help-23

[short sparrow]

damn, howd u do it

[toxic stratus]

the construction is using one of the quintessential properties of sups/infs over the reals

[solar hazel]

i did it how i explained already lol

idr how i "came up with it" back then

[short sparrow]

oh alr

[short sparrow]

can u elaborate pls

[toxic stratus]

this is one of the key properties of sups over sets of real numbers

you use this exact property to construct the sequence you want

[short sparrow]

what sequence do i want tho

[toxic stratus]

something that converges to sup A

the above property says there are elements in A which are arbitrarily close to sup A

[short sparrow]

yeah i get that

[toxic stratus]

this is the same as saying a sequence converging to sup A exists

[short sparrow]

okay

[toxic stratus]

thats all there is to it

[short sparrow]

can u help me find out what to do with it

cause im not so sure myself

[toxic stratus]

layla has explained the core parts of the solution

[short sparrow]

but where's a or epsilon relevant in any of that

[toxic stratus]

thats how you conjure up new elements for your sequence

written exactly here

[short sparrow]

hmm okay ill try connecting it all now..

[short sparrow]

but how does $$
\text{sup}A - \frac{1}{n} \leq a_n \leq \text{ sup}A
$$ "formally" prove that $a_n$ converges to supA?

[flat frigate]

Levens

[toxic stratus]

do an epsilon-N proof

[short sparrow]

with that? how

[toxic stratus]

typically i would take not 1/n but 1/10^n

so

[toxic stratus]

so then the question is

if you want an element of the sequence within say the 59714th decimal place of sup A

then you can go along 59714 elements of the sequence

and find the 59714th element

and by the way its been constructed

itll be within the 59714th decimal place of sup A

[toxic stratus]

so basically to do an epsilon-N proof

[short sparrow]

wait why -k

[toxic stratus]

?

[short sparrow]

oh oh

lol mb

ok im still reading give me a sec

[toxic stratus]

usually you do if epsilon > 0 then there exists N such that ...

im just saying in the specific case epsilon is 0.0.....01 you can do something like this

and actually thats all you need

[short sparrow]

thanks.. for infA, can I choose $\epsilon = \frac{1}{n} +10$?

[flat frigate]

Levens

[toxic stratus]

for inf you just switch all the inequalities around

[short sparrow]

oh

[toxic stratus]

or if you want to think of it as decimal places

[safe radish]

@short sparrow Has your question been resolved?

[short sparrow]

thank u snow, helped a lot

[safe radish]

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[safe radish]

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[warm dune]

two families rent a helicopter. The cost for the first family of 5 was $520. The cost of the second family of 3 was $432.
Write an equation to represent the cost per person and the base cost.

[safe radish]

@warm dune Has your question been resolved?

[trim swan]

"Write an equation to represent the cost per person and the base cost."

This is what the directions say?

I think a system of equations makes more sense

@warm dune

[warm dune]

yeah like f(x)=mx+b

[trim swan]

Ah, okay. But first we need to find m and b

And we can do that by solving a system of equations

[warm dune]

oh

how do i do that?

[trim swan]

Do you know what I mean by a system of equations?

[warm dune]

would i do 5x + b = 520?

[trim swan]

Yeah, except I'd say 5m + b = 520

m and b are what we want to find

[warm dune]

oh ok

[trim swan]

That's one equation, what's the other one?

[warm dune]

3m + b = 432?

[trim swan]

Yeah

So

Do you know how to solve the system for m and b?

[warm dune]

b = 520-5m?

and b = 432-3m?

[trim swan]

looks good

[warm dune]

5m + (520-5m) = 520

[trim swan]

Close, but you substituted one equation into itself

You want to substitute one into the other

[warm dune]

ohh

5m + (432-3m) = 520?

[trim swan]

Yep

[warm dune]

2m + 432 = 520
2m = 88
m = 44

is that right?

5x44=220

520-300=220

so the equation is f(x) = 44x + 300?

[trim swan]

Yep 👍

You can check it yourself by seeing that f(5) gives 520 and f(3) gives 432

[warm dune]

yeah your right

[warm dune]

thank you very much

[trim swan]

Sure thing 👍

[warm dune]

.close

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[frail wraith]

hello?

[safe radish]

@frail wraith Has your question been resolved?

[obsidian bone]

Its correct.

[frail wraith]

Ima submit this and if Im wrong itll be on you

I really. need the points, no pressure

[obsidian bone]

If you intend to submit that, you should define what F is

[frail wraith]

F is the function that represents the matrix.

[obsidian bone]

Yep. Then thats fine

[frail wraith]

Thanks, if theres nothing more.

[safe radish]

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[safe radish]

@frail wraith Has your question been resolved?

[safe radish]

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[solemn vault]

Can you do a quick check??

[warm rose]

Is 10/18 x 9/17 = 10/17?

And 10/18 x 8/17 = 80/153?

[solemn vault]

Oh ye

Expect that is everything fine??

