help-39

[random ermine]

do u know what a vector space is?

[tacit sedge]

Yes

[random ermine]

do u know what a basis is

[tacit sedge]

Yes it’s the vectors that span that space right

I think

That follow certain rules

[random ermine]

they also have to be linearly independent

[tacit sedge]

Yeah

I def knew that one

[viscid shale]

quick example, on my personal exam on laplace they asked us to derive the formula of heaviside step function on our own using the integral.

In reality its a quite simple idea on how to get the result, but you have to give it some thought.

[random ermine]

...

[tacit sedge]

Ahh gotcha okay u comparing it to derivatives really helped

Lemme quickly overview my topics

[wet osprey]

Why is a basis important/interesting/useful?

That’s pretty much the next question you want to know after knowing the definitions

[tacit sedge]

For spans?

Okok

Hmm ok I guess definitions can wait

I just usually like to understand the way

Why

[wet osprey]

For the laplace or the linear algebra stuff?

[tacit sedge]

Yes usually bc linear is SO helpful and I feel like

[random ermine]

i would suggest u do some concrete questions, u seem a bit confused

[tacit sedge]

My understanding of how basis span and everything relates is so bad

Okok

Yeah true

[wet osprey]

I asked “x or y” and you answered yes

[tacit sedge]

LMAO

[tacit sedge]

For both

I meant

bc I think I just lack in understanding how they all connect

which I guess I don’t have time for

[viscid shale]

Laplace just barely ties Calc/Analysis and Lin.Algebra because as a tool, it comes from the concept of Linear Transformations.

And for Diff.Eq we use it to transform calculus problems into algebraic problems which are arguably easier to manipulate.

[random ermine]

so the laplace transform operator is a linear map

on the vector space of what type of functions

[viscid shale]

In actual truth, we (mostly engineers or other stem majors) ignore what "S" is because we do the direct and then come back through the inverse.

In reality laplace transforms ties a real valued function to a complex valued function of frequency

Which is quite similar to fouriers.

[tacit sedge]

Ohh

I know fourier kinda

Hmm

[random ermine]

...

[tacit sedge]

Okay imma work on vocab a lot

Stop

[viscid shale]

anyways, id say just practice how to use laplace to solve DEs

[random ermine]

wait can t be complex in this

[viscid shale]

s is the complex value

[tacit sedge]

Thank u.

[random ermine]

oh yeah i meant s

[viscid shale]

by formal definition s is a complex number of the form (a+wi) or sum like that

[random ermine]

t is dummy

[viscid shale]

Which encode amplitude and frequency iirc

so w goes from 0 to 2pi, etc...

[random ermine]

you mean arctan(w/a)

or arg(s)

[viscid shale]

whatever, totally out of the question rn.

[tacit sedge]

LMAO

[viscid shale]

yeah, for DEs theres pretty much no use to knowing S is a complex number in frequency domain

[tacit sedge]

Thanks for acc giving me like advice

I know it’s my fault for studying late

[viscid shale]

np

Wikipedia has a pretty complete table of properties and solutions:
https://en.wikipedia.org/wiki/Laplace_transform

4. Properties and Theorems
5. Table of selected Laplace transforms

[tacit sedge]

Thank you!! I’ll read up and watch a video to go through

[random ermine]

will you have DEs in ur exam

[tacit sedge]

Yeah for sure here’s my uh topics

[toxic field]

almost everything revolves around laplace transform

[random ermine]

there are many sources online

[toxic field]

very vast topic

[random ermine]

i don't think more sources will help

[random ermine]

what about probability

oh this is like linalg and DEs mixed

[tacit sedge]

I think I’m pretty good at de like I have a good understanding of second order and higher order non homo and homo

yes

[toxic field]

ofc nothing can peak probability 😅

[random ermine]

same

[random ermine]

no i mean probability has nothing to do with laplace

[toxic field]

oh no it does

[random ermine]

how

[toxic field]

there's a topic in laplace tranform named fourier transforms

[random ermine]

yes ok, and characteristic function i see

[toxic field]

many applications in probablity are solved by fourier transform

[dense junco]

wsg

[pearl pond]

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[nocturne oxide]

how do i go about these questions

[dusk willow]

form the equation first

[nocturne oxide]

i did but i am not getting anywhere

[dusk willow]

then what have you tried? show your work thus far

[nocturne oxide]

the only way i can think of is hit and trial after this and form a relation using values that satisfy but its going to be tedious

...

[dusk willow]

then I don't think I'm equipped enough right now to try my hand at this problem. I apologize

[nocturne oxide]

[dusk willow]

but what I would try in your position is, from the last line shown in your picture, write the equation in terms of b
then, since a, c are both less than 10 and have to be distinct, do some testing with diff values of a and c

[nocturne oxide]

last line is ac= b(2a+9c)

[dusk willow]

exactly the line I was referring to, yes

[midnight haven]

i think its supposed to be 1/10a

[dusk willow]

oh yeah I didn't even notice that

[placid wharf]

this part is a bit wrong, rearrange it to form 1/b = 2/c - 9/10a

then do hit and trial

as we know a,b,c <10

[nocturne oxide]

ty!

i was banging my head on a sum that i copied wrong

[placid wharf]

no prob

[nocturne oxide]

ty @placid wharf @dusk willow @midnight haven !!!

i did like this and managed to find values that satisfied

!close

[placid wharf]

yayy

its .close

[nocturne oxide]

ty again lmao

.close

[pearl pond]

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[pearl pond]

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[lucid sequoia]

hello! could someone please help guide me with this problem

[lucid sequoia]

this is the first question

[autumn fossil]

u can use the fact that
lim(f(x) - g(x)) = lim(f(x)) - lim(g(x)) (if both limits exist)

so u just need to find lim(f(x)) and lim(g(x))

[urban tapir]

oh this is pretty easy!

[lucid sequoia]

ohhh okay

well in f(x) x approaching 0 is two from both sides

but how do i find g(x)?

