help-0

[hardy mirage]

Actually I a always confused by coordinates, is is (x,y)?

So it is point C

[oak chasm]

It needs to be less than 5 from (5, 5).

Another way to solve it is to draw a circle of radius 2 with center (2, 2) and a circle of radiul 5 with center (5, 5). The intersection will be where the allowed points are.

[alpine sable]

pls ping

thats it?

[ocean seal]

SubGui

[astral dagger]

then you may find a way to calculate it, but first did you understand why the integral is like that?

[alpine sable]

i think so

how do u set it up for v?

part b

@astral dagger

[astral dagger]

you need to parametrize the surface somehow, then calculate the partial derivatives of the "vector" wrt to the new variables, then set the integral: (\iint_S,\rm{d}S=\iint ||X_u\cross X_v||,du,dv)

[ocean seal]

SubGui

[alpine sable]

im not sure ;/

[astral dagger]

read it again

slowly

[astral dagger]

as you want to calculate the area of a surface restrained to the quarter of circle, you need to parametrize it first

or maybe there's no need

I also have solved a question related to this same subject today and got wrong

because it wasn't needed to parametrize, but let's see

[vapid seal]

Hii

Please prove

[alpine sable]

can i ask another

q

[astral dagger]

go ahead

[astral dagger]

go to another room, please

this one is already in use

[vapid seal]

Yeah sure

[alpine sable]

is this room in use?

[astral dagger]

yup, just waiting for her to send the question

good

[alpine sable]

sorry

im here

[alpine sable]

for this did u get 30415.74

i have a hw prob and im v confused

what do i do here @astral dagger

[astral dagger]

well, this is also about physics

[alpine sable]

LOL IK IM SO BAD

at physics

almost failed in HS

it wantsvector valued funcs

[astral dagger]

but well, I think you should assume velocity vector tangent to the trajectory

then decompose in two components, a vertical and horizontal ones

[alpine sable]

how?

[astral dagger]

then maybe the horizontal one is constant, and the only one changing with time being the vertical

[astral dagger]

when it reaches the maximum height, v_y component is equal to zero

and starts pointing downwards after that

[alpine sable]

interesting

[astral dagger]

but for that, you may call helpers, I'm also not that good in physics

[alpine sable]

;/

[alpine sable]

did u get this

[astral dagger]

I haven't calculated it by hand

but it turns out not being that complicated

[alpine sable]

if u do it could u verify <

3

[astral dagger]

trust Wolfram

there's no need to verify if the integral is correct

[alpine sable]

i plugged it in

and i got pi in the answer

idk how it got pi thats y

[shut knoll]

how would i approach this

im at sin(x) = -1/2

it wants an answer in radians

[alpine sable]

@shut knoll trig value table just google

So it is -pi/6

[placid zinc]

You'll need to memorize your unit circle

[alpine sable]

Because 1/2=pi/6

@shut knoll

No don’t memorize

Just look it up lol

[shut knoll]

it didnt work

i think it wants it in the interval [0, 2pi]

apparently 7pi/6 is an answer too but how do i get that

is there a way to find 7pi/6

without just memorizing that

[alpine sable]

Just add 2pi until you get 7pi/6

?

[spiral mason]

the unit circle repeats itself, so for any angle A in radians, there is also an angle A + 2pi that is a solution

-pi/6 + 2pi = 7pi/6

they probably want 7pi/6 because it's positive

[placid zinc]

You are confusing yourself if you are trying to avoid unit circle arguments

[spiral mason]

and positive angles are generally preferred

[shut knoll]

i see

[alpine sable]

You had to get it explaine

In the book

[willow wadi]

do u get

29.8 km/h N 13.4° E

<@&286206848099549185>

☝️

[alpine sable]

hello

$\int _0^{\pi }\left(cos\left(x\right)+4\right)^2dx$

[ocean seal]

cup

[alpine sable]

can i use u-substitution to solve this integral?

[willow wadi]

bro read #❓how-to-get-help

[alpine sable]

and would it be easier than just doing it the "normal" way

[willow wadi]

this channel is taken

[alpine sable]

well

this channel was dead for a good

10+ minutes

sorry

[willow wadi]

u cant take the channel until its dead for 30 min

[glass lichen]

No, expand then apply identities

[alpine sable]

Ok so when should I use u-sub

[glass lichen]

When the derivative of u would be present usually

If the integrand was -sin(x)[cos(x)+4]^2 for example

[alpine sable]

Ok so its when a function is multiplying itself by its derivative?

[alpine sable]

Thank u

Thats my question

[thin wolf]

need some help with q2, not sure if i done it correctly

[alpine sable]

how do i know if i should use it or not?

[thin wolf]

it has imaginary solutions ^

[willow wadi]

is that a test

[thin wolf]

is there an ongoing question? my bad 🙏🏻

[thin wolf]

it’s a homework

[willow wadi]

look at the 3

[thin wolf]

yea it’s taken from past year papers

[willow wadi]

should have cropped it well

if u had to cheat

[jagged imp]

Why are you so insistent on this being a test

leave it be

[thin wolf]

no no it’s homework, it’s just questions taken from past year papers that

[jagged imp]

theyve provided justification for the numbers

[thin wolf]

in my school we constantly do worksheets that contain marks because 99% of these questions are from sch papers, if not then past GCSE papers

[gaunt magnet]

@glass lichen ayo wassup wanna help me

[thin wolf]

it’s also for the teachers to gauge how well we do

[gaunt magnet]

what

is gcse

[gaunt magnet]

Oh

,w 5x^2+2px+p-3=0

[ocean seal]

Results provided by WolframAlpha

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

Requested by Usman#0988

[median tendon]

@alpine sable did u ever get the help for the projectile question?

[alpine sable]

no:(

But I think it’s Suzie q

Not sure

[median tendon]

do you know what kind of relationship a projectile follows when only subject to gravity?

like is it cubic?, or linear? and so on

remember the question does not say there is wind resistance, so the projectile will follow a nice polynomial

ping me if u do answer, im working in another window and wont see your answer

[grim shadow]

bro its said that only for thats on the projectile is gravity

so no wind resistence

[median tendon]

i know

thats what i said?

[grim shadow]

and even if there is wind resistence they didnt give function for that

[median tendon]

they dont need to give a function, a projectile will follow a quite simple polynomiall (not gonna say which one cause im trying to let xoxo think about it)

when only subject to gravity

simple physics

[grim shadow]

nah i was talking about wind resistance

they need to give a function for that if thats the case

[median tendon]

wind resistance would make it very complex and most probably unsolvable without more info

[grim shadow]

nope its easy

there are several problems in ie irodov

u can check if u want

its basically caluculus

[median tendon]

you would need the speed of wind atleast....

