help-39

[viscid shale]

We usually dont write it like that, but there isnt nothing wrong with it

you were right with this one

[lean plume]

ok

so basically

after

we gotta multiplby

eq 2 x 2

@viscid shale

[viscid shale]

mb

[viscid shale]

You probably want to define B or P in terms of the other

[lean plume]

ok

can you explain trigonometry to me

[viscid shale]

No clue what that has to do with the current problem, but yeah, go ahead

[pearl pond]

@lean plume Has your question been resolved?

[lean plume]

bro im so cooked bro

im under so much pressure rn

i dont understand trigonometry AND the topic we just did

AND i have a math exam

@viscid shale

[viscid shale]

what do you need to learn, trig functions and geometry?

[lean plume]

no

trigonometry and simultaneous equations with worded problems

https://youtu.be/PUB0TaZ7bhA?si=xdYEuI2YICP0hEFF

im watching this rn but dont understand jack SHIT

@viscid shale

[viscid shale]

oh, that

[lean plume]

yes

yo lowkey

@viscid shale

would i be good if i just memorize the formulas

[viscid shale]

if you know how to apply them, maybe

You got

Rule of Sine and Cosine

SOHCAHTOA

[lean plume]

ye

[viscid shale]

Pythagoras

Adjacent and Opposed Angle rule

Sum to 180

And prob a few more related to triangle centers

[lean plume]

how to do

@viscid shale

[pearl pond]

@lean plume Has your question been resolved?

[pearl pond]

@lean plume Has your question been resolved?

[pearl pond]

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[hasty junco]

How do i do number 1

[plush bramble]

Plug the expansion of sin(a-b) into 1/sin(a-b)

[hasty junco]

Is this the answer?

or can it be simplified further

[plush bramble]

Yes

[plush bramble]

Can't tell without your work

[hasty junco]

Is it simplified to those?

*this

[plush bramble]

This looks right

[pearl pond]

@hasty junco Has your question been resolved?

[pearl pond]

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[versed remnant]

Can someone help me out with this ode :p
y''-4y=(x²-3)sin 2x?

[versed remnant]

I found the yc thingy, but for yp I let
Yp = (ax²+bx+c)(sin2x + cos2x)

Then Yp'' = (ax²+bx+c)(-4sin 2x -4cos 2x) smthing anything

Now , here we get 2 terms

[sharp vigil]

you need a separate polynomial for each i think

[limber nimbus]

you should put in coefficients for sin and cos

[versed remnant]

-8Ax²sin2x and -8Ax²cos 2x Which is distributing me

[limber nimbus]

yeah in general you just need more terms

[versed remnant]

If I multiplied both brackets , I would sill end up with 1 coefficient , so I didn't think it would matter

[limber nimbus]

No I mean sin and cos need their own polynomials

those won’t necessarily factor the way you’ve written it

[versed remnant]

I'm lost

[limber nimbus]

right now you have p(x) [ sinx + cosx ]

[versed remnant]

Ye

[limber nimbus]

but that assumes both sin and cos have the same polynomial

That could very well not be true

The proper form is p(x) sinx + q(x) cosx

this does not reduce to your form, because p(x) and q(x) don’t have to be proportional

[versed remnant]

what would I assume q(x) to be

[limber nimbus]

the same quadratic

Just with diff coefs

So you would have 6 coefs total

[versed remnant]

This seems it would be lengthy af

[limber nimbus]

yeah it will be unfortunately :(

[versed remnant]

Ok I'll try it that way D:

Wait is that the case for all sin and cos?

Cuz the professor told us that this way would be fine, I think

[limber nimbus]

this might be more straightforward with laplace transforms but I don’t know if you’ve seen those yet / are allowed to use them

[versed remnant]

Ok , got it

I wonder if I can salvage what I have written

[versed remnant]

This exercise is before that

Ty both

.close

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[flat raven]

Hello, in my problem, there is a magician and people choose a random 9 digit number. The magician then tells them a set of instructions, of which he may repeat intructions or not repeat any at all. The trick is that no matter what 9 digit number these people choose, they will always get the number 2025. The instructions are: a) add 2 to the number. b) add 5 to the number. c) substract 2 from the number. d) substract 5. e) multiply the number by 2. f) multiply by 3. g) multiply by 5. h) take all digits of the number and reverse them, for example 456 becomes 654. i) cross out the last three digits of the number. What I've tried: so far, I've tried crossing out the last three digits two times until I get a number in the hundreds, but unless that number literally ends up as 100, I cannot just reverse the numbers and then multiply by 5 until I get the solution. Additionally, I cannot use the add or substract instructions to get a number with two zeroes because these instructions have to apply to all numbers. Considering there are no divide instructions, I think I'm supposed to force all numbers to be 1 or zero at some point, but I have no idea how to. My only other idea is to multiply the number in a way that I get a 6 or 3 digit number that I can cross out, and get zero, but have no idea how to multiply in a way that no number goes over this number of digits.

[flat raven]

At no point can the number be negative. So far, I've tried instructions in this order: i, i, e, g, i. Doing this, I always get a number from 1 to 9. I have no idea how to make them into the same number though no matter instructions.

The number must also be a positive whole number in between all instructions, so that rules out getting 0 as the number where they all meet

[spiral coyote]

9digit no.?

[flat raven]

yes

so it cannot begin with 0

[spiral coyote]

And you have to do what?

Order those instructions in a way that you always get 2025.

[flat raven]

Yes

[spiral coyote]

Repetition allowed?

[flat raven]

yes, and not all instructions have to be used

[spiral coyote]

Ohkayy thtts trickyy

[flat raven]

I think I'm getting closer though. I got to the point where I have numbers from 1 to 9, which I multiply by 5, which I then multiply by 2 to get numbers 10 to 90, then add 5

which gives me numbers 15 to 95, all numbers ending in 5

although that might be useless but if I change the digit order and multiply by 2 and five three times and cross the last three numbers, I should get five?