[tacit sedge]

i hope so

[lucid sequoia]

oh jk i see it on the table

[bitter herald]

What can you deduce from the table given you

[lucid sequoia]

its 2 right as well?

[bitter herald]

Btw I have the slides thing now if you still want them

[bitter herald]

Yes

[tacit sedge]

sure!

[lucid sequoia]

so it's just zero?

[bitter herald]

Yeah

[lucid sequoia]

ohh okay thank you! not sure if you know ap requirements but would you know a way to specifically word this in the text box, or should i just say
f(x) approaching 0 is 2 g(x) approaching 0 is 2
2-2=0

[light apex]

is the ans 0

[light apex]

yuh

[autumn fossil]

"f(x) approaches 2 as x approaches 0" would be the correct way to word it

[light apex]

just plug and play

lol

[autumn fossil]

or just write down the limit

[lucid sequoia]

yes! the answer is 0

thank youuuuu

[autumn fossil]

$\lim_{x\to0}\left(f\left(x\right)-g\left(x\right)\right)=\lim_{x\to0}f\left(x\right)-\lim_{x\to0}g\left(x\right)=2-2=0$

[light apex]

yezzir u can see from graph that f(0) is alr 2 and from the table u can see g(x) approaches 2 as x approaches 0

ez

[jolly parrot]

MathIsAlwaysRight

[light apex]

yo bro how do i learn the syntax lowk

[autumn fossil]

its pretty intutive

[light apex]

like the basic syntax

🤔 🤔

[autumn fossil]

put ur equation between $$

[jolly parrot]

MathIsAlwaysRight

[light apex]

lets move to #latex-testing

[lucid sequoia]

aww yesss thank you so much

[autumn fossil]

seems like u really want to have 0 channels, lol

[lucid sequoia]

ok heres b

are we just finding x approaching the values given ?

like x approaching 1 , 2 , 4 , 6?

and it has to be a limit from both sides right?

[autumn fossil]

yep

[lucid sequoia]

nice! im getting the hang of this

okay this is c

and i would say yes according to the graph but how would i use correct notation

[autumn fossil]

continuous means that

1. the limit as x -> -3 exists
2. the limit is equal to f(-3)

so find the limit as x->-3 and then verify that it = f(-3)

[lucid sequoia]

ohh okay so i would say, "the limit as x approaches -3 is -1 on both sides" and how would i verify its equal to f(-3)?

[autumn fossil]

look at the graph

what's f(-3)?

What's the y-value at x = -3?

[lucid sequoia]

-1?

[autumn fossil]

nope, there is an empty circle at x = -3. What does that mean?

[lucid sequoia]

that its discontinuous?

[autumn fossil]

Well, yea, kinda

but more importantly, it means that there is a "gap" at x = -3

and that f(-3) isn't -1 like you'd suspect

f(-3.00001) is probably very close to -1, but f(-3) isn't -1. The empty circle means that there is a gap

and then the full circle above it means that the graph contains a point (-3, 2)

that is, f(-3) = 2

[lucid sequoia]

wait so how would i implement that in my answer?

so it isnt continuous?

[autumn fossil]

well, what is f(-3)?

[autumn fossil]

indeed

The limit = -1

f(-3) however = 2

[lucid sequoia]

ohh okay

[autumn fossil]

so its not continuous

just to check ur understanding, whatss f(1)?

[lucid sequoia]

1? lol

[autumn fossil]

yes

[lucid sequoia]

cuz theres an empty circle at 0

nice thank you!

[autumn fossil]

yep, exactly

[proper nova]

What's f(2) then?

[proper nova]

?

[lucid sequoia]

um it dne?

[proper nova]

Cool

[proper nova]

You got any more questions?

!done

[pearl pond]

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

[lucid sequoia]

okay thanks so much, heres the last part of the problem

noo sorry, im just jotting down notes

okay so im basically finding the derivative for g at x = 0

how would i do that using the table tho?

[still hamlet]

which values of x in the table are closest to 0?

[lucid sequoia]

-0.001 and 0.001

would i find the slope using those?

[still hamlet]

yep you use those two

[lucid sequoia]

okay so after i get the slope using the y2-y1/x2-x1 formula, is that basically "a difference quotient" or is there another way to use it

[still hamlet]

I'm pretty sure that's all you do

[lucid sequoia]

aah okay nice! thanks guys!

.close

[pearl pond]

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[lucid sequoia]

hey guys! could someone help guide me with this problem?

[lucid sequoia]

i know g(x) but im not sure how to find f(x) based of the graph

[toxic lichen]

you're not supposed to find f(x)

you're supposed to read off specifically the values of f(0) and f'(0) for use with the product rule

[lucid sequoia]

then how would i compute k(x)?

[toxic lichen]

you wouldn't

you would, once again, use the product rule

-# btw, your name is "evan" but your pronouns are she/her. is this on purpose?

[lucid sequoia]

oh yes! i just like that nickname

its a shorter version of evangeline lol

[toxic lichen]

anyway

yeah i will keep saying this: you need to write out the product rule, in terms of the functions f and g written as f and g,
and without trying to replace either one with a formula. not even g!

[lucid sequoia]

oh? just write it out?
like f'(x) * g(x) + f(x) * g'(x)?