[alpine sable]

I think it’s suzie q not sure

[median tendon]

and temeprature

[alpine sable]

Wind and temp and all don’t matter

[median tendon]

since desity will depend on that

ok lets go back to the original problem xD

[alpine sable]

Nothing matters except time and height

[grim shadow]

look u literally only need bulk modulus of that gas

[median tendon]

yes, now that graph, what kind of equation is it?

just the general form

[grim shadow]

with only gravity its basically a parabola

[median tendon]

do you know?

I know u know xD

[grim shadow]

like y=ax^2+bx+c

xd

[tawny totem]

Hey

https://math.stackexchange.com/questions/4217090/prove-that-the-twisted-convolution-of-f-and-g-lies-in-l2cn

[median tendon]

ok @alpine sable have u learnt that before?

that its parabolic

[grim shadow]

bro this channel is occupied

[grim shadow]

i feel this question is pretty easy but there's just too much to read

[median tendon]

'easy' is relative, to you it might be easy if you've covered alot of physics, to someone just learning, it might be hard

[grim shadow]

hmmmm true

wait let me read

it may take 10 mins xd

[median tendon]

also this is very wrong xD

[grim shadow]

how??

[median tendon]

an easy counterexample, something that has a very large surface area would resist much more than something that has a smaller surface area

[grim shadow]

so??

[median tendon]

The question gives not enough information to solve

with wind resistance

[grim shadow]

you know how do they predict the trajectory of bombs dropped from fighter jets in real life??

have u studied research level mechanics

[median tendon]

they definitely know the dimensions of the bombs

[alpine sable]

mmm no clue

[grim shadow]

bro look

[alpine sable]

yeah i feel like this is a level easy difficulty question in physics

problem is

im BAD at physics

[median tendon]

ok well now you know i guess

[alpine sable]

i think u use the position and velocity vectors

[median tendon]

does that help you solve the problem?

or do u still need more guidance

[alpine sable]

i think its suzie q so if thats correct then i think im good

if not then idk

[grim shadow]

literally bulk modulus is the only thing u need to know

[median tendon]

I'm not too sure what a suzie q is tbh

but know that the projectile motion follows a quadratic is all you need to know to solve that question

[grim shadow]

i cant explain it to u why coz u havent studied that level of physics but trust me

[median tendon]

care to explain how a couch will follow a different trajectory than a coin?

if all you need to know is "modulus of gas"

[grim shadow]

u forgot the bulk??

[median tendon]

and not knowing the dimesions of the projectile itself

[grim shadow]

different dimensions wont change the trajectory that much

be it couch or coin

if u know bulk modulus eqn u can still find the trajectory

the surface area effect is already accomodated in bulk modulus eqn

[median tendon]

something pointy will be subject to less resistance for example, dimensions play a huge role in the projectile planning...

[grim shadow]

i know but u know the def of bulk modulus??

how it is defined or what is universal eqn of it

[median tendon]

ohh its calculated with volume?

[grim shadow]

not that elementary one

def of bulk modulus changes much later on

universal bulk modulus eqn gives relation b/w temperature effect, surface area effect, even things as small as dipole-dipole attraction b/w gases and material of projectile

basically literally every damn thing that we know till now in physics

[median tendon]

what is b/w?

[grim shadow]

between

[median tendon]

"surface area effect"

[grim shadow]

yep

that also

[median tendon]

so you will need to know the surface area..

the dimesions of the projectile

[grim shadow]

obviously

but u need it only when ur finding it very accurately

if u want approx then u dont even need dimensions

crazy, right??

but all these things are experimental

[median tendon]

without dimesnsions and material of the projectile, i dont think u can get good accuracy

throw a coin and throw a paper

[grim shadow]

ok if u throw a coin and a paper from 50k feets

[median tendon]

even half a meter

[grim shadow]

what will be the difference b/w their landing points

max to max, mb a 100 ft

[median tendon]

depending on speed of wind, pretty large error

[grim shadow]

nah max error cannot be more than 500 ft

[median tendon]

and that is proven?

[grim shadow]

but look compared to what height u threw it, thats pretty low

[grim shadow]

its not theoritical but more like an experimental proof

we still dont know why is there so low error b/w a coin and paper

[median tendon]

Is there a paper on this?

[grim shadow]

i think an experiment was conducted around 1997 or smthing but idk

they threw a coin and a bed ( i dont remember tbh) from 20k feet

[median tendon]

coin and bed I can believe

[grim shadow]

and calculated result and experimental result had an avg error of 0.91 %

[median tendon]

but paper is weird

[grim shadow]

it wont change that much

[median tendon]

btw this is only dropping right?

[grim shadow]

yep

imagine giving a velocity of 50 m/s to both of them

[median tendon]

actually shooting something at a distance is where error will be insane

[median tendon]

vertical or horizontal?

[grim shadow]

well look

they were thrown from a helicopter

so it cant remain stationary at that heights

wind speed is too high at those heights

actually u wont even know if ur under a velocity at that height

[median tendon]

the helicopter would be able to tell you

satlellite tracking + imu sensors + other shit I'd imagine

[grim shadow]

mb but the question is would it be constant

[median tendon]

no

[grim shadow]

bro its 1997 we're talking about

[median tendon]

oh

[grim shadow]

but more interesting things happened when they tried to calculate the trajectory of fluid

they got a fucking 7th order differential eqn

and its still kind of uncalculated till now

[median tendon]

thats expected, just the flow of water is computationally expensive to calculate

[grim shadow]

expensive?? more like impossible

[median tendon]

depends on the problem

but yeah massive area of research

[grim shadow]

hmm interesting but not my field of research

[alpine sable]

<@&286206848099549185>

(a-b)^2 = a^2-2ab+b^2 = a^2+2ab+b^2-4ab = (a+b)^2-4ab

(a+b)^2=a^2+2ab+b^2 = a^2-2ab+b^2+4ab = (a-b)^2+4ab

there u go

@alpine sable

[alpine sable]

https://tenor.com/view/anybody-leah-rilke-the-wilds-can-someone-help-me-i-need-help-gif-19528434

[burnt pond]

no i'm not talking about the pi symbol, I mean the n

[glacial hedge]

yes....

n just represents the current cycle its on

n*pi

for n is any integer

its on it's nth cycle

[burnt pond]

oh

i've never seen that notation before

do you have any links about it?

