help-27

[near jolt]

maybe best to think about these pairs through examples like 24 or 63 or 64

[primal shadow]

this is slightly off topic but if i worded this as an answer would i say "the lockers left open have an odd number of divisors out of 1,2,3,4 and 5" or is the last part with the "out of.." wrong

[near jolt]

why out of 1,2,3,4,5? there are 100 students

[urban needle]

The easiest way to find them are to follow the perfect square numbers from 1 to 100

[primal shadow]

because it would be irrelevant if the number was divisible by 12 for example because it only matters if its divisible by 3 4 2 and 1?

[near jolt]

but we are tallying all possible divisors? 12 could be a divisor of the number 24, for example

[primal shadow]

i seee

[near jolt]

7 is toggled twice, but the number of divisors out of 1,2,3,4,5 is odd

so you see, there is a mistake with this only consider 1,2,3,4,5 characterization

[urban needle]

I think 🤔 BCOS only perfect square have odd numbers divisors

[near jolt]

yeah thats what we are trying to guide them to do

[primal shadow]

i see is ee

so is it only all square numbers ?

[near jolt]

its important to see why though

[primal shadow]

i do understand i think

[near jolt]

can you try proving it?

[primal shadow]

because usually numbers have an even number of divisors

and because square numbers have 2 of the same divisor

then it becomes odd

because you subtract 1

[near jolt]

why so?

[primal shadow]

because prime numbers have 1 and themself so thats 2 and composite numbers have 1 and themself and have to be even because the 2 divisors will be different if its not square

sorry if thats hard to understand

like 21 has a factor of 7 but it also needs another factor because it sint square

the other one being 3

[near jolt]

I don't think this argument is formal enought to even begin judging the correctness

[primal shadow]

sorry

[near jolt]

27 has a factor of 3 and no other prime factor

yet its not a square

[primal shadow]

but i didnt mention prime factors

i meant just regular

1 3 9 27

[near jolt]

ok but then how do you show that its even though?

normally you'd pair up the divisors, but I don't see you pairing them up

[primal shadow]

ohh

because 27/9=3 and 27/3=9 so it's a pair

?

[near jolt]

yes!

The key thing to note is that this pairs every divisor that is strictly less than $\sqrt n$ to a divisor that is strictly greater than $\sqrt n$

[woven radish]

qwertytrewq

[primal shadow]

wow

i understand it now

[near jolt]

then can you explain the argument?

[primal shadow]

because when you divide it by a divisor you get a pair of 2 factors like 16/2=8, and then the next divisor is 4 which is the square root after which would just give the pairs in reverse like 16/8=2, and since every number has at least one pair, like prime numbers have 1 and themself, only square numbers will have an odd number of divisors because they have 2 of the same

sorry if the explanation is like

rambly

idk how to really do them

or if i was explaining the right thing

so i tried to do both

[near jolt]

yeah that is roughly the argument, one need to justify that the pairing is unique: in the sense that if (a,b) is a pair and (b,c) is a pair, then a=c. So in this way every pair are disjoint. But all those are not too hard to write down the details of.

and that this pairing gives us a way to count all the divisors (except when it is a square, and the pairing contains a pair (a,a) so we are counting a twice)

[primal shadow]

i see

so would it only be square numbers that are left open?

[near jolt]

i think so!

[primal shadow]

thank you !!!!!

[near jolt]

np

[primal shadow]

also the part with the square numbers was very cool

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[kindred finch]

simple thing to solve but I’ve never learned about it

[wicked turtle]

what's the question? find the shaded area?

[kindred finch]

yea

[full cloak]

do you know the area of a fan?

[wicked turtle]

try expressing it as the area of the circular sector minus the area of the triangle

[kindred finch]

?

[full cloak]

like how to calculate the area of a fan?

[kindred finch]

are you being sarcastic or

[knotty sage]

section of a circle...

[full cloak]

^

[kindred finch]

yeah idk these terms sorry

ok so

[full cloak]

it is ok, do you know how to calculate the area of the fan

[kindred finch]

would it be safe to say that the sector of the triangle is 60

wait

sector of the circle is 120

sorry

nah nvm, I don’t even really know where to start

[potent tusk]

area of a sector bro

[wicked turtle]

well can you find the area of the whole circle?

the sector is 1/6th of the circle, right?

[kindred finch]

yes

Pi r squared?

[full cloak]

(pi)r^2 is the whole circle

[kindred finch]

ok

[full cloak]

what is 1/6th of that

[kindred finch]

24 pi

correct or no

[full cloak]

yeah that is right

[kindred finch]

if we are keeping pi as pi and not doing it as a nunber

ok

[full cloak]

now you have the area of the sector, what do you need to find out the shaded area?

[kindred finch]

find out the triangle

and minus it from that?

[full cloak]

good, do you know how to do that?

[kindred finch]

hold on sorry

1/2 base x height

[full cloak]

alright

what is the base?

[kindred finch]

12

all sides are 12

I think

[full cloak]

alright prove that to me

[kindred finch]

Equilateral triangle

[full cloak]

good

[kindred finch]

60 60 60

[full cloak]

you have base at 12

do you know the height?

[kindred finch]

no

[full cloak]

how do you solve for that then?

[kindred finch]

Just putting it down here so we don’t have to scroll

Split it in half

[full cloak]

alright

[kindred finch]

And then

[full cloak]

ok good

[kindred finch]

306090

[full cloak]

good

[kindred finch]

So 6 root 3

?

[full cloak]

yes nice

now you have base and height, you know the area of the triangle

finally, what is the area of the shaded region?

[kindred finch]

But wait before that, sorry but how would I do 6 root 3 x 6

[full cloak]

you dont know how to do that?

[kindred finch]

Weird numbers are fine right

?

[full cloak]

6root3 is basically 6Xroot3 X6

yeah weird numbers are right

does the question ask you to round or give exact answer?

