help-4

[harsh jolt]

And questions like that

[inner mirage]

Yeah it only works when there is exactly 1 root in an interval, emphasis on exactly one

[harsh jolt]

Alr man cool thanks a lot

[inner mirage]

Np

[harsh jolt]

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[vague gulch]

Not sure if this counts as math but this is some chemistry calculations I’m really confused with! 😭 Could I possibly get some help with how I should approach the calculations for the last column?

[vague gulch]

Oh wait I read the rules, this may be off topic

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[faint cave]

Any advice on how to do this problem?

So far I notice that if you connect all the midpoints you either have a parallelogram or trapezoid:

But that is all I got ;-;

I am thinking to consider what happens when angle ACB and angle ADB is 90 degrees,

[vale dock]

@faint cave Has your question been resolved?

[faint cave]

<@&286206848099549185>

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[hasty nest]

I'm solving for the system of equations using substitution because y is already isolated but the quadratic that I got can't be simplified without using the quadratic formula and the video I'm watching for help doesn't tell me what to do if this happens.

[stark wedge]

well then you use the quadratic formula lol

[hasty nest]

I know but idk if I'm allowed to use a calculator with this problem

I also just wanted to double check and make sure what I did so far is right too

[stark wedge]

just leave it in surd/radical form lmao

[hasty nest]

Okay I'll try that

thx

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[pallid smelt]

how would anyone go about doing this?

[neat locust]

Let $z=\left(\left(x^2-1\right)^2-2\right)^2-3$

[rocky lotus]

SWR

[pallid smelt]

yeah then subtract 4

z = +-2

[neat locust]

Not quite. Well, I suppose you could, but not the idea I had in mind

[neat locust]

You mean z, not z²

[pallid smelt]

my mistake

yeah

[neat locust]

But you seem to get the idea

$z=\pm 2$

[rocky lotus]

SWR

[pallid smelt]

what was ur other way of doing it?

[neat locust]

Now, let $y=\left(x^2-1\right)^2-2$, so $y^2-3=\pm2$

[rocky lotus]

SWR

[pallid smelt]

i'm doing that bit rn

yeah since i don't actually need to know the roots but i just need to know how many

[neat locust]

So, what are you getting so far?

[pallid smelt]

i am on 4 roots w x^2 - 1 = 4 of them

and so then u get an extra 2 from that

and hence yk we got 6 roots

so b ig

[neat locust]

six roots huh?

[pallid smelt]

thats what i got..

wait 8

my mistake

[neat locust]

x^2 - 1 = 4 of them
Not sure what you mean by this

[pallid smelt]

8

wait not

[neat locust]

Looks good

[pallid smelt]

i'm an idiot

wait maybe i'm not

[neat locust]

[pallid smelt]

i got 7

not 8

bc one of the roots turned out to be 0

[neat locust]

I haven't verified all the way tbh

Did you get a repeated root?

[pallid smelt]

and +-0 is 0

[pallid smelt]

at 0 i believe it is repeated

so i got 7

[neat locust]

Ah I suppose you are right

good work

[pallid smelt]

yeah cheers

thanks for the help

[neat locust]

you faced a contradicting opinion and held firm

That's worth a lot

[pallid smelt]

i was thinking of doing it that way

but i then like doubted it anyways

but thanks for all the help

[pallid smelt]

yh cheers

have a goodnight

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[ebon thorn]

How do we use Cauchy to prove f(x) = 0 for all x in the interval?

[stark wedge]

do you have to specifically use this criterion?

[ebon thorn]

I don't know any alternatives

[stark wedge]

i can think of a method that's simpler conceptually

[ebon thorn]

Monotonicyty doesn't say anything about what we are trying to show

[ebon thorn]

What's that?

[stark wedge]

ok so you know that integration is monotonic in the sense that if f <= g pointwise then int[a,b] f <= int[a,b] g

right

[ebon thorn]

Yeah

[stark wedge]

so what i recommend here is to go via contradiction: suppose f isn't identically 0. then there's a point x_0 in [a,b] with f(x_0)>0

construct a piecewise linear function that will serve as a lower pointwise bound for f but whose integral will be positive

it may help to try to draw a graph

[ebon thorn]

What about using the epsilon-delta definition?

That seems simplier

[viscid spade]

Contraposition is how id do

[ebon thorn]

Superior way:

[stark wedge]

this literally begins the same way i suggested????

[ebon thorn]

It isn't.

[stark wedge]

suppose there exists c with f(c)>0

they suppose the same thing that i do

[ebon thorn]

Just one piece.

That isn't the same

[stark wedge]

my construction would have proceeded in a similar way, only that i would have made the lower bound function identically zero away from the point

[sharp whale]

ann said that it begins the same way she suggested, not that its the same proof all around

[stark wedge]

rather than constraining to the interval [c-delta, c+delta]

[stark wedge]

yeah but who doesn't like pointing out how wrong Ann is :))))

[sharp whale]

you hear the tone from this guy and youre taking him seriously?

[ebon thorn]

Doesn't matter

[sharp whale]

at least think your words through before you say them

[ebon thorn]

Cope

Still wrong

[sharp whale]

was just about to say youre focusing so hard on being right

Id go with this proof too instead of constructing a piecewise but look at how much youre talked yourself as

[ebon thorn]

I think you need to slow down on the copium

[sharp whale]

please learn to understand people

[ebon thorn]

Mabe subliminate that into something more productive

[sharp whale]

theres one change I could do for this proof

the definition for continuity only really works if domain intersect (c - delta, c + delta) is also an interval

so that we can guarantee a nonzero area over it

presumably we should be able to assume that this interval exists since f is continuous? but

personally I used a different method to doing this which used FTC so its not that much better

@ebon thorn so which direction are you going to take proving this problem in?

