ug-maths-physics

[rancid schooner]

How’s the campus food?

[vestal swift]

kant's principle of humanity :)

[stuck rapids]

give us some advice

cmon

[vestal swift]

least of my worries tbh

[rancid schooner]

Show campus food pics

[vestal swift]

more worried abt my grades

[vestal swift]

crysto knows haha i always post on ig :)

[rancid schooner]

See #the-food-network

[stuck rapids]

got miss our malaysian food mou

HAHAHAHHA

[vestal swift]

i miss bean sprout chicken :)

[rancid schooner]

Post some here. On #the-food-network

[stuck rapids]

wht wht

[rancid schooner]

Singapore has good chicken rice.

[stuck rapids]

? food as an advice

[rancid schooner]

Enjoy your food :)

[stuck rapids]

owhhhhh that one ahhhh

[vestal swift]

damn bot has good taste haha nice burger

[rancid schooner]

With an unusual name

[vestal swift]

ipoh food :)

[stuck rapids]

he is indeed rich

HAHAHAHAH

[vestal swift]

coz he is smart

[stuck rapids]

yi pou gai

[rancid schooner]

That’s my only square meal for the week

[stuck rapids]

sure or not

[rancid schooner]

Channels here getting very quiet

[stuck rapids]

wht about the aussie breakfast

rm 42

[rancid schooner]

That was a few weeks ago

[vestal swift]

fuiyoh wtf

[rancid schooner]

Or a few months ago

inflation

[vestal swift]

for 2

why is it n and k

[stuck rapids]

but that means u r rich either ways u k

[vestal swift]

k is fixed or ?

[vestal swift]

but u still can afford :)

[stuck rapids]

else why would u spend 40 over ringgit for a meal

[vestal swift]

i come here eat grass

jk

[stuck rapids]

maybe u ordered a drink also

[rancid schooner]

k is already a factor of n!

[stuck rapids]

light it up lez partyyy

https://tenor.com/view/crazy-dance-funny-dance-michael-scott-dance-danse-marrante-debile-gif-21385827

[rancid schooner]

So k|n!
And k|k
So k|(n!+k)

[vestal swift]

ye but need to prove by induction

i digest sia

wait so base case is 2! + k is divisible by k ?

[rancid schooner]

It’s given that k is less than or equal to n

[vestal swift]

wait so for 3!+k then it must be divided by 2, 3?

then 4!+k , must be divided by k=2,3,4?

shitz

[rancid schooner]

yes

well not at the same time

but for any given k

choose any k, and it works for that k

[little widget]

I am as confused as u i do not understand how tf k is a list

[vestal swift]

so do I have to do a mathematical induction inside a mathematical induction :)

or use strong MI

i still dk how strong MI works

[rich wraith]

found this in another server

find A

[little widget]

Very sussy diagonal matrix

Also very sussy pattern

[rancid schooner]

That would be Inception not Induction 😄

[tawdry gull]

there is double math induction

...which i have no idea how to do, i just know it exists

[rancid schooner]

https://en.m.wikipedia.org/wiki/Inception

[long aurora]

stupid q but might find the answer from a physics student, i was wondering, are air particles affected by acceleration of free fall? do they experience g (9.81) ?

[long aurora]

good movie?

[rancid schooner]

Yes very good. Recursive.

[stark wave]

Yes but the dominant force will be electro / nuclear depending on the context since gravity is very weak

[verbal nova]

Yes, it has all the good actors and the best director too

[little widget]

Idk but assume gravity constant

Assume acceleration constant

Assume zero friction

Assume all strings are inextensible

Assume u can fuck with the coordinate system while the linear algebra teachers have a seizure

[rancid schooner]

You know your movies well

He’s a very good director.

[long aurora]

So if let’s say i tried to calculate the displacement of an air particle assuming it’s moving vertically downwards, would it be wrong to use acceleration as 9.81

[long aurora]

In physics everything is assumptions

[rancid schooner]

If all air particles drop to the ground then people would suffocate standing up.

[long aurora]

HELP I LAUGHED

[stuck rapids]

😆😆😆😆

Wtf HAHAHAHAY

[rancid schooner]

And no gravity

[rancid schooner]

What’s the answer?

[rich wraith]

no idea

no one replied to it on the other server either🥲

[rancid schooner]

If it’s A^20 + B^21 then it is possible to get a “nice” answer for A and B.

[verbal nova]

Yes

[rancid schooner]

Otherwise perhaps not.

[rancid schooner]

Try drawing the time graph for Inception as well as for Memento :)

[verbal nova]

oh god

[rich wraith]

this is the reply I got

"People suggested that the matrix is top triangular so there is a pattern when multiplying by itself"

[rancid schooner]

Yes it clearly is top right triangular but what is the final answer?

[rich wraith]

I don't think anyone knows

[brazen rampart]

halp plz im cri

[rancid schooner]

Nice question

[rancid schooner]

What if it’s A^20 and B^21?

[vestal swift]

@rancid schooner :')

[rancid schooner]

Sentence on the left doesn’t seem to make sense. 🤔 @tawdry gull ?

[vestal swift]

I don't understand also HAHAH

For the circle one it's a subspace right?

[rancid schooner]

Ask the lecturer!

[tawdry gull]

When proving subspace, u need to prove the 3 conditions are met right. He/she is asking to prove that it is NOT a subspace, then we have to show it doesnt satisfy at least one of the conditions

[little widget]

@rancid schooner im quite confused if we can use the fact that detA = detA^T and det(AB) = det(A)(B) how does it equal zero??

[rancid schooner]

It’s not equal to zero. You’re supposed to fill in the answer :)

[little widget]

Why did they set it equal to zero then lol

Yeah if it isnt seems trivial

Oh lord im fucking dumb

Thats the place to fill it in i see

[rancid schooner]

😊

[little widget]

Thanks for the observation

I dont have the keenest eye

[rancid schooner]

It’s not very intuitive. They should just leave it blank.

[little widget]

No they prolly did

Op did not 😭

[rancid schooner]

||doesn’t go through (0,0)?||

[tawdry gull]

Yes

[rancid schooner]

😊

[rancid schooner]

This is the numerical solution. But there doesn't appear to be a nice analytical solution.

[little widget]

Ehhh

Thats lame

Everything was lined up so beautifully too

The 20 and the 21

[rich wraith]

no one tried it

[rich wraith]

fr

[little widget]

The diagonal matrices

Wth man

Troll q

[rancid schooner]

Try it sometime. The solution is quite nice.

