help-27

[rocky marlin]

ok ima do the other question now

[devout snow]

@rocky marlin Has your question been resolved?

[rocky marlin]

@potent dirge

[potent dirge]

Yep. It’s just gonna be the first graph, but flipped

[rocky marlin]

ah ok thanks

[rocky marlin]

.close

[devout snow]

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[devout snow]

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[heavy current]

say that a metric space $X$ is a length space if for any $\varepsilon > 0$ and $x, y \in X$, we have a path $\alpha$ connecting $x$ and $y$ such that $\ell(\alpha) < |x - y| + \varepsilon$, where $\ell$ is the length function.
\
say that a metric space $X$ is geodesic if for any $x, y \in X$, we can find a geodesic connecting $x$ and $y$, where by a geodesic we mean a map $\gamma: I \to X$ such that $|\gamma(s) - \gamma(t)| = |s - t|$ for any $s, t \in I$.
\
I want to show that any proper length space is geodesic.

[woven radish]

higheг!

[heavy current]

here's what I've done so far

fix $x, y \in X$ and let $\varepsilon_n > 0$ be a monotone sequence st $\varepsilon_n \to 0$. since $X$ is length, we can choose paths $\gamma_n$ from $x$ to $y$ so that $\ell(\gamma_n) < |x - y| + \varepsilon_n$. WLOG, we can take $\gamma_n$ to be parametrized proportionally to their arc lengths, i.e. $\ell(\gamma_n |_{[s, t]}) = c(t - s)$ for some $c \in \R$ and any $s, t \in I$.

[woven radish]

higheг!

[heavy current]

now for any $\varepsilon > 0$, take $\delta = \varepsilon/c$ and get that for any $n$ and $s, t$ such that $|s - t| < \delta$, we have $|\gamma_n(s) - \gamma_n(t)| \leq \ell(\gamma_n |_{[s, t]}) = c|t - s| < c \delta = \varepsilon$, so $\gamma_n$ is equicontinuous

[woven radish]

higheг!

[waxen niche]

define proper?

[heavy current]

closed + bounded = compact

hm, hold on

as I was writing up the next section, I realized I've made a mistake

my apologies, I return when I've fixed this up, since it's preventing me from continuing

.close

[devout snow]

Channel closed

Closed by @heavy current

Use .reopen if this was a mistake.

[heavy current]

.reopen

[devout snow]

✅ Original question: #help-27 message

[heavy current]

agh, I guess I'll repost everything again, since that'll be easier to read

[woven radish]

higheг!

[heavy current]

with all of that context provided, I want to prove that this subsequence \gamma_n is a geodesic path from x to y to conclude

but uh, I'm having a fair amount of trouble with that, and I'm sorta not even sure what to do from here

[heavy current]

I messed this up too, since c depends on n generally, but I've fixed it by choosing a different delta instead

[heavy current]

okay, I know that $\ell(\gamma_n |_{[s, t]}) = \ell(\gamma_n) |t - s|$, so because distance is always less than or equal to path length, $|\gamma_n(s) - \gamma_n(t)| \leq \ell(\gamma_n) |t - s|$, so by taking the limit as $n \to \infty$, I get $|\gamma(s) - \gamma(t) \leq |x - y| |s - t|$ because $\ell(\gamma_n) \to |x - y|$

[woven radish]

higheг!

[heavy current]

if I can just show the other inequality, I'll be done by reparametrizing \gamma

more work to do

I think.. I got it!

by the triangle inequality, $|x - y| \leq |x - \gamma(s)| + |\gamma(s) - \gamma(t)| + |\gamma(t) - y|$; by the last thing I wrote, this implies that $|x - y| \leq |x - y|s + |\gamma(s) - \gamma(t)| + |x - y|(1 - t)$, so by rearranging we have $|t - s| |x - y| \leq |\gamma(s) - \gamma(t)|$

[woven radish]

higheг!

[heavy current]

that gets us the other inequality, so by reparametrizing like $\widetilde{\gamma}(u) = \gamma(u/|x-y|)$, I'm done, I think

[woven radish]

higheг!

[heavy current]

good grief, this was hard

thank you @waxen niche for choosing to take a look at my mess of a question

and thank you @crystal dawn @gloomy aurora @uncut crow for the moral support

I'll close this in a few mins if I can't find any more errors

.solved

[devout snow]

Channel closed

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[devout snow]

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[ripe pivot]

Integration by parts help

[ripe pivot]

Hi, I have this problem that i need to use integration by parts, im just slightly confused about the I = integral lnx/x how does that work?

[tender cobalt]

the more sensible approach is u-sub, do you know how you'd do this with u-sub

[ripe pivot]

which method is u-sub im unfamiliar with the terms

u and du?

u would be lnx and du would be 1/x correct?

[tender cobalt]

u' would be 1/x

du/dx would be 1/x

du would be 1/x dx

[ripe pivot]

kinda lost on what u meant

this is a method i was taught

[tender cobalt]

you wrote down "du would be 1/x"

thats not correct

you mean du would be 1/x dx

[ripe pivot]

1/x dx

[tender cobalt]

you forgot the dx, yea

so its not 1/x, its 1/x dx, thats what I meant

[ripe pivot]

gotchu

[tender cobalt]

you can see here we should get (ln x)^2/2 + C out of this

now lets see how IBP does this

first, IBP was used to turn that top line to the next line

do you understand this step?

[ripe pivot]

i get the first part but idk what happened in the seccond part

[tender cobalt]

ok look at the first line again

it says I = ∫ (ln x)/x dx right?

[ripe pivot]

yup

[tender cobalt]

so what do you think ∫ (ln x)/x dx ** + 1** would be?