I'm not sure if it's foolproof always

[pearl pond]

@flat raven Has your question been resolved?

[spiral coyote]

Will multiplying by 2025 do anything?

[flat raven]

I'm not sure, there are multiple solutions but I actually just managed to solve it so this is closed!

[pearl pond]

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[gleaming musk]

im trying to prove this theorem (to learn it ig, its not obvious to me why it would be true immediately) but the only natural intuition to me is to extract a decreasing subsequence like I would in R, but obviously trying this doesn't work here.

Could someone point at how I would go about this, or even just some basic intuition

[gleaming musk]

my idea is to somehow show that we can extract a subsequence such that for some index N, all the terms in the subsequence from that point onwards have a "good enough" approximation by their first N-terms in the expansion a_k p^k such that truncating the leftmost digits afterwards doesn't affect it

[tropic saddle]

a stackexchange comment says this:

This can be done very much as you would show using decimal expansions that every sequence of real numbers between 0 and 1 has a convergent subsequence, using Cantor diagonalization for example.

[gleaming musk]

i haven't done decimal expansions or cantor diagonalisation (altho i dont think the latter really is required since they say 'for example)

But im under the impression the proofs for convergent subsequence from a sequence is generally done by taking 'peaks'

which was what i tried to do but that doesn't seem to work for p-adics

[tropic saddle]

a different classic way to prove bolzano weierstrass is bisection

[gleaming musk]

would i "p-sect" in this sense then?

like partition into p-disjoint sets

then in the same way (at least from how i remembered it), using pigeonhole principle at least one of these sets must have infinite elements

at least intuitively this still seems to make sense
because every time im "p-secting" im fixing more and more of the lower index a_k p^k terms to agree

ok ig i will try my best to put it down into something a bit more sensible,

because 0<= a_0 <= p-1, theres p-choices
then by PHP, at least one of these choices for last digit a_0 must have infinite elements and we form our first subsequence from that.

Then in the same way, a_1 has p-choices in the same way and again by PHP we must be able to extract an infinite subsequence again, then just repeating this iteratively we must be able to extract an infinite subsequence that agrees up to any arbitrary amounts of coefficients and this infinite sequence of digits must define a single p-adic number that is the limit the subsequence is convergent to?

[gleaming musk]

well maybe it makes more sense to build the subsequence by taking a random element from our new subsequence at each step of the iteration so that x_k has k-1 digits agreeing and the k-th digit just being arbitrary

[toxic lichen]

yeah that's basically it

[pearl pond]

@gleaming musk Has your question been resolved?

[gleaming musk]

Having some formal language definitely helps a bit more as well, wasn’t familiar with residue classes till I saw ur message so thanks a lot

🙏thx

[pearl pond]

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[brazen blaze]

Hi

[brazen blaze]

how do i calculate the reaction time to avoid my mom slippers

[toxic lichen]

uhhhhhhhhh

[brazen blaze]

😔

im literally hiding rn

[toxic lichen]

if your mother is physically abusing you like that, you have bigger shit to worry about than calculations

[brazen blaze]

she is hunting me with hr slippers

[toxic lichen]

but also im sorry you are being put through that at all

[brazen blaze]

.close

[pearl pond]

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[inner granite]

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

@brazen blaze

But, since air resistance is still considered...

[brazen blaze]

i mean her slipper defying the physics..

[inner granite]

I think you might need to do more math.

[brazen blaze]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[inner granite]

I'm sad to hear that. I'm praying for your survival.

[toxic lichen]

trust me, the slipper is not defying physics. your mother is just abusive enough to hurl it at high speeds.

[brazen blaze]

i need someone to help me asap

😭

[toxic lichen]

we can't help you, we are literally thousands of km away

[inner granite]

Is it soft or rough?

[toxic lichen]

sorry

[inner granite]

I meant is she really intending to hurt you or are you really hurting?

I can't do anything tho...

[random ermine]

@brazen blaze are you being serious or is this a joke

[brazen blaze]

no

im serious

fucking serious

brb i gotta switch the place

[spiral coyote]

What Happened

[wheat rain]

I suggest you get external help or contact emergency services if this is a life-threatening situation.

[brazen blaze]

.close

[pearl pond]

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[pearl pond]

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[gilded jacinth]

Hello, how would one go on to solve this differential equation?

[gilded jacinth]

so far, ive split it up like this

[verbal whale]

What about writing 6•1/2 as 3 at least? 😅

[toxic lichen]

nooooooooooooooooo

how COULD you!!!!! that's too unbureaucratic!!!!!!

it doesn't follow correct procedure!!!!!!!

[iron basin]

thats 6 point half

[gilded jacinth]

I thought about this and decided not to do it. Now i have done it

[wheat rain]

wait

why did the 1 + 3t join the big fraction?

[proper nova]

How about writing $3 \cdot t$ as $3t$

[jolly parrot]

1 divided by 0 equals Infinity

[toxic lichen]

why did you decide not to do it

[gilded jacinth]

Its a long story man

[gilded jacinth]

Idk

[toxic lichen]

anyway you're gonna have to expand that bracket on the top

[proper nova]

dani

[wheat rain]

I mean, from the first image to the second. that 1 + 3t should remain outside the big fraction no?

[toxic lichen]

better yet, actually show us where tf this function even came from so that we don't have to wonder why it's in such a stupid form

[gilded jacinth]

I have changed now

[gilded jacinth]

2 sec

[proper nova]

Lol

[proper nova]

use integration

[toxic lichen]

not when the thing is in such a messy form 😭

[gilded jacinth]

holy shit google translate destroyed it

[cedar sentinel]

GT has never been good at preserving mathematical details when translating unfortunately

[proper nova]

At least expand the brackets out first, and then seperate by parts

Like a+b/c = a/c + b/c

[gilded jacinth]

Like this?

[proper nova]

Yeah, expand the brackets

[gilded jacinth]

All of them? Like say( 1/100)+3t/100?

