel, so mentioning rectangle separate from trapezoid is necessary

c. You should attempt and it'd not prove too difficult. 
d. No, show that you can have a quadrilateral with two opposite right angles but the consecutive angles are not supplementary, thus not making it a trapezoid```

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@prime swallow Has your question been resolved?

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[spice oriole]

How to find average from this

[midnight plank]

@spice oriole Has your question been resolved?

[fresh sparrow]

maybe a translation would be helpful? i don't think many people here speak icelandic

[spice oriole]

YOU CAN SEE 4 GROUPS

EACH HAVING 15,20,10,18 STUDENTS

AND EVERYONES AVERAGE HEIGHT

AND I NEED TO FIND AVERGAE HEIGHT OF ALL STUDENTS

[gusty falcon]

(1.62 * 15 + 1.48 * 20 + 1.53 * 10 + 1.4 * 18)/(15 + 20 + 10 + 18)

[spice oriole]

I KNOW THAT.

[heady plume]

!done tyvm baii

[spice oriole]

AHHHHHHHHH

[midnight plank]

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

[spice oriole]

ARGGGGGFHHHH

OOOOO

AAAAAAAAAAAAAA

[heady plume]

@spice oriole close if you're opening another channel

.close ||damn ragequit||

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[last slate]

Hello

[last slate]

"The set V is closed under addition... 2 by 2 matrices as the end result"

What do they mean by that?

[sage helm]

It means that if you add two 2x2 matrices you will always get a 2x2 matrix

[runic hamlet]

if we take two 2x2 matrices and add them, we still get a 2x2 matrix

[sage helm]

In general if you're performing an operation on a set and the resultant remains in the set then you can say that operation is closed on the set

[runic hamlet]

and if we take a 2x2 matrix and scale it, we also get a 2x2 matrix

[last slate]

What do they mean by closed then

Oh

Thanks

.close

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[last slate]

.reopen

[midnight plank]

✅

[last slate]

,rccw

[last slate]

Example 6

Are we supposed to say h(x)= f(x) +g(x)

And kf(x) = p(x)

[subtle blaze]

What do you mean

[last slate]

And hence the set V is closed under addition and scalar multiplication because both operations give another function

[burnt flame]

yes

[last slate]

Why didn't they mention "V is closed "?

[subtle blaze]

Another function (in V)

[subtle blaze]

V is a set

Sets can’t be closed

Only operations can be closed

[last slate]

Mb didn't write the whole sentence

"V is closed under addition and scalar multiplication"

[subtle blaze]

Well idk you cut off part of the paragraph

Also it’s just giving you examples of vector spaces

[last slate]

So it's not necessary to write that sentence always ?

[subtle blaze]

Why would they need to write it

They are giving examples of vector spaces

[last slate]

,rccw

[last slate]

They wrote it for the first 3 examples

[subtle blaze]

You’ve cut off what comes next

Maybe it does say it

To talk about a vector space you need only provide the set and the operations

You might want to prove that what you’ve given is indeed a vector space

But you also just trust them I guess

[last slate]

They started to prove axioms after that

[runic hamlet]

if you wanna talk about operations its a good thing to say that they are indeed well defined

aka the codomain works

[subtle blaze]

Then that’s where they talk about closure

[last slate]

I'm lost

[midnight plank]

@last slate Has your question been resolved?

[runic hamlet]

what exactly is the point of confusion currently

also, what is on the next page

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[winged snow]

What's the volume and other formulas for an isosceles triangle rotated around it's base? What's it even look like

[zenith stone]

visualize what happens when an isosceles triangle is rotated around its base

(hint: it forms a cone)

[dusty portal]

you can also just consider a line being rotated around the y-axis; for a general equation y=a-x for 0<x<a

[heady plume]

it does not form a cone tho?

[dusty portal]

what

[winged snow]

I don't thinks it's a cone

[dusty portal]

define "base"

and/or draw a picture of what you mean

[winged snow]

ABC AB - base BC AC equal sides

[heady plume]

how does rotating this isosceles triangle about AB make this a cone

although you can make the resultant into two cones

[winged snow]

And now rotate triangle around AB

I don't think it's cones though idk

[dusty portal]

if you rotate it "sideways"

like, revolve it

from left to right

[heady plume]

OP asked to rotate about AB :| not about altitude to AB

[dusty portal]

tetrahedron?

[heady plume]

it's two cones uff

[winged snow]

It should be like a pyramid but the edges are curves like a circle

[dusty portal]

wtf

this too confusing for me bro

😭

[heady plume]

drop altitude from C to AB at X and you'll see that it's two cones, one with CX radius forming cone with top B, and one with CX radius forming cone with top A

[heady plume]

huh? it's not pyramind omg ._. you're revolving

[winged snow]

Oh

Aha thanks

It was weird I couldn't find an image online

.done

.close

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[dark warren]

idk how to find but if i had to guess is y= 25ish and x=10ish

[buoyant yoke]

,rccw

[flat plinth]

Do you know which properties of angles are associated with parallel lines?

