help-13

[subtle quartz]

thank you!!

[crimson sedge]

no problem

you can zoom in if you want

[subtle quartz]

thank you for taking the time to help me

[crimson sedge]

:))

[subtle quartz]

:))

.close

[cedar kiln]

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[dire geode]

Why did your exponent change from-3/2 to-1/2

[wraith dagger]

bob420

bob420

bob420

[cedar kiln]

@atomic pecan Has your question been resolved?

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[marsh warren]

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[plain sun]

So I understand I have to split it into [0,a] and [a, infinity]

since it is continuous on [0, infinity) it is uniformly continuous on [0,a]

so if x and y are in [0,a] i can show it is uniformly continuous and if x and y are in [a, infinity) I can show it is uniformly continuous

and then take delta as min of both of those two

but I'm confused as to what happens if x is in [0,a] and y is in [a, infinity)

[cerulean sail]

Hint: a is in the middle and is contained in both of those intervals you mentioned…

[plain sun]

I feel very lost

should I take something like |x-a| and |y-a|?

reverse triangle inequality?

[cerulean sail]

Well, to make things a bit easier, maybe assume (wlog) that x is in [0, a] and that y is in [a, ♾️)

Using the fact you know you’re uniformly continuous on each of those two intervals, maybe try and make corresponding statements, and see if you can get to |x - y| < delta implying |f(x) - f(y)| < eps

[cerulean sail]

(This might be useful!)

[plain sun]

Um so should I split x-y less than delta x-a+a-y?

[cerulean sail]

That’s the idea, consider x to a and a to y

You know you’re uniformly continuous on each of those bits, and that each corresponding difference (a - x) and (y - a) is less than the whole difference (y - x) = |x - y|

[plain sun]

Okay so if I take x-a less than delta and f(x)-f(a) less than epsilon/2

Then do a similar thing for the other one

Take min delta

And arrive at f(x)-f(y) less than epsilon

Is that right?

[cerulean sail]

That’s kind of the idea, yep! Few minor comments but yep

[plain sun]

Minor comments?

Like details I should fill?

Or something I didn’t consider

[cerulean sail]

Because you're uniformly continuous on $[0, a]$ and also on $[a, \infty)$, you know if you're given a $\epsilon > 0$, and that if $x$ and $y$ are both in $[0, a]$ or both in $[a, \infty)$, there's that $\delta$ (being the minimum), that if $\abs{x - y} < \delta$ then $\abs{f(x) - f(y) } < \frac{\epsilon}2$

but note of course that $\frac{\epsilon}2 < \epsilon$ - that might be handy to keep in mind

[wraith dagger]

@cerulean sail

[cerulean sail]

More like something to make the final argument really smooth!

You have that you've found your delta such that wherever x and y are, if you had |x - y| < delta, you're forced to have |f(x) - f(y)| < epsilon, as a result of uniform continuity on each of the "bits"

[plain sun]

I didn’t understand

[cerulean sail]

Basically you want your final argument to not need to change depending on where x and y are

When you find your delta, you want to be able to say that after that point, as long as the x and y’s that you have are within distance delta of each other, then f(x) and f(y) are within distance epsilon of each other

Earlier bits of the argument depended on where x and y were a bit

[plain sun]

Doesn’t taking delta as the min of the cases take care of that ?

So I take delta as min of all three cases

Is that it?

[cerulean sail]

It does, but the idea is that you start your "prep" first, then use that afterwards, like

[plain sun]

Oh right

To make the argument flow

Like chose your delta then you’ll get your epsilon

[cerulean sail]

We have $f$ and know that it is uniformly continuous on $[0, a]$, and also on $[a, \infty)$, so:
\begin{enumerate}
\item given any $\epsilon > 0$, there is a $\delta_1 > 0$ such that when $x,y \in [0, a]$ and $\abs{x - y} < \delta_1,$ we have $\abs{f(x) - f(y)} < \frac{\epsilon}2$,
\item given any $\epsilon > 0$, there is a $\delta_2 > 0$ such that when $x,y \in [a, \infty)$ and $\abs{x - y} < \delta_2,$ we have $\abs{f(x) - f(y)} < \frac{\epsilon}2$
\end{enumerate}
Let an $\epsilon > 0$ be given and let us choose $\delta = \min{ \delta_1, \delta_2}$. Then given $x, y \in [0, \infty)$, there are three cases:
\begin{enumerate}
\item We have $x,y \in [0, a]$. Then [...stuff...] $\abs{f(x) - f(y)} < \epsilon$.
\item We have $x,y \in [a, \infty)$. Then [...stuff...] $\abs{f(x) - f(y)} < \epsilon$.
\item We have, without loss of generality, $x\in [0, a]$ and $y\in [a, \infty)$. Then [...stuff...] $\abs{f(x) - f(y)} < \epsilon$.
\end{enumerate}

[wraith dagger]

@cerulean sail

[cerulean sail]

The first two points are basically immediate from the fact that eps/2 < eps, it's just the last one that you need to fill in from the above really!

[plain sun]

Sorry I’m taking time I’ll just try and think about it😅

[cerulean sail]

Sure sure sometimes it's a bit tricky, but of course noting that for uniform continuity your delta should not depend on where x and y are, or what they are, hence me trying to be careful with laying it out!