[tawdry gull]

what did u use to find the solution?

oh and also, could u please check my hw6 q2b

my friend (an IMO guy) and i didnt find a way to solve it yet

[rancid schooner]

Numerical search 😄

[tawdry gull]

wow

[rancid schooner]

Looks challenging! Will take a look when I have some time.

[tawdry gull]

yes thank you 🙏

[rancid schooner]

On Excel! 😄 Wolframalpha is not very good at this type of problem.

[tawdry gull]

interesting

[rancid schooner]

Anyway I found that the value of the leading diagonal elements is very close to: 2^ln(20/21). But not exactly so.

It should be c where c^20 + c^21 = 1.

The two edge elements are in the ratio 20:21.

[main topaz]

Excel?!?!?

[rancid schooner]

Yes. Using the solver on Excel.

Excel can handle simple matrix operations.

Anyway if you modify the problem to
A^20 + B^21 = (given matrix on RHS)
then it’s easier to solve analytically.

[rancid schooner]

How would you have gone about solving it numerically?

[main topaz]

No idea tbh 😅

[rancid schooner]

You could write some Python code but it would take too much effort.

[main topaz]

Pretty out of sync with math atm, college apps destroying the few brain cells I have left

[rancid schooner]

Anyway I just wanted to see that a solution exists.

Good luck. Where are you applying?

[main topaz]

Just submitted UCAS a couple days ago (oxford, warrick, UCL, imperial, lse) for a maths major

Working on ED for US atm

[main topaz]

Prolly not even gonna go there considering I can't pay for uni unless I get aid/scholarship and it's pretty tough in the UK to get one

[rancid schooner]

Nice. Hope it goes well!

[main topaz]

Thx

[idle rain]

Gave +1 rep to AlexV#1811

[rich wraith]

"IF P, then Q" is true when P is false and Q is true

I don't get how :/

[rancid schooner]

So it must be false

[vestal swift]

Vacuously true

Something like "if elephants can fly, then I am Superman" hahahha

[rancid schooner]

by inspection it looks like f(x) = ln(x) or something similar.

[tawdry gull]

f(x) is bounded

[rancid schooner]

no, it's stated that f(x) → +∞ as x → ∞

[tawdry gull]

eh, it didnt

[rancid schooner]

which is true for f(x) = lnx

[tawdry gull]

its the sequence that goes to +infty

[rancid schooner]

can we make up a sequence that follows ln(x)? i.e. xₙ = n and f(x) = ln x?

[tawdry gull]

yeah sure

but f(x) can be anything

as long as it is cont and bounded

[rancid schooner]

ok bounded. first line.

so some sort of asymptotic function then

maybe a logistic function

[tawdry gull]

it could also not converge

[stuck rapids]

Why do proofs even exist

😂

[little widget]

if anyone can solve this itd be sick

i keep seeming to get 15.696N

[vestal swift]

Why tf they expecting a math major to do this shit

[little widget]

wow i like option D

no no dont put in A choose D

[rancid schooner]

Logic

Prove that you exist

[little widget]

alex u fancy some physics this morning/afternoon

veri trivial perhaps

[rancid schooner]

Prove by contradiction that they don’t.

[vestal swift]

Nope

The answer is C

:')

D makes more sense to me too

[little widget]

well obviously

its a mcq

why would both be correct

in before bamboozle tho

[vestal swift]

Wuh

????

[little widget]

you can pick A

or D

but D is A

theyre the same answer

[little widget]

philosophy channel la?

[vestal swift]

But d got contradiction no?

[stuck rapids]

Bruh like

[rancid schooner]

The politicians won’t be very happy with B

[little widget]

Oh god

I feel like this q is gaslighting me rn

Its like A is applying itself to D

AHHH

[vestal swift]

LITERALLY I DONT GET WHY ITS C

[rancid schooner]

It’s a logic prank by the lecturer

A circular reference.

[little widget]

I hate ur lecturer

[stuck rapids]

Both a and c are correct

Hahahahaha

[rancid schooner]

how did you get that?

[rancid schooner]

since xₙ → ∞ as n → ∞,
(xₙ + B)/xₙ = 1 + B/xₙ → 1 as n → ∞, i.e.
(xₙ + B) → xₙ as n → ∞.
So, as n → ∞,
f((xₙ + B) → f(xₙ), i.e. f((xₙ + B) - f(xₙ) → 0.

Is this valid?

[tawdry gull]

that is given f(x) converges right?

or r u assuming f can be either converge diverge?

[rancid schooner]

no assumption

except that f(x) is continuous

and bounded, as given in the question

[tawdry gull]

i see

hmm

wheres the use of boundedness there?

[little widget]

Im dumb

I was looking at the force being applied to block B not A

[tawdry gull]

we also have to show an existence of a sequence that will make it converge to 0

[rancid schooner]

how about {xₙ} = B ln n?

Then, xₙ + B = B(1 + ln n) → B ln n (= xₙ) as n → ∞.

[tawdry gull]

hmm i dont think so

suppose f(x) = sin(x^2) and B=10

[rancid schooner]

Well it would work in the case where f(x) has period B.

[tawdry gull]

f(x) may not have a period

so thats the problem 😭

[rancid schooner]

Just pointing out the one instance where it would work :)

[tawdry gull]

yeah haha

if it was periodic then it would surely work

[tawdry gull]

@rancid schooner

i got the answer

do u want to take a look?

[rancid schooner]

Not yet 😄

[tawdry gull]

okay sure haha

[rancid schooner]

Let me take a look again when I have some time :)

[tawdry gull]

do u want a hint at least?

[rancid schooner]

Haha ok

[tawdry gull]

use MCT and IVT

did i say that already?

[rancid schooner]

Ok similar to what was suggested in the question

[tawdry gull]

ah okay haha

[rancid schooner]

I think you’re supposed to prove that a sequence exists,not necessarily find a particular sequence, is that right?

[vestal swift]

Fellow math gods

How to develop math intuition to solve math problems :')

N the proofs

Coz

I suck :')

And I wanna die coz I'm not cut out to study math

[tawdry gull]

exactly

[tawdry gull]

umm, there are visuals for some math concepts

which would help

[stuck rapids]

why do proofs even exist

what on earth

[rancid schooner]

Otherwise how would you know that 1+1 = 2?