[proper nova]

Expand the 4t first

And then make the numerator non fraction

And then seperate again

[gilded jacinth]

ive gotten it to this

[proper nova]

I mean like: $\frac{\frac{4t + 12t^2}{100}}{100} = \frac{4t + 12t^2}{10000}$ then seperate

[jolly parrot]

1 divided by 0 equals Infinity

[gilded jacinth]

i see

[gilded jacinth]

do i get this by multiplying the 4t into the paranthese?

[proper nova]

Yep

[gilded jacinth]

Ok

[gilded jacinth]

i dont get how it can become 10000

its obviously 100*100, but what allows this to happen?

[proper nova]

a/b is just a * 1/b

So the LHS is just $\frac{4t + 12t^2}{100} \cdot \frac{1}{100}$

[jolly parrot]

1 divided by 0 equals Infinity

[proper nova]

That's why

[gilded jacinth]

I see

So what do i do with this now?

Do i start integrating?

[green spoke]

If u wanna u can separate the fractions too so that they become separate terms

Like $\frac{1}{100} + \frac{3t}{100} - (\frac{t}{2500} + \frac{3t^2}{2500})$

[gilded jacinth]

Is it possible for me to begin integrating now, on this basis?

[jolly parrot]

AviJoe273

[green spoke]

Ofc
Very

Integrate them separately, integrate them under one banner, put the coefficient inside, outside, the result is the same

[gilded jacinth]

What rule should i follow to integrate the fractions that are not 1/100?

[jolly parrot]

AviJoe273

AviJoe273

[gilded jacinth]

@green spoke yea

[green spoke]

@gilded jacinth

[green spoke]

K

So for any $\int{t^n dt}$ where $t$ is your variable and $n$ is a constant,
The integral results in $\frac{t^{n + 1}}{n + 1}$.

[jolly parrot]

AviJoe273

[green spoke]

Other than if n = -1 well that turns out to be the natural log of $t$, also known as $\ln{t}$

[jolly parrot]

AviJoe273

[green spoke]

Got it?

[gilded jacinth]

Okay, thank you

Yes

[green spoke]

Noice

[green spoke]

Yw

[gilded jacinth]

I think this is it, for now

.close

[pearl pond]

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[swift whale]

confused on part a but i can do part b

[swift whale]

how do i find part a

[pearl pond]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[swift whale]

1

[limber nimbus]

we can think of line integrals of vector fields as “dot products” of the field against <dx, dy, dz>

What would the field be then

[swift whale]

soo the vector field would be like <y,x,1> ?

[limber nimbus]

-x but yup

[swift whale]

right

thank you

.close

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[hushed nimbus]

Hi, I was wondering if someone could help me confirm whether the answer to this problem is zero. I wasn’t sure how to approach it during the exam, and now that I’m reviewing it, I think the answer might be zero.

[dense jasper]

,w limit of (3x^2+x)/(2x^2+5x^3) as x approaches infinity

[jolly parrot]

Results provided by WolframAlpha

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

Requested by civil_service_pigeon

[dense jasper]

you can check like this in #bots

[pearl pond]

@hushed nimbus Has your question been resolved?

[pearl pond]

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[blissful brook]

for b

[blissful brook]

why wouldnt this work

nvm i see the mistake

.close

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[craggy bone]

It is 0 .

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[plucky carbon]

Can someone explain where to go from here

[plucky carbon]

sorry for my handwriting if you need clarification just ask

here are my options I guessed and got it right based off what I have but I want to know why

<@&286206848099549185> please help a brotha out I'm just trying to understand where the 2 is disappearing to

[prisma kernel]

Can you please send the question from where it was?

[prisma kernel]

Alright

Wait why is 2 outside

[plucky carbon]

cause I do sqrt of 4 * 11 and sqrt of 4 is 2

[prisma kernel]

it's (-b +- D)/2a, not -b +- (D/2a)

[plucky carbon]

so it goes on outside

[prisma kernel]

No I'm talking about the -b

[plucky carbon]

cause that's part of the quadratic formula

[prisma kernel]

Try to write out formulas before using them, that helps

[plucky carbon]

I have a picture of it

[prisma kernel]

write it and send me an image please

[prisma kernel]

alright send it please

[prisma kernel]

Alright, now substitute your values here, and remember to extend the fraction to -b too

[plucky carbon]

I just did and I don't see whats wrong with my original

[prisma kernel]

Okay see

give me a minute

[plucky carbon]

ok

[prisma kernel]

This is basically what you're doing

[prisma kernel]

instead of this

[plucky carbon]

its the same except the d ? what does d represent

[prisma kernel]

Ahhh, well where I'm from we take b^2 - 4ac as D and solve it separately

[plucky carbon]

haven't seen that i've been learning off yt videos

specifically this one https://youtu.be/kW1uzR7fOdA?si=bPfgyq_kwG6SDzvq

[prisma kernel]

No problem man, but do you get your mistake?

[plucky carbon]

I truly don't I followed the formula I don't see what I'm missing you said that you solve d separately but what does that mean separately from the denominator ?

am i cooked ? ts is supposed to be basic

[prisma kernel]

No no try to get what I'm saying

[prisma kernel]

This

is what you're doing

[prisma kernel]

This

is what you're supposed to do

you get it

[humble root]

What doctor is trying to say is that your working out is right, but there is an issue with the way you wrote the quadratic formula.

[humble root]

If you look at your formula here, notice that everything is over the 2a.

[humble root]

Is this the case in your working out?

[plucky carbon]

yea

[crystal dew]

you neglected to put -b in the fraction as well.

[humble root]

[crystal dew]

your 2 is outside the fraction when it should not be

[plucky carbon]

ahh I see

this doesn't really answer my question though since I was already doing this

my question was where does the 2i go when I divide 6 by everything in the numerator

[plucky carbon]

cause the 2i isn't in the options so its going somewhere

[humble root]

How do you simplify 2isqrt(11)/6?