[plain sun]

So if we take the third case

And take x-y less than delta

Where delta is the min of delta 1 and delta2

Then x-a is less than delta

Because a is in between x and y

So f(x) - f(a) will be less than epsilon/2

And then use a similar argument for f(y)-f(a)

[cerulean sail]

(absolute value of that - you don't know how f behaves)

[plain sun]

Yeah absolute value

Sorry I’m using my phone so it’s a bit of a struggle to type it all out

[cerulean sail]

Understandable, some of the stuff I had to move to computer to type out haha

[plain sun]

Do I have to add anything else?

I’ll finish the argument out completely when I write it

[cerulean sail]

Not really - just normal triangle inequality from there

[plain sun]

Okay!

Thank you so much

[cerulean sail]

No problem

[plain sun]

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[frank tangle]

what am i doing wrong?

apparently its 4sqrt(3)

[waxen mortar]

When you attempted to multiply both sides by $\sqrt{3}$, you forgot the 8 on the left. Your second line should read $8\sqrt{3}+x\sqrt{36} = 8x$

[wraith dagger]

Shenzao

[waxen mortar]

Try again from there.

[frank tangle]

ahhh

fuck youre right

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[rough latch]

How would you do pt b?

[subtle harbor]

think of the cases for starting with each number, and adding on an additional number

and the restriction that they must be in increasing order

it may be easier to think backwards (from the last digit), but i think its exactly the same

just think of approach with starting with a number, and the number of possibilities the second number has and how that limits the range for 3rd number...so on

i.e. if you start with 0, your second number must be in a range from 1 to 5 (do you see why?)

[cedar kiln]

@rough latch Has your question been resolved?

[subtle harbor]

and even thinking as a first position, you should realize that first position has to be a range of numbers too (not the full range!)

look what happens when first position is 4 :)

[cedar kiln]

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[lavish oyster]

i dont know what this is, i dont know where to start, i never learned this, and why are the numbers not simplified

[thorny shadow]

What numbers aren't simplified?

If you didn't learn it then unlucky, look up the general formula for the area of a regular polygon

[lavish oyster]

like why is everything in square root form and not caried out

[thorny shadow]

7 is a prime number, you cant simplify the square root of 7

The square root of any prime number is irrational so there's not really anyway to simplify those numbers anymore without losing accuracy

[lavish oyster]

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[winged wharf]

My drawing is a little different from the reality, is it problematic or is it okay?

[winged wharf]

should I have done a second derivative?

[cedar kiln]

@winged wharf Has your question been resolved?

[winged wharf]

<@&286206848099549185>

nvm asked a friend

.close

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[cedar kiln]

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

[thick cipher]

What percentage of people replied no?

[distant fern]

35%

[thick cipher]

Okay

[distant fern]

how do i change that to people?

[thick cipher]

What percentage of people replied yes?

[thick cipher]

(It's not 35%)

[distant fern]

oh

oh 40

60%

YES I GOT IT

20 PEOPLE

thanks!

[cedar kiln]

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[frigid belfry]

Does the dot product of a vector with itself equal the sum of the squared components even if the basis is not orthogonal?

[granite knoll]

what does basis have to do w/ it?

[potent fractal]

Let e1, ..., en be an orthogonal basis. Then the standard dot product in this basis is defined as x^T y.
If we have a different basis and C is the matrix that changes coordinates, then in this basis dot product will be (Cx)^T Cy = x^T C^T C y
This is equal to x^T y if and only if C^T C = E, so in other words when C is orthogonal.
So the answer is no

[cedar kiln]

@frigid belfry Has your question been resolved?

[frigid belfry]

perfect, thanks

[cedar kiln]

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[rancid pollen]

I know I have to find an upper/lower bound or prove there is one

But I don’t know how to

[dire geode]

Try just odd or just even n

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[south granite]

Anyone able to help

With problems like these

[dire thorn]

Recall exponent rules, specifically when there is multiplication in exponent. Also recall when you can cancel log and exponents

[civic eagle]

rewriting it with fractions might help here.

[south granite]

Oh wait

I just realized

Is it

5^3log5^3

[dire thorn]

I think u got right idea but u wrote it wrong

[cerulean sail]

(And you can keep going, if you meant log base 5 of 3 there…)

[cedar kiln]

@south granite Has your question been resolved?

[cedar kiln]

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[shrewd blaze]

Heyyy

[shrewd blaze]

How do you solve something like thiss?

I'm reviewing my Papers and Still dont know how to do it

I have an exam in a few hours

[austere hull]

using sohcahtoa

[shrewd blaze]

I know but howw

Oh i know howw

Waitt

I got it

.close

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[sharp oyster]

im not sure where i went wrong

it is a linear alg problem, but ik i took a multi variable calc approach

[solid juniper]

[1,1,0] is not what you wanted

[sharp oyster]

uh oh

what did i want

pls elaborate 🙏🏼

[dire geode]

Show the entire problem

[sharp oyster]

that is the entire problem

including this

mb

[cerulean sail]

inner product defined how?

[solid juniper]

define by...?

