help-10

[merry vault]

that means that the limit of the sequence is inf{a_n}

[royal basin]

yeah, sure it does.

[grizzled shore]

No I just meant I haven’t seen limit of a set even in probability

[merry vault]

ok so with how inf is defined, that means the limit could be an element of the sequence or maybe not right

or am I misunderstanding

[grizzled shore]

See how it says ℕ

[merry vault]

I proved with induction that a_n > 2 for all n in N

[merry vault]

yes

[grizzled shore]

Infinity is not in ℕ

[royal basin]

the limit of a sequence need not appear as a term in it even once

it CAN but it doesn't need to

[merry vault]

okay thanks haha

.close

[obtuse pebble]

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[merry vault]

.reopen

[obtuse pebble]

✅

[merry vault]

which means the sequence has a min and min = inf ?

[grizzled shore]

Yes

[merry vault]

so I assume all sequences with n in N have either inf=min or sup=max

[grizzled shore]

For similar reasons to max and sup

[royal basin]

???

[grizzled shore]

What

[merry vault]

[royal basin]

...

[merry vault]

nvm

[royal basin]

you're overthinking it so hard tbh

[grizzled shore]

Why does n in N imply there exists a min/max

[merry vault]

oh it could also just be infinite in both ways right

my bad

but I mean

[grizzled shore]

n in N means n is always finite

[merry vault]

ok I don't understand that

don't you count up infinitely?

[royal basin]

what even are you asking at this point

this entire exchange has confused me severely

can you repeat the question you want to ask, please

[grizzled shore]

There are an infinite number of terms, but you never get to the limit

There’s no term that is the limit in the sequence

Well there could be but on accident

[grizzled shore]

But that goes back to this point

[merry vault]

Sorry I don't understand because there's lim n->infty (a_n)

[grizzled shore]

This evaluates to a number

[merry vault]

Let a_n be some sequence with n in N. If a_n has a lower bound, it's inf a_n = min a_n. If a_n has a upper bound, it's sup a_n = max a_n.
Is that right?

[grizzled shore]

But $a_{\infty}$ doesn’t exist

[warm shale]

Frosst

[merry vault]

Ah I see what you mean now

I think

because you count up forever

infinity is not a number

anyway

don't worry I was just thinking about stuff randomly

it's not important

.close

[obtuse pebble]

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[vale forum]

So uh I've been going through my textbook and I'm confused on why it does something at one point.

[vale forum]

It's gonna take a while to type this so bear with me...

This is the question

It explains it in 3 steps

I understand the first step, which is...

Now I'm a bit stuck on the second bit...

I understand the formula r*e^iθ = (cosθ + isinθ)

but why does it multiply by

[warm shale]

Silly Little Twat

[vale forum]

instead of $e^{-i\frac{\pi}{n}}$

[warm shale]

Silly Little Twat

[vale forum]

Which is the actual congugate

For reference, here is the 3rd step

I see pi/2n is used in the final answer, but I can't see how it came up with pi/2n

Maybe I'm tired and missing something really simple and obvious 🤷‍♂️

I've been trynna figure it out but I couldn't so I'm going to take a tiny rest from it, ping me when you get it please

[obtuse pebble]

@vale forum Has your question been resolved?

[vale forum]

<@&286206848099549185>

woah whats with the super reactoin

scary#

the question

[timid silo]

no idea sorry

[vale forum]

ok

[warm canopy]

Just because your end goal involves π/2n

So in step 2 they're making sure all the angles are π/2n

[vale forum]

genius

thanks

That's so obvious I'm doubting my existance

.close

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[mint moat]

why is this not e

why is this d and not e
Image

[cinder lintel]

it is d because if you have number

less than -2

the square of those numbers

is also greater than 4 so function will exist

[mint moat]

why can't it be e

[steel parcel]

wdym?

because x can be a negative number

it works with - 1, -2, -3 and so on

[mint moat]

it doesn't work with -1

@steel parcel it's an imaginary number if -1

[steel parcel]

How

oh I se

numerator is a semicircle, x can be any number but anything between (-2, 2)

[mint moat]

wait i just don't get whats the point of the absolute value

[steel parcel]

turns any negatives into positives

|-2| = 2
|2| = 2

[mint moat]

OHHHH

tunnel vision

thanks a lot @steel parcel

[steel parcel]

nws, close this ticket before using the other one

[mint moat]

.close

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[fathom flicker]

I don't know if I am being a bot rn or not but I don't get why this is wrong

I tried applying unit step to kill it after t=3 but that didn't work either

shouldn't this just be 5* L(1) = 5*1/s =5/s

[elfin burrow]

,w integral 0 to 3 5e^(-st) dt

[warm shale]

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Requested by tushar#0942

[fathom flicker]

hey atleast I had the 5/s part

I guess since it doesn't go off to infinity the upper bound matters

normally it would get killed

because it is a transient

I see

ty Tushar

.close

[obtuse pebble]

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[grizzled shore]

.reopen

[obtuse pebble]

✅

[grizzled shore]

.coose

@fathom flicker

[fathom flicker]

.clsoe

.coose

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[warm canopy]

.texas

[obtuse pebble]

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[ripe oracle]

Hey I have this exercice:
Show that the matrix
$M =\frac{\sqrt{13}}{13}\begin{matrix} 2 & 3 \
3 & - 2
\end{matrix}$ is orthogonal. Is it the matrix of a rotation or a bilateral symmetry? If it is a rotation, calculate its angle. If it is a symmetry, calculate its axis.

[ripe oracle]

I've already done the first part by showing that multiplying the transpose of the matrice times itself gives the identity matrix, now for the second part, how should I proceed? What I though of is to calculate the eigen values then the eigein vectors to diagonalise the matrix and determine the transform from its diagonalised version

[warm shale]

froud

[kind hawk]

often it is enough to compute trace and det

[ripe oracle]

how so?

which one should I do first and to find what?