I + 1?

[ripe pivot]

yeah?

[tender cobalt]

yea

so if you have - ∫ (ln x)/x dx

wouldnt that be - I

[ripe pivot]

true

[tender cobalt]

yea thats all

the trick here is that IBP may give you the same integral back

this isnt an issue, because you get "integral = something else with that integral in it"

but look at it like I

I = (ln x)^2 - I

you can solve for I like an x now

[ripe pivot]

this part i completely understand

[tender cobalt]

is it the missing + C?

[ripe pivot]

no not that

is it basically reassigning the value of I?

[tender cobalt]

no

look at what changed

to call it a "reassignment" you need to be saying "the values change"

no they dont

did IBP change the value of the integral? no

[ripe pivot]

wait could i explain what i mean

[tender cobalt]

well we dont need a further explanation than what the steps are showing you

theyre just algebra, if you can trust the algebra then youre forced to trust the steps

[ripe pivot]

algebra is fine but where does lnx^2 come from in the second part thats the only thing im stuck up on

didnt take IBP so im kinda confused

[tender cobalt]

didnt you say you got the IBP

[ripe pivot]

not ibp

i meant this method ig

[tender cobalt]

this is IBP

are you talking about this (ln x)^2?

the one that IBP made?

[ripe pivot]

yes that

[tender cobalt]

yes this is just IBP

there are a few ways people show this

here, you forgot a dx again

dv = 1/x dx

[ripe pivot]

actually this wasnt me 😂

[ripe pivot]

okay i see

[tender cobalt]

∫ u dv = uv - ∫ v du

∫ (ln x) (1/x dx) = (ln x)(ln x) - ∫ (ln x) (1/x dx)

(ln x)(ln x) is (ln x)^2

[ripe pivot]

yup

[tender cobalt]

both integrals are the same I = ∫ (ln x)/x dx

[tender cobalt]

with this formula, you plug in the u, v, du, dv, and have the left side get rewritten to the right side

that way it turns ∫ (ln x)/x dx to (ln x)^2 - ∫ (ln x)/x dx

so other than the missing dx, do you get it now

[ripe pivot]

yeah i got it thank you

[tender cobalt]

np

[ripe pivot]

the missing dx is in dv correct

just to be sure

im not lost

[tender cobalt]

yep, just like earlier

du = 1/x dx when you did u-sub

[ripe pivot]

okay thanks bro

[tender cobalt]

yep

[ripe pivot]

🙏

.close

[devout snow]

Channel closed

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[devout snow]

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[hollow bolt]

<@&268886789983436800>

[devout snow]

Channel closed

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[red hornet]

please @ me (I asked class mate he didnt answer)

[celest lantern]

Is that the previous question?

[red hornet]

yes

ive asked many people

[celest lantern]

Yup correct

Now cancel out sin 45

You multiply sin 45 to both side.

[red hornet]

makes sense now

[celest lantern]

No worries.

[red hornet]

whats next

[celest lantern]

Well show me what you got?

[red hornet]

do i move 5 to rhs?

[celest lantern]

Correct, now you see $\frac{x+3}{\sin{30}}(\sin{45})$

[woven radish]

Mercury (ヤフォダ)

[red hornet]

yes

[celest lantern]

It is the same as: $\frac{1}{\sin{30}}(\sin{45}) \cdot (x+3)$

[red hornet]

should we make sin, into their numbers?

[woven radish]

Mercury (ヤフォダ)

[celest lantern]

Right?

You basically take of (x+3) from the numerator.

[red hornet]

i dont see a reason to

[celest lantern]

Then it is still the same to: $\frac{\sin{45}}{\sin{30}}(x+3)$

[red hornet]

i follow logic

[woven radish]

Mercury (ヤフォダ)

[celest lantern]

Now, can you calculate $\frac{\sin{45}}{\sin{30}}$?

[woven radish]

Mercury (ヤフォダ)

[red hornet]

yes

[celest lantern]

Please do.

[red hornet]

sqrt 2

[rain summit]

yep

[celest lantern]

Correct.

Then would it be $\sqrt{2}(x+3)=(2x-5)$?

[woven radish]

Mercury (ヤフォダ)

[red hornet]

yes

[celest lantern]

Can you expand and solve for x?

[red hornet]

now i see what happened last time

[rain summit]

-# ||definitely square this thing and solve a quadratic (jk you shouldn't do that)||

[red hornet]

the other person made a mistake and said sin 45/sin 30 is 2sqrt 2

[rain summit]

yes!

[rain summit]

it's sqrt 2

[celest lantern]

Coder is a false information giver.

[red hornet]

maybe its an accident

i think best of people

[rain summit]

anyways you can solve for this right

[celest lantern]

Well, anyways, can you focus in expanding it.

[celest lantern]

Wait, what?

[red hornet]

wait i see a mistake\

[celest lantern]

Expand this: $\sqrt{2}(x+3)$

[woven radish]

Mercury (ヤフォダ)

[red hornet]

found ma mistake

do i move 5 to rhs

<@&268886789983436800>

thank u

did everyone dip?

@celest lantern ?

[celest lantern]

Correct.

[west ridge]

How?

What nonsense you told.

[celest lantern]

Now: $2x- \sqrt{2}x=3\sqrt{2}+5$

[red hornet]

i like to think best of people

[woven radish]

Mercury (ヤフォダ)

[red hornet]

it was an accident

[celest lantern]

Can you continue here?

[west ridge]

Let's see

[red hornet]

uhh what, that defeats all my logic

and everyone dipped again

[celest lantern]

You just move all x to one side.

And non-x to other side.