[kindred finch]

Exact

62.353822907

Is what I got

Which I don’t think is right

[full cloak]

^that is not how you notate exact answers

if you do exact answers:

never multiply anything by pi (aka leave 24pi as it is

[kindred finch]

Ok

[full cloak]

never break a square root (aka leave 18root3 as 18root3)

[kindred finch]

So the answer is 6 root 3 x 6 - 24 pi =- shaded region?

[full cloak]

sorry shouldnt be 18root3

[full cloak]

yes

[kindred finch]

Ah ok thanks so much

[full cloak]

um

do not leave shaded region as negative

[kindred finch]

I know

It was typo

[full cloak]

do A=24pi-36root3

[kindred finch]

It kept auto correcting to 13 something

Thank you

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[lyric widget]

Find the equation of the normal to the parabola y²=-8x at (2, 2) in the standard form.

basically this question came in our last year annual exam and i am convinced this question is flawed, the point doesnt even exist on the parabola, further more the exam marking notes state a totally flawed approach of basically derivating the parabola and then getting the slope at a place where the parabola doesnt even exist, honestly i just want someone else to verify this for me, but i am like 99% sure the examiners messed up

[lilac moat]

is there a tangent line of the parabola passing through (2,2)?

[lyric widget]

nope, the parabola is on the left side of the origin

while the point 2,2 is on the right side

the red line is the slope point line from 2,2 and the slope from the parabola at 2,2 doesnt exist

[drifting mauve]

hmmm you could make a normal line that contains (2,2)...

though it would be a bit problematic because you dont know the intersection

i.e. normal point to find the slope

you could find it maybe...

[lyric widget]

exactly

[drifting mauve]

so did you do as much?

[lyric widget]

heck, the exam probably doesnt even have enough space for all that working

[drifting mauve]

😢 it is what it is

[lyric widget]

did u take a look at the "better response" image

[lyric widget]

.

[drifting mauve]

💀 y=2? huh?

[lyric widget]

well i have my my exam tomorrow, this was from previous year's past papers, i am practicing on them

i hope smth like this doesnt come 😭

[drifting mauve]

well erm

good luck on

analytic geometry then 💀

the y=2 thingy

is faulty

not faulty, it just has no basis

wrong solution

its wrong

[lilac moat]

this has two slopes for two possible tangent lines though

i switched x and y because why not

[drifting mauve]

to convince yourself its wrong find the tangent using that normal

[lyric widget]

yea and i dont think our syllabus covers finding normal from parabola which contains (x,y)

[drifting mauve]

😢 its just dy/dx and -dx/dy

parametrisation helps

tremendously

[lyric widget]

i tried asking gpt for help and it gave a super long and complicated answer

[drifting mauve]

yeah ts problem is too long relatively speaking

and the given solution js

wrong

[lilac moat]

yeah definitely we should do that

[drifting mauve]

❎

[lyric widget]

anywayss thanks alot guys

[drifting mauve]

but you do understsnd how to obtain the actual solution right?

[lyric widget]

honestly speaking, no 😭

[drifting mauve]

you use an arbitrary point which has the slope of normal equal to slppe with (2,2)

thats it

then you get the slope and then you use (2,2)

[lyric widget]

how do i work that out 😭

[drifting mauve]

erm

you should paramterise first

y²=4ax

as

(at²,2at) for all real t

but i think it might not be

pertinent for you

[lyric widget]

yea we havent even been taught this concept

i think imma move onto other practice questions

[drifting mauve]

erm okay then

just know that the

[lyric widget]

thanks alot btw

[drifting mauve]

solution js wrong

💔

[lyric widget]

🥀🥀

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[uneven thistle]

Hi i need help with this

[uneven thistle]

<@&286206848099549185>

[devout snow]

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[opaque nova]

Hi y'all, I'm super stuck on this problem and was hoping someone could give me a hand:
Let ( I ) be an ideal in a commutative unital ring ( R ). Define
[
\widetilde{I} := { r \in R \mid r^n \in I \text{ for some } n \in \mathbb{Z}{>0} }.
]
Prove that ( \widetilde{I} ) equals the intersection of all prime ideals of ( R ) which contain ( I ); that is,
[
\widetilde{I} = \bigcap{\substack{P \supseteq I \ P \text{ prime}}} P.
]

[woven radish]

theaveragejoe6029

[devout snow]

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[dusky apex]

Can anyone help me with question 1? (if someone could check my other answers that would be appreciated but i know that it might be alot so i understand if no)

[knotty sage]

for number 1 you add the elemnt in place a_11 with the element b11 and a12 b12 and so on

[dusky apex]

So the answer is c. I?

[knotty sage]

yep but feels weird not writing it as a matrix tbh

[dusky apex]

xd

Thank you

How do i close this again i forgot

[knotty sage]

.close ,

[devout snow]

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[knotty sage]

oh

lol

i forgor my role does that

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[craggy matrix]

How do I find x and y?

[devout snow]

Please don't occupy multiple help channels.

[knotty sage]

sine rule it or cosine rule it either one will work

[craggy matrix]

I am in 9th grad geometry, we have Not learned that yet 😭

[normal lagoon]

It should already have been covered.

Specially with this kind of problem.

[dense arrow]

https://www.mathsisfun.com/algebra/sohcahtoa.html this website is very helpful

also has interactives to see the wavey bits :3
at least as far as i remember ill find it

[normal lagoon]

SOHCAHTOA was in our Grade 8 classes.

[dense arrow]

what country are you in?

[craggy matrix]

what country do you do school

[craggy matrix]

ty

[topaz beacon]

how about we stop comparing grades and just do the problem

[dense arrow]

https://www.mathsisfun.com/algebra/trigonometry-index.html ii think this is the main one where u can access the wavey bit

[normal lagoon]

Philippines. Although it was at the end of my grade 8. (Before the curriculum change a few years ago.)

[craggy matrix]

if u dont mind, what age is that?

[normal lagoon]

Anyway SOHCAHTOA should solve for the missing side lengths

[topaz beacon]

do you know the law of sines?