[kind hawk]

do both

they tell you the sum and product of the eigenvalues

from that you can already argue some stuff

without actually having to diagonalise

[ripe oracle]

so after calculating the trace and the determinant what do I do with the resulting scalar values?

[kind hawk]

what are the possible types of orthogonal transformations and which eigenvalues do they have

[ripe oracle]

I'm sorry I don't quite get the reasoning...

what operations are to be applied

can you give an example

[kind hawk]

if the matrix is a rotation

what do you know about the eigenvalues

[ripe oracle]

I don't know, I can't quite make the link between eigen values and matrices in general

[kind hawk]

do you know what eigenvalues are

[ripe oracle]

yes the roots of the caracteristic polynomial or $det(M-\lambda I ) = 0$

[warm shale]

froud

[kind hawk]

technically correct but not the definition you should think of

[ripe oracle]

also each eigen value is associated to an eigen vector v such that $f(v) = \lambda v$

[kind hawk]

ok

[warm shale]

froud

[kind hawk]

so an eigenvalue belongs to a vector which is only scaled

do you know that eigenvalues of orthogonal matrices have absolute value 1?

[ripe oracle]

no I didn't, orthogonal matrices are matrices which have orthogonal column vectors right?

[kind hawk]

orthonormal column vectors

[ripe oracle]

so orthogonal unit column vectors (magnitude 1)?

[kind hawk]

yes

[ripe oracle]

ok i see, what does the property with the absolute value 1 state?

[kind hawk]

wdym state

I wrote it down

[ripe oracle]

I mean what does it implicate?

so every eigen value of an orthogonal matrix is 1?

[kind hawk]

no

[ripe oracle]

oh -1 and 1?

[kind hawk]

no

unit circle

[ripe oracle]

what do you mean, something similar to $a^2 + b^2 = 1$ or am I lost?

[kind hawk]

complex numbers

[warm shale]

froud

[ripe oracle]

oh

so 1 + i?

[kind hawk]

no

[ripe oracle]

or 1 + \lambda i ?

[kind hawk]

that has abs value sqrt(2)

a+bi where a^2+b^2=1

[ripe oracle]

okayyy that's what I thought

[kind hawk]

or exp(i phi) where phi is some real number

or cos(phi)+i sin(phi)

[ripe oracle]

polar coordinates right?

[kind hawk]

yes

[ripe oracle]

Ok I see, however we don't have this property to learn for the exam, this hint was given for the exercice: The eigenvalues are −1 and 1. An eigenvector of eigenvalue 1 is located on the symmetric axis, while an eigenvector of eigenvalue −1 is orthogonal to it. To know the axis of this orthogonal symmetry, it
It is therefore sufficient to know an eigenvector of eigenvalue 1, would it be possible to explain what they're talking about here

[kind hawk]

....

ok

[ripe oracle]

It isn't the only way to proceed, since I didnt understand the hint, I wanted to see if they were any other ways, thus asking about your way

[kind hawk]

let's think about what the axis of symmetry means

those are the points which do not change under the transformation

or in other words, only get scaled with the value 1

aka eigenvalue 1

[ripe oracle]

I see

[kind hawk]

meanwhile those vectors orthogonal to the axis get mirrored. aka flipped. aka multiplied with -1. aka eigenvalue -1

[ripe oracle]

I don't understand

why are we considering the orthogonal vectors and why only the value -1?

[kind hawk]

well that's what reflecting over the axis does

[ripe oracle]

what happens with the vectors that aren't orthogonal? and I don't get why -1 multiplies vectors here

[kind hawk]

the vectors which are not orthogonal also get reflected but thats not just scaling with some number in that case. aka not an eigenvector

if v looks into some direction, then -v looks into the opposite direction

[ripe oracle]

yes

how is the axis of symetry built with eigen values and vectors?

[kind hawk]

.

[ripe oracle]

ok, lets say we have eigen values -1 and 1 and eigen vectors (-1, 0, 2) and (1,-3,4) what would be the axis of symetry?

[kind hawk]

well lets stay 2 dimensional

[ripe oracle]

yeah sorry

[kind hawk]

the eigenvector with eigenvalue 1 gives the axis

the eigenvector with eigenvalue -1 is orthogonal to the axis

[ripe oracle]

why orthogonal and not colinear?

[kind hawk]

because that's how mirrors work

eigenvector means that it stays on the same axis

that only happens if it is orthogonal to the symmetry axis

[ripe oracle]

I don't understand

which axis? since the eigen vector with eigen value 1 is the axis

[kind hawk]

eigenvector of a transformation means that it gets only scaled. which means that it stays on its own axis

[ripe oracle]

ohhhh okay, so the transformed vector is colinear to the original vector?

[kind hawk]

if its an eigenvector, yes

[ripe oracle]

ok I see

[kind hawk]

either in the same direction if the eigenvalue is positive. or in the other direction if it is negative

or 0 if the eigenvalue is 0

(and lets ignore complex eigenvalue for now cause that is impossible to imagine anyway)

[ripe oracle]

yeah, but you said that the -1 eigen vector is orthogonal, I still dont get that?

[kind hawk]

if you have a mirror and something gets reflected so that the mirror image is collinear

that something has to be orthogonal to the mirror

[ripe oracle]

Ok, I understand, but what is the mirror in this case?

[kind hawk]

the axis of symmetry

[ripe oracle]

what's the connection with the eigen vectors?

like how is it related, that's what I dont get

[kind hawk]

the eigenvectors of eigenvalue 1 are the axis of symmetry

they dont change

the axis of symmetry is the thing that doesn't change

[ripe oracle]

ok, that's what I have in mind

I really don't see why the left vector (-1v) should be orthogonal to the right vector (1v) and not to the y unit vector...