@warm rose

[warm rose]

Yeah looks fine

[solemn vault]

Ok thx

.close

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[bold void]

help

[bold void]

im stucked

at this problem

help

[thin bridge]

what problem

[bold void]

HEY

HERE

im stucked for like 20mins

[safe radish]

@bold void Has your question been resolved?

[rigid inlet]

@bold void still stuck?

[safe radish]

@bold void Has your question been resolved?

[safe radish]

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[wild trout]

Can someone explain all 3 of these questions? The process to get the roots of the function.

[wild trout]

,rotate

[sonic spruce]

Okay so

[wild trout]

It all equal to 0

[sonic spruce]

If you let x^2=y for first it should be doable

Same way

[wild trout]

Then I put the biggest ^ number in brackets to be ^2?

[sonic spruce]

x^5 = y

And x^(1/3) = y for third

The main takeaway is

[wild trout]

Part b has 4 roots apparently

I need x E R

[sonic spruce]

Can you do that?

Solve for y

[wild trout]

All of it equals 0

[sonic spruce]

There should be 2 solutions for y

And further 4 solutions for x

[wild trout]

(X²)²-12x²+32

[sonic spruce]

Ye

[wild trout]

X=0 x=0 x=4 x=8?

Hmm

How do i get the roots?

I'm confused 😕 can you help me factorise?

[sonic spruce]

Setting x^2= y here, we get y^2-12y+32=0

Can you factorise that ?

For y?

[wild trout]

Ohh

(Y-4)(y-8)

[sonic spruce]

So y is?

[wild trout]

Is that correct?

Y is 4 or 8

[sonic spruce]

Yessir

So x^2 = 4 or 8

Right cause y= x^2

Right?

[wild trout]

Sorry

My mom called me

Yeah

[sonic spruce]

So x is?

[wild trout]

So u square root them

[sonic spruce]

Yessir

[wild trout]

X is 4, can be negative 4

[sonic spruce]

No

Square root?

[wild trout]

I mean 2 and negative 2

[sonic spruce]

And?

When x^2=8

What about then?

[wild trout]

Sqrt 8 and negative 8

[sonic spruce]

Which is also?

But yes

Good job

The next ones are the same logic

[wild trout]

Ty

Makes much more sense now

[sonic spruce]

Can you do the next one?

[wild trout]

Yeah

[sonic spruce]

Great:)

[wild trout]

Ty

[sonic spruce]

Also also

What does you name mean?

Username i mean

[wild trout]

It's just a code

[sonic spruce]

Okay okay

Have a good day

[wild trout]

You too

Tysm

[sonic spruce]

Np:)

Close the channel when you are done

[wild trout]

.close

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[dusky marsh]

sasa..

[sonic spruce]

Lol this channels gonna close soon

[safe radish]

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[remote kernel]

Things got worse rq huh

I could loop forever and ever here

What went wrong?

[quasi bison]

...what even is going on here?

2x+y = 5
4x+z = ?

is this EXACTLY how the problem was given to you?

[remote kernel]

Substitution chaos

[remote kernel]

Yes

[lean otter]

new variable just dropped

[quasi bison]

then there's so, so, so little information.

you literally cannot say anything meaningful.

[remote kernel]

What?

[quasi bison]

if all you know about the variables x, y and z is that 2x+y = 5, then you cannot say anything about the value of 4x+z

[remote kernel]

Oh

Im gonna try and find the original equation

I got online

[quasi bison]

should have gotten your hands on that to begin with tbh

[lean otter]

im guessing you mistook some number with ? or something

[remote kernel]

Gosh dangit i forgot a y

💀💀💀💀💀💀💀💀💀💀💀

Let me try again

[lean otter]

yeah that changes things

a tinsy weensy bit

[quasi bison]

...

well i mean like. yeah.

but also taking this sort of thing directly from Google is sus

[lean otter]

never thought google maths practice was a thing

[remote kernel]

Epic

[lean otter]

😭

where is the y again

[remote kernel]

10 is right

This next one is interesting

[lean otter]

fun fact, you could've just simply multiplied the whole original equation by 2 and that would've saved you a lot more time

[remote kernel]

Which one? Theres two of them?

[lean otter]

[lean otter]

2x + y = 5

multiply by 2

4x + 2y = 10

you are done

[remote kernel]

Oh

Theres a trick

I forgor the y again

Lol

[remote kernel]

Ann thats a two btw

Not a z

Lol

[quasi bison]

i gathered as much

[remote kernel]

Anyways with 2 instead, would it still be solvable?

Anyways

This one is interesting

And yk i have an evil devious idea

[quasi bison]

yes, and lex even already told you how.

[remote kernel]

The forbidden technique

[quasi bison]

this is really bad on many levels

[remote kernel]

Why?

[quasi bison]

the first of which is that you actually went out of your way to write $2 \divisionsymbol 3$ instead of $\frac{2}{3}$ (and likewise in two other places)

and the second line is simply nonsensical

[flat frigate]

Ann

[quasi bison]

granted, the problem itself was also written poorly; it should have been $$\frac{2}{3} x - \frac{3}{2}x = \frac{1}{12}$$

[flat frigate]

Ann

[remote kernel]

How do you know its (2/3)x and not 2/3x?