[dense arrow]

https://www.mathsisfun.com/sine-cosine-tangent.html ahah i found the interactive @craggy matrix

same website different page

[normal lagoon]

Oh nice. Never realized this site had stuff like this.

[lost laurel]

alternatively ,drop a perpendicullar from the vertex including the angle of measure 100, and then find the new angles

[supple knot]

Can you close one of your channels

[dense arrow]

aus schooler, i only remember learning trig in like year 9, year 8 intro was like a single lesson teacher mentioned offhand

[normal lagoon]

^

[topaz beacon]

this is by far the easiest method

[lost laurel]

can we just focus on the problem please

[lost laurel]

This if you don't know what garlic suggested

[dense arrow]

did you work it out?

[craggy matrix]

im trying to understand 😭

[dense arrow]

this is what whatawonderfulworld was saying btw

[dense arrow]

sohcahtoa is just a formula of ratios that applies to every right angle triangle

[restive river]

i can recommend a nice yt video lol

[dense arrow]

https://www.youtube.com/watch?v=g8VCHoSk5_o

[craggy matrix]

like this?

[dense arrow]

i love organic chemistry tutor

[lost laurel]

yup

[dense arrow]

yes

🎉 then u just have to apply the formula

[devout snow]

@craggy matrix Has your question been resolved?

[craggy matrix]

Quick question, does this affect the original equation, or are they 2 completely different equations?

[lost laurel]

wdym

[craggy matrix]

Does tje [octire I just sent, affect the 1st picture?

[devout snow]

@craggy matrix Has your question been resolved?

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[gritty rock]

how did the answer pi/9 come to be?

[autumn girder]

!xy

[devout snow]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[autumn girder]

Ok

Do you know how we measure angles in radians?

[gritty rock]

yep

like my teacher teaches it so its like u simplify the fraction

and so you get 2pi/9

and thats 2 360 rotations which means the coordinate stays in place

but pi/9 isnt a value on the unit circle which ig is why im confused

[autumn girder]

$\pi = 180^{\circ}$

[woven radish]

@autumn girder

[gritty rock]

oh mb 2pi on the unit circle is 1: 360 degree rotation

[autumn girder]

Yep

[gritty rock]

but that means the coordinate still stays in palce

[autumn girder]

Yes

[gritty rock]

so does that mean any value that isn't on the unit circle is the reference angle? like im js confused as to why the answer is pi/9 because pi/9 isnt a value on the unit circle

[autumn girder]

It isn't a value that you've learned, it is (most definitely so) on the unit circle

[gritty rock]

pi/9?

[autumn girder]

yea

Do you know how to convert radians to degrees?

[gritty rock]

yeah but only if the value is a value on the unit circle, but like i am pretty sure that it isnt

at least to the degree that we are learning

[pseudo basin]

conversion between radians and degrees is just like any other unit conversion

[autumn girder]

It is there, it's not a value you've learned though

[pseudo basin]

pi radians = 180 degrees

it does not need to be any special number

any more than a conversion from inches into feet would need to be any special number

[autumn girder]

Yea we just use this relation to convert

$\pi rd = 180^{\circ} \implies \frac{\pi}{9}rd = \left(\frac{180}{9}\right)^{\circ}$

[woven radish]

@autumn girder

[primal ferry]

or in other terms, unitary method

[pseudo basin]

with all due respect if you know how to work with fractions then you know how to convert pi/9 radians into degrees

[gritty rock]

okay yeah i think i was js confused as to why we had pi/9

[primal ferry]

you just have to get out of that "syllabus" mindset if you truly want to learn, and most of the time it probably DOES tie in with your curriculum

[autumn girder]

Because that's what the author put in question

[gritty rock]

okay great thanks

[autumn girder]

Cool

!done

[devout snow]

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[gritty rock]

.close

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[dusky apex]

Can anyone check my answer on these questions? If its too much you can do as much as you can and i can just come back later thank you (i know the first one is correct because of someone who helped me here)

[lost laurel]

2 is right as is 3,4

[dusky apex]

Thank you

[pseudo basin]

wtf is option d doing in question 1

why did they put a fucking circle there

[dusky apex]

I swear i want to ask the same question

[pseudo basin]

anyway your answer is correct

[dusky apex]

I dont even know what the fuck that is

[pseudo basin]

for 1

[dusky apex]

Thanks

Just 5 to 10 left I'll come back later if you guys are too busy

[primal ferry]

just checked they all look correct

this is so funny to me

but anyways @dusky apex

good job

[dusky apex]

Thank you

[dusky apex]

Lol

[dusky apex]

Thanks

[primal ferry]

!done

[devout snow]

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[dusky apex]

Ill close now

[primal ferry]

<3

[dusky apex]

.close

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[spring oasis]

Let ( B = {v + w; (-1, 1, 1); (1, 2, -4)} ) and ( B' = {2v; w; (1, 0, 1)} ) be bases of (\mathbb{R}^3).
Find ( v, w ) and ( u ) in (\mathbb{R}^3) such that:
\begin{itemize}
\item The coordinates of the vector ((0, 0, -4)) in the basis ( B' ) are ((1, 2, -4)).
\item The coordinates of the vector ( u ) in both bases are ((1, 1, 1)).
\end{itemize}

[woven radish]

938c2cc0dcc05f2b68c4287040cfcf71

[spring oasis]

(0,0,-4) = 2v + 2w - 4(1,0,1)

(0,0,-4) = 2v + 2w + (-4,0,-4)

(0,0,-4) - (-4,0,-4) = 2v + 2w

(4,0,0) = 2v + 2w

(2,0,0) = v + w

[zenith fern]

Can someone help me with this question

|x - 2| ^ (log_2(x ^ 3) - 3 * log_x(4)) = (x - 2) ^ 3 , then the sum of values of x ≥3 are

[spring oasis]

u = v + w + (-1,1,1) + (1,2,-4)

u = (2,0,0) + (-1,1,1) + (1,2,-4)

u = (2,3,-3)

(2,3,-3) = 2v + w + (1,0,1)

(2,3,-3) - (1,0,1) = 2v + w

(1,3,-4) = 2v + w

(1,3,-4) - 2v = w

w = (2,0,0) - v

(1,3,-4) - 2v = (2,0,0) - v

(1,3,-4) - (2,0,0) = v

v = (-1,3,-4)

(2,0,0) = (-1,3,-4) + w

(2,0,0) - (-1,3,-4) = w

(3,-3,4) = w

.close

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[mighty kiln]

finite set are always closed

ohh i got the reasoning now

{0} is finite set and our supreme set is complex set

so compliment of closed set is open

thanks

.close

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[keen sundial]

"everywhere I have a positive, i will have a negative"

"everywhere i have a negative, i will have a positive"

what does my lecturer mean?

