help-23

[dense sage]

oh oka

cool

[left gyro]

I thought your syllabus directly taught you that equation

[dense sage]

um

not to my knowledge

[left gyro]

in that case theres still something easier we can do

originally we have the tangent line (y - h(k)) = h'(k) (x - k)

[dense sage]

but i feel like the equation ur giving me is gonna be easier to use

[dense sage]

wha

[left gyro]

the

with h(k) at x = k, your point is at (k, h(k)) with slope h'(k)

[dense sage]

yessir

ohh that makes sense

[left gyro]

now if you move the - h(k) to the other side, you get the y = h'(k) (x - k) + h(k) equation

this is a common way to write the tangent line equation

[dense sage]

oh and since h(k) is even then h(-k) is the same

so c = c

[dense sage]

omg cool

[left gyro]

not necessarily

[dense sage]

oh wait wha

[left gyro]

youre forgetting that its not in y = mx + b form

its got an (x - k) there

we'll need to address that

[dense sage]

ohh ye

[left gyro]

also, usually the tangent line equation is shown online as $y=f'(a)(x-a)+f(a)$ for a tangent line at $(a, f(a))$

[flat frigate]

mtt

[left gyro]

commonly the function is f, and the point's x-coordinate is either a or c

if youre memorizing the equation, the convention is this one

[dense sage]

oh oka

[left gyro]

we're using the equation for our purposes, so our function is h and our coordinates are (k, h(k))

[dense sage]

okaoka

[left gyro]

now with y = h'(k) (x - k) + h(k),
you can expand (use distributive property on that h'(k) (x - k)

[dense sage]

so h'(k) x - h'(k)k +h(k)

[left gyro]

yep

[dense sage]

okaa

[left gyro]

now the slope is h'(k)

and the y-intercept is h(k) - k h'(k) as shown by what you typed

[dense sage]

okaa

[left gyro]

now you can put in -k in and see what happens

[dense sage]

okaa

y = h'(k) x - h'(k)k +h(k)
y= m x + c

h(k) is even so h'(k) is odd and h'(-k) = -h'(k)

h(k) is even so h(-k) = h(k)

y= h'(-k) x - h'(-k) (-k) +h(-k)

y= - h'(k) x - h'(k) (-k) + h(k)

=- h'(k) x + h'(k) (k) + h(k)

so y = -mx + c

is that right

[left gyro]

unfortunately it isnt

[dense sage]

oh no :c

did i do smth weird

[left gyro]

first youre beginning with y = mx + c instead of y = mx - c

[dense sage]

oh oop

there

mb

[left gyro]

next, it looks like you assumed h' is even

h is even, so h' is odd

[dense sage]

wait where

did i

[left gyro]

around the - h'(-k) (-k) part, the next line says - h'(k) (-k)

[dense sage]

oh shoot didn't carry the negative in

y= h'(-k) x - h'(-k) (-k) +h(-k)

y= - h'(k) x + h'(k) (-k) + h(k)

neg neg

right

[left gyro]

yes

[dense sage]

so y = h'(k) x - h'(k)k +h(k)
y= m x + c

h(k) is even so h'(k) is odd and h'(-k) = -h'(k)

h(k) is even so h(-k) = h(k)

y= h'(-k) x - h'(-k) (-k) +h(-k)

y= - h'(k) x + h'(k) (-k) + h(k)

=- h'(k) x - h'(k) (k) + h(k)

so y = -mx + c

[left gyro]

yep

[dense sage]

YAYYY

THANK YOUUU

[left gyro]

np

[dense sage]

lifesaver!!!

.close

[safe radish]

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[whole acorn]

A test consists of 100 questions. A student gets 4, -1, or 0 marks, if he answers a question correctly, wrongly, or leave it blank, respectively. How many different total marks of the test are there?

[safe radish]

@whole acorn Has your question been resolved?

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[marble summit]

can someone help me with vector

[marble summit]

If AB = u, and AF = v. Determine vector of DF

here's my guest
DC = FA = -AF = -v
CO = BA = -AB = -u
DF = DC + 2CO = (-v) + 2(-u)

is that correct?

i need someone actually know the answer, and i have some question

[true horizon]

This is a regular hexagon?

[marble summit]

yes

[lime nexus]

seems right

im no mod but i dont think ur suppose to ping for helpers until 15mins later...

[marble summit]

rite? but i see another answer with DO and OF then include DE and so on

[marble summit]

i didnt know that, sorry

[lime nexus]

i think DF= -v -2u, which u have gotten

it makes sense

[lime nexus]

it should eventually end up with -v -2u as the final result

[wanton schooner]

u should end up with -2u-v whichever way u do

[lime nexus]

theres many ways u could go about for it

the way u did it is just i would say the fastest way?

[marble summit]

yeah

so it work with another side?

[lime nexus]

mhmm

[marble summit]

like DO or DE

[lime nexus]

bc anyway u go, u will eventually reach -v -2u

[lime nexus]

u can but itll be quite longer

ur way is faster and more convenient

[marble summit]

so it will be a triangle?

[lime nexus]

sorry?

[marble summit]

if im not right

like this D to O and D to E

and last one is O to F

[lime nexus]

if u do that, u would be finding DE

[marble summit]

ahh

i see

[lime nexus]

it would be from E to F*

[marble summit]

so yeah, why is it reach to C?

[lime nexus]

D to O to E to F will give DF

[lime nexus]

huh?

[marble summit]

D to C is like why is it line up to C and not to
D to F instead and count it normally ( but still i dont get it with hexagon thing)

[lime nexus]

in order to get from D to F, u can go from D to C to O to F, and it will give DF

[lime nexus]

sorry i dont get what ur trying to ask

[marble summit]

bad thing discord mobile cant draw

[lime nexus]

r u trying to ask why we do this?