[quasi bison]

you mean how i knew it wasn't $\frac{2}{3x}$?

[flat frigate]

Ann

[quasi bison]

i had to make an assumption. you were studying linear equations so i assumed the equation you posted here was also meant to be linear.

[remote kernel]

Yeah that

Isnt 2/3x linear too?

I mean no exponents= automatically linear

[quasi bison]

no

2/(3x) has x in the denominator

you cannot have that in a linear equation

[remote kernel]

Hmmmm may i ask why?

[quasi bison]

allowing x into the denominator puts you in the wider class of rational equations, which can get considerably more complicated in their behavior than linear ones

i mean for example $\frac{1}{x} + x = 5$

[flat frigate]

Ann

[quasi bison]

this is not linear anymore

[remote kernel]

Ohh thats scary

[remote kernel]

May i ask about the 2nd line?

[quasi bison]

you can try to ask me but i will be unable to answer except to repeat that it is nonsensical

[remote kernel]

Alright....

I mean technically nothing was wrong?

Normal tranposing rooles

[quasi bison]

no,

there is no such thing as "transposing"

you did some symbol manipulation that ultimately does not mean anything

like

ok no nevermind

there is nothing more i can say here.

[remote kernel]

I guess i just butchered math so badly lol

[quasi bison]

yes

yes you did

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[dark pike]

here is my process for finding the derivative of 1 over x

[dark pike]

apparently the end result should be negative 1 over x squared instead of positive 1 over x squared

why is this?

[fresh remnant]

From step 3 to 4 you dropped the negative sign

You keep it

[dark pike]

surely it should be -x^2 instead of -{1 over x squared} though right?

ah

wait

yeah i think i get why now

[final halo]

The minus out front and the minus in the power aren't interacting in any way

Perhaps you thougut they were cancelling somehow

[dark pike]

nah i was just being silly and not knowing where to move the negative sign but its the same if its on the bottom or on the outside

alright thanks you two 🙂

.close

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[crude star]

is inf^0 = 1?

[final halo]

No

Infinity is not a number

[crude star]

I rmbr seeing a post where one said that

got it doe ty

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[reef galleon]

can i add and remove pi from something like cos(3pi-x) like if its in a form of +2kpi sorry if my vocabulary is bad i study maths in french

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@reef galleon Has your question been resolved?

[safe radish]

@reef galleon Has your question been resolved?

[reef galleon]

<@&286206848099549185>

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@reef galleon Has your question been resolved?

[clear blade]

you can add and subtract 2pi because cosine is periodic

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[lean otter]

do you know how to change 12cos²(a)

in sina idk

[pseudo scroll]

sin^2 A + cos^2 A = 1

[lean otter]

ah ok ty

.closer

.close

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[safe radish]

@fiery escarp Has your question been resolved?

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[lean otter]

<@&286206848099549185>

[lean otter]

nvm

[uncut cloak]

Nice

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[worn cairn]

I am completely lost on this question right now, Three vectors, x,y, and z satisfy x+y+z=0 . Calculate the value of x. y+y.z+x.z if lxl =2 , lyl=3 and lzl=4

[worn cairn]

I've tried applying properties and manipulating the formula, but nothing works and i don't know what to do

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[worn cairn]

<@&286206848099549185>

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[lethal creek]

honestly i just need help working through it bc i have no idea how to solve it

then i have 1 more like this

[solemn vault]

Find such numbers when you multiply them it becomes 60 and when you add them it becomes 23

[lethal creek]

oh goodness

you multiplied a and c right

[worthy hemlock]

Yes

[lethal creek]

k

oh 20 and 3!

[worthy hemlock]

Here's a video on how to factor when a isn't equal to 1
https://www.youtube.com/watch?v=f0GqRY1ORbQ&ab_channel=BrianMcLogan

[lethal creek]

ok thank you

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[sick matrix]

A 43-foot rope is tied to the top of a 31-foot vertical cliff face. When pulled so that it just reaches the ground, what is the angle of elevation to the top of the cliff?

[sick matrix]

This is how I interpreted the angle

[sudden birch]

U can construct a right triangle with this (triangles are every which is awesome, I know). What does it look like? The height of the cliff and the rope belong to which of the legs and hypotenuse?

Oh yeah

Now apply trig stuff. Soh Cah toa stuff

[sick matrix]

I did Tan -1 (43/31) but it got marked as wrong.
So I'm trying Cos-1(31/43) But I don't know if I'm interpreting the triangle in the problem properly.

[thin bridge]

the triangle is correct, you choice of trig function is not

[sudden birch]

^^

Hint:

31 is a side that’s opposite to the angle

41 is the hypotenuse

[sick matrix]

So Sin-1 (31/43)?

[sudden birch]

One is the o and the other is the h in soh Cah toa. Yes u r correct

[thin bridge]

yes

[sick matrix]

Thats weird, we didn't use Sin at all for any of these problems.

[sudden birch]

Sin = opposite side / hypotenuse

[sick matrix]

Ye I know.