[dark warren]

uh

perchance

do u have an example

[flat plinth]

Alternating angles =

[dark warren]

like

what they are

or is there smt else

like alt angles are just on diff sides right

like alt interior are switched and on the inside

i think

or do u mean they are congruent

[flat plinth]

Alternating angles are opposite sides of the adjoining straight line

[dark warren]

yea

ik those if thats what u mean

[flat plinth]

That particular one applies to your question

[dark warren]

so like

3x=4y

ive never done it on triangles

only like transversals i think theyre called

[flat plinth]

The triangle doesn't matter

[flat plinth]

No

Always try to find a Z or N shape

If you're struggling to see it

Since you're parallel lines are horizontal it's a Z shape

[dark warren]

can it be backwards z or not

[flat plinth]

It can be

[dark warren]

ok i see it

[flat plinth]

Lovely

[dark warren]

so 2y?

[flat plinth]

No

[dark warren]

💔

[flat plinth]

Which 2 angles are =?

[dark warren]

so its not 3x=2y or 3x=4y

[flat plinth]

Nope

[dark warren]

5x-20 and 3x?

oh

if thats right x=10

[flat plinth]

Exactly

[dark warren]

gotcha

[flat plinth]

[dark warren]

tysm man

[flat plinth]

Np

[dark warren]

have a good one

.close

[midnight plank]

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[mint ravine]

I don't see what it means by "the sum of the four numbers on each of the five segments" in this problem:

[mint ravine]

Unless I'm overthinking it or smth

what 4 numbers are they talking about

[slender walrus]

on each straight line

e.g the horizontal one
a + 1 + 4 + c

[mint ravine]

oh okay i understand it now

thanks

.close

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[grand pond]

Result:

0

[midnight plank]

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[gritty hatch]

ight

[gritty hatch]

https://media.discordapp.net/attachments/905286165316403271/1330345829243944960/20250118_200011.jpg?ex=679241da&is=6790f05a&hm=2de29d5172115a95bbebfcf6e04c6d5f6ae4110d403eb302c65f30cd0fc671b7&=&format=webp&width=456&height=608

No more wasted thought, I will complete my self-development by removing all the unneccesary things.

Anyways, enough of that.

[gritty hatch]

So, I have a question.

[gritty hatch]

1. Why did they divide and multiply b by 2?
2. What is the purpose of doing so and the operation they hoped to achieve?
3. If they hoped to achieve something what is the purpose that thing serves?

• Give 2 separate explanations.
• Explain with clarity and simplify your reasoning and account for possible lack of knowledge I may have.
• Is this a neccesary operation?
• What would occure were this operation ommited?

https://tenor.com/view/divider-globe-world-space-sci-fi-gif-2881399313018008423

[gritty hatch]

If I expand (x+b/2)^2 it gives 5, so they should've been able to create (x+b/2)^2 whether or not they multiplied and divided. Did they do it for some weird technical purpose?

[heady plume]

2. The purpose of doing so is to make it easier for the reader to comprehend and follow what is going on.
3. If you're able to get insights from this middle term manipulation, you'd not only understand the derivation for the quadratic formula and re-derive it yourself if necessary, you'll also get insights into completing the square method which will help factorize quadratic later on in cases where you're not able to use quadratic formula.

It is not a necessary operation and one can simply assume the reader has skimmed through the prerequisites before getting into the topic of quadratic equations, specifically the quadratic formula, and omit the step by writing x² + bx + c = (x + 0.5b)² + c - 0.25b², leaving it on the user to derive the explanation.```

[gritty hatch]

• If I were to simply omit the division and multiplication and add (b/2)^2 to the equation as they did and go straight into (x+b/2)^2 would that work?

Though I will attempt to understand the things you wrote prior.

[formal blade]

Can you explain why it would work?

If so then sure.

[gritty hatch]

Can you explain why it would not?

[formal blade]

The purpose of this derivation is to prove the quadratic formula

The onus is on the person doing the proving to justify why they can do each step.

So, the answer is sure, you can omit that if you can justify why the proof still works.

I think instead of doing that you should take responsibility for the way you say things rather than pushing blame onto other people for how you make them feel.

Best to drop this

You are here for math

No I'm not going to accept this as just solely being my fault for misinterpreting you.

Multiple people have complained to me about this issue regarding you.

I don't care how you interpret it.

We aren't debating it.

I am telling you nicely to be considerate

This is me warning you to not do something.

Back to the math, the purpose of this exercise is to walk you through a proof of the quadratic formula.

[gritty hatch]

I will delete for further clarity

anyways, continuing

[formal blade]

Sort of a tangent but this proof is very old and has been done in many ways by many different people so you can find variations of it if you try as well.

[gritty hatch]

• Can you further describe the purpose of writing the division and multiplication as it pertains to the form pursued?
• Can you further describe the form and how it works and is achieved?

