help-39

[outer hare]

hmm

ill get back to this after i learn some more proofs lmao

[warm current]

In logic, every statement we make must be either true or false. We're looking at the whole statement "if P then Q". We are no looking at just P, or just Q. That's where a lot of people seem to really get stuck.

Let P and Q be two statements, let R be the statement "if P, then Q". If P is a false statement, then R is true statement regardless of whether Q is true or false. Consequently, this means "if P, then not Q" is also a true statement

I can actually give you an example related to set theory, which I think might clear some stuff up

For any set $A$, $\emptyset\subseteq A$. The statement $x\in \emptyset$ is always false by the definition of the empty set. So, the statement "if $x\in\emptyset$, then $x\in A$" is always true. Hence, $\emptyset\subseteq A$

[jolly parrot]

SWR

[warm current]

Does that make sense, @outer hare?

[outer hare]

@warm current im so sorry I was redoing my proof

[outer hare]

thats really interesting

I mean it holds

but in this case x just does not exist right

[warm current]

This is where we start to get into first-order logic

[outer hare]

is this a proper proof?

[outer hare]

hmm?

[warm current]

$x$ is what is known as a $\textbf{free variable}$ with the existential quantifier "for all": $\forall$. This means $x$ can be $\textbf{anything}$. What we say in first order logic: \
\"For any $x$, if $x\in\emptyset$, then $x\in A$". \
\
You can also write this more compactly in logic notation:
$$\forall x(x\in\emptyset\to x\in A)$$

[outer hare]

this is like the second proof ive ever done btw lmao

[jolly parrot]

SWR

[warm current]

all good

[outer hare]

so for all can imply to something that doest exist

also also for my proof, is the abstract x not implying a for all

like do i read it as for all x in A in mty head

[warm current]

not quite, it gets a little abstract here

[warm current]

it is

[warm current]

There's two ways you can think of it

Maybe three

Here are some equal ways to define subsets, all of these mean exactly the same thing

Plain English: Every element of $A$ is also in $B$\
Using set inclusion: If $x\in A$, then $x\in B$\
Same as above: $x\in A$ implies $x\in B$\
Using the universal quantifier: For all $x\in A$, $x\in B$. In symbols: $\forall(x\in A)(x\in B)$\
Using the universal quanitifier, but in stricter logical language: For all $x$, if $x\in A$, then $x\in B$. In symbols: $\forall x(x\in A)(x\in B)$. This is technically the most rigorous way to write it. All of the above are just this, but in fewer words.\
\
The $\forall x$ by itself can be thought of as "for any $x$ in our universe $U$".

[jolly parrot]

SWR

[outer hare]

I like the for all / for any def

it makes it a lore more clear

as to why the prof works

[warm current]

In pure logic, subset is defined as so:
$$A\subseteq B\Leftrightarrow \forall x(x\in A\to x\in B)$$

[jolly parrot]

SWR

[outer hare]

im proving the second law which is not in the book

so when we assume any abstract X

x

I can read it as for any x in set A

like in the case of my proof

[warm current]

yup

[outer hare]

Wait

I got another question

one sec

For case 2

where I underlined the or

should it be or/and

[warm current]

or

[outer hare]

It could be and too tho right

like or / and

x could be in both

[pearl pond]

@outer hare Has your question been resolved?

[prisma elk]

it could be and

[outer hare]

Ok yeah thats what im thinking

[prisma elk]

but just keep it OR

[outer hare]

does writing or imply that or should i specify

ah

[prisma elk]

an element that is in the intersection of two sets means it is also in the union of the two sets u agree? @outer hare

[outer hare]

yes

[prisma elk]

but the inverse isnt true generally u agree? @outer hare

[outer hare]

An element not in the intersection can be a part of the union?

[prisma elk]

no i meant if an element is in the union, doesnt mean it has to be in the intersection

[outer hare]

Yeah I agree

[prisma elk]

so when u said x not in A or x not in B

and u suggest that it can be replaced by and instead of or

[outer hare]

No, I mean like

[prisma elk]

i said it could but not necessarily true

[outer hare]

or/and

like write both

[prisma elk]

or is enough

[outer hare]

ok, I was not sure if it implied the and condition

[prisma elk]

not generally true, so OR is enough for you 🙂

[outer hare]

ok that makes life a lot easier lmao

tyty

[prisma elk]

@warm current can i open a help channel to ask you about how first order logic and second order logic came up, seems like you know enough about that hahah?

[warm current]

Ill bbl

[pearl pond]

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[prisma elk]

sure ping me pls when ur available

[warm current]

Actually. I more very little about them

[pearl pond]

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[sharp smelt]

Trying to switch the order of integrals here

[finite geode]

wheres this from?

https://www.youtube.com/watch?v=LUvynduoUX0

[sharp smelt]

My book

[finite geode]

found a video on it

first video on yt

[sharp smelt]

thanks

[finite geode]

im not sure if giving yt links is allowed here

[sharp smelt]

just wanted my work verefied

[finite geode]

but eh its good enough ig

[sharp smelt]

It's allowed and common

[finite geode]

good to know

[plush bramble]

Drawing a picture of the domain usually helps

[sharp smelt]

I was thinking it would be $\int_{0}^{2} \int_{0}^{y^3} \frac{dx dy}{y^4+1}$

[jolly parrot]

What a wonderful world it is !

[plush bramble]

Yea the video confirms

[sharp smelt]

Thanks

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[outer hare]

In this case, would the image just be c,d

[versed mica]

do you know the definition of the image of a function

[outer hare]

yes

Essencially its a set of the outputs if im not mistaken

[versed mica]

yep

[outer hare]

so this holds?

