help-39

[kind rampart]

in other words, you can simplify a fraction if both the numerator and denominator have a common factor

do you have any further q's?

[pastel valley]

Yeah makes sense, but not rn, I am prepping to do the Aleks so be ready to see me again lol

[kind rampart]

if youre done, you may close the channel with .close

[pastel valley]

Mb

.close

[pearl pond]

Channel closed

Closed by @pastel valley

Use .reopen if this was a mistake.

[pearl pond]

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[junior anchor]

Can someone explain what intuitionism is and how it's different from formalism

[brisk steeple]

https://en.wikipedia.org/wiki/Intuitionism

[junior anchor]

So is it just constructivism

[brisk steeple]

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

[autumn fossil]

for comparison, you could also try
https://www.britannica.com/science/philosophy-of-mathematics/Logicism-intuitionism-and-formalism

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Contemporary_schools_of_thought

[junior anchor]

I mean I am reading ts

But like

I'm kinda confused

Which is why I asked here

[brisk steeple]

What are you confused about?

[junior anchor]

Like is it just about not accepting the law of excluded middle

Like if you try to outline the details of differences between formalism and intuitionism

Because rules of inference exist in both schools of thought right?

<@&286206848099549185>

[rotund nymph]

intuistionistic logic rejects both (axiomatic) lem and double negation elimination

[junior anchor]

Isn't that the same thing

[rotund nymph]

not that i know of

[junior anchor]

Huh

[rotund nymph]

i think theres a different definition of negation

[junior anchor]

Double negation elimination aka "if it ain't not false, it must be false" just is a restatement of lem

[junior anchor]

Oh?

[ivory basin]

It is the same

You can formulate classical logic using either, they are equivalent

[junior anchor]

So like my question remains

[ivory basin]

unfortunately I'm not a philosopher so I can't help you much here

[brisk steeple]

Just ask ChatGPT

[junior anchor]

!noai

[pearl pond]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[errant solstice]

actually, this may be a good question for #foundations?

[junior anchor]

[brisk steeple]

Not for mathematical tasks, but ok for the philosophy of mathematics

[junior anchor]

!no

[junior anchor]

Please never suggest use of LLMs to anyone, and definitely not in help channels

[split void]

If that were so, OP wouldn't have taken the pain to open a help channel

[wicked edge]

Have you asked your professor, or your teacher, OP?

Intuitionism is like a constructive creation of the human mind instead of relying on existed truth.

[junior anchor]

I'm just studying on my own

[wicked edge]

Formalism is considered to be the field using symbols strictly, rather than relying on existed truth.

[wicked edge]

How nice.

[verbal rover]

is there anyone who can help me out with statistics, i need to study for a test i have tomorrow which will make or break whether i can come back to the school im at next year so I'm really stressed right now

[errant solstice]

!occupied, please open an available channel like #help-21｜아리스킨충1.

[pearl pond]

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

[verbal rover]

oh sorry

im new to this server

[junior anchor]

Yeah I saw this, but this seems empty to me, like what does that mean, like why can't formalism be regarded as a constructive creation of human mind?

[verbal rover]

got confused

[autumn fossil]

I too know very little about this, but I don't think that you can reduce those views just to the axioms / rules of inference they accept / don't accept. They are much more broad than that. They also view mathematics as a whole differently, and "mathematical truth" has different meanings in them.

To me, intuionism is the view that mathematics is a mental construct of individuals rather than some objective stuff. Truth then corresponds to being able to make a constructive proof, and falsity means being able to refute it. From this view, it pretty much follows that LEM, double negation and contradiction arent valid proof techniques. (You cant just say A or not A without proving A or refuting it)

Formalism on the other hand is the view that mathematics is just about manipulating some strings according to some rules. Those strings don't necessarily have inherent meaning, but one can choose to assign meaning to them.

[wicked edge]

Well I am not specialised in this field much, that's what I have learned so far.

The barriers between them are vaguely pointed out.

Could the foundation channel explain this?

[junior anchor]

Hmm

Also I had another question

[wicked edge]

Ask away!

[junior anchor]

what quantifies the "power" of a certain foundation of mathematics

[wicked edge]

What do you mean "power" here?

[autumn fossil]

Like the "expressive" power of some axiomatic system? E.g. in the sense of being able to embeed arithmetic in it?

[junior anchor]

I can see two things

One is how much of what is true with the axioms outlines is provable in the system

And second is how much truth is expressible in the system

[junior anchor]

Yea one of the things

Like I think not all ZFC proofs are expressible as programs right

[autumn fossil]

https://en.wikipedia.org/wiki/Ordinal_analysis

this could be relevant (but i know practically nothing about it, i only heard about it when i did logic)

[junior anchor]

Dude I try to learn one thing (1) and now I have 89 wiki tabs open and have learnt nothing

I started out trying to learn STLC

Anyway thanks for yalls help!

I'll have to sit and ponder more

Thanks! @autumn fossil @wicked edge

.close

[pearl pond]

Channel closed

Closed by @junior anchor

Use .reopen if this was a mistake.

[wicked edge]

No worries!

I didn't help much, but good luck.

[wicked edge]

How are you going to know which tabs to find then?

Wrong reply ;-;.

[junior anchor]

I will have to keep notes

I am trying to create a graph of how one concept connects to another

I usually don't take notes

[wicked edge]

Interesting.