[vestal swift]

Where ah besides trusty old 3b1b :')

[vestal swift]

Smh

[vestal swift]

Common sense? :)

[rancid schooner]

Must start from axioms

[verbal nova]

Guys

I have a discrete math mid term tonight

And I cant stop wondering how this subject is actually useful

Plz someone tell me any famous use of discrete math

[rancid schooner]

Discrete maths is fun

Without it we wouldn’t be able to count anything

[verbal nova]

😭

I cannot detect the fun at all man

[tawdry gull]

useful in cs

especially cs deals with discrete things

[main topaz]

uhh considering this chat is quite active could someone please help with this MAT question?

[rancid schooner]

(e)

[rancid schooner]

||Simplify to (sin y)(1 - sin y) = (sin x)(1 - sin x)||

|| then solution is x=2n𝜋+y, 2n𝜋-y for integer n ||

nice question

[main topaz]

i got till here but couldnt think of the next step

[main topaz]

thanks

[idle rain]

Gave +1 rep to AlexV#1811

[rancid schooner]

😊

The other element of the solution (not required in the question) is ||sin x + sin y = 1||. Visible when you plot the graph.

[rancid schooner]

Combinatorics. Which is fun as well.

Like the CROCODILE question 😄

[verbal nova]

you are no longer my friend

[rancid schooner]

What discrete maths topics are you doing?

[verbal nova]

very first chapter, propositions, inferences, predicates, quantifiers, intro to proofs

[rancid schooner]

That sounds rather dry

[vestal swift]

Yea p much same here

I failed my discrete math midterms :')

[verbal nova]

😭

stop scaring me bro

if you wanna pass then you study

[vestal swift]

Dw you'll pass

[verbal nova]

studying is necessary for passing

[vestal swift]

Necessary but not sufficient

[vestal swift]

Just don't panic like I did

And you'll do fine

[verbal nova]

either you study or dont pass

you cant pass and not study

let the argument be that if you wanna pass then you study and you didnt study therefore you didnt pass

Modus tollens moment

true for all students

[vestal swift]

Tf

Feels like contradiction

But idk

[rancid schooner]

Hold a study session

[main topaz]

uhh second part??

do they mean like ||the point (a,b) say lies in S such that it itself is its nearest point?||

[verbal nova]

[rancid schooner]

That would be one nearest point

Perhaps S is the subset of points on a circle with centre (a,b).

[tawdry gull]

the subset has a non unique solution that is nearest to point (a,b)

as alex mentioned, a boundary of a circle with centre (a,b) would work

since there are infinitely many points equidistant to (a,b)

so there is technically no "nearest point"

or an empty set would work too, an empty set by definition is a subset of any set

[main topaz]

then what about iii) wont a circle with centre a,b work for that?

[tawdry gull]

oh yep

thats for iii)

or an ellipse centred in (a,b) would too

for ii), any open set would work actually

as long as it doesnt contain (a,b)

[rancid schooner]

Where is this from?

[main topaz]

MAT 2019

uhh sorry but I still dont get (ii)

[tawdry gull]

the open set?

[main topaz]

x^2+y^2>1 would be considered an open set right?

[tawdry gull]

yes

just make sure it doesnt contain (a,b)

so u can make (a,b) the centre

(x-a)^2 + (y-b)^2 > 1

[main topaz]

AHHHH i got it know, they meant there's no one point that is nearest right?

[tawdry gull]

no points at all

not even 1 or 2 or more

completely none

[main topaz]

wait no

[rancid schooner]

So that’s the empty set

[main topaz]

in this case all points on the boundary of the region will be closed to a,b no?

closest*

[tawdry gull]

except it has no boundary

because it is an open set

:)

[main topaz]

oh fudgee

[main topaz]

thx

[idle rain]

Gave +1 rep to The_Pierrot#7889

[tawdry gull]

gotcha

[rancid schooner]

So for any point that is close you can find a closer one

[main topaz]

sheesh mat is hard

[tawdry gull]

exactly

hence no nearest point

[rancid schooner]

Like an infinity pool

[main topaz]

and thx @rancid schooner

[idle rain]

Gave +1 rep to AlexV#1811

[rancid schooner]

😊

[main topaz]

i dont think i'll be able to prepare for mat in 10ish days

[tawdry gull]

do ur best

try watching vids for it too

znotes got a playlist on that too btw

[main topaz]

college apps and school exams are killing me

[main topaz]

ooo will check it out

[rancid schooner]

Watch Oxford Mathematics Plus videos on YouTube by James Munro.

[tawdry gull]

oh i see

[rancid schooner]

A nice MAT question was this: what is the maximum value of (ax+by) where a, b are positive and x² + y² <= 1?

[main topaz]

no other restrictions on a and b?

[rancid schooner]

No

[verbal nova]

statistics and probability are so sexy

its so hot how everything connects

[verbal nova]

Guys how does Universal Generalization work 😭

how does the argument P(c) conclude that ∀xP(x)

[stuck rapids]

大可不必。。即时逃走

😂😆

[verbal nova]

Finished the midterm

Went good

[vestal swift]

Could u send me the questions if you have them?🥺

[rancid schooner]

Congratulations!

Which university/country/course?

[verbal nova]

Concordia, Canada, Data Science

[rancid schooner]

Nice. Good luck with the course!

[vestal swift]

Uni mathematical induction next level compared to furthermaths 😭😭😭

[tawdry gull]

technically induction can be done as long as it proves along the integers so... :D

there is also math induction on a chessboard

[rancid schooner]

classic example of recursion

and also classic example in a course on theory of algorithms

[main topaz]

Tower of Hanoi does bring back epic memories of grade 8 me cracking the optimum algorithm my second try of playing

[vestal swift]

Can't relate, in grade 8 I was still dumb af

Can anyone help me out with this? Proving it is a subspace, I have a LA quiz tmr😭😔

Idk how to prove that this satisfies closure under scalar multiplication

.
This was my approach:
S={(a-c,b-c,0); a-c∈R, b-c∈R}
We can rewrite it as
S={(x,y,0); x,y∈R}, where
x=a-c
y=b-c
this is a x-y plane in R³ which contains the origin

[vestal swift]

Idk how to do exactly this 😭 help pls🥺

[tawdry gull]

let v be in S, then tv is also in S

for any real number t

u will get ta(1,0,0)+tb(0,1,0)+tc(-1,-1,0)

if t>=0 then it immediately follows that tv is in S

if t<0

[vestal swift]