[glacial hedge]

uh...

[glacial hedge]

I can't think of any right now... but I'm pretty sure a lot of trigonometry videos/textbooks cover the n*period

[vale wigeon]

i think y'all are focusing too much on the wrong things

[glacial hedge]

^

[vale wigeon]

$n\pi$ can be read as ``some integer multiple of $\pi$''

[ocean seal]

Ann

[vale wigeon]

it could stand for π, it could stand for 2π, it could stand for 3π, it could stand for 17π, it could stand for -42069π, etc...

[glacial hedge]

i'm not gonna lie... but that's pretty much what I said xD

[vale wigeon]

point is, the values of tan repeat themselves every π units

thus any solution of tan(x) = c will usher in a whole family of solutions, with adjacent ones π units apart

[burnt pond]

yep ok

[jagged rapids]

Can someone help me formally prove this:
show that there is no set V such that every set is a member of V.

Basically the set of all sets paradox, I mean I have the intuitive idea

<@&286206848099549185>

[vale wigeon]

,rcw

[willow wadi]

can someone please

check my work

helper pls

[vale wigeon]

.

[willow wadi]

Hi helper may u check my work please

[velvet citrus]

Bob, the proprietor of Midland Lumber, believes that the odds in favor of a business deal going through are 8 to 5. What is the (subjective) probability that this deal will not materialize? (Round your answer to three decimal places.)

how do i solve this?

[little lotus]

8/5 > 1, the odds cannot be greater than 1

[jagged imp]

that's not what odds are

[velvet citrus]

that doesnt make sense

[jagged imp]

The idea of odds here is that for every 8 deals that go through, 5 don't

[velvet citrus]

so then how do i figure out the probability that they wont

[jagged imp]

idk what subjective probability is personally so I can't really help. If it just means the probability that a business deal doesn't go through, then its 5/(8+5)=5/13.

[velvet citrus]

that is probably right

[uncut depot]

so i have an equation

x - 2y = 5
2x + 3y = 10``` and the answer give in the book is `x = 5` and `y = 0`, but there are many solutions for the equation, is there a *rule* or something like that to say that only a particular number must be used for `y` or `x` to solve the equation to get `x = 5, y = 0` or is the book just choosing an answer from the list randomly

[forest drum]

Is this channel busy?

[uncut depot]

yep

[alpine sable]

Umm, there are three rules to do that

[uncut depot]

what are they?

[alpine sable]

You can use substitution method

But the easiest one is eliminating one of the variables.

@uncut depot

[uncut depot]

well, i used substitution

but those are methods

[alpine sable]

Multiply the first expression so you'll receive 2x-4y= 10. Then, substract it so that you'll get the value of y.

[uncut depot]

they say solve it graphically, they ask us to not use elimination or substitution

does that make sense

[alpine sable]

Oh, I'm really sorry.

[uncut depot]

for what?

[alpine sable]

I haven't noticed that 😔

[uncut depot]

oh its fine

[gray isle]

Did you graph your two lines?

[alpine sable]

For graph, you have to assume a value of x, and then find the value of y with respect to that equation @uncut depot

[uncut depot]

so my doubt is, is there a rule for that assumption

[alpine sable]

No.

Take easy ones.

Like if x=0

If x=1

And so on...

[uncut depot]

i assumed y = 2 i got x = 9 but the book says x = 5, y = 0 does that mean the book is just using one of the examples?

[uncut depot]

ummm, yes?

[alpine sable]

Yeah

[uncut depot]

oh okay, then fine

thanks for ur help

[gray isle]

the solution will be where your twoines intersect

[uncut depot]

i see

[alpine sable]

Yeah

[uncut depot]

but there are multiple solutions, so i got confused

[alpine sable]

Bruh.

[gray isle]

how are you getting multiple solutions?

[uncut depot]

because

wait

[gray isle]

Can you show your graph?

[vale wigeon]

because two straight lines can intersect at two points without coinciding, somehow

[uncut depot]

well

i lied

[vale wigeon]

why did you lie?

[alpine sable]

Look, there are two equations.

You have to assume values of x and y. And then make the graph.

[gray isle]

lying doesn't help anyone here

[uncut depot]

see if i assume x = 1 i get one solution

[alpine sable]

Hmm, nice.

[uncut depot]

if i assume x = 2 its a different solution

isn't it?

[alpine sable]

Do you mean a straight parabola? XD

[vale wigeon]

can i see the equations again?

[gray isle]

you're not using the word "solution" properly here

[vale wigeon]

also, who said anything about assuming anything?

[alpine sable]

Solutions will just be relative offf.

Take 2-3 solutions and Mark the points on the graph.

[uncut depot]

oh ok

[gray isle]

you're not using the word "solution" properly here

[uncut depot]

2x + 3y = 10```

[uncut depot]

❓

[vale wigeon]

that was addressed to ryuzaki, not to you.

[uncut depot]

oh okay

[vale wigeon]

anyway, what seems to be the problem here?

you're asked to solve this system graphically?

[uncut depot]

yes

[vale wigeon]

so you know that you're supposed to graph each equation

[uncut depot]

i was asking if there are rules for assuming

[alpine sable]

That's not assuming actually, I said that to make him understand how to find the values.

[vale wigeon]

why ASSUME anything????

[uncut depot]

uhh

[vale wigeon]

why do you need to assume anything?

[alpine sable]

I'm not saying to assume anything smh.

[vale wigeon]

i'm asking azorfus now.

[uncut depot]

idk

[vale wigeon]

exactly.

[alpine sable]

Take values of x which can be easily solved.

[vale wigeon]

do you know how to graph equations of straight lines?

[uncut depot]

yes ig i know

[vale wigeon]

"i guess"?

[uncut depot]

yes

[vale wigeon]

well then

are you able to graph x - 2y = 5?

[uncut depot]

yes

[vale wigeon]

then do it and share the result.

[alpine sable]

Like in this first equation, if x=0, you can clearly say that the value of y is -5/2

[uncut depot]

i see

[vale wigeon]

what ryuzaki is trying to say is that the point (0, -5/2) lies on the line x - 2y = 5.

but the way they're phrasing it makes it sound like (0, -5/2) is the solution to the entire system, which it very clearly isn't.

[alpine sable]

If x=1, then the value of y should be -2 I guess. And then plot the points in the graph (X,Y)

[vale wigeon]

but ryuzaki, azorfus just confirmed that they knew how to graph equations of straight lines.