[toxic lichen]

yes but even more specific

you need to find k'(0) after all, so these x's need to be 0 too

[lucid sequoia]

ohh ok then i would compute for the corresponding function with 0
like look for f'(0) * g(0) + f(0) * g'(0)

[toxic lichen]

"compute for __" is rather a strange turn of phrase

[lucid sequoia]

alrightyyy thank you imma compute rn

?

oh

[toxic lichen]

pressed enter too soon

[lucid sequoia]

well i guess find hehe

[toxic lichen]

anyway, yeah. you will need the values and also the derivatives of both f and g at 0

[lucid sequoia]

alrightyy thank you!!!

hey guys! i would use quotient rule right?

hello! i figured the prev one out dw. but can someone briefly explain how to find the derivative of

$g(x) = \sqrt{x^2 - x + 3}$

[jolly parrot]

evan

[lucid sequoia]

like after i get

$g(x) = (x^2 - x + 3)^{1/2}$

[jolly parrot]

evan

[lucid sequoia]

what do i do?

chain rule right? but how

[glacial charm]

now use chain rule followed by power rule

[lucid sequoia]

can you help me with that please

[glacial charm]

substitute the expression inside the square root

[lucid sequoia]

im sorry! i dont know how to do that

[glacial charm]

do you know how chain rule works?

[lucid sequoia]

uhh i believe so, it's used for functions like f(g(x)) but i still confused on how to apply it

[bitter herald]

First figure out what g(x) and f(x) are meant to be in what you sent

[lucid sequoia]

i think i recall you find the derivative of the inside?

[glacial charm]

$\frac{dy}{dx}\ =\ \frac{dy}{du}\ \cdot\ \frac{du}{dx}$

[jolly parrot]

Xerxes

[lucid sequoia]

wdym

if you mean the question a) i found f(0), f'(0), and g(0)

im just struggling on the derivative for g'(x)

[glacial charm]

if you want to understand like this then

$\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)\ =\ f'\left(g\left(x\right)\right)\cdot g'\left(x\right)$

[jolly parrot]

Xerxes

[lucid sequoia]

ohh yess ok

[glacial charm]

these both are the same thing

[lucid sequoia]

oh okay, let me process this rn

how do i compute out

$\frac{1/2}{u}^{-1/2}$

bru

[jolly parrot]

evan

[glacial charm]

ok i get what you mean

[lucid sequoia]

haha thank you

[glacial charm]

but it is the derivative wrt u right what do you want to compute?

[lucid sequoia]

wait what does "wrt u right" mean

[glacial charm]

you substituted x^2 - x + 3 as u right?

[lucid sequoia]

yuh

[glacial charm]

and then differentiated it (as u) wrt u?

[lucid sequoia]

is wrt "with respect to"

[glacial charm]

yeah

[lucid sequoia]

oh yeah i was just confused on that part

yes

[glacial charm]

$\frac{d}{du}u^{\frac{1}{2}}\ =\ \frac{1}{2}u^{-\frac{1}{2}}$

[jolly parrot]

Xerxes

[lucid sequoia]

yes yes thats what i did

[glacial charm]

now all is left to differentiate u wrt x to complete the chain rule

[lucid sequoia]

how do i do that

[glacial charm]

.

remember what you substituted as u?

[lucid sequoia]

yes

$x^2 - x + 3$

[glacial charm]

yeah just differentiate it wrt x

[jolly parrot]

evan

[lucid sequoia]

like get 2x - 1?

[glacial charm]

yes

[glacial charm]

and also dont forget to put x^2-x+3 back in here

[lucid sequoia]

$\ \frac{1}{2}{(x^2 - x + 3)}^{-\frac{1}{2}} \cdot {2x -1}$

[jolly parrot]

evan

[lucid sequoia]

yeah?

[glacial charm]

yes this is your derivative

[lucid sequoia]

oh! then i could just sub 0 for x now right

[glacial charm]

if you want to find the derivative at x = 0 then yes

[lucid sequoia]

yeah this is the og equation

[glacial charm]

ok

[lucid sequoia]

okay thank you

heres the last question

i dont think ive had a lesson on this yet, how would i make f'(x) equal to f'(x)

[glacial charm]

well graphically if you plot f'(x) and h'(x) all the points where the two graphs intersect are the values of x which satisfy f'(x) = h'(x)

[lucid sequoia]

? im sorry! i dont know what you mean

im confused how do i find the derivative of f(x)

[glacial charm]

recall what a derivative is

[lucid sequoia]

instantanous slope

[glacial charm]

or?

[lucid sequoia]

thats the only def i know

[glacial charm]

rate of change of quantity wrt another quantity?

[lucid sequoia]

oh? yeah i didnt know that

[glacial charm]

anyway both are correct

[glacial charm]

you find the slope of the function in the interval (-1,2) because x lies in this interval for that equation

[lucid sequoia]

ohh my god i totally skipped over that part yes okay

thank you

[glacial charm]

you're welcome

[lucid sequoia]

okay so i found the derivative as 2

then what do i do once i find the derivative of h(x)?

[glacial charm]

simply equate them both

[lucid sequoia]

ohh okay so i got

$5e^{x} - 9cosx$

[jolly parrot]

evan

[lucid sequoia]

for h(x)

then i simply just

$5e^{x} - 9cosx = 2$

[jolly parrot]

evan

[lucid sequoia]

but then how do i go from here?

[glacial charm]

it has become one horrible equation

well we cant directly solve for x because they are different functions

[lucid sequoia]

lmao! but its right? like i didnt make any mistakes?

i feel it shouldve been more straightforward since this is only practice

[glacial charm]

but we know x lies between -1 and 2 so we have use trial and error substitute x as some number which lies in -1 and 2 and just check if we get 2

[glacial charm]

nah i dont think so

[lucid sequoia]

hm? idk how to do that

okay! at least im on the right track

[glacial charm]

wdym its pretty straight forward just check the equation's result for various values which lie in -1 to 2

[lucid sequoia]

alrighty!

how woudl i do that?

like plug in -1, 0 , 1 , and 2 ?

[glacial charm]

those arent the only values which lie in (-1,2) but you can try any value

ig if such equations arise then calculators should be allowed like doing this with a graphical calculator is much better

[lucid sequoia]

lol yeah this what the start of the equation says

my graph. calc is lwks brokennn, do you know how to do this with desmos?

oh wait, dont i jus use a slider

[glacial charm]

yeah i would crash out if such equation forms and graphic calculators are forbidden

but yes please use a graphical calculator

[lucid sequoia]

yess i will but i just have desmos rn

[glacial charm]

ok then use desmos

[lucid sequoia]

okay i inputed that into desmos but i feel it looks so weirddd

[glacial charm]

that is not how desmos works

[lucid sequoia]

do i make it equal to 0?

