7 levels of Calculus!

[fleet mauve]

I covered as many topics as possible to justify the vastness and versatility of calculus. (Ofc, I have still missed some parts tho).

Upto level 6, the questions are original and mostly mine. But for lvl 7 I picked 2 textbook questions as I am not good enough at those advanced topics.

@modest ravine @nova mason Dinner ready.
Enjoy the feast.
Rn pls solve and tell answers u get. I will post solutions later.

[modest ravine]

Yeahh!!

I'll do it when I get to school but my inital notes are, 3.6/7.14 Contour integral, some solid of revolutions, a ton of big words I don't know and will look up, some physics stuff

area moment of inertia is pretty cool, the efficient engineer has a video on it which I liked.
https://www.youtube.com/watch?v=Bls5KnQOWkY

[fleet mauve]

@nova mason Not interested?

[nova mason]

dude

you just posted this 30 minutes ago

i'll look at some point

[fleet mauve]

Ok

[autumn pollen]

Q14 is a contour integral

Q15 is oog

[nova mason]

idk what camphor is

[autumn pollen]

It’s like a substance

That turns into gas at room temperature

Very gassy

[nova mason]

||(1 + by)y'' + by'y' = 0
byy'' + by'y' = -y''
(byy')' = (-y')'
byy' = c - y'
(by + 1)dy = cdt
by^2/2 + y = ct + d
blah blah yap yap solve the quadratic
||

ok

[autumn pollen]

I don’t like that integral

[fleet mauve]

Oog?

[autumn pollen]

Keyhole contour the hell out of it

[autumn pollen]

(Idk what the question is even saying)

e^{3ix}/(x^2+1)^2

Real part of that

[fleet mauve]

No need to solve the quadratic. Just need to dubstitute boundary conditions.

[autumn pollen]

Poles at -i, i are of degree 2??

Idk what the formal term is

Anyways

Fah I forgot

Formula for residue at a with nth degree

,w formula for calculating residues

[narrow dock]

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Requested by inverse_cupid

[fleet mauve]

Guys. Idk residue theorem. I only know Cauchys integration theorem.
Can I solve all contour integration problems with that alone.

[autumn pollen]

Wait

,w residues (e^(3ix))/(x^2+1)^2

[narrow dock]

Results provided by WolframAlpha

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

Requested by inverse_cupid

[autumn pollen]

There’s a random 2

I need to search shit up

$\Res(f,a) = \lim_{x\to a}\frac1{\Gamma(n)}\frac{d^{n-1}}{dx^{n-1}}(x-a)^nf(x)$

d/dx e^{3ix}/(x+i)^2

[fleet mauve]

Q15 is from Calculus of variations.

[narrow dock]

ℝανιølι

[autumn pollen]

Ik that much

Idk how to solve

[fleet mauve]

Oh ok

[fleet mauve]

@autumn pollen When does Cauchys integration theorem fail?

If not, why do we need a new theorem viz. Residue theorem

[nova mason]

well you need to get y in terms of t..

[nova mason]

when the domain has a hole

[fleet mauve]

Hole means pole?

[nova mason]

sure you can see it that way

if the function is specified to have complex output

(not infinity)

then you need a hole

[fleet mauve]

For 1 pole, we can use
$\oint \frac{f(z}{(z-z_0)^n} = \frac{2\pi i}{(n-1)!} f^{n-1}(z_0)$

[narrow dock]

MS Riddle

[modest ravine]

you do the contour integrals yet

[modest ravine]

No real point of using gamma, in complex analysis there are just things we can't do

ie, say for negative base logs, we can do it for like log -3 -3^2, but if you try to do fractions it breaks because of how complex logs work

[modest ravine]

just do Re(I) or Im(I)

[crude oar]

first time ive laughed out loud in a long time

[fleet mauve]

Ugh.. what

[modest ravine]

Why lmfao

[crude oar]

one bad integral experience

[modest ravine]

Interesting

but hey, most simple numerical integrators have the same time complexity

[fleet mauve]

What numerical integration are u talking about.

[fleet mauve]

Did u understand the solution?

[autumn pollen]

I didn’t see it yet :p

[fleet mauve]

Ravi? You too Indian?