[marble summit]

yeah, wait i'll check on my computer 1 minute

here

this is what am i trying to ask, why we dont do this

[lime nexus]

if u do that, how are u able to find the vector for it?

itll be too hard with what we are given with right?

[marble summit]

ahh

thats why

so we can't?

[lime nexus]

vectors is like if u wanna get from a certain point to anohter point, u can use vectors where u follow the given paths until u reach ur point

[lime nexus]

there is no values we can use to find the red line

wait

[marble summit]

i mean, firstly when i try to find the vector was like that

[lime nexus]

for the green parts, notice how CO is the same as AB but in the opposite direction, since we know AB = u, then CO is -u, and it happens twice, so thats why we minus u twice

[marble summit]

i was like nuh uh, this is got me confusing

[lime nexus]

and then we add the 3 vectors to get DF= -v - u -u

hence DF = -v -2u

[marble summit]

is there any other way?

[lime nexus]

try reading this and see if it makes sense

[lime nexus]

how so?

if u want to try doing it with the red way, i dont think it’s possible as we cant find what red vector is since there is no vector and values we can work with from the question

ig qns like these, to solve it u would want to try find a path to get from where u want to go, by following other vectors and adding them up!

[marble summit]

i see, so we just follow the instruction of the question? and try find a path

[lime nexus]

mhmmm!

[marble summit]

like find a similiar AF and AB and determine the vector ( consider the direction )

okok i got this, thank you boss

appreciate it

[lime nexus]

yes, do note, these "similar" vectors only work or like AF is only thr same as CB is bc theyre parallel, and have the same direction or opposite direction

so like AF and AO arent the same vector bc it goes into 2 diff directions that arent opposite directions and arent parallel at all

[marble summit]

mhmm yea it makes sense, its like a puzzle then

[lime nexus]

altho u can see that AF and CD are parallel and only thing different about the direction is its going opposite sides

so thats why i deicded to make a path thru D to C to O to F, so i also dont need to deal with vectors going in the direction of AO

[lime nexus]

yess

[lime nexus]

no worries 🔥

[lime nexus]

yuhhh

does it make sense now?

[marble summit]

yessir

[lime nexus]

hurray

[marble summit]

do you mind, if i add you to ask more question?

[lime nexus]

suree

[marble summit]

appreciate it, man i wanna learn something

[lime nexus]

send it thru!

ill try to help

[marble summit]

yes pls, and how do i close this help section?

[lime nexus]

OH LOL

.close

but it only closes if u type that

goodluck!!

[marble summit]

oki, accept my friend request, ive already sent it to u

[lime nexus]

uh okay?

[marble summit]

.close

[safe radish]

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[cerulean wolf]

is it "valid" to show lim x->inf of sin(x)/exp(x) = 0 by claiming sin(x) will always be in range -1 to 1 for all real input and then that e^-x goes to 0 and since any real "a" * 0 is 0 it's all 0? or some similar reasoning

[cerulean wolf]

or would real proof use the Taylor series or something like that

[peak estuary]

you want the squeeze theorem

-1/e^x <= sin(x)/e^x <= 1/e^x

[short topaz]

i mean it depends on what u mean by "valid", like yeah, that is the intuition for why it tends to 0

so like if ur doing i.e. a physics/engineering course, that would probs be sufficient

[short topaz]

however if ur doing something like analysis that's basically it, but i would write it out properly

[short topaz]

like this ^

[cerulean wolf]

but it's not rigorous right

[cerulean wolf]

got it

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@cerulean wolf Has your question been resolved?

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[half briar]

Can we do this together: THere are 4 races in total which have a winner and 10 participants. What are the possible total results if we only look at races won

[half briar]

basically after each race the winners we pick reset

and the order does not matter i guess

WE got 10 participants so: A B C D E F G H I J

Basically asking how many possibilities we can build out of those. Like ACDJ for example

this formula?

$$\binom{10+4-1}{4}$$

[flat frigate]

Jaeger

[digital sparrow]

no wait

as there is a winner

each race will have atleast 1 participant

[half briar]

what

its each race same 10 participants

[digital sparrow]

like you use this one one group might have 0 objects

[digital sparrow]

this one is valid when groups might have 0 objects in them

for this ques that is not possible as

"each race has a winner"

[half briar]

yeah? but every time it resets

next race new outcome

[digital sparrow]

so you just subtract the number of "fixed" objects from the top

in this case 10 + 4 - 1 - 4

[digital sparrow]

wdym resets?

[half briar]

they do a race

one winner

next race same person can win again

[digital sparrow]

no i dont think that is the case

more like 10 ppl divided into 4 groups with no repetition

then one winner out of each group

[digital sparrow]

is this exactly the question statement?

[half briar]

yeah it was the question. 10 participants and we do 4 races. Only wins matter. What are the possible chances of the winners

Can be anything like AAAA AABC etc

[digital sparrow]

i mean it is kinda vague

like how are the races conducted?

and what does "only wins matter" mean

[safe radish]

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[outer lance]

Someone help me with derivatives please. Ive been stuck looking at my page for hours because I cannot seem to understsnd

[crystal cloud]

send the sum

[outer lance]

Both E and F

For E I understand that it equals to sin (x+y)

But why isnt the derivative cos(x+y) • y'

using chain rule

[crystal cloud]

no it will be cos(x+y)

[outer lance]

But I thought you had to use chain rule and multiply the inside afterwards

The answer is -1 btw

[crystal cloud]

nah

are u sure

[outer lance]

Yes

[crystal cloud]

lemme get a paper hodl

[outer lance]

I got that

[crystal cloud]

i got it

solved it

[outer lance]

ok how pls

[outer lance]

Oh wait im stupid

I forgot derivative x is 1

[outer lance]

Could you help me with F) as well

I can show you my work but my answer is slightly different for some reason

[crystal cloud]

what is the answer

yeah show me the work

[outer lance]

cos x-2sinx2x-2y /2x

[crystal cloud]

what

[outer lance]

answer

[crystal cloud]

can u send a piv

itll be better pls

idk if it is correct.