[cerulean sail]

Yea - "normal" norm of the coordinates won't be the same as the norm of the polynomial wrt that ip

[sharp oyster]

so how would i solve that problem then

[cerulean sail]

Pretty much work out $\sqrt{\ip{p}{p}}$ as per here

[wraith dagger]

@cerulean sail

[sharp oyster]

what does that mean

[dire geode]

Yes this is very important to the problem

[sharp oyster]

bc if so oop

rip

[solid juniper]

this took a twist i was not expecting

[sharp oyster]

huh

😭

[cerulean sail]

I mean, the "usual" norm of that does work out to be the same thing sure, though I wouldn't think about it like that

[sharp oyster]

how should one think about this problem then

[solid juniper]

i was not expecting that random ass norm

[sharp oyster]

what do you mean by this 😭

[cerulean sail]

Same, was thinking there's be some integral inner product

[sharp oyster]

nope

[solid juniper]

it's nothing dw lol

[sharp oyster]

lol ok

[sharp oyster]

^

[dire geode]

🍑

[sharp oyster]

oof

[solid juniper]

nice delete timing

[cedar kiln]

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[inland mason]

I dont get this

[dull oxide]

Do you know what an arithemetic series is?

[inland mason]

Yes

Oh wait

I think I did geometric series

Anyways, I still cant do it

😭

[dull oxide]

!show

[cedar kiln]

Show your work, and if possible, explain where you are stuck.

[inland mason]

Oh okay

It's kind of messy but,

[rocky kettle]

ok

so lets do part a

an arithmetic sequence means common difference

$$(2x+9)-(x-7)=x+16$$

[wraith dagger]

everything_addict

[rocky kettle]

@dull oxide then ill leave it to u

[inland mason]

yeah?

Oh

😭

[rocky kettle]

then

take the next difference

$$(15-3x)-(2x+9)=-5x+6$$

[wraith dagger]

everything_addict

[rocky kettle]

$$6-5x=x+16$$

[wraith dagger]

everything_addict

[rocky kettle]

i think u can do it from here

apply the same to the next qn

[inland mason]

Oh okayy!

Tysm!

I will try it myself

.close

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[azure dock]

what did i do wrong

[opal schooner]

why are you treating 11 as the constant for the x term when that has no x

[azure dock]

im using the quad formula

you gotta use b

[opal schooner]

$ax^2 + bx + c = 0$

[wraith dagger]

Triaxyz

[opal schooner]

there is no term multiplied by just x here

that just says $x^2 + 11 = 155$

[wraith dagger]

Triaxyz

[azure dock]

oh

i didnt notice

that there was no x

[bold hinge]

You don't even need the quad formula

[opal schooner]

^

[bold hinge]

11 - 155 = ?

[azure dock]

-144

[bold hinge]

then x^2-144 = 0

x² - 12² = 0

[azure dock]

dont we need 3 numbers

to state a, b, c

[bold hinge]

b is 0

[azure dock]

ohhh

so if there isnt a number stated it means its 0?

[bold hinge]

Yeah

[azure dock]

this made sense

thank you so muchh

.close

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[jade edge]

How would I factor this?

[inland sky]

write the factors of 70

[austere surge]

if you're referring to k^2 + 12k - 70, list the factors of -70 and pick the ones that add to positive 12

[jade edge]

Yeah I tried that

It didn’t work for me

[austere surge]

lemme see it then

[inland sky]

Then use the quadratic formula

[jade edge]

Ah ok got it

[austere surge]

oh yeah that'll do it

[jade edge]

Ty

.close

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[meager violet]

I have this problem and I am completely lost on what is happening
I'm also trying to look at the answer key given but i don't understand where the 2 is coming from and how the limits of the integral are changing

[inland sky]

Show the entire problem

[meager violet]

i'm only trying to look at 11 right now

[inland sky]

ok do you know u substitution?

[meager violet]

yes

i dont understand where the two is coming from and then the limits change from -3 to 3 to 0 to 3

.close

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[weak bobcat]

Is the answer to the first one -5

[worldly walrus]

Waddup

[weak bobcat]

OH AASSUL

[worldly walrus]

LOL

Lemme try to help

I think u right

[weak bobcat]

Lmao alright

[inland sky]

Do you know what the inverse means?

[worldly walrus]

@weak bobcat

[cedar kiln]

@weak bobcat Has your question been resolved?

[cedar kiln]

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[crimson sedge]

help

[crimson sedge]

whats the diff between "linear and irreducuble quadratic factors" and "linear factors which may be complex"

would this be the irreducible one

HELP

MEEE

[cedar kiln]

@crimson sedge Has your question been resolved?

[crimson sedge]

help

[cerulean sail]

You can't be linear and quadratic at the same time - where are you getting that statement from?

[crimson sedge]

ik im asking the difference

what is the one shown on the image

[cerulean sail]

There's no such thing as "linear and irreducuble quadratic factors"

[crimson sedge]

yes there is

its on my

study sheet

[cerulean sail]

That's what I'm saying - show your study sheet

There's linear factors, and irreducible quadratic factors

[crimson sedge]

ok

[cerulean sail]

[linear factors are almost always irreducible]

[crimson sedge]

sorry i had to airdrop it

[crimson sedge]

its this

[cerulean sail]

They mean this - you're factoring such that your factors are either linear, or irreducible quadratic

[crimson sedge]

so which is the one i showed u

is it linear?

[cerulean sail]

Something like $x^3 - x^2 + x - 1$ factors to $(x - 1)(x^2 + 1)$, the $(x-1)$ is linear and $(x^2 + 1)$ quadratic irreducible

[wraith dagger]

@cerulean sail

[crimson sedge]

oh ok

[cerulean sail]

But you can also factor that as $(x - 1)(x - i)(x + i)$, the factors there are complex and linear

[wraith dagger]

@cerulean sail

[crimson sedge]

yes ok

thank u

[cerulean sail]

Those factors are all linear - but note that all real numbers are complex numbers

[crimson sedge]

yes yes ok

bye

love u!