[thin bridge]

use whatever the problem calls for

[sudden birch]

Just making shots

Shore

[sick matrix]

46.13

[safe radish]

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[crude ledge]

A 125-foot tower is located on the side of a mountain. In addition, a 170-foot-long guy wire is attached to the top of the tower which is anchored at a point 55 feet downhill from the base of the tower.

What is n ∠A

[worthy hemlock]

Apply law of cosines

[crude ledge]

How?

I’m completely lost on it

😅

[worthy hemlock]

Do you know law of cosines?

[crude ledge]

Partially

[worthy hemlock]

Maybe this will help
https://www.youtube.com/watch?v=3wGQMyaWoLA&ab_channel=ProfessorDaveExplains

[crude ledge]

This one right?

Being the law of cosines

Would be writing

170^2 = 55^2 + 125^2 - 2(55)(125) cos C

?

[worthy hemlock]

Not quite

[crude ledge]

Wrong placing?

[worthy hemlock]

Notice how the angle you are finding is equal to the opposite side. Like in this diagram, side c is across angle c

[worthy hemlock]

So here, is 170 across from angle a?

[crude ledge]

No … it’s 125

[worthy hemlock]

Yes

[crude ledge]

A = 170
B = 55
C = 125

Right?

[worthy hemlock]

Technically the angle you are looking for is angle A, so across from angle A is side A

I just noticed that your problem labeled that angle as A

And not C

[crude ledge]

Yeah!

I’m guessing that would change

The number

Right?

[worthy hemlock]

You would just need to swap some numbers around

[crude ledge]

125^2 = 55^2 + 170^2 - 2(55)(170) cos C

?

[worthy hemlock]

Yeah

[crude ledge]

Thank you!!! A LOT

[safe radish]

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[lean otter]

Uhhh where would I start with this problem, the limit goes to 0, and I’m pretty sure I can’t do the integral test, all I’ve learned is p-series, integral test, comparison test, divergence theorem, and smth else, I havent learned the ratio test tho, any help?

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@lean otter Has your question been resolved?

[lean otter]

<@&286206848099549185>

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@lean otter Has your question been resolved?

[stray bluff]

u can use comparison test

[lean otter]

O okay I’ll try Ty

[safe radish]

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[lament flare]

hello can someone help me with this?

[safe radish]

@lament flare Has your question been resolved?

[marsh walrus]

whatd you try?

[lament flare]

The letter c

[marsh walrus]

what do you mean?

thats what you think is the answer?

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@lament flare Has your question been resolved?

[lament flare]

yes

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[flint python]

this is my global minimum when it asks for the coordinates for the x value would i write it between my domains like for exmaple (x,-20) where x is 20<x<40

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[halcyon stump]

Can anyone have alternative solution for this question. Q) If x=a(cost+tsint) and y=a(sint−tcost), find dx^ 2/ d ^2y ​

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[eager cloak]

who can help me

[safe radish]

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[noble sail]

if a man has to work 5 days in 31 days, without working 2 consecutive days, how many arrangements are possible for his days of work?

[untold sky]

if they have to work for 5 days, how many days must they not be working?

then can you simplify the problem in terms of arranging working and non-working days?

[noble sail]

i dont think this can work

i tried it

[untold sky]

it can

starter: how many ways can you fit the first working day into the block of non-working days?

[buoyant shadow]

it works but you need to try it the right way, it's a trick

[noble sail]

wdym

there are 5 working days

[untold sky]

you get that the man has 26 non-working days

how many ways can you insert 1 working day into the 26 non-working days

then, while satisfying the condition that the man is not working 2 consecutive days, a second working day

[noble sail]

26 for the first one, then 24 for the second one...

no

but i dont see the point

this is very difficult

[untold sky]

the point is that this generalizes for any number of working days :p

instead of computing directly, try to fit working days between non-working days

[noble sail]

why? why are we fitting working days into non working days? to avoid having 2 consecutive days of work? if yes how are we achieving that

[buoyant shadow]

there's 27 slots when you have 26 days

it goes in between

[noble sail]

how?

each day represents one slot no?

[buoyant shadow]

no, they are flexible like that

you imagine slots in between

and unimagine them back after choosing

[noble sail]

I dont get what you mean

[buoyant shadow]

like if you have two non working days and you add a working day
.X.X.
the dots are where you can fit it

WXX, XWX, XXW

and this solves the problem in one step

if you;re slotting many working days at once, they will automatically be non consecutive

it's a trick, iv just hoped you'd invent it

[noble sail]

I still have a hard time understanding this

[noble sail]

here the X represents non working days?

[buoyant shadow]

yeah

[noble sail]

and the dots my possibilities?

for work

[buoyant shadow]

you keep them both, you add a W in between

yes

[noble sail]

so i am attempting to fit 5 W between 27 X

fuck

i see

very clever

[buoyant shadow]

when you add a second day, the dots will literally stay like that

so it's a one step choice

[noble sail]

but you removed the dots

i dont get why the dots are necessary

we're just starting with 27X and then seeing how many ways we can fit a W between 2 X right?