[ebon thorn]

Mine

[sharp whale]

do you need help on this problem then

[ebon thorn]

I got my answer

[sharp whale]

.close sounds good

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[tight lark]

help

[tight lark]

hi

so for the vertex form

if it were to be (x + 8)^2

i mean

(x + 6)^2 + 18

would it be

(-6, 18)

[broken orchid]

👍

[tight lark]

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[wooden dock]

Hi

[wooden dock]

Can someone help me to solve

[broken orchid]

What’s the question

[wooden dock]

$x=\frac{12*ln(x)}{ln(8)}$

[oak crane]

\frac

[rocky lotus]

Nicolas Matheisen

[wooden dock]

At this Point I am stuck😭

[lament copper]

you can use base change formula

$\log_ba=\frac{\log_ca}{\log_cb}$

[rocky lotus]

cis(x) = cos(x) + i sin(x)

[lament copper]

where c is a positive number not equal to 1

[wicked zenith]

Maybe use the transformation $y = e^x$ and refactor to get $$e^y = ky : k = \frac{12}{\ln 8}$$

[rocky lotus]

Shuba

[wicked zenith]

might be easier to solve than logs (I hate logs, and love exponentials XD)

[wooden dock]

Ahhhh jeahhhhh 100 points tyyyy

[lament copper]

$\frac{x}{12}=\frac{\ln{x}}{\ln{8}}$

[wicked zenith]

hint: you'll need the lambert W function

[rocky lotus]

cis(x) = cos(x) + i sin(x)

[lament copper]

$\frac{x}{12}=\log_8x$

[rocky lotus]

cis(x) = cos(x) + i sin(x)

[wooden dock]

This is my goal

By the way the Original task is

$8^x=x^{12}$

[lament copper]

oh

in that case

[rocky lotus]

Nicolas Matheisen

[lament copper]

why did you convert to log

[wooden dock]

Cause I try to understand the lambertsche W func

Or I did. it wrong?

I am confused😰

[fickle rose]

this is equivalent to solving (e^{\frac{x\ln2}{4}}=x)

[wicked zenith]

Use y := ln(x)

Then you can rearange into -y e^(-y) = - ln(8)/12 which the Lambert-W function can solve

It's generally a good idea, when presented with an equation of logarithms, to turn those logs into exponentials

logs are terrible to work with imo

[wooden dock]

Perfect ty I got it

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[proud vault]

what up

[proud vault]

this is the hw help a brotha

[west cloud]

!status

[vale dock]

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

[proud vault]

my bad

i think i know what im doing but im not sure if im doing it right

2

[west cloud]

Then can you please show your work?

(Or what your idea is)

e.g. have you found the zeros?

[proud vault]

i found y

idk how to find x unless its the same as the earlier problems

y=-18

[fickle rose]

recall that is a Polynomial has a factor (P(x)=(x-r)Q(x)) then (x=r) is a root of (P(x))

[rocky lotus]

PajamaMamaLlama

[proud vault]

do i replace x w negative 18

[fickle rose]

so we have (P(x)=4(x-6)(x+1)(x+3)) right?

[rocky lotus]

PajamaMamaLlama

[fickle rose]

we have that (x-6), guy right?

[proud vault]

uh yes

is that like part of a data point?

[fickle rose]

that's part of the polynomial

since this guy is multiplied by the rest of the polynomial

then x=6 is a root, I'll give you this one

[proud vault]

is it at all like this?

do i redo the equation with x=6

[fickle rose]

yes

set each factor to 0

[proud vault]

6,-1,-3

and then plot those on x axis?

and plot 0,-18?

on y

[fickle rose]

,calc 4(-6)(1)(3)

[rocky lotus]

Result:

-72

[fickle rose]

how did you get -18?

[proud vault]

oh i didnt do 4 i just did -6x1x3

mb

[fickle rose]

ahh yeah that 4 is very important

but yeah that'll scale is correctly

then you got your polynomial!

[proud vault]

cool thats ez asf

this is fs wrong it doesnt y-intercept at -72

o maybe it does

nvm it was right

the next unit lwk look scary lol

[spring jackal]

i believe in u

[proud vault]

whered did the 0 come from

i lied it doesnt look hard at all

look light tbh

[spring jackal]

you have to include all the powers of x in the division, so they added 0x

synthetic division is pretty light fr

[proud vault]

bc of the 7x^2?

this whole unit has been ez tbh

just takes a while

[spring jackal]

no, like the original polynomial is technically $$-9x^4 + 10x^3 + 7x^2 + \textit{0x} - 6$$

[proud vault]

oh bc its has to go 4 3 2 1

[spring jackal]

yee exactly

[rocky lotus]

haseeb

[proud vault]

can i flip there little diagram

bc you have to subtract upward

right?

[spring jackal]

no harm in writing it a different way, if it makes sense to you

personally i think about it as adding downwards

[proud vault]

i lied they adding its js bc it was negative

[spring jackal]

o

[proud vault]

yea it makes sense i was just being stupid

whered they get 1

oh nvm

if you can’t read my atrocious hand writing lmk

[autumn whale]

hmm

when you're synthetically dividing by x-r, you use x = r

i believe you used x = 2 in this case (which is incorrect)

[proud vault]

so -2?

[autumn whale]

yeah

try that

[proud vault]

cool i got it

i got 2x^2 + 0x + 5 + 14/x-2

how do the variable correlate

[autumn whale]

what

wdym

[proud vault]

whaats my q and d

i

d is -2

r is 14?