[uncut depot]

so i do that another time and then i draw the graph?

[vale wigeon]

there is no need to explain it again.

[alpine sable]

I think I havent seen the version of him that he's confirmed he knew how to draw straight line graphs.

[vale wigeon]

i just posted a screenshot.

from mere minutes ago no less.

unless azorfus LIED to us again?

[uncut depot]

i did not lie

[vale wigeon]

well then

[uncut depot]

we need two points for a line ryt?

am i wrong.

[alpine sable]

2-3

[vale wigeon]

you're not wrong.

it takes two points for a line.

[uncut depot]

one point is (x,y) so we need another solution?

[vale wigeon]

taking a third point is merely a bonus and might help with keeping the line more straight.

no you've gone and overthought this again azorfus

if you're graphing x - 2y = 5 then you can (and should) temporarily forget about everything else and focus on that equation and that equation only

[uncut depot]

so i get x and y for that equation?

[vale wigeon]

i don't know what you mean by this

[uncut depot]

I'll think about it first

[alpine sable]

Do you know how to find x and y?

[uncut depot]

yes i do

do you want me to find them?

[alpine sable]

Then what are you supposed to mean by asking this?

Yeah

Smh

[uncut depot]

The part im confused about it

solution of that equation is the point (x,y) ryt (im asking this so i don't get confused another time)

[vale wigeon]

"the point (x,y)"

it's unclear what you mean by this

[uncut depot]

?

wait i'll try to draw it out

sorry, it took me forever to do this

lets say x = 3 and y = 4

[alpine sable]

Fine. It would get better with practice.

[uncut depot]

so (3, 4) is a point ryt?

this thing

[alpine sable]

No.

[uncut depot]

then what is it called

[alpine sable]

I'll better suggest you to revise how to solve these questions properly.

[uncut depot]

oh okay

oh now it makes sense, thanks anyways

[alpine sable]

You're welcome😊

[marsh vector]

Please someone

Check if this is correct

I failed the most easiest question

Im a fucking idiot... I failed the exam 😭

[surreal meadow]

looks good

[marsh vector]

Are you forreal????

ITS CORRECT??

Omg

I literally ran out of tears

[icy trail]

$cosec x + cot x = 3$
find $cosecx - cotx$

[ocean seal]

querty

[icy trail]

could anyone help with this? :)

do i need to solve for x in the first equation then put those solutions into the second?

[alpine sable]

Unfortunately, it's not.

[marsh vector]

Fuck my life

[alpine sable]

As √-27 is an imaginary number @marsh vector

[alpine sable]

Work hard, you'll surely get it next time 😊

[icy trail]

its a cube root though?

[marsh vector]

Im gonna kill myself not even joking

Im so dumb

😔

Whats the right answer??

[surreal meadow]

its not the square root ryuz

its the cube root

helem you're correct

[marsh vector]

omg. You just made my day man

[icy trail]

my calculator says it's positive 4/15 - this is a lie

[surreal meadow]

its negative

[icy trail]

nvm

yes it is negative

[marsh vector]

-4/15 or 4/-15?

[surreal meadow]

same thing

[marsh vector]

Same thing????

[alpine sable]

Oh, damn I thought he has cut some part looll

[marsh vector]

Im literally the happiest person on earth rn

[alpine sable]

The handwriting.

Rip

[surreal meadow]

yes $-\frac{4}{15}=\frac{-4}{15}=\frac{4}{-15}$

[ocean seal]

maximo

[marsh vector]

My hand was shaking when i was writing that

I literally ran out if tears

[hexed viper]

You need to care less

It’s just numbers

[hearty nest]

if we have a polygon of all sidelengths 1, and say there are two rabbits r1,r2 at some vertices. r1 can take jumps only of length 1 from one vertice to other and r2 can take jumps of length 2, how to formally prove that they will eventually meet ?

[marsh vector]

Wow one question i got the answer correct even though it was all wrong

Lmfaoooooo

[hearty nest]

they both move in the same direction

[icy trail]

prove for a general polygon with like n sides?

[hearty nest]

yeah,but ig number of sides doesn't matter

so let's just assume it has n sides

[hexed viper]

Do they always jump clockwise?

[hearty nest]

yeah same direction

[hexed viper]

Their relative distance changes by 1 unit

So after n jumps they’ll be back again

You can even see that, n jumps they’ll both be back at the start

The 1 unit guy goes round once and the 2 unit guy goes round twice

Even if they start on different places they’ll meet because their relative distance changes by 1

[hearty nest]

they may or may not start from the same vertex, they are at any two vertex initially

oh yeah got it , we can assume r1 is at rest so r2 moves by length 1 so they will surely meet

[surreal meadow]

how many practice quizzes do you have

start with the practice problems and do enough of each section until you feel like you understand the material well enough

[placid zinc]

Hard to know without knowing how ready you feel. I assume you need to sleep eventually haha

[surreal meadow]

then move forwards to the next topic. finish with the practice quiz to test your overall retention

🤷🏻‍♂️ you could just skip the studying

of course, so just do a bit of each section to see what you need to work on and start with that part

what's the format of the quiz?

just write the proofs?

[swift temple]

nice pfp

[surreal meadow]

then i'd do exactly what I said, skim through the proofs to see which you retained the best and worst

then focus on the ones that need the most work from you

but i'd finish with the practice quiz since you only have the 1 copy

[surreal meadow]

🙂

[opal wyvern]

If the first root is the square root, and the second is the cube root, what's the third

[prime badge]

nothing

[vale wigeon]

what are you talking about?

[surreal meadow]

tesseract root

[vale wigeon]

$\sqrt[4]{x}$ is not called the third root. it's called the fourth root.

[ocean seal]

Ann

[upper escarp]

,help

[ocean seal]

A brief description and guide on how to use me was sent to your DMs!
Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!

[vague jolt]

I honestly have no idea how to approach this question

Practice question

[vale wigeon]

$A = {0, \varnothing, {0}}$

[ocean seal]

Ann

[vague jolt]

Yessir

[vale wigeon]

...don't call me sir

[vague jolt]

Yes Ann

[vale wigeon]

i suppose they expect you to write out all the elements of P(A) and not fuck up any braces as you do so...