[west ridge]

Yes. I've been trying to explain this to him

[red hornet]

ok that bring my logic back

[west ridge]

Not spreading false information

[red hornet]

ik ur not

[west ridge]

x(2-√2)=3√2+5

[red hornet]

factoring?

[west ridge]

x = (3√2+5)/(2-√2)

[red hornet]

buddy step by step

i need to understand why we do each step

[west ridge]

See I took x common from LHS

Understood?

[red hornet]

i get this

[celest lantern]

Now you divide all by 2-sqrt2.

To remove it from x side.

[west ridge]

The divide by 2-√2 in both sides

[west ridge]

Yes

[red hornet]

this is what i dont get

[west ridge]

Then rationalisation

[red hornet]

ohh that says divide i thought it says = silly me

[west ridge]

We divided both sides by 2-√2

Then we multiply and divide only rhs by 2+√2

We get $2^2-(\sqrt{2})^2$ in denominator

[woven radish]

Coder decoder

[red hornet]

shoot i flipped em

[west ridge]

And $(3\sqrt{2}+5)(2+\sqrt{2})$ on numerator

[woven radish]

Coder decoder

[red hornet]

im supposed to multiply and divide?

[west ridge]

Yes

[red hornet]

are we doing long division?

[west ridge]

Its what I was taught in 9th and 11th

No we are removing $\sqrt{2}$ from denominator as it makes calculating it difficult

[woven radish]

Coder decoder

[red hornet]

cant we just cancel em out

[dusty valley]

just foil

[west ridge]

Then whats the point?

There is no point then

[red hornet]

since we have square root 2 on numerator and denomerator, dont they cancel?

[red hornet]

do yk what ur talking about

[dusty valley]

yeah

[west ridge]

How? Thats not what it worked like

[dusty valley]

oh wait

[red hornet]

u sure?

[dusty valley]

whats even the problem bc the bot is spitting out random stuff

sorry im new'

[red hornet]

can i please have coder as my helper

thanks

[west ridge]

The bot renders the complex mathematical equations which are not on keyboard

[red hornet]

ok so what was my next step

[wintry socket]

You got the answer

[west ridge]

Multiply the terms in numerator using standard methods

[red hornet]

that isnt the answer

so 2*-2?

[red hornet]

no i didnt

[west ridge]

What 2^2-2 is denominator

[celest lantern]

This is correct.

[red hornet]

where?

[red hornet]

is that my final answer?

[west ridge]

And (3√2+5)( 2+√2) is numerator

[red hornet]

can u show me in latex?

[celest lantern]

$\frac{(3\sqrt{2}+5)}{(2-\sqrt{2})}$ You are here, right?

[woven radish]

Mercury (ヤフォダ)

[west ridge]

$$(3\sqrt{2}+5)(2+\sqrt{2})/2$$

[red hornet]

correct

[celest lantern]

Then multiply $2+\sqrt{2}$ to the numerator and denominator.

[woven radish]

Coder decoder

Mercury (ヤフォダ)

[west ridge]

We've past that step

[red hornet]

ohh so we can get the rid of the square root?

[celest lantern]

Yes.

[celest lantern]

Really?

[west ridge]

Yes

[red hornet]

are we?

[west ridge]

Yes multiply the numerator and done

What you got?

[red hornet]

multiply n with d?

[west ridge]

No n1 with n2

[lusty leaf]

Can u send the question again

[red hornet]

this n?

i cant really weve been working on it here only

[lusty leaf]

Ok np

[lusty leaf]

If u dont get i can help

[west ridge]

Multiply 3√2+5 with 2+√2

[west ridge]

thanks

[red hornet]

6 sqrt 14

[lusty leaf]

Np

[west ridge]

Ok divide that by 2

[celest lantern]

It's pinned.

[west ridge]

And now we'll use Wolfram

,w 3√14

[woven radish]

Results provided by WolframAlpha

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

Requested by codingdecoder

[west ridge]

Something is incorrect in you multiplication

[lusty leaf]

That triangle question?

[celest lantern]

The one with x-variables.

[west ridge]

Then solved

[west ridge]

If you're multiplying correctly

[lusty leaf]

Ohhk

[red hornet]

i see what happened

[west ridge]

Got it it is 5+3√2

Not 3√(2+5)

[west ridge]

Add brackets

[red hornet]

to?

[west ridge]

(3√2+5) and( 2+√2)

,w (3√2+5)(2+√2)

[woven radish]

Results provided by WolframAlpha

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

Requested by codingdecoder

[celest lantern]

Such a mess here.

[red hornet]

where?

[west ridge]

Check in with the answers

[celest lantern]

No, keep going.

[west ridge]

See it its right

[red hornet]

we got the correct answer

its just in diff format

[west ridge]

Done what else

[red hornet]

that all for today

thank u

[west ridge]

Welcome

[red hornet]

one question

whats the rule called when u multiply n by d to remove square root 2?

[celest lantern]

Rationalisation.

[red hornet]

thanks

.close

[devout snow]

Channel closed

Closed by @red hornet

Use .reopen if this was a mistake.

[celest lantern]

No problem.

[devout snow]

Available help channel!

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[frail osprey]

hi i was solving a sum, and this was one of the steps i had to do. why does this work?

[frail osprey]

i dont really understand this concept

I went and determined g^-1(x) (which took me a lot of time) and solved the equation

[placid minnow]

like you dont understand the way it was done or why it is done

[lost laurel]

also isn't this wron anyways

[frail osprey]

why it was done

[lost laurel]

fg^{-1}(x)=9 , so gf^{-1}9=x

[placid minnow]

cause instead of spending 10 minutes finding the equation for (g^{-1}(x)) and then solving for (x), you can plug 9 into (g) to get a value. Plug that value into (f).The result is your answer for (x).