[autumn whale]

the quotient is the big answer you get

[autumn whale]

yeah

[proud vault]

wym big number

[autumn whale]

almost not quite

[autumn whale]

the polynomial that you got as your answer

[proud vault]

oh the whole equation under the line

[autumn whale]

your divisor should be a linear term

not just a number

[proud vault]

are they just asking for me to write it in the format its alr in

[autumn whale]

yes 👍

[proud vault]

oh wtf

what do i do with 0x

is it just not included

or do i put 2x+x+5 etc.

[autumn whale]

you dont include it

because 0x = 0 and theres no point in writing an unnecessary 0

[proud vault]

what did i do wrong

no way this is correct

[autumn whale]

seems fine

[proud vault]

yea it was right

what abuot this

[autumn whale]

do you have your work

[autumn whale]

ah

[proud vault]

oh

[autumn whale]

you forgot to account for 0x^3

[proud vault]

its the exponents

[autumn whale]

you need a 0 between your 1 and -2 in your setup

[proud vault]

ohhhhhh

im dumb i messed up 2. things and only fixed one

now i gotta restart😭

[autumn whale]

happens to the best of us

[proud vault]

how do i rotate

nvm

its supposed to be -3

this is really starting to piss me off

[spring jackal]

0 + (-4) = 4?

[proud vault]

if i get this wrong im gonna to punch my monitor

[spring jackal]

this looks good to me idk what could have happened

maybe a typo when inputting? 😬

[proud vault]

how does this work

new section time

.close

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[hybrid gyro]

Can someone please show me how to do this.

[lone furnace]

can you solve a cubic equation

[hybrid gyro]

no I dont know this stuff

pleeeaaasee

[lone furnace]

cool, neither do i

[lament copper]

where is this problem from

[hybrid gyro]

its a practice quiz

[lament copper]

for what unit

[hybrid gyro]

its advanced functions

[lament copper]

um

maybe rational root theorem?

[hybrid gyro]

yes that it

[lament copper]

then because we know the height, we can set h(t) to 980

so we need to solve $980=-2t^3+3t^2+149t+410$

[rocky lotus]

cis(x) = cos(x) + i sin(x)

[lament copper]

just move the 980 to the other side and use rational root theorem

[hybrid gyro]

i know that part but i’m not sure exactly how to do the rational root thearem

is it possible for somebody to send a picture of the answer so i can analyze it and try and figure out the steps

[lament copper]

basically the roots are in the form $m/n$ where m is a factor of 570 and n is a factor of -2

[rocky lotus]

cis(x) = cos(x) + i sin(x)

[lament copper]

and u use guess and check

so for example, one possible root is 2 because it can be expressed as 2/1

and you have to check all of them

until you get one

[vale dock]

@hybrid gyro Has your question been resolved?

[vale dock]

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[gilded horizon]

Honestly not sure where to go from this

[west cloud]

first, remember that the cos is the x value and the sin is the y value

-21/29 isn't quite the y value of that terminal point

[gilded horizon]

wait i mistyped LOL

[west cloud]

mhm

[gilded horizon]

okay and tan would be sin/cos right

[west cloud]

yep!

[gilded horizon]

so how is -21/20 incorrect

[west cloud]

well…your sin value is -20/29 and your cos value is 21/29; what do you get if you divide that?

@gilded horizon

the division is not quite right

[gilded horizon]

im getting -580/609

simplifies to -20/21

.solved

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[west cloud]

minor note: you could also have canceled out 29 from the start

[gilded horizon]

ah i see, thank you for the help

[river shale]

Minor note: you can always reopen a channel if you wanna replenish anything 🙂

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[rough talon]

prove that $6(6a^2+3b^2+c^2)=5d^2$ does not have a nonzero integer solution

[rocky lotus]

skissue.in.a.teacup

[rough talon]

i remember my teacher said something about infinite descent?

[green urchin]

Ideal for this typa shit

It's pretty hectic sometimes tho

[rough talon]

this was the cleanest i got it to, idk how to continue this though

if this just breaks the possibility of infinite descent i wouldnt be that surprised cause i subtituted alot here

[green urchin]

Hmm , you get any common gcd of the fours?

That's insta kill

[rough talon]

wdym

[green urchin]

The gcd

Take any solution (a,bc,d)

Is there any prime or anything dividing all four of them?

[rough talon]

i couldnt get anything of that sort no

[green urchin]

In that case, do you get individual factors of those 4?

[rough talon]

uhh

6|d, 3|c, b and c have the same parity

[green urchin]

No infos on a?

[rough talon]

no

wait hold on i made a mistake

[green urchin]

Look it's clear that d² has a 6 in it, so there should be another 6 of pair there

So 6|d, all good

$2a^{2}+b^{2}+3c_1^{2}=10d_1^{2}$ that's where it boils down to

[rocky lotus]

Licht

[vale dock]

@rough talon Has your question been resolved?

[green urchin]

Aw shucks man

We see $b$ and $c_1$ are of the same parity, let both of them be odd. Then $b^{2}+3c_1^{2}$=$4(mod 8)$.
Now,
$10d^{2}=2$ or $0 (mod 8)$
$2a^{2}= 2$ or $0(mod 8)$ so that's a contradiction

[rocky lotus]

Licht

[green urchin]

@rough talon so b and c both are even

[rough talon]

interesting

[green urchin]

Yes

And it gets down to

[rough talon]

if we can get 2|a then infinite descent works now

[green urchin]

Easier to devise a contradiction here

It's pretty simple looking

[rough talon]

is it not +6c_2^2?