[formal blade]

For the first point arya explained why that was done pretty well imo

If you have a more specific question about it that might help. But the point is just that it's something you can do algebraically that leads us to the form we want at the end.

You can DO anything you like in a proof so long as you aren't violating some logical/mathematical rule.

But not all things lead you to the conclusion and some things are complicated and make the proof hard to follow.

[quiet hinge]

the purpose of writing division and multiplication is to achieve the form that is being pursued

[gritty hatch]

• Can you describe forms, their purposes, the merits and functions of them, and why we pursue them?

[formal blade]

Might help to clarify this question

It's a bit vague is all

[gritty hatch]

• If I ask repetitively it is because I may attempt to amass multiple explanations to a singular thing as one alone may fail at times. (it would work it would just take longer)

[heady plume]

Rather than looking at the same expression, looking for multiple explanations to the exact same thing, you might want to look at possible varieties of the same expression and fluid but countably many ways to deal with them so as to increase your proficiency on the topic.

In techno language, I advise you to move on and start practicing questions on quadratic factorization instead of sitting on the 2/2 = 1 issue and debating philosophical insights for the same.

[gritty hatch]

tbf philosophy is over unless expressed in a direct, clear, and precise format with a more clearly expressed question (it was actually psychology not just philosophy as I was attempting to repair a broken part of myself)

But, can you define what others exist or what in particular I should learn?

[formal blade]

The big picture pov for why you'd wanna do this is that it leads you to a way to test whether an equation of the form Ax^2 + Bx + C = 0 has solutions and to give those solutions in one single formula.

The actual technique to doing this is literally just completing the square in general

[gritty hatch]

close

[formal blade]

I'm not totally meaning to answer your question

[quiet hinge]

wdym by "close"?

[formal blade]

Just explaining why people care about this specific derivation and how it relates to completing the square

Do whatever you want with what I said lmao

[gritty hatch]

If you did that but for the 2(b/2)^2 and why its done as it pertains to that formula it might explain it

[quiet hinge]

the explanation is in the equations

[formal blade]

The first response to you explains basically everything.

[gritty hatch]

sure

[formal blade]

"We want the perfect square trinomial identity. Might as well put it in a form that looks like that."

[gritty hatch]

??? and

that isnt the question

unless its blunt

[formal blade]

Well then clarify the question?

Are you lost on why we want to do that?

[gritty hatch]

ahem, why do we w a n t the trinomial identity? what do you gain from that?

yes??????

[formal blade]

Because we can solve by subtraction and square roots that way

I.e., because immediately after that I can solve for x explicitly

Which is what I did in the non-cropped image lol

[formal blade]

Coincidentally that is also roundabout what I am explaining here too lol

[gritty hatch]

but i can do that anyways

dont you see how this explanation fails?

because anything you say i can do with your method i can do with mine

and you havent presented a way in which itd fail

[radiant roost]

do you agree that (x+b/2)^2 = x^2 + 2(b/2)x + (b/2)^2 ?

[gritty hatch]

no

and yes

[radiant roost]

you can prove it by expanding

[gritty hatch]

i agree that (x+b/2)^2 = x^2 + bx + (b/2)^2 ?

not that 2(b/2)x

[radiant roost]

oh

[gritty hatch]

simutaneously, I do agree with this, just that this is redundant and doesnt do anything

it is equal, but not useful

[radiant roost]

but it's equal

[quiet hinge]

can you stop switching between whether you do agree or not

[gritty hatch]

it doesnt "do" anything it expands to b not the simplified form

[radiant roost]

i don't really need to convince you that it's useful. if you think it's equal, i'm happy enough

[small helm]

they did this to complete the square of x+b/2 to make it easier to find the value of x

its basically the quadratic formula but with extra steps

[gritty hatch]

they did not

as they simply created (b/2)^2 by inserting it into the formula afterwords

turning the original B into that form is redundant

[radiant roost]

it's basically an intermediate step
because if we expand, we see that intermediate step
so when factoring, it's like expanding backwards. we can pass through the intermediate step as we go backwards.

[gritty hatch]

I dont see that intermediate step when going backwards

or forwards

[radiant roost]

you see it when going forwards

you will see it if you use FOIL

[gritty hatch]

oh i might know what it is now

erm

and if im right i f####g hate this

[gritty hatch]

do they do it to exemplify it and extrude off of the base?

basically

[radiant roost]

[gritty hatch]

do they do it so that the original number converts to a form most similar to the number

they are trying to add to the equation?

[radiant roost]

what

[gritty hatch]

and if so why isnt substitution like just adding whatever you want without a need for a base-number/prior ammount to make it based on?