Example: If (g(x) = x + 2) and (f(x) = 3x), and you are told ((f \circ g)^{-1}(x) = 9): Just calculate ((f \circ g)(9)).(g(9) = 9 + 2 = 11).(f(11) = 3(11) = 33). So, (x = 33).

[woven radish]

thecrumbeler2

[frail osprey]

is it?

i really dont know lol

[frail osprey]

huh, how does this work? im gonna require this to strike me during an exam

is this basically the same as just taking g(x) on both sides?

functions are so confusing man i cant get my head around them

[placid minnow]

you give an input they give an output

whatever happens in between you dont care about unless you are working within the definition

[frail osprey]

fair enough. so can i just generalse this problem type? whenever i have f(g^-1(x)) = SOMETHING, i can just do f(g(SOMETHING)) = x

?

[placid minnow]

no

Move (f) to the other side by making it (f^{-1})

[woven radish]

thecrumbeler2

[placid minnow]

(g^{-1}(x)=f^{-1}(9))

[woven radish]

thecrumbeler2

[placid minnow]

To get (x) alone, move (g^{-1}) to the other side by making it (g).\(x=g(f^{-1}(9)))

[woven radish]

thecrumbeler2

[placid minnow]

you can only do the direct swap if the entire group is inversed together

[frail osprey]

OHHH

i see

[placid minnow]

((f \circ g)^{-1}(x) = 9) applies to both

[woven radish]

thecrumbeler2

[frail osprey]

awwesome

[placid minnow]

(f(g^{-1}(x)) = 9) applies to only g

[woven radish]

thecrumbeler2

[frail osprey]

so in this case, i just plug 9 into the original function and i get x

[frail osprey]

in this case i cant, but i can make it gf^-1(x)

right?

[placid minnow]

ye

it would be (g(f^{-1}(9))) tho

[woven radish]

thecrumbeler2

[placid minnow]

meaning you find (f^{-1}(9)) first, then plug that result into (g)

[woven radish]

thecrumbeler2

[frail osprey]

makes sense

any idea where i could just like find a few of these sorta sums to practice so i can make sure i got it correctly lol

[placid minnow]

this is like one of the few times i would suggest AI so it can give you feedback

this is a very simple topic so ai wont mess up

[frail osprey]

fair enough

just plain chat gpt works? or anything more specialised

[devout snow]

@frail osprey Has your question been resolved?

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[jaunty moat]

Don't really know where to start on this

[true pollen]

Start by making a sketch and put all the information you have on it. How far do you get?

[jaunty moat]

...

.close

[devout snow]

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[wispy warren]

hello I need help

[wispy warren]

actually it's y = -49e^-0.5x + 6x^2 - 24x + 50

[celest lantern]

Well you need to find the differential equation.

What have you done already?

[wispy warren]

y' + ay = f(x)

is what it should look like I think

[celest lantern]

You need to diffentiate it.

[celest lantern]

Differentiate this.

[wispy warren]

okey one second

[celest lantern]

Take your time.

[celest lantern]

Correct.

We notice that 24.5 = -0.5(-49).

Then you can rewrite as: $y' = -0.5(-49e^{-0.5x}) + 12x-24$.

[woven radish]

Mercury (ヤフォダ)

[celest lantern]

But from the given question we already knew that: $-49e^{-0.5x} = y - 6x^2 + 24x - 50$

[woven radish]

Mercury (ヤフォダ)

[celest lantern]

Can you substitute in?

Then you can just expand and rearrange.

[wispy warren]

yes it's the correct answer according to the key

sweet

thank you so much

.close

[devout snow]

Channel closed

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[celest lantern]

No worries.

[devout snow]

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[clever sphinx]

Prove that 2^k>2k for n>=7

[clever sphinx]

I just need help with the inductive step

[flint talon]

can you show what you've done so far?

[clever sphinx]

Im not sure how to show that 4k>2k+2

[dark sable]

well, k__>__7

4k>2k+2 you can subtract 2k from both sides and show 2k>2

so for k>1, 4k must be greater than 2k+2

[clever sphinx]

Ohh ok

[dark sable]

also I don't think this is right

2(k+1) is not 4k

[clever sphinx]

I multiplied everything by 2

[dark sable]

oh

that works yeah nvm

[clever sphinx]

From my assumption

[clever sphinx]

Thanks this was really useful though

What about this example ?

[dark sable]

uhhh try cube rooting rather than doing that math

you would get cuberoot3 * k>k+1
cuberoot3 * k-k>1 => k(cuberoot3-1)>1 now u just gotta err

prove that that's true

[clever sphinx]

Is this ok

[dark sable]

well u kinda have to erm, still find if whatever k value ur considering is greater than that

[clever sphinx]

It’s n>=4

[clever sphinx]

Idk how to rationalise
That but it’s value is 2.26

[dark sable]

r u allowed calculators

[dark sable]

it's not rationalizableish ig but you can multiply its conjugate

[clever sphinx]

Yes in the exam but not usually for this purpose

[dark sable]

a^3-b^3=(a-b)(a^2+ab+b^2) ---- oh no just use the calculator

[dark sable]

use the calculator, don't suffer my faith

[clever sphinx]

Ok thanks

So nice it’s true for k>2.26 and n>=4 it’s true for al valid k

So that’s an adequate proof ?

[dark sable]

prolly

[clever sphinx]

Meh

If it’s wrong I’d only lose 2 marks max anyways

.close

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[devout snow]

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[velvet geode]

yea i think this is a crappy explination

[molten glen]

Set

Chapter?

[velvet geode]

the x ∈ C and x ∉ A doesnt make sense

[velvet geode]

sets...