[versed mica]

so where are you getting d from

[outer hare]

oh

[versed mica]

did you mean b

[outer hare]

yes

Sorry :D

[versed mica]

all good boss

[outer hare]

but yeah just to make sure im not overcomplicating it lmao

tyty

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[jolly parrot]

938c2cc0dcc05f2b68c4287040cfcf71

[stoic imp]

I mean

[pearl pond]

@stoic imp Has your question been resolved?

[tiny lintel]

Did you find it ?

[stoic imp]

on it

[tiny lintel]

What dimension should ker f be ?

[quick plover]

Probably 3rd dimension

Seeing as that's the one we live in 🤣

[tiny lintel]

https://tenor.com/view/based-gigachad-chad-gif-23901955

[tiny lintel]

@stoic imp

[stoic imp]

dim ker f <= 2

unless they mean proper subspace

[tiny lintel]

Proper subspace

[stoic imp]

if they mean the proper subspace then dim ker f = 1

[tiny lintel]

Cause it’s strictly included

Ok !

Now what is ker f ?

Look at your second condition

[stoic imp]

could be possible they mean $$\subseteq$$

[jolly parrot]

938c2cc0dcc05f2b68c4287040cfcf71

[stoic imp]

ker f = im g

[tiny lintel]

I don’t think so. Your teacher usually is good with notations

[tiny lintel]

Really ? What is dim Im g

[stoic imp]

dim 2

[tiny lintel]

So that can’t be ker f cause ker f dim 1 …

[stoic imp]

my bad

[tiny lintel]

So what’s ker f ?

[stoic imp]

find ker(g) n im(g) that is ker f

[tiny lintel]

Yaaaaaas

I can see it from here

Try (1,1,1,1)

[tiny lintel]

So (1,1,1,1) belongs to both right ?

[stoic imp]

ye

[tiny lintel]

So that’s ker f

Ok

Now Im f is simple

You take (1,1,1,1) (0,0,0,1) and (1,1,0,0)

[stoic imp]

Im f = <(1,1,0,0),(1,1,1,0),(0,0,0,1)>

[tiny lintel]

Niiice

Now that you have ker f and Im f

[stoic imp]

1111 is linear combination of (1,1,1,0),(0,0,0,1)

[tiny lintel]

Yeah but I prefer to have it since it’s the intersection

I think both work (dim 3)

[tiny lintel]

Choose wisely now

[stoic imp]

1. f(1,1,1,1)=0
2. f(1,0,0,0)=(1,1,0,0)
3. f(0,1,0,0)=(1,1,1,0)
4. f(0,0,1,0)=(0,0,0,1)

(1,1,1,1) is in Imf because is generated in <(1,1,0,0),(1,1,1,0),(0,0,0,1)>

(1,1,1,1) is in ker f because f(1,1,1,1)=0

or is there any other way you would have define it

[tiny lintel]

There are many ways.
Yours is correct and efficient and it’s all that matters

[stoic imp]

yeah thank you Shaki as always

I will continue practicing

.solved

[pearl pond]

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[cunning pulsar]

hii can someone double check my work and answers?

[vale meadow]

seems good to me

[cunning pulsar]

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[compact shoal]

hi! basically just looking for opinions on my geometry notes, since this is kind of my first time bothering to take any, primarily want to make sure that i'm getting the points across clear and consice and that there isnt any misinformation that might mess me up later 🙃 and if there's any way to word something better let me know :p

[lusty pier]

Hi! Before doing this assignment, can someone please recommend some videos to watch to do these questions mathematically before i attempt to program it 🙂 ty!

[compact shoal]

😬 sorry

[lusty pier]

OH NO im sorry

[open rivet]

!occupied

[pearl pond]

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).

[compact shoal]

he sent it like a few seconds after i sent mine, no big deal :p

<@&286206848099549185>

[unborn shuttle]

use powerpoint

[compact shoal]

you arent the first person to yell at me about using notepad, mb it was just easily accessible 😕 but that's not really what i was asking, no?

[unborn shuttle]

ok you can use google slides but ok

sorry

[compact shoal]

yeah but im asking opinions on the NOTES that i actually made, not the program i used to write them :v

[unborn shuttle]

ok

[unborn shuttle]

• why you put this on help, maybe can be better just move to dicussion?

[compact shoal]

i didnt want to risk getting lectured about using the wrong channel there either but okay then

.close

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[ancient schooner]

You have a string and some beads of three different shapes, as follow: 
• You have 5 round beads, 3 of which are red and 2 of which are blue. 
• You have 6 diamond shaped beads, each of the colors violet, blue, green, yellow, orange and red. 
• You have 2 triangular beads, one of black color and another one of white color.

<attachment1>

You are arranging the beads in a string as per the design given in the above figure for making a decoration piece. Now based on the beads of different shapes and colors and the design which you are following, in how many ways can you prepare the decoration piece? Show the necessary steps and calculations in favor of your answer.

Premutations, Combinations problem! The thing I first want to clearify is that I shouldn't think of changing the initial bead orders right? Like, I shouldn't consider the case whether the first tirangle bead can be moved after the round beads or not, or the group of round beads or a single round bead can be moved to other position.

How should I approach it?
Case1: First consider the round beads? How many unique ways I can arrange 3 of them among the 5?
Case2: Consider the diamond beads? How many unique ways I can arrange 3 of them among the 6?
Case3: Triangle beads? 2! well only 2 of them are given.

And multiply Case1, Case2, Case3?

[pearl pond]

@ancient schooner Has your question been resolved?

[pearl pond]

@ancient schooner Has your question been resolved?