[buoyant shadow]

well yeah they don't carry information

just visual aid

[noble sail]

this is correct?

[buoyant shadow]

i can add them back WXX → WX.X

and it recreates the pattern before removing

[noble sail]

Ah i see the point of the points

[buoyant shadow]

you would think it's literally stars and bars but I don't see a way to make it like that

[untold sky]

it is, a+b+c+d=2 nope, a+b+c+d+e=22

then stars and bars

oh actually might be 22

[noble sail]

what made you think of this emthod

i want to understand how we went on to think about this

[buoyant shadow]

someone probably spoiled it to me i don't remember

[noble sail]

so there is nothing that can help me in future problems, some pattern to spot

i was helpless in front of this one

[buoyant shadow]

so how does it work?
I can see like, we're distributing 26 in 4 stretches
and then also into 5, twice

and then also into 6

[untold sky]

no it's 5 bars, give me a second, my brain is dying .-.

[fair hound]

it's 5 bars yeah

[buoyant shadow]

but how

you're allowed to have one group empty, or the other one or even both

[untold sky]

ok it's 5 bars and 22 stars, that took too long to wrap my head around

[fair hound]

by making a change of var

yep

[buoyant shadow]

ahh

yes

it clicked

thank you both

[fair hound]

you have 5 bars and 26 stars
but 4 of the bars can't directly follow each other, so you sub these 4 bars by a shift of 1

and you get 5 bars and 22 stars

[buoyant shadow]

26−4 yeah

[noble sail]

is this something i should know :/

[buoyant shadow]

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

yes please :D

[noble sail]

great, a wiki article

[untold sky]

it's a neat trick that will save you a lot of time if you get it into memory

when i was doing math comp and undergrad combinatorics we had a saying "if it looks too hard transform it into stars and bars"

[noble sail]

looool

[buoyant shadow]

yeah that's lame i know, i don't know youtube links

[noble sail]

lool no i wasnt sarcastic i spend most of my time on wiki

but wait didnt we actually transform this problem to make it easier

like it isnt easy, but we seem to have added a spot to make it easier

no?

our month became 32 days

of which we need 31

I just cant see how this overwhelming problem is suddenly a simple one

and instead of picking the days of work and non work... we instead fixed the days of non work and simply bothered with our days of work right?

[untold sky]

the not-so-rigorous answer is that you transform the problem in a way that you won't be counting in a way where previously arrangements drastically changes count formula

if you try to brute force by saying "okay day 1 can be a working day, then 3,5,7,9, then if you fix 1,3,5,7 you have 23 more choices" you get massive nested sums

[noble sail]

wdym previously arrangements drastically cahnges count formula

That 1,3,5,7 and 1,3,5,8 are drastically different?

[untold sky]

if you try to count by "if work day 1 is here how many ways can i place work day 2, then how many ways can i place workday 3"

you can immediately see that the number of ways to place say, workday 3 depends on placements of workday 2 and workday 1 in a not very nice way (and gets much worse for days 4/5)

so you want to either 1. make the dependency less nasty or 2. hopefully skip the dependency thing altogether

[fair hound]

another way to see it for dazzle:
a sequence of working and non working days is a sequence of w and x, where w are working and x non working
you have 5 w and 26 x
but two w can't be next to each other
so we make a little transformation: we couple 4 of the x with 4 of the w, to form couples Y = (x, w), and now Y can follow w and Y can follow each other
now we have 27 symbols: 1 w, 4 Y, 22 x where w and Y are of the same object "working day to place in the sequence"

[noble sail]

yes

[untold sky]

the method frog introduced, where you insert work days into the 26 non-work days, makes the dependency less nasty (you just get 27 * 26 * 25 * 24 * 23, and you can determine how many ways to insert more)

stars and bars skips that dependency altogether, by observing the structure:
there must be at least 4 non-work days between the first work day and the last work day, since no work days can be consecutive

therefore when we say 5 bars and 22 stars, it's equivalent to treating there being only 22 non-working days, and removing the constraint that workdays have to be non-consecutive, because we implicitly add 4 more non-work days, one after each of the first 4 work days

[noble sail]

yeah thats a cool way to put it

[noble sail]

yeah i mean just how did the method frog used removed dependency what did he notice

[noble sail]

The structure is a very cool thing ill look for in future problems

just incredible

[untold sky]

what frog's method does is it ignores the order of the work days

because when you try to brute force, you label work days "first work day", "second work day (after the first)", etc

by inserting workdays between non-workdays you're saying one of the workdays can be here, .. etc

[noble sail]

Ah

I see why this would simplify a ton the work

[untold sky]

also that method to count under the restriction of no two identical items together is one of the earlier tricks you'll learn in combinatorics probably in the first lecture or two

worth remembering because it's easy to apply

[noble sail]

Okay

how did you get so good at combinatorics

you seem to have understood it to its core

[untold sky]

did math comps back in middle/highschool and read a decent amount of brilliant/art of problem solving

[noble sail]

waaw

whats brilliant/art of problem solving

can i access them?