Ye? 🥺

[tawdry gull]

let u=-t so that u>0
ta(1,0,0)+tb(0,1,0)+tc(-1,-1,0)
= ua(-1,0,0)+ub(0,-1,0)+uc(1,1,0)
= uc(1,0,0)+uc(0,1,0)+(-ua,-ub,0)
= (uc+ub)(1,0,0) + (uc+ua)(0,1,0) + (-ua-ub,-ub-ua,0)
= (uc+ub)(1,0,0) + (uc+ua)(0,1,0) + (ua+ub)(-1,-1,0)

all ua, ub, uc are nonnegative

so its in S

[vestal swift]

Uh wait why not (uc+ua)(1,0,0) ah

[tawdry gull]

because top of third vector have -ub

so have to +ua to have -ub-ua

[vestal swift]

Hold on lemme digest digest 🥺

[tawdry gull]

oke

[vestal swift]

Oh

Omg

[vestal swift]

Thank u LIFESAVER

[idle rain]

Gave +1 rep to The_Pierrot#7889

[tawdry gull]

okeh 👍

[verbal nova]

what is this shit bro?

what language

[vestal swift]

Subspace

[vestal swift]

Math :')

Linear alg

[verbal nova]

great, now Im scared of studying linear algebra too :(

Why cant math just be calculus

[tawdry gull]

cuz calculus is not enough 😭

[vestal swift]

Exactly

Don't worry you didn't pick maths like me :')

[verbal nova]

whatever bro

❤️ ❤️ ❤️ ❤️ ❤️ ❤️ ❤️

[tawdry gull]

miller miller

[rancid schooner]

What is that object to the right of the whiteboard?

[vestal swift]

Idk tbh

Did I choose the wrong course :")

[rancid schooner]

Don’t believe everything you read

[rancid schooner]

Ask the lecturer

[stuck rapids]

senpai should guide us

[rancid schooner]

Ask @tawdry gull 😄

[stuck rapids]

why not u

HAHAHAH

[vestal swift]

But how u so sure math is right for me :')

I'm like dying in LA

I can't write proofs

Which is the essence of math 😞

[rancid schooner]

Just copy from sample solutions as an exercise and you’ll get the hang of it

[stuck rapids]

i struggle with proofs as well

😦

[vestal swift]

So your advice is copy solutions to master proofs ah

I copy alr still shit

😔

[rancid schooner]

You’ll get better :)

Host a study session!

[vestal swift]

Bro I can't even teach myself

[vestal swift]

@rancid schooner @tawdry gull :')

[rancid schooner]

Probably EᵀM or something like that

[tawdry gull]

Ci -> Ci + c Cj ?

[vestal swift]

4 marks :')

[tawdry gull]

other marks come from explanation :)

[vestal swift]

Idk how :)

I got 0.5/4 :)

[tawdry gull]

ME is basically the rows of M multiplied to columns of E

which basically means column i of M becomes Ci+c Cj

because in E, column j has a c in it

[little widget]

Yeah idk why people dont talk about this

People jus say “just do the proof bro try not to look at solutions”

[rancid schooner]

Learn by imitation. The first few times at least.

[vestal swift]

But what if the qs are so diverse

That you can't find any pattern in it

[little widget]

Well ur not trying to find a pattern

[vestal swift]

Yeah but idk how to justify :')

[little widget]

More like developing a thought process

[vestal swift]

Idk how to get that man :')

[rancid schooner]

What do they teach during tutorials?

[little widget]

Eh itll take time

Im shit at it too

[rancid schooner]

Isn’t that what it’s for?

Some universities have a maths cafe.

[little widget]

Yea i hope in ur proofs class its not just

The types of proofs

End of course

[little widget]

We have a maths club and if u join free coffee not sure if that counts 🤣

[rancid schooner]

Free coffee! How nice 😊 ☕️

[tawdry gull]

we have an elite math class

which... is not easy to get in

[rancid schooner]

Is that in the first year?

Do they give free coffee?

[little widget]

Man they call it “elite”

[vestal swift]

tutorial just discuss problem sets

I rlly think I'm not cut out for maf :')

[tawdry gull]

wine and cheese party exists

[tawdry gull]

https://www.math.hkust.edu.hk/ug/chern_class/

[rancid schooner]

Now I feel like having wine and cheese 🍷 🧀

[rancid schooner]

Are you a participant?

[tawdry gull]

nope

only chern class students get to do it

which im not

or maybe not yet

who knows

i dont have high expectations

[rancid schooner]

Not yet.

Anyway you can still buy wine and cheese for yourself.

Maybe we can have a ZNotes wine and cheese study session 😄

[vestal swift]

doobee's dumb qs

[jagged quarry]

help needed

[verbal nova]

X/Y = 5/7
Y - X = 24
Solve da equation

[jagged quarry]

Needs to be solved by some ratio method

The topic is ratios, percents and proportions

[verbal nova]

What grade are you in btw?

[jagged quarry]

-_-

BS

This from 101 buisness maths

[tawdry gull]

@rancid schooner differentiation

[rancid schooner]

Wow who would have thought differentiation would be so complex 😄

[tawdry gull]

wouldnt be fun if differentiation is just the way it was

hahaha

try doing 1b

its nice haha

[rancid schooner]

cos² (πx/2)?

[tawdry gull]

no....

it must be decreasing haha

[rancid schooner]

it is decreasing on [0,1] :)

[rancid schooner]

(cos² x)/x?

For the last part

[rancid schooner]

The optional part on Hermitian Polynomials looks interesting

[tawdry gull]

1b must be decreasing on the whole domain

[tawdry gull]

So it’s valid for 1a

[tawdry gull]

Haha yess

[rancid schooner]

😊

[rancid schooner]

Monotonically decreasing?

[tawdry gull]

yes, not strictly tho

do have fun with the questions haha

[rancid schooner]

Please host a study session for this channel

[stuck rapids]

elloooo..how to approach this question after finding the solution space of S xD

[tawdry gull]

is it just M={v1,v4}? cuz pivot is in column 1 and 4

so rank(S)=2

[blazing wraith]

can someone help me plsss

[idle rain]

Help needed <@&597104882482413589>

[verbal nova]

Arent the absolute extremes just where dy/dx = 0?

wait, one sec

[unique zodiac]

I thought it was asking for the max. and min. values the graph can yield

If you're looking for the point where dy/dx=0 its quite simple: x = -2.4

But theres ofc more to it than that bcuz its 10 marks TT

[verbal nova]

No wait

[unique zodiac]

thats quite dark aha

[unique zodiac]

mmm icic

[verbal nova]

pngs :/

[unique zodiac]

:(

[verbal nova]

the abs maxima could be -1 or -5 too

[unique zodiac]

yeah

Well in that case, u have one local max, we can check if its a min/max later.