[brave sluice]

yeah seems like you're on the right track

[ancient schooner]

I get 1680 ways of doing it, does it sound too low?

[brave sluice]

i got the same answer

[pearl pond]

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[split berry]

unsure how to solve this practice question

[tiny lintel]

Where did you start

[split berry]

tbh idek how i would begin with this question

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[rain river]

Suppose I have sequence of natural number wheres $A_{i} = i$. If I erase all number that divisible by 10 and the new sequence is $B_{i}$. What is $B_{200}$ and what is the formula to find $B_{900}$?

[regal herald]

use {}

[jolly parrot]

カザミ

[rain river]

Thanks

I only know there's 20 numbers that must be skipped from 1 to 200, so 200+20 = 220, but 220 is divisible by 10 so 221

[warm current]

btw you only need to use {} for multiple characters

e.g. $A_i$ and $A_{i}$ are equivalent.

[jolly parrot]

SWR

[regal ether]

$$B_{i} = m-\left \lfloor \frac{m}{10} \right \rfloor $$

[jolly parrot]

御華 Mr. Yu

[regal ether]

one is removed every ten numbers, so $$\left \lfloor \frac{m}{10} \right \rfloor $$ of the numbers are removed

[jolly parrot]

御華 Mr. Yu

[regal ether]

The symbol ⌊x⌋ means round down

⌊1⌋=1, ⌊9.8⌋=10

[rain river]

I see

[regal ether]

⌊x⌋ is the largest integer not greater than x.

[rain river]

B200 = 200 - [200/10] = 180?

I thought the number is adding up since B_10 is 11 and B_20 is 22

[regal ether]

guh my fault thats how many numbers in Bi

[rain river]

Nope!

[regal ether]

$$200 = m-\left \lfloor \frac{m}{10} \right \rfloor $$

[jolly parrot]

御華 Mr. Yu

[rain river]

Thanks

[regal ether]

Let x=⌊m/10⌋, that is, x is the result of m/10 rounded down. Therefore, we can rewrite m as:
m=10x+r
where r is the remainder, satisfying 0≤r<10 (because the remainder after m divided by 10 is at most 9)
Substitute into the equation:
(10x+r)−x=200
9x+r=200
Since 0≤r<10, we can rewrite the equation as:
9x≤200<9x+10
Solve the first inequality:
x≤200/9≈22.2
Since x is an integer, we take x=22.
Substitute into the equation 9x+r=200
r=2, m is 222

[pearl pond]

@rain river Has your question been resolved?

[rain river]

Where is r = 2 comes from?

[pearl pond]

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[buoyant relic]

part c confuses me because can i not just use IBP after getting up to $\sqrt{4-x^2}$?

[jolly parrot]

Michele

[buoyant relic]

my course is yr 12 level maths and trig subs are not taught in it

and unless i'm out of my depth is IBP not easier at my level?

nvm

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[unkempt yacht]

hello i'd like my solution checked please

[pearl pond]

@unkempt yacht Has your question been resolved?

[west sapphire]

(a) looks fine

so does (b)

[unkempt yacht]

alright thanks

.close

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[next lark]

INCREDIBLY stupid question but, why is u just 3/2 and 4u is 1/2?

[versed mica]

they just distributed the sqrt(u)

exponent rules

x^a * x^b = x^{a+b}

[next lark]

ohh because they're both u variables

gotcha gotcha thank you

[versed mica]

you’re welcome

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[bitter lodge]

What plane

[pearl pond]

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[earnest portal]

discuss the meaning of multi variable functions and their extrema with examples. find the extrema value of (x,y) =x^2+3xy-5y^2, subjected to 2x+3y=6

[pearl pond]

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[honest spear]

.close

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[next lark]

need some help understanding this

[next lark]

it should be integration, however i dont ujnderstand how it transition from the first line to the second line

[spare lark]

Antiderivative of x^n

Is (n+1)x^{n+1}

Replace n by -2

[next lark]

OH

the negative sign blew over my head I'm actually a dumbas 😭

.close

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[round galleon]

does anyone know if i can make another function have this exact tangent without having to calculate that much

[round galleon]

.close

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[pearl pond]

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[rapid wagon]

Is there a 2x2 matrix that is invertible and diagonalizable on R but not diagonalizable on C?

[rapid wagon]

I guess not since anything diagonalizable on R is also diagonalizable on C ?

Is there a real matrix 2x2 that is invertible and diagonizable on C but not diagonizable on R ?

[regal heath]

C is a bigger set than R that includes R soo

[spare lark]

any diagonalization over R is also a valid diagonalization over C, so no.

[rapid wagon]

Got it.

[rapid wagon]

What about the other way?

There should be one but Idk how to find an example

[regal heath]

there can be complex eigenvalues

so i'd say yes

just make a quadratic with complex roots

then try to make it a 2x2 matrix

considering the quadratic is the det

[rapid wagon]

So if I do this and find that my eigenvalues are complex numbers and it is diagonalizable (aka arithemetic multiplicity equal geometric multiplicity) then its automatically diagonalizable over C but not over R?