[autumn pollen]

Yeah lol

I live in America though

[fleet mauve]

Did u try anything other than level 7 in the paper?
Why is everyone interested in Q14 alone😅

[autumn pollen]

Well, I looked at them, but honestly I didn’t try to solve them

[fleet mauve]

Why? Not interesting or too hard or too busy?

[autumn pollen]

I was genuinely just not in the mood T-T

Too burnt out

[fleet mauve]

Oh ok. U are still doing undergrad?

[autumn pollen]

I’m in high school :p
But I’m doing analysis

[autumn pollen]

Also for Q5, is the region bounded by the parabola as well?

[fleet mauve]

Oh. Impressive calculus knowledge for school level.

[autumn pollen]

Thanks :p

[fleet mauve]

No. The triangular region formed a tangent, X and Y axes

[autumn pollen]

Ohhh I get it

Alright

Q10 I literally don’t want to solve

Idk Q12 and Q15 :p

[fleet mauve]

Q10 has a neat trick that reduces the 2nd order non linear DE to 1st order LDE

[autumn pollen]

Idk that trick

[modest ravine]

Numerical integration, like Riemann sums or trapezoidal rule

[crude oar]

Hmm nvm

Fairly certsin i know what the trick is though

just kidding I was right it is ||u=y’|| :3

sorry for the ghost ping ravi!

the answer should be ||y+b/2y^2=1/L(1+b/2)x||

Not bothering to express it

[tidal trail]

|| 7 a) (pi)(e^(-3)) ?||
||b) x - (pi/2)sinx ? ||

[fleet mauve]

@tidal trail I provided solutions in this pdf btw.
Both ur answers are correct

[fleet mauve]

Thats the final answer.

[crude oar]

k thx

didnt realize there werr solutions below 💀

[pseudo heron]

6767676767

[fleet mauve]

😆 😆 😆 😆 😆 😆

[glacial jacinth]

vro , where u been studying?

[autumn pollen]

On my own

[glacial jacinth]

i mean in uni?

[autumn pollen]

I’m not in uni yet

Which is why my knowledge in Calc is funny

[glacial jacinth]

so ur age?

below 18 or abv?

[autumn pollen]

Below

I just do math for fun

[glacial jacinth]

how much years?below

[autumn pollen]

I’m 17

[glacial jacinth]

sure?

r u lying?

[autumn pollen]

Yeah, I’m actually 148 years old

[glacial jacinth]

that's ryal!

Actually which age

[autumn pollen]

17

[glacial jacinth]

sure

?

[autumn pollen]

Yes

I’m 17

[glacial jacinth]

Anyways

that's good

did you know atleast what a chain rule is

[autumn pollen]

What T-T

[glacial jacinth]

[autumn pollen]

Yeah…

[glacial jacinth]

good

[autumn pollen]

This

Smh

[glacial jacinth]

if u r solving that Q10 or others in it ,that means you know Callculus very well

good

keep going

but but

Maths is love

[autumn pollen]

I’m doing analysis

Calculus got boring

[glacial jacinth]

Real?(analysis)?

[autumn pollen]

Yes

Real analysis

Rudin

[glacial jacinth]

then you may should already finished calc thn

[autumn pollen]

Hm, I skip around

[glacial jacinth]

but u said ur calculus is bit shaky and loosy ,then how ya doing real analysis vro

[autumn pollen]

I know enough to do elementary real analysis

I like, can’t exactly solve crazy integrals or huge differential equations

Or can’t exactly tackle any calculus problems that bring in abstract measures or other weird stuff, but that isn’t covered in Baby Rudin, so :p

[glacial jacinth]

u good bro,

be the next Einstenium in the periodic table!

[modest ravine]

I mean, understanding calculus and calculations are different

You should be able to understand like solids of revolution but I don’t think some of the methods matter calculation wise

[autumn pollen]

I understand Coochie sequences

[modest ravine]

Yes

Ravi what’s up with you and Cauchy sequences

[fleet mauve]

How much were u able to solve.
And yeah Q10 is quite a challenge for 17 year olds.

[glacial jacinth]

in dreams

[fleet mauve]

?

[glacial jacinth]

[polar tusk]

@fleet mauve shall i attempt it? idk if i can do it, but ill try nonetheless

@glacial jacinth remmember me from the Math server? I was banned from it; if you search my whereabouts you will find it too.