[kind hawk]

everything always goes through the origin

how is anything in that image related to reflecting along an axis

[ripe oracle]

indeed, my bad, gimme a sec

[kind hawk]

I would draw something but I'm on mobile

[ripe oracle]

yeah, no worries I'm going to try from my side

something like this?

where is the orthogonality?

[kind hawk]

so far you only drew the axis of symmetry

[ripe oracle]

Oh, the north-east vector is 1v and the south-west is -1v, in this case -1v is not orthogonal to 1v, I dont understand

[kind hawk]

well it's not

[ripe oracle]

ok so that's exactly what I don't comprehend here

[kind hawk]

I don't really understand how you think that this is the image you produce

[ripe oracle]

what do you mean?

from what I understand, -1v should be the red or green vector in this image

since it is orthogonal to the black 1v vector

[kind hawk]

no

what does -v have to do with the eigenvector of eigenvalue -1

v and -v both have eigenvalue 1

every vector on that line has eigenvalue 1

[ripe oracle]

no I meant -1 x v the eigen vector of eigen value -1

[kind hawk]

if you call the green vector w, then the red vector is -w

aka multiplied by -1

aka eigenvector with eigenvalue -1

aka orthogonal to v

which is your black vector to the north east

[ripe oracle]

yes exactly so why aren't both eigen vectors on the black line?

[ripe oracle]

the eigein vector of eigen value 1

[kind hawk]

there is more than one eigenvector for one eigenvalue. every vector on the line is again an eigenvector

[ripe oracle]

oh right

[kind hawk]

v, 2v, 17v etc

[ripe oracle]

oh, but isn't the only allowed scalar values 1 and -1?

[kind hawk]

what

[ripe oracle]

I'm really super confused here I think

so eigen vectors are vectors that are multiplied by a scalar after transformation and eigen values are the scalar values which multiply the eigen vectors, right? Why do we have 2 and 17 as scalar if the eigen values are -1 and 1

[kind hawk]

2v gets transformed to 2v

aka multiplied by 1

[ripe oracle]

ohhhhh, my bad

[kind hawk]

17v gets transformed to 17v

[ripe oracle]

I didn't see it like that

ok back to the drawing, where are situated the eigen vector lines?

[kind hawk]

the line through the black vectors is the axis of symmetry. all the vectors on that line are eigenvectors with eigenvalue 1

all the vectors along the line through the red and green vectors are eigenvectors with eigenvalue -1

cause their mirror image is again on the line, just in the opposite direction

[ripe oracle]

ok perfect, that's what I thought

so now back to the exercice hint, what do I have to do from here?

[kind hawk]

you want to calculate the axis of symmetry (cause apparently you are already given that it is not a rotation)

so you have to compute the eigenvectors belonging to eigenvalue 1

I hope you know how to do that

[ripe oracle]

ok, yeah, I solve $(M-1\cdot I)(x,y,z) = (0, 0, 0)$, right?

[warm shale]

froud

[kind hawk]

2d

but yes

[ripe oracle]

yeah I was going to modify it 😂

ok great, so I solve that and then? I determine what type of transformation the vector is?

[kind hawk]

then you are done

[ripe oracle]

Yep, Indeed I just give the vector and that gives the transformation axis right?

[kind hawk]

yes

[ripe oracle]

ok great, also a last question why can't the symmetry axis be the eigen vector of eigen value -1?

[kind hawk]

the axis of symmetry doesn't change

eigenvalue 1

[ripe oracle]

?

why can't it be -1?

[kind hawk]

eigenvalue -1 means that it gets changed

from w to -w

[ripe oracle]

but the axis of symmetry changes the vectors directions anyways , no?

[kind hawk]

no

[ripe oracle]

oh sorry, yeah I get it

[kind hawk]

the mirror image is again the vector itself

[ripe oracle]

Yeah I understand 😄

Great, well thanks a lot for your help, have a great rest of the day !

[kind hawk]

u2

[ripe oracle]

thanks

.close

[obtuse pebble]

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[obtuse pebble]

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[opal sonnet]

Can someone help me with this

[opal sonnet]

The question is in blue

[umbral zinc]

,rotate

[obtuse pebble]

@opal sonnet Has your question been resolved?

[warm shale]

nebula40

[tender stratus]

@opal sonnet I think you can finish the last step

[opal sonnet]

did you multiply by cos ?and where did the csc come from?

[tender stratus]

yes, and 1/sin(t) = csc(t)

[opal sonnet]

and how do i get rid of the cos on top

[tender stratus]

well the top product is just cot^3(t)

[warm shale]

nebula40

[opal sonnet]

ohh ok

got confused a bit

but all good now

thank you

[tender stratus]

yeah maybe I should have turned csc to sin instead to be clearer

[tender stratus]

no problem

[obtuse pebble]

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[wary grove]

Hey all, it's quite embarassing but I need help with catching up to math to prepare for statistics..I'll put it into questions to explain what I need help with
(for example 3/2 is a fraction, 3 over 2)

1. Find the slope and write the equation of the line containing the points (6, -2) and (0, 5)

2. Solve and graph each inequality. Graph each.
   a. | 4x-6 | < 4
   b. | x + 3/2 | > 3/2
   c. -2x + 3y > x + 9
   d. -y > 3x - 1

[wet moss]

do you need help with both 1 and 2?

[wary grove]

yes please

[wet moss]

ok so do you know how to find the gradient of a line between 2 points?

or the "slope"

[wary grove]

m is the slope which is before the x right?

[wet moss]

yeah

[wary grove]

I dont think I know how to find it on #1 when it comes to the 2 points though or maybe I'm overthinking it. but for #2 I'm guessing each number before X is the slope?

[wet moss]

not always

the number before y must be 1

and on the other side to the x

but to find the gradient between 2 points its the change in y/change in x

[wary grove]

Alright so how would I apply this to the first problem, wouldn't I have to do it in reverse or something?