[green urchin]

Oh yeah

True

We had a three there

OH

It's easier now

[rough talon]

$$2a^2+b^2+3c_1^2=10d_1^2$$
$$2a^2+4b_1^2+12c_2^2=10d_1^2$$
$$a^2+2b_1^2+6c_2^2=5d_1^2$$

[rocky lotus]

skissue.in.a.teacup

[green urchin]

Yeah

$a^{2}$and $5d^{2}$ has the same parity

[rocky lotus]

Licht

[green urchin]

d_1 I mean

[rough talon]

maybe you can do the same thing?

[green urchin]

Yeah that's where I'm going to

Maybe infinite decent will work , but I don't have a constraint relating to 5 yet

So they won't factor out either or it's gonna be a hassle

Wait we do have something interesting, we can get back to the equation maybe

[rough talon]

yeah it was simmilar just it needs to be a 3 instead of a 2, and you want a factor of 6 on the lhs

[green urchin]

Can you see that no matter what it is $2b_1^{2}+3c_2^{2}$ is divisible by 8?

[rocky lotus]

Licht

[green urchin]

Three case bash

[green urchin]

6*

[rough talon]

huh?

[green urchin]

Do three case

One is b and c both odd, then b and c both even and leaves remainder 4, and then b and c both even which leaves remainder 0 mod 8

[rough talon]

(0,2)+(0,3,4)
so possibilities are 0,2,3,4,5,6

[green urchin]

Look carefully we have a 2

Oh well it works if b_1 and c_2 has the same parity

😔

[twilit plume]

bro why u here

[vale dock]

@rough talon Has your question been resolved?

[green urchin]

@rough talon your infinite decent

[rough talon]

👀

[green urchin]

Yuh check it , was hectic

[rough talon]

damn i see

thanks!

.solved

[vale dock]

Channel closed

Closed by @rough talon

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[vale dock]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[lime bone]

hello

[glass kelp]

Hi

[cloud coral]

[lime bone]

im a bit of overwhelmed

[stark wedge]

(remove the "of" here)

do you have a question?

[lime bone]

What are the ratio of the standard deviations

a1, and d are real constant

I need someone to answer, then I can throw you a lot of question based on it

I believe I will learn more about standard deviation through the process

🐯 :It is 2, for the interval of <x2> is twice that of <x1>

[lime bone]

So the standard deviation is actually the interval of any sequence?

🐯 :not quite so, it is because both of <x1> and <x2> are arithmetic sequence.

does it change anything?

<@&286206848099549185>

help me please

please

please

plesae

lend a hand

[green urchin]

Take a chill

Explain your situation

[lime bone]

^

[green urchin]

SD is a real number, how are you getting an interval?

Its just sqrt var [x]

[lime bone]

🐯 :do you know, in arithmetic sequences, the middle term will always be the means of the sequences?

[green urchin]

Yeah

[lime bone]

Does it imply anything?

[green urchin]

Imply what, and how are you getting a relation with variance?

and youre avoiding my question

where is the interval

[lime bone]

🐯 :indeed, it would be much easier for you to interpret it, if you take some examples of arithmetic sequences with sample numbers.

🐯 :for it can spare you doing the laborious calculation

[green urchin]

@stark wedge sorry for the sudden ping, but i'm not getting anything out of this convo, mind to take over?

maybe im too dumb for this chat

[stark wedge]

uh why me

[green urchin]

You were there

[green urchin]

.

i have no idea what this dude is talking about

[lime bone]

Ann, can you explain why in arithmetic sequences.
If the interval of one sequence is twice that of the another.

then their ratio of sd will be 2:1 as well

[lyric sundial]

!original

[vale dock]

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

[lime bone]

that's the original mr. Alberto Z.

[green urchin]

how many sequences are there

[lyric sundial]

Apart from the photo being blurred, where did you copy that from?

[green urchin]

Exactly lol

[lyric sundial]

Original means the picture of the textbook where you picked it from

[green urchin]

I dont think this dude is getting it

[lime bone]

i want an explanation

[lyric sundial]

Lol

[lime bone]

because everyone seems to accept it

it is not that intuitive IMO

[green urchin]

😭 I heard of that for the first time

[lime bone]

It is true however

[lyric sundial]

So unintuitive that noone is knowing what you're talking about

[green urchin]

@lime bone wait

[lyric sundial]

Yeah me too

[green urchin]

does interval mean common difference?

?

[lime bone]

I hate it when my teacher acts like it is a common knowledge

fk

[green urchin]

wait twin

[lime bone]

sure

[green urchin]

do you mean common difference

like a, a+d..

[lime bone]

yes sir

[green urchin]

the d

Cool

[lyric sundial]

Also, SD comes up with random variables, why are you talking about them with arithmetic sequences?

[green urchin]

Now I understand

[green urchin]

No its ok, the distribution is in AP

[lyric sundial]

Ahhhhnnnn

[lime bone]

those datas are arithmetic sequences

It is not that fking intuitive

[green urchin]

Two distributions

[lime bone]

I hate it

[green urchin]

cool

It isnt

True

[lime bone]

u guys better give me an intuitive explanation

so to claim that it is intuitive

I can do the computation on my head

and find it is ture

true

but fk

it is not intuitive

[green urchin]

dyk that the variance of an ap is given by $\sigma=d\cdot {\frac{n^{2}-1}}{12}}^{0.5}$

[lime bone]

because you have to do some computation

to get that to be true

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

I do not want to see any number

an intuitive explanation

is that exists

*does

.close

[vale dock]

Channel closed

Closed by @lime bone

Use .reopen if this was a mistake.