[regal heath]

it would be diagonalizable but the result will include imaginary numbers in the matrix

would be my guess

[rapid wagon]

Yesyes

Say my quadratic is x^2+1

Matrix can be (0,1)
(-1,0)

It is invertible too

Nice

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[sharp smelt]

import matplotlib.pyplot as plt
print ("The reccurance relation is a_{n+1}= 640-0.2a_n")
i=640 #intital dosage
r=0.2 #decay rate
while i+r*i>=100:
  i=i+r*i
  print(i) 

why does this code not print any of the outputs

[sharp smelt]

this is the question

It just results in an iverflow

nvm, was able to fix it

now how do I get it to identify at which iteration it terminiated

My best bet is if i<100 then ( something I can’t figure out)

[autumn fossil]

you could just make a variable that'd count the number of iterations of that loop

[sharp smelt]

nvm,,got it with the help of someone

yeah, using ctr

right

[autumn fossil]

yep

[sharp smelt]

how does this look

I think It could use some formatting improvements though

[vestal tapir]

round(n,3)

round the number without making it a string

[sharp smelt]

what

how

[vestal tapir]

print(round(i,3), ctr)

[sharp smelt]

h

got it

[vestal tapir]

i guess that's not better

you still don;t get alignment

[sharp smelt]

how does this look

[pearl pond]

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[narrow bear]

Rewrite the following statement using quantifiers: 'Every integer has a cube root.'

Would this suffice as an answer?

[narrow bear]

.close

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[toxic fractal]

i'd say "exists cuberoot of x in R"

[tropic saddle]

when writing the symbol $\sqrt[3]x$ you are kinda already assuming that it is a thing

[jolly parrot]

Denascite

[tropic saddle]

\exists z in R: z^3=x is better

@narrow bear

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[pearl pond]

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[next lark]

so im pretty sure you'd have to find the area of the curve twice and then add it together right?

[bitter lodge]

You must first show that both functions are odd

Which is honestly annoying

I recommend just doing
$$\int_a^b |y_2 - y_1| \dd{x}$$

[jolly parrot]

King Leo

[next lark]

huh

you lost me 😭'

[quick star]

show that x^3 - 16x is an odd function then you can claim "area of the curve twice..."

[bitter lodge]

Essentially, if your ap reader has had a bad day, you will get docked points for just doubling the integral on [-4, 0]

• either you show that both functions are odd
• or you do two separate integrals

[next lark]

im in calc 2 😭

[bitter lodge]

In college or high school

[next lark]

college

[bitter lodge]

Oh then idk

[next lark]

honestly I was just gonna do two separate integrals

[bitter lodge]

But if college finals are anything like ap exams, youre running a massive risk by merely doubling the integral. You can protect yourself by proving both functions are odd

[cinder bane]

the concept of odd and even function is an algebra concept by the way

[bitter lodge]

I think they said "im in calc 2" bc i talked about ap reader

[next lark]

^

[cinder bane]

fair, cause I also don't know that.

you understand the concept of odd functions though right?

[next lark]

it's been a while but yeah

currently what i got jotted down

apologies about the horrendous scribble writing

[cinder bane]

yeah we say a function f(x) is an odd function if and only if f(-x) = - f(x)
and it has the property of being symmetric about the origin point
so the integration from -a to 0 of f(x) is equal to the negative of the integration of 0 to a of f(x)

[next lark]

so something like x^3 would be odd right?

[cinder bane]

yeah but you need to know why
because f(x) = x^3, f(-x) = (-x)^3 = -x^3, -f(x) = -x^3
because f(-x) = -f(x) then f(x) is an odd function

[next lark]

basically the signs arent the same on the input and output?

or am i simplifying it too much

[cinder bane]

what do you mean

basicaly if you showed that f(-x) = - f(x)
then you showed that f(x) is an odd function

[next lark]

yeah

I get that

[cinder bane]

ok so in your case you will need to prove that f(x) (let it be y1 - y2 from the gragh) is odd by showing the f(-x) = - f(x)

[next lark]

mk

[cinder bane]

which you would indicate that:
$\int_0^a f(x) dx = - \int_{-a}^0 f(x) dx$

[jolly parrot]

Sherif Player

[cinder bane]

so you say that you want just the absolute area without the negative part so you will take the integration of one half then double it to get the shaded area

[next lark]

ohh ok

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[midnight haven]

Okay, so I wrote a set of axioms to define this extremely large finite number and I’m feeling like something is wrong with it, but I’m not entirely sure. Reading the axioms might take a while so before I even do anything I want to make sure that if this takes a minute that won’t be a problem. (It would be five screenshots too)

[midnight haven]

Sorry, give me one second to get the screenshots, I accidentally put my first and last name in one of them and didn't realize, I'll only be like another 2 minutes

okay, so the first one is the actual number and the rest are the axioms

It should all be in order

[pearl pond]

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@midnight haven Has your question been resolved?

[midnight haven]

Are you still there?

<@&286206848099549185>

[midnight haven]

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[pastel haven]

4a, i have other methods for this but they dont work

[pastel haven]

The one that says x = 8

my methods dont work for this

pls help

[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.

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[pearl pond]

@pastel haven Has your question been resolved?

[pastel haven]

ps: ping me if u want to help

[pearl pond]

@pastel haven Has your question been resolved?

[pastel haven]

<@&286206848099549185>

[plush bramble]

, rotate

[jolly parrot]

Couldn't find an attached image in the last 10 messages.

Feb 1, 2025, 10:49 PM

[pastel haven]

bruh

ima just close

.close

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[rose sentinel]

How can I do 42 a and b

[pearl pond]

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[toxic trout]

Guys im not sure how to find which one would have the largest horizntal distance? we did lab 3 but we used the same heights but different initial speed so im not sure how that helps me find this question

[agile valley]

do you know the 2 dimensional projectile motion formula?

[toxic trout]

x-x0 = [Vo^2 * Sin(2theta)]/g

[agile valley]

• y0 to account for starting height

for simplicity, lets assume that x0 = 0, so the projectile begins movement at x = 0

[toxic trout]

like on the right side

[agile valley]

yes, your equation only applies to projectiles launched at an initial height of 0

[toxic trout]

ok wow. go on

[agile valley]

this question does depend on the value of y initial, so do you have a value for that?