[glacial jacinth]

how did u got banned?

and there r 2 maths servers

is the server u r talking abt is the one which i hva tag of!?

and wht u want to attempt?

these questions?

[fleet mauve]

Ofc

[polar tusk]

the tag one

[glacial jacinth]

o, but how

[polar tusk]

some internal reasons; basically my approach of taking doubts ig

[glacial jacinth]

wtf, anyways,

You can attempt them

if u have basic knowledge of atelast calc 1 , did u?

[polar tusk]

surely i will; yes

[glacial jacinth]

which class u r in?

[polar tusk]

passed grade 12

now waiting for exam results and other comp. exams

[glacial jacinth]

great

but can u yourself even looked at those questions , and asked yourself and even tried few of them and see if u can solve thm?

[polar tusk]

till 3-4 i can manage, as the stuff of volume aint taught to us in grade 12 nor competitive exams

but when i learn them i can definetly try

[glacial jacinth]

so u would gonna skip them for now and focus on ur comming competetive exams now?

[polar tusk]

yeah definetly

i am a contributor on MSE. Are you a contributor too @glacial jacinth ?

[glacial jacinth]

k , i will not encourage u ..... enjoy! but sure try them one day !

[glacial jacinth]

wha?

[polar tusk]

Math Stack Exchange - MSE. https://math.stackexchange.com/

[glacial jacinth]

o got it

[glacial jacinth]

umm, Why solve questions from this? ain't book r good than these Amthematics websites?

[polar tusk]

no reason for that. i would like to contribute my knowledge there, and learn more in-depth, you know. Books are okay, but MSE seemed as a way to proffesionalize math and related fields

[glacial jacinth]

i never studied mathematics from webs ,expect sometimes for calculus , but moslty studied less from them , ...

[glacial jacinth]

its jsut a name

If u can't even solve those pdfs questions then how this

sorry i m not encouraging u

But dont be too obessed over a platform

if that platform tomorrow got crashes , thn?

[polar tusk]

i never partake in every problem; my level of problems do come up there and i scrape away by doing them.

[polar tusk]

your point is true.

[glacial jacinth]

kk,ik

But bro , in which subject u being loved the most with?

remove any exams and marks in ur mind

[polar tusk]

i had seen people gaining less rep than me by contributing yesrs upon years(i joined back in Nov 25') (def they are inconsistent or work lesser than me)

[polar tusk]

definetly math. hence MSE

i know that all these sites are going to get busted one day or the other, due to the advent of AI, but I never believe that point to be true

and the signs were seen back from 2018 when the ques coming on the site became low

[glacial jacinth]

Sorry i didnt understand your this sentence perhaps

[glacial jacinth]

r u been in this site over 5+ years?

[polar tusk]

leave it; it was a error on my end

[polar tusk]

no. I joined back in nov 25'

i read the analytics and meta of the site hence knew about it

[glacial jacinth]

k

[glacial jacinth]

So which exams r u gonna contribute to?

or i mean would likely to be a participant in?

[polar tusk]

in local engineering exams for CSE and AI

maybe even electronics too

[glacial jacinth]

So u love Cs and maths at the end

thats good

k

[polar tusk]

tell me about youself @glacial jacinth

[glacial jacinth]

and maybe physics ryt?

[polar tusk]

not to a high degree but somewhat ig

[glacial jacinth]

SO u love finnaly wht ?

its ambiguous

[polar tusk]

basically math

[glacial jacinth]

technically makes sense

[polar tusk]

cs is also math and coding

[glacial jacinth]

yeah ,most probably , but not whole

almost like 40% maths , and 60 % is tht own field(that particular subject's field) logics and topics ,just being done by the help with maths!

[polar tusk]

somewhat ig

[fleet mauve]

@polar tusk How many did u solve

[polar tusk]

i didnt start yet lol; will report back with progress when i do

[polar tusk]

shall i dm you?

[glacial jacinth]

k , but for ?

[pseudo heron]

speaking of 7, TREE(3, 7) =

in ATN ofc . .

and TREE(7, 4) =

[fleet mauve]

@polar tusk U havent reported yet

[polar tusk]

aint that free uk

will try to do em and reply

but dont expect it within a week or so

maybe like a month

[fleet mauve]

👍 👍

[polar tusk]

i am extremely busy in my own works and exams

so will take a long time to do em completely