[outer lance]

The answer is (cos x- 2sinx2x- 2y) / (2x)

idk where the 2y or 2x comes from

[crystal cloud]

mine is correct

[outer lance]

why would this not work

[crystal cloud]

no i did not

[outer lance]

derivative of 2xy = 2y' no?

where did you get the x and 2y from

[crystal cloud]

its 2 dy/dx

[outer lance]

He has the right answer idk how

[crystal cloud]

no i wrote the positive first and

then the negative term

[outer lance]

Can you show your work maybe

[crystal cloud]

mine is correct 😭

[outer lance]

well its somehow correct

[crystal cloud]

no om

wait guys

its aarya lol

[outer lance]

Yeah can you show how you got it

[crystal cloud]

here

[outer lance]

Thats wrong tho

[crystal cloud]

u j have to write sin first then thr cos term

[outer lance]

is x times dy/dx not just 1?

because derivative of x = 1?

why wouldnt 2xy = 2 (dy/dx)

[crystal cloud]

it is 1

[crystal cloud]

correct

buddy cmon i solved it i dont get what ur cribbin on

[outer lance]

Yeah but my answer key says theres a -2y and x somewhere

[outer lance]

You got it wrong bru

Ive said this like 100 times

[crystal cloud]

IDIDNOT

[outer lance]

Yes you did man the answer key dont lie

[crystal cloud]

OH I FORGOT PRODUCT RULE?

shit im so so so so sorry

[outer lance]

wait so the derivative of 2xy = 2xyy' or what

Im still lost

[crystal cloud]

cuz yall did not use rhe term

[outer lance]

Oh wait

Product rule

I see it

I got it thanks

In the answer there is no negative

Is there another negative sign im missing for I)?

i

also how is the apart of the derivative of sin^4(2-x)^-1 just cos (2-x)^-1

why wouldnt it be like cos^4 or smt

Its like (2-x)^-1

so it would be 3sin^4(2-x)^-1

[safe radish]

@outer lance Has your question been resolved?

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[graceful widget]

N1 and N2 are norms

how do i prove that Cn exists?

Cn>0

<@&286206848099549185>

[steep lily]

I'm thinking about this, but you might want to try in #real-complex-analysis

[graceful widget]

are you doing it now?

[steep lily]

Yes, though I might need some time

[graceful widget]

ok im gonna wait for you thanks

[plucky elk]

What's your book's notation for $\R_n[X]$ mean

[flat frigate]

riemann

[graceful widget]

polynomial with real coeff and degree at most n

[plucky elk]

yea that's important to tell

[graceful widget]

ok

[safe radish]

@graceful widget Has your question been resolved?

[steep lily]

(I was unable to figure it so far and I have to go, see my previous advice)

[graceful widget]

ok thank you

@plucky elk hello just in case

[plucky elk]

no

if you're not doing anything, i suggest doing it for small n like 1 and 2 to see how both sides evaluate to try to find a pattern

[graceful widget]

you mean we do it together

[plucky elk]

no

what

do your own homework

[graceful widget]

but i dont know how to do it

[plucky elk]

.

do you know what p looks like for n=1 ?

[graceful widget]

aX

a real number

[plucky elk]

no

[graceful widget]

a then

aX + b

[plucky elk]

what's a polynomial with degree at most 1

yes aX+b

plug that into both sides

[graceful widget]

ok i will do it but just a question, is Cn changing depending on n?

[plucky elk]

C_n is notation for a constant that depends on n

[graceful widget]

ok

[graceful widget]

on the left side it's a²/3 + ab/2 + b² and the right side it's a²/4 + ab + b²

[plucky elk]

probably gonna take more than just n=1 to find the pattern

try n=2

[graceful widget]

no i was wrong in my first answer

a²/3 + ab/2 + b² and the right side it's a²/4 + ab + b²

i see that it is smaller

[safe radish]

@graceful widget Has your question been resolved?

[safe radish]

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[golden granite]

How do I know which interval makes it true

[lean otter]

u know multiplying two positive numbers or two negative numbers is > 0. if one is positive and the other is negative then it is < 0

[golden granite]

So you plug in the solutions into the problem above?

[hidden gyro]

If a number is less than zero then it is ||negative||

[golden granite]

Right

So is it the top and bottom

[hidden gyro]

no

[lean otter]

ok easier example, (5)/(-3) is < 0 and (-5)/(3) < 0

but (5)/(3) > 0 and (-5)/(-3) > 0

[golden granite]

ok

[lean otter]

so u need [x+2 <=0 and 4x > 0 ] or [x+2 >= 0 and 4x < 0]. do you see why?

[golden granite]

Because if it’s zero in the denominator then it’s undefined

[lean otter]

ok that is also true

but i meant that n/d <= 0 if [n<=0 and d>0] or [n>=0 and d<0]

it means that one number is positive (+) and the other is negative (-), so the multiplication is negative (+- -> -) idk if i explain myself correctly

[golden granite]

Yeah I got you

And if they’re both negative it’s pos

[lean otter]

yes

so basically u need (top >= 0 and at the same time bottom < 0) or ( top <= 0 and at the same time bottom > 0)

[golden granite]

Does that always apply

[lean otter]

i think ur not getting how im trying to explain it

+/- is negative and -/+ is negative

+/+ is positive and -/- is positive

[golden granite]

I know

[lean otter]

okay how would you solve it

[golden granite]

Look at the 2 intervals that result in a positive fraction

and use those number

[-2,0)

[lean otter]

oh i didnt see the whole image

[lean otter]

u didnt even need what i yapped, and btw i think u meant negative cuz u want the fraction <= 0

but yes, that happens in [-2,0) because the image says -1/4 <= 0

so u already know the answer

[golden granite]

yeah

lol

[safe radish]

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[sand garnet]

How do I find the mean?