In the range x = [-5,1], the graph is always negative (because for terms to the power 2/3, you're squaring it and then taking the cube root so its a positive number. Then you multiply by x which is negative)

If you have a term s.t. f(x) = 0, that will be the absolute maximum.

From the dy/dx equation, theres a certain point where we cant evaluate dy/dx, try using that value in f(x) and see what u get

bruh u spoiled it for him

[verbal nova]

Ah shit, lemme hide it

||Looks like absolute Max is where y = 0 which looks to be -4
and absolute Min is f(-5)||

@blazing wraith

[stuck rapids]

Okok tq tq! I thought it was way more complex

[unique zodiac]

For the absolute minimum, we can check that when dy/dx=0 is a turning point, it is the local minimum. To find if its the abs minimum, compute x=-2.4

Then, using the graph of the derivative, you will find that the point where dy/dx = 0 is local for a certain domain, now check if thats true for the remaining domain (by subsituting the most extreme value of that domain ie. endpoints, and then compare)

If ur not going to use a calculator, its helpful to write dy/dx which ive computed earlier as: 1/(x+4)^1/3 * (5x+12)

[blazing wraith]

@verbal nova @unique zodiac thank you so much! ill have a look and if thers any questions ill ask you guys again :D

[idle rain]

Gave +1 rep to Brave#5303, ukan#9821

[blazing wraith]

but then, how did you get 2x/(x+4)^(1/3)

i cant understand the way to derivative, i thought outside bracket then only inside

[unique zodiac]

Oh, here I'm using the product rule, so bcuz u have (3x)*(x+4)^1/3 , u have to use the product rule

[verbal nova]

you forgor A2 math?

[blazing wraith]

haha kinda LMAO

[unique zodiac]

💀

moment

For the first part im sure uk how to get, for the second part its: 3x * 1/3 * (x+4) * (1/3 - 1) * (d/dx x+4)

[verbal nova]

@blazing wraith hows data science 😎

[blazing wraith]

awesome for now but some places kinda drowning cuz bad lecturer

me trying to study by myself haha

[verbal nova]

uni life 101

every subject is killing me

[blazing wraith]

yea im bad at calculus

[verbal nova]

whats the subject called?

[blazing wraith]

calculus 1

[verbal nova]

I see I see

for some reason we starting with multivariable calculus

[blazing wraith]

whats that??

i will study calculus 2 in other semester

forgot which one

[verbal nova]

3D calculus

[blazing wraith]

damn :o

so is this correct?? or is there anything missed out

@verbal nova @unique zodiac

[verbal nova]

actually from the graph I sent, the abs max appears to be f(-4) = 0

[unique zodiac]

u got the min. value correct which is cool

but yeah the abs max is (-4,0)

ur right in assuming that if the turning point isnt the local max/min, then one of the endpoints would be

Here we only have one turning point which is a local minimum, then one of the endpoints would be the abs. minimum (if the turning point isnt). It doesnt tell us anything ab the abs maximum.

[blazing wraith]

how would you know the abs max is (-4,0)? what calculation can help you to get it?

[unique zodiac]

Yea lme guide u there

Heres sm food for thought

If u show that f(-2.4) is a minimum, isnt it weird that f(-5) is somehow lower?

Bcuz u didnt find any other turning points, so its quite strange that f(-5) is lower than f(-2.4) isnt it

Now think: How is that possible, between -2.4 and -5, how did the derivative suddenly flip signs? Where could it have happened? (Hint: Look at the expression for dy/dx)

[verbal nova]

Ig making a graph

what 💀 bro

[unique zodiac]

https://tenor.com/view/cats-graph-up-awesome-gif-23011592

[verbal nova]

just make a graph

[unique zodiac]

chii in the exam

[verbal nova]

https://tenor.com/view/look-at-this-graph-nickelback-gif-12683147

[blazing wraith]

HAHHAHAHAHAA

[unique zodiac]

Unless u cn use a graphing calc in the exam, u have to get it analytically somehow

[blazing wraith]

i have no idea 😭

[unique zodiac]

dy/dx = 1/(x+4)^1/3 * (5x+12)

here its not possible to get a derivative at the point x+4 = 0, x = -4

But the point on the graph does exist, so u have to see what happens there

[verbal nova]

You guys took F Math?

[blazing wraith]

no :(

[verbal nova]

in AS we had a whole chapter for making graphs

[unique zodiac]

yea but u dn need to make a graph here

Okay tbf if it does help u then its fine

But b careful and see if graphs are an accepted form of working

[blazing wraith]

do u mean like we find the valyue of x at here, that make this equation 0 , ie:(x+4)^1/3

[verbal nova]

is the derivative at x = -4 0?

no its infinity ooof

[unique zodiac]

What I meant to say was, find the value of x such that the derivative doesnt exist

So bcuz dy/dx = 1/(x+4)^1/3 * (5x+12), we can see that if the denominator is zero, the derivative wont exist

So this only happens when (x+4)^1/3 = 0, x = -4

Its a little tricky to explain, but bcuz u can show that at x=-2.4, we expect the y-values around it to be more than f(-2.4) right?

But somehow f(-5) is less than that, so that means that there is a turning point that we didnt find, OR there is a discontinuity in the gradient

In this case, we find a discontinuity and the point where the gradient is discontinuous will be a max/min

In this case its a max.

I hope that helped, this q is a bit big brain if ur doing it strictly w/o graphs

[blazing wraith]

ohhhhh

thank you so much!

[unique zodiac]

u get it now? based

https://tenor.com/view/cat-gif-24874327

[blazing wraith]

lemme digest, but i should be able to get it

means discontinuity gradient will also be either max and min

right

[unique zodiac]

I meant that it could possibly be a max or min, it depends on the circumstance

As I said its a little difficult to explain

[unique zodiac]

Next time u encounter a q like this, think ab this thought process

Its important to consider gradient discontinuities

[blazing wraith]

yeaaa thank you again @unique zodiac

[idle rain]

Gave +1 rep to ukan#9821

[blazing wraith]

alrightttt

[stuck rapids]

Hi guys xD does anyone know how to approach this question for Tp M subset.. This is the second part of the question.. M is a vector space for 2 × 3 matrix in this case

The thing for me is.. the AP=PA condition doesn't make sense to me cuz AP= 2 by 2, while PA should be 3 by 3

I also thought of alternative approach like A=P^-1PA, but since neither of them is a n by n matrix, this won't work

[vestal swift]

why here so quiet :(

also found this in a coding textbook but i'm curious about the maths approach to it :P

[idle rain]

Help needed <@&597104882482413589>

[quasi sequoia]

Let's say you have vector u v and w. u=(-1,2,0) v=(4,1-3) and w=(10,-2,-6). How do you know when to use Gaussian elimination to find if they are dependent or independent

[unique zodiac]

What is M in this case?