[junior anchor]

So I often end up doing some mumbo jumbo

[junior anchor]

[wicked edge]

https://tenor.com/view/sleepy-cat-cat-sleepy-cat-yawn-cat-yawning-cat-cuddling-gif-15776729946187212353

[junior anchor]

[autumn fossil]

Well, if you want a simpler but not so complete answer, then you could just claim that a system A is weaker than system B, if you can embeed A in B (for the expressiveness)

For the proof power, (which i think can be reasonably compared only in systems with similaar expressive power) system A is weaker than system B, if everything provable in A is provable in B

those "weakers" are not strict btw, and it only orders them partially, not totally

[junior anchor]

Yea these partial orderings

There will be always incomparable systems

[autumn fossil]

which, i guess, is to be expected

[junior anchor]

Yeah,

[wicked edge]

[junior anchor]

[wicked edge]

[junior anchor]

[pearl pond]

Available help channel!

Send your question here to claim the channel.

Remember:
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[placid steppe]

Hi guys!!! I need help with the last part of this question (ii, b). I feel like im missing a bit of knowledge on what to do.

[random ermine]

do you get how matrices relate to linear transformations

[placid steppe]

You multiply coordinates by the matrices to get the transformed points

[random ermine]

yes

what is the matrix for rotating twice

so rotate by theta once

then rotate by theta again

so do B twice

[placid steppe]

Ye k

I see

[random ermine]

the matrix is B^2

[placid steppe]

Let me see

[random ermine]

in general, you multiply matrices X and Y to get XY, which represents do Y first then X

this is a new linear transformation whose matrix is the product XY

[random ermine]

so composition of functions corresponds to matrix multiplication

[placid steppe]

Kk

Why is n 3

[random ermine]

wait i'm getting there

[placid steppe]

Kk thx

[random ermine]

...

so if you rotate something twice by theta degrees what is that equivalent to?

[placid steppe]

The original position?

[random ermine]

no

if you rotate something by 90 degrees

[placid steppe]

Oop sorry

[random ermine]

then 90 degrees again

what do you get

[placid steppe]

Oh the reflection?

[random ermine]

yes, 180 degrees

[placid steppe]

Kk

[random ermine]

if you rotate something by 30 degrees then 30 degrees again, what do you get

[placid steppe]

60

[random ermine]

yes

[placid steppe]

Kkkkk

[random ermine]

so B^2 is rotating by 2*theta

[placid steppe]

Ohhh ok

[random ermine]

and B^n is rotating by n*theta

this makes it easy to find B^n

[placid steppe]

Makes sensee

[random ermine]

what did you get for ii.a

[placid steppe]

240

[random ermine]

alright

so here theta=240

[placid steppe]

240 n is equal to identity

[random ermine]

mhm

[placid steppe]

I see

Wait

So 240n = .....?

It should be 720 if I go backwards

[random ermine]

backwards?

[placid steppe]

Knowing the answer. How do I know that it is 720

If that makes sense😭

[random ermine]

you check each n one by one

if n=1 implies 240n=240, not the identity

what if n=2?

[placid steppe]

I domt think I get it

What is the identity

[steep saddle]

no transformation

[random ermine]

identity matrix

[steep saddle]

1 along main diagonal 0 everywhere else

[placid steppe]

Ye, tis might seem stupid but I js took that as 1

Like

240n = 1

I don't know 😭

[random ermine]

what happens if you rotate by 360

[placid steppe]

Sorry yall✌️✌️

[placid steppe]

Tye points come back to the original position?

[random ermine]

yes so u get identity

so you want 240n = multiple of 360

[placid steppe]

Oh I see

LHH MK

I get it 😭

I difnt see the integer part mb

Tyy

Sorry for thetrouble lolz kk

.close

[pearl pond]

Channel closed

Closed by @placid steppe

Use .reopen if this was a mistake.

[pearl pond]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sinful jasper]

I am confused cuz i just learnt intregration with u sub and with parts

[sinful jasper]

Now i am trying to learn integration with trigonometric sub

And im confused tbh

i know that if it is sqrt a2 - x2 it is x = asin

sqrt a2 + x2, x = atan

sqrt x2 - a2, x = asec

but im stuck at thsi question tbh

[flint basalt]

well, which of these that you listed fits into this question?

[sinful jasper]

Well non of them, no? because the requirment of sqrt() isn't there?

[lofty glade]

it is one of those 3, promise

[sinful jasper]

Im confused isnt sqrt supposed to be one of the req for the x to be 1 of them?

but if sqrt is not, then its the 2nd one

the atan

[smoky gull]

be careful tho x^2 has a coeffiicient!

[lofty glade]

true

[sinful jasper]

yes
it becomes 2^2 + (sqrt(3)x)^2

[south sail]

Try putting x=2/√3 (sin theta) you will get 4 cos^theta in denominator. Maybe that help a little bit

[sinful jasper]

What?

[sinful jasper]

I did this for atan
then would i just do 4 + 2tanTheta?

or am i missing smth

[south sail]

The question asked for the suitable trigonometric substitution. I may be not correct but maybe that substitution can help u out

[sinful jasper]

im so confused icel to you, cuz idek what trigo stuff i have to memroise all i know is those 3 cases?

[smoky gull]

uhm?

[sinful jasper]

everything else im actually so confused on

[sinful jasper]

yeah exactly im confused....

[smoky gull]

you can take the three out!

as in, pull out three from the denominator

that would give you the form you desire,
a^2 + x^2

[smoky gull]

.

need another hint?

[sinful jasper]

How would i pull 3 out?

like

[south sail]

Break 4 into 1 + 3

[smoky gull]

well divide the numerator and denominator both by 3

[sinful jasper]

ye but that would make the 4 inton 4/3

isnt that worse?

[smoky gull]

okay sure

[smoky gull]

nope!

[smoky gull]

all we need is this form.

[sinful jasper]

ok 1sec also while we are here what formulas / trig idneities / trig tuff i need to memroise for trig sub, i know those 3 cases but what else?