[toxic trout]

well i thoguht the point was that it varies but in my lab data we did expirements where it was always 0.99 that is the height it was shot from

[agile valley]

y initial is constant between a and b, but the distance each projectile travels depends on yinitial as well as its angle of release

when the projectile hits then ground in either a) or b), what will its y value be?

[toxic trout]

0

[agile valley]

ah wait

[agile valley]

this is the formula for one dimensional motion

[toxic trout]

oh no

[agile valley]

so a particle being dropped would be a good example of that

but here, a particle is being launched, meaning it has an x and y velocity and position at any given point in time. Do you know of an equation that can equate the relationship between x and y?

[toxic trout]

hmm i thoguht that was what the sin theta did

are ytou sure you know about physics or are u just winging it right now

[agile valley]

the general formula for 2d motion is y = xtan(theta) - gx^2/(v0^2 cos^2(theta)) + y0

[toxic trout]

i dont htink youre right

cuz look they give this

[agile valley]

huh

[toxic trout]

for a previous question

i think since its just the y componetn is just gravirty or something that the fomrula siplfiies so whatever u just gave is unnecessary

[agile valley]

this formula is only accurate if the particle were to have the same starting and final y position

you see how the particle is launched from, and hits the ground?

[agile valley]

in the case that the initial and final height is equal, yes the equation above reduces to this. However, because the projectile is launched from a different height than its final height, 0, we cannot make any simplifications

[toxic trout]

ok I see the light

[agile valley]

do you see how we can solve for x, horizontal distance, given that we know values y, v_0, and theta?

[toxic trout]

ok man you gave me a lot to ponder ok bless you buddy

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[night lake]

what was the process for getting this result? i don’t understand

[agile valley]

when I factorize, I look for shared terms

what's common between all three terms?

[night lake]

pgr

[agile valley]

so we would factor pgr out of that expression

think of it as reverse distributing

[night lake]

i still don’t get it

why does the 3 disappear

[agile valley]

we are factoring out one multiple of p out of p^3

[midnight haven]

sorry to disturb you but can I ask an out of syllabus question?

[rigid mist]

open a new channel sir

[cobalt hinge]

You've already made a new post. Ask your question in there.

[midnight haven]

so we need friends here to talk to about anything?

[cobalt hinge]

!occupied

[pearl pond]

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).

[plush moss]

You already have your own channel

Ask in there bruh

[midnight haven]

when?

[cobalt hinge]

No. Just follow directions.

[midnight haven]

I don't get any directions

[plush moss]

https://discord.com/channels/268882317391429632/908078008609431674

[night lake]

ohh i get it

[plush moss]

Click on this link @midnight haven

[night lake]

theyre cancelling out each other

right

[agile valley]

yes, in a way

[night lake]

wheres the 1 coming from tho

[agile valley]

what happens when you factor out one multiple of pgr from pgr?

[night lake]

you’re left with 0

which is 1

[agile valley]

careful

[night lake]

oh

one multiple of pgr

[agile valley]

if we are left with 0 that would imply that

0 * pgr = pgr, which is not true

instead, we are left with 1, because 1 * pgr = pgr

[night lake]

ohh so we factor it out but theres still one left

[agile valley]

think distributed division. If you are factoring out pgr from pgr + p^2g^2r + p^3g^3r

you are asking yourself for each term

what is this term divided by pgr?

that is the factored term

[night lake]

i get it now

thank you, have a wonderful day

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[iron stream]

Hii

[iron stream]

How do I proceed?

[bitter lodge]

This looks a lot like a taylor series. Do you know what that is

[iron stream]

Yeahh

But ig it just looks like 🤧

[bitter lodge]

What is this supposed to be?
$$\frac n{n^2 + 2.1^2}$$
$$\frac n{n^2 + 2 \cdot 1^2}$$

[jolly parrot]

King Leo

[iron stream]

seems like arctanx derivative

[bitter lodge]

Btw i just want to warn you, i probably cant help you work out the entire thing

[iron stream]

Ok np

[meager frost]

$\lim_{n\to\infty}{\frac1n\sum_{k=1}^n{f\left(\frac{r}{n}\right)}}=\int_0^1{f(x)dx}$

[jolly parrot]

GoldBarley

[iron stream]

Oh riemann sum ?

[iron stream]

Idk !!

[meager frost]

Seems to have a similar pattern, you can try this.

[meager frost]

That's supposed to be f(k/n), sorry 😅

[iron stream]

Yeahhh, the question is unclear too !

Idk why they use such notations in exams !

[meager frost]

If that doesn't work, then I'm out of ideas 😕

[iron stream]

It's working, thank you! I thought it was a decimal point, so I didn't even think about it !

TYSM TYSM !

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[iron stream]

.reopen

[pearl pond]

✅

[iron stream]

?

[meager frost]

I was just saying "Good for you" 😅

.close

[iron stream]

.solved

[pearl pond]

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[tidal stone]

Hi

[tidal stone]

May someone help me with this problem (i redrew this on MS paint)

[sharp tree]

m?

Length?

[tidal stone]

Angle

[hearty sphinx]

Angle between D and F?

What doesn’t make any sense

That

[tidal stone]

Hm

Lemme recheck

[sharp tree]

mDG = 1?

Thats make 0 sense

[limpid lily]

!status

[pearl pond]

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

[tidal stone]

idk where to begin

am trying to ask my teacher what he meant by M

[limpid lily]

OK, is angle y 30 degrees?