[crystal dew]

realize that $\frac{2i \sqrt{11}}{6}$ is the same as $\frac{2 \sqrt{11}}{6}i$

[jolly parrot]

Hanako(x, y); ∂(fox)/∂x

[crystal dew]

can you continue?

[plucky carbon]

yes 2/6 is 1/3

[plucky carbon]

I didn't know this thanks

[crystal dew]

you can always split up a fraction into multiple fractions multiplied together so long as they multiply back to the same thing

[plucky carbon]

ok I think I see now but why does the denominator get turned to 3 as well on the right side cause you have 1/3 + or - the sqrt of 11 / 3 does it just automatically change when there is a fraction or something ?

[crystal dew]

sorry, I don't quite get what you mean

[plucky carbon]

here why does the numerator change from 6 to 3

[plucky carbon]

cause we have it as 6 here

[crystal dew]

you have 2 in the numerator

[plucky carbon]

sorry I meant denominator

[crystal dew]

what's 2/6?

2/6 = 3?

[plucky carbon]

1/3

[crystal dew]

there we go

[crystal dew]

you noticed it from the start

[plucky carbon]

yeah but thats just the left side

[crystal dew]

there's a 2 and a 6 as well on the sqrt term!

[crystal dew]

check:

[plucky carbon]

I think I'm just a monkey and complicating it thanks for your help !

[crystal dew]

nps

[plucky carbon]

.close

[pearl pond]

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[pearl pond]

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[worn finch]

do u know how this person lost 3 marks

[compact ridge]

$\frac{3}{3 \sqrt 3} = \frac{1}{\sqrt 3}$ or you could write that as $\frac{\sqrt 3}{3}$ also

[jolly parrot]

south

[worn finch]

but that cant lose u 3 marks

for one mistake

[compact ridge]

oh, that's a major misreading of the question

it's the angle in between A and B

[worn finch]

YEP I FIGURED IT OUT

[compact ridge]

no worries!

[worn finch]

wait i am wondering

after u find the angle

what do i do next ( i know the side lengths as well)

will there be 2 sets of sin/tan we need to find

[compact ridge]

you can just take sin and tan of that angle

your diagram is actually on the right track, given you know the two other angles on the straight line already

[worn finch]

its a right angle tho

[compact ridge]

yep

[worn finch]

for the sin the opposite would be the hypotenuse?

[compact ridge]

yes

[wild fable]

south w the clutch

[worn finch]

wait what 😭 i did not know this was possible

@compact ridge then would tan be undefined

[compact ridge]

yep!

you'd get something like o/a = 1/0

[worn finch]

there wouldnt be any a possible right

because its a right angle

[compact ridge]

yeah and that's the same thing as adjacent = 0

[royal timber]

how does that make sense shouldn't it make a triangle with 8 and 6

[compact ridge]

it's the angle in between those 2 vectors

[worn finch]

I see what u meannn

[compact ridge]

you actually don't need trig

you can recognise that y = 1/sqrt(3) x and y = -sqrt(3) x are perpendicular

slopes multiply to -1

well for the last part you kind of need trig but yeah

[worn finch]

😭 this lowkey frying my brain

[royal timber]

could u explain how u got the angles

is it like sin 90 = 1

[compact ridge]

properties of 30-60-90 triangles

[pearl pond]

@worn finch Has your question been resolved?

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[pearl pond]

Please don't occupy multiple help channels.

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as this action is irreversible, and this channel may abruptly lock.

[pearl pond]

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[lyric helm]

Simon is due north of a tall totem pole and walks 75m, 40º west of south. From his new position, the totem pole is due east, with an angle of elevation of 17º. Determine the height of the totem pole, to the nearest tenth of a metre.

could someone help me draw the diagram? im a bit confused on the due east and west of south stuff cuz i havent taken physics yet

[strong relic]

what have you tried

[lyric helm]

um I drew this but this doesn’t look right at all so idk

[strong relic]

do you know what north south east west is

[lyric helm]

yes

[strong relic]

so what does 40 degrees west of south mean

[lyric helm]

no clue i just dont get it means by west of south

[strong relic]

assuming north is upwards, south means pointing down

west is to the left

[lyric helm]

ohhh

[strong relic]

so 40 degrees west of south just means 40 degrees to the left pointing down

[lyric helm]

is it like this?

[strong relic]

40 degrees from south

[lyric helm]

oh

so does west of south mean from south to west

[strong relic]

yes

[lyric helm]

ohh

and what does due north and due east mean

like directly north and east?

[strong relic]

yes

[lyric helm]

ok but what does it mean when it says due east with an angle of elevation of 17

[strong relic]

imagine youre standing on the ground

and the totem is to your right

if you tilt your head 17 degrees up then you will be able to see the top of the totem pole

[lofty spoke]

chicken butt!

[lyric helm]

ohh

[lyric helm]

is it supposed to look like this?

oh wait sorry

[strong relic]

make sure you read the question

he walks 75 meters

[lyric helm]

is it the line for 40º

thast 75

?

[strong relic]

yes

[lyric helm]

ok so do i just draw a line from the end of that line to the top of the totem

wait what

ok nvm im confused

[strong relic]

so he walks 75 meters along the line you drew

the totem is directly to the right from the new position

[lyric helm]

ok wait sorry idk what u mean

[pearl pond]

@lyric helm Has your question been resolved?

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[sharp smelt]

Would like my answers to part b and c checked

[sharp smelt]

for (b), we need to show E((X_n)^r)≠0; r≥2

[toxic lichen]

ok so calculate $E(X_n^r)$ lol

[jolly parrot]

Ann

[sharp smelt]

ye, just wanted to be sure that's all I had to do

ie there's no catch

for (c) , 1/n^2 ->0

.close

[pearl pond]

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[terse mountain]

67

[toxic lichen]

<@&268886789983436800> troll

[slow oak]

don't shitpost in help channels thanks

[pearl pond]

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[static wolf]

Find function f:Z -> Z such that
f(n+1) = nf(n) +(n-1)f(n-1)

[pearl pond]

@static wolf Has your question been resolved?