[mortal meteor]

Do you know about mod?

[buoyant narwhal]

Using the absolute value function takes the positive value for any given x. For instance, if we have $x\in (-\infty,0]$ then the absolute value sign will give us -x (so we get a positive instead of a negative). When we have $x\in [0, \infty)$ we will get a +x (to retain positive. So I guess what your prof is sayin is that if you take a positive absolute value you will always get a positive number, and if you use a negative absolute value you will always get a negative number. (Obviously 0 is neither so it is a common point).

[woven radish]

PSCell

[drifting mauve]

$y=\abs{x}\geq 0$ so wouldn't it make sense for $y=-\abs{x}\leq 0$?

[woven radish]

parabolicinsanity

[drifting mauve]

but the absolute value "operator"

is

dependant on what input you have

$x\in R^- \implies \abs{x}=-x$ and $x\in R^+ \implies \abs{x}=x$ and erm $\abs{0}=0$

[woven radish]

parabolicinsanity

[devout snow]

@keen sundial Has your question been resolved?

[keen sundial]

sorry let me have a read

[keen sundial]

modulus?

[keen sundial]

but -|x| is smaller than 0

beccause it will be a negative number unless x itself is a negative

[devout snow]

@keen sundial Has your question been resolved?

[devout snow]

@keen sundial Has your question been resolved?

[devout snow]

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[woven radish]

lyzyno

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[urban otter]

Dont have a clue

Tried adding the 2 integrals together

But then 0 clue

[lilac crescent]

You're trying to find the maximum value of f ?

[urban otter]

Yes

[thick ledge]

Yo I haven't solved properly

But isn't C and D logically

Wrong

Cuz 2 is not in the domain of f(x)

Or f'(x) assuming the function is differentiable

[urban otter]

Yeah that was a trick option

Apparently

[restive river]

What's the answer?

Is it a

?

[devout snow]

@urban otter Has your question been resolved?

[urban otter]

Yes

[devout snow]

@urban otter Has your question been resolved?

[tawny pewter]

yeah question is weird imo, from the way the function is defined f(2) is out of it’s domain, and for option D to make sense f must be differentiable, not only continuous

[devout snow]

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[unborn atlas]

How do I do 9

[olive snow]

I feel like its wierdly written

[unborn atlas]

Yeah

[unborn atlas]

what’s the first step?

[olive snow]

I'd identify where x and y belong first

[unborn atlas]

How so?

[outer eagle]

you could start by making a drawing to understand what the text is talking about, it's always a good idea to have a picture in mind

[cinder bobcat]

and have you learned green's theorem?

[unborn atlas]

This isn’t a greens theorem problem

[unborn atlas]

I did

[outer eagle]

do you understand over what you are integrating?

[unborn atlas]

Yeah

But im parametrizing it wrong

[outer eagle]

how are you parametrising it?

[outer eagle]

I thought this was exercise 9?

[unborn atlas]

Oh 11

My bad

[outer eagle]

sorry but I'm confused by the wording of the exercise, they say area but tell you the equations of 2 surfaces
they probably mean the intersection of x+y+z=1 with x²+y² ≤ 1

[unborn atlas]

Yes they do

[outer eagle]

as such (x,y) move in a circle, z is just the height z = 1-x-y of the plane, probably better not to use it as a parameter

[unborn atlas]

Okay

But then what

[outer eagle]

you parametrise the surface and use the formula for the area

[unborn atlas]

Okay but

Now what

[outer eagle]

have you written down everything?

[unborn atlas]

Yeah

[outer eagle]

what are you stuck with?

[unborn atlas]

Solving ds

dS

[outer eagle]

have you tried calculating it?

[unborn atlas]

Yes how do I

[outer eagle]

there's a procedure to compute it, do you know it?

[unborn atlas]

Yes which one am I using

None work

There’s 3

[outer eagle]

can you show your work?

[unborn atlas]

I did

There’s no work to show

I can’t cross it with anything

I can’t take the gradient

And I can’t do explicitly

[outer eagle]

if you take the surface as the graph of the function z = f(x,y) = 1-x-y over the circle x²+y² ≤ 1 you can use the expression √(1 + |∇f|²) dxdy

which is the 2nd you've written

[devout snow]

@unborn atlas Has your question been resolved?

[unborn atlas]

How do you integrate that

[outer eagle]

Did you calculate it?

[restive river]

@nathan

This seems like an exact equation

@unborn atlas

And you probably have to integrate the function along the square

[outer eagle]

Its not exercise 9

It's 11

[restive river]

Oh it's not?

[outer eagle]

They miswrote the number

[restive river]

Ohh sorry

Surface integrals huh?

Reminds me of greens theorem

I will look into greens theorem tomorrow

[devout snow]

@unborn atlas Has your question been resolved?