[radiant roost]

i can't understand what you're saying

[gritty hatch]

basically

yk how they have

(b/2)^2

right

[radiant roost]

yes

[gritty hatch]

and how thats based on

B

[radiant roost]

yes

[gritty hatch]

but divided by 2 and squared

so if B is the base

[gritty hatch]

when you attempt to "make" the perfect square which is this

[heady plume]

then let us all see you attempt to factorize a^4 + 4b^4 ^-^

[radiant roost]

(b/2)^2 is not the perfect square

[gritty hatch]

you would go through the operation seen right here

actually no...

wait

[heady plume]

it'll be indeed amazing when you use something "redundant" to go through one step of factorization

[gritty hatch]

2(b/2) is not apart of that

F

well b/2 is but not the multiplication

[gritty hatch]

!

what do you mean it isnt the perfect square

[radiant roost]

i mean the perfect square we're trying to make is (x+b/2)^2

(b/2)^2 is a perfect square too but it's not the one we're trying to make

[gritty hatch]

if 10div2 squared you get 25 whos factors add to 10 and square to 25

this proves its the PSquare

a

[gritty hatch]

ok

[radiant roost]

yeah "a" perfect square

i mean, it could be a fraction

[gritty hatch]

wdym

[radiant roost]

it could be 1.234 or something

[gritty hatch]

and that would matter why?

[radiant roost]

well i'm just not sure what you mean by "(b/2)^2 is a perfect square"

it's expressed as the square of the quantity b/2, that's for sure

a "perfect square" usually refers to an integer

[gritty hatch]

elaborate

why must it be an integer

[radiant roost]

it's just a definition

[gritty hatch]

if it has factors that add to the number and multiply to itself

h o w is it not a perfect square

function>form

[radiant roost]

i guess it is a perfect square in the real numbers

[gritty hatch]

snez~1

https://www.youtube.com/watch?v=XAGqbFsk_XE&list=RDMM&index=3

https://media.discordapp.net/attachments/1015759484708208792/1045835008012714017/a_3a6633f55a6525e045c618e50b97782a12.gif

ight

1 sec

[gritty hatch]

why does that matter

all you do to make that is add the perfect square onto it

the only time you cant do it is if X has a coefficient

but even then you can just divide

[radiant roost]

i don't know why it matters

it matters because we're completing the square

if it doesn't matter to you that's fine

[midnight plank]

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[midnight plank]

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[mint ravine]

Can someone please help me on this question

[mint ravine]

I think the answer might be 8 but I need someone to prove it

<@&286206848099549185>

[violet storm]

!15m

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[mint ravine]

can you please help me on this question

[delicate sage]

How did you get 8?

[mint ravine]

I tried tracing the longest path the ant could take

but i think

that's wrong

[delicate sage]

It's 8

I just wanted to know the path you traced

Wait lemme sketch

[mint ravine]

ok

[violet storm]

i can definitely see a path thats 9, im trying to convince myself whether 10 is possible or not

[mint ravine]

oh yeah i can see one too

i don't think 10 is possible because of all of the paths would be taken

[delicate sage]

How is 9 possible?

[delicate sage]

Can't use the topmost vertex

[cedar coral]

you can do 12

[violet storm]

it's only using it for the beginning and end of the path

[mint ravine]

how

[cedar coral]

oh wait, 12 is if you do go on top vertex twice

[delicate sage]

Ohh !!

[cedar coral]

yeah, if you can only be on top vertex at start and end it's 9

[mint ravine]

so the maximum units is 9

ok

you can't go through the top vertex unless it's at the start or the end

[delicate sage]

I didn't see ! I overlooked the Q

[mint ravine]

no worries lol

[violet storm]

ok so one way you can prove that 10 is not possible is:
each non terminal vertex has to have an even number of segments connected to it, since one is arriving and another is leaving. but only two segments from the top half can be connected because of the top vertex limitation, so the most connections two of the middle segments can have is 2, while the other two middle segments can have 4

[cedar coral]

thing is, if you can only be on top vertex at start and end, it means that not all vertexes have 4 edges connecting them.
Top vertex would only have two, and whichever vertexes connect said two "removed" edges would have 3 edges connecting them.
Graph theory says the ones with 3 edges can either be only traversed twice, or need to be the endpoints. Since they cant be the endpoints, you cant traverse that 10th edge

[violet storm]

ya but i dont think you really need to bring in any graph theorey terminology here, can be thought of in simpler terms

[midnight plank]

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[mint ravine]

Can someone please help me on this question

[buoyant yoke]

are repeated steps allowed?

ah nvm, he cant move back anyway

[dusty portal]

what have u tried

[buoyant yoke]

try a 2x2 or a 3x3 board first and look for a pattern

[mint ravine]

there are 2 ways for him to reach the top corner in a 2x2 set up

[buoyant yoke]

and what about 3x3?

[mint ravine]

these are the two ways btw

[buoyant yoke]

there are 3, you can also go right then up

[dry merlin]

there are 3 ways, are there not?

[mint ravine]

i thought he has to go upward first and then to the right

so there are 3 ig

[buoyant yoke]

it says the moves are up, right, or upwards and to the right

which are 3 distinct moves

[mint ravine]

i think there are 8 ways for a 3 by 3

[buoyant yoke]

and how did you get that?