[vague jolt]

But isn't P(A) a function

[vale wigeon]

P(A) is the powerset of A

[vague jolt]

How tf do I know what's in P(A) without knowing what P is

[alpine sable]

P isn't just a function, it's known notation for "all possible subsets of this set"

[vale wigeon]

i just told you... P stands for powerset

[vague jolt]

I know we sent our message at the same time sorry

[vague jolt]

Wait so

Isn't the answer just

[alpine sable]

What Ann said? Yes

[vague jolt]

A combination of {0, empty set, {0}}

Oh so it's

A very very simple question

Damn

[vale wigeon]

yes it is

[alpine sable]

You did say it was a practice question

Probably just checking you know definitions

That's literally everything

So, in approaching questions like these, the first step is to double check you actually understand everything that appears in the question (notation, terminology) - if you don't, you fill in those gaps before attempting to answer

Otherwise you're guessing

[vale wigeon]

i think you might have missed something here

such as the empty set, which ought to be a member of the powerset of anything

[sage summit]

isn't 0 usually defined as the empty set ?

then 0 is 0

[vale wigeon]

i didn't say that was the only thing you missed

i didn't check if you missed anything else

[alpine sable]

Or rather, what does {∅} mean?

So, the powerset of ∅

It's the only thing I can think of that would use that notation, does it belong in the powerset of A though?

Oh yeah, I suppose if ∅ was an element of A to begin with, {∅} would be part of the powerset

If you're proving by contradiction you should assume the opposite and find the fallacy, so assume it's rational

[vague jolt]

Oh yeah mb

Okay hold on this questions shouldn't be that hard

Rational means that it can be represented in the form of x = a/b where a, b are integers and b =/= 0

So I should probably work off of that definition of the set of rationals

[alpine sable]

The proof by contradiction serves the purpose of making use of some property gained by the assumption and finding a fallacy. The property you have there is a very good starting point

https://www.askiitians.com/forums/Mechanics/an-object-moves-in-the-xy-plane-with-an-accelerati_117626.htm

Isn't part (c) option A?

Because v_x is always increasing, and regardless of what v_y is, v_y^2 ≥ 0 always

So the magnitude will be increasing

[alpine sable]

What is the matrix representation of the laplacian operator on differentiable functions from $R^2$ to R?

[ocean seal]

Cauchy

[alpine sable]

<@&286206848099549185>

Can anyone check the link above?

may someone please tell me for "6b" how the value of h=0.01 finds the gradient at the x-coord "x=-1"? how do i determine which x-coord it will find

@alpine sable What was your answer for 6a?

[rigid smelt]

it doesnt gives you the slope of the tangent

it just gives you an estimate

[alpine sable]

@alpine sable okay, so now make a line through x = -1 using that slope

Then plug in h = 0.01 to find the slope

for its slope

You're making the line through the point x = -1... the value of h does not matter for that

h decides how close to the derivative the estimate will be

oh right

and plug in 0.01 for h

Yeah

You're estimating the slope of the curve at x = 2, up to some accuracy depending on how small h is

[wicked drum]

can anyone help me with a differential equation question pls??

I tried everything I could for the past hour and now i am desperate pls help

[vague coral]

1/M dM/dt = 1 - S

and you use logarithm

[wicked drum]

i tried doing that but it has to be done so that the only variable M(t) and S(t) is related to is t

[vague coral]

integration

[vale wigeon]

this looks mighty complicated

can you share some of the things you've tried?

systems of nonlinear ODEs rarely, if ever, have nice formulas for their solutions...

[vague coral]

you will have $$ log|M| = t - \int S dt $$

[ocean seal]

Herels

[vale wigeon]

herels, i'm almost certain this won't lead anywhere.

[wicked drum]

i've tried @vague coral 's way and tried doing dM/dS thinking it might help and some other algebraic non sense. I am so lost cause I missed some classes

[vale wigeon]

i mean ok, maybe it is possible to combine these equations into a 2nd order ODE in S alone

S' = -S + MS
differentitate both sides: S'' = -S' + M'S + MS'
S'' = -(-S + MS) + (M-MS)S + M(-S+MS)
= S - MS + MS - MS^2 - MS + M^2S...

hm

no, looks like this is no good either...

[wicked drum]

arghh im stuffed haha

idk what else can be done to try to solve it

[vague coral]

I'm thinking about matrix here

[vale wigeon]

herels, this is a nonlinear system of equations.

[vague coral]

yes I know

but I'm trying to find something

[wicked drum]

i dont think anyone mentioned matrix in the class but idk

[vague coral]

$$\frac{dM}{dt} - M = -\frac{dS}{dt} - S$$
can we do something here ?

[ocean seal]

Herels

[wicked drum]

you could make a (dM + dS)/dt but im not sure how that helps

[vague coral]

we will have $\frac{d}{dt}(M+S) = M-S$

[ocean seal]

Herels

[vague coral]

hmmmmmmmmmmmmmmmmmmmmmmm

I was tempted to do a u-sub xd

[wheat shore]

The perimeter of a triangle is 84cm and its area is 336cm^2. If one of its sides is 30cm. Find the length of other remaining other sides.

help me

[swift roost]

Hi.. can someone help me out with these two questions T_T stuck on it for the whole day.. if possible can solve with solution tq :3

I have the answer but I can’t work out the solution

[tight locust]

,w 120 = a + b, b = 6.5a

[ocean seal]

Results provided by WolframAlpha

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

Requested by EndTimes#8053

[vague jolt]

Is this channel being used right now?

[tight locust]

go ahead ig

if 36 divides a^2 then 6 divides a

[vague jolt]

Yeah

I spent some time thinking of a counter example and I couldn't find any

So I'm sure it's true

Wait

I'm an idiot

Wait no I'm not

I'm pretty sure this is true

[tight locust]

a^2 is a perfect square so each prime of its decomposition must be raised to an even power

so if 6^(2n) divides a^2 then 6^n must divide a

[vague jolt]

By any chance do u know what theorem this comes from

So I could read more abt it from my notes

[tight locust]

it's just basic arithmetic idk what the law is called

but let's say a = (xyz). then a^2 = (xyz)^2 = x^2y^2z^2

yeah partially

but what i just stated is a basic rule of exponents

read what i just wrote

yes

[vague jolt]

And each prime in the decomposition of a^2 needs to be raised to an even power?

[tight locust]

yes

by definition

otherwise it's not a perfect square

[vague jolt]

Okay yeah that makes sense

But where does 6^(2n) come from

[tight locust]

6^(2n) = 36^n

[vague jolt]

Why is 36 being raised to the power of n

[tight locust]

because we know that 36 is a factor

so some multiple of 36 must divide a^2

[vague jolt]

Ah 36 is a multiple of a^2

But wouldn't it be 36(n) instead of 36^(2n)

[tight locust]

36 is a factor of a^2

[vague jolt]

Actually no it can't

Huh

Oh it can't because

a^2 is a power right?