[devout snow]

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[rustic jetty]

Given that
\begin{align*}
\cos x + \cos y + \cos z &= 0, \
\sin x + \sin y + \sin z &= 0,
\end{align*}find
\begin{align*}
&\tan^2 x + \tan^2 y + \tan^2 z - (\tan^2 x \tan^2 y + \tan^2 x \tan^2 z + \tan^2 y \tan^2 z) \
&\quad - 3 \tan^2 x \tan^2 y \tan^2 z.
\end{align*}(Assume $\cos x,$ $\cos y,$ and $\cos z$ are all non-zero, so $\tan x,$ $\tan y,$ and $\tan z$ are all defined.)

[woven radish]

Dork9399

[rustic jetty]

from the two equations i got

$e^{ix} + e^{-ix} + e^{iy} + e^{-iy} + e^{iz} + e^{-iz} = 0 \ e^{ix} - e^{-ix} + e^{iy} - e^{-iy} + e^{iz} - e^{-iz} = 0$

[woven radish]

Dork9399

[rustic jetty]

im not exactly sure what to do from here

or how to approach the expression

[olive snow]

The find seems to have some simplification

[rustic jetty]

yeah i could say

$e^{ix} + e^{iy} + e^{iz} = 0 \ e^{-ix} + e^{-iy} + e^{-iz} = 0$

[woven radish]

Dork9399

[lyric hornet]

(e^{ix}+e^{iy}+e^{iz}=(\cos(x)+\cos(y)+\cos(z))+i(\sin(x)+\sin(y)+\sin(z))=0+0i=0) (by Euler's formula)

without loss of generality assume (x=\theta,y=\theta+\frac{2\pi}{3},z=\theta+\frac{4\pi}{3})

[woven radish]

PajamaMamaLlama

[lyric hornet]

just an idea ¯_(ツ)_/¯

[rustic jetty]

.close

[devout snow]

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[devout snow]

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[native tartan]

say i have a function e.g f(x)=5cos2x-1 for the range of a one to one function from 0<x<pi/2. how would i deal with the cos2x when trying to get the inverse of the function. so far i am at (y+1)/5=cos(2x)

[native tartan]

and then i would also want to find the new domain and image set/range for this new inverse function

[olive snow]

well

If its the inverse function

[native tartan]

is it just 1/2arccos(_) bc it feels wrong to do that

[olive snow]

f and f^-1, you will have Im(f) = Domain(f^-1) and domain(f) = Im(f^-1)

[lyric hornet]

if you have no domain issues it's completely correct to do that

[olive snow]

The thing is that you have 2x between 0 and pi so there is no real problem to this since arccos(cos(x)) = x for x in 0,pi

[native tartan]

so when solving a normal equation with cos 2x and i try to divide by it it just always go to arccos

and would it go to 1/2arccos

[olive snow]

Here yes

[native tartan]

and is arcos the same as cos^-1

[lyric hornet]

wdym exactly? 🤔

[lilac moat]

in general for inverting a trig function, if the domain doesnt match the usual one

[lyric hornet]

to some cos(2x)=something we take arccos then 1/2 it to solve for x

[lilac moat]

you need to define a customized inverse, and say, like

[lyric hornet]

chartbit!

[lilac moat]

"here cos‐¹(y) is defined as y=cos x for x in blah, y in blah"

if regular arccos doesnt work

[native tartan]

ok so i am fine doing what i have done

and then i just gotta try get my head around the domain and image set when its inversed

can i work through an example of a function i make up as helps if i work through it and make sure im doing it right

so f(x)=2cos(2x) domain: 0<x<pi/2 range: -2<y<2

inverse of that would be:

y/2=cos(2x)

1/2arccos(y/2)=x

f(x) inverse = 1/2arccos(x/2)

inverse domain: -2<x<2 inverse range: 0<y<pi/2

is that correct?

[lilac moat]

yes

[native tartan]

!close

.close

[devout snow]

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[native tartan]

thanks guyss

[devout snow]

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[wispy geyser]

am i right with A here

[wispy geyser]

the one with four petals

<@&286206848099549185>

[wicked arch]

I think so

Looks like 4cos$\theta$

[woven radish]

Alex

[wicked arch]

But I might be wrong

Polar is my worst section in calculus

[wispy geyser]

same

[wicked arch]

Oh wait

[wispy geyser]

okay ill go with this

[wicked arch]

I'm wrong

[wispy geyser]

oh shoot okay

[rotund umbra]

only A and B have (r,theta) = (4,0), and only A has (r,theta) = (-4,pi/2), so it's A

[wicked arch]

oh

so i was right then

[wispy geyser]

i got it i understand

just submitted

was right

[wispy geyser]

dont doubt urself bro

😭

[wicked arch]

i suck at polar though

[wispy geyser]

nah i feel you

im just learning this

and its already complicated

[wicked arch]

the AP Calc BC test was today and it was on an FRQ and I probably didn't get many points on it

[wispy geyser]

man ik thats stressful

prayin u pass man

[wicked arch]

I feel pretty good about everything else though

[wicked arch]

Haha I think I'm gonna pass I just want a 4 or 5 so I get credit for Calculus 2

[wispy geyser]

you got this bro

your in 11th too?

[wicked arch]

Yea I'm a junior

[wispy geyser]

feel like i shoulda done ap

missed out

im in honors pre calc

[wicked arch]

can you not do calc next year?

[devout snow]

@wispy geyser Has your question been resolved?

[devout snow]

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[serene fossil]

it looks easy for me but I just don't get the logic behind it, can someone help?

[faint gorge]

you can try to think of it as when do you have a max of |f(theta)| on 0 < theta < pi in terms of absolute value because radius is not negative

[serene fossil]

huh alr

lemme process that message 😭

[faint gorge]

notice that since f(theta) < 0 you actually have (f(theta),theta) = (-f(theta),theta+pi)

[serene fossil]

oh i didnt see that

wait, if that function is a circle, doesnt that mean that all points are spread out equally from the center?

so that would rule out a and b

[faint gorge]

yes, however the center here is not the origin, and when it comes to distance we are interested for distance from a point to origin

and the argument here is, that the maximum deflection of f(theta) in absolute value occurs at x = pi/2

[serene fossil]

ah i get it

i graphed out the 2 functions, is this supposed to be a correct representation?