[smoky gull]

doesnt matter what "a"becomes

[smoky gull]

there is list for sin-1 (x), cos-1(x), tan-1(x), sec-1,csc-1, and some logarithm ones

[sinful jasper]

I got 1/(4/3 + (4/3 * tantheta)^2)

[smoky gull]

basically the major ones you need to memorise are
1/(a^2 +- x^2)
1/sqrt(a^2 +- x^2)
1/sqrt(x^2 -a^2)
sqrt(a^2 +- x^2)
sqrt(x^2 - a^2)

[sinful jasper]

+-??

[smoky gull]

plus minus

as in both cases

[sinful jasper]

Ah okay

😭

[sweet junco]

man, if only there was an easy way to do this (the sweet and humble triangle)

[smoky gull]

-# which country or grade btw

[sinful jasper]

Ireland 1st yr

im doing cs major but i have to take 2 math modules

2nd module is easiest thing

1st one is pain

cuz i hate trigo

Anything with trigs is just bs for me

[smoky gull]

oh sorry then i dont know much

[sinful jasper]

1st yr uh uni

[sweet junco]

well too bad so we will now be talking about triangles if you don't want to memorize the identities and stuff(depends on what u want)

[smoky gull]

but you are missing smth

[sweet junco]

isn't 1/(a^2-x^2) partial fraction-able

[smoky gull]

ye but still "formula" makes it faster

converts into ln

[smoky gull]

check you dx!!

[sinful jasper]

estimating area using left endpoint approximation, anti-derivatives, the fundamental theorem of
calculus, indefinite and definite integrals, u-substitution, integration by parts, trigonometric
substitution, partial fractions, applications of integral (length of curve, area under/between curves(s),
volume of revolution using both disk/washer and shell methods), numerical integration.

this is the topics i have

for this module

[steep stag]

yes but there's also the key identity that:
[\frac{d}{dx}\left[\frac{\operatorname{arctanh}\left(\frac{x}{a}\right)}{a}\right]=\frac{1}{a^2-x^2}]

[sinful jasper]

im just stuck at stupid trig

[jolly parrot]

Stumpman

[sweet junco]

teach a man a formula, they can easily forget the formula, teach a man to derive the formula, they can remember for a lifetime

[smoky gull]

mhmm okay lets first do this qsn

[smoky gull]

is this the rip off of give a man a fish 😭😭

[sweet junco]

yes

[smoky gull]

lmao

[sinful jasper]

😭

[smoky gull]

@sinful jasper

[sweet junco]

it works tho

[smoky gull]

..

[sinful jasper]

forgot that even exists

💀

So my demo is fine now?

denom*

[smoky gull]

ye

[sinful jasper]

so x = 4/3 * tantheta

Wait what is even the derivative of tan? uh is that smth i gotta memroise cuz they only give us this in the exam 1 sec

nvm i mgith be trippin

tan is sec^2?

[south sail]

2/√3 tan theta

[sinful jasper]

oh

ye

ty

[south sail]

You are welcome

[sinful jasper]

xD

ggs

trust me bro

[south sail]

Wtf

[sinful jasper]

xDDD

[south sail]

Can't believe people are still falling for this trap

[sinful jasper]

istg

like fr

[smoky gull]

anyway did you sub out dx?

[sinful jasper]

Wait i see smth at the bottom

wait nvm

im blind

[smoky gull]

,rotate

[sinful jasper]

,rotate

[lofty glade]

almost there...

[smoky gull]

4/3?

[sinful jasper]

Omg

[smoky gull]

16/9?

[sinful jasper]

i keep forgetting

1 sec

its 4/9

and 2/3

[smoky gull]

no

a^2 is 4/3

a is 2/sqrt(3)

[sinful jasper]

What

wait 1 sec

[smoky gull]

.

[sinful jasper]

this was correct no?

[smoky gull]

no

[sinful jasper]

except the 4/3 part

where it should have been 2/3

[smoky gull]

4/3 should be outside ()^2

[smoky gull]

2/sqrt(3)

[sinful jasper]

..... okay now im confused

cuz i was told that was correct

wait

4 + 3x ^2

now

dvided by 3

[smoky gull]

it isnt

[smoky gull]

okay

[sinful jasper]

so 4/3 + x^2

[smoky gull]

nice

[sinful jasper]

Ok there

that was the issue

it is sqrt of 2/3

[smoky gull]

now compare it with

[smoky gull]

bro😭😭

[sinful jasper]

what

[smoky gull]

sqrt of (4/3)

[sinful jasper]

im clapped mb gng

sob

[smoky gull]

= sqrt(4) / sqrt(3)

[sinful jasper]

sqrt 4/3

[smoky gull]

2/sqrt(3)

[sinful jasper]

there

yup

okay okay we getting somehwere 1 sec

let me write in my notes rq#

so then

its x = 2/sqrt3 * tantheta

and x^2 is 4/3* tan^2 theta

so its 1/(4/3 + 4/3*tan62 theta)

and i dont know the trig identity but it defo loosk like 1 so we take out the 4/3

so its this 1 sec

Oh wait the dx as well

[south sail]

Are you really in uni?

[sinful jasper]

😭 ???

what's up with that question

[smoky gull]

still stuck?

[twilit crane]

sorry for disturbing, did you know $1 + tan^2 x = sec^2 x$

[jolly parrot]

夜靈

[sinful jasper]

.

[south sail]

U are on the right track. Keep solving

[smoky gull]

okay good!!

[smoky gull]

now you can use this!!

[sinful jasper]

Now I know the bottom bit is an identity, ite just i never bothered to memorise identities, so I needa memorise wm now, do tou mind sending me a list rhar is essential for trig sub

[smoky gull]

am in 11th rn so may not be sufficient for your level

Would be better to ask uni student

#math-discussion btw

[sinful jasper]

Where r u? USA?

[smoky gull]

ind🥀

[floral narwhal]

the essential ones are the 1+trig^2 or 1-trig^2

because of the way trig sub works you’ll almost always be subbing in those situations

tan^2 +1=sec^2
sec^2 -1 = tan^2

[sinful jasper]

So the pythag identities?