[glacial charm]

5e^x - 9cosx and 2 are two seperate equations

[lucid sequoia]

ahhh im sorry im struggling

okay how would i do it

[glacial charm]

.

you write two different equations one for y = 5e^x - 9cosx and one for y = 2 and see the points where the graphs intersect thats your solution

[lucid sequoia]

ohhh okaykay thank you!!

alright i have to bounce but ill come back if have anymore questions

.close

[pearl pond]

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[raw night]

Help to show that
f(f^-1(B)) included in B

[raw night]

f: X -> Y

B subset of Y

First i put y to be any element in f(f^-1(B))

Then when we recall the definition of the image it gives
That
There exist an element x in f^-1(B) such that y=f(x)

[feral sedge]

is f a bijection?

ah nvm this is true

[naive zinc]

There is no need for f to be

[feral sedge]

ok

[naive zinc]

Then what is the question, you are done

[raw night]

Not necessary

[feral sedge]

so y = f(x) for some x in f^-1(B)

[raw night]

Yea

Im stuck here

[feral sedge]

what does the definition of f^-1(B) tell you about x

[raw night]

f(x) in B

[feral sedge]

right

[raw night]

But the problem

Is the existence quantifier

I mean i always find problem with this one

[feral sedge]

well all of this is happening after you already selected x right

so this is all inside the existence quantifier

[raw night]

But not all x

In f^-1(B)

we select specified x

[feral sedge]

well you started with y right

[raw night]

Yes

[feral sedge]

the end goal is to show y in B

[raw night]

Right

[naive zinc]

a=b, b in some X, then a is in X

[raw night]

Wut

[naive zinc]

y=f(x)

[naive zinc]

^

[raw night]

Your answers always confuse me lol tbh

[raw night]

Then?

y in B?

[naive zinc]

Then what is the question?

[raw night]

What

[raw night]

.

[naive zinc]

?

[raw night]

??

[naive zinc]

You know the definition of subset, like a set is contained in another set?

[raw night]

Yea?

[naive zinc]

What is it?

[raw night]

Forall x (x in A => x in B)
Or
Forall x in A such that x in B

We need

To prove that

But

What’s point of what you say

[naive zinc]

So, you are trying to show that for any y in f(f^-1(B)), we have y is in B

[raw night]

Yeah i know?

[naive zinc]

And ???

What do you expect further?

[raw night]

And what

A contained in B

C’mon

[naive zinc]

You got y is in B. What next do you expect to happen

[raw night]

[naive zinc]

You finished your proof and you are pointing at your last step saying “how can I solve this”, you know how confusing it sounds to others?

(f(f^-1(B))=B cap f(X) by the way)

[raw night]

What
I didn't finish

[naive zinc]

Goal

[raw night]

Im still stuck in
There exist x in f^-1(B) such that y=f(x)

[naive zinc]

Your result

[raw night]

How this inplies that y in B

[raw night]

Yes why

[naive zinc]

y=f(x), f(x) is in B

[raw night]

What

[naive zinc]

I am done again. I really can’t figure out your mindset

[raw night]

You just deal with it superficially.

Why
(There exist x in f^-1(B) such that y=f(x) )implies that y in B?

[naive zinc]

^

[raw night]

But for some x

[severe quarry]

We want to show $f(f^{-1}(B)) \subseteq B$. \begin{enumerate}[\bf 1)] \item Let $y \in f(f^{-1}(B))$. Thus, there exists $x \in f^{-1}(B)$ so that $y = f(x)$. {\color{red} You are here!} \item But since $x \in f^{-1}(B)$, this means $y = f(x) \in B$. \end{enumerate}

[raw night]

That’s why i hate existence quantifier
It always confuses me

[jolly parrot]

Kepe

[naive zinc]

^

[raw night]

Well this true but for some x

[severe quarry]

Yes, but we don't care about the x at all. What we want to show is y in f(f^(-1)(B)) => y in B

What we do in the middle of that doesn't matter, what x we pick or whatever

[naive zinc]

You really don’t understand c=d, d is in some set, then c is in the same set?
7-4=3, 3 is odd, therefore 7-4 is odd

[severe quarry]

What matters is the very beginning and the very end. We picked some arbitrary y in f(f^(-1)(B)). What we conclude is that for this arbitrary y, y in B.

[raw night]

I know

But

Is there a property
We were able to move from the statement I reached until f(x) in B

@severe quarry I

[severe quarry]

Ah, you mean the step from 1) to 2)?

[raw night]

Is there an equivalent statement for
There exist x in f^-1(B) such that y=f(x)?

Or in general there exist x in B ,P(x)

Is there equivalent statement?

Maybe I look stupid to you, but I want to understand how to deals with the existence quantifier. That's why I want to know how we moved from this statement to the result.

[severe quarry]

I mean, it's really just noticing what x in f^(-1)(B) means. The set f^(-1)(B) is the set of elements that get sent to B by f. So if x is inside of that, x gets sent to B by f, per definition.

[raw night]

I know bro

x in f^-1(B) iff f(x) in B

I know this one

But

I have problem with (y=f(x) for some x)

[severe quarry]

and $x \in f^{-1}(B) \implies f(x) \in B$.

[jolly parrot]

Kepe

[raw night]

Conclude that f(x) in B
I feel it's a hasty reasoning

[raw night]

Lol ik

Fuck
I don't know how to ask the damn question

You didnt understand me

[severe quarry]

It's good that you are thoroughly thinking about this, you mean the second step here, right?