[mint ravine]

and so on

these are 3 ways

and i continued using this pattern and i got 8 ways

what did you get?

[dry merlin]

what about this pattern?

[mint ravine]

sure yeah that works

[mint ravine]

is that lichess?

[dry merlin]

this?

[dry merlin]

yes

[mint ravine]

yes, i tried that too

these too

i think there are a lot

there might be 9 actually

<@&286206848099549185> please help

[dry merlin]

i got 11

[mint ravine]

how about this

[dry merlin]

oh yes i hadnt considered that

and conversely,

[mint ravine]

i think that's all

so there are 13 for a 3 by 3 set up

[dry merlin]

14 or 13?

[mint ravine]

i meant 13, sorry typo

and there are 3 ways for a 2 by 2 set up

[dry merlin]

think we should try 4?

[mint ravine]

there seems to always be one more way compared to a multiple of the side length

[dry merlin]

That one more way is the pure diagonal path

[mint ravine]

so for an n by n set up, there is nx+1 ways , where x is an integer and n is the side length of the board

[mint ravine]

yeah let's try it

[dry merlin]

Doesn't seem that simple...

[buoyant yoke]

x=n! ?

[dry merlin]

Wouldn't fit with 3

would it?

[buoyant yoke]

you got 13 right?

ah nvm

[dry merlin]

yes?

[heady plume]

2. Consider one diagonal movement: 6Rs 6Us 1D
3. Consider two diagonal movements: 5Rs 5Us 2D
...```

[dry merlin]

im having so much fun rn thank you ramennoodle for this beautiful puzzle

[buoyant yoke]

perhaps try to look at your sequence of moves

[mint ravine]

np lol

[buoyant yoke]

up up right right, up upright right, etc

[polar mortar]

You know there's 3 for the 2x2
For the 3x3 You have a 2x2 in the top right corner, call it
A B
C D
There's 3 ways to get to C, which is the starting point of your original 2x2. So 3 * 3 gives you 9 ways to get to B
There's 2 ways to get to A without touching C, From A there's only 1 way to B. So there's 2 * 1 more ways
Similarly, there's 2 ways to get to D without touching C, so there 2 * 1 more ways.
So 13 ways.

This feels like a binomial or pascals triangle situation. Pascal's tetrahedron?

[dry merlin]

If we consider the simpler case where we only have the rectilinear directions to go, the problem is elementary

[buoyant yoke]

for every up, there must be a right
and upright is and up and right combined

there are n-1 choices of up or upright or right

[heady plume]

can you proceed from here?

[dry merlin]

Then the number of ways is $\frac{(2n-2)!}{((n-1)!)^2}$, where n is the number of sides

[grand pond]

RadMeerkat62445

[mint ravine]

and for 3 diagonal movements: 4R 4U 3D and so on

[heady plume]

Yes

the number of ways to arrange those alphabets is what you have essentially

[dry merlin]

Ok so now its a permutation puzzle

[heady plume]

A sum of multinomial coefficients

[mint ravine]

So there are a max of 7 diagonals

[heady plume]

Yes or to be precise, 7 diagonal movements. Hence 7Ds

[dry merlin]

$$\frac{14!}{7!7!} + \frac{13!}{6!6!1!} + \frac{12!}{5!5!2!} + ...$$

[heady plume]

.

[grand pond]

RadMeerkat62445

[mint ravine]

this problem reminds me of pascal's tetrahedron

would considering that help?

[heady plume]

are you referring to that in search for a closed form answer?

[mint ravine]

yes

Would considering it like this help

I got the answer 48639

[heady plume]

,w 14!/(7!7!) + 13!/(6!6!) + 12!/(5!5!2!) + 11!/(4!4!3!) + 10!/(3!3!4!) + 9!/(2!2!5!) + 8!/(6!) + 1

[mint ravine]

how did you get that expression

[dry merlin]

It's a permutation and combinations thing

[heady plume]

Yep no, we're way past what you got

or perhaps not

[grand pond]

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[heady plume]

That's right ig

I'm kinda for not being able to think of a closed form

[mint ravine]

thanks for helping

idk how 6th graders taking the same course as me are able to solve these questions lol

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[heady plume]

oh they're advanced :o

[dry merlin]

hey he's on the right track

[mint ravine]

im only in 7th grade, so it's not that bad

[dry merlin]

is this the pascal's tetrahedron?

fucking hell man you're doing great!

[midnight plank]

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[mint ravine]

yes, that was what i was thinking

[dry merlin]

no you're on the right track def

[mint ravine]

thanks bro you too

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[dry merlin]

huh? your answer's right @mint ravine

.reopen

there is no closed form answer as far as i can find using OEIS

you're looking for the central delannoy numbers

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[spark nebula]

how to find the inverse of a matrix

[sharp coral]

https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/02%3A_Matrices/2.07%3A_Finding_the_Inverse_of_a_Matrix

the usual algorithm for calculating by hand (described in the link above) is to augment with the identity matrix and find the rref

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[gray jackal]

I need help figuring out which is a and which is b

Since there’s two different equations, and they both confuse me

[last slate]

Isnt the focus major axis + c?