[versed pendant]

Area/x

This is the definition of mean, and area is the sum of every y

[halcyon dagger]

thats for discrete

continuous requires integral

[versed pendant]

Yup. 👍

[halcyon dagger]

int[0,10] (x*f(x)) dx

[safe radish]

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[native robin]

A ball of mass $m$ dropped from a distance $d$ above a plate attached to a spring with spring constant $k$ compresses the spring a distance $x$.

[flat frigate]

Jash

[native robin]

Derive an expression for x, the distance the spring compresses as a result of the interaction with the ball.

[odd crest]

energy conservation

[native robin]

do i have the right setup here:

$mg(d+x)=\frac{1}{2}kx^2$

[flat frigate]

Jash

[supple shore]

make a scheme of the situation maybe you will see it better, I don't really understand here

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[shy wing]

Trying to study for my test that's coming up, and on the review this problem came up which is very confusing. It's part of the autonomous equations, but the book and lecture don't really cover too much of it, or what "absent harvesting" means, or what a condition would mean towards the problem.

So what I do know: The logistic equation I have been taught goes by:

dx/dt = kx(M-x)+-h

where k is a constant based on the yearly renewal of the population, M is the carrying capacity, x is the population.

I know generally, expanding out the equation and setting up the quadratic equation is helpful, as I'd want the radical to end up being 0. Now, a few issues. It says absent harvesting, but then it says the lake is opening to fishing. So I'm assuming the equation is now what it is, but +-h, where I am solving for h?

As for y(0), I think that means that when t=0, y=1280? I'm honestly not sure with this one, because the lecture or homework didn't really go over conditional statements.

If anyone knows a good way to explain this problem to me, that'd be greatly appreciated.

[raven heart]

I mean you seem to have a decent idea of what to do here, there's just one little thing

Now, a few issues. It says absent harvesting, but then it says the lake is opening to fishing. So I'm assuming the equation is now what it is, but +-h, where I am solving for h?
Yes, but it's not +-h, it's just -h. h is a harvest rate, it's a rate of fish you're removing out of the lake

[shy wing]

I see, it's specified it's 100% going to be harvesting, so it's -h in this scenario

[raven heart]

then the goal is to see how high you can push h so that the equation still has an equilibrium

i.e. what's the highest value of h for which kx(M-x)-h = 0 has a solution ?

or to use the notations in the question replace kx(M-x)-h by 0.06y(1-y/3600) - h

[shy wing]

Okay, I see. I went ahead and set everything up

Lemme send a screenshot

Now is that my answer, or do I have to do something with this answer

This is as far as I usually get with this problem

[raven heart]

there's a ton of small errors here and there but you fall on your feet in the end at least

also +57 not -57

[shy wing]

Where are the errors

Ah I see that now

Any other ones other than the small one at the end?

Ahh I left the +- sign in as a habit

So what I did on the third step of the run is I changed all the signs

which is why h comes out at a positive number

[raven heart]

most of these signs don't matter in the end tbf

[shy wing]

Yeah would probably have been helpful for me not to continue writing the +- like that lol and yeah it doesn't matter too much, I just did it to keep the lead term positive

Made it look better

But anyways, now what should be done?

[raven heart]

you're done

gg

@shy wing

[shy wing]

Okay, so that's what I thought this entire time

[raven heart]

actually

yeah I thought the initial condition didn't matter

but it does

[shy wing]

Okay that's the missing link

I do not know what the initial condition means in relation to the question

I can only assume it just means the initial population at the start of harvest is 1280

[raven heart]

yes

the idea is

here's your function for some h < 57, you get two equilibrium points (approx 800 and 3000 here)

problem is

if the initial population is less than 800, what happens to the population ?

[shy wing]

h = 38.7?

But if the initial population is less than 800, then the population would be on an incline

[raven heart]

incline ?

you're sure about that ?

this graph is dy/dt as a function of y

so if y is less than 800, dy/dt is negative

[shy wing]

I'm not sure

[raven heart]

well I just told you no

[shy wing]

okay so questions

What's h=38.7 in this scenario

Oh wait

That's probably the answer

[raven heart]

some random value

don't look into it

[shy wing]

Okay

[raven heart]

also it's not the answer

[shy wing]

Ah yeah I see now

I did my calculation wrong

When it is less than 800, it is on a decline

[raven heart]

this is what I've been drawing here, not the actual evolution of the population

[shy wing]

Oh

[raven heart]

the f(y) plot at least

[shy wing]

Okay let me try again

[raven heart]

but you should be able to infer the phase line from that

[shy wing]

Is 800 half-stable

[raven heart]

unstable

[shy wing]

Ahh yes

i see it now

Originally I did not connect the phase line because I'm used to it being a vertical line

But if I start at 0, it is negative, or decreases when it's less than 800, and increases when it is greater than 800

which is unstable

Is that correct?

[raven heart]

yes

so the problem is that if you start your population at < 800, the population will go straight down to -infinity

i.e. you don't reach an equilibrium

[shy wing]

Yes

So maybe instead

This looks like it's in the form of a typical population equation

Would this ones equilibriums be 0 and 3800? Or do I calculate them in the quadratic equation

[raven heart]

for h = 0, yes that's the equilibriums

[shy wing]

Ahh yes you're right

[raven heart]

but you still want to do something w/ the quadratic equation

[shy wing]

So I need to use the quadratic for the new equilibriums

[raven heart]

yes

[shy wing]

And is this for h=57?