Like a is an integer

[tight locust]

every prime factor of a^2 must be raised to an even power

[vague jolt]

Yeah makes sense

Kind of

Hmm

[tight locust]

a = (p1p2p3...) so a^2 = (p1p2p3...)^2 = (p1)^2 * (p2)^2 * (p3)^2 * ...

[vague jolt]

Yeah I get that

36 | a^2 => 6^2 | a^2 => 6 | a right?

Like is that the logic?

[tight locust]

yes

[vague jolt]

I feel like this isn't a reasonable proof though

Idk

Am I wrong for thinking that?

[tight locust]

why not?

use your words

[vale wigeon]

36 | a^2 => 6^2 | a^2 => 6 | a right?
Like is that the logic?

do we know that m^2 | n^2 implies m | n though

this jump feels the tiniest bit dodgy to me

[vague jolt]

I don't think it does

[vale wigeon]

i could be wrong about it and it could actually be true, but still.

[sage summit]

it's true in Z

[vague jolt]

They are all integers

[vague coral]

@wicked drum a friend of mine solve the problem

[vague jolt]

So ig it's true

[tight locust]

it has to be true. when a number is square rooted then the powers of its prime factors are halved

[vale wigeon]

is there a non-FTA proof of it though

[sage summit]

sounds unlikely

[vale wigeon]

bruh

@acoustic ingot channel busy please move

[acoustic ingot]

sry bout that

[sage summit]

but I think in this case you can just show that if n = 18k+a with 1 <= a <= 17 and a not 6 or 12 then n² is not a multiple of 36 because n² = 324k²+36ka+a², and so you check all 15 values of a² and check they are not multiples of 36

wait uuh

[vale wigeon]

you want to say that if 6 doesnt divide n then 36 doesnt divide n^2

i.e. contrapositive

[vague coral]

@vale wigeon for the system of differential equation, if you want to see the solution, I have it, I can send it to you if you want

[noble sinew]

euclids lemma?

[vale wigeon]

12ka + a^2 should never be zero mod 36 for a in 1:5 and k in Z i htink

hm

[sage summit]

and obviously only k mod 3 matters

so that's a finite computation

[vale wigeon]

would still be nice to have a non-FTA proof for m^2 | n^2 => m | n though

this just FEELS like something that should be provable without nuking it using FTA

[noble sinew]

.

[sage summit]

what's euclid lemma again ?

[noble sinew]

if p divides ab then p divides a or p divides b

[sage summit]

/:

genius

that works for this case

but not for the general case Ann is talking about sadly

[vague jolt]

So I can literally go 6^2 | a^2 => 6 | a?

[sage summit]

I think ?

6 is not prime

but 3 and 2 are

[vague jolt]

Huh

So I should go

3^2 * 2^2

[sage summit]

you go 6² | a² => 3 | a² => 3 | a

do the same with 2

and then argue that (3 | a and 2 | a) => 6 | a

[hexed viper]

I think you need a powerful theorem as it doesn’t hold in general rings I believe

[vale wigeon]

can we prove p^(2m) | n^2 implies p^m | n

[hexed viper]

It’s got to use some ring theoretic property of Z

[vale wigeon]

because if we can prove that then my general case follows i think??

[vague jolt]

Okay wait this going a bit too fast LOOOOL

[sage summit]

the second implication is euclid's lemma

[vague jolt]

The question is to prove 36 | a^2 implies 6 | a

If a and b are ints

What is the first thing I should recognize as soon as I see this

That a^2 is a perfect square so it can be decomposed into primes with even powers?

[sage summit]

you should hit it with the fundamental theorem of arithmetic yeah

[vague jolt]

What I stated is derived from FTA right?

[sage summit]

what you stated ?

[vague jolt]

That a^2 is a perfect square so it can be decomposed into primes with even powers?

[sage summit]

yes

[vague jolt]

Okay and then my next step should be

To decompose 36 right?

Which can be decomposed into 6^2 = 3^2 * 2^2

[sage summit]

well somehow you will have to conclude that 2 divides a and 3 divides a because if not you would not have 36 divides a²

[vague jolt]

Ah wait

So we have

[sage summit]

yes

yes

4 and 9 divide 36 and 36 divides a² so 4 and 9 divide a²

where does that last implication come from

[vague jolt]

Ur right

[sage summit]

and the first one too

[vague jolt]

I'm not sure

But I know the logic is correct

But where does it come from

I have no idea

[sage summit]

use euclid's lemma

or the fundamental theorem

to show them

[vague jolt]

I'm lost again

[sage summit]

so far you have showed that 36 | a² => (4 | a² and 9 | a²)

and you didn't use any deep result

[vague jolt]

Okay

So I'm at 4 | a^2 and 9 | a^2

[sage summit]

so you will need to use something to end

like euclid's lemma

euclid's lemma should be your best friend

[vague jolt]

I just saw Euclid's lemma right before u said that lol

So we got

2^2 | a^2 and 3^2 | a^2 => 2 | a^2 and 3 | a^2 and according to Euclid's Lemma if p is prime and p | xy then p | x or p | y

So 2 | a * a and 3 | a * a => 2 | a and 3 | a

But now how do I get 6 | a

[sage summit]

you compute the least common multiple of 2 and 3 ?

[vague jolt]

OH

SHIT

YEAH

THAT MAKES A LOT OF SENSE

SINCE WE KNOW THAT BOTH 2 AND 3 ARE DIVISIBLE INTO A

THAT MEANS THAT A IS AT LEAST EQUAL TO THE LOWEST COMMON MULTIPLE OF 2 AND 3

WHICH IS 6

HOLY SHIT

MATH IS SO SICK BRO LOOOOL

[sage summit]

[vague jolt]

Sorry

I actually like math a lot it's just always hard for me to get started studying

And the best way for me to learn is to talk to someone like this

But it's kinda hard to get people to have a convo like this with me lol

So thanks

[sage summit]

it's fine lol

[alpine sable]

is the channel free?

[vague jolt]

Yeah take it

[alpine sable]

ok

thanks

can i know what trans. prop. of ~= is supposed to stand for?

[sage summit]

transitivity

[alpine sable]

oh

thanks

[sage summit]

if angle A = angle B and angle B = angle C then angle A = angle C

dark magic

[alpine sable]

thanks

help anyone?