[untold sky]

https://artofproblemsolving.com/community

https://brilliant.org/home/

[noble sail]

are they free?

[fair hound]

artofprobemsolving is

[untold sky]

aops is, brilliant no

[noble sail]

and do they explain core intuitions?

or just solutions

[fair hound]

depends tbh, aops has problems + people who gives solution

[noble sail]

like you did

[fair hound]

some people are clearer than others

[noble sail]

okay i see

thanks a lot this was super helpful

.close

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[misty belfry]

statistics question: Can you have an estimate with no bias and high variance?

[misty belfry]

doesnt "no bias" imply ur estimate is the same as the population parameter?

so for example if i want to estimate the mean of something using a sample data what does it mean for my sample mean to have bias zero?

[fair hound]

no bias means the expected value of your estimate is the real expected value for your parameter

[misty belfry]

and in that case is it possible to have no bias and also high variance?

[fair hound]

yes, the impossible task is to have both no bias and low variance

[misty belfry]

why would it have high variance? high variance means if i get another sample data i would get a different result right?

and if i keep repeating the results will differ largly

[fair hound]

high variance means the probability of getting data far from the expected value is high

like a flat distribution instead of a condensed one

[misty belfry]

but doesn't no bias mean that what im estimating will be close to what is real?

nvm is ee

[safe radish]

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[night sparrow]

How to prove AD bisect angle BAC?

[fair hound]

AD is diameter so ABD and ADC are right triangles and share two sides since BD = DC

[night sparrow]

angles in semi circle?

[fair hound]

I don't understand what you're trying to say

I'm talking about similar triangles

ABD and ADC are congruent

[night sparrow]

i see

thx i can solve it now

.close

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[hidden rose]

how would i go about multiplying a rotation martix for the Xs and Ys here? i havent learned aboht matrices yet but im trying to understand what im supposed to do when given such a thing

[hidden rose]

just out of pure curiousity

[obsidian bone]

https://www.mathsisfun.com/algebra/matrix-multiplying.html

[hidden rose]

so im supoosed to multiply the x with every single one of the matrix elements in Rx?

is it supposed to be a matrix with xyz coords and then i would need to multiply it like 2 matrices would be multiplied

[obsidian bone]

$\begin{bmatrix} 1 & 0 & 0\ 0&\cos{\theta}&\sin{\theta}\end{bmatrix}$

[hidden rose]

$\matrix{x, y, z}$

like this?

[obsidian bone]

Right

[hidden rose]

ah so its literally just 2 matrices being multiplied

that was the only thing i didnt undestand, whether it was supposed to be just x multiplied, but that would be weird and not propper, so i was confused as to what exactly to multiply

thx

.close

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[crisp lodge]

hey guys
I was tasked with solving the diff equation, f''+f=(cosx)^2
I managed to arrive at the solution f=1/2-1/6(cos(2x))

[crisp lodge]

however, im not sure if those are indeed all the solution, not am I sure how to compute it to account for all possible solutions

can anyone calrify this for me ?

[white relic]

help

[solid shell]

Go to #❓how-to-get-help, this is someone else’s channel

[safe radish]

@crisp lodge Has your question been resolved?

[safe radish]

@crisp lodge Has your question been resolved?

[safe radish]

@crisp lodge Has your question been resolved?

[safe radish]

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[lean otter]

when given a function like this

how do I find the zeros?

[glass carbon]

notice that it's always negative

[lean otter]

i meant the root

it wont touch at y = 0

thats for sure

[glass carbon]

u want to solve

[lean otter]

yep

[glass carbon]

?

[flat frigate]

Modus

[glass carbon]

or what

[lean otter]

yes

[glass carbon]

then think about it - can it achieve positive value? (I mean function)

[lean otter]

the x can be positive

[glass carbon]

well, other question, can sqrt( something ) be negative?

[lean otter]

i solved it on a calculator, just wanted to know how the process is with such functions

[lean otter]

no

if i remeber

[glass carbon]

yes, it can't

so

it can be 0 or positive

[lean otter]

yes

[glass carbon]

let's consider it's even zero, then if we subtract 4 we'll definitely get negative value

[raw pivot]

what about complex values

[glass carbon]

wait, you're asking about complex? ;d

[raw pivot]

just generally asking

[glass carbon]

but still

it works

[lean otter]

substract 4 from what

isn't it add ?

[glass carbon]

from 0 or negative value

in other words you have:

$$\sqrt{4-x^2}=-4$$

[flat frigate]

Modus

[glass carbon]

you've said square root is always NON-NEGATIVE

on LHS u have negative 4

so is there any x such that equation is true?

[lean otter]

this equation is false

[glass carbon]

yes

this means function - 4 - sqrt(4-x^2) has no zeros

[lean otter]

ok with functions that dont touch at y = 0

how do we do it ?

i imagine this parabole touches -4

or something

y = - value

@glass carbon do I do trial and error to see which x value the function stops ?

[nova creek]

.