[cerulean sail]

Awww

[cedar kiln]

@crimson sedge Has your question been resolved?

[cedar kiln]

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[fathom raft]

I might be a bit tunned visioned but I just fail to see how here applying the chain rule gives that result

[fathom raft]

the part how they get theta dot in the equation and why it is not just the first part ar0exp(a*theta)

only theta and r is time dependant

[cedar kiln]

@fathom raft Has your question been resolved?

[potent fractal]

Yeah so by the chain rule theta should be differentiated

So essentially what they wrote is $\frac{dr}{dt} = \frac{dr}{d\theta} \frac{d\theta}{dt}$

[wraith dagger]

EQUENOS

[fathom raft]

oh yea obviously 🤦‍♂️ thanks a lot 🙂

❤️

.close

[cedar kiln]

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[native shuttle]

im confused on how to go about finding the correct val

i know that we want the limmit of a_n to approach 0

the answer key is telling me that my left end point is incorrect

[dusk hemlock]

There are some theorems of convergence, you need to rely on those

[native shuttle]

mm im not sure which ones exactly apply here?

[gentle pecan]

I might try the ratio test

I'm going to give it a go to see if I can get an answer

[subtle harbor]

you can use substitution to show lim n->inf ln(n)^p/n^4 goes to 0 for all p

and hence just apply alternating series theorem

[gentle pecan]

Yeah Ratio Test is definitely inconclusive Whoopsie lol

[subtle harbor]

let n=e^u

@native shuttle

[subtle harbor]

also you can clearly see it works for p=0, so you can see your answer is wrong already

(-1)^(n-1) 1/n^4

[native shuttle]

$\frac{ln(n)^p}{n^4}$

[wraith dagger]

arm

[subtle harbor]

$\frac{\ln(n)^p}{n^4}$

[wraith dagger]

🫎 A Certain User(Moosey) 🫎

[subtle harbor]

$\lim_{n \to \infty} \frac{\ln(n)^p}{n^4}$

[wraith dagger]

🫎 A Certain User(Moosey) 🫎

[native shuttle]

right i agree and i intially thought of (0, inf) but the answer key says is wrong

[subtle harbor]

-inf,inf

p can be negative

i assume

[native shuttle]

if its negative then we will have

[subtle harbor]

it doesn't put any restriction on p

[native shuttle]

n^4/ln(n)?

[subtle harbor]

no...?

p is only applied to ln(n)

[native shuttle]

sorry im buggin

[subtle harbor]

chartbit ur always lurking

watchin me

[native shuttle]

why can p be negative?

[subtle harbor]

think about it

$\f{1}{\ln(n)^{p}n^{4}} \leq \f{1}{n^4}$

[wraith dagger]

🫎 A Certain User(Moosey) 🫎

[native shuttle]

oh rightt

ln just comes to the bottom

n^4 stays

[subtle harbor]

yes ;)

[native shuttle]

ok that makes sense yeah that i agree with

ty moosey!

[subtle harbor]

np :)

[native shuttle]

.close

[cedar kiln]

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[shrewd gale]

How is OB 4?

[shrewd gale]

So OD is 6?

And BD is 2?

they didn't show it in the solution at all that's why I'm confused 💀 man geometry sucks

So true, anyways thank you

.close

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[crimson sedge]

Hi

[crimson sedge]

I need help with this equatiob

Sorry i dont understand

Yes true

Ohhh

So its

3,2 = 150 ÷ m

Im guessing the formula i should use is

M = vertical distance divide horizontal distance

So the vertical distance given is 150

Yea 3 over 2 is the gradient which is why i placed it first

3 150
_ = _______
2 m

Like that

Cause the base of the pyramid is the horizontal distance

Hello?

Yes ik

Lets replace the m with x then

Yea

Ahh ok

Give me 5 minutes

Is it 225?

Heyy??

I keep getting 224

225

What is the actual answer then?

And how to calculate for the actually answer

[wraith dagger]

Mycobacterium

[crimson sedge]

Where did the x come from?

[wraith dagger]

Mycobacterium

[crimson sedge]

Thats complicated-

[wraith dagger]

Mycobacterium

Mycobacterium

Mycobacterium

Mycobacterium

[crimson sedge]

So its 100 ×2 ?

Hey u there?

So its 200?

Ok this migth sound crazy

But i just checked the answers

The answer aint 225 nor 200

Its 141.42

Close.

!closw

!close

End.

.close.

[cerulean sail]

.close by itself should do it

[cedar kiln]

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[cedar kiln]

Please don't occupy multiple help channels.

[subtle harbor]

<@&268886789983436800>

.close

[cedar kiln]

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[winged idol]

My teacher gave me hw and i cant solve this one

For what values ​​of the real parameter k is the inequality true for any real x?

[tropic oxide]

знаеш ли в общия план как се решават квадратните уравнения и неравенства?

[winged idol]

Да, обаче сега като има к и х не ми е особено ясно как трябва да стане

[tropic oxide]

знаеш ли за дискриминантата?

[winged idol]

Да

[tropic oxide]

извинявам се за късния отговор, че сега обядвам

[winged idol]

Няма проблем

[tropic oxide]

идеята е ето такава

за да бъде неравенството вярно за всяко x, трябва съответното уравнение да няма корени

[winged idol]

Все пак не разбирам

Обикновенните уравнения ги решавам без проблем

Обаче сега като има и к и х като неизвестни не знам какво да направя

[tropic oxide]

тука x ти е неизвестното, а k пък е параметър

тоест като сложиш за k някакво число, става квадратно неравенство относно x

дискриминантата му трябва да е отрицателна

значи я намираш като функция от k

[cedar kiln]

@winged idol Has your question been resolved?