If A is 3x3 then AP = 3x3 matrix and PA = 3x2
Hmm thats odd

[unique zodiac]

Well, if one is obviously a scalar multiple of another, then you dont have to use gaussian elimination. Otherwise you do

Its not like u need certain criteria to use gaussian elimination

You can apply it on any 3 vectors to see if they are linearly dependant or not

[quasi sequoia]

oh ok, thanks

[unique zodiac]

this is kinda brainfarting me XD

[stuck rapids]

The whole question b is just odd hahaha

Ap=PA is clearly unsolvable? They can't be equal?

[quasi sequoia]

Can anyone explain how vectors u = (4,-3,0) and v = (5,0,2) are coplanar, they are not on the same plane, one has a z value, but Ive been told that 2 vectors of R3 that are brought back to the origin are always coplanar. What does that mean?

[idle rain]

Help needed <@&597104882482413589>

[tawdry gull]

It’s possible that they’re equal

But thats not the main focus of the question

[stuck rapids]

Hmmmm

But to show that Tp is closed under addition.. Oh wait ya

[tawdry gull]

jia you

u can do it

[stuck rapids]

在死路了hahah

[tawdry gull]

😩

[stuck rapids]

I think I got it.. But it has to satisfy that condition given isn't it

[tawdry gull]

mhmm

[stuck rapids]

Hmmm

[tawdry gull]

||(A1+A2)P=(A1)P+(A2)P=....||

[stuck rapids]

Owhhhhhh

🫣Now it makes sense danggg

[stuck rapids]

Tq tq!

[tawdry gull]

:)

[stuck rapids]

@rancid schooner and u didn't give any tips before hahahaa

🫣

[rancid schooner]

Because πerrot would come up with a good answer 😄

[stuck rapids]

U would too

🫣

[rancid schooner]

still learning :)

[stuck rapids]

Rite

[rancid schooner]

Machine learning 😄

[stuck rapids]

Hmm how to find the image of matrix HAHAHA

[rancid schooner]

Put it in front of a mirror

[stuck rapids]

Nice joke

😂

[unique zodiac]

wha idgi, ur getting a 3x3 matrix on the lhs but a 2x2 matrix on the rhs

[unique zodiac]

If u look at the parametric definition of a plane: r = p + λd1 + μd2
p is a point, d1 and d2 are directions. You can take any 2 vectors and they can be coplanar, choosing an arbitrary point

[vestal swift]

Samee so is part b a subspace or nah?

If it's not then one of the axioms must fail right? (First axiom in this case)

Since Tp is a subspace iff it satisfies the 3 axioms isn't it 😭

[ebon rose]

can someone help with this one

[stuck rapids]

Yeah that's wht I thought..

[stuck rapids]

Yeah the first condition zero vector thingy is not satisfied.. If not wrong

[rancid schooner]

Please do a study session for university matrix maths

[tawdry gull]

Try asking ur prof abt it

[stuck rapids]

I think it is ahhahaa

Due date in 6 mins

[verbal nova]

Is b a function?

[brazen rampart]

\

can you help me figure out what i did wrong
the actual answer based on the solution sheet doesn't have the /e^t but i don't see how to negate that

[tawdry gull]

no because undefined when n=pi

functions need to be

1. defined in the domain
2. has only one output

[verbal nova]

Is 1/x a function?

[tawdry gull]

for x =/= 0

[verbal nova]

Ok so a function has an output for all values in the domain

[tawdry gull]

yes

and it can only output 1 value

[verbal nova]

Alright, thank you!

[idle rain]

Gave +1 rep to The_Pierrot#7889

[young marten]

one to one or many to one

only

[late frost]

wait

math & phy combined ?

[verbal nova]

Yes, they the same

[stark wave]

so is philosophy

[late frost]

so No Phy role?

[stark wave]

There's a UG physics role but they both lead here

[late frost]

ah

IC 👀

[verbal nova]

XD i was have a friend who took philosophy elective their beginning syllabus was so similar to logical propositions in Discrete Meth

All mathematicans are philosiphers but not all philosiphers are mathematicans

[late frost]

true fact

[verbal nova]

Is the inverse function even possible???

[idle rain]

Help needed <@&597104882482413589>

[verbal nova]

I assumed there was and got [1/3, 2/3)

But now im wondering if it is injective or even surjectuve

[jagged quarry]

Needs to be solved using matrices

How to do this using matrices?

[idle rain]

Help needed <@&597104882482413589>

[late frost]

probably I guess? Cuz I saw this equation during my IIT ¯_(ツ)_/¯

[long aurora]

Bro didnt u start AS this year

[late frost]

I Study JEE Too

[long aurora]

But u werent able to solve an AS trig question yesterday…

[late frost]

cuz trig is said less
not a lot

only basic he said

@long aurora u were guys like made me confused

[long aurora]

I see

[late frost]

I started in June 13th

[late frost]

$\frac{d}{dx}e^{tanx}=$

[round siren]

Unknown Member

[idle rain]

Help needed <@&597104882482413589>

[late frost]

what's the answer?

[main topaz]

$e^{\tan x} \sec ^2 x$

[round siren]

Acid

[main topaz]

The derivative of $e^{f(x)}$ is $e^{f(x)} \cdot f'(x)$

[round siren]

Acid

[late frost]

Ty

[wary jungle]

btw do you guys have any youtube channel recommend for discrete math? Also help my peanut brain xD I think c and d is equivalent but I am not sure about the rest (Question 4)

[verbal nova]

D

[wary jungle]

nice same as the answer I chose 0.0

does a,b and d works too?

[verbal nova]

a and b are possible but not equivelent

[wary jungle]

oo ok

[stuck rapids]

I have a question xD.. Is there a way to compute the determinant of 6*6 stochastic matrix?