(using unitary method)

[tight locust]

a + b + c = 85342
a = 3b/8
b = 5c/2

[vague jolt]

Is there a bijection between N and (Q X Q) ?

In other words, is Q x Q countable?

[vale wigeon]

yes there is

[vague jolt]

So Q X Q is countable?

[vale wigeon]

yes, Q^2 is countable.

[alpine sable]

ty

[hexed viper]

Its easy

Yeah

Since Q is countable, two copies is countable. You just alternate from counting the left one and the right one

[vale wigeon]

euler2

[hexed viper]

Do a diagonal thing

[vale wigeon]

we aren't talking about the union of two disjoint copies of Q

[hexed viper]

Zoom zoom

[vale wigeon]

we're talking about the product of two copies of Q

@vague jolt is your teacher draconian about that exact method being used?

[vague jolt]

We're talking abt a cartesian product

[vale wigeon]

does your teacher award you zero points for using any method other than this?

[vague jolt]

Uhhh as long as it's a cardinality theorem it should be fine

[hexed viper]

It’s bijective with NxN

And count that using a snake

[vale wigeon]

euler2 seems so eager to help that i think i'll let them do the talking

and dip

[vague jolt]

Tf is a snake

[hexed viper]

A zig zag

[tight locust]

what is A1

just 0?

[vague jolt]

Yeah

0

[hexed viper]

Step 1: prove N is bijective with NxN
Step 2: use the bijection between N and Q to show NxN is bijective with QxQ
Step 3: compose the compositions

[vague jolt]

Actually

0 is just 0

[tight locust]

does N include 0 in this scenario

[vague jolt]

Nah N doesn't count 0

For us the first natural number is 1

Bruh where did u guys go

[hexed viper]

I told you what to do

[vague jolt]

I don't get it

:(

[hexed viper]

Which step can’t you do

[vague jolt]

How do I prove N is bijective with N x N LOOOL

When N x N is a cartesian product

[hexed viper]

Construct a bijection

[loud frigate]

How do I solve -3 * 5^x = -6

[ancient creek]

Start by isolating 5^x on one side

[left knot]

can someone help explain how this formula was derived? from https://en.wikipedia.org/wiki/Argument_(complex_analysis)

[left knot]

<@&286206848099549185> I've looked up the tangent half-angle formula, but all it is over there gives some function in terms of theta, but I want it in terms of a and b, or x and y.

I'm just trying to understand this explicit function of arg(z), and I'm getting nothing.

[trail mantle]

How do u do find the solution do this ODE

[left knot]

I'm still stuck with my question...

any help at all would be appreciated

all of this is just to have an accurate function to convert from cartesian to spherical coordinates, am I wasting time on a dumb method? I've been scratching my head for one day now 😅

the really annoying part is that arctan doesn't take into the quadrant that the coordinate is, which is why I have to go through all this trouble

I know that it is true, but I just don't get how to derive the explicit formula from the half angle tangents

[bitter sage]

hey im trying to understand is the anwser to this 2 or d.n.e

[warped mist]

Hey there, im trying to find two polynomials, a(x) and b(x), such that

[alpine sable]

the sum of the two polynomials is going to be one

a(x)(x^2 + 1) + b(x)(x^2 - 3x + 2) = 1

[warped mist]

I got to that part, and I understand there are multiple solutions

But is there a method to find these?

Could I just set b(x) = 0 and continue from there?

[alpine sable]

you'd still be left with x^2 + 1, which isn't as the question asks

[warped mist]

hmmm

[night owl]

Does anyone here understand circuits well enough to answer questions about resistors?
DM if you do.

[sly mantle]

just ask. whoever can & wants to answer will do

@night owl

[night owl]

Why is it that resistors in a series circuit have identical voltage. I thought I had an explanation in my head, but it contradicted the ability of the total resistance in a parallel circuit to be less than any of the resistors.

[sly mantle]

resistors in a series circuit have identical voltage
who said this?

[austere bridge]

resistors in series with each other will have the same current, but certainly not the same voltage

[strange urchin]

It's 2

The function isn't defined at that point, but the limit doesn't need the function to be defined

[night owl]

My bad. I was referring to the current.

[sly mantle]

apply conservation of charge to each point in the circuit. as a result current must be constant throughout

[austere bridge]

My answer to your question would be, then, how could they not have identical currents? Since they're in series, there's nowhere else for the charge to go. If they weren't all equal, then charge would build up somewhere

[night owl]

But how is it that they halve their charge when flowing through the first resistor? Why isn't it that they reduce after the first and then as a result end up weaker when passing through the second?
I don't want the law of conservation as an answer because it seems like magic to me. When people answer with the law, I just imagine that the current somehow knew there were more resistors and that it needs to divide their charge.
I want to know why the law of conservation is a thing as well, since it is playing a part in my misunderstanding.

[austere bridge]

The charge doesn't get halved when flowing through the first resistor. It's voltage gets halved after going through the first resistor (assuming there's two resistors with equal resistance)

[night owl]

That still doesn't answer my other questions and it brings up more. Like it does reduce after the first but it depends on what's ahead? That's exactly like the magic thing I talked about, isn't that supposed to be false?

[austere bridge]

The answer to that is electromagnetic waves

https://electronics.stackexchange.com/questions/19759/how-does-the-current-know-how-much-to-flow-before-having-seen-the-resistor

reading (I think) should be able to help you

[night owl]

ok I'll try it.

[cloud portal]

assume we'd sell 5 oranges a day @ $1, 2 oranges a day @ $2, and 1 orange a day @ $3
and i have 10 days, and 30 oranges. what is the combination of pricings that will sell all of the stock and get the highest amount of profit?```

very simple but i can't figure it out

[alpine sable]

how could I solve this

I drew the diagram but I don't know if there are any relations since I don't know if CM is a angle bisector or no

and nor do I know whether the line AB with M on it has anything special in this

[noble sinew]

given that you have to sell all of the stocks one way is just to find integer solutions to
5 * x+2 * y+1 * z=30 given that x>=0 and y>=0 and z>=0 and x+y+z<=10
and then just checking the amount sold for each combination

[ionic jewel]

,w 5x+2y+z = 30 over the positive integers

[ocean seal]

Results provided by WolframAlpha

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

Requested by 兔子#6562

[noble sinew]

this is assuming that you are not allowed to sell for example 4 oranges on a day for 1 dollar each which you might be allowed to do tbh