[cedar kiln]

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[cedar kiln]

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[rugged tusk]

can someone check whether it is correct?
My goal is to calculate marginal PDFs. Im specifically unsure about the integral bounds.

[tropic oxide]

!redir

[cedar kiln]

This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

[mortal hemlock]

🤓

[tropic oxide]

do you have something constructive to add?

or did you just want to say sth pointless

[mortal hemlock]

i just thought it was funny :)

[tropic oxide]

it was not.

[mortal hemlock]

thats subjective

[tropic oxide]

off to #discussion we go

[mortal hemlock]

anyway, the marginal pdfs look fine

assuming T = [0,1]

[rugged tusk]

i found out that it is incorrect

the bounds are incorrect

[mortal hemlock]

.

[rugged tusk]

T is that triangle

[mortal hemlock]

aha

[rugged tusk]

is sum of marginal PDF always equal to 1?

[mortal hemlock]

the integral of the marginal PDFs is always one

because the total probability is 1

across the whole space

but T might not be the whole space

[rugged tusk]

this is my newer solution

it still does not seem right

if i plug in certain values into marginal PDFs it can be greater than 1

[mortal hemlock]

at a certain point it could be more than 1, yes but the total area (integral) should not exceed 1

[rugged tusk]

hmm, dont get it why it can be more than 1

is there any good way how to verify corectness of marginal PDFs?

[cedar kiln]

This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

[tropic oxide]

<@&268886789983436800> doubled down misgenderer

[cedar kiln]

@rugged tusk Has your question been resolved?

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[hoary vessel]

$known that:
\begin{pmatrix}d\e\f\end{pmatrix}\times\begin{pmatrix}g\h\i\end{pmatrix}=
\begin{pmatrix}j\k\l\end{pmatrix},

how do we solve:
\begin{pmatrix}a\b\c\end{pmatrix}
+\lambda\begin{pmatrix}d\e\f\end{pmatrix}
+\mu\begin{pmatrix}g\h\i\end{pmatrix}
=\gamma\begin{pmatrix}j\k\l\end{pmatrix}

[wraith dagger]

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

[hoary vessel]

can someone help?

[cedar kiln]

@hoary vessel Has your question been resolved?

[cedar kiln]

@hoary vessel Has your question been resolved?

[cedar kiln]

@hoary vessel Has your question been resolved?

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[burnt kite]

can anyone help me with this question

[tropic oxide]

!occupied

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[tropic oxide]

(sorry, the other person got here a second sooner)

[turbid isle]

oh

[burnt kite]

oh sorry bro

there are other free rooms

[tropic oxide]

!status

[cedar kiln]

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

[burnt kite]

2

part b

[tropic oxide]

show current progress

[burnt kite]

lemme show

I haven't done anything in part b

I think I've done part a

actuall,y nevermind

hold up

I used to euclidean algorithm to figure out the common denominator between the 2 numbers

which was 4003 and I confirmed it with an online calculator

[cedar kiln]

@burnt kite Has your question been resolved?

[burnt kite]

@tropic oxide hey you still therr

[tropic oxide]

probably not

[burnt kite]

:(

!close

.close

[cedar kiln]

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[near aurora]

Every side of the hexagon is equal

[near aurora]

I alr found fc its 12

I need the area of the hexegon

[feral coyote]

Wait, how AC not equal AE?

[near aurora]

No

Diagram is a bit wrong

Fc cuts the hexagon into tei trapezoids

[slow jewel]

This seems a bit different...

[near aurora]

This was the original one that i drew

Have any new ideas?

[feral coyote]

Are you sure that AC = 13 and AE = 10?

[slow jewel]

Wait yeah AC should be equal to AE

[near aurora]

Yes

[slow jewel]

If it's a regular hexagon

[near aurora]

Its not

[slow jewel]

You said it was

😭

[near aurora]

Thats why we got it wrong?

Every side of the hexegon is equal

[slow jewel]

But then AC should be equal to AE if it is a regular one

[near aurora]

Its drawn badly i guss

That the problem

Thats what the problem says so i guss its not regular then

Ok ill do it later

.close

[cedar kiln]

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[rugged cape]

I need help with a)

btw AB is (5, 2)

the column vector

do i have to use the concept of parallel vectors here?

k(x,y) and (x,y) are parallel

what should I use as k?

is this D?

[feral coyote]

Aren't there 2 solutions?

[feral coyote]

yup

[mortal hemlock]

yes

[rugged cape]

oh so

like

i think of the vectors above having the same coordinates

[sick bear]

hello please help someone.. One letter is missing in the following sequence of the English alphabet. What kind? C D G K R?

[rugged cape]

like

i just thought of a reflection of AB above

moving 5 boxes left from C and then 2 boxes down

thats where D lies

is my way correct?

[cedar kiln]

@rugged cape Has your question been resolved?

[rugged cape]

<@&286206848099549185>

what about this btw?

its right according to the ms

but im not sure about my way of finding it

[rugged cape]

like this

[cedar kiln]

@rugged cape Has your question been resolved?

[cedar kiln]

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[cedar kiln]

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[silver fable]

if x(t) means x in function of t
what does f(x) mean ??