[floral narwhal]

mostly

so now you can simplify further

[sinful jasper]

Anything else or meh?

[floral narwhal]

well for any situation involving trig imo it’s best to also know the half angle identities so you can reduce the power of like a lone sin^2 or cos^2

$\sin^2{x}=\frac{1-\cos{2x}}{2}$
$\cos^2{x}=\frac{1+\cos{2x}}{2}$

[jolly parrot]

dgtdx

[floral narwhal]

these have other variations ofc but you can derive them all basically

[sinful jasper]

😭

Im cooked

Anyways ler me solve the question 1 sec

[sinful jasper]

Alr i got

$ sqrt(3/2 * theta + C $

[smoky gull]

huh?

[sinful jasper]

$sqrt(3/2 * theta + C$

[jolly parrot]

🥀 repenting_slave 🥀

[smoky gull]

bro istg

[sinful jasper]

ok the bot

is weird

i will send a pic

[smoky gull]

u fry me😭

2 should be outside the sqrt😭

[sinful jasper]

it is

i wrote it wrong

dw

😭

[floral narwhal]

your latex needs a lot of work lol

[sinful jasper]

ik sob

[smoky gull]

thank god 😭

[smoky gull]

...

[sinful jasper]

what 😭

i double checked everythin

[smoky gull]

,w integrate 3/(4+3x^2)

[jolly parrot]

Results provided by WolframAlpha

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

Requested by u_seen_yaj

[sinful jasper]

i havent finished it fully

[smoky gull]

why is there a 6 in the denom

[sinful jasper]

cuz that is what i am about to ask

[floral narwhal]

wait why did you do trig sub for this anyway

[smoky gull]

??

[sinful jasper]

look at the top

the sec^2 cancel

what is left is

1/(4/3) * 2/sqrt(3)

[smoky gull]

should be 2 in denom man

[sinful jasper]

it is in the denom?

at the end

[smoky gull]

What

[sinful jasper]

6/4*sqrt(3) is sqrt(30/2

[smoky gull]

I thought thats a six

[floral narwhal]

could have been solved directly just with antiderivative of 1/(1+x^2)

trig sub not necessary here

[smoky gull]

okay and how do you do that

[smoky gull]

you do this using trigo sub

[sinful jasper]

i just plucked 6/4sqrt(3) in calc which gave sqrt(3)/2

[floral narwhal]

this is entirely unnecessary to do though

[sinful jasper]

cuz thats how its simplified

[smoky gull]

yeah no its correct go on i misread

[smoky gull]

okay lmk ur method then

in #math-discussion tho

[sinful jasper]

Anyhow now what i was told is cuz it's converted to theta, we gotta get i back to X

and idk how to do that part

[smoky gull]

see,

x = a tan(theta)

a is 2/sqrt(3) here

[smoky gull]

so we can just rewrite as x/a = tan theta

then take tan-1 on both sides

[smoky gull]

-# these come from trigo subs

[floral narwhal]

again

this is a formula that you can use

to skip that step

[smoky gull]

Bro read the qsn

it clearly says

use trigo sub

and show the steps

what are you on

[smoky gull]

also lets avoid spamming here

[smoky gull]

@sinful jasper

[floral narwhal]

yeah no im good

[sinful jasper]

x = atan theta?

so uh

1 sec

let me see

[smoky gull]

yeah isnt tht the sub we used org?

[smoky gull]

take your time no hurry!!

[sinful jasper]

that is correct

ye i just dont get how to convert it, we used that sub but where did u get that tan -1 would help..

[smoky gull]

see,

i saw tan on one side

and this should click that if we have any trigo(theta) = c

we can simply inverse the trigo-1

there are some conditions but since we had org subbed in the values asw its all good

[sinful jasper]

oh

i though u meant tan - 1

like an actual -17

😭

i was so confused

[smoky gull]

ohh😭

tan^-1

or arctan

sorry we call it tan-1 here😭

[wicked edge]

Or inverse tan.

[sinful jasper]

ye xD

nws

[wicked edge]

But it just a convention, not a big deal.

[sinful jasper]

Should be correct

[sinful jasper]

I will do 1 more question from it then I will move on to the next topic
Partial functions, dont even know how does a question like that look but I will use rhe chemistry tutor guy ig and see if I undersrand him

[smoky gull]

okey!!

[pearl pond]

@sinful jasper Has your question been resolved?

[sinful jasper]

Its a bit messy

[sinful jasper]

If you cant undersrand tell me i will re write

[smoky gull]

no i understand

[sinful jasper]

Ik I should have subbed in properly like did it step by step but once I know the method I usually just skip steps

[smoky gull]

your x is 5sectheta

not 5sectheta tan theta

[sinful jasper]

Oh ye

Oops

[smoky gull]

5sectheta tan theta . dtheta = dx

[sinful jasper]

Now I can first like take out rhe sqrt

Then uh

1 sec

I can make the second half it the denom to be 5(sec - 1)

Not inverse but actually-1

[sinful jasper]

And then I can replace it with tan2

@smoky gull is it just converted back now?

[sinful jasper]

The 4 is ^4 not 4theta

nvm im wrong

[smoky gull]

oh sorry

[smoky gull]

wait

what happened to the sqrt in denom

it should have become 5tan theta??

sec^2 theta -1 = tan^2 tehta

[sinful jasper]

no i am stupid

i forgot its 25sec2theta - 25

idk how i thought i could remove sqrt that

💀

[smoky gull]

blud cancelled 's' in sinx/cosx type shi

[sinful jasper]

xD

shhh

ur dreaming

this never happened

[paper terrace]

we all watched it

you are not alone in this chat hehe

[sinful jasper]

CAP

you are dreaming

Stop lying!!!!!