I have to prove that the eigenvalue is 1,since a stochastic matrix should have an eigenvalue of 1 and have a stable distribution

[tawdry gull]

all the best 🙏

row operation to the matrix would help

[rancid schooner]

Interesting topic

Please host a study session and share what you have learnt

[stuck rapids]

Yeahhh xD

[rich wraith]

which axiom justifies that if x = 1 then x + 1 = 2?

or in general which axiom justifies 'doing the same thing to both sides'

[tawdry gull]

field axiom

[rich wraith]

Thx👍

[idle rain]

Gave +1 rep to The_Pierrot#1126

[stuck rapids]

https://en.wikipedia.org/wiki/Singular_value_decomposition very interesting topic which came out in my assignment

[blazing wraith]

how to do 1b

[idle rain]

Help needed <@&597104882482413589>

[blazing wraith]

i need help plssss

[rancid schooner]

It’s greater than the integral of x² from 1 to 3 which is 26/3.

[blazing wraith]

i dont get it, can you elaborate further, by using the theorem pls @rancid schooner

[rancid schooner]

From the graph in part a the √ graph is above the x² graph. So its integral cannot be less than the integral of x².

[blazing wraith]

so how could we proof that the integration will be more than 26/3?

[rancid schooner]

Because the graph is higher

[blazing wraith]

yeah so??

i knowww then??/

[blazing wraith]

neverminddd

i got it

[wary jungle]

Can anybody explain what does Y means?

[verbal nova]

f(x) = y

Set X is the domain

Set Y is the range

[tawdry gull]

basically, words

Y is the set of words using the english alphabet

doesnt have to be english words

it can be "aksdghasjgkdash"

thats also in Y

the point is that it's non-null

[verbal nova]

ohhhhhh

[wary jungle]

Oo ok got it thank you!

[wary jungle]

Can anybody help? I don't get this 2 questions

[rancid schooner]

Q6: int(1000/7) + int(1000/11) - int(1000/77)

[wary jungle]

oo ok thank you!

[idle rain]

Gave +1 rep to AlexV#1811

[rancid schooner]

😊

[wary jungle]

Guys, may I know why f(x) is not onto? |B| <= |A|

I thought it should be

Or f(x) should be integer? And not N?

[tawdry gull]

f doesnt map to all the naturals

a quick example, there is no such x in N such that 3x-2=2

2 is in the Naturals, yet we cant find a natural number x such that 3x-2 becomes 2

so it is not onto

[minor schooner]

It will be a quotient not a natural right??

[tawdry gull]

yes exactly

[minor schooner]

For (iv) it does not exist because all natural numbers are +ve but intergers cld be either

[wary jungle]

Ohhh got it thank you!

[idle rain]

Gave +1 rep to The_Pierrot#1126

[verbal nova]

https://www.youtube.com/watch?v=ngCos392W4w
I highly recommend this video for problem solving, I used this videos techniques yesterday by testing out multiple cases and finding a pattern in my discrete math exam in a question that is not even related to recursions

[blazing wraith]

i need help with this

[idle rain]

Help needed <@&597104882482413589>

[rancid schooner]

Volume = Cross sectional area multiplied by distance travelled by centroid

[rancid schooner]

https://youtu.be/uvSrUG0yYQY

Hopefully this is not true

[vestal swift]

Math gods

How y'all do dis

[late frost]

WTH?

math error 😭

[young marten]

why did he sniff a textbook😭

perving out on a book ffs

[verbal nova]

If we ever end up doing that, we are long gone

[rancid schooner]

Add (2-√3)²⁰²² then subtract 1 to get the integer part.

[vestal swift]

Wait why though?

[rancid schooner]

That's to find the integer part. Standard A-Level binomial expansion question.

To find the first digit to the left of the decimal point, first let 𝛼 = 2+√3 and 𝛽 = 1/𝛼 = 2-√3.
Note that 𝛼 and 𝛽 are roots of x²-4x+1=0.

Is that enough of a hint?

Would you like me to continue?

[vestal swift]

N^2022 + (1/N)^2022 - 1 ? Nani?

[rancid schooner]

Will put the rest in spoiler below.
||Let Sₙ = 𝛼ⁿ + 𝛽ⁿ, and note that Sₙ = 4Sₙ₋₁ - Sₙ₋₂, and S₀ = 2, S₁ = 4.
Computing Sₙ mod 10 iteratively for successive values of n shows that
Sₙ mod 10 = [2,4,4,2,4,4,2,4...] for n=0,1,2,...
i.e. Sₙ = 2 if n=3m, and 4 otherwise. Since 2022 = 3(674), therefore S₂₀₂₂ mod 10 = 2.
Also, since 0< 𝛽= 2-√3 <1, therefore 0<𝛽ⁿ<1.
Hence the digit of the left of the decimal point is 1. ||

[rancid schooner]

Yes, but it's a very big number :)

Extra credit: What is the digit to the right of the decimal point of (2+√3)²⁰²²? 😄

[vestal swift]

Why mod 10 though?

And the beta thing, since we're calculating alpha^2022 right 😢

[rancid schooner]

To get the last digit

[vestal swift]

But S_2022 is α^2022 +β^2022 right? How does dividing by 10 give the left digit next to the decimal point of α^2022?

[rancid schooner]

α^n on its own is messy so we pair it with β^n to get an integer which is easier to compute

[rancid schooner]

Yes. But since β is between 0 and 1 β^2022 is also between 0 and 1 (and much smaller than β).

We know α^2022 + β^2022 is an integer. So the integer part of α^2022 must be 1 less than that integer.

[vestal swift]

Oh so something like finding the floor value of α^2022 ah 😂

[rancid schooner]

Well the floor is the integer part :)

the integer part is lava 😄

[vestal swift]

Im guessing 9 HAHHAHA coz I kept pressing my calc :)

[rancid schooner]

Your calculator is unlikely to be able to handle that haha

[vestal swift]

And coz the beta thing gets closer to zero for higher powers of beta :D

[rancid schooner]

Have you read my spoilered solution?

[vestal swift]

Yes bc I forgot all my further maths stuff

:(

Thanks math god 🙇‍♀️🙇‍♀️🙇‍♀️ @rancid schooner

[rancid schooner]

😊

[vestal swift]

Wait no rep point thing?

Hahah

[rancid schooner]

Haha maybe they removed it or perhaps you have to reply

[vestal swift]

Thank u maths God🙇‍♀️🙇‍♀️ I'm relying on you to carry my gpa when my tutor graduates 😂🥺

[idle rain]

Gave +1 rep to AlexV#1811

[rancid schooner]

😊

No no for that you would need @tawdry gull 😄

[vestal swift]

Yes both of u 😭

I had to retake foundation of maths coz my midterms score was below avg 😭😭

[rancid schooner]

Which topic is this question under?