[faint gorge]

the blue one, yes

[serene fossil]

ok

[faint gorge]

the circle is flipped because remember when i said for all points, when r < 0 then (r,theta) = (-r,theta+pi) so to make r positive

[serene fossil]

i see

so the pi/2 becomes a 3pi/2

[faint gorge]

yes

[serene fossil]

and that therefore is the farthest

[faint gorge]

yes

[serene fossil]

thanks!

how do i close this, tysm

.close

[devout snow]

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[devout snow]

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[lost laurel]

Let $a$ and $b$ be integers , not both zero. denote $d$ as the generator/the smallest element of of the subgroup $\Z d = \Z a +\Z b$. Prove:
\begin{enumerate}
\item d divides a and b
\item if an integer $e$ divides $a$ and $b$ it also divides $d$.
\item There are integers $r$ and $s$ such that $d=ra+sb$
\end{enumerate}
Define $S= \Z a+\Z b ={n \in \Z \mid n= ra+sb}$
\~\
\begin{enumerate}
\item As $a \in \Z a$ and $0 \in \Z b$, it follows $\exists k \in \Z$ such that $k_1d = a \implies d \mid a$. Similarly as $b \in \Z b$ and $0 \in \Z a, \exists k_2 \in \Z$ , such that $k_2d =b$. so $ d \mid b$. Therefore $d$ divides $a$ and $b$.
\item As $a \mid e$ and $b \mid e$, it follows that it belongs to $\Z a$ or $\Z b$. Thus It belongs to $\Z d$ too( By definition). Therefore $d \mid e$
\item $\textbf { This is essentially asking me to prove bezout's lemma, right ?}$
\end{enumerate}

[woven radish]

What a wonderful world !

[faint gorge]

1. messed up the definition of k
2. looks good
3. yes

[lost laurel]

Hmm, should be k_1,k_2, right

[faint gorge]

yes

[lost laurel]

Cool

I'll now try to prove bezout's

[faint gorge]

tbh i dont think you have to prove it actually, but apply it

[lost laurel]

I can just say the previous two together define $d$ to be the $GCD$ of $a,b$ and thus by bezouts this follows?

[woven radish]

What a wonderful world !

[faint gorge]

yes, also 2. could be more thorough

and you wanted to say e | d

[lost laurel]

yea

[lost laurel]

Can I close this now, then?

[faint gorge]

sure

[lost laurel]

thanks

.close

[devout snow]

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[devout snow]

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[cinder plover]

im jus curious

[cinder plover]

what do mathgematicians do

math majors

is it fun

[misty crest]

https://tenor.com/view/squidward-spare-change-spare-some-change-begging-poor-gif-18999842

[cinder plover]

no offense to me it sounds miserable

because like

idk

[misty crest]

it is

i cant lie

[cinder plover]

what you do

[misty crest]

i’ve considered switching to engineering multiple times at this point

😆

[misty crest]

cry usually

[cinder plover]

who hires math majors anyways?

just wondering

[misty crest]

a lot of them work in defense, finance, software jobs, actuaries etc

[lost laurel]

If you're very good, unis

[misty crest]

worst of all possible worlds is teaching

[lost laurel]

Best for some

[misty crest]

academia low-key sucks

[lost laurel]

Teaching + research

[misty crest]

i realized that this semester

don’t want that

looks horrendous

poor compensation for some of the hardest work there is

[lost laurel]

It is fun, yes. But very challenging too

[cinder plover]

i see

[lost laurel]

Most of us will maybe prove one major result in our life, and that;s a huge maybe

[misty crest]

no

[lost laurel]

after years of work

[cinder bobcat]

if you're a math major and don't have a minor or anything, finding a job is gonna be really hard... speaking from experience

gotta have some sorta secondary focus

[misty crest]

are you american

[cinder bobcat]

unless you're teaching i guess

[lost laurel]

I'm going to minor in sociology, let's see where that takes me ( Another mostly academic field)

[cinder plover]

that sounds tuff

[cinder bobcat]

that sounds pretty random lol, know what you're gonna do?

[misty crest]

it seems like all of the "math jobs" like defense, finance etc look for phds

[lost laurel]

I've done one course ( Anthroplogical thoery), really enjoyed it and got a B ( 8 grade points), so going to do it just for fun

[misty crest]

and a phd is just such a resource sink

takes like fucking 6 years

horrible pay

[cinder plover]

i see

that sounds rough and

[misty crest]

are you in engineering?

[cinder plover]

isnt it mjust

proof reading stuff

[cinder plover]

yea

mech

whats the highest math

id have to take

?

[misty crest]

yea like reading through rudin really made me question what i’m even doing

[misty crest]

idk like pdes?

no proof garbage

[cinder plover]

oka

[lost laurel]

Probably differential equations, maybe basic Tensor alegbra ( I think it's used in mechanical )

[cinder bobcat]

if you're doing engineering bachelor's, the highest you'd do is probably differential equations or multivariable calc

[misty crest]

maybe complex variables

if your uni offers it

[lost laurel]

yea , a little "Complex analysis " maybe

[cinder bobcat]

most engineering schools offer comprehensive courses that cover the random math topics like complex analysis

[lost laurel]

@storm raptor probably knows better( They're doing engineering + math)

[misty crest]

fourier series too

[primal ferry]

what about acturial though

[misty crest]

this doesn’t

i’m taking my actuary exams this summer

[primal ferry]

good luck!!