[wet moss]

well you have 2 points, (6,-2) and (0,5)

the 6 and 0 are the x coords and the -2 and 5 are the y coords

now how much has y changed between those 2 points

[wary grove]

my bad I was occupied, I'm lost 😅

is that using the y2-y1 formula thing?

[wet moss]

yeah

[wary grove]

rise over run i believe?

[wet moss]

yeah exactly

[wary grove]

OH okay thats a refresher

so -7/6 because y2-y1/x2-x1 is 5-(-2)/(0-6) which is -7/6

[wet moss]

yeah

now do you know the formula y -y1 = m(x-x1)?

[wary grove]

no but I believe I just saw that on a previous question

So that makes it y - (-2) = -7/6(x-6)?

[wet moss]

yeah

then just rearrange for y

[wary grove]

Oh wait so is that good or? is my y1 wrong?

[wet moss]

no

its right

[wary grove]

oh okay, now how about question #2, i was just soaking in every step

[obtuse pebble]

@wary grove Has your question been resolved?

[obtuse pebble]

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[wary grove]

.reopen

[obtuse pebble]

✅

[wary grove]

actually i'll get back to it late

[rigid condor]

I tried to solve this problem by using branch of logarithm, but then i realised to use the branch of log we need to make sure 0 doesn't in the domain and in my case 0 is in the domain. Can anyone help me with this?

[obtuse pebble]

@wary grove Has your question been resolved?

[obtuse pebble]

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[lapis ether]

i need help

[obtuse pebble]

Please don't occupy multiple help channels.

@lapis ether Has your question been resolved?

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[lapis ether]

.reopen

[obtuse pebble]

✅

[obtuse pebble]

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[obtuse pebble]

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[swift coral]

I have this problem
I can't figure out if f|u ^-1 (O) is open in X or the subset topology of U.
Or if that makes a difference

[fickle turret]

Maybe one could argue that if f wasn't continous then there exists a sequence a_n with lim a_n = a, but lim f(a_n) is not f(a) and make a contradiction with that

But that doesn't look like something that is worth 4 points

[swift coral]

4/20 points

[fickle turret]

How much do you need to write for 4 points from your experience?

[swift coral]

3/4 of a page?

ah wait we have a convention that neighborhoods are all open neighborhoods

so I think that means the distinction doesn't matter

[fickle turret]

Yes, we had this convention, too

[swift coral]

since f|u^-1(O) intersection U (U a neighborhood) is open in X

[fickle turret]

What is O?

[swift coral]

oh I meant to say O is some open set in Y

I want to do f^-1(O) is open for all open O in Y

[fickle turret]

Okay

[swift coral]

now f|u^-1(O) open by continuity of restriction to specially given U.
if it's open in U it means it is some open set in X intersected with U, but U open therefore- the result of that intersection is an open set in X.

it's also a subset of f^-1(O)

and I think it's fine then

we can just union them all

[fickle turret]

Like you are saying that for all x in f^(-1)(O) we can find an open subset U where x is continous in. And then f^(-1)(O) is the union of all those?

[swift coral]

yeah

I mean union of all of f|u^-1(O) actually

[fickle turret]

But doesn't that only show that f^(-1)(O) is a subset of this union?

[fickle turret]

What's u here?

[swift coral]

subscript of U

[fickle turret]

And U?

[swift coral]

so for some x there is some neighborhood U

where f cont

[fickle turret]

How do you know that U is actually a subset of f^(-1)(O)?

[swift coral]

I am not unioning U

I am unioning f|U^-1(O) subset of f^-1(O)

[fickle turret]

Do you mean the intersection of f^(-1)(O) and U? Because the way you write it one could interprete it as a ristricted domain, but then U was part of O

[swift coral]

yeah

it could be that way too

U isn't part of O?

O is in Y

U is in X

[swift coral]

so I argue f|U^-1(O) open in X
then that f|U^-1(O) subset f^-1(O) obviously
then union them for all x in O.
and we are done.

[fickle turret]

Wait: So you are saying x is in O, so x is actually in Y not in X?

[obtuse pebble]

@swift coral Has your question been resolved?

[obtuse pebble]

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[azure anchor]

pt. 1

[azure anchor]

pt. 2

does pt 2 look right so far? im checking my work from a matrix calculator but its saying my answer for x_4 is wrong.

[obtuse pebble]

@azure anchor Has your question been resolved?

[nocturne minnow]

I think I found the mistake

Here

[azure anchor]

-4r_2 + r_3?

[nocturne minnow]

-4R2 = [0 -4 16 -4 | 280]
-4R2 + R3 = [0 -4 16 -4 | 280] + [0 4 0 -4 | 20] = [0 0 16 -8 | 300]

[azure anchor]

omg 280 not 210

okay let me continue now with that fix

thanks

.close

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[nocturne minnow]

I'm assuming that fixed everything?

[honest yacht]

Hey can someone explain what all those like rational, irrational, real, natural, and whole number mean? I can't figure out how to apply them to my homework

[warm canopy]

definition of whole number isnt very standard and may be different in whatever you're learning from

[polar fossil]

"whole number" is almost never used

[honest yacht]

Thanks for explaining:)

.close

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[honest yacht]

.reopen

[obtuse pebble]

✅

[honest yacht]

I thought I understood but I guess not? Bc like I thought the number would just be like an integer but
It said that was wrong?

[warm canopy]

you selected rational, and only rational

[obtuse pebble]

@honest yacht Has your question been resolved?

[obtuse pebble]

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[honest yacht]

What other ones does it work with

[obtuse pebble]

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[mellow mason]

hey

[fathom flicker]

@mellow mason Ask your question here

[mellow mason]

ok i need help with a biology assigment

[tight thunder]

this is a math server

but go on i guess?