[raven heart]

what is ?

[shy wing]

Sorry

In the quadratic equation, am I assuming the harvest rate is what I found?

So the radical would be 0?

[raven heart]

radical = 0 just tells you whether there are equilibriums or not

we want something more precise than that though

I'll increase h a little bit on my graph and you'll see

ok h=53.4

you still have 2 equilibriums, approx 1400 and 2400

what happens if you start at 1280 now ?

[shy wing]

Population is declining still

Ooh idea

[raven heart]

yeah and we don't want the population to decline when we start at 1280

[shy wing]

If the population is 1280 at y(0)

Would I find h by setting the equation = 0 when = 1280?

And then run my quadratic using that harvesting number for that population to find my equilibriums?

I see the point now though

I want it to be in equilibrium with 1280

[raven heart]

you want 1280 to be between the two equilibriums yea

[shy wing]

Oh

So what does that mean

[raven heart]

the quadratic formula gives you the two equilibriums

[raven heart]

or at least you want it to be bigger than the smaller equilibrium

[shy wing]

Okay

[raven heart]

otherwise you end up with a "goes to -infinity" situation

the phase line doesn't change very much really, it's just the equlibrium points that move

[shy wing]

I'm just confused on the quadratic formula part

h=57 would obviously set the radical to 0 and obviously give me 228/0.12 +- 0

Which is reflected in the graph as the vertex

1900

but that means if I'm at 1280, it's another -infinity issue

[raven heart]

yeah

so you don't want h=57

you'll want something smaller

you want h such that you end up in the 3rd situation

[shy wing]

Oh

If 228/0.12 is 1900, then the radical needs to be equal to (1900-1280)^2?

so that 228-sqrt()/0.12 is 1280, and then I can find 228+ using that same number?

[raven heart]

228-sqrt()/0.12 is 1280
ok yeah

[shy wing]

I'm sure that's a method, is that the best method to deal with this though?

Felt a little weird to me

[raven heart]

this is prolly the most natural way

"I want 1280 to be between the two equilibria, so I solve for the two equilibria and find the biggest h for which 1280 is between the two"

it's not like there's a completely wild step going on here

[shy wing]

Okay

Ah wait I did my logic incorrect

If 1280/0.12 is 1900, then I simply need the minus term to be 620 but I have to account for the 0.12 on the bottom

So the radical needs to be like 74.4

[raven heart]

,calc 0.12*(1900-1280)

[flat frigate]

Result:

74.4

[raven heart]

yea

[shy wing]

So with that, I set up my equation 51984-912h = 74.4^2. Solve for h, and I got that my value of h is 50.930, or 51 fish of harvesting?

[raven heart]

yeah

[shy wing]

Okay

Yesss

Thank you thank you thank you

That was a tricky one but I feel like you helped me understand it

I hope you have an amazing rest of your day and I appreciate all the help you gave me

🫡

.close

[safe radish]

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[safe radish]

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[full quarry]

Just need someone to verify if I’ve done these correct

[safe radish]

@full quarry Has your question been resolved?

[full quarry]

<@&286206848099549185>

[full quarry]

@helpsrs

<@&286206848099549185>

.close

[safe radish]

Channel closed

Closed by @full quarry

Use .reopen if this was a mistake.

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!

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[spiral saddle]

let $f : \mathbb{V} \to \mathbb{V}$ such that $M_{B}(f) = \begin{pmatrix} 1 & 4 & -5 \ 2 & 1 & 0 \ 0 & 3 & 1 \end{pmatrix}$ and let $B = {v_1, v_2, v_3}$ and $B' = {-v_1 + v_2 - v_3, v_1 + 2v_3, v_2 }$ bases of $\mathbb{V}$. Find $M_{B'B}(f), M_{BB'}(f)$ and $M_{B'}(f)$

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[torpid elbow]

Helpers

<@&286206848099549185>

[mystic adder]

e is 2.71 if I'm correct? then you need to calculate it

I'm not sure

[safe radish]

@spiral saddle Has your question been resolved?

[spiral saddle]

<@&286206848099549185>

[safe radish]

@spiral saddle Has your question been resolved?

[spiral saddle]

<@&286206848099549185>

[spiral saddle]

<@&286206848099549185>

[spiral saddle]

<@&286206848099549185>

[safe radish]

@spiral saddle Has your question been resolved?

[spiral saddle]

<@&286206848099549185>

[final wasp]

e ≈ 2.71828

normally your teachers want you to use either 2.72 or 2.718

[tender sleet]

what

[spiral saddle]

help

let $f : \mathbb{V} \to \mathbb{V}$ such that $M_{B}(f) = \begin{pmatrix} 1 & 4 & -5 \ 2 & 1 & 0 \ 0 & 3 & 1 \end{pmatrix}$ and let $B = {v_1, v_2, v_3}$ and $B' = {-v_1 + v_2 - v_3, v_1 + 2v_3, v_2 }$ bases of $\mathbb{V}$. Find $M_{B'B}(f), M_{BB'}(f)$ and $M_{B'}(f)$

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[tender sleet]

idk sry

[spiral saddle]

is oki

[wild cape]

how are the M's defined?