I got ton 10/625sec^3(theta)

Omg im.stupid

I forgot the 24

25*

Okay forget about the 5, cuz I can divide the 10 by 5 which leaves 2 at the top and 625 at bottom

[smoky gull]

how did you rven get sec^3 in denom

[sinful jasper]

well in the sqrt we had

25sec2 - 25

so 25(sec^2 - 1)

[smoky gull]

oh right mb

[sinful jasper]

which is tan2

which turns into 5tan

and the 5 goes with the 10 at the top

so its 2 at the top

and tan with tan and 1 sec with the sec at top

is my workings correct so fR?

[smoky gull]

yeah

no it isnt

you forgot

a 5

[sinful jasper]

i just said that

above

iforgot the 5 in the writing

[smoky gull]

ohshoot

yeah its correct

[sinful jasper]

but the 5 cancels with the 10

creating a 2

[smoky gull]

yeah here

mb

[sinful jasper]

Wait now im kinda of confused cuz now

i have theta = sec-1(x/5)

so do i make it sec^3(sec-1(x/5))???

😭

[smoky gull]

you cant sub it back in yrt

[sinful jasper]

okay

[smoky gull]

you havent performed your integration

[sinful jasper]

oh brev

wait what how

cuz i did botht the top and bottom?

[smoky gull]

@sinful jasper you sure ur qsn is right?

[smoky gull]

subbing wont save you from integration

[smoky gull]

you hv to integrate at some point of time no matter how much you sub

[sinful jasper]

i just changed y into an x

and i removed the -5 and -77

[smoky gull]

yeah thats fine

okay so far so good then

[sinful jasper]

do i just take out the 2/625

and integreate the

1/sec3

[smoky gull]

yeah

[sinful jasper]

isnt sec like the opposite of tan? idk tbh i forgot or is that cot?

[smoky gull]

cos..

[sinful jasper]

okay so 1/sec is cos

so oppsoite of sec is cos

opposite of tan is cot

opposite of sin is csc?

[smoky gull]

ye

[sinful jasper]

okay good to know

i just took a guess icel 💀

[smoky gull]

can you
here can you simplify the sec into smth better?

[sinful jasper]

so by oppsoite we mean sin = 1/csc and vice verse and for all of em

[sinful jasper]

ye

1/sec3 theta is

cos3

[smoky gull]

nicee

[sinful jasper]

and we know the derivative of sin is cos

so its

sin3 for integr

[smoky gull]

no

[sinful jasper]

wa

[smoky gull]

you cant do that right away

[sinful jasper]

oh

the 3

the cubid

uh

sin4/4

is that corret?
sin^4(theta)/4

[smoky gull]

first lets rewrite cos^3 theyta into sin^2 theta and cos theta

[smoky gull]

no

[sinful jasper]

wait what?

😭

[smoky gull]

that would have worked only when it would have been simply theta^3

[smoky gull]

here you have to do double substitution ggs🥀

[sinful jasper]

😭

[smoky gull]

dw ill hekp u out

[sinful jasper]

never seen more gayer shi then this bro

😭

[smoky gull]

Oka

So lets

try to recall, cos^2 + sin^2 =1

so cos^2 = 1 - sin^2

now you can see where am going with this kinda

rewriting cos^3 theta as cos theta( cos^2 theta)

subbing, we get
costheta (1-sin^2 theta)

[sinful jasper]

alr 1 sec let me get the rules here

[sinful jasper]

i need to memorise these sob

i think for trig sub thats all i need to memo no?

[smoky gull]

now, we can assume (1-sin^2 theta) as another var t

[smoky gull]

mhm prolly for now

[smoky gull]

try seeing what happens after this

[sinful jasper]

either ways i only got math for this yr and next yr only 1 math module

okay imma sub it 1-sin

1 sec

[smoky gull]

still a lot of maths

[sinful jasper]

might have to go to ind

to teach me the ways

cuz i never met an indian that is not smart at math

💀

1/1,000,000,000 chance smh

so it becomes

cos(1-sin2)

so its cos - cossin2

yeah im stuck icl to you

[sinful jasper]

ok

so cos()t

so for a var we take it outside?

correct?

,w integrate 3/((x^4)*sqrt(-25+x^2))

[jolly parrot]

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[sinful jasper]

gg

[pearl pond]

@sinful jasper Has your question been resolved?

[smoky gull]

,w integrate 3/((x^4)sqrt(x^2 -25))

[jolly parrot]

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[sinful jasper]

okay im confused

the 3/625 i have outside

insdie the integration we have

cos3

which we said we can chance into

cs2 = 1- sin2

so cos(1-sin2)

souh

cos - sin2cos

we can integrate cos into sin?

but sin2cos?

[plush bramble]

have you learned u sub

[sinful jasper]

yes

lovely, so i do a u sub inside a trig sub

lovely days innit?

xD

[sinful jasper]

but i can take the cos out and make it intoa sin correct? so it becomes 3sin/625 times intgeral of -sin2cos?

or actually -3sin/625

times integral of sin2cos

I got -sin^4(theta)/625

i have had to do smth wrong xD

[plush bramble]

Show your full work if you want someone to look at it

[sinful jasper]

it was above but im rdoing the whole question cuz i realsied smth

@smoky gull

I finally get it

Whoever created it trigo, I hope nothing but pain for them

I do finally undersrand them tho so xD

Imma do 1 more than do arlwast 2 more topics

Exam is rhis friday xD

[smoky gull]

YESSS

lets goo

w

[smoky gull]

😭🥀

[smoky gull]

oh damn atb!!

[sinful jasper]

Tyty

[sinful jasper]

Partial functions, applications of integral, area under between curves, volume of revolution using both disk, numerical integration left

Are these easy to learn xD @smoky gull ?