[raw night]

Yes

I'm thinking about
Existence quantifier
That its one value

Or

[severe quarry]

\begin{enumerate}[\bf 1)] \item [y \in f(f^{-1}(B)) \implies \exists_{x \in f^{-1}(B)}: y = f(x).] \item Independantly (completely ignore {\bf 1)} at first): [x \in f^{-1}(B) \implies f(x) \in B.] \item Now combine with {\bf 1)} to get the following chain: [y \in f(f^{-1}(B)) \implies \exists_{x \in f^{-1}(B)}: y = f(x) \implies y \in B.] \end{enumerate}

[raw night]

I think i find the question

Does the statement
There exist x in f^-1(B) such that y=f(x) implies x in f^-1(x)?

[jolly parrot]

Kepe

[severe quarry]

The last part of that contains a typo probably, I don't think that's what you want to ask

Can you correct that

[raw night]

y in B?

Wait

Is that y=f(x) important step?

[severe quarry]

You meant to say "x in f^(-1)(B)", right?

Does the statement
There exist x in f^-1(B) such that y=f(x) implies x in f^-1(B)?

Is that what you want to ask?

[raw night]

Yes

I dont think that’s true

The two statements are completely different

[severe quarry]

The left side states "there exists x in f^(-1)(B)". The right side of the implication is "x in f^(-1)(B)".

So it is true trivially

[severe quarry]

But like really, here just take 1) and 2) as completely separate statements

And at the end, in 3) you can combine them

[raw night]

x in f^-1(B) why its true?

[severe quarry]

1). Or do you mean in your question?

Your question is \ \li \ Is [\Big(\exists_{x \in f^{-1}(B)} : y = f(x)\Big) \implies \big(x \in f^{-1}(B)\big)] true? \ \li \ The answer is yes. If the left side is {\bf not} true, then the {\bf statement} is true anyways. An implication that begins with "false" is always true (check truth table of $\implies$).

[jolly parrot]

Kepe

[severe quarry]

But this question is independant of the exercise. We know the left side is true because it follows from y in f(f^(-1(B)) which we assume.

Yep

And if the left side is true, it's also true.

Because we literally state "x is from f^(-1)(B)"

And then we ask "Is x in f^(-1)(B)?"

I'm not using this. I'm using that the exists quantifier says "there exists an x in f^(-1)(B)"

And on the right, we check, for the same x, if it is in f^(-1)(B)

If it makes it clearer to you; [y \in f(f^{-1}(B)) \implies \Big(\big(\exists_{x \in f^{-1}(B)}: y = f(x)\big) \wedge \big(x \in f^{-1}(B) \implies f(x) \in B\big)\Big)][\implies y \in B.] But the $x \in f^{-1}(B) \implies f(x) \in B$ is always true, not strictly implied by the left side.

[jolly parrot]

Kepe

[severe quarry]

In your question? Because on the right, we are referring to the same x that we got the existence of on the left.

Anyways, I feel like this is kind of overthinking this LOL

[severe quarry]

Usually you would do this and be happy

Everyone thinks about stuff a bit differently so it can be hard to communicate what you mean sometimes

[severe quarry]

But yeah, are you happy with this?

Yes

We pick y at the beginning.

It's one value from now on in the rest of the reasoning.

Yes but then you go to the start of the reasoning again

At the beginning we fix an y

Only one

Say I want to prove that the square of an even number is always even.

I pick one even number n, some arbitrary one.

Then I prove that n^2 is even

And then I'm done, right?

We take an arbitrary value, yes. That means, just some value you'd like, doesn't matter. If we prove the statement for it, we showed it for every value in the set, because the value we picked didn't matter

In the even square example, I pick some arbitrary even number.

I show that the square is even

And then I showed that the square of every even number is even

Because what I picked was arbitrary

Could have been 2, could have been 4, ...

Yep, and this does prove it for every possible value. Think about the even square example

Does that make sense?

y is an arbitrary value, yes

And x is some other value that gets picked afterwards, depends on y

x is not random anymore

It depends on y

No, x is not random

It depends on y

For x, we don't just say "let x ...". We say there exists an x

Saying "let" means you pick an arbitrary value. Saying "there exists" means you find a value depending on the previous construction (not arbitrary!)

Maybe for all the arbitrary y values, the same x gets picked.

x depends on y

So maybe there is just one x that's really good and fits for all the y

So it's really not a "pick an arbitrary x"

[cinder flower]

hi kepe

[severe quarry]

Hi layla

The idea is that f^(-1)(B) could be some big set, but because we say "there exists", x is always dependant on y. So maybe x is the same value for all the y. Then that big set doesn't get covered by the x we take

f^(-1)(B) could be big, not just including 1

No

We don't care about if x gets picked as every value in the set or not eventually

We just care about y being arbitrary

Because our goal is to prove every element in f(f^(-1)(B)) is in B

So we have to let y in that first set be arbitrary

Otherwise there might be some elements that don't lie in B

Say we want to prove that $\mathbb N \subseteq \mathbb Z$

[jolly parrot]

Kepe

[severe quarry]

We pick an arbitrary element n from N.

And we show that n in Z.

Then, because what we picked was arbitrary, every element in N lies in Z

and so N c Z

Yeah

Yes

y in B

that's the result we get at the end

Yeah

And that holds for every y

Yeah, choice of x isn't important. Only y needs to cover every element in the set. We are just happy with "there exists an x fulfilling a property (depending on y)" after we pick y

np

[pearl pond]

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[pearl pond]

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[barren owl]

The revenue from a student album can be calculated using the formula I = 600x - 2x² where I denotes the revenue and x the ticket price in kronor. Which ticket price gives the maximum revenue?