[spiral saddle]

\textbf{Property (base changes in the matrix of a linear transformation):}

Let ( B_V ) and ( B'_V ) be bases of a vector space ( \mathbb{V} ) and ( B_W ) and ( B'_W ) be bases of a vector space ( \mathbb{W} ). If ( f : \mathbb{V} \to \mathbb{W} ) is a linear transformation, then:

[
M_{B'W B'V}(f) = C{B'W B_W} \cdot M{B_W B_V}(f) \cdot C{B_V B'_V}.
]

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[spiral saddle]

like this i think

[wild cape]

this doesnt tell me how M is defined

well ik the concept but the order of subscripts isnt universal so i'd like it clarified

ah this suggests subscripts go right to left

[spiral saddle]

like we need to derive the change of basis matrix formula for each M i think¡

[wild cape]

ok so how do we use this to find M B'B

just an equation, no explicit numbers yet

[spiral saddle]

yes but I dont even know what C is

someone said identity matrix but im not sure

[wild cape]

just write it down and compute it later

its good write the solution symbolically before computing anything

[spiral saddle]

to write down the formula for M B'B I need to know what C is doe

but that is my issue, finding the formulas for the matrix representations

[wild cape]

yes but you can just write down an equation using C

[spiral saddle]

oh ok

,, M_{BB'}(f) = C_{BE} \cdot M_{EB}(f) \cdot C_{BB'}

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[spiral saddle]

like maybe like this but im not sure tbh

[wild cape]

whats E

[spiral saddle]

like standard basis

[wild cape]

V is a general vector space so standard basis is nonsense

we only have B and B'

[spiral saddle]

oh right im sorry

,, M_{B'W B'V}(f) = C{B'W B_W} \cdot M{B_W B_V}(f) \cdot C{B_V B'_V}.

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[spiral saddle]

domain and codomain are V

,, M_{B_VB'V}(f) = C{B_V B_V} \cdot M_{B_V B_V} (f) \cdot C_{B_V B'_V}

oh my, I messed up wait a moment

[flat frigate]

938c2cc0dcc05f2b68c4287040cfcf71

[spiral saddle]

no idea tbh

this stuff is confusing as hell

[wild cape]

general tip for math, always try to understand the form of a rule instead of the exact letters

i think the best way to understand change of basis is the right to left order of subscripts

back to the general rule

https://cdn.discordapp.com/attachments/903481106819612714/1297067604803981362/1187916384341078070.png?ex=671493d7&is=67134257&hm=96902563634714231b1fc987df1f3bc601c70a1481e8f2316c9583e8655e3784&

[spiral saddle]

what do you mean right to left subscripts}

[wild cape]

change of basis tells us there are two ways to compute f(x)

the first way is direct via $[f(x)]_{B'W}=M{B'_WB'V}(f)[x]{B'_V}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

agree?

[spiral saddle]

oh my

[wild cape]

have you seen this property?

[spiral saddle]

I have never seen this no

what is it ?

[wild cape]

do you know what [x]_B means

[spiral saddle]

coordinate of x with respect to basis B?

[wild cape]

yes

[spiral saddle]

is this the first way ?

[wild cape]

this property is a generalized version of standard matrix for linear maps R^n->R^m

[spiral saddle]

right to left subscripts?

[wild cape]

ill get to that later

[spiral saddle]

okok

[wild cape]

standard matrix for linear map f:R^n->R^m satisfy f(x)=Ax yes?

[spiral saddle]

sure

linear transformation can be expressed as matrix multiplication i think

[wild cape]

but now for f:V->W the matrix product no longer makes sense

[spiral saddle]

mmm

[wild cape]

we compensate by using coordinates

[spiral saddle]

how

[wild cape]

hence this

you can see it still roughly follows the form of f(x)=Mx

[spiral saddle]

like for M_{B'_W B'V}(f)[x]{B'_V}

what is that with words

[wild cape]

well back here its "image=matrix*input"

[wild cape]

here its "coords of image=matrix*coords of input"

[spiral saddle]

matrix change of basis input basis of domain B' with domain W and output basis B' with codomain V of linear transformation f multiplied by coordinates of x with respect to basis B'

something like that?

is a mouthful

[wild cape]

my phrasing is more succinct, having all those variables kinda clutters the meaning

[spiral saddle]

sure but why B'_V

like output basis of the matrix representation must match

in order to be multiplied

[wild cape]

x is in V so it only makes sense to use a basis of V for coordinates

tbh i can restate the rule without the primes, no difference

[spiral saddle]

yes

[wild cape]

the primes only exist to properly state change of basis

$[f(x)]{B_W}=M{B_WB_V}(f)[x]_{B_V}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

ok is this clear now?

[spiral saddle]

sorta

like is a complex topic

[spiral saddle]

what now?

[wild cape]

if its still not fully clear i would go back to these two messages

[spiral saddle]

I hate it the subscripts V and W make everything like 10 times more confusing

[wild cape]

.

[wild cape]

.

but you can do that later

ok we're back to this

https://cdn.discordapp.com/attachments/903481106819612714/1297067604803981362/1187916384341078070.png?ex=671493d7&is=67134257&hm=96902563634714231b1fc987df1f3bc601c70a1481e8f2316c9583e8655e3784&

this tells us there are two ways to compute f(x)

the first way is direct via $[f(x)]_{B'W}=M{B'_WB'V}(f)[x]{B'_V}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

the second way is indirect via [[f(x)]{B'W}=C{B'WB_W}M{B_WB_V}(f)C{B_VB'V}[x]{B'_V}]

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[spiral saddle]

bruh

wait a secons

you modfied the formula??

[wild cape]

i rewrote the RHS using the change of basis rule

[wild cape]

.

[spiral saddle]

LHS aswell

[wild cape]

i didnt

[spiral saddle]

M_{B'w B'v}(f) vs

[f(x)]_B'w

like what happened with B'v output basis

[wild cape]

??

[spiral saddle]

ohh you modified the RHS by multiplying with

okay that's way better

[x]_{B'v}

okay now what?