If the function has no zeroes

Then it has no zeroes

Full stop

[lean otter]

this is the function here, it stops at x = - 2

x = 2

[nova creek]

But it never touches y = 0

[lean otter]

ok so I have to trial and error with the y

[nova creek]

No

It will never touch y = 0

[lean otter]

i know

[nova creek]

You're asking for the zeroes of that function

There are none

[glass carbon]

maybe formulate your question properly

Are you thinking about y-intercept or what

domain maybe??

[lean otter]

the points -4, -2 ; -4, 2

ok domain

thats the one that gives these points

it works

thanks

.close

[safe radish]

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[mortal kernel]

Can someone help me with this? My professor wants it done with Plurality with Elimination AND Pairwise Comparison .

[safe radish]

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[mortal kernel]

.close

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[sharp wharf]

I don’t really understand what the function g does

By g(f) is it referring to a general function f in B_n?

But then what is f(1)… representing?

[obtuse jackal]

f is in Bn yes

It's based on the idea that an n-periodic function is uniquely defined by its first n values

So this associates the function to these values

And proves it's a bijection

[sharp wharf]

Could you explain this a bit more, I’m not following when you say by it’s first n values

[obtuse jackal]

Say you want to find f(k).
You can just use euclidian division to write k = qn + a
Then because f is n-periodic, f(k) = f(a), with a in [1, n[.
Therefore f is entirely defined by its first n values. The rest is found by repeating it

[sharp wharf]

I see, thanks. Essentially this g function is just mapping f to its n tuple so f(1)…f(n) before it repeats

What is f and f over score though?

I also don’t really get how it is injective

Since take for example a function f with f(1) to f(5) as (2,4,6,2,8,)it has period 5 but, f(1)=f(4)

[obtuse jackal]

You don't want f to be injective. You want g to be injective

[obtuse jackal]

They took f, f_ as the two functions for injectivity. I agree it's a very weird notation

[sharp wharf]

I see on the very last line shouldn’t it also be f over score(p)

But I don’t really see how I would efficiently come up with this in an exam haha, are there any tips to do so?

[obtuse jackal]

Yes. That proves f = f_, and thus that g is injective

[obtuse jackal]

Honestly this is pretty straightforward I'd say

How do you describe periodic functions ? First by their period. That gives you the union of the Bn

Then you know that it is sufficient to prove each Bn is countable (or uncountable since it's an open question)

[obtuse jackal]

Then a simple understanding of how periodic functions behave lets you find this function very quickly and guess that it's a bijection. Then checking it isn't difficult either

So if you understand the material, I wouldn't say there's a single hard step for coming up with this proof. I came up with the same thing before looking at the solution and took me a few seconds. Which is not meant to be boastful, but to argue that it's quite a straightforward proof for a countability problem

[safe radish]

@sharp wharf Has your question been resolved?

[safe radish]

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[fiery escarp]

<@&286206848099549185>

[safe radish]

@fiery escarp Has your question been resolved?

[tropic schooner]

What's the first part of the question that you find confusing?

[fiery escarp]

I computed the wronskain determinant and got -3

So I’m not sure how to get the dimensions

[tropic schooner]

As far as I can tell, the Wronskian determinant (that I had to look up, because I hadn't seen it before) isn't necessary to solve this problem.

[fiery escarp]

I was thinking the dimensions of the subset h is 3

[tropic schooner]

I do get that dim(H) is 3.

[fiery escarp]

And for part b I said it is a basis for p2

And for part c I got some fractions then o I don’t know what to do

[tropic schooner]

What is dim(P_2)?

[fiery escarp]

The question doesn’t say the dimensions of p2

[tropic schooner]

Hm. I guess I can explain it a different way too. You said that the three polynomials are a basis for P_2, what does that mean?

(I'm looking for a particular idea, or an equation, that follows from the definition of what a basis is.)

What is the span of a basis?

Why did you say for part b that it is a basis for P_2?

(As in, what led you to decide that.)

[fiery escarp]

Here’s my working out

Wait so it doesn’t spam p2

So it’s not a basis for p2

[tropic schooner]

FYI it's -4, not 4, for the matrix you wrote in the middle there.

What's the standard basis for P_2?

[fiery escarp]

O

So I messed up

I can’t be asked anymore

[tropic schooner]

In this case, it does span P_2.

The standard basis for P_2 is {x^2, x, 1}. It shouldn't be too hard to convince yourself that span({x^2, x, 1}) will give you every polynomial of degree at most 2. Therefore dim(P_2) = 3.

How I solved the problem: Since dim(H) = dim(P_2) = 3 and H is a subspace of P_2, this means that H = P_2. This makes answering b and c easy.

[safe radish]

@fiery escarp Has your question been resolved?

[fiery escarp]

Does this help with solving c

[tropic schooner]

I... can't deduce what you're doing for most that image.

[fiery escarp]

Oh

[tropic schooner]

The only thing I could recognize was the matrix made of the three polynomial's coefficients (with 4 instead of -4).

[fiery escarp]

I'm confused how to answer c

[tropic schooner]

If you have that H = P_2, and the three polynomials are a basis of P_2, what is a basis of H?
If you have that H = span({three linearly independent polynomials}), what is a basis of H?