[cinder plover]

i dont even know what a ctuary exma is but

it sounds like

high intellectual stuff

[misty crest]

no

it’s like math nerds who work at insurance companies

[primal ferry]

pretty much 😭

[cinder bobcat]

I'm in actuarial science, it's a good field. you need to take a lot of extra exams (outside of school) to become one

[misty crest]

which exams did you take

[cinder plover]

oh

[cinder bobcat]

I've passed P and FM, taking SRM in 2 weeks

[cinder plover]

huh

that sounds

interesting

[cinder bobcat]

nice thing about that field is if you pass more exams, you automatically get a raise

so if you like taking tests and you like math, that's the way to go lol

[misty crest]

how is the pay

are you fresh out of undergrad

[cinder bobcat]

uhhh base pay is almost always at least 70k, depends on how many exams you have

[primal ferry]

photomath dying out before i got good enough at math is something i will always be pissed about

my brother did photomath duing covid and it was actually really convenient

[lost laurel]

That's pay I'd die for tbh

70k is enough to live a lavish lifestyle here

[misty crest]

it’s like poverty line in some american places lmao

[cinder bobcat]

yeah really depends lol

[primal ferry]

it's upper class in other parts of the world

[cinder bobcat]

but highly recommend if you don't hate stats and probability

[misty crest]

is it even worth taking graduate pure math classes

like i can load up in the next few years

get into graduate courses

but what’s the point

[cinder bobcat]

for fun :p but no technical reason to if you're not going into a graduate program

[devout snow]

@cinder plover Has your question been resolved?

[devout snow]

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[serene rampart]

are the options wrong or is my approach wrong?

[devout snow]

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[serene rampart]

oops had to delete because the image wasn't that good

there are only 4 ways such that tulip and rose always together

and total number of wways = 6!

6!-4*4!*2 seems to make sense to me

4! ways of arranging other flowers, and 2 for arranging the rose and tulip

[serene rampart]

Sorry, i meant to write R instead of S on the other cases

[thin fern]

I think this sounds right

lemme think

[serene rampart]

alright

[thin fern]

Yeah I think you're right

[lost laurel]

@serene rampart please open a new channel

this can be locked at anytime

[lost laurel]

see this

[serene rampart]

Okay

.close

[devout snow]

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[hidden sapphire]

How do you find period on sin graph

[hidden sapphire]

Is it max to max?

[supple knot]

Yes

[hidden sapphire]

Ok thanks

.close

[devout snow]

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[heavy current]

could I have this proof reviewed?
#1368757937954099342 message

[finite obsidian]

This works I think. I had a harder time reading the bit about M_1 not being embedded though

I think it works, just the bit about that not being embedded is stinky

[heavy current]

hmm, do you have any suggestions?

[finite obsidian]

Well you say U isn’t homeo to an open subset of R^2, but it’s a 1d manifold so that wasn’t the goal?

And you can remove a point and make the line {y=0} disconnected

So I don’t think the argument about not being embedded quite works

[heavy current]

ah wait, you're right

ack

[finite obsidian]

Plot M_1 fr

[heavy current]

Art suggested something with connectivity and/or connected components

so I'm trying to use that

[heavy current]

it's a self-intersecting curve

[finite obsidian]

Yeye, there’s points that removing would disconnect but also points that wouldn’t

Which is not very good

[heavy current]

hmm

can I argue via number of connected components?

like deleting (1, 0) gives me 4 connected components if I choose U small enough

and it was homeo to an open subset of R, it should only disconnect into 2 components

[finite obsidian]

Ye it turns from 1 into 4 pieces for all small enough neighborhoods

And that doesn’t happen in R

[heavy current]

so this is a good argument then

I just need to write it a little more formally

[finite obsidian]

But that means M_1 can’t be an immersed manifold I think?

Since it’s not a manifold at all?

[heavy current]

yeah

wait, no

[finite obsidian]

Certainly the image of an immersion you gave, I think, but it’s not a manifold

[heavy current]

immersed submanifolds need not be topological manifolds

I was just complaining about this in #math-discussion an hour ago

[finite obsidian]

Well it’s the image of the immersion of the thing you gave but what’s exact conditions to be an immersed manifold

[heavy current]

because the topology on an immersed submanifold needn't be the subspace topology

[finite obsidian]

The immersion kinda thing isn’t an injection at all right?

Might be on each component of the domain, but the two parts both hit the intersection?

[heavy current]

subset of a manifold endowed with a topology and smooth structure st the inclusion is a smooth immersion, but the topology doesn't have to be the subspace topology

the important thing though is that Lee has a propsition where he says that the images of injective smooth immersions are immersed submanifolds

in my case, M_1 should be the image of G which is smooth and injective and has nowhere vanishing Jacobian

so immersed via the prop

[heavy current]

this thing

[finite obsidian]

Is G injective tho

[heavy current]

why wouldn't it be

[finite obsidian]

Doesn’t it hit the self intersecting point twice?

[heavy current]

no

I deliberately excluded -1 from the domain so that it doesn't

it hits that point at t = -1 and t = 1

if no more -1, then nothing to worry about

and its image is still M_1

[finite obsidian]

Certainly not a connected domain, which idk if you require that

[heavy current]

idt Lee's prop minds that

[finite obsidian]

But if excluding that makes you injective then yeah sure

[heavy current]

it's an open subset of R, so its an open submanifold of R

so we're good

[finite obsidian]

In this topology, it thinks it’s disconnected though I think

[heavy current]

wdym?

what is "it"?

[finite obsidian]

The M_1

Which is terrible

[heavy current]

I mean, does it matter if M_1 is disconnected, if it's not gonna be a submanifold anyways?

[finite obsidian]

True

[heavy current]

[finite obsidian]

But yeah that should be fine

Immersed submanifolds a stinky definition

[heavy current]

alright, I'll go write that argument about connected components a bit better

[finite obsidian]

Yeyeye

[heavy current]

but why are they not necessarily manifolds grr

anyhow, tanq Sharp

.solved

[devout snow]

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[dry leaf]

I'm not sure where did I make a mistake

[devout snow]

@dry leaf Has your question been resolved?

[dry leaf]

<@&286206848099549185>

Hello ┬─┬ノ( º _ ºノ)

<@&286206848099549185>

Please somebody help I'm getting stuck every time I solve this way

[devout snow]

@dry leaf Has your question been resolved?

[devout snow]

@dry leaf Has your question been resolved?