[pearl pond]

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[jolly parrot]

Renato

[bitter herald]

any ideas?

[stoic imp]

no

[bitter herald]

ok do you know how to find the critical points

as a first step

[stoic imp]

no

[bitter herald]

the critical points are where $f_x = 0$ and $f_y = 0$

[bitter herald]

so, begin with computing $f_x = \pdv[f]x$ and $f_y = \pdv[f]y$

[stoic imp]

so x = 0 and y = 3

@bitter herald

[bitter herald]

ok nice

now classify the critical points using the second derivative test or through observation

do you know how to do that

[stoic imp]

no

[bitter herald]

compute f_xx and f_yy

and f_xy too

[stoic imp]

can you explain why

@rotund ferry

[rotund ferry]

No sorry I misread just now

[stoic imp]

you can check with wolfram

everything should be correct unless i had a brain fart

[rotund ferry]

Seems all correct to me

Well I think you should know how to figure out whether a critical point is an extrema or minima based on the 2nd derivative

[stoic imp]

you think wrong

if i would know everything i wouldn't be asking for help, no?

[rotund ferry]

Okay calm down

If (x, y) is an extrema, what do you think the second derivatives will be?

[stoic imp]

im am calmed, i just don't know what is so unreasonable about not knowing

[stoic imp]

idk

[rotund ferry]

What would it be in 1 dimensional calculus

[stoic imp]

either positive or nega

[rotund ferry]

What exactly tho?

[stoic imp]

you mean the second derivative evaluated at the critical points?

[rotund ferry]

Yes

[stoic imp]

well what do you want me to say

if f(p) = 0, then f'(p) < 0 or f'(p) > 0

a turning point

the second derivative test idk what it measure maybe convexity instead of concavity

[rotund ferry]

It measures whether it is concave or convex yeah

[stoic imp]

what

[rotund ferry]

Have you never learnt about optimization in calc 1?

[stoic imp]

yes but i kinda forgot

[rotund ferry]

Then you should go and review it

[stoic imp]

preuni was dark days

[rotund ferry]

Especially about the second derivative test

[stoic imp]

why

[rotund ferry]

Cause you don't know what the second derivative test even is when you're already in fudging multivar?

[stoic imp]

at the end of the day single variable intuition is nothing when you have multiple directions

[pastel umbra]

Try to graduate from "intuition"

[pastel umbra]

You need to be familiar with this from a definitional point of view more than just from a feeling

Else MVC is gonna compound on you

[stoic imp]

alright thank you for the help

instead of explaining it you guys prefer to rub it on my face

[pastel umbra]

In fairness to you @stoic imp , I believe the wrong terminology has been used

"extrema" are either "minima" or "maxima", which is what the second derivative is testing for

[rotund ferry]

Oops just realized that typo

[stoic imp]

everytime i open a channel on mvc i get kids saying i get back to calc1 unless is a proofy exercise

[pastel umbra]

Now, in the univariable case, it's pretty simple to get, so let's just check this to make sure you've got it down

Suppose I have a function f(x), and I find a critical point at h (testing f'(h) = 0)

How then do I check what sort of critical point it is, using the second derivative?

[stoic imp]

can you just say what the second derivative test is

[pastel umbra]

Without trying to sound mean here, genuinely, this [the second derivative test for single variables] is from Calc 1

Are you unfamiliar with this, though? /gen

Cos we can go over it rl quick

[stoic imp]

we already got that is from calc 1

[stoic imp]

yes without the rubbing would be awesome

[pastel umbra]

Find the second derivative, and evaluate it at this point x = h

If f''(h) > 0, this graph is trying to turn upwards, meaning this crit. point is a local minimum

If f''(h) < 0, for the opposite reason, this is a local maximum

[stoic imp]

what?

what does the first derivative does then?

[stoic imp]

i thought this was first derivative test

[pastel umbra]

The first derivative was just to check the slope itself, which is how the graph behaves

The second derivative is checking how this slope behaves

[stoic imp]

wdym

[pastel umbra]

oh ffs this is a clear png

[stoic imp]

what about the first derivative test

[pastel umbra]

The first derivative's just to find where this point even is

[stoic imp]

so this is wrong

[stoic imp]

i do remember using this quite a lot, i just don't understand first derivative test

[pastel umbra]

Matcha didn't disagree with you here, for the record - idk why you then said "what" in response, though 😅

[stoic imp]

because i don't remember what concave vs convex

was

[pastel umbra]

Though, you're testing for p where f'(p) = 0, and checking whether f''(p) < 0 or > 0

-# recall that f(p) = 0 is how you solve the roots of f, not the critical points

[stoic imp]

🤣

i see, ya

[pastel umbra]

nice - is this then clear why the second deriv. is useful at all though?

We're gonna need this understanding to make the MVC case make sense

[stoic imp]

yes

[pastel umbra]

cool

Now, the MVC case with two variables... well, we've got two directions to have to differentiate in, don't we

the x-direction, and the y-direction

[stoic imp]

well also xy/yx direction or what

[stoic imp]

i guess? idk like now we have partial derivs

[pastel umbra]

lemme check rl quick, one sec

[stoic imp]

shit got nasty

[pastel umbra]

ye

We're gonna bump this up; we're still gonna apply the same principle, but there's now no one "second derivative" of a bivariate function

imma just paste this here a moment...