[wild cape]

now this is where the understanding of the rule comes from

we apply the effect of each matrix to [x] going right to left

and the subscripts guide us

the first product is $C_{B_VB'V}[x]{B'V}=[x]{B_V}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[spiral saddle]

right to left

[wild cape]

the second product is $M_{B_WB_V}(f)[x]{B_V}=[f(x)]{B_W}$

[spiral saddle]

okay

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

the last product is $C_{B'WB_W}[f(x)]{B_W}=[f(x)]_{B'_W}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

this verifies the indirect way of computing f(x)

remember, the understanding of change of basis is the right to left effect of matrices guided by subscripts

use this whenever writing down the appropriate change of basis for a problem

[wild cape]

actually i think its even better if i show this in a diagram

@spiral saddle

[wild cape]

im going in a bit so ill leave you this

[wild cape]

what you specifically do for problems is pick the matrix product so that the basis subscripts line up correctly

[spiral saddle]

ok

I will come back tomorrow to finish this, thank u fir the definitions

like I feel so tired I was falling sleep

[wild cape]

for example $M_{B'B}(f) = C_{B'B} M_{BB}(f) C_{BB}$

[flat frigate]

𝓡𝓞𝓚𝓔𝓣𝓣𝓞─୨ৎ─❥ ♡ <𝟹❤

[wild cape]

no prob

when you wake up, please go over all of this convo to make sure it sticks

[wild cape]

and try to understand how i applied my tip to write this

[robust river]

I also tried to explain that stuff yesterday

Explaining is harder than it seems 😅

[safe radish]

@spiral saddle Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

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.

[split fulcrum]

Confused on what to do next

[split fulcrum]

,rotate

[safe radish]

@split fulcrum Has your question been resolved?

[lime nexus]

use ref sheet for normal distribution, in their 30s is essentially between 30-40

notice how its also between z score of -1 and 1

also hi again lol

and in 40s would be ppl between age of 40-50 which if u keep drawing the curve itll be z score of 1 to 3

[split fulcrum]

😭 this normal distribution shit killing my confidence

[lime nexus]

NOO U GOT HTHS

BEELIVE

[lime nexus]

try attempting the qn knowing these facts, maybe itll make it a lot clearer

[rich elm]

universal hsc suffering experience

@split fulcrum u got this

[lime nexus]

ur doing hsc too?

[rich elm]

yeh

[lime nexus]

ohhh nice

[split fulcrum]

bro i actually hate this so much bruh i tried doing some financial maths questions from the previous hsc papers too and i cant do shit 😭 💀 its joever

[rich elm]

chill chill dont stress

do you have a routine/working out process for finance questions?

they tend to fall similar to each other in some way or the other, being AP or GP mostly

dont ask me about FV and PV stuff, i just guessed those questions when i did adv last year

(bad student)

[split fulcrum]

no not really except i try to use the recurrence relation stuff but i think i have a hard time understanding whats going on in the question

[split fulcrum]

but im confused between 30 to 40 isnt that like 10 standard deviations cuz the question says a std of 5 yrs

[lime nexus]

yes

remember the rules

between -1 and 1 is 68%

-2 and 2 is 95%

and so on

[rich elm]

35 is mean, if thats what u are confused

[lime nexus]

use this fact to ur advantage!

[rich elm]

z score tells u how many standard deviation a particular 'event' is from the mean

[lime nexus]

pick up ext ur never gonna see financial again lol /kidding

[split fulcrum]

so like

1 standard deviation from the mean is

36?

[lime nexus]

percentage?

[split fulcrum]

like 36 years old

[split fulcrum]

but the people aged between 30-40s isnt that 10 standard deviations but the normal distribution curve only goes to like 3 standard deviations

[lime nexus]

nah i think u mistaken it for the years between it

one standard deviation = 5 year difference

so ig 10 year difference is 2 standard deviations away from each other (technically)

[rich elm]

thats 2 standard deviations from in total

but they are respectively 1 standard deviation from the mean

1 z-score = 1 standard deviation

z-score could be thought of as the 'unit' for how many standard deviations

[split fulcrum]

ohhhhh ok

[split fulcrum]

so ok using this

so 30 to 40 is 68%?

[lime nexus]

yuhhh

[split fulcrum]

so um how do we use this to help us?

[lime nexus]

since we know now that ppl in their 30s makes up 68% of the ppl in the league

u can use that fact to find the exact amount of ppl in the 30s now right?

we know the total is 10000 ppl

and 68% of that is ppl in their 30s

so what do u think u can do to find the exact amount of ppl in their 30s?

[split fulcrum]

so 10,000 * 68%

[rich elm]

si

👍

[split fulcrum]

6800 people in their 30s

to get from 40 to 50 thats 2 standard deviations?

[lime nexus]

yussss

[rich elm]

not quite

[split fulcrum]

so from the mean its 3 std

[lime nexus]

specifically which 2 stdanrd deviations

[rich elm]

think about it geometrically

draw out the normal distribution curve if need be

think about which part of the graph is inclued in the 40-50

range

[split fulcrum]

uh

[lime nexus]

ill suggest finishing that curve

[split fulcrum]

i dont understand

[lime nexus]

remember one standard deviation is 5 years

is it 50 years after the 40?

[lime nexus]

notice this graph here (its on ref sheet) the 0 essentially means that at thay point it is the mean, the 1 from 0, means its one standard deviation to the right, ig u can think of the negative as in the other direction, its one stdanrd deviation to the left, the 2 means 2 standard deviations to the right of the mean and so on yeah?

[split fulcrum]

yeah

i just dont get why from the mean, 50 years isnt 3 standard deviations

[split fulcrum]

so from the mean to 50 years is 49.85% of data right?

because you halve 99.7

[lime nexus]

yus

the 2nd standard deviation should be 45, not 50

as each stadnard deviation is 5 years right?