[tidal stone]

or mDG = 1

[tidal stone]

Yes

[limpid lily]

OK, so let's look at the biggest triangle, DEF.

What's angle EFD in degrees if FDE is 90 degrees and DEF is 30 degrees?

[tidal stone]

60

I think

[limpid lily]

What's EDG?

[tidal stone]

uhm, 180?

[limpid lily]

No, angle EDG, not the triangle.

What are the angles in triangle EDG?

Two of them are already known and one is unknown.

[tidal stone]

E is 30
D is 60
G is 90?

[limpid lily]

Right.

What's angle FDG?

[tidal stone]

F is 30, D is 60, G is 90

[limpid lily]

No, you already got angle the angle at F ^

You should write down the angles on the diagram as you get them.

Or copy the diagram to paper and write them there.

[tidal stone]

but thats EFD

[limpid lily]

Right, it's also DFG and GFD.

Trace that out with your finger.

Start with E, go to F, then you change direction by 60 degrees, then go to D.

Same thing here. Start with G, go to F, then you change direction by the same 60 degrees, then go to D.

Did you trace it?

[tidal stone]

I'm very confused.. what are we trying to do even?

[limpid lily]

I'm trying to show you something about angles.

Did you trace with your finger?

[tidal stone]

I did

What i'm i supposed to see?

[limpid lily]

Well, an angle is sort of the amount you turn.

Did you turn the same amount in both cases?

[tidal stone]

Yes

[limpid lily]

OK, then the angle will be the same for both.

Since you found that it was 60 degrees when looking at the largest triangle, it's also 60 degrees for the smaller triangle.

[tidal stone]

So its like similar triangles?

[limpid lily]

No, it's the same angle in both cases.

When we look at the large triangle, angle EFD is 60 degrees.

When we look at the leftmost small triangle, angle GFD is also 60 degrees since it's the same angle.

If you have trouble seeing that they're the same angle, start at E, go to F, turn, go to D. Then start at G, go to F, notice that it's the same turn to go towards D.

[tidal stone]

Well.. it's the same turns because we both turned it 60 degrees, how would that prove anything?

[limpid lily]

No. I didn't mention anything about the number of degrees.

I mentioned tracing with your finger.

You'll see that the amount of turn is the same in both cases.

[tidal stone]

Oh well

How do we continue from there?

I'm not even sure how DFG, GFD or such is a singular angle, it doesn't make much sense to me since i dont even know what 'mDG =1' even means

[pearl pond]

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[tidal stone]

.close

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[midnight haven]

Could someone share a playlist to watch alongside the book Introductory Real Analysis?

[midnight haven]

Introductory Real Analysis by A.N.Kolmogorov

[sharp smelt]

you have RA?

[midnight haven]

no lol

extra class

also hi!

[midnight haven]

got any playlist?

[sharp smelt]

Not really

sorry

[midnight haven]

these kind of questions do come under Real analysis right?

[glacial sequoia]

sure

what's an odd positive rational

[midnight haven]

no odd

typo

[glacial sequoia]

lol alr

[midnight haven]

got any playlist ?

[glacial sequoia]

Not particularly

I self studied through this stuff in Abbott, and it's quite doable

[midnight haven]

Abbott?

[sharp smelt]

A book

a really nice book

Understanding analysis

[midnight haven]

dang

that book was available too

but it didnt have "intro" in it so choose this one 😆

[plucky python]

not really

you'll find them in introductory uni classes

[midnight haven]

Which branch of mathematics does this come under then?

[plucky python]

as in like it's more just introductory maths rather than some branch of mathematics

if you insisted on putting it under some branch, potentially set theory?

[pearl pond]

@midnight haven Has your question been resolved?

[midnight haven]

I just want a book and hopefully a playlist which would give me an idea on how to prove those kind of stuff

[pearl pond]

@midnight haven Has your question been resolved?

[pearl pond]

@midnight haven Has your question been resolved?

[midnight haven]

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[whole snow]

Could someone translate that into a more simple form?

[sharp tree]

Can u translate it for me?

[whole snow]

What do you mean?

[sharp tree]

Convert whatever font u use to something more readable

[west sapphire]

The definition is not correct, it should stipulate that $x_n \neq x_0$. It's not even necessary that $f$ is defined at $x_0$.

[jolly parrot]

Bungo

[inland ivy]

well the sequences are in the domain, so if x0 isn't in the domain then x_n != x_0 automatically

[west sapphire]

but if x0 is in the domain, you can't let your x_n take on that value

otherwise a function like f(x) = 1 for x=0, and f(x) = 0 elsewhere, would not have a limit as x->0

[inland ivy]

ah, you are right

[whole snow]

So what does it say in plane english?

[west sapphire]

if you have a sequence of "x" values that approach (but do not equal) x0, then the corresponding "f(x)" values approach c

that's about as simple as it can be stated

[pearl pond]

@whole snow Has your question been resolved?

[whole snow]

So basically if I have a sequence of values that approach a number x0 with a limit, the corresponding f(x) values of that sequences have the same limit c?

[pearl pond]

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[cobalt hinge]

Given the rose curve $r=4\sin(3\theta)$, find the 3 points where $|r|$ is a maximum. Show that the tangent line is perpendicular to the line segment connecting the point to the origin.\
What do they mean in the first line, "$|r|$ is a maximum"?

[jolly parrot]

;(

[west sapphire]

the three values of theta that maximize |r|

[cobalt hinge]

But what is |r| in this context?

[west sapphire]

r = 4sin(3theta), so |r| = 4|sin(3theta)|

[cobalt hinge]

Oh, ok.

I'm still confused.