[stoic imp]

yes

[stoic imp]

this

the det of the hessian

[pastel umbra]

ye

You can check why later, but for now let's just accept that we use this D function in place of our f''

[stoic imp]

we need a positive determinant and fxx determines if it's local max or local min

Hopefully determinant of the hessy it's positive definite

[pastel umbra]

hessy

You'll need all three second-order derivatives (fxx, fxy and fyy)

[stoic imp]

I computed them

[pastel umbra]

Which I think you acc do have

ye

[stoic imp]

we evaluate them at the critical point

or what

(x,y)=(0,3)

[pastel umbra]

ye

let's use this formula then

(meaning no offence here, but imma trust your computation of this result btw ^)

[pastel umbra]

Nah I don't doubt you here

[pastel umbra]

(I just mean it's rly cold here and i wanna get some art done atm lol, so I'm trusting your functions are correctly diffed lol)

[pastel umbra]

I can help with the conclusion

[stoic imp]

p = (x,y) = (0,3)

,w 8e^(x^2+y^2-6y) + 16x^2 * e^(x^2+y^2-6y) at x = 0 and y = 3

[jolly parrot]

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[stoic imp]

,w 16xye^(x^2+y^2-6y) -48xe^(x^2+y^2-6y) at x = 0 and y = 3

[jolly parrot]

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[stoic imp]

,w (e^(x^2+y^2-6y))*(8+y(2y-6)-24(2y-6)) at x = 0 and y = 3

[jolly parrot]

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[stoic imp]

fxx(0,3) = 8/e^9
fxy(0,3) = 0
fyy(0,3) = 8/e^9

,calc 8/e^9

[jolly parrot]

Result:

9.8727843269344e-4

[stoic imp]

this is bad @pastel umbra

[pastel umbra]

Do you really need to evaluate this with decimals?

[stoic imp]

xd

[pastel umbra]

Here, use your algebra

[pastel umbra]

All we need to do is check for positivity, negativity or zeroness

[stoic imp]

,w -(8/e^9)^2

[jolly parrot]

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[stoic imp]

det of H(0,3) = is -9.7 x 10^(-7)

(0,3) is a saddle point

[pastel umbra]

Your conclusion from that seems right

[stoic imp]

but

[pastel umbra]

Your derivatives might be wrong, though

[stoic imp]

what about the relative extrema

[stoic imp]

do you got something to believe they are wrong

[pastel umbra]

cos the graph itself looks like a minimum

[stoic imp]

the graph itself?

[pastel umbra]

(0, 3) refers to the bottom point of this "cup"

ye

[stoic imp]

looks like a minimum?

[pastel umbra]

Well I mean this (circled) doesn't look like a saddle to me

,w Hessian of f(x,y) = 4 exp(x^2 + y^2 - 6y)

[jolly parrot]

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[pastel umbra]

,w 64 e^(2 (x^2 + (y - 6) y)) (2 x^2 + 2 y^2 - 12 y + 19) at x = 0, y = 3

[jolly parrot]

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[pastel umbra]

64 / (e^18) is what you should have for D(0,3)

[stoic imp]

I got this

[stoic imp]

.

[pastel umbra]

...ah - your minus is in the wrong place

It should be inside the fraction, I reckon

[stoic imp]

oh ye

nice reckon

whatever that means

[stoic imp]

alright okay 👍

[stoic imp]

,w (0 - 8/e^9)^2

[jolly parrot]

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[pastel umbra]

(I reckon that = creo que)

[stoic imp]

got it

alright okay so

[pastel umbra]

Right so, D is positive

Check fxx

[stoic imp]

.

fxx(0,3) > 0

[pastel umbra]

ye

D > 0 and fxx > 0

So we have a...?

[stoic imp]

local minim

[pastel umbra]

yeee

[stoic imp]

now what

[pastel umbra]

wdym now what

[stoic imp]

what about the saddle points

did we prove none exist

[stoic imp]

interpreting the results

[pastel umbra]

ya hemos terminado, no?

[stoic imp]

what about b)

[pastel umbra]

oh wait that was (a), shit

[stoic imp]

did we proved none saddle point exists

[pastel umbra]

ye

[stoic imp]

yes or no

[pastel umbra]

cos the only crit point gave us a minimum

[stoic imp]

then we are done with a)

cant I get help with b)

[sonic patrol]

What is b

[last summit]

aqui

[stoic imp]

this ^

[pastel umbra]

this one

lool

"Determine, if they exist, the absolute extrema of f restricted to A [defined]"

[stoic imp]

I appreciate the help with a) and with explaining the second derivative test without any rubbing

[pastel umbra]

well, this is the second restriction

er....

[sonic patrol]

Ah cool

You can split into interior and boundary I guess

[stoic imp]

idk if we need lag multi for dis

[stoic imp]

care to elaborate

[pastel umbra]

-# I've pointed this out to you before that that phrasing can come across as rude

[stoic imp]

boundary like the frontier

[pastel umbra]

I mean, the minimum is still there, so we have that

That is what boundary means, yes

[sonic patrol]

So it’s a generalisation of how it works in 1D

[stoic imp]

what?

[sonic patrol]

Suppose you want to extremise a function defined on a closed interval [a, b]

[pastel umbra]

This - this can come across as rude; please refrain from phrasing your confusion like this.

[stoic imp]

okay, so then what would be the procedure in single variate

[stoic imp]

I dont see why it can come as rude

[pastel umbra]

Es como "te importa explicarlo?"

[sonic patrol]

One way you can be an extremal point is if the derivative is 0

[stoic imp]

is more like te importa explicarte

[sonic patrol]

But another way you can be an extremal point is if you’re at “a” or “b”

For example, say you’re extremising f(x) = x^2 on [-2, 2]

Then f’(x) = 2x, and that’s 0 only when x = 0

So the only critical point is x = 0, where f(0) =0

[pastel umbra]

Point is, it's ruder than e.g. "podría explicar más" - "Could you explain in more detail?"

[sonic patrol]

And that is the minimum, sure

But the maximum occurs at the ends of the interval, with x = 2 or x = -2

The derivative there is nonzero, but it’s still a maximum point

[stoic imp]

yes, -2 and 2 are candidate for critical points is what you saying, irregardless if f'(2) or f'(-2) = 0

[stoic imp]

yeah, I see what you mean, how does this intuition relate to 3d space

[sonic patrol]

Yep!