[midnight plank]

@gray jackal Has your question been resolved?

[gray jackal]

No clue

[last slate]

So

Basically a is actly js a^2

And b is actly js b^2

rather m is m^2

So c is js c

[gray jackal]

Hmm

I mean like does the order matter of which is a and which is b

[last slate]

not really if u understand it

But for your curriculum if you’re required to memorize which is a which is b then ya

But usually a is the major

[gray jackal]

Ok

So m is a ?

Or is m b

[last slate]

M should be a

Because based on the focus it would be (0, +-c)

Final ans B

M = 25

a = sqrt25

[gray jackal]

Ok I get it ty

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[fervent kiln]

average grade of this kid is 6,27?

30% 7,3
50% 4,8
20% 8,4

[fervent kiln]

'weighted average'

[snow pumice]

It is

[fervent kiln]

ok ty

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[leaden matrix]

Is anyone able to explain this? I've looked all over to try and find an explanation but I cant find anything

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[wind arch]

Hi I need help with my linear algebra homework. I know that the answer is no solution but I cannot find what part I got is wrong?

[gusty glen]

double check R1 = R1-R2, your fourth step

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[wind arch]

Thankyou !

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[gusty falcon]

i'm confused why u can't place the observer to the spotlights

[gusty glen]

what do you mean

[gusty falcon]

from right to left, you'd have blue spotlight, white spotlight, and then the observer

[gusty glen]

i’m confused about what your question is here

[gusty falcon]

why is such a diagram not right

[gusty falcon]

yeah well that gives us the wrong answer

what answer do you get?

[cerulean solar]

Looks right enough

[delicate sage]

Why ?

[gusty falcon]

it's not my question so i guess u can share answers

[delicate sage]

Nvm I didn't calculate

[gusty falcon]

the answer u get is

[heady plume]

something like this?

[gusty falcon]

9.4 or smth right

so my diagram is right?

[delicate sage]

Yeahh I think it is

[gusty falcon]

it's the same as you guys

yeah then

,w tan(45 deg) = h/(n + 6) and tan(70 deg) = h/n

[grand pond]

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[gusty falcon]

right?

[heady plume]

,w 6 tan 70°/(tan 70° - 1)

[grand pond]

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[gusty falcon]

yeah well

[heady plume]

Yes apparently

[gusty falcon]

that answer is not right

apparently

[heady plume]

wow

[gusty falcon]

because think about it

if u put the observer in the middle

you get a different answer

4.4 or something

not my question but

i guess that was the point

so is there anything in the question

that suggests that people can't be in the opposite corner?

😭

or is this an ambiguous question

[gusty glen]

which corner the observer is on shouldnt matter

[heady plume]

Lmao you seem to be having a tough time dealing with ambiguous questions

[gusty falcon]

googled it

and this solution is valid too

😭

[delicate sage]

Aren't you the same person, who questioned the ambiguousness of that series problem ?

[gusty falcon]

😭 😭 ikr

i think i keep doing that

[gusty glen]

😭

[gusty falcon]

i swear i'm not trying to ebe annoying

it's just that it's all ambiguous????

😭

[cerulean solar]

😭

[gusty falcon]

anyway so

[cerulean solar]

You need to file a complaint

[gusty falcon]

i'm not going insane right

and this question is objectively ambiguous?

[gusty glen]

seek a manager

[gusty falcon]

alright thanks guys

[heady plume]

yes it is indeed ambiguous. they should not complain if you interpreted the observer's position differently if they didn't specify

[gusty falcon]

tbh our diagram

is the more natural interpretation

the other one is just 😭 idk not that natural

but yeah

i guess it can be anywhere

okay anyway bye thx

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[remote halo]

guys please could someone please explain to me what is here meant by more general while_rule as i checked the end of note 6.3 also attached in the post it is basically called about the differnece between the partial correctnes and total correctnes so how is that relevant to the general while loop ?

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[remote halo]

<@&286206848099549185>

[hollow sail]

s mean sum right?

well use the loop invariant proof so

[midnight plank]

@remote halo Has your question been resolved?

[unique stream]

as above mentioned loop invariance

propose 0<=i<=n

and s=i*(i+1)/2

before entering the loop the conditions hold

Assume the given statement holds for some loop cycle with given s, i

then perform the induction step and show that the conditions hold for the next loop cycle as well with s', i'

clearly i' = i+1 and s' = s+i'

invariance is maintained

by induction this holds for all loop cycles

Does the algorithm terminate? Yes, i' is strictly larger than i by 1 and there is a finite amount of iterations as i iterates from 0 to n, ensuring total correctness

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[novel ocean]

how do i isolate x

[unique stream]

multiply with the denominator that contains x

then bring the other x over

factor out x

[novel ocean]

ah

ok wait

how do you factor out x😭

[sleek cloud]

xy+xz = x(y+z)

[novel ocean]

Ok

[novel ocean]

wait how

[unique stream]

-x

on both sides

[novel ocean]

ah ok

[novel ocean]

liie this?