[molten glen]

B is bigger than A,
so removing B from C removes more elements,
therefore C−B is smaller (or equal) compared to C−A.

[velvet geode]

B can be equal to A asw 🤷🏼‍♂️

'⊆' isnt rlly a thing in my book

they didnt explain proper nd improper sets

they jst put the '⊂' everywhere

[molten glen]

If A=B, then:

C−B=C−A

and equality automatically implies

C−B⊆C−A

[celest lantern]

Aren't they the same?

[velvet geode]

nah

[molten glen]

No

[primal ferry]

a venn should help you visualize this very clearly

[molten glen]

One means = or sub set othe rmeans only subset

[velvet geode]

A ⊆ B means A is either a subset, OR equal to B
thus A is a improper subset IFF it is equal to B

[celest lantern]

Interesting.

[primal ferry]

do you know what a contrapositive is

[molten glen]

Bro the soln is wrong

[velvet geode]

nah i understood the question, did an example asw nd yea. im jst saying THIS explination is crappy right? or am I missing something

[molten glen]

And ur answer is correct

[velvet geode]

yep

[molten glen]

I think printing mistake

[primal ferry]

i don't feel so

the => means this implies that

[velvet geode]

might be

[velvet geode]

ik dude 😭

the x ∈ C and x ∉ A doesnt make sense
@primal ferry

[molten glen]

https://www.teachoo.com/2088/574/Misc-5---Show-that-if-A---B--then-C---B---C---A---Chapter-1/category/Miscellaneous/

[primal ferry]

if x is a part of C and not a part of B, since A is a part of B then x is not a part of A

[molten glen]

Just assume set

[velvet geode]

dis better

[primal ferry]

hence every element in C-B is also in C-A

[primal ferry]

i mean they just didnt linebreak

[velvet geode]

hmmmmmmmmmm

makes sense

[primal ferry]

they wrote the same thing

[velvet geode]

whts linebreak?

[primal ferry]

in typesetting it's when you press "enter" to go to the next line
so something
like this

they wrote everything in one line

[velvet geode]

hmmmmmm

nvm i thought its some 'sets' thing

i think i get it

yea now it feels obvious

thanks sam

.close

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[dapper quail]

https://youtu.be/XkY2DOUCWMU?si=116tiQr3p08KMezr

[dapper quail]

im trying to understand this video but i dont understand how the matrix multiplication is generalized as a transformation

at 6:35

the next video explains how in higher dimensions its just an extension of this same idea of transforming it but i dont feel like i understand it in the first case

[celest lantern]

Think the matrix as a machine.

If you have a point x, then Ax (The machine A) gives x another location.

[dapper quail]

ya

i understand that its a transformation and that it changes in a specific way regardless of wethers its 2x2 or 3x3 i just dont know how to generalize the computation

[celest lantern]

Would be good to think as a linear combinations?

Because the output is the sum of the columns of the matrix though.

[dapper quail]

hm

[celest lantern]

Well if you have two matrices multiplying, you are taking each elements of first row multiplying the elements of first columns of others.

Then sum them up.

[devout snow]

@dapper quail Has your question been resolved?

[lyric hornet]

I don't know if this perspective helps at all, but in the (2\times 2) case:
[\begin{bmatrix}a&b\c&d\end{bmatrix}\cdot \begin{bmatrix}x&z\y&w\end{bmatrix}=\begin{bmatrix}\begin{bmatrix}a&b\c&d\end{bmatrix}\cdot\begin{bmatrix}x\y\end{bmatrix}&\begin{bmatrix}a&b\c&d\end{bmatrix}\cdot\begin{bmatrix}z\w\end{bmatrix}\end{bmatrix}=\begin{bmatrix}ax+by&az+bw\cx+dy&cz+dw\end{bmatrix}]

Or more generally, for matrices (A) and (B) if (B=[\mathbf{b}_1\quad\mathbf{b}_2\quad\ldots\quad\mathbf{b}_n]) then:
[AB=[A\mathbf{b}_1\quad A\mathbf{b}_2\quad\ldots\quad A\mathbf{b}_n]]

So we're applying the transformation (A) to all the vectors of (B).

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[dapper quail]

hm

yeah it does

what do you mean by b1 b2 though

[lyric hornet]

if (B=\begin{bmatrix}a&b\c&d\end{bmatrix}) then (\mathbf{b}_1=\begin{bmatrix}a\b\end{bmatrix}) and (\mathbf{b}_2=\begin{bmatrix}c\d\end{bmatrix})

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[dapper quail]

ah i see

so if it was two 3x3 matrices you would just seperate one of the matrices into 3 vectors by columns?

[runic prawn]

the matrix on the right, yes

[dapper quail]

i see

can you guys help me practice?

[covert root]

practice what?

[dapper quail]

matrix multipliaction

like more abstract questions with it

[covert root]

Ask a specific question

[lyric hornet]

<@&268886789983436800>

[trail eagle]

Oh we're jumping from multiplication to eigenvectors

[lyric hornet]

the key thing here is if (A=P\Lambda P^{-1}) then (A^n=P\Lambda ^n P^{-1})

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[lyric hornet]

and since (\Lambda) is a diagonal matrix this makes life really easy

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[dapper quail]

oh right that comes from (diagonal matrix)D = SAS-1 right (i use different letters but its the same thing u manipulated that)

[dapper quail]

oh so you dont actually have to do multiplication in this?

[lyric hornet]

well you will, because you still have to compute (P\Lambda^{100}P^{-1})

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[lyric hornet]

but to actually do the raising to 100th power, no you don't have to multipliy (A) to itself 100 times lmao

[woven radish]

ΠαϳαμαΜαμαΛλαμα

[dapper quail]

i see

[dapper quail]

.close

[devout snow]

Channel closed

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[devout snow]

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[devout snow]

Please don't occupy multiple help channels.