[split fulcrum]

ohh

yeah

[lime nexus]

so if it is 40 years at the 1st stdanrd deviation, u need to add 5 to get the the 2nd stadnard deviation right?

and another 5 to the 3rd

[split fulcrum]

yes

[lime nexus]

so far so good?

[split fulcrum]

yeah

[lime nexus]

so first of all we know that terms 6800 ppl in their 30s

we want to find how many ppl r on their 40s

with the info we have rn we are able to calculate the percentages of ppl in their 40%

[split fulcrum]

idk if this is right

,rccw

[lime nexus]

take a look at the curve and tell me between which deviations are ppl aged in their 40s r between

[lime nexus]

wiht this, what is the % of ppl in their 40s

[split fulcrum]

15.85%

[lime nexus]

yessss

now with that

u can use it to find the ppl in their 40s yeah

which isssss

u tell me

[split fulcrum]

1585 people

[lime nexus]

yuhhh

now back to the qn

we want to find the difference in amount of ppl right

so what do u do now?

[split fulcrum]

subtract 6800-1585 = 5215

[lime nexus]

yuhhh

woop woop

i think thays the answer

dont ahve a calculator on me rn

[split fulcrum]

@junior smelt hello

[lime nexus]

eating dinner

[split fulcrum]

ye it is

[split fulcrum]

lmao at 11pm

[lime nexus]

shhhh

[split fulcrum]

so part B how do i do that

[lime nexus]

what do u notice about it

[split fulcrum]

is this just the portion of people who are 50 or over technically

[lime nexus]

(i got this part of the qn wrong before its okay)

[split fulcrum]

[lime nexus]

tbh i thought it was that too

but

it is

but isnt

[split fulcrum]

what 😭

[lime nexus]

read it

very carefully

[split fulcrum]

percentage of people aged 40 or over who are also 50 or over

[lime nexus]

im not a fan of thr wording

[split fulcrum]

what does this mean

do they want me to find like

the number of people in between

or something

[lime nexus]

nope

its like

out ouf the people whos aged 40+, how many of them are 50+

(dont be like me who just thought i just needed to find how many ppl are over 50+) and divided it by 10000

[lime nexus]

well the % of it

[split fulcrum]

but thats just the amount of people who are 50+ cuz technically theyre 40+ 😭

[lime nexus]

the question is basically asking, find the % of ppl aged 50+ but u divided by by the % of ppl aged 40+

[lime nexus]

as much i would like to say yes it isnt

[lime nexus]

ie % of ppl aged 50+ OUT OF % of ppl aged 40+

does this kinda make sense

[split fulcrum]

yes and no

[lime nexus]

which part doesnt

[split fulcrum]

so it wants to know like

[lime nexus]

ill try to explain

[split fulcrum]

out of the people who are 40+ it wants to know how many of those 40+ people are also over 50+

[lime nexus]

yessss

so note

[lime nexus]

this

[lime nexus]

isnt the smae as this

[split fulcrum]

ya i realize now

[lime nexus]

yeyeye

bro they put this qn in my trials but changed the numbers

and part b

i got a 0

bc i just thought isnt it just the amount of ppl over 50

[lime nexus]

out of the total amount of ppl in the league

so sad

anyways

[split fulcrum]

lmfaoo

[lime nexus]

try giving it a go

[split fulcrum]

so people aged 40 or over have z>1

[lime nexus]

correct

[split fulcrum]

so that means that

34% of people have z>1?

[lime nexus]

r u sure>

[split fulcrum]

actually

no im not sure

[lime nexus]

how did u get 34%

[split fulcrum]

is it 16%

[lime nexus]

mhmm

do yk why its 16

do i assume u get 34 from dividing 68 by 2?

[split fulcrum]

yeah

[lime nexus]

remember its only 68% if its between -1 and 1 right

so if u half it

only half of it is 34% ie from 0 to 1 or -1 to 0 is 34%

notice how from 0-1 it is indeed 34%, but that saying its not z>1 right?

[split fulcrum]

yeah

[lime nexus]

its z<1 or more specifically 0<z<1

odes that make sense?

[split fulcrum]

ye

[lime nexus]

yess

another way is that

0 is the middle point

everything to thr right of it is 50%

everyone to the left of it is 50% too

overall making up 100%

so if yk from 0 to 1 is 34%

z>1 is the rest of it

u can just do 50-34

to get z>1

[split fulcrum]

ohhh okay

true

that makes sense

[lime nexus]

yayy

[lime nexus]

this way also works

js longer

time is precious in the hsc!

anyways

we now know that 16% of the total makes up z>1 aka ppl aged 40+

what would u do next?

[split fulcrum]

i multiply it by 10000?

so 1600 people have z>1

or are 40+

[lime nexus]

yes and wha would that number mean

oops

[lime nexus]

correct

[lime nexus]

yes

u got one part done

so now we need to find how many ppl are >50 right?

[split fulcrum]

yeah

[lime nexus]

ur nearly there!

[split fulcrum]

so we know like

there are 1585 people in their 40s

[lime nexus]

yeah?

[split fulcrum]

and there are 1600 people who are over 40

[vale oriole]

lock in

[split fulcrum]

bro go do extension 2 😭

[vale oriole]

lock in bro

[split fulcrum]

stfu i am

im doing

stats

[vale oriole]

lock in

🔒 in

[split fulcrum]

ur 4u exam is on monday too

u lock in

[split fulcrum]

does that mean i can take the inbetween of people over 40 and people who are 40

[vale oriole]

lock in pls

[lime nexus]

could u explain it more?

[split fulcrum]

okay wait let me draw a diagram

[digital sparrow]

im gonna need some context

[lime nexus]

i think i can see what u might be trying to do but thers a shorte way

[lime nexus]

read this again

[split fulcrum]

like if we can subtract this difference