[inland sky]

f(x) is a single-valued function that takes in values of x

x(t) is a single-valued function that takes in values of t

[silver fable]

so this statement is wrong ?
x(t) means x in function of t

[raw gulch]

f(x(t)) = g(t)

superposition of functions

[static herald]

I dont understand the highlighted green part

[silver fable]

!occupied

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[inland sky]

x as a function of t, yes

that's the way it is usually stated

[crimson sedge]

f(x) is a constant

f is the function

[silver fable]

so x is also a function ? in the case of x(t)

[inland sky]

no, it's the input

you input values of x into this function, and it spits values out

[inland sky]

Yes, in that case x(t) is a function of t

it's just a notation thing

x(t) usually means the x-coordinate as a function of time

[silver fable]

isn't x(t) a constant xd

x is the function ?

[inland sky]

no, it could be varied

x(t) = 0.05t

[silver fable]

i don't understand the difference between y=f(x) and x(t)

[inland sky]

Your confusion lies in how these are used. You usually won't see f(x) and x(t) show up in the same place. x(t) is for physics (measuring position as a function of time) and f(x) is for any general function on the x-y plane used in all of math

[crimson sedge]

the difference is Letters

that's all

[inland sky]

we could say g(x), h(x), h(t), doesn't matter

[raw gulch]

$x\left( t \right)=sint\text{ }\text{ and}\text{ }f\left( x \right)=x^{2}\text{ then}\\f\left( x\left( t \right) \right)=\sin^{2}t$

[inland sky]

the outside tells you the name of the function, the inside what you input

[crimson sedge]

[
\m fx \q \m c d \q \m \pi \sigma
]

[crimson sedge]

all of those are functions evaluated at some point if you wanted them to be

[wraith dagger]

Joanna Angel

[raw gulch]

for what to spend so much time on it

[silver fable]

if x(t) = t we can call x a function ??

[crimson sedge]

yes

[raw gulch]

x(t) is a funciton of t

[inland sky]

Yes. At t=5, x=5

[silver fable]

so x(t) and f(t) are the same thing ?

[inland sky]

If they are defined to be

I could say

x(t) = 2t

and

f(t) = 3t

[raw gulch]

both are functions of t, but not necessarily the same, like x(t) = sint, f(t) = exp(t)

[silver fable]

ok thnx guys

.close

[cedar kiln]

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[silver fable]

Does x(t) mean that x can be written as an expression of t ?

[inland sky]

x is an expression in terms of t

for example

x(t) = 5t describes the position of an object as a function of time. At 5 seconds, the position is x=25.

[hollow minnow]

Yeah and it can also be a constant

[silver fable]

Does f(x) mean that f is an expression of x then ?

Like f=5x+1

[inland sky]

Indeed

the convention is f(x)=5x+1

[silver fable]

So x=25 is wrong here

x(5)=25 is the correct way to write it

[hollow minnow]

I mean you can say, for example, when t = 5 seconds, x = 25 (units)

that's fine.

[silver fable]

When x=2 f=11 ?

That looks weird

[inland sky]

It's usually y

in most early algebra contexts, f(x) and y are interchangeable

f(x) = 2x + 1 has a point (2, 5) or x=2 y=5

[silver fable]

Yeah usually f(x) can be written as y=f(x)
How can i write x(t) that way too ?

[inland sky]

For specific points, not defining the function

don't say x = 2t + 1 imo

but you can say at t=2 x=5

[silver fable]

Ok but how can i write x(t) like y=f(x)

Can i choose any letter ?

Like z=x(t)

[raw gulch]

y = x(t) or even x = x(t)

notaiton x = x(t) , shows that x is a function of an argument t

and in same time, x its value

[silver fable]

Understood

Thnx

.close

[cedar kiln]

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[cedar kiln]

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[frail warren]

Hi
How do we evaluate something to the power of a fraction like 9^1/2??

And how do we do sth like log(base5)(input square root of 5)???

[inland sky]

,rotate ccw

[inland sky]

9^(1/2) = sqrt(9) = ?

[frail warren]

Oh so 1/2 is the same as sqrt(9)?

[inland sky]

Yes

Do you know what log is "asking"?

log_5^sqrt(5) is asking what x solves 5^x = sqrt(5)

Now that we know sqrt(5) is the same as 5^(1/2), what's the answer?

[frail warren]

1/2

Oooo nice

So what would something to the power of 3/2 be?

[mortal hemlock]

$a^{\frac{b}{c}} = \sqrt[c]{a^b}$

[wraith dagger]

Flappie

[frail warren]

Ohhhhhh

That makes sense

Nice

Thank you!!!!!

[mortal hemlock]

nice

with that property you should be able to solve them quite quickly

[frail warren]

Yup yup

How do I close this channel?

[mortal hemlock]

use .close

[frail warren]

.close

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[crimson sedge]

bit confused here, a^2=e?

[crimson sedge]

google somehow isn't helping and is showing me much more advanced stuff

[humble pivot]

yes

[crimson sedge]

alright thanks

[humble pivot]

the 2^2 equals 4 thing is just to show that doubling amplitude doesnt double energy, it quadruples it

[crimson sedge]

i see

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[haughty flume]

find the missing sides

[haughty flume]

i dont know where to start

except maybe creating an equation?

[mortal hemlock]

can you tell what the triangles ahve in common?

[haughty flume]

so would i create a ratio equation

for example 4/w = 9/6

ah yes thank you

🙏

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[twilit forum]

The map scale is 0.25 in = 8yd. How far is it from the rollercoaster to the dolphin show? (line on map is 7 in from roller coaster to the dolphin show)

[civic eagle]

that map scale is 1/4 inch. whats 8 * 4 to get yards per inch?