Would it be a global or local maxima?

Or is it w.r.t. petals?

[rough forge]

w.r.t. petals you wanna find the greatest distance from the origin to a point on a petal

you just need to figure when 4|sin(3theta)| is maximized

also is theta restricted?

[cobalt hinge]

I don't know.

[rough forge]

if it says 3 points where |r| is maximum it suggestes there are only 3 petals

[cobalt hinge]

Alright.

[west sapphire]

both, in this case (there are multiple values of theta that yield the same maximum |r|)

try plotting 4|sin(3theta)| as a function of theta

or just reason about what values of sin(3theta) will maximize this

[pearl pond]

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[pearl pond]

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[dreamy lintel]

Is this an empty channel

[dreamy lintel]

I can ask word problems right?

[plush bramble]

[dreamy lintel]

A car company charges 70 dollars as a base price which includes the first 100 miles you drive but any miles over 100 costs 40¢ and you have payed 212$ solve for M

M is all the miles driven

What would an equation for this look like

[cobalt hinge]

Ok. What do you think we should do to model this relationship?

Hint: ||Linear equation||

[dreamy lintel]

I’ll write an example of the type of equation

27z+18=10

What type of equations are those called

[upper drum]

linear eqn in single variable

[dreamy lintel]

Thanks

[misty adder]

Hey, correct me if I'm wrong, but wouldn't this involve a piecewise with linear equations?

[dreamy lintel]

I’m not sure would this work as an algebraic equation?

[misty adder]

Wouldn't the graph look something like this:

[misty adder]

Yes

[dreamy lintel]

So I was confused. I’m supposed to this in an algebraic equation

[misty adder]

So for the first 100 miles, the price stays at $70 constantly, producing a straight line. After that, it becomes a linear equation. It increases by 40 cents for every mile OVER 100, so try to think of a piecewise equation for this

first branch is when x <=100, and second branch is when x > 100

x being the miles driven

Answer:

[dreamy lintel]

But wouldn’t the 0.4 be distributed to 100

Making it 25

Not 25 mb

40

[misty adder]

Sure you could simplify it

then it would be 30+0.4x

And since we know that the total cost was 212, we solve using the 2nd branch because anything over $70 means it's over 100 miles

So the final equation you're solving is 212 = 30+0.4M

[dreamy lintel]

I messed up with explaining this so I’m just going explain a different way

[misty adder]

ok

[dreamy lintel]

Thanks for being so patient

[misty adder]

no problem

[dreamy lintel]

The original equation I made for this was 70+0.4M=212 but that’s not right because that just gets us 355 but the answer is 455 and I don’t know were the 100 would go

[misty adder]

whats wrong with that equation is that it starts adding money from 0 miles

youd have to do

70+0.4(100-M)=212 so that it starts adding money after 100 miles

[dreamy lintel]

Ok

[misty adder]

if you solve that youll get 455 i believe

[dreamy lintel]

But then wouldn’t you have to distribute the 0.4 first?

[misty adder]

Yeah thats part of the solving process

[dreamy lintel]

Ok I’ll solve then get back to you

[misty adder]

oh sorry i messed something up

I meant M-100

.

70+0.4(M-100)=212

I have to go but here's the solving process:

Dm me if you have any further questions and ill try to get back to you asap

[dreamy lintel]

Thanks

[pearl pond]

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[mild blaze]

dont u just do 2x-3?

[midnight haven]

Yes normally but from first principles you have to use that formula and show the working

[rough forge]

f(x+h) you replace each x with (x+h)

use brackets

[rough forge]

hmm I don't know of any currently

your best bet is to expand, simplify and hope that you can factor h

in the numerator

[pearl pond]

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[pearl pond]

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[cobalt hinge]

Solve, $\tan(3x)=3\tan(x)$.

[jolly parrot]

;(

[rough forge]

there is the obvious solution

could apply tan addition theorems to get there tho

[cobalt hinge]

Which is?

$\tan(2x+x)=\frac{\tan(2x)+\tan(x)}{1-\tan(2x)\tan(x)}$, something like this?

[jolly parrot]

;(

[rough forge]

yes

[rough forge]

well i think you might come up with it yourself

at least if you were to draw them that's where it should click which i am implying

[pearl pond]

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[silent bramble]

im back lol

[silent bramble]

late start to the day but its ok!😭

im gonna stay here just in case I have questions

[midnight haven]

lol

what are you studying?

[pearl pond]

@silent bramble Has your question been resolved?

[silent bramble]

the course is cryptography

right now we are doing lots of mod stuff

[prisma elk]

go ahead

[pearl pond]

@silent bramble Has your question been resolved?

[plush bramble]

Please don't open a channel till you have a question

.close

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[midnight haven]

Hello

[soft vapor]

hey

[soft vapor]

yeah?

[midnight haven]

Im not able to completely understand what R is here

[reef harness]

R - A is the complement of A

[midnight haven]

Can I say it's the universal set ?

[reef harness]

So R must be the universal set

[midnight haven]

Nice

Thankyou

.close

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[chrome plank]

This doesn't look right. r_hat is the normalised direction vector for the r coordinate.

This makes sense. It's telling us how to calculate the radius from an origin from 3d cartesian coordinates (x_hat, y_hat, and z_hat)

simply put, you can apply the chain rule.

[pearl pond]

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[midnight haven]

To Prove:-
$f^{-1}(A \cup B)= f^{-1}(A) \cup f^{-1}(B)$

[midnight haven]

Is this okay?