[mellow mason]

yea ik i couldnt find any other servers

sorry

[tight thunder]

is this a math related biology question

just ask it

[mellow mason]

its a picture ill send in a sec

[fathom flicker]

There is a biology server in #old-network

"if you have trouble finding the information in the reading, feel free to use google to find any remaining notes"

Have you tried this?

[mellow mason]

i dont know what too look up

if that makes sense

[fathom flicker]

For example if you have trouble filling in the first column I would google,
"What is the function of nucleic acids as macromolecules?"
"What is the function of proteins as macromolecules?"
etc...

[mellow mason]

ok what about examples of it

nvm i could prob look that up too

[obtuse pebble]

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[woeful gyro]

please can someone help get the answer or give me the answer

[obtuse pebble]

@woeful gyro Has your question been resolved?

[woeful gyro]

<@&286206848099549185> help please

[tepid mountain]

So what are these values on the top of the table

[woeful gyro]

what do you mean

the sr?

[tepid mountain]

No I mean the numbers

That are next to each other

[woeful gyro]

its 2/3x-40 and 1/4x+12

[tepid mountain]

So the x has 2 values ?

I mean it seems that without knowing what is that row
The question can not be solved

[woeful gyro]

no its jus yu have to solve for one

the row is the answers when yu solve for x

[tepid mountain]

What I want to say is that.
The question needs some number in it to be solvable

So the only number given here is that row of numbers.

So it is important to know what that row of numbers represent

I don't think that it presents the equations of x

[woeful gyro]

so how would should i do to get x

do i solve 2/3x-40

[tepid mountain]

Oh I am sorry I understand that now

So let's start
First of all we need a little drawing that will explain the question a lot

[woeful gyro]

like this

@tepid mountain

[tepid mountain]

I am with you

[woeful gyro]

i drew it what's next

@tepid mountain

@supple pumice could yu help me

.close

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[topaz bone]

im stuck on this problem

[obtuse pebble]

@topaz bone Has your question been resolved?

[topaz bone]

<@&286206848099549185>

[obtuse pebble]

@topaz bone Has your question been resolved?

[topaz bone]

.close

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[serene jetty]

assuming you have the given differential equation (1), dividing both sides by dy gives (2) and integrating both sides to solve gives (3). My question is, what exactly does integrating with dx in denominator mean?

[serene jetty]

my intuitive answer is it makes no sense to try to integrate with a dx in the denominator, and i suppose this example is not great because it can just be re-arranged to get rid of dx and dy from denominator, but the core of my question still remains

[obtuse pebble]

@serene jetty Has your question been resolved?

[compact shadow]

dx/x=dy/6y

6In|x|+C=In|y|

There is no such thing as 1/dx

dx,dy… things like that, are not numbers, you can treat them as symbols for now, the precise definition appears in differential geometry, they are actually cotangent vectors. For now you can view them as symbols

[obtuse pebble]

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[gray kindle]

does taking the log of 30(0.5)^x/47 simplify to 30x/47 log(0.5)

[swift coral]

((yeah, we are building up f^-1(O) as an open set))

[gray kindle]

????

[teal turret]

$30 \log \left[(0.5)^{\frac {x}{47}} \right]$

This ?

[gray kindle]

like if i wanted to take the log of 0.5 specifically

[teal turret]

That?

[daring rock]

I thought it meant either $30(0.5)^{x/47}$ or $\frac{30(0.5)^x}{47}$ lol

[warm shale]

tatpoj

[daring rock]

and then log of one of those

[teal turret]

Yea the first one I had originally and they said no

[daring rock]

@gray kindle it would be helpful if you could take a picture so we can see how it's written

[gray kindle]

30 on the outside

log base 10 of 0.5

sorry

[daring rock]

Can you post the full problem? There isn't really any situation when you'd have an expression like that and take a log of just part of it

[warm shale]

Stephen

[gray kindle]

.close

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[fading charm]

hellooo may I ask if this is correct?

[fading charm]

I used sine and cosine to solve it, I don't know if its true

[glossy basalt]

let's check

,w 67*sin(39 degrees)

[warm shale]

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[glossy basalt]

y is correct

,w 67*cos(39 degrees)

[warm shale]

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[glossy basalt]

x is correct

[fading charm]

i see thank youu

.close

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[molten lintel]

How can I do this question with Newton's divided difference?

[molten lintel]

This is my solution, is it correct?

[obtuse pebble]

@molten lintel Has your question been resolved?

[molten lintel]

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[buoyant valve]

i dont understand

[obtuse pebble]

@buoyant valve Has your question been resolved?

[timid silo]

Can i post a physics question here

Mechanics

[glossy basalt]

firstly

[obtuse pebble]

Please read #❓how-to-get-help

[glossy basalt]

secondly #old-network

[glossy basalt]

is seems clear.
is it the partial fraction part that you don't understand?

[buoyant valve]

yes thats the part

[glossy basalt]

i see, usually we will set that fraction into
A/s + (Bs+C)/(s²+1)

and find A B C by comparing coeff

[buoyant valve]

im not sure how do the comparison part despite watching some of the yt videos

[glossy basalt]

i see

first we have
s-1 = A(s²+1) + (Bs+C)s

all good till here?

[buoyant valve]

yes

[glossy basalt]

then, we expand
0s² +1s -1 = As² + A + Bs² + Cs
then we have
0s² +1s -1 = (A+B)s² + Cs + A,
then we have
A+B = 0
C = 1
A = -1

is it okay?

[buoyant valve]

i am very lost

sry

[glossy basalt]

which part

don't be sorry, it's part of learning

[buoyant valve]

i understand the As^2 part by why does 0s^2 +1s -1

why does that happen

[glossy basalt]

it's because

s-1

we have 1 s

we have -1 1

and 0 s²

since originally we dont have s² term

for easier reading, we add a s² term by multiplying it by 0

so we will have an s² term with coeff (0)

[buoyant valve]

okay? i think i sort of understand

[buoyant valve]

what do u mean by this part tho

[glossy basalt]

that is why i wrote
0s²+1s-1

in this polynomial s-1
we have one s
we have negative one 1
and zero s²

[buoyant valve]

ohhh

alright i got it now

so afterwards i just substitute it in to get the ans?