[flat rivet]

i need help with my math

[sturdy fiber]

!status

[devout snow]

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

[flat rivet]

idk what to do

[sturdy fiber]

did you find the volume of the sphere?

[finite fable]

The volume is the same. Can you equate both of them?

[flat rivet]

which one do i talk to

[sturdy fiber]

astor

[flat rivet]

simpler terms pls

[finite fable]

notice the question says. both have same volume

right?

[flat rivet]

oohh ye

so what

[finite fable]

so the volume obtained by formula of cone will be equal to volume obtained by formula of sphere

right?

[flat rivet]

yes

[finite fable]

can you equate them. and tell me what constants cancel out?

[flat rivet]

bro what

[sharp adder]

-# not to intterupt but ithink there is an extra step first i belive..

[finite fable]

umm. 1/3 pi r^2 h = 4/3 pi r^3 right?

[flat rivet]

yes

[finite fable]

you are talking about radius of sphere

right?

[finite fable]

so 3 cancels out? and pi as well?

by cancels

[flat rivet]

ye

[finite fable]

so in the end it becomes r^2.h = 4r^3

[flat rivet]

ya

[finite fable]

r^2 h is of the cone
4r^3 is of the sphere
can you tell me radius of sphere

[flat rivet]

ofc i can

6

[finite fable]

now do you know height of cone?

[flat rivet]

20

but the sphere lowk isnt that tall

cone is way taller

so idk

[finite fable]

so 20 times r^2 = 4 times 6^3 right?

[flat rivet]

yes

[finite fable]

so can you figure out, r from here?

[flat rivet]

yessir

[finite fable]

that isn't drawn to scale

[finite fable]

ok!

!done

[devout snow]

If you are done with this channel, please mark your problem as solved by typing .close

[flat rivet]

.close

[devout snow]

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[solemn holly]

part d

why cant i treat it as the distance between a point and a line

.close

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[loud spade]

how do I solve this I just picked randomly

[runic prawn]

dy/dx = infty

[loud spade]

?

[runic prawn]

this is where there's a vertical tangent

[loud spade]

what is infty

[runic prawn]

infinity

[loud spade]

i still dont get how this relates to my question

so just find

dy/dx

then equal that to infinity

?

what does that mean

[runic prawn]

yes

[runic prawn]

?

do you know what dy/dx means

[loud spade]

dy/dx = 1-2x^2/2y

how do I set that to infinity

[runic prawn]

just set it

write =infty

[loud spade]

do I solve for x?

[spiral raptor]

<@&268886789983436800>

[runic prawn]

for (x,y)

[loud spade]

x= rootinfinty2y+1/2

so hows that an answer

can you provide more context

/elaborate

[native gyro]

Bruh

Lemme handle this

What is dy/dx?

[runic prawn]

that makes it 0

[runic prawn]

what

[loud spade]

the slope

derivative

[runic prawn]

you're trying to find the points where there's a vertical tangent

[loud spade]

ok

[native gyro]

What is it equal to

Im trying to help you bruh

[loud spade]

uh

idk

oh

thats equal to

1-2x^2/2y

[native gyro]

Use only x

[loud spade]

well the question has y

[native gyro]

Yea substitute y in

[loud spade]

wdym

[native gyro]

,, y=\sqrt{x-x^3}

[woven radish]

Roy

[loud spade]

ah

so you solved for y for the normal equation

ok

then?

[native gyro]

Derivate

[loud spade]

1/2(x-x^3)^-1/2 *1-3x^2

[native gyro]

Yes

Now at what value of x does this approach infinity?

[loud spade]

how do I know?

idk

[native gyro]

Other way, when does this not have a value?

[loud spade]

itll always have a value no?

theres no root

theres no fraction

[native gyro]

No you divide

,, by \sqrt{x-x^3}

[woven radish]

Roy

[native gyro]

you see that -1/2 power, right?

[loud spade]

I divide the derivative by the original function?

oh right yeah

lemme rewrite on a paper rw

rq

[native gyro]

Ok

[loud spade]

0?

[loud spade]

1-3x^2/2*rootx-x^3

[native gyro]

Yep

[loud spade]

so answer is D?

[native gyro]

Wait

[loud spade]

wait no

C

does it have only one vertical tangent

[native gyro]

We have to check if root has no values or not

[loud spade]

how

[native gyro]

Although i guess

Because if x-x^3 is smaller than 0, the root value is imaginary

So it does have values

So yea answer is 1 if im right

[tall walrus]

Yeah

[loud spade]

answer is A

[runic prawn]

!noans

[devout snow]

The purpose of this server is to help you learn; please don't ask for direct answers. Ask for guidance, explanations, or feedback instead.

[celest lantern]

Giving answers straight away is not a smart move.

[native gyro]

I walked him through

Also, its nosols

[loud spade]

well alright

[runic prawn]

where did you get this from

[loud spade]

so what do i do to get the answer

[loud spade]

answer sheet

[runic prawn]

i'm gonna leave that to roy

but it's the same thing i was saying

[celest lantern]

But Core is OP btw.?

[loud spade]

what did you mean equal it to infinity

[loud spade]

yeah?

[native gyro]

The function has a vertical tangent when the value limits to infinity or negative infinity

[runic prawn]

wdym wdym

[loud spade]

he wanted me to literally equal it to infinity

yo can you show me

[runic prawn]

no

[loud spade]

what you wanted me to do

[celest lantern]

Anyways let just stick to the problem.