[vital estuary]

reread the problem statement

@midnight haven

[midnight haven]

hi

shit

To Prove:-
$f^{-1}(A \cup B)= f^{-1}(A) \cup f^{-1}(B)$

[midnight haven]

sorry

[vital estuary]

oh ok

[midnight haven]

there is a solution provided by the book,but i want to know if writing it this way is also okay

[vital estuary]

what does the book say?

[midnight haven]

Theorem 1

they used "but then"

which made absolutely no sense

[vital estuary]

but then could be replaced with so

[midnight haven]

Yeah,that makes it sound way better

[midnight haven]

so is this okay?

[vital estuary]

you proved one direction of the equality

[midnight haven]

we are supposed to proved the converse also ?

[vital estuary]

yes

[midnight haven]

i thought they proved the converse for no reason lol

thanks

.close

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[worthy quiver]

i need help

[worthy quiver]

does anybody have any ap physics 1 momentum frq problems. specifically around 4 to 5 parts

and maybe a short answer queestion in the format of:

Student A reasoning on a ballistic pendulum situation
Student B reasoning on a ballistic pendulum situation
What part of each students reasoning is correct and incorrect?

[pearl pond]

@worthy quiver Has your question been resolved?

[pearl pond]

@worthy quiver Has your question been resolved?

[devout pagoda]

#help-0 help me guys

[pearl pond]

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[restive bluff]

.close

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[boreal igloo]

How do u find a regression for a square root function of the form f(x)=a sqrt of x on desmos?

[bitter lodge]

I havent read this but it might help
https://help.desmos.com/hc/en-us/articles/4406972958733-Regressions

[boreal igloo]

Oh ok thanks

.CLOSE

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[warm elbow]

I feel like this proof isnt strong at all

specfically constructing a new rational number

What would be a better idea of constructing another number in Q\Z in (a,b) given that m/n in Z

[pearl pond]

@warm elbow Has your question been resolved?

[pearl pond]

@warm elbow Has your question been resolved?

[pearl pond]

@warm elbow Has your question been resolved?

[midnight haven]

You can consider the succesion {m/n + 1/k} that converges to m/n and so there exists k such that m/n + 1/k is less than b

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[random geyser]

is this good or

[random geyser]

I get the basic concept but idk if that's what truth functional actually is

because correct reasoning works even if the variables are changed but it's assuming the premises are true

[pearl pond]

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[pearl pond]

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[rapid wagon]

How did they get to the RHS?

[rapid wagon]

nvm

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[amber jewel]

Hello. Can someone help me?

[sand oak]

What is ur question

[fierce totem]

Easiest way is to parameterize point B then minimize distance AB using differentiation

B(x,y) = B(x,√(3x+1))

[amber jewel]

I need to find shortest lenght between point A and point B.

[quasi sundial]

Do you understand the geometric implications when you're dealing with the shortest distance between a curve and a point?

The line AB would be perpendicular to the tangent at B

[fierce totem]

AB² = (x - 3)² + [√(3x+1)-1]²

[amber jewel]

yes. But how I calculate the tangent?

[fierce totem]

I dont think this function is hard to find absolute minimum

[amber jewel]

then I derivate it

[fierce totem]

Ye

[sand oak]

Are we given the coordinates of B?

[amber jewel]

no

[fierce totem]

Look at the image

B is on the given curve

[amber jewel]

this is wrong. On the image g is shortest at about 1.92

[fierce totem]

Yep

[fluid axle]

yeah you forgot a + mate

sqrt((x-3)^2 + (f-1)^2)

[amber jewel]

Yeah you're right. Thank you very much

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[solemn violet]

I just found out about the property that Ax=lambda(x) where if lambda=0 then that eigenvector is the kernal. By extension does that mean that all non zero vectors in the kernal have to be a eigenvector with a eigenvalue of zero?

[tropic saddle]

yes

for all nonzero vectors x in the kernel, Ax=0 = 0x

[solemn violet]

Ah ok. Thx for the quick response

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[lucid hazel]

i am confused on how to relate the raduis

radis

so gear K has a diameter of r/2 , and the 3 gears have a raduis of 2r

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[lucid hazel]

can someone help with this, i know the torques must be equal so the lever is balanced.

So F1D1 = F2D2

but i am still confused

[regal herald]

call the length of each partition as l, and the weight of P as W

in the first case for example we have 2lW=4lF1

try do that for the rest of them then see how they can all be related

[pearl pond]

@lucid hazel Has your question been resolved?

[lucid hazel]

what about II and III

why in II the directiion of the force changed

[regal herald]

to keep it balanced?

if it was going up in II for example the rod would just turn clockwise

[lucid hazel]

i am kinda confused

isnt P pushing down

[regal herald]

yeah

[lucid hazel]

the weight force

[regal herald]

its on the other side of the pivot

[lucid hazel]

so F2 should push upwar

[regal herald]

no

[regal herald]

^

[lucid hazel]

so if they are on the same side, they puch upward but if the different side opposite direction

[regal herald]

balance a pencil on your finger, if you press down on one side, you have to press down on the other side to keep it flat

or you can press upwards on the same side

the direction of the forces just depends on which side of the pivot it is

[lucid hazel]

so the 2nd one is 2f = 2W

and the 3rd one 3f = 3w?

[regal herald]

no

[lucid hazel]

why

[regal herald]

try to follow how i did it

in II
how far from the pivot is F2

[lucid hazel]

2

[regal herald]

it isnt

[lucid hazel]

you mean fulcrum by pivot`?

can you show me please

[regal herald]

yeah

im referring to the triangle

theres not too much to show, its just counting

its one partition away

like in I, i said it was 4l away

in II its just l away

[lucid hazel]

so f2 and f3 are 1

or the distance is 1