[glossy basalt]

yep

[buoyant valve]

tysm that helped alot

[glossy basalt]

then you'll have a nice partial fraction

[buoyant valve]

tysm

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[brisk arrow]

given f(x)= (x-3)(x-6) / (x-1)(x-6)(x-8)

[brisk arrow]

can i considered f(x) as polynomial in the range of 1<x<6

[polar fossil]

no not really, but what do you mean by "consider it as polynomial"

[brisk arrow]

in the range of 1<x<6, f(x) is continuous

[polar fossil]

yes (that doesn't make it a polynomial though lol)

[brisk arrow]

i guess polynomial has a strict definition then

however i have one more question

how to know whether f(x) is continuous or not in the range of 1<x<6

is it provable? through mathematical process

[polar fossil]

well, polynomials are continuous, and f(x) / g(x) is continuous whenever f and g are continuous and g ≠ 0

[brisk arrow]

ohhhh

[brisk arrow]

that's lit

why f(x) is not polynomial in the range of 1<x<6

i mean it is continuous in that range

what makes it disqualified as a polynomial

[polar fossil]

well, a polynomial is specifically a function of the form $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$

[warm shale]

hayley!

[polar fossil]

um one feature that f exhibits on (1,6) that polynomials don't is that it has unbounded derivative [slope] on a bounded set

[brisk arrow]

"unbounded" and "bounded," im not sure what these words mean in math

[polar fossil]

eh

don't worry about it for now

here let me graph the thing

,w graph (x-3)(x-6) / ((x-1)(x-6)(x-8))

[warm shale]

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[polar fossil]

see how it goes just about vertical?

polynomials don't do that except at infinity

[brisk arrow]

I do, but how do you notice the graph gonna looks like this without putting it into wolfram. Can i spot the characteristics just on the expression

[polar fossil]

well, most salient is the fact that the expression isn't shaped like a polynomial

because it's got that big ol fraction bar

making it what's called a rational function

[brisk arrow]

from that, i can deduce f(x)/g(x) is not polynomial, when f(x) isn't 0 and deg g(x) >=1

[polar fossil]

yeah with the exception of if g goes away after cancelling common factors

like $\frac{x(x+4)}{x+4} = x$ is a polynomial (with a hole in it) even though it started with a fraction bar

[warm shale]

hayley!

[brisk arrow]

but f(4) is not defined. that, as you said, with a hole in it.
So i think it isn't a polynomial, because polynomial is continuous

[polar fossil]

true, good point! it would not be a polynomial on the real line

however, if you restricted the domain to something like (-4, 4) then it would act like a polynomial (and really would be one on that domain)

[frosty river]

You're confusing rational fractions in the sense of the field of fractions of the ring of polynomials and rational functions in the sense of quotient of polynomial functions. They're not the same object although they can be written similarly

This is true in R(X)

$$\frac{X(X+4)}{(X+4)}=X$$

[warm shale]

Categorist

[frosty river]

But

$$\frac{x(x+4)}{(x+4)}=\begin{cases}x & \text{ if } x\neq -4,\ \text{undefined} & \text{ if } x=-4.\end{cases}$$

[warm shale]

Categorist

[frosty river]

You have to take care whether you are working with algebraic fractions vs. rational functions

[obtuse pebble]

@brisk arrow Has your question been resolved?

[obtuse pebble]

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[viscid spindle]

Test for convergence of power series √n.x^n/(√n^2+1)

[errant lark]

Is that $\frac {\sqrt{n}\cdot x^{n}}{\sqrt{n^{2} +1}}$ ?

[warm shale]

Enemagneto

[obtuse pebble]

@viscid spindle Has your question been resolved?

[viscid spindle]

Yes

@errant lark

[errant lark]

What have you tried?

[viscid spindle]

Dalembert ratio test

But it's giving 1

[errant lark]

well, that's inconclusive.

[errant lark]

Can you show your working?

[obtuse pebble]

@viscid spindle Has your question been resolved?

[obtuse pebble]

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[restive pendant]

$\int^{\pi/2}_0 \ln{(\sin{x})} ,dx$

[warm shale]

bigpufik

bigpufik

[restive pendant]

maybe the complex form of sine?

<@&286206848099549185>

[frosty river]

What is the question?

[restive pendant]

i need to calculate this

[timber pine]

could be

im not sure

but the integral invovles complex numbers

[obtuse pebble]

@restive pendant Has your question been resolved?

[pseudo tendon]

use kings rule

[restive pendant]

you mean cauchys theorem?

that int(a+b-x)=int(x)?

[pseudo tendon]

yes

that'll prove the equality of two integrals then you may look into how to use properties of log

[restive pendant]

yeah but i get ln(sinxcosx)

And then the euler form?

[pseudo tendon]

no you can just use different log properties and get to the result

for euler form afaik you convert it right away

[restive pendant]

I dont see it

[pseudo tendon]

you know sin's complex form right

[restive pendant]

yup

So i use that then?

[pseudo tendon]

put that in and then use linearity of integral and log properties to get separate integrals

[rancid mural]

use sin2x = 2sinxcosx and put its value

[obtuse pebble]

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[hard citrus]

can someone help me with this question?

[hard citrus]

we're not meant to integrate the ln(x-1) as we haven't learnt that yet

[royal basin]

as in you havent learned integration at all?