[vestal swift]

It was my friend's discrete maths midterm

[rancid schooner]

That’s ok. That means you’ll have a stronger foundation :)

You’re not taking discrete maths?

[vestal swift]

MH1812 ahaha

Next sem

Different code

MH1301

[rancid schooner]

Looks like a flight number

[vestal swift]

This is designed for the comp sci students

[vestal swift]

And this is for the maths students like moi but I'm too dumb 💀

[rancid schooner]

Did your friend solve it?

[vestal swift]

One of them did but they on vacation now

Still waiting for their reply ahaha

[rancid schooner]

Why are you doing your friend’s exam even though it is over? For practice?

Presumably exam means in-person exam.

Anyway hope that was helpful :)

[vestal swift]

Curiosity😂

[vestal swift]

Yessir

[vestal swift]

Yep

[rancid schooner]

Now try the extra credit part :)

[vestal swift]

Not 9 ah

:(

[vestal swift]

Btw out of curiosity, how to be absolutely sure that the pattern doesn't break tho?

[rancid schooner]

Yes 😄⭐️

[rancid schooner]

What do you mean?

The 2,4,4?

[vestal swift]

Yess

[rancid schooner]

It’s cyclic. So you just have to prove it for one round.

[vestal swift]

Prove by induction or?

[rancid schooner]

Don’t think that’s even necessary

[vestal swift]

So how to go about proving it ah :(

[rancid schooner]

f(2,4) = 4
f(4,4) = 2
f(4,2) = 4

That should be good enough

[rancid schooner]

See above

[vestal swift]

Hmm ok thank u sifu amongst sifu🙇‍♀️

[vestal swift]

Thanks rep +1

[idle rain]

Gave +1 rep to AlexV#1811

[vestal swift]

😂😂

[rancid schooner]

It was a fun question 😄

[rancid schooner]

😊

Here’s an idea. Why don’t you do a reverse study session? You livestream your attempt and then people can give suggestions.

[vestal swift]

I'm too dumb to even think of a solution in the right direction😭

[rancid schooner]

You can just ask for suggestions for the next step

[late frost]

is Limits done for u guys?

[vestal swift]

Is gigachad asking moi? 😂

[late frost]

Nani?

[vestal swift]

How to even get this role 😂

[rancid schooner]

dQ and Sasuke are friends?

[rancid schooner]

What if the expression is
(√ 2 + √ 3)^2024?

[late frost]

I'm boosting this server -.-

[tawdry gull]

@rancid schooner

very interesting question

[rancid schooner]

Looks tricky 😄

[young marten]

no way out of all people you find smthn tricky

[verbal nova]

He'll have to use 1.1 % of his power now instead of 1%

[tawdry gull]

Question origin: ||Tokyo university math entrance exam for medical students||

[rancid schooner]

||are you thinking of switching to medicine?||

[tawdry gull]

No thanks 🙏

[late frost]

😂

[minor schooner]

If this is a medical exam qn

What are engineering qns like 💀😭

[tawdry gull]

Medical ones are the hardest

Harder than math major’s even

[rancid schooner]

Mathematical Medicine

[minor schooner]

But why

[verbal nova]

Lmaoo I have a friend, first year medical, and he asked me a couple of times to help him in probability questions

Bayes theorem and combinatrics and shit

[tawdry gull]

They’re meant to do difficult work under time pressure

U dont want medical students who couldnt think properly under time pressure dont u?

[minor schooner]

That wld be a mess💀

[verbal nova]

Lmaoo i saw the video just yesterday

[tawdry gull]

:v

[upper socket]

hey

do you guys know where can i find university physics book by young and freedman

[tawdry gull]

i have it, do u still need?

[rancid schooner]

young and free physics 🤔

@tawdry gull studies physics at university as well?

[tawdry gull]

required course 🤷‍♂️

[rancid schooner]

Just think of it as applied maths.

Experimental as well?

[tawdry gull]

yes

[rancid schooner]

Yes to experimental?

[tawdry gull]

we have experiment course for it too

[rancid schooner]

Interesting

[idle rain]

Help needed <@&597104882482413589>

[wary sluice]

Basically, I've done A levels and started undergrad maths

They got something like let's say MATH A and MATH B

So MATH A I exempted since I did a levels and I am on MATH B

Math a was the prerequisite for MATH B and it seemed to just be A levels

....

To get to the point

I've done integrals

Buts what's this thing with E and then n --> infinity????

By E I mean the sigma sign

[tawdry gull]

Sum to infinity?

Probably talking about riemann sums

[rancid schooner]

He’s probably referring to the epsilon-delta definition of a limit.

[stuck rapids]

seems so

[verbal nova]

Can you send a pic of the question?

[vestal swift]

just need a small hint w this :') @mathgods

[rancid schooner]

Triangle inequality

[vestal swift]

Yea thought of that too but like I think something’s off here, idk how to get there

[tawdry gull]

$| \sum f(x_i) \Delta x| \leq \sum |f(x_i)| \Delta x$

[round siren]

The πerrot

[vestal swift]

OH FCK HOW COME I DINT THOUGHT OF THAT T-T

[tawdry gull]

or u can use the fact that $-|f| \leq f \leq |f|$

[round siren]

The πerrot

[vestal swift]

this one also eventually have to use reimann sum?

[tawdry gull]

nope

just integrate all sides and conclude into one single inequality

[vestal swift]

but how to move the modulus outside tho

for the lhs

[tawdry gull]

u dont

u take modulus of all sides

lol

after u integrate

[vestal swift]

hmm ok i try try first thank uu

[idle rain]

Gave +1 rep to The_Pierrot#1126

[stuck rapids]

Calculus

Horrible

[vestal swift]

but how to integrate modulus f ah :(

[vestal swift]

ikr p sure this sem my gpa gonna drop until cannot afford economy rice T-T

[vestal swift]

er and also for this right just ignore the lim n-> infinity for reimann sum ah

and just say approximation? idk sounds weird tho

[stuck rapids]

That 比喻🤸‍♂️😵

[tawdry gull]

https://math.stackexchange.com/questions/284322/prove-or-disprove-that-int-ab-fx-mathrmdx-geq-big-int-ab-fx

[vestal swift]

means modulus this part but on the RHS the modulus technically has no effect? coz all the areas are positive?

[tawdry gull]

f can be negative?

[vestal swift]

Uh huh?