[native gyro]

Yea

[loud spade]

aight guys chop chop im even more confused now

[runic prawn]

why is that so hard for u to do

[celest lantern]

"Chop chop".

[loud spade]

whys it so hard for you to elaborate

my bad

💔

[runic prawn]

we r talking about the problem

[loud spade]

my exam is in 12 hours

[runic prawn]

there's not much i can elaborate

[loud spade]

I got dy/dx

set that to infinity

what next

[runic prawn]

solve

[loud spade]

I got 0

[runic prawn]

what's 0

[loud spade]

x = 0\

at x = 0 it goes to infinity

[loud spade]

.

[runic prawn]

peace out ✌️

[loud spade]

come back 🥀

[runic prawn]

the vertical tangents are at y=0

so you set the cubic to 0

and there are 3 solutions

[loud spade]

okay

so can you explain

what the infinity

you wanted me to do is

[native gyro]

Ok i realised

We are right

The question is when is rootx-x^3 = 0

But its not only 0

There are 2 more values

[loud spade]

alright

what does it mean

to set an equation

to infinity

[native gyro]

Try to solve $x-x^3=0$

[woven radish]

Roy

[loud spade]

yeah there are two more solutions I know

but before that

he wanted me to set

dy/dx = infinity

[native gyro]

No i said let y=infinity

[loud spade]

not you

him

sure

what does it mean to set

[native gyro]

Then idk

[loud spade]

y = infinity

nvm bro

.close

[devout snow]

Channel closed

Closed by @loud spade

Use .reopen if this was a mistake.

[runic prawn]

y=0

not infinity

[loud spade]

so I try to find something that makes the whole thing undefined

thanks

[runic prawn]

yes

[runic prawn]

is this sarcasatic

lol

[loud spade]

?

no i was serious

thank you

[runic prawn]

alr mb

[devout snow]

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[fierce rune]

i think im a little confused on connected spaces. could someone critique this please

[fierce rune]

Suppose $(A,d_A)$ is a metric subspace of $(X,d_X)$, and assume the closure of A, $\bar{A}$ is disconnected. Then $\exists$nonempty, disjoint, open $U_1,U_2\subset\bar{A}$ such that $$U_1\cup U_2=\bar{A}$$. Since $U_1,U_2$ are open, and openness is preserved under union, $\bar{A}$ is open, which is a contradiction. So $\bar{A}$ is connected.

[woven radish]

lyric

[fierce rune]

but this is obviously wrong

[lusty sapphire]

Non connected subspaces can simultaneously be both open and closed. That is a side of effect of partitions

[fierce rune]

ah of course a set has to only be open or closed

😅

thank you <3

.close

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[potent dove]

Hello! Why would this be wrong? I’ve been told it’s (2a+3) squared instead. Why?

[wicked turtle]

try multiplying out your answer, see that it doesn't match the original

[polar chasm]

(a+b)^2 = a^2 + 2ab + b^2

[obsidian nebula]

what is 3 squared

[potent dove]

Oh shit not squared lol

My fault one sec

[potent dove]

I meant (2a+3)^2 would be the answer

I’m feeling really dumb right now I’m kind of dumb at math

[polar chasm]

that is the correct factorization

[obsidian nebula]

its fine everyone have stuff that they are not good at

[potent dove]

Tysm for the help!!

I gotta go keep factoring it wish me luck lol

[spiral raptor]

!done

[devout snow]

If you are done with this channel, please mark your problem as solved by typing .close

[polar chasm]

good luck

[potent dove]

.close

[devout snow]

Channel closed

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[potent dove]

Ty!!

[devout snow]

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[loud spade]

can anyone help me w this rq

[devout snow]

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

[loud spade]

1

but which ones b and which ones a

and how do I even figure out the function

[lunar harbor]

really this is just knowing [Riemann sums](https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum\) though.

Better image that actually tells you what $\Delta x$ is explicitly. Which factor do you think represents $\Delta x$ and which do you think represents $f(x_i)$?

[woven radish]

Civil Service Pigeon

[loud spade]

ok so all the answer choices can work for the boundaries

since b - a = 3

our a has to be 2?

b -2k/n

b - 2 = 3, so b is 5

[lunar harbor]

a doesn't really have to be anything because you can "shift" the region.

[loud spade]

cant tell what function it is tbh

[lunar harbor]

Why don't you answer this

[loud spade]

3/n represents delta x

3k/n + 2 represents xi

[lunar harbor]

mhm

[lunar harbor]

Now as I said here, a doesn't really have to be anything.

But most of the options have a as zero

oh actually I'm blind

two of them have a=0 and two of them have a=2

we'll let's just start with a=0 first

[loud spade]

alrighty

[lunar harbor]

So you said that
$$\frac{3}{n}=\Delta x, \qquad \left(\frac{3k}{n}+2\right)^2=f(x_i).$$
Now if we operate off of $a$ being zero, what is $x_i$?

[woven radish]

Civil Service Pigeon

[loud spade]

its gonna be (3k/n)^2?

idk

[lunar harbor]

Read the image and how it defines $x_i$

[woven radish]

Civil Service Pigeon

[loud spade]

its gonna be the same function tehn

3k/n +2

• 0

[lunar harbor]

[loud spade]

yeah?

[lunar harbor]

How are you getting $\frac{3k}{n}+2$ from that

[woven radish]

Civil Service Pigeon

[loud spade]

uh

isnt a

added onto it

so it doesnt change

so its (3k/n + 2)^2

?