[stoic imp]

I will think it through but I dont see why people would get offended from caring to elaborate

the problem here is that we have two inequalities or intervals

y >= 0 and 2y+x^2 <= 9

[sonic patrol]

So in general, “extremal point => critical point” only works for points in the interior

For points in the boundary it might not be true

[stoic imp]

okay

what is the boundary here

2y + x^2 = 9

y = 0

[sonic patrol]

Yes, either of those equations

[stoic imp]

can I get some handhold

with this exercise

[sonic patrol]

So, we’re trying to find extremal points/values of f

An extremal point might be in the interior, or it might be on the boundary

[stoic imp]

alright

[stoic imp]

we already found one extremely point in the a) part

[sonic patrol]

Mhm

[stoic imp]

(0,3) is a local minimum of f(x,y)

[sonic patrol]

We should make a distinction between extremal and critical points

Do you know the difference between those?

[stoic imp]

critical points if and only if f'(p) = 0

?

[sonic patrol]

Yes, exactly!

And extremal point?

[stoic imp]

Just f"(p) ≠ 0

????

[sonic patrol]

Nope

An extremal point is a point that is a local minimum or maximum of f

[stoic imp]

this

[sonic patrol]

it’s where the value of f is “extreme”, hence the name

[stoic imp]

yes

all of this highly related to extreme value theorem in single variate

[sonic patrol]

The main theorem we use is that:
“If an extremal point is in the interior, then it is a critical point”

[stoic imp]

alright

now coming back to the 3d genetalisation

how does all this relate to my thingie

[pearl pond]

@stoic imp Has your question been resolved?

[sonic patrol]

So the same sort of thing is true here

[stoic imp]

alright okay

[sonic patrol]

We have $f(x, y) = 4 e^{x^2 + y^2 - 6y}$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

And in both parts, we want to find extremal points of f, right?

[stoic imp]

in both parts, you mean in a) and b)

[sonic patrol]

Yes

[stoic imp]

or interior and boundary

[sonic patrol]

I mean a and b

[sonic patrol]

Because of this theorem, an extremal point is either a critical point or a boundary point

Does that make sense?

[stoic imp]

fair nuff

[sonic patrol]

Now, in part a, we identified all the critical points of f

In fact there’s only one

(0, 3)

[sonic patrol]

Just to check - if we define $A = {(x, y) \in \mathbb{R}^2 : y \geq 0, 2y + x^2 \leq 9}$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

Is $(0, 3) \in A$?

[stoic imp]

well

3 >= 0 ✅

6 + 0 <= 9 ✅

[stoic imp]

yes

[sonic patrol]

Cool, so A contains 1 critical point

What about the boundary points of A?

Said differently, what is $\partial \space A$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[stoic imp]

what?

care to elaborate

[sonic patrol]

It’s a notation for the boundary of A

[stoic imp]

how do I found them

[sonic patrol]

E.g. in 1 dimension, $\partial [a, b] = {a, b}$, since these are the boundary points

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[stoic imp]

2y + x^2 = 9

y = 0

[sonic patrol]

So, A is defined using the inequalities $y \geq 0$ and $2y + x^2 \leq 9$

To find the boundary points, you set one of these to an equality

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

One option is $y = 0 \land 2y + x^2 \leq 9$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

The other option is $y \geq 0 \land 2 y + x^2 = 9$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

Can you try solving these?

[stoic imp]

oof

[stoic imp]

can I get some handhold or minimum guidance

[sonic patrol]

Try this first

[stoic imp]

x^2 <= 9

[stoic imp]

dude

[sonic patrol]

uh don’t call me dude

[sonic patrol]

Mhm, and then solve this

[stoic imp]

srry, miss

but uhh how am I

supposed to

x^2 <= 9

I see

[sonic patrol]

One option is to sketch f(x) = x^2

[stoic imp]

x^2 - 9 <= 0

[sonic patrol]

And see what range of x it is <= 9

[stoic imp]

difference of squares

[stoic imp]

alright I like dis

so x^2 has range [0, ∞+)

[stoic imp]

Just 0 <= x <= 3

[sonic patrol]

nope!

[stoic imp]

right, miss?

[sonic patrol]

You missed some values

E.g. (-1)^2 <= 9

But -1 isn’t in the range [0, 3]

[stoic imp]

yes

-3 <= x <= 3

[sonic patrol]

That’s correct

[stoic imp]

right, miss?

[stoic imp]

alr

[sonic patrol]

You can also do this and obtain $(x - 3)(x + 3) \leq 0$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

For that to work, one has to be >= 0 and the other has to be <= 0

That works for $-3 \leq x \leq 3$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[stoic imp]

alright okay

got it miss, now?

[stoic imp]

.

[sonic patrol]

So, if we say $B = {(x, y) \mid y = 0 \land 2y + x^2 \leq 9}$, then $B = {(x, 0) \mid -3 \leq x \leq 3$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

Yes, so try solving this

[sonic patrol]

This is one part of the boundary of A

[stoic imp]

shit got harder

[stoic imp]

alright I think I can do it tho

[sonic patrol]

good luck!

[stoic imp]

2y + x^2 = 9 and y >= 0
2y = 9 - x^2
y = (9- x^2)/2
(9-x^2)/2 >= 0
9-x^2 >= 0
9 >= x^2

the range of x^2 is [0, ∞+)

we want x^2 <= 9

[sonic patrol]

Yep, and you already solved this, right?

[stoic imp]

yes, that's what I was thinking

[sonic patrol]

You get -3 <= x <= 3

[prisma geode]

sorry but where is the question?

[sonic patrol]

So, the other part of the boundary of $A$ is $C = {(x, \frac{9 - x^2}{2}) \mid -3 \leq x \leq 3}$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

Does that make sense?

[stoic imp]

sure

[stoic imp]

check the pinned message, should be there

[sonic patrol]

Now we have a complete description of the boundary points of A

[stoic imp]

I think so, yes miss

[sonic patrol]

Notice that while A is a two-dimensional region, its boundary is a 1-dimensional curve

[stoic imp]

fair enough

[sonic patrol]

Next, we need to figure out how f behaves on the boundary of A

[prisma geode]

yea i cant read that anyway i think you got the help you need

[sonic patrol]

Let’s focus on the B part - what does $f |_B$ look like?