[rocky lotus]

Licht
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[green urchin]

.reopen

are you fucking dumb twin

Let me write the number

why tf are you crashing out

[lime bone]

I have to stay productive

I have to move on to the next question

I can dwell on one too long

I only want the perfect score

[green urchin]

Then dont fucking post it in a math channel if you have that much less time and dont have the interst to make it intuitive to you

Be quiet

and listen

[lime bone]

I have faiths in you guys

[green urchin]

Yeah so listen

dont crash out

The variance of an ap with n terms is given by this

dyk that?

@lime bone stay with me

[lime bone]

.close

[green urchin]

do you know this?

wdym close do you know it or not

[lime bone]

it is solved

it is true

[green urchin]

oh cool

bye then

[lime bone]

but there's no intuitive explanation

[lyric sundial]

...

[green urchin]

Crazy

[lyric sundial]

Indeed

[noble anvil]

What is this weird typing

[green urchin]

get a load of this mf

😭

[slate folio]

god this guy is just

[green urchin]

Im crying

[glass kelp]

[green urchin]

what industrialised education has done to us

[vale dock]

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[zenith osprey]

are there any realistical ways to prove this? i've tried doing it through trigonometry, but it still didn't make any sense to me

[green urchin]

try this, f(x)=sgnx and g(x)=-sgnx. Now clearly f(x)g(x)=-1 and so its limit exists at x=0

whereas they individually do not at x=0

[zenith osprey]

thank you. this helped

.close

[vale dock]

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[lament flicker]

my question is how does showing that d(n+1) > an or d(n+1) < an correlate to showing a_(n+1) > a_n or a_(n+1) < a_n

[frank salmon]

Is that imo?

[lament flicker]

yeah

i mean the problem is shortlisted meaning it doesnt make into the actual imo

[vale dock]

@lament flicker Has your question been resolved?

[frank salmon]

Can i have the pdf

?

Or where can i find it

[lament flicker]

you can search the question online

they have plenty of it

if you are wondering the book im reading it is modern olympiad number theory

[frank salmon]

Thanks

[vale dock]

@lament flicker Has your question been resolved?

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[modern crater]

heeeeeeeeeeeelp

[modern crater]

https://tenor.com/view/cat-gif-12450451651611168773

just copied the latex output since discord messes up the _s after u type them

[waxen scarab]

So it is only given that Xn+1 = Xn + sqrt(Xn) - 1
without a base value?

[frank salmon]

Yeah thats weird we should atleast get the value of Xo

[vale dock]

@modern crater Has your question been resolved?

[vale dock]

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[lethal cloak]

Need help in proving that the integral of 1/(x^2+ax) is equal to 1/a ln(x/x+a) +c

[upbeat swift]

the images are taking forever to load

[upbeat swift]

I can see now so

[tardy orbit]

unfortunately you can't decompose 1/(x^2 + ax) into partial fractions like that

[upbeat swift]

huh

$$
\int \frac{dx}{x^2 + ax}
$$

$$
\frac{1}{x^2 + ax} = \frac{1}{x(x+a)}
$$

$$
\frac{1}{x(x+a)} = \frac{A}{x} + \frac{B}{x+a}
$$

$$
1 = A(x+a) + Bx
$$

$$
\text{For } x=0 \implies 1 = Aa \implies A = \frac{1}{a}
$$

$$
\text{For } x=-a \implies 1 = -Ba \implies B = -\frac{1}{a}
$$

$$
\int \frac{dx}{x^2 + ax} = \int \left( \frac{1/a}{x} - \frac{1/a}{x+a} \right) dx
$$

$$
= \frac{1}{a} \int \left( \frac{1}{x} - \frac{1}{x+a} \right) dx
$$

$$
= \frac{1}{a} (\ln|x| - \ln|x+a|) + C
$$

$$
= \frac{1}{a} \ln \left| \frac{x}{x+a} \right| + C
$$

[rocky lotus]

Ren🍓

[tardy orbit]

!nosols

[vale dock]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[upbeat swift]

my point being it is possible to ?

you just said it is not

[tardy orbit]

i stand corrected

[lethal cloak]

It took me like hours of thinking how to solve but I could not come up with a correct way so I took it here

I did not know it was this obvious but this helped me, thank you so much for the time

[upbeat swift]

your approach was on the right track tho

just a bit of try and you would have solved it

[vale dock]

@lethal cloak Has your question been resolved?

[vale dock]

Channel closed

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[vale dock]

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[lucid furnace]

anyone could make me understand ts?

[vale dock]

@lucid furnace Has your question been resolved?

[west cloud]

maybe you should try #algebraic-geometry

[vale dock]

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[steady glacier]

Question 2.1 and 2.2 are giving me trouble

[steady glacier]

So far for 2.1 I've found M to be
(x; -2x-4)

A i don't know how to find Y in terms of x

and 2.2 is undoable if i don't know that

[viscid hull]

You realise that we are talking of a family of circles here right?

[steady glacier]

Uhm no

i thought it was just 1 circle

[final adder]

is the circle tangent to the y-axis?

[steady glacier]

That's what i was thinking

[viscid hull]

Yea, we are talking about all such circles which touch the y axis and pass through (-1,-1)

[steady glacier]

But it doesnt make sense

[steady glacier]

Oh

[viscid hull]

And you basically have to trace this family

[steady glacier]

hmmm okay

[viscid hull]

As in, find an equation which all centres of the circles in this family will satisfy

[steady glacier]

That sounds complex

[viscid hull]

Similarly for the points on which these circles will touch the Y axis

[viscid hull]

It's not

[steady glacier]

So if i find distance BM and AM and equate them will it give me the correcr answers?