[lunar harbor]

I'm asking you about $x_i$, not $f(x_i)$.
$$x_i=\underbrace{a}{0}+\underbrace{\Delta x}{\frac{3}{n}} \cdot i$$

[woven radish]

Civil Service Pigeon

[loud spade]

ok i might be dumb

my bad

[lunar harbor]

So what is $x_i$

[woven radish]

Civil Service Pigeon

[loud spade]

0+ 3/n * i

[trail cairn]

Sry to disturb but aft this help, can u help me @lunar harbor ?

[lunar harbor]

mhm

just simplify that rq

[devout snow]

Please do not ping individual helpers unprompted.

[trail cairn]

Alr

[loud spade]

3i/n

f(xi) is (3i/n + 2)^2

[lunar harbor]

correct

so what is f(x)?

[loud spade]

uhh idk how do you get it from that? just remove the i?

[lunar harbor]

$$f\left(\frac{3i}{n}\right)=\left(\frac{3i}{n}+2\right)^2$$

[woven radish]

Civil Service Pigeon

[lunar harbor]

Agreed?

[loud spade]

yeah

f(xi) -> xi = 3i/n

f(xi) = (3i/n + 2)^2

[lunar harbor]

mhm

does this rewrite make what f(x) is more clear?

[loud spade]

just that 2 and ^2 got added

yo

im actually tweaking

[lunar harbor]

yes

$f(x)=(x+2)^2$

[woven radish]

Civil Service Pigeon

[lunar harbor]

So what is the integral with $a=0$ that's equivalent to the Riemann sum

[woven radish]

Civil Service Pigeon

[loud spade]

a = 0?

B?

[loud spade]

ohhh

0 + 2^2?

f(b) - f(a)

thats 4

[lunar harbor]

Remember that finding $f$, $a$, etc. was all to help us convert the sum to an integral

[woven radish]

Civil Service Pigeon

[lunar harbor]

we found all of those based on the sum

put it all together and give me the integral

[loud spade]

but we dont know necessarily that a is equal to 0 right??

idk

[lunar harbor]

Like I alluded to earlier, it doesn't really matter what the lower bound is since you can find everything else by shifting/substitution

For example,
$$\int^{2}{1} f(x) \dd{x}=\int^{3}{2} f(x-1) \dd{x}.$$

[woven radish]

Civil Service Pigeon

[lunar harbor]

So we just chose to find the integral with bounds $0$, $3$ first to get us started

[woven radish]

Civil Service Pigeon

[lunar harbor]

and if that ends up not working, we can convert to an integral with bounds $2$, $5$ relatively easily

[woven radish]

Civil Service Pigeon

[lunar harbor]

and/or you can go through the whole process again for bounds of 2, 5 as an exercise on your own

[loud spade]

yeah but how do we know that equals our riemann sum

im so confused ngl

[lunar harbor]

What we've been doing this entire time is pattern matching the definition of an integral as the limit of Riemann sums

the link (or maybe one of the associated ones that stems from it, idr) gives visual intuition behind this

the tldr is to split your region into a bunch of rectangles and let the number of rectangles tend to infinity

We've established that $f(x)=(x+2)^2$ and took $a=0$ to obtain this. Since $\Delta x=\frac{b-a}{n}=\frac{3}{n}$,
$$b-a=3 \implies b=3.$$
and so the corresponding integral is $\int^{3}_{0} (x+2)^2 \dd{x}$.

[woven radish]

Civil Service Pigeon

[loud spade]

ok yeah

but I still dont get how we find the answer

[lunar harbor]

Well as we can see none of the options with bounds of 0, 3 work

so as I said, we should convert to bounds of 2,5 by means of a substitution

[loud spade]

why dont they work though

[lunar harbor]

because they're literally different integrals?

[loud spade]

right

[lunar harbor]

What substitution do you think would send 0 -> 2 and 3 -> 5?

[loud spade]

we just need to get something that b-a equals to three

ok ok

[lunar harbor]

Mhm. Because $\Delta x=\frac{b-a}{n}$ in general, and is $\frac{3}{n}$ in this case.

[woven radish]

Civil Service Pigeon

[loud spade]

A looks the same?

[lunar harbor]

What does this mean

[loud spade]

wait no

we have toi do

everything all over again

with A = to 2

and check

right

[lunar harbor]

You could

Or as I said earlier, just do a substitution

[loud spade]

u = x +2 would make our new boundaries

2 to 5

[lunar harbor]

good

and what is your new integral upon doing that substitution

[loud spade]

int 2 to 5 (x +2)^2

[lunar harbor]

what happened to changing the integrand under the substitution

[loud spade]

huh

[lunar harbor]

$$\int^{x=3}{x=0} (x+2)^2 \dd{x}=\int^{u=5}{u=2} \underbrace{(x+2)^2 \dd{x}}_{\text{needs to be in terms of } u}$$

[woven radish]

Civil Service Pigeon

[loud spade]

u = x + 2
du/1 = dx

int 2 to 5 (x + 2) ^2 du

looks the same

[summer summit]

the point he's making is that what you have written here is in the form "f(x) du" whereas you want it to be in "u(x) du"

what you are trying to do is integrate x with respect to u

for example, x du would be xu + c, whereas u du would be u^2 / 2 + c

[loud spade]

ahh

but whys it over 2

[summer summit]

why is what over 2?

u^2 / 2?

[loud spade]

yeah

[lunar harbor]

Dumping this channel on you since I gtg

[summer summit]

just an explanation, it doesnt necessarily have to be the exact same for this question

[loud spade]

ok ok

[summer summit]

$\int x \dd u = xu + c$, and $\int u \dd u = \frac{u^2}{2} + c$

[loud spade]

so just substitute u to x

[woven radish]

Krish

[summer summit]

not directly, remember you have u = x+2

so replace "x+2" with u

[loud spade]

yeah

int 2 to 5 u^2 du