[barren owl]

Appreciate any help

🤞

Lmk if anything seems unclear

[steep saddle]

are you supposed to use calculus

[barren owl]

Yes

[steep saddle]

cool so what defines a maximum

[barren owl]

Uh

A point

Where it goes from growing to receding

[steep saddle]

think about how the slope would need to change

to make a point a maximum

[barren owl]

Needs to go up and down

Right

[steep saddle]

ok

in other words it goes from positive slope to negative slope

[barren owl]

Idk the correct terms to use sorry 😭English ain’t my first language

[barren owl]

Yes

[steep saddle]

ie the slope would be 0 at some point

when is the slope 0

[barren owl]

When it’s not positive nor negative

[steep saddle]

ok

but like

solve it

slope = derivative

[barren owl]

Do I need to use the second derivative aswell

[frigid swallow]

The gradient of a point is the value of the first derivative of its function at this point

You should have an idea of how the revenue against ticket price curve should look like

To make things easier

Do you?

[barren owl]

Sure

[frigid swallow]

Do you have any idea how it would look like

[barren owl]

A maximum point?

[frigid swallow]

No the curve itself

How does it look like

[barren owl]

Uh no

I don’t know

[frigid swallow]

The 600x - x^2

This is a quadratic function correct

[barren owl]

Ye

[frigid swallow]

Quadratic functions are parabolas

Yes?

[barren owl]

The U looking things right

[frigid swallow]

Yes

[barren owl]

Yeah

[frigid swallow]

Basically the parabola either looks upwards or downwards depending on the sign of x^2

[barren owl]

I see

[frigid swallow]

If its positive then the parabola looks up if negative it looks down if thay makes sense

[barren owl]

It does

How does that helps me solve it though

[frigid swallow]

This function is negative yes so its gonna look down

Ill tell u in a sec

[barren owl]

Yeah

[frigid swallow]

Okay each parabola has a turning point

Either maximim point or minimum point

[barren owl]

Mhm

Where it goes positive to negative or vice versa

[frigid swallow]

Gradient wise yes

Okay in our scenario sinxe the parabola is looking down it will look like this

Making our turning point the maximum

Its maximum y level wise which is the revenue

So it corresponds to maximum revenue

[barren owl]

I see

[frigid swallow]

Which is what the question wants

Now whats the gradient of a turning point

[barren owl]

Wdym

[frigid swallow]

Each point has a gradient right?

A slope

[barren owl]

Oh you mean the curvature?

[frigid swallow]

Idk what you mean by curvature its usually referred to as gradient or slope

[barren owl]

If your talking about what I think you are then it is 0

[frigid swallow]

Yes correvt

[barren owl]

Sorry English isn’t my first language so idk a lot of the terms

[frigid swallow]

Yeah no worries

Okay now u know that the point where the gradient or curvature is 0 is the maximum revenue

[barren owl]

Yes

[frigid swallow]

How do you usually get the gradient of a point if u have the function and an x value

[barren owl]

Derivative?

[frigid swallow]

Correct

You get the first derivative of the function then substitute with the x value

Now you have the function but instead of the x value u have the gradient

So youre gonna find x instead of finding the gradient

[barren owl]

How so

[frigid swallow]

First derivative = gradient

Can you find the first derivative

[barren owl]

Ye

[frigid swallow]

Okay do that

[barren owl]

600-4x

[frigid swallow]

Yea

And the gradient is equal to?

[barren owl]

The derivative

[frigid swallow]

No like value

Of the point we want to find

[barren owl]

This?

[frigid swallow]

Thays the derivative

You also have the value of the gradient

Of the maximum point

Correct?

[barren owl]

0?

[frigid swallow]

Yes

[frigid swallow]

So using this 600-4x=0

Yes?

[barren owl]

Mhm

[frigid swallow]

Now find x

[barren owl]

So now we solve that

[frigid swallow]

Yep

[barren owl]

150

[frigid swallow]

Yea

Now u have the x

But tou want the revenue

Which is the y

What will you do now

[barren owl]

I assume we plug in 150 as x

Although I’m not sure

[frigid swallow]

Yes thats correct you plug the value of x you obtained in the function of revenue in terms of x

[barren owl]

Do k plug it in to the function or the derivative

[frigid swallow]

The derivative would just get tou zero

Thats how you got 150 in the first place

[barren owl]

Oh right

[frigid swallow]

You want the revenue which is the function itself

Revenue = 600x -2x^2 right?

[barren owl]

Mhm

[frigid swallow]

You have the x

So you can now substitute and grt the revenue

[barren owl]

45000?

[frigid swallow]

Correct

[barren owl]

Is that it

[frigid swallow]

Yes

[barren owl]

So the answer is 150 since they’re asking for X right

[frigid swallow]

X is the ticket price

I is the revenue

They want the maximum revenue

[barren owl]

Got it

[frigid swallow]

I = 600x -2x^2

[barren owl]

Thank you so much sir

[frigid swallow]

So 45000 is your answer

Np

[barren owl]

Have a great day

🙏

[frigid swallow]

You too

[barren owl]

.close

[pearl pond]

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[tranquil tiger]

A1= 280, corresponding B1= 280.13
another value A2= 285, corresponding B2= 285.14

these are fixed values from tables, and i need to interpolate for A=283 (which is between A1 and A2), and hence find the corresponding B

[sly garden]

!original

[pearl pond]

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

[tranquil tiger]

is it thermodynamics

i am finding enthalpy corresponding to the temperature

in question temperature given is 283 K. and we need enthalpy to put in the formula, so we find that value (corresponding to 283K) from the reference table (which are fixed values)

[sly garden]

this sounds like physics. which is cool but not a thing i can help with sorry :/

i’m not good at physics

[tranquil tiger]

its just basic interpolation, which is used by making some similar traingles

its not related to phyics, thats why i was not saying it earlier

its like

[sly garden]

oh it's a regression kinda situation

ok i can work with that

can you send a screenshot of the table you currently have

[tranquil tiger]

x1= 280 ; y1= 280.13
x2= 283 ; y2 = ?
x3= 285 ; y3 = 285.14

the reference table has fixed value of 280 and 285, but to solve question i need value if 283, which is not in table. so we need to do something called interpolation.

i used to do some years ago, but i dont know now

[sly garden]

assuming a linear relationship you can use FORECAST.LINEAR

ye interpolation is the word exactly, i just wasn't sure if there was any additional info

[tranquil tiger]

!close

.close

[pearl pond]

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[pearl pond]

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[robust lichen]

i just wanna know what the question is from an another perspective

[robust lichen]

d^2 x / dx^2 + 4x = 2025^ t or 2024^+ ??