[floral narwhal]

it's not too bad

although sometimes visualizing the graphs can really help when you think about like volumes and stuff

some people have trouble visualizing it but thats all really

partial fractions is pretty simple as long as you dont have some insane power in the denominator

but with simple things like difference of squares it's easy

[pearl pond]

@sinful jasper Has your question been resolved?

[pearl pond]

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[lean plume]

how do i do this

[flint basalt]

well, first of all what is the formula for the sum of the interior angles in any polygon of n sides?

@lean plume ^

[lean plume]

(n-2) x 180

[flint basalt]

yes, so what is your sum?

[lean plume]

nvm i got it

its 110

degrees

SIR

[flint basalt]

that's right

[lean plume]

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[desert cliff]

How is that 3, -1 and 8?

[desert cliff]

Shoudnt it be -3, 1 and -8?

[sharp vigil]

what are a1, a2, and a3?

[abstract oriole]

The points are written like x-a so x-3 is x-(3) and x+1 is x-(-1)

[desert cliff]

Variables

[sharp vigil]

yes but they must be coming from a standard formula of some sort

[desert cliff]

I’ll send the full thing

[mystic geode]

$\vec{r} = r_{0} + t \vec{v}$

[jolly parrot]

InterGalactic

[mystic geode]

it is r_0 vector

which has a_1, a_2, a_3

[abstract oriole]

@desert cliff

[desert cliff]

Can you go step by step from what we are given im a bit confused

[abstract oriole]

See all the things are equal to each other so let them be equal to the same thing lambda

[mystic geode]

I gave you a rundown of it in https://discord.com/channels/268882317391429632/903486480075354182

[abstract oriole]

Then write x y z in terms of lamba by that relationship

To get 3 linear equations

Which can just be converted into vector form

[desert cliff]

Wdym

[abstract oriole]

Like z-8=lambda means z= lamba+8

[sharp vigil]

so your teacher writes
[ \pmat{x \ y \ z} = \pmat{3 \ -1 \ 8} + \lambda \pmat{5 \ 2 \ 1} = \pmat{a_1 \ a_2 \ a_3} + \lambda \pmat{v_1 \ v_2 \ v_3} ]
this is really 3 equations,
[ \begin{cases} x = a_1 + \lambda v_1 \ y = a_2 + \lambda v_2 \ z = a_3 + \lambda v_3 \end{cases} ]
which if we solve for lambda in each of them we get
[ \begin{cases} \lambda = \frac{x - a_1}{v_1} \ \lambda = \frac{y-a_2}{v_2} \ \lambda = \frac{z - a_3}{v_3} \end{cases} ]
in other words
[ \lambda = \frac{x - a_1}{v_1} = \frac{y - a_2}{v_2} = \frac{z - a_3}{v_3} ]

[jolly parrot]

clôud

[desert cliff]

Yes I understand but isn’t the final part in the Cartesian form

The question said to convert to vector form

[sharp vigil]

well i showed how you would go from vector form to cartesian form

but you go from cartesian to vector form just by recognizing a1, v1, a2 and so on in the cartesian equation

if you don't like that method you can start by setting each of them equal to lambda and then solving for x, y, and z manually

[desert cliff]

What’s the best shortcut

[sharp vigil]

do you see how the first equation you were given is literally just the last equation here with specific values plugged in for a1, v1, a2, and so on?

[desert cliff]

Yes

[abstract oriole]

Or you can just say that in x-a1/v1= y-a2/v2 =z-a3/v3 the line goes through (a1,a2,a3) and it's direction ratios are (v1,v2,v3)

[sharp vigil]

so if you recognize those you can just plug them into the first equation here

[desert cliff]

could you show me the solution or is it the same from the screen?

But I do get the gist of what your saying

[sharp vigil]

that's what the solution seems to have done

[desert cliff]

Ah I get it

[desert cliff]

Got it

.close

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[fair oasis]

from what i understand for a transformation,
it's onto if all vectors have images?
and it's 1 to 1 if for all vectors that have images, each vector maps to their own image?

[dense jasper]

Onto means that every vector in the target space is mapped to (aka no possible outputs are left out)

One-to-one means that every input vector maps to a unique output (aka no two different input vectors ever share the same image)

[fair oasis]

so basically for all b's in Ax = b, there's an x for it?

[fair oasis]

and so i think

Ax = Ay if and only if x = y?

[dense jasper]

yes

[dense jasper]

yes

[fair oasis]

thank you!

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[desert cliff]

What is the question asking?

[desert cliff]

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[pearl pond]

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[spare rivet]

not sure how to find the maximum possible value here

[spare rivet]

is it some manipulation? the sqrt(33)(y-5) looks weird

[wicked edge]

Well, I will just tell you what I think of, don't take into account if this is false.

If I let A = tan x, and given x from -90 to 90, then A must be anywhere from negative to positive infinity.

Then, you can just let the function y, in term of t. So that you can solve quadratically.

Does it make sense @spare rivet ?

[spare rivet]

hmm

alright ill give it a try

[pearl pond]

@spare rivet Has your question been resolved?

[spare rivet]

ok i got it

thanks

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[wicked edge]

No worries.

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[tough viper]

Hi I do know this is physics but I don't really have anywhere else to ask for help so I hope someone could help me out
So for the ball question I can use 0.15tan30
Essentially treating the dotted line as the weight to find force of airflow
However why for the rod I can't take the dotted vertical line as the weight and do 400tan37

[tough viper]

(I can solve it btw) just unsure why the method doesn't work for 2nd qn

[errant solstice]

||because the weight of the rod acts about the rod's center, while F acts at point Y. they're not perpendicular components of the same force.
(if I'm not mistaken.)||
ignore this first, and go with Bacter.