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[stoic imp]

f restricted to B?

[sonic patrol]

Yes

[stoic imp]

well

[stoic imp]

when y = (9-x^2)/2, x is between -3 and 3

no idea what to say reallt

[sonic patrol]

Oh that’s the C part

The B part is when y = 0

And x is between -3 and 3

[stoic imp]

yes f Harpoon B, <=> y = 0 and x in [-3,3]

something like dat ?

[sonic patrol]

Lemme guide you a bit more

If we take a point in B, it’s of the form (x, 0), where x is in [-3, 3]

So to figure out f |_B, we want f(x, 0) for x in [-3, 3]

So, what is f(x, 0)?

[stoic imp]

4e^x^2

[sonic patrol]

Yes!!

Let’s take a step back for a sec

We started by trying to extremise f on A

For the interior points, we could set f’ to 0

For the boundary points, we had to figure out what the boundary looked like

Now that we’re on the boundary, though, we’ve reduced the problem by 1 dimension

B is a one-dimensional curve, and we want to extremise 4 e^(x^2) on it

So we apply the same procedure

For the interior points of B, we differentiate and set the derivative to 0

For the boundary points of B, we evaluate f

[sonic patrol]

Start with this step

[stoic imp]

well

8xe^x^2

= 0

x = 0 is the crit point

@sonic patrol correct miss?

[sonic patrol]

Yep!

[stoic imp]

well

[sonic patrol]

So (0, 0) is another critical point

[stoic imp]

alright

[sonic patrol]

Then do this step

[stoic imp]

f(3) f (-3)

you mean I evaluate f?

or f"

[sonic patrol]

f itself

The boundary points of B are (-3, 0) and (3, 0)

So they get added to the list of candidate extremal points

So far, our list of candidate extremal points consists of (-3, 0), (0, 0), and (3, 0)

Now you repeat the process for the C part

Find what f(x, (9 - x^2)/2) is

For the interior points you differentiate and set equal to 0

And you also check the boundary points

[stoic imp]

but is harder

miss

[sonic patrol]

Yes but you can do it

[stoic imp]

well

[stoic imp]

how you got this?

oh wait B = {(x,0)∈ R2 | -3 <= x <= 3}

[stoic imp]

sure

extremely points include critical points (0,0)

[stoic imp]

I will try

C = {(x, (9-x^2)/2) ∈ R^2 | x ∈ [-3,3]}

the boundaries huh

so what is f(x,(9-x^2)/2)

help

f(x,(9-x^2)/2) = 4.e^(x^2 + (9-x^2)^2 × 1/4 - 3(9-x^2))

@sonic patrol miss I might need some high level hint here

the process for single variable functions was simple

f'(x) = 0

I see

,w differentiate 4.e^(x^2 + (9-x^2)^2 × 1/4 - 3(9-x^2))

[jolly parrot]

Results provided by WolframAlpha

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Requested by renatoxbz

[stoic imp]

this shit is 0 when x = 0 or when x = 1 or when x = -1

so (0, 9/2), (1, 4), (-1, 4)

are the critical points of f(x, (9-x^2)/2)

[proper nova]

Let me guess. Given $f(x, y) = 4 \cdot e^{x^2 + y^2 - 6y}$

[jolly parrot]

1 divided by 0 equals Infinity

[proper nova]

Your espanol made me confused

What's a) mean?

[stoic imp]

check the pins

here's the translated

[sonic patrol]

Nice

[sonic patrol]

This looks correct!

Now you have the complete list of candidate extremal points

(0, 3) in the interior

For the boundary, (-3, 0) and (0, 0) and (3, 0) and (0, 9/2) and (1, 4) and (-1, 4)

All you have to do now is evaluate f at each point and compare the values

[stoic imp]

did you checked (0,3) is in C?

[sonic patrol]

You checked this here

[stoic imp]

and C is a subset of A?

[sonic patrol]

Yes, in fact it’s a subset of the boundary of A

$C \subset \partial A$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

The idea that we’ve been using is

To extremise a function over A, it suffices to extremise it over $\partial A$ and $\text{Int}(A)$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

And to extremise over $\text{Int}(A)$, you find critical points

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[stoic imp]

wait

[sonic patrol]

So this gives a kind of recursive algorithm for extremising functions

[stoic imp]

boundary of A and interior of A

[sonic patrol]

Boundary of A and interior of A, yes

[stoic imp]

alright

[pearl pond]

@stoic imp Has your question been resolved?

[stoic imp]

ok

[stoic imp]

I think so

[stoic imp]

ok

[stoic imp]

if I do that

then I find the extreme values and find the local maximas and local minimas and global maxima and global minima

[sonic patrol]

mhm

[stoic imp]

I find it tough still

but I think I kind of get it but

like, re arranging the core algorithm, what would be steps?

like, A is what? a restriction?

a set of points

A is a region

[sonic patrol]

so, we have a domain $A$, and a function $f : A \to \mathbb{R}$

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

where A is some set of points, a region as you said

our goal is to extremise f over the set A

we do this recursively as follows:

1. Break up $A = \text{Int}(A) \cup \partial A$, into the interior and boundary

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

2. Extremise $f$ over $\text{Int}(A)$ by setting $f' = 0$ and finding the critical points

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

3. Extremise $f$ over $\partial A$ recursively

[jolly parrot]

Pseudo (Cat theory #1 Fan)

[sonic patrol]

note that in step 3, the boundary of A has dimension 1 lower than the dimension of A

so the problem does get easier as we continue

[stoic imp]

yes it does

[stoic imp]

I appreciate the help miss

[sonic patrol]

[stoic imp]

let me write this down in my notebook, compare the extreme values and evaluate them on f and see which one is the smallest one that will be my global maxima