[unique stream]

when you bring the denominator over, the entire left side needs to be in brackets

(left side) * (x-16/ln(5)-10) = x

and then you can do -x

[novel ocean]

ok

wait sorry i gtg

thanks tho

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[vapid cedar]

I would like to try and integrate the gamma function manually. I want to be able to do gamma(any real) with high accuracy somehow using integrals.

I started off with the integral for gamma with a slight conversion (Instead of a x-1 above the t, it is just x, it is the same concept I believe)

Starting from there, I tried doing part integration using the u, du, dv and u method.

I saw a pattern and decided to create a summation based on it. HOWEVER, one of my greatest problem is, example if you expand it and try evaluating the boundaries you would get undefined for if x is smaller than 0 and if x is an integer you would get 0?

Here is a short snippet of the summation being expanded I just forgot to put the boundary evaluation 0 to infinity.

[vapid cedar]

I want to try and integrate it and get some infinite series in which I can use to evaluate gamma of any real.

not just a whole number

Please note, I am slightly new to integrals and may have done something wrong, or maybe there is a method I don't know of or how to implement it.

[dreamy tapir]

damn

[vapid cedar]

<@&286206848099549185>

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[twilit field]

I'm, really confused

[radiant roost]

are you meant to use calculus?

or like, geometry

[twilit field]

calc

nvm got this

So I split this into 4 region

Is that right

[radiant roost]

what regions did you split it into?

[twilit field]

x>0,y>0

each quadrant would be integrated seperately

[radiant roost]

that should work

$\int_0^1 \int_0^{1-x}z(x)dydx$

[grand pond]

Axe

[twilit field]

eh?

[radiant roost]

something like this for the first quadrant?

[twilit field]

$\int_{0}^{1} \int_{0}^{1-x} (3-3x) dx dy$

[radiant roost]

[grand pond]

What a wonderful world!

[twilit field]

The rest will be the same, but with the upper bound of the first integral changed

like $\int_{0}^{1} \int_{0}^{x-1} (3-3x) dx dy$

[grand pond]

What a wonderful world!

[radiant roost]

will that come out negative?

[twilit field]

It will yeah

[radiant roost]

i think you can combine these two integrals

$\int_0^1 \int_{x-1}^{1-x}$

[grand pond]

Axe

[twilit field]

Do the properties carry over

from single var calc

[radiant roost]

your dx and dy are swapped

[hollow sigil]

so basically just goon and solve the problem

[radiant roost]

yeah use the 4 integrals if you're not confident to combine them

[heady plume]

$I = 2 \int_{-1}^1 \int_{0}^{1-|x|} \int_0^{3-3x} \dd z \dd y \dd x$

[twilit field]

wait, what

[heady plume]

should work

[twilit field]

Thanks

[radiant roost]

here, it should end with dy dx and not dx dy

[twilit field]

Ah right

[heady plume]

symmetricity about y = 0 allows me to rewrite integral on -y = integral on y, so I skipped the -ve y so your cuboidal equation is |x| + y = 1

[twilit field]

Got it

thanks

I was thinking some part of the volume would be give by $\int_{1}^{2} \int_{0}^{\frac{1}{x}} (x+1) dy dx$

[grand pond]

What a wonderful world!

[heady plume]

that is very trippy (@_@;) I don't see how y = -1/x (which lies entirely in 3rd quadrant) has an intersection between x = 1 and x = 2

[twilit field]

yeah

which is why I chose y=1/x

[heady plume]

then again, if only one side is bounded by y = 1/x, the other side is unbounded.

[twilit field]

so the volume is $\int_{1}^{2} \int_{0}^{\frac{1}{x}} (x+1) dy dx + \int_{1}^{2} \int_{\frac{-1}{x}}^{0} (x+1) dy dx$

which feels sus

[grand pond]

What a wonderful world!

[heady plume]

ntm where's dz ?

[twilit field]

ntm?

[heady plume]

not to mention

[twilit field]

does dz have to be there if we're using a double integral

[heady plume]

for volume dV = dx dy dz

dy dx is just the area

[twilit field]

yeah

but whayt ive written is equivalent, pretty sure

[heady plume]

oh you integrated dz beforehand, I see.