[tardy reef]

what was your idea behind using 0.15tan30 ? Like which physics concept did you use before arriving at this trig expression?

[errant solstice]

<@&268886789983436800>

[timber cape]

bruh

2 mr beast scams withing half an hr

[errant solstice]

<@&268886789983436800>

another one??

[slow oak]

they come in waves

[tough viper]

I think of it as if there was no air flow the ball would be vertical so 0.15N is also the parallel component then after that I just created a right angle triangle to find force of air flow

[tardy reef]

yea, the thought is correct, but the actual concept that I am looking for you to name is static equilibrium

so what you wanna do is balance the forces and their moments.

Fortunately for you, the force balance is unnecessary as the tension in the string takes care of that, and calculating that is pointless for the specific problem

so all you wanna do is consider moment balance

Now luckily, as chiaki said, the forces were acting at the same point in your first problem and in the second problem, the forces are acting at two different points

so things worked out even when you directly wrote the expression the way you did

However, since the situation is slightly different in this case, you should start from scratch and first write the moment balance equation and then use that to establish the relationship between the weight and the applied horizontal force

[tough viper]

Oh yes the weight acts at the midpoint of the rod but I don't get how it affects why I can't do 400tan37
If there was no F wouldn't thr perpendicular distance from weight to Pivot X just be 0 so it wouldnt matter

Or should I think of taking the weight as the vertical component only works if the force applied is directly on the center of gravity of the weight?

[tardy reef]

perpendicular distance from weight to Pivot X just be 0.
well, the reason for that is that the rod would be vertically oriented in that case. Application of F changes that, so the distance of the weight vector from the rod is non zero

[tardy reef]

force of weight always acts vertically (which is the direction of gravity ofc)

and center of gravity is specifically defined to be a singular point where the weight acts as an equivalent singular force vector, so thats not an issue

[tough viper]

Oh so the forces have to act at the same point for Me to just take weight as vertical component

[tough viper]

Ohh

[tardy reef]

no, thats wrong interpretation. Forces have to act at the same point, and that itself doesnt change much. However, in special cases it might simplify the expression. Thats all. But you shouldnt bother remembering these special cases. What you should know is how the physics works in a general case, and you should be able to simplify using algebra on the go once you establish the equations.

So in this problem, draw a free body diagram and then moment balance (through the pivot coz that eliminates the moments of the irrelevant reaction forces)

[errant solstice]

I think in general for any physics problems, drawing an FBD never goes wrong.

[tough viper]

Hmm okay I do sort of it understand it now
Thanks a lot!

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[stoic imp]

how do I calculate the arclength of the equation x^2 = y^3 with -1<=x <=1

[rough forge]

Do you know the arclength fomula

[stoic imp]

yeah

can you show it to me please

[rough forge]

So you dont know it

[stoic imp]

I do know it

but this is not the arclength that I am looking for I think

because the one that I am looking for the is the norm of the parametrization of x^2 = y^3 correct?

[rough forge]

Intuitively, it's just Pythagoras, and then you let the hypotenuse shrink very small ds²=dx²+dy²

but you don't have a parameterization

[stoic imp]

I can make one out of this

[rough forge]

ok sure

[stoic imp]

r(t) = (t^2, t^3)

@rough forge

[stoic imp]

yeah this is the arclength formula, but I was looking for the arclength formula for a parametrization dawg

@rough forge

[rough forge]

not quiet

[stoic imp]

care to elaborate?

[rough forge]

(x,y)

[stoic imp]

yeah so?

i dont understand your point

[rough forge]

what you have would satisfy x³=y²

[stoic imp]

how so?

wdym by this?

[rough forge]

how did you come up with your parameterization

care to elaborate

[stoic imp]

yes

I can elaborate

[rough forge]

||because x=t² => x³=t⁶ and y=t³ => y²=t⁶ => x³=y²||

[stoic imp]

say (x,y) = (t1, t2)
then t1^3 = t2^2
so t1 = t^3 and t2 = t^2

meaning r(t) = (t^3, t^2)

[rough forge]

but thats different from what you stated

also I don't understand your reasoning

say (x,y) = (t1, t2)
then t1^3 = t2^2

[stoic imp]

where are you confused? @rough forge

[rough forge]

you have x²=y³ not x³=y²

[stoic imp]

what do you recommend then?

[rough forge]

you just messed up your exponents

[stoic imp]

say x^2 = y^3 passes through (1,1) and (4,8)

so we can find the direction vector

(4,8) - (1,1) is the direction

so the direction vector is (3,7) and one of that we pass through is (1,1)

so the equation is
L(t) : (x,y) = k.(3,7) + (1,1)

so (x,y) = (3k + 1, 7k + 1)

so y = f(x) = 7k + 1 with x = 3k + 1 so k = (x-1)/3

basically we get that
y = f(x) = 7/3 * (x-1) + 1

[rough forge]

what the helly

[summer imp]

You computed the secant line between two points. What does this have to do with your problem? You first need a working parametrization.

[rough forge]

Thanks I am speechless

[stoic imp]

can you guys help or not?

[rough forge]

I was just waiting on what you had to say

[stoic imp]

I am trying to parametrize x^2 = y^3

[rough forge]

When you have x²=y³ and choose x=t^a and y=t^b (some powers of t) then you get t^(2a)=t^(3b) or 2a=3b => a=3b/2 so you can choose b=2 for example

[stoic imp]

this is nonsense, no offense dawg clearly a=b=1

[rough forge]

aright mate

i am leaving

[summer imp]

Well clearly not because then x^2 = t^4 and y^3 = t^9. They gave you a systematic way to find a correct parametrization.

[stoic imp]

i edited

[summer imp]

So t^2 = t^3 for all t?

Still doesn't work

[stoic imp]

why

[summer imp]

Because you're suggesting that (t,t) satisfies t^2 = t^3, which isn't the case.