[twilit forum]

32 but wouldnt i divide 7 by 0.25

[civic eagle]

yeah but look, now we know for 1 inch we have 32 yards

how long is the line again? 7 inches, now what?

[twilit forum]

32x7

[civic eagle]

good

[twilit forum]

which is 224

[civic eagle]

so how far is it from the rollercoaster to the dolphin show

[twilit forum]

224 yd

[twilit forum]

thank you so much

[civic eagle]

np, good luck

[twilit forum]

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[formal pasture]

in the last three steps

where did cos pop from

like did they just add a denominator???

help

[grand forge]

They just

Do it for simplifying purposes

[formal pasture]

u can do that

just add a denominator

whenever u want

what

[grand forge]

we know sin(2x)=2sin(x)cos(x)

yes?

[formal pasture]

so i can add

any den

i want

as long

as it proves

other side

?

[grand forge]

Well if u have to prove the other side

than u have to make it the way they want it

[formal pasture]

yea i changed other side

i didnt do what they did

[grand forge]

you shouldnt do that

[formal pasture]

why

it worked out

[grand forge]

you should mainpulate either one side or the other

yea ofc it will

Why dont you just times both sides by 0?

that also works

[formal pasture]

why 0

[grand forge]

coz youll get 0 = 0

[formal pasture]

o

[grand forge]

Thats why u focus on one side

[formal pasture]

i see

[grand forge]

But yes

that is a hard spot

[formal pasture]

yea 😭

ty

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[glass moth]

Can someone explain vacuous truths to me? If someone states that: All the mobile phones in this room are turned off, and there are no mobile phones in the room, how can this statement be true at all, why wouldn't it be false?

[flint plinth]

you cannot exhibit a counterexample of a phone in the room that is turned on

so it's vacuously true

[upper abyss]

F → T is a true statement

Or, even F → F

[glass moth]

Wouldn't it be neither true or false?

[upper abyss]

There's no third option

Not in the logic we're using

[grand forge]

if i said all the elephants on mars are pink, this would be 'true' as there are no elephants on mars so therefore the codition of it being pink or not does not apply to any elephants

on mars

[upper abyss]

You might see this as a difference between our common idea of "if, then" and the logical symbol "→". They are a little different in this case

[glass moth]

Wouldn't it be reasonable to say something as vacuously false?

[upper abyss]

No, simply because → isn't defined that way

[glass moth]

So you could define another school of logic inwhich it replaces vacuously true with vacuously false

[upper abyss]

Yeah you could!

Or, you could define a logic where there is a third option

But the average mathematician is communicating in a logic where vacuous truth happens

[glass moth]

What if you defined two statements:
Statement P: x and y are even
Statement Q: x + y are even
The third statement would be If P is true, then Q is true

what exactly does the third statement mean?

or atleast what is the formal definition of it?

[celest seal]

depends on your exact formalism, but typically we just say that $P \implies Q$ is equivalent to $\neg P \lor Q$

[wraith dagger]

Namington

[celest seal]

so either P is false, or Q is true (or both)

to use your example statements P and Q, we're saying that, whenever x and y are even, we know that x+y is even as well

[glass moth]

Oh ok

[celest seal]

in other words, either it is not the case that x and y are both even, or x + y is even

[glass moth]

Weird way to define things

[celest seal]

well we do this because its convenient

let's say i want to prove "when x and y are even, x+y is even too"

in other words, P → Q

do i care about what happens when P is false?

no; it's a statement about when x and y are even

if P is false, we take the statement to be automatically true

[glass moth]

So you are basically generalizing right?

[celest seal]

no

we're just saying "the entire implication is automatically true when its premise is false"

in other words, "when proving the statement, we can assume the premise is true"

if i want to prove "when x and y are both even, x+y is even", the first step of my proof will be "assume x and y are both even"

this is valid because, if the premise is false, the statement is automatically true

otherwise this first step wouldn't be valid!

we'd have to consider the case where x or y is false!

which would be very silly

an analogy i like to use is

imagine we have a politician running for office

and she says "if I am elected, then I will cut taxes"

this is an implication statement: politician elected → politican cuts taxes

so let's consider the possible statements

True → True: the politician is elected and cuts taxes. clearly, the politician was being truthful

True → False: the politician is elected, but does not cut taxes. clearly, the politician lied

so True → True is True, True → False is false

but what if the politician wasn't elected?

well, then no matter what happens to the taxes, we can't really say the politician lied, can we?

she wasn't elected

in other words, False → True and False → False are both True

regardless of whether taxes get cut or not, the politician told the truth

[glass moth]

We can't say the politician was telling the truth either, because there would be no basis for the truth in that case

[celest seal]

well, in standard logic, we usually assume every statement is either true or false

(this is known as the law of the excluded middle)

so "the politician's statement was neither true nor false" isn't really a logically valid answer

it wasn't false, hence it's true

the point is that we care about being able to express mathematically useful statements

[glass moth]

More so there isn't enough information

[celest seal]

not really relevant

i mean, there is enough information

the premise is true, therefore the implication is true

thats just how it's defined

this may be a bit unintuitive at first but once you do more math you'll understand why this definition makes things more convenient

again, the basic idea is that it allows us to prove P → Q via hypothetical, i.e. by assuming P and proving Q

[glass moth]

Wouldn't it just be easier to assume P to be true from the start?