[hard citrus]

no I have

[royal basin]

or just for logarithms

[hard citrus]

but we haven't specifically learnt integrating ln(x)

[royal basin]

you can write y = ln(x-1) as x = 1 + e^y

[hard citrus]

hol up

oh yeah I see it

ohhhhhhh

so you change it from x to y

so our range is 0-3?

[tepid mountain]

Did you hear of
Integration by parts ?

[hard citrus]

nope

[tepid mountain]

It kinda requires integration by parts to solve for the integration of ln(x)

But for now do you know the boundaries of the integration you will make ?

[hard citrus]

yeah

0 and 3

since we swapped it to the y axis

[tepid mountain]

No

[hard citrus]

what

[tepid mountain]

You will integrate with respect to x

[hard citrus]

i mean for this

we haven't learned to integrate ln(x) yet

[tepid mountain]

Yeah forget about it for now

We want to know the limits of the integration

[hard citrus]

right

[tepid mountain]

$$\int_{a} ^{b}$$

[warm shale]

Sherif Player

[hard citrus]

2 and e^3+1

[tepid mountain]

We want to know a and b

In the integration we will do

Do you know how ?

[hard citrus]

i know a and b

idk how to integrate ln(x) so I don't see how I am supposed to go from there

[tepid mountain]

What are they ?

[hard citrus]

this

[tepid mountain]

No it is
0 and e^3 + 1

[hard citrus]

oh right

[tepid mountain]

You want the area under the y=3 curve - ln(x) curve starting from 0 to the point of intersection to

So the equation now looks like this
$$\int_{0} ^{e^{3} + 1} (3) - \int _{0} ^{e^{3} + 1} (\ln(X - 1))$$

[warm shale]

Sherif Player

[tepid mountain]

Oh wait the boundary of the second one isn't 0

Because we want to start from x-axis

So it will be 2 as you said

So the equation now looks like this
$$\int_{0} ^{e^{3} + 1} 3 dx - \int _{2} ^{e^{3} + 1} \ln(X - 1) dx$$

[hard citrus]

🤔

[warm shale]

Sherif Player

[tepid mountain]

So we want to integrate ln(X-1)
So you will need to know about integration by parts to do it
But for now
You should know that
$$\int (\ln(x)) dx = \ln(x) \times x - x + c$$

[hard citrus]

right

[warm shale]

Sherif Player

[tepid mountain]

So the indefinite integral of ln(X - 1) is
$$\int \ln(x - 1) dx = ln(x-1)\times (x-1) - x + c$$

[warm shale]

Sherif Player

[tepid mountain]

That is the integral

Now you should be able to solve it. right ?
@hard citrus

[hard citrus]

i think so

[tepid mountain]

So it will be this
$$3x |{0}^{e^{3}+1}dx - ( (x-1)\ln(x-1) - x) |{2}^{e^{3}+1}dx $$

[hard citrus]

bruh

my brain cannot comprehend that

[warm shale]

Sherif Player

[tepid mountain]

So that is it
Now we just want evaluate it

[hard citrus]

i don't even know how to sub that in

im fried

[tepid mountain]

You don't know about this
$$x|_{0}^{3}$$

[warm shale]

Sherif Player

[hard citrus]

yeah how does it become 3x

[tepid mountain]

Hmm

Do you know about integration?

$$\int 3 dx = 3x + C$$

[warm shale]

Sherif Player

[hard citrus]

yea

[tepid mountain]

So let's get to this?

[hard citrus]

how did the 3x go to the outside though

isn't it meant to be inside the bracket

[tepid mountain]

Evaluating this will give this
$[3\times(e^{3}+1) - 3\times 0] - [(e^{3}+1 -1) \times \ln(e^{3} +1-1) - (e^{3} +1) - [(2-1) \times \ln(2-1) - 2]]$

[hard citrus]

oh

its just 3 *

alright

[tepid mountain]

Oh half of the picture is cut away lol

[warm shale]

Sherif Player

[tepid mountain]

That is better

$$(3e^{3} + 3) - [e^{3} \ln(e^3) - (e^3+1) - [ 1 \times \ln(1) - 2]]$$

[warm shale]

Sherif Player

[hard citrus]

right

[tepid mountain]

$$ 3e^3 + 3 - [ 3e^{3} - e^{3} - 1 + 2]$$

[warm shale]

Sherif Player

[tepid mountain]

$$3e^{3} + 3 - [2e^{3} + 1]$$

[warm shale]

Sherif Player

[hard citrus]

ah

[tepid mountain]

$$e^{3} + 2$$

[warm shale]

Sherif Player

[hard citrus]

so e^3 + 2

[tepid mountain]

Yes

[hard citrus]

holy shit

that was long

thanks a lot for being patient though

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[tepid mountain]

No problem

[hard citrus]

.reopen

[obtuse pebble]

✅

[hard citrus]

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[high mirage]

sorry forgot basic math
ax^2 + bx + c
for b^2 - 4ac >/</=0
how do u know whether it is > or < or =?
yes i know its based on number of solution but how u even know how many solution there is?

[solar trellis]

by knowing this

[harsh remnant]

We don't know

It's based off what your function is

[high mirage]

u^2 -8u + 12 (example)

= (u-6)(u-2)

does this mean 2 solution so discriminant > 0?