[dry leaf]

<@&286206848099549185> please help

[rare kernel]

Did u get the acceleration of m1 m2 and m3

[dry leaf]

Yes, but my answer is wrong

[rare kernel]

Huh

Let's call acceleration of m3 a

What's acceleration of m1 and m2 in terms of a

[dry leaf]

I need to know what's the mistake in my solution because I already have answers like these

[rare kernel]

Ok I can't really understand ur solution

[dry leaf]

I've isolated the m2 and m3 pully , so they weigh 5g

And we've got tension 2T

And a1 should be equal to that which is produced by the net force

on that isolated system

[rare kernel]

Can u show me a diagram marking all the tensions and acceleration?

[dry leaf]

https://cdn.discordapp.com/attachments/903486480075354182/1371734684630913076/IMG20250513115315.jpg?ex=682436fa&is=6822e57a&hm=9591dd61afb596135e408af741d59af672cd1273e34cda020785357301647d69&

This

[rare kernel]

Dude

On the pulley itself please

[glossy dew]

a1m1 = 2T - m1g

assuming you meant 2T as the tension force

[dry leaf]

I've that in the second image

[glossy dew]

you did the same mistake in all the equations if it was that way

[dry leaf]

m1=1 so that is what I've written

[glossy dew]

hm

oh yea

[rare kernel]

I don't see a clear relatiom between accelerations

[glossy dew]

5g - 2T = 5a1

[rare kernel]

If u could answer my questions it would make this a lot easier

[dry leaf]

The thing is that I already have the answers that way, with using pseudo force, relations, etc

[rare kernel]

Pseudo force???

U only use that when the reference frame is accelerated

Why would u use that here

[dry leaf]

Yeah, the second pulley will experience pseudo force when we calculate it in its own frame

[rare kernel]

💀 alr then I guess wait for someone else to help...I can't work with this method

[dry leaf]

I'm not using pseudo force if you understood that

[glossy dew]

anyways

like I said

i still have a slight feeling that you messed up the second law somewhere

[glossy dew]

like this one

[dry leaf]

Yes that's a mistake I know actually but won't make a difference to 0

0/5=0

[glossy dew]

and why is 2T = 5g

[dry leaf]

I don't know, nor I've written that exactly

It's when you know a1 =0

I just need to know where's the mistake

[rare kernel]

a1=0??

[glossy dew]

then the mistake is when you assumed 2T = 5g 😂

[dry leaf]

https://cdn.discordapp.com/attachments/903486480075354182/1371734684630913076/IMG20250513115315.jpg?ex=682436fa&is=6822e57a&hm=9591dd61afb596135e408af741d59af672cd1273e34cda020785357301647d69& here

a1 turns out to be zero

[rare kernel]

Is that the correct answer?

[dry leaf]

Where do you see that written this way exactly

[dry leaf]

Nope

[glossy dew]

because you just replaced 2T with 5g..

[rare kernel]

Dude this is not how u solve pulley questions 😭

[glossy dew]

5g - 2T = a1
5g - 5g = a1
a1 = 0

[dry leaf]

Because you see I have calculated the value of T and 2T is 5g

[glossy dew]

oh

hm

once again

you used the equation

3g - T = a3

[dry leaf]

But what's wrong is what I want to know because if you don't write any false statements in science then you can't have a false result

[glossy dew]

pretty sure it should be 3g - T = 3a3

[dry leaf]

Can you pinpoint where I've written this

See carefully there's a division by 3

[glossy dew]

this one?

[dry leaf]

Yeah my bad but still won't get the answer

[glossy dew]

im not really sure if this 'isolated system' idea would work

ill check

1 min

[dry leaf]

Again why

@glossy dew are you still working on this

[glossy dew]

yeah hold on

is this hc verma by chance

[dry leaf]

Yes this is from hc verma book

I've solved this many times in the past after looking at the solutions 😂 but my natural approach always gets me stuck

[glossy dew]

another probable mistake btw

both blocks need not experience a force of T

[dry leaf]

And I want to correct the mistake in my natural approach

[dry leaf]

How

It's the same string

[glossy dew]

nvmnvm

im getting too confused

ill come back after a while

[dry leaf]

Same, btw I must ask how the lower pulley moves from rest even after zero force

[glossy dew]

well

im getting the answer using constraint relations

now to see for isolated systemss

[glossy dew]

acceleration can be anything

but net force should be zero

[restive river]

HC Verma?

[dry leaf]

And how's the acceleration produced

[dry leaf]

Yes

[restive river]

Nice nice

What's the issue?

Tried constraint motion?

[glossy dew]

spooky wizard magic

we can't really say until it's constrained

the physics we are using now is mere speculation to understand

whats going on

[restive river]

Uhh someone let me in too?

[glossy dew]

no one's blocking

the op says they can't get the answer using their natural thought flow

[restive river]

It's a lot to read🥲

[glossy dew]

and is asking why they can't take the two lower blocks as an isolated system and add another equation

[restive river]

Okk I'll need to read the whole thing

[glossy dew]

im getting an equation with 5g - 2T on the LHS after using pseudo force and simplifying in terms of relative accelerations

[restive river]

Wait wait wait

One thing

How is he equating t on LHS and a on rhs??

Force and acceleration mustnt be equated like that

Or am I missing something?

[glossy dew]

you aren't

op has a very vivid interpretation of physics

[restive river]

Uhh ok?

But how are the equations working?

[glossy dew]

they arent

[restive river]

So my call is the whole thing is wrong?

[glossy dew]

and now the op just wants to know if the isolated system method will actually work

[restive river]

Isolated system?

If u talk about the bottom pulley it's never isolated

Coz it has a force being acted upon by thee upper pulley

[glossy dew]

yeah

but it sounds like an interesting idea

[restive river]

If u balance the force and then think about it, you can call it isolated

But I must say it's all meaningless discussion until the equations are correct

[dry leaf]

Yes you are missing a mistake

[restive river]

What am I missing?