[viscid hull]

Ok but at the end they have given an additional piece of information

[steady glacier]

That was my initial thought and it gave me 2 x values, which is probably what you mean by a family of circles?

[viscid hull]

That M lies on the line 2x+y=-4

[steady glacier]

Ahhhh

[viscid hull]

So among this family of circles , we are talking of one member

The member who's centre lies on 2x+y+4=0

[steady glacier]

Ohh so i have to find out which X i found satisfies the straight line equation

[viscid hull]

Do you know the general equation of a circle?

[steady glacier]

Yes

[viscid hull]

So this equation represents all circles possible in 2d plane

Now we have to slowly isolate this family, and subsequently this member

[steady glacier]

I found (x+1)²+(y+2)²=1 and another one

[viscid hull]

So we substitute the information we have into the equation

[viscid hull]

The general equation is
(x-a)²+(y-b)²=R²

[steady glacier]

Yes

[viscid hull]

For a circle with centre (a,b) and radius R

[steady glacier]

Mhm

[viscid hull]

So we now have a family passing thru (-1,-1)

[steady glacier]

Mhm

[viscid hull]

What will the equation become then?

The equation of family of circles passing thru (-1,-1)

[steady glacier]

(-1+1)²+(-1+2)²=1

Is that what you're asking?

[viscid hull]

.

[steady glacier]

Okay lemme redo

[viscid hull]

.

Here the Radius is not known

[steady glacier]

(x+1)²+(y+1)²=r2

[viscid hull]

(-1,-1) isn't the centre

[steady glacier]

Oh

Ohhhhhhhhhh

the distance from the center to B

is what you want

Basically

Wait

f im confused

[viscid hull]

Don't be

We can say something about a, b, and R

[steady glacier]

Okay

[viscid hull]

Now that we have the info that it passes thru (-1,-1)

Sub in x=-1, y=-1

[steady glacier]

Okay

(-1-a)²+(-1-b)²=r²??

[viscid hull]

Yes

[steady glacier]

How does that help

[viscid hull]

We got one piece of info

[steady glacier]

Hmm okay

[viscid hull]

Now we can substitute the other piece of info

That it's tangent to Y acis

Axis

How do you think we'll do that

[steady glacier]

Okay we can use maybe A?

ermmm

Maybe im rushing lemme think

We can distance AM is equal to the radius?

[viscid hull]

Correct

AM is the radius

[steady glacier]

Lemme calculate

[viscid hull]

But you still haven't incorporated the information

Into the equation

[steady glacier]

Hmmmmm

[viscid hull]

When a circle is tangent to Y axis, it's centre's x coordinate will be?

[steady glacier]

Ermmmmmm

positive?

[viscid hull]

Try to visualise the circle

It passes thru (-1,-1)

[steady glacier]

Okau

[viscid hull]

And tangent to Y axis

[steady glacier]

i see it

[viscid hull]

So it must be on the left side of Y axis right

[steady glacier]

holdon lemme picture it

Ys

yes

its on the left side

[viscid hull]

So in the general eq

[viscid hull]

.

What can you say about R and a

[steady glacier]

They will be equal??

Idk

[viscid hull]

Yes!

Njce

[steady glacier]

Wait

why

what

wait

OHHHHHHH

Wait

i see it now

but how does that help

[viscid hull]

Because tangent is Y axis

That's why R=a

[steady glacier]

Yes

okay i see it

[viscid hull]

Do u get why

[steady glacier]

Yes

its like

[viscid hull]

Nice

[steady glacier]

in line with it

[viscid hull]

So now we have two pieces of info in

About a, b and R

[steady glacier]

We have a=R and we have x and y values to sub in

and b is uhhh

Idk

[viscid hull]

a=R

(a+1)²+(b+1)²=R²

[steady glacier]

Yes

[viscid hull]

Simplify the second equation using first equation

[viscid hull]

Last thing we need is to isolate the circle from this family

[steady glacier]

Okay

so using maybe

[viscid hull]

For that we use the last piece of info

[steady glacier]

the straight line

Wait what??

[viscid hull]

.

[steady glacier]

Ohh

so we find M

using the straight line?

[viscid hull]

Yes

[steady glacier]

wait

M is x; -2x-4

[viscid hull]

That basically gives us
2a+b =-4

[steady glacier]

Ohhh you're making x and y into a and b

[viscid hull]

Yea, here x is the x coordinate of the centre

[steady glacier]

and the other x and y are the same

[viscid hull]

Yea

Instead of using x twice

It's simpler to take a,b

[steady glacier]

That makes more sense, where I'm from they don't do that so i was kinda mixed for a second

Lemme sub it into a and b here real quick

[viscid hull]

So we have

a=R

(a+1)²+(b+1)²=R²

2a+b+4=0

[steady glacier]

Okay lemme simplify so i can send it in one msg

I got

x=-1 and x=-5/2

now are both of them correct or do i use the answers in the strsight line to verify

[viscid hull]

Yes, now remember

We deduced earlier

[steady glacier]

Mhm

Oh that a must be equal to r

Yes i did that

and got 4x²+14x+10=0

So after that

i can resub in my a to see if it gives me the same as r?

[viscid hull]

We get two values of x right

So correspondingly we get two values of y

Which make two circles

[steady glacier]

I see

[viscid hull]

Make those two circles

And check if they pass thru (-1,-1)

[steady glacier]

Ok

[viscid hull]

If they don't, then discard that circle

I think you got the basic idea now

Just verify

There may be some algebra required here and there

I gtg rn sorry💔

[steady glacier]

So for both circles

no problem

[viscid hull]

Someone will help u out

[steady glacier]

are both circles x=r?