[stoic imp]

why\

it is true when x = y = t surely

@summer imp

[summer imp]

It's not true when x=y=t.

Because, well, x=y

And you need x^2 = y^3.

[rough forge]

How can you square one side and cube the other

[stoic imp]

x=0, x = 1 satisfy

[summer imp]

But you want to parametrize the whole curve x^2 = y^3, not two points of it...

[stoic imp]

what do you recommend then

?

[summer imp]

If you can't see intuitively what it should be, then this

[stoic imp]

why do you grab b = 2

[summer imp]

Because you need a,b such that 2a = 3b.

Any such pair works.

[stoic imp]

sure

so r(t) = (t^3, t^2)

@summer imp @rough forge

you here?

[rough forge]

dude

i reacted with the emote

now continue with the formula

[stoic imp]

but the formula you gave me was for non parametrizable curves

there is an arclength formula for parametrizable curves

[rough forge]

look on the right

[stoic imp]

alright so the integral of the norm of the parametrization

[rough forge]

yes

[stoic imp]

what is alpha what is beta @rough forge

[rough forge]

the bounds for t

[stoic imp]

what are my bounds for t

[rough forge]

if -1=<x=<1 then figure t

[stoic imp]

?

[rough forge]

how was x(t) defined in your parameterization

[stoic imp]

x = t^3

[rough forge]

what follows

[stoic imp]

-1 <= t <= 1 ?

somewhat somehow

@rough forge @summer imp

[rough forge]

not somewhat

t³ is strictly increasing so thats why

-1 <= t <= 1
follows

[stoic imp]

yeah

@rough forge @summer imp what now?

[summer imp]

Well now you have your bounds for the arclength formula, which you've already seen and used before.

What do you need help with?

[stoic imp]

im still unsure how to join every puzzle piece together @rough forge @summer imp

[rough forge]

Yeah it's like everytime is your first time doing math

[summer imp]

Look at the formula that was given to you or the one you're using in class. Compute what you need to compute.

It's not any different than the last times you've done it.

I can assure you the definition you're using didn't change since last time.

[stoic imp]

,, \text{arclength(shit)} = \int_{-1}^1 \norm{\varphi(t)} \ dt = \int_{-1}^1 \sqrt{t^6 + t^4} \ dt

[jolly parrot]

Renato

[stoic imp]

where $\varphi$ is my parametrization \

[summer imp]

Wrong definition.

[jolly parrot]

Renato

[stoic imp]

care to elaborate?

@summer imp

[summer imp]

This is neither what star gave you as a formula, nor is it the formula you've seen.

[rough forge]

You didn't differentiate the components of your param

[stoic imp]

how so ? @summer imp @rough forge

[rough forge]

because the arc length formula requires you to do so

[stoic imp]

pretty sure the left hand side of the equality is the standard for arclength of a parametrizable curve

correct?

[summer imp]

It should be phi'.

[stoic imp]

what?

[summer imp]

When computing the arclength you're integrating the magnitude of the velocity

Not the magnitude of the position.

[stoic imp]

so its $\int_{-1}^1 \sqrt{(2t)^3 + (3t^2)^2}$

@summer imp

[jolly parrot]

Renato

[stoic imp]

@rough forge

[rough forge]

[summer imp]

What's the velocity vector of your parametrization?

[stoic imp]

I am not a physicist dawg

[summer imp]

...

[rough forge]

What's the tangent vector of your param

[stoic imp]

$\text{tangent vector(\varphi)} = \frac{\varphi '(t)}{\norm{\varphi(t)}}$

[rough forge]

your are taking the norm of phi' not phi

[jolly parrot]

Renato

$\text{tangent vector(\varphi)} =  \frac{\varphi '(t)}{\norm{\varphi(t)}}$
```Compilation error:```! Missing $ inserted.
<inserted text> 
                $
l.60 $\text{tangent vector(\varphi)}
                                     =  \frac{\varphi '(t)}{\norm{\varphi(t)}}$
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.```

[rough forge]

<@&268886789983436800>

bot

[stoic imp]

pretty sure there is no mistake in the latex

but somehow there is a mistake I cant see

[rough forge]

i am sure you are missing the bigger picture now

we just want to know what's phi'(t)

[stoic imp]

@summer imp wdym?

[summer imp]

It's fine just ditch the latex. What is phi'(t)?

[rough forge]

\text{tangent vector}(\varphi) = \frac{\varphi '(t)}{\norm{\varphi ' (t)}}

[stoic imp]

yeah what is the problem?

@rough forge @summer imp

[summer imp]

What. Is. phi'(t). ?.

[stoic imp]

is the derivative of the parametrization of the parametrizable curve

[summer imp]

I'm not asking for the definition.

You have a parametrization

I want you to write down what phi'(t) is

For that parametrization

[rough forge]

In other words: Apply what you already know

[stoic imp]

yeah

[summer imp]

yeah

[stoic imp]

my parametrization phi(t) is just (t^3, t^2)

so if we differentiate that shit assuming phi is C1 we get that @summer imp @rough forge

3t^2, 2t

[summer imp]

What's the magnitude of that?

[stoic imp]

that magnitude is just sqrt((3t^2)^2 + 4t^2)

thats the norm of my two dimensional vector in R2 @summer imp @rough forge

[summer imp]

Good.

So that's what you're integrating.

[stoic imp]

what about it

@summer imp

[rough forge]

integrate

[summer imp]

What do you mean what about it?

Now you have your integrand, you integrate it to get the arclength.

[stoic imp]

how do I integrate this shit?

[rough forge]

For integration I can give you a hint: ||factorize t² and substitute inside the sqrt||

[stoic imp]

,, \text{arclength of } \varphi(t) = \int_{-1}^1 \sqrt{3t^2 + 2t} \ dt