[ionic jewel]

idk why Wolfram didn't find any solutions

[noble sinew]

,w 5 * x+2 * y+1 * z=30 given that x>=0 and y>=0 and z>=0 and x+y+z<=10

,w 5x+2y+1*z=30, 1x>=0, y>=0, z>=0, x+y+z<=10

[ocean seal]

Results provided by WolframAlpha

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

Requested by ScapeProf#9427

[ionic jewel]

4,4,2 is the solution

[noble sinew]

There are more

[ionic jewel]

ah i guess checking them is good

1x + 2y + 3z maximum

18, 14,20, 15, bas

looks like x = 5, z = 5 for your $20 is the best

although you could make more by just not selling your whole stock :thonk:

[noble sinew]

its x * 5+2y * 2+ 3z

[cloud portal]

hold on

i dont want to do any form of manual input

i just did a simultaneous equation consisting of the following combination:
5x + 1.5y = 30,
x + y = 10,

where 1.5 (the y coefficient) is the average of the remaining amounts sold per day, and a recursive check continues doing the equation with other values

[ionic jewel]

oh yeah you sell 5 oranges for that price, so 5x + 4y + 3z, you right

makes so much more sebse

,w max of 5x+4y+3z given 5x+2y+z=30, 1x>=0, y>=0, z>=0, x+y+z<=10

[noble sinew]

what where does this come from? And you are also allowed to sell your stock on less than 10 days

[ocean seal]

Results provided by WolframAlpha

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

Requested by 兔子#6562

[ionic jewel]

yeah idk how to do it all in one Wolfram query

someone who knows the proper notation probably could

[cloud portal]

sure but that wouldnt lead to max profits

[noble sinew]

It could in theory

but anyway what you posted isn't correct

[cloud portal]

sure but not in this current example

it isn't yeah i just checked

i have to do it programmatically

and the issue is that the price lists change, so it doesn't necessarily have to be 3 prices, it could be 5, or 10

so i can't do a specific equation

i most definitely can't do random number tests

[noble sinew]

here is the formula for the price

[cloud portal]

right but that formula still needs solving

you'd have two equations

[noble sinew]

and what is the problem? Any program can solve this over the integers

[cloud portal]

5x + 2y + z = 30

x + y + z = 10

well <= 10 actually, yes

i can't programmatically solve it that way, i have to have a direct method of approach either manually or in an equation form

how does wolfram figure it out

[ionic jewel]

ur either gonna need to brute force it or learn number theory looks like

[cloud portal]

how does number theory factor in here

[ionic jewel]

that's where the "over the integers" stuff gets taught

diophantine equations

[cloud portal]

ok so double checked, selling the maximum amount of stock using the average sales of the remainder works fine

the only issue is that i need to integrate the algorithm to account for the maximum profit

i think actually the concept here might be an optimal price weight, where the optimal price is calculate beforehand and the recursive loop attempts to have the price of the sold stock approach the optimal price

[alpine sable]

I have linear algebra question

***The following vectors are given:
|f1>= ξ|e1 > +|e2>
|f2>= |e1 > +ξ|e2> +|e3>,
|f3>= |e2> +ξ|e3>

Where |e1>, |e2> and |e3> are orthonormal and ξ is a real number. For which values of ξ are |f1>, |f2> and |f3> linearly independent? For which values are they linearly dependent?***

Can someone help me out this this? I am wondering if I need to do this:

c1|f1> + c2|f12> + c3|f3> = 0
is this what i have to do? the problem is that i will end up with four scalars c1,c2,c3 and ξ
is that fine anyways?

[sly mantle]

yes, solve the linear system c1|f1> + c2|f12> + c3|f3> = 0. along the way we must split into appropriate cases of the value of ξ, seeing which lead to all c's being 0 (independent) and which lead to at least one of the c's being nonzero (dependent)

[alpine sable]

very well, thank you

[livid knot]

Can someone help me understand this differentiation question please

[slender forge]

This is a power rule with chain rule @livid knot

[livid knot]

Ok thanks

[slender forge]

Try to brute force it and see if u can get it or see where u get tripped up at

[livid knot]

alright, ill let you know if i have an answer or get stuck

[glad girder]

Can anyone solve this...

[noble sinew]

start with placing A and B

[halcyon pulsar]

Help.

[alpine sable]

do i need to have the e's as common factors btw?

[vague coral]

the complex cube root of unity are the number :
$$ \omega_k = e^{2ik\pi/3}, k \in {0, 1, 2}$$

[sly mantle]

wdym

[ocean seal]

Herels

[alpine sable]

i came up with the following:
c1ξ + c2 = 0
c1+c2ξ+c3 = 0
c2+c3ξ = 0

[halcyon pulsar]

Appreciated.

[crude moss]

pls

[alpine sable]

c1ξ + c2 = 0
c1+c2ξ+c3 = 0
c2+c3ξ = 0

just reposting because i replied to myself and all

[sly mantle]

i'll repeat we must eventually do this

split into appropriate cases of the value of ξ, seeing which lead to all c's being 0 (independent) and which lead to at least one of the c's being nonzero (dependent)
i'm not really checking algebra done on the way

[thorn swallow]

man

[alpine sable]

i dont really know where to go from this, im rather stuck

should i solve the three equation system i displayed above?

[sly mantle]

yes

[sharp patio]

i'm reviewing my notes for riemann sums, but i'm a bit stumped from looking at this. It all makes sense to me up until the line that the red arrow is marking.

A few things I'm confused about is
Why 8/n^3 from the line before becomes 8/n^2 and how the denominators 3,2, and 6 are decided to be used
Basically I'm just confused how
8/n^3 sigma k^2 becomes
8/n^2 (n^3/3* n^2/2 * n/6)
I understand that you have to expand it and then take the limit but i can't quite visualize it

[shell widget]

,w sum k^2 from k=1 to n

[ocean seal]

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Requested by Sup?#7960

[shell widget]

Just use this

and also this

,w sum k from k=1 to n

[ocean seal]

Results provided by WolframAlpha

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

Requested by Sup?#7960

[shell widget]

Factor out n from each factor, so you have (n^3)/6 [1 + (1/n)] [2 + (1/n)]

We factor out n so that we have n^3 and then it cancels out with n^3 in the denominator

Same for second term, factor out n, we have n^2 (1/2) [1 + (1/n)]

sum of 1 from 1 to n is just n, and n/n = 1

[sharp patio]

i don't quite see it yet but probably rereading it a couple of times will work.thanks for the help!