[celest seal]

what do you mean?

i dont see how thats different from what i said

[glass moth]

Well I mean, we don't include the case where if P is false and where Q is True, and we only include the cases where P is true

[celest seal]

that seems like a far more complicated thing to prove suddenly

now whenever i want to show P → Q, i also need to show P is satisfiable?

what's the advantage of this?

[glass moth]

can you give an example of where vacuous truths are more convenient?

[celest seal]

sure, consider the definition of a function on a set

given sets A, B, a subset S of A × B is a function if, for each a in A, there is exactly one pair (a, b) in S for some b in B

we typically write f(a) = b and call A the "domain", B the "codomain"

allowing vacuous implications to hold allows for functions on an empty set to exist

which is mathematically useful in basically every theorem about functions, e.g. the number of functions from a set A to a set B is |B|^|A|

if not for this we'd have to add special "...unless the set is empty" or "...unless the function is not a valid function" modifiers to every single theorem like this

which just wastes space on the page and adds an extra step to all of our proofs

for no clear gain besides avoiding slightly counterintuitive behaviour in certain edge cases

another example is basically any combinatorial argument, e.g. the combinatorial proof of the binomial theorem

you'd no longer be able to prove the binomial theorem for arbitrary (a+b)^n combinatorailly without jumping through hoops

you'd have to specifically force a, b ≠ 0

or otherwise add special handling for a = 0 or b = 0

this isnt like, a big deal

it doesnt break anything mathematically

but you can see why this is a useful convention to adopt that simplifies some theorem statements and proofs

[glass moth]

I'm confused, I haven't learnt about the binomial theorem yet

[celest seal]

alright, let me give an example of something you may have seen

have you seen the classic proof that √2 is irrational?

[glass moth]

yep

[celest seal]

alright; fundamentally, this proof relies on the fact that, if a number (say x) is rational, then we can write x = m/n for m, n integers (n nonzero)

this is true for ALL x

if x is rational, then x = m/n for some m, n integers

it's an implication

so we're allowed to use this fact (the entire fact) without knowing whether x is rational or not

the proof then proceeds by letting x = √2

if √2 is rational, then √2 = m/n for m, n integers

and then it shows that letting √2 = m/n for m, n integers leads to a contradiction

but we know that the statement "if √2 is rational, then √2 = m/n for m, n integers" is true

so if the conclusion (√2 = m/n for m, n integers) is false, for the entire statement to be true, we know the premise must be false — i.e. it must be a vacuous truth in this case

in other words, it lets us conclude that √2 is irrational

if we did not have vacuous truth here, we wouldn't be allowed to start with our claim "if √2 is rational, then √2 = m/n for m, n integers"

since this claim would not be true

"if x is rational, then x = m/n for some m, n integers" would only be true for certain x, not for √2

and thus we could not use it in our proof

[glass moth]

yeah I see what you mean

[celest seal]

this would force us to use far more contrived methods to prove the irrationality of √2

but the proof makes intuitive sense; it isn't some mathematical trick

well, it is, but not some off-the-wall trick

it's one where each step can be followed

so if our logical language couldn't express the proof's argument, that would be a flaw

by allowing vacuous truths, we patch this flaw

ultimately it is an arbitrary choice

we could say P → Q is false whenever P is false, instead of saying it's true whenever P is false

or we could even say it takes on a third truth value

but i think you'd find that implicational statements would be far less useful if we did this

or at least, far clunkier to use in proofs

[glass moth]

But what you said sounds circular, because in the initial assumption we already assumed that √2 is irrational making the statement "√2 is rational" false

[celest seal]

no, our point is that we KNOW:

[√2 is rational] → [√2 = m/n for m, n integers]

and we can PROVE:

[√2 = m/n for m, n integers] is false (it leads to a contradiction)

so we have:

[√2 is rational] → [√2 = m/n for m, n integers]
NOT [√2 = m/n for m, n integers]

therefore we MUST have

NOT [√2 is rational]

for these to hold

(this strategy is known to logicians and philosophers as "modus tollens")

we never assume that √2 is irrational, we assume that it's rational and show that it leads to a contradiction

[technically you can phrase it noncontradictively but im oversimplifying]

---

to restate:
the point is that we know that

P → Q

is true

but we can prove that Q is false (i.e. P → Q is actually P → FALSE)

we know that TRUE → FALSE is false, so if P → Q is true while Q is false, that means P must be false (so we have FALSE → FALSE, which is true)

[glass moth]

Oh ok

wow

[celest seal]

it's easy to get "lost in the weeds" of notation here

as you practice more, it'll become more familiar

[glass moth]

I get it now

[celest seal]

for now i'll just say "accept that FALSE → [anything] is true for now, and a few months from now, you'll understand why this simplifies things"

but hopefully you can see some of the motivation for why we say this

again you COULD come up with some operation (say » or something) that works like → except without vacuous truth

so TRUE » TRUE is true, TRUE » FALSE is false, and FALSE » [anything] is undetermined (or false, if you want to obey LEM)

but i think you'd find that → is more useful than » in more situations

[glass moth]

could you elaborate more on that last sentence?

[celest seal]

sure

we have

P → FALSE

P is either true or false (since √2 is either rational or irrational), so let's see what value of P makes P → FALSE work

from the definition of →, we know that:

TRUE → FALSE is false
FALSE → FALSE is true (vacuously)

so if P → FALSE is true