[snow sail]

my guess would be $x(t)$ and that $\dv[2]{x(t)}{t} + 4x(t) = 2025^t$

[jolly parrot]

jan Niku

[snow sail]

hard to know if thats an x or a u or what

[robust lichen]

that would make sense

its x in cursive

can tell that

[snow sail]

fancy

you got it from there?

[robust lichen]

yeah

.close

[pearl pond]

Channel closed

Closed by @robust lichen

Use .reopen if this was a mistake.

[robust lichen]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[robust lichen]

@snow sail small doubt

when inserting it into the formula

do i put sin -2t or sin 2t

does it even consider sign

[snow sail]

well, you can get lucky here

your c_2 is arbitrary right

[robust lichen]

yeah

[snow sail]

sin(x) is actually odd

so sin(-x) = -sin(x)

maybe you see how thats useful here

[robust lichen]

yeah

oh

it wont matter

[snow sail]

[robust lichen]

ill put 2t

imma solve completely then close this

.close

[pearl pond]

Channel closed

Closed by @robust lichen

Use .reopen if this was a mistake.

[robust lichen]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[robust lichen]

why are we multiplying by x
where the fuck did that x come from at the end

should i stop questioning what is and just swallow it?
@snow sail

jan
january

[snow sail]

soon

[robust lichen]

soon

[snow sail]

if you know anything about linear algebra

you can wave it away by

[robust lichen]

i shouldnt question it

[snow sail]

that its a way to manufacture another linearly independent solution

[robust lichen]

like i usually do with everything

[snow sail]

no not at all

[robust lichen]

im going to keep this ticket open until i sleep

[snow sail]

i can see if i can find a good link

[robust lichen]

"it is what it is" basically

[snow sail]

no, not really

it has to do with the fundamental set of solutions

the differential operator can be seen as a linear operator on a vector space made up of functions

that brings with it some pretty strong theory

it can just be involved

[robust lichen]

i see

[snow sail]

https://tutorial.math.lamar.edu/classes/de/FundamentalSetsofSolutions.aspx

i wonder if reduction of order is enough thats also not very satisfying

VOP maybe

I think just reading about the fundamental set would probably get you there

sorry i havent worked out a really convincing simple argument in my head yet

[robust lichen]

maybe after i write my finals

[robust lichen]

its fine

[snow sail]

if you are super curious, I'd ask in #odes-and-pdes

after finals, or whatever

thats where all the smart people hang out

[robust lichen]

ill do that later

i just need to get this over with

its nearly 6 am

[snow sail]

if youre doing more math, it can come together later

once linear independence is a more fleshed out concept once its bashed into you by like

analysis or algebra

ill stop babbling

[robust lichen]

probably 3rd sem

[robust lichen]

yappster

next sem is going to be my own personal hell with chemistry and BEEE

[snow sail]

hmm well it will go by faster than you think i bet

[robust lichen]

things have been going fast lately

time moves fast when you do something interesting

[robust lichen]

@snow sail CF = C1 cos 3x + C2 sin 3x

For PI from what i just learned its going to be

sin 4x / (D^ + 9)

a = 4 so we input f(4) so its gonna be

sin 4x / 25 right

[snow sail]

👀

if you just want to check your answers the bot is wolfie

which is just mathematica light

its great with diff eq like these

[robust lichen]

i have no idea what any of that means

[snow sail]

you do it like this

,w y'' + 9y = sin(4x)

[jolly parrot]

Results provided by WolframAlpha

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

Requested by janniku

[robust lichen]

oh wolfram

[snow sail]

yea

[robust lichen]

SEVEN

[snow sail]

[robust lichen]

oh

nvm

so wait

un nvm

FAH

makes sense

not understanding the rule then

[snow sail]

is it not just MUC

so $y_p = A\sin 4x + B \cos 4x$

[jolly parrot]

jan Niku

[robust lichen]

nooo

3x you mean

[snow sail]

no, its 4x

to match the forcing term

[robust lichen]

huh

[snow sail]

right, because the 3 comes from the complementary, unforced part

[robust lichen]

ye

[snow sail]

we just take y'' + 9y = 0

get those

[robust lichen]

this the rule

[snow sail]

now the forced parts, will look like some LC of these

I'm not sure the rule you mean maybe you assume that the cos part will drop off

what is PI?

[robust lichen]

particular integral

coz the general solution is complementary func + PI

[robust lichen]

when Q(x) = sin 4x then PI = sin 4x / f(D)

[snow sail]

its been a long day, im sure i could track you out if it werent the end of a long week haha

[robust lichen]

imma see if it means like

[snow sail]

MUC works out, for me, here, i think

i dont have a rule

maybe #odes-and-pdes

keeping a single channel open for all questions is like

[robust lichen]

havent done MUC yet so

[snow sail]

well i cant answer every question

[robust lichen]

too bad youre stuck here with me

https://tenor.com/view/muhehehe-gif-11037525145585928229

[snow sail]

I'm not stuck

[robust lichen]

50 50

[snow sail]

if you want i can ask if anyone knows wtf youre talking about with the PI thing

[robust lichen]

😂

wait

imma just

put it in that channel

[snow sail]

sure, ill ask too

or, in general, opening a new channel for each doubt is better method to get fast help, i think

there are plenty of channels to be cycled in i wouldnt worry about being fast

anyways

[robust lichen]

ive asked

can i horde this channel

lol

.close

[pearl pond]

Channel closed

Closed by @robust lichen

Use .reopen if this was a mistake.