[viscid hull]

Hope u got some idea tho

[steady glacier]

I did thanks

[viscid hull]

Uh, the x coordinate of the centre is -R where R is radius yes

[steady glacier]

wherever you're from y'all do complicated ah maths ngl

I see

Thank you

[vale dock]

Channel closed

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[steady glacier]

.close

[vale dock]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[spring warren]

i havent been able to solve this question if somebody could help please
It's about the different number base systems

[vale dock]

Please don't occupy multiple help channels.

[spring warren]

oh im sorry i didnt know

[viscid spade]

Close the previous channel and its ok

[spring warren]

but my doubt isnt asnwered yet 😦

i was stuck in 2 questions

[viscid spade]

One at a time

[spring warren]

alright ill keep that in mind

.close

[vale dock]

Channel closed

Closed by @spring warren

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[supple plaza]

I wanna help if you'd tell me what's written cuz I cant understand

[spring warren]

basically a number is expressed in the 11 base system

and its 3 digits

when the same number is expressed in base 4, those 3 digits are reversed

like if the number was abc in base 11, it is now cba in base 4

[supple plaza]

so u have to find the number

right?

[spring warren]

yes

but in base 10 (decimal system)

[supple plaza]

ok wait I'll see if I can understand

ok

[spring warren]

alrightt

[spring warren]

.reopen

[vale dock]

✅

[supple plaza]

sorry I cant

[icy sigil]

A number, when written in base 11, has 8 digits. When written in base 4, it's digits are reversed. Find the number in base 10.

Wait

3 digits not 8

I read it wrong

That's the question

[spring warren]

yes

[icy sigil]

Idk how to solve it though

[spring warren]

aw man : (

[final adder]

well we know that $121a+11b+c=16c+4b+a$

[rocky lotus]

Frances

[final adder]

wait hold on

[spring warren]

ah yes i didnt know what to do after this

[vale dock]

@spring warren Has your question been resolved?

[modern rune]

now you know that a b and c must be digits in a certain range

that narrows things down a lot

[modern rune]

is this even possible what

[spring warren]

okayy

cud u make the cases then

like they can lie in any range from 0 to 3

[spring warren]

well apparently yes 😭

[modern rune]

yeah but you can make some inferences about the range

like for instance if a is 1 then it's not possible to create a 3 digit number in base 4 thats large enough

so a must be 0, which means the base 11 number isn't 3 digits but okay whatever

like that you can put a range on what b and c can be

[spring warren]

ohh

yea that makes sense thank you

.close

[vale dock]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

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[serene grove]

I Need Help determining Growth Rate for Population Growth Math

Okay, so if we take a growth rate of 1.0012 as a given set rate for humans in a medieval/ancient society, that is our baseline. Humans need two people in order to reproduce, society is usually made up of around 50-50 male and female, they gestate for nine months, primarily only have one offspring at a time, females are primarily reproductively active between 14 and 35 only, and they have an average life expectancy of 25-35 (due to the outlier data of high infant mortality dragging down the metric).

All that is the baseline, now how might I try to address the variables in this equation as we have it above for species that:
only need one to reproduce
gestate for longer or shorter amounts of time
live for shorter or longer amounts of time
have more offspring per reproductive cycle
have less females than a 50-50 split

Essentially I need help finding or creating a formula to find growth rate that does not include final population. One that includes:

of individuals who can reproduce

gestation time, aka possible births per individual per year
offspring per birth
infant mortality
likelihood to reproduce in a given year

I need help making a an obtuse and slightly complicated algebraic equation. I suck at mathmatics, so all help greatly appreicated

[wicked zenith]

Define the states each member of society can be in. Then define how states transition (i.e. non-pregnant woman becoming a pregnant woman). Then draw a graph of this model. Then you can use linear algebra to construct an equation describing how this system evolves through time, given an initial condition vector. (i.e. each node in the graph is represented by a value in a vector, the edges of the graph are represented in a matrix).

[serene grove]

I'm sure that is incredibly helpful information, but I'm very sorry I dont know what any of that means

here's what I have so far:

reproductive individuals X likelihood to reproduce X possible pregnancies in a year X avg offspring per birth

What I've yet to figure out how to include is:

possibility of death of the reproductive individual
and infant mortality

[wild parrot]

.

A Parameter-Based Population Growth Model, you defined it, but here is an obtsuse interpretation:

Nomenclature

• $P_F$: Proportion of the population that is female.
• $A_F$: Proportion of females that are of reproductive age.
• $P_A$: Proportion of an asexual population that is of reproductive age.
• $L_B$: Likelihood of a reproductive individual giving birth in a given year.
• $T_G$: Gestation time in years.
• $N_O$: Number of offspring per birth (litter size).
• $T_L$: Average life expectancy at birth in years.

Deterministic Model For our Params

Assumptions:

• Rates are constant averages.
• Environmental factors are not considered.

Formulation:
The intrinsic growth rate, $r$, is the per capita birth rate, $\beta$, minus the per capita death rate, $\delta$.
$$r = \beta - \delta$$The birth rate $\beta$ for sexual reproduction is:$$\beta = P_F \cdot A_F \cdot L_B \cdot \left(\frac{1}{T_G}\right) \cdot N_O$$For asexual reproduction:$$\beta = P_A \cdot L_B \cdot \left(\frac{1}{T_G}\right) \cdot N_O$$The death rate $\delta$ is:$$\delta = \frac{1}{T_L}$$

--

Stochastic Model

Assumptions:

• Births and deaths are independent, random events.
• The time step, $\Delta t$, is small enough that the probability of more than one event is negligible.