[solar trellis]

yes

[harsh remnant]

Yes

[high mirage]

ok what if the question give this u^2 = 8u + 2k

[solar trellis]

but normally you use the discriminant to find the no. of solutions

[high mirage]

how do u know how many solution

[solar trellis]

depends on k

[high mirage]

wait but isnt the number of solution for u to solve the discriminant

[harsh remnant]

Equal to 0? = One sol

[solar trellis]

no, it just tells you the sign

[harsh remnant]

Less than 0? = No sol

Greater than 0? = Two sol

[high mirage]

the question is phrased, explain why u^2-8u+2k has no solution if k>8

[solar trellis]

in general the discriminant is easier to calculate

[high mirage]

that is what i want to know

[solar trellis]

ok find the discriminant

[high mirage]

(-8)^2 - 4(1)(2k)

which is 64-8k

but idk whether its > or < or =

[solar trellis]

if k>8 then you know

[high mirage]

huh wtf but its only k>8 at the final, like for this one there is negative which means the sign must flip

so like how do u know what if its alot more flipping/complicated eqn?

its very hard to reverse do it

[solar trellis]

wdym

if k>8 then

-8k < -64

so 64 - 8k < 0

[high mirage]

but u are doing it reverse r

rn

what if the equation is too complicated to reverse do it

then how do u figure out

or wait

is it cuz it says

K>8 has "no solution" so discriminant < 0?

since disc<0 is no sol

[solar trellis]

you can go from no sol ==> discrim < 0 ==> k < 8 then reverse the steps

[high mirage]

ohhh

i see ok

tyty

thanks alot for ur help

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[pliant bone]

How do I get the right thought process / ideas / solving ways for math Olympiad level questions?

The math Olympiads first round will soon start in my country and I've tried practicing some old math olympiads tasks in the past few days and I am not able to find the right ways to solve them.

I watched many videos about math Olympiad tasks and I can mostly understand how it is solved and it makes sense but when I try some myself I never seem to get the right beginning of the solution and just sit there wondering how I can start solving them.

[spice citrus]

If you can't solve any problems, even after spending a lot of time on it, maybe you should try some easier problems first

[proven zephyr]

We also have a math server focusing on math olympiads btw

[sacred root]

Yes

[pliant bone]

oh my bad

[sacred root]

https://discord.gg/mods

[pliant bone]

thx

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[proven zephyr]

You can check the resources from the the training channel below the mathematical olympiad section

in that channel

[pliant bone]

thank you

[obtuse pebble]

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[little pelican]

Looking for help for question a b and c, don't understand what the formula means so an explanation would be greatly appreciated! Thank u

[lone stag]

Hi! For part a, try substituting the variable d with the values 1 and 2 to obtain the first two terms.

Ah, but here you have n and d as variables. Does the problem give you more information?

$u_n = 12+7n$ This formula would make a bit more sense, since it tells us how to obtain the $n-$th term, given the position $n$.

[warm shale]

Drenitor

[hollow dove]

it does look like a typo in the question

the common difference means the difference between consecutive terms is the same, and it wants you to find this difference

so if we had the sequence 1,3,5,7,9,..., the common difference would be 2

assuming the formula is supposed to be u_n = 12 + 7n, you can try write out a few terms and spot the common difference

or look at the formula, and see what happens every time you increase n by 1

[little pelican]

well n is just the placement of the term and then d is another variable

basically what people call "x"

[little pelican]

nope thats all it says

[little pelican]

so if i substitute d as 1 do i just do n as 1 too

cuz if yes then u_1 would be 19 and u_2 would be 26

common difference is 7?

and then for question c the 25th term is 187?

could anyone help me double check that

please just dm me and help me double check cuz im going to close the ticket first cuz im gonna go sleep

@hollow dove if u dont mind helping me thanks :)

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[lethal pulsar]

Can someone help me with this function

[royal basin]

image inbound?

[lethal pulsar]

So I have to prove that the given point is right

[royal basin]

,rccw

[lethal pulsar]

Do i take the y point and put at the place of f(x)?

[royal basin]

wording misstep, but yes.

you calculate f(-1) and see whether or not it equals 3

[lethal pulsar]

Thx was not sure

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[sharp hare]

Hallo

[sharp hare]

Tryna solve this one. I tried putting in a+ib on z's place, but it feels like I just complicated it further

[cloud igloo]

Translate it

[sharp hare]

Solve the equation P(z) = 0

[cloud igloo]

Equate it to zero I guess?

[harsh remnant]

Is that i?

[sharp hare]

yes

[harsh remnant]

Oh no

That's bad

[cloud igloo]

Equate it to 0

Move I to the other side

[sharp hare]

right

[cloud igloo]

Idk

[sharp hare]

so we have (2-i)*z = -i

[cloud igloo]

Divide both sides by (2-I)

[sharp hare]

i did that, but its not the correct result

somehow

nvm ur right

holy shit why did they write the answer in such a stupid way

[cloud igloo]

Ez

[hidden compass]

Show your work @sharp hare

[hidden compass]

Because complex numbers in the standard form don't have imaginary unit appearing in the denominator, hence you have to rationalize

[hidden compass]

,rcw

[sharp hare]

yeah i think because they want me to times it with the conjugated

[hidden compass]

Yes, you need to do that if you have some i in the denominator

[cloud igloo]

Simplify it

[hidden compass]

$$\frac{-i}{2 - i} = \frac{-i}{2 - i} \cdot \frac{2 \textcolor{green}{+} i}{2 \textcolor{green}{+} i} = \frac{-i \cdot \left(2 + i\right)}{\left|2 - i\right|^2} = \frac{1 - 2i}{5} = \frac{1}{5} - \frac{2}{5} i$$

[warm shale]

Alberto Z.

[hidden compass]

Yep, exact

[sharp hare]

Thanks guys

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[hexed apex]

just checking this is true right

[hexed apex]

I expressed arg(-1+it) in terms of t

[royal basin]

yes this is cts

[hexed apex]

what's cts

[royal basin]

continuous

i was lazy

[hexed apex]

ah okay

wow

that wasnt even five seconds lol

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[versed stratus]

What do you think

[gilded needle]

20x, of course

[versed stratus]

explain why you think so

[cloud igloo]

<@&268886789983436800>

[versed stratus]

multuplication is just repeated addition

[cloud igloo]

Why are you entertaining this?

[slim cove]

alright

.close

[obtuse pebble]

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[versed stratus]

usually when someone posts such a question, they don't rpely. So. I thought it was genunine this time

[obtuse pebble]

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