Hint that gives away what I tried to get you to remember/say: ||A basis of V is a set of linearly independent vectors (in this case polynomials) that span V.||

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@fiery escarp Has your question been resolved?

[safe radish]

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[white crow]

so for (a), I got 1 as my gcd using the euclidean algorithm. How do I apply that to question (b)? Do I just solve for s or t?

[misty coral]

when you worked out your gcd

you got something like a=bq+r right

and r is 1

[white crow]

mhm

[misty coral]

you can think of it as going backwards

if r = a-bq

[white crow]

37 x 2 + 1

oh

[misty coral]

eight

right

so express one in terms of 37 and 75

[white crow]

so 1 = 75 - 37 x 2?

[misty coral]

exactly

,calc 75-(37x2)

[flat frigate]

The following error occured while calculating:
Error: Undefined symbol x2

[misty coral]

oop

,calc 75-(37*2)

[flat frigate]

Result:

1

[misty coral]

seems legit

[white crow]

but how do I know what the values of s or t are based on this?

[misty coral]

you've done it

1=37(-2)+75(1)

[white crow]

ohh

its because you subtracted

[misty coral]

for the negative? yes

[white crow]

right

thanks for the help

.close

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[pastel kite]

What would 5x+6=56 be?

[foggy salmon]

um

first u want only x on 1 side

im guessing u r solvin forx

so we can start with -6 on both sides

what do u get after that?

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@pastel kite Has your question been resolved?

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[plain gate]

i answered iii) only

im just unsure if thats correct

im 99% sure it is

[turbid raven]

They are all discrete I think @plain gate

[stray bluff]

yeah seems right

[stray bluff]

how ?

[turbid raven]

Well it's not how much gas it is, it's the number of litres

[trim swan]

Which can be any positive real number, not just integers

same with the age and weight, they are continuous

[turbid raven]

Idk the task seems extremely ambiguous.

[trim swan]

I disagree

I think you could go real brainiac and count the number of molecules of gasoline and say it's discrete

But that seems counter to the spirit of the question

And the age and weight are unquestionably continuous

Looks good

[plain gate]

.close

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[white crow]

So i did (a) and (b), got 1 as the gcd for (a), and s = -2 and t = 1 for (b)

[white crow]

for question (c), I did 1 == 75(1) + 37(-2) (mod 75) and since 75 mod 75 is 0 I'm left with

1 == 5(-2) (mod 75)

oh wait

is 10 supposed to be the answer

[drowsy moss]

where did the 5 comes from?

[white crow]

that's a good question why did I write 5

uh

let me go fix it

ok so

what i have now is

1 == 37(-2) (mod 75)

[drowsy moss]

yeah

so what's the inverse of 37 (mod 75)?

[white crow]

-74?

[drowsy moss]

nope

what's the definition of an inverse (mod n)?

or really any multiplicative inverse

[white crow]

37(x) == 1 (mod 75) right

[drowsy moss]

yeah

[white crow]

well 37 and 75 could be any number

[drowsy moss]

so what's the inverse of 37 (mod 75)

[white crow]

oh

-2?

[drowsy moss]

yes

[white crow]

one question

[drowsy moss]

or 73 if you want to make it positive

[white crow]

why is -74 mod 75 1?

[drowsy moss]

two numbers a and b are equivalent (mod n) if n | a-b

well, 75 | 1 - (-74)

[white crow]

oh

makes sense now

[drowsy moss]

another way to think about it is a and b are equivalent (mod n) if a = b + kn

[white crow]

ah

and k is integer

[drowsy moss]

yep

[white crow]

right

thanks for the help

.close

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[opaque sorrel]

@toxic stratus hi

for the characteristic equation is it just lambdaA - I

how do i find lamda

[drowsy moss]

characteristic equation is det(A-Lambda*I) = 0

[toxic stratus]

hi

[opaque sorrel]

o

[opaque sorrel]

nihao

where do i spawn in lambda tho

or is it an arbitrary value

oh wait

it is

[drowsy moss]

arbitrary

[opaque sorrel]

cuz the equation roots are the eigenvalues

oh

[toxic stratus]

lambda is just some number

like

you get a polynomial

[opaque sorrel]

ohhh

lemme do it

thx

did u miss me

[obsidian bone]

Two snows?

[opaque sorrel]

yes

[toxic stratus]

itll just look like that

a polynomial

[opaque sorrel]

yesmaam

the roots are the

[toxic stratus]

e-vals

[opaque sorrel]

egyenvalues right?

[toxic stratus]

ye

[opaque sorrel]

nice

lemme do it rq

wait

its det(A - lambdaI) = 0

can it be

det(lambdaI - A) = 0

[toxic stratus]

doesnt really matter since its just a possible sign change

[opaque sorrel]

o

[toxic stratus]

if you do lambda first the coefficient in front of the largest power of lambda will always be 1

[opaque sorrel]

how do i type lambda in latex

[toxic stratus]

so idk

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