[twilit field]

yeah

[heady plume]

that looks correct

[twilit field]

Thanks!:D

[heady plume]

I'm pretty sure you can just leave the other integral be and twice the first integral coz symmetricity

Hmm

very bad 3D graph ||geogebra sucks at 3d geo||

[twilit field]

use Demose

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

[heady plume]

wow that's indeeeeeed cool

$V = 2 \int_1^2 \int_0^{\frac{1}{x}} (x + 1) \dd y \dd x$

[twilit field]

this is an even cooler problem

[heady plume]

feels like I'm doing my math hw

[twilit field]

Wdym

$V = 2 \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \int_{0}^{\sec(x)} (1+y^2) dy dx$

[grand pond]

What a wonderful world!

[twilit field]

Thanks

[heady plume]

!show

[midnight plank]

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

[twilit field]

Why the

😭

[heady plume]

crying is as infectious as laughter >.< it's fun to cry when doing math !!
Fun Fact: ||If you laugh when doing math, you're probably a psychopath|| - Tzu Sun [Art of Living]

/j lol

[twilit field]

hmm

The smaller sector would be the part between π/6 and π/2

no?

[heady plume]

Yes

So basically 0 ≤ x ≤ √3

[twilit field]

hmm

yeah

but the limits of y are going to cause issues

[heady plume]

No, cause the lower limit of y is the straight line y = x/√3, upper is the circle

[twilit field]

wait what

oh right

so $\int_{0}^{\sqrt{3}} \int_{\frac{x}{\sqrt{3}}}^{\sqrt{4-x^2}} \sqrt{4-x^2} dy dx$?

[grand pond]

What a wonderful world!

[twilit field]

Thanks

Here how would I decide the order

[heady plume]

deciding order is secondary, first you set up the integral !

[twilit field]

the limits are already given

so $\int_{2}^{\infty} \int_{0}^{2} \frac{1}{ (x^2-x)(y-1)^{2/3}}$

[grand pond]

What a wonderful world!

[heady plume]

$\int_{2}^{\infty} \int_{0}^{2} \frac{1}{ (x^2-x)(y-1)^{\frac{2}{3}}} , \dd y \dd x$

[twilit field]

hmm, yeah

[heady plume]

well, you seperate the integral lol

$= \left(\int_{2}^{\infty} \frac{1}{x^2 - x} \dd x \right)\left(\int_{0}^{2} \frac{1}{(y - 1)^{\frac{2}{3}}} \dd y \right)$

[twilit field]

eh

what

how

[heady plume]

y is independent of x no?

[twilit field]

yeah

[heady plume]

isn't int (dy/ ... ) just a constant?

[twilit field]

oh right

yeah

[heady plume]

m genius ||not||

[twilit field]

what

[midnight plank]

@twilit field Has your question been resolved?

[midnight plank]

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[last slate]

jus the square one, idk if im correct

i jus put "integrate (1-cbrt(y))^2 dy from 0 to 1

[last slate]

and calculator says 0.1

anyone can check for me?

Hellloooooo 😭 jus some basic integration helppp

<@&286206848099549185>

<@&286206848099549185>

<@&286206848099549185>

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[remote halo]

in this case the type of induction is complete induction ? also as i know the invariant shall satsify three proprties as mention only two properties are proved what about the last one regarding to the termination ? Thanks anyway for your asnwer. i wanna add something so in this cas we have proven the inilization which is the base case in the induction and the maintance which is the inductive hypothesis and the inductive step ?

[midnight plank]

@remote halo Has your question been resolved?

[remote halo]

@unique stream

[midnight plank]

@remote halo Has your question been resolved?

[misty gorge]

[mod ping dealt with]

[remote halo]

.close

[midnight plank]

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[jolly roost]

Can someone explain why this is wrong?

[latent elk]

the force on the cart is 4 m/s?

[jolly roost]

sorry

edited

@latent elk

[latent elk]

Huh?

Isnt it impulse the change of velocity over time?

So it should be 4 * 1 = 4 N*s?

[jolly roost]

oh the units are wrong

[jolly roost]

yea thats supposed to be the right answer

but whats wrong with my approach

[latent elk]

a = dv/dt = (4-0)/4 = 1

F = ma = 1 * 1 = 1

[latent elk]

Then this, impulse = 1 * 4 = 4

Are u sure its the area under the graph?

OHHH

@jolly roost Influenced by STEAD force

Your graph should look like this, a rectangle not triangle

[jolly roost]

so what if it was an increasing froce (linearly)

[latent elk]

Well, the problem then need to provide more data

It is wrong for us, to assume the force is changing at the first place

[jolly roost]

what is wrong with what I did if the question was phrased

"A smart cart is initially at rest and then is influenced by an increasing linear force. The cart has a mass of 1 kg. At 4 seconds, the velocity of the cart is 4 m/s. What is the impulse? "

[latent elk]

unsolvable then

[jolly roost]

how come?

[latent elk]

Cause we dont know how the force is changing

We can only know at t=4 that F = 4N

[jolly roost]

but its changing linearly

[latent elk]

If it at least gave us something like, at t = 0, F = 2N

[latent elk]

Linearly with what angle?

[jolly roost]

but its at rest, so at t = 0 then f = 0

no?