Formulation:
The probability of a change in population size $N(t)$ during a small time interval $\Delta t$.

• Probability of a Birth:
  $$P_{\text{birth}} = N(t) \cdot \beta \cdot \Delta t$$
• Probability of a Death:
  $$P_{\text{death}} = N(t) \cdot \delta \cdot \Delta t$$
• Probability of No Change:
  $$P_{\text{no change}} = 1 - P_{\text{birth}} - P_{\text{death}}$$

oops

[rocky lotus]

Sandcrawler
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[wild parrot]

bad render

[serene grove]

Omg

Yes

A parameter based model is exactly what I’m looking for

[wild parrot]

You need to run Monte Carlo with this to gain good insight

and perhaps use confidence intervals to better understand your outputs

[serene grove]

What is Monte Carlo?

what are confidence intervals

I'm a History Major I never took anything past math 111

[wild parrot]

As you can see this is a stochastic model, which means there is an inherent uncertainity baked into this model. When something is uncertain you cannot determine it, you can say 2 = 2 but you can't die roll = 1

it is either 1,2 or ... 6, this equation is like that

to interpret it, essentially you simulate it or run the eqaution a 1000 times. Why 1000? It's a good number and because of various statistical thereom, we know it's good enough

[serene grove]

How would you set up a parameter based population goeth model using the parameters you set above?

at the top of that pic

[wild parrot]

It's not that hard to do for a history major too

[wild parrot]

hmm as in?

[serene grove]

A Parameter-Based Population Growth Model, you defined it, but here is an obtsuse interpretation:

Nomenclature
$P_F$: Proportion of the population that is female.
$A_F$: Proportion of females that are of reproductive age.
$P_A$: Proportion of an asexual population that is of reproductive age.
$L_B$: Likelihood of a reproductive individual giving birth in a given year.
$T_G$: Gestation time in years.
$N_O$: Number of offspring per birth (litter size).
$T_L$: Average life expectancy at birth in years.

[rocky lotus]

DuFresne

[serene grove]

this

how would I set this up as an equation where I can input numbers to reach an output?

I'm sorry if I'm difficult to help. Again I can barely do arithmetic let alone complicate algebra

[spark vale]

that's a really insightful message you came up with

[wild parrot]

N_0 = Initial pop
Deltaat is a step size = 1/365 is small enough if you're calcuating rates per year

I'll set M = 1000
FOR run from 1 to M:
Initialize current_population = N_0
Initialize current_time = 0
WHILE current_time < T_max:

at each step we caluclate prob of birht and prob of death where R is a random number between (0,1)

IF R < probofdeath:
current_population = current_population - 1
ELSE IF R < (probofdeath + probofbirth):
current_population = current_population + 1

[serene grove]

I just copied what was said above.

will this channel stay open if I dont reply for like an hour? I'm at work and I need to get to some stuff, but I'm definitely coming back. This is the closest I've ever gotten to cracking this nut and getting a good equation to determine pop growth.

[wild parrot]

store the current pop in current time in a list

[spark vale]

sorry i was replying in a caustic manner to the chatgpt copy paste

not replying to you

[serene grove]

Oh lord I'm lost 🥺

I'm sorry guys

[wild parrot]

Don't be put off by his caustc reply

just run the sim first and see what kind of values you are getting and fine tune the values, so it is realistic enough. If you're using it for some clinical work or something research oriented, use better well defined model already published

[serene grove]

Not for anyhting published

its for worldbuilding

however Idk how to "run a sim" nor how I'd format the equation in one... sorry

[vale dock]

@serene grove Has your question been resolved?

[vale dock]

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Send your question here to claim the channel.

Remember:
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• After 15 minutes, feel free to ping <@&286206848099549185>.
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[near pewter]

guys help no.15, is there a correct answer??

[near pewter]

cs what i did is 7(x²-y²)

and its not on the choices

kind of confused

[glass kelp]

,rccw

[cloud epoch]

yes there is an answer

[stark wedge]

yes there is

[glass kelp]

Factor out 7 first

[cloud epoch]

thats what she did

[glass kelp]

Yes there

[stark wedge]

there's also a lot of trap answers

[near pewter]

yeahhh

im so confused lmaoo

[glass kelp]

Difference of square next?

[stark wedge]

who?

[glass kelp]

Op

[cloud epoch]

do you know how to rewrite $a^2 - b^2$?

[stark wedge]

OP is a she 💀

[rocky lotus]

TheVibe

[glass kelp]

Oh💀

[cloud epoch]

I changed my bad

[near pewter]

i forgot abt that..

[steel igloo]

D is the answer right

[glass kelp]

It’s the factoring of this

[glass kelp]

!noans

[vale dock]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[glass kelp]

Shhh

[steel igloo]

Oh sorry my first time here

[near pewter]

can u explain it to me TT

[cloud epoch]

the formula is $a^2 - b^2 = (a+b)(a-b)$

[rocky lotus]

TheVibe

[cloud epoch]

which is just something you need to remember

so try using that in your problem

[near pewter]

GUYSSS

I THINK I GOT IT

D. is right cs i literally forgot to factor our the x² and y²

[cloud epoch]

D is right indeed, nice

[near pewter]

and if i do foil method the answer will be x² and y²

AIGHT TY, IM KIND OF TIRED LMAO ITS 3:57AM HERE JWHSJAHSJAH

[glass kelp]

GMT+8 mentioned

Yes, u should go zzz

[near pewter]

.close

[vale dock]

Channel closed

Closed by @near pewter

Use .reopen if this was a mistake.

[vale dock]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[dusky pulsar]

Brain fart lowkey , needed help in finding AB in terms of AC