[snow sail]

you can do whatever you want, the channels are here to use

[pearl pond]

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[spring crystal]

whys this have 2 solutions im confused

or its just showing 2 diff ways to do it

[cinder flower]

confused about what

the final answers look the same

[spring crystal]

so they did it twice for fun

[waxen agate]

Showing different methods

Seemingly, the first method gives more marks?

[spring crystal]

why does this not contain f(2,-1,1) in front

[waxen agate]

You treat the surface as a graph z = f(x,y) rather than as a level surface F(x,y,z) = 0.

[spring crystal]

whats that even mean

so if it doesnt say F(x,y,z) then i dont use f?

[waxen agate]

There are two common ways a surface is given. Either z = something involving x,y or something involving x,y,z = constant. You can use the tangent plane formula for the former, and the gradient method for the latter.

[pearl pond]

@spring crystal Has your question been resolved?

[pearl pond]

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[frank violet]

Can I get a hint on 2.

[waxen agate]

Check a_n - a_{n-1}

[frank violet]

I have done that for part 1

What should I do next

[spiral pivot]

The form of the question makes it seem to me as if the sum never reaches 1, rather than being specific to anything about 2022, which is common in these sorts of questions. If that's the case, then you might be able to do a comparison test of each terms against (for instance) 1/((n)(n+1))

[midnight haven]

Looks like a good one

[spiral pivot]

(because 1/(n)(n+1) is a well known series that sums to 1.)

[frank violet]

Oh that's smart, I'll try

[pearl pond]

@frank violet Has your question been resolved?

[frank violet]

I'm trying

[waxen agate]

I think it’s more intuitive to go with the telescoping sum route

[urban tapir]

hello everyone! I recently started my own server (for socializing) and im not trying to advertise it right now but i need suggestions on how to improve the already existing server!

[waxen agate]

This isn’t a maths question and so doesn’t belong here.

[dusk willow]

not a good place to ask. probably try asking in #chill

[urban tapir]

oh wth i thought this was the general chat! im so sorry

😭

[brisk thorn]

For n1 you have to do Un+1 - Un and show its positive? Just asking

[waxen agate]

Uh yeah

[brisk thorn]

Thank you

[frank violet]

I dunno how to make it telescopic tho, I did tried

[waxen agate]

Consider [ frac1{ a_n - 1 } - frac1{ a_{n+1} - 1 } ]

[jolly parrot]

クーリー

[frank violet]

Okay HOW

How did you see this

$1-\frac{1}{a_n-1}$

[jolly parrot]

Fionna The Unemployed

[frank violet]

So we just have to show diverge

[waxen agate]

I assume this is meant to be a_{2023} - 1?

There’s no need to consider the limit. We’re dealing with a finite sum.

[frank violet]

he said it might not relate to 2022 so I'm showing that this sum has upper bound of 1

That's why I just use n instead of 2023

[waxen agate]

Ag

Then yes

[waxen agate]

This just proves every partial sum is bounded above by 1

[frank violet]

okay so I'm comparing a_n with n, cuz lim 1/[n-1] is 0

Using induction

Let assump a_k>k

[waxen agate]

I mean

a_{n+1} = a_n((a_n-1)^2 + 1) >= 2a_n

That should do it

[frank violet]

but how can I show a_n is unbound with that

I meant we have to show that a_n grow extremely large so a_n when n big enough is close to infinity right

then 1/[a_n - 1]~ 0

[waxen agate]

It means a_n >= 2^n for all n by induction

[frank violet]

Ohh

Oh yeah

Damn I didn't see that

one question, do you know any books writing about sequences like this? maybe a workbook

[waxen agate]

Sequences and series by Titu Andreescu & Gabriel Dospinescu

Titu Andreescu has lots of really good workbooks.

[frank violet]

TYSM

.close 💙

[pearl pond]

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[north talon]

Hi! can somebody help me out with this geometry problem
translation: A cube with side length of 2 units is placed on top of a cube with sidelength 6 positioned at the center of the top face. What is the radius of the sphere that just encloses both of the cubes?

[north talon]

i dont really know how to start ;-;

[rotund ferry]

I think you might wanna use coordinates

[north talon]

or maybe im just severely overthinking this😭

can we assume that the sphere is tangent to each vertex? since its symmetrical

[still hamlet]

i think you only need the diagonal of the larger cube

[north talon]

why?

is the center of the sphere on the diagonal?

[rotund ferry]

I'm pretty sure we can safely assume that the sphere is tangent to :
The 4 bottom points of the large cube, and the
4 top points of the small cube

[still hamlet]

oh right that's my bad

[rotund ferry]

If we start using coordinates :
Label the front left bottom vertex of the large cube (0, 0, 0)
We should know that the center of the sphere is on the line x=3, y=3

Then coordinates will solve this problem easily

-# Hopefully

[north talon]

wouldnt placing our origin at the center of the big cube be easier? since the center of the sphere and both cubes should be colinear

due to symmetry

[rotund ferry]

I mean that's up to you

[frank violet]

the base of the 6 units square is inscribe the circle in that same plane, the same for the top base of 2 units

So like the line passes through center of the sphere also passes through the intersections of diagonals in both bases

cus both planes are parallel

[rotund ferry]

-# That's basically what I said

But alright

[frank violet]

But then just consider 2 corners of each bases that in the same plane with the line passes through center of sphere like I said

[north talon]

is it 3 sqrt(3) then?

[frank violet]

We will have a regular trapezoid

[frank violet]

I dunno I haven't done any calculation

[north talon]

bleh

[rotund ferry]

Probably unless you made any arithmetic mistakes

[north talon]

okay, cool

3d geometry suckss😭

[frank violet]

haha

[rotund ferry]

Coordinates will kill almost everything tbh

[frank violet]

Wait

[north talon]

i guess yeah, but that ruins the spirit of it :p

[frank violet]

Almost*

Check this out

3D geometry

[north talon]

nuh uh