help-19

[quasi falcon]

Not really, it means that the x is multiplied by some factor.

Resuming:
$\ A\sin Bx$ has a range of $[-A,A]$ and period $\frac{2\pi}{B}$

[clever fjord]

∫ᴄ 𝐅·𝑑𝑟 = ∬ʀ ∇⨯𝐅 𝑑𝐴

[quasi falcon]

why multiplying the inside makes the period smaller?

If you think about it, 2x grows twice as fast as x, which means that functions of the form f(2x) have their input increase twice as fast as the regular f(x). Visually speaking, this "squishes" functions horizontally.

Since the period of sin x is 2pi, double the rate means that you require half the input to reach the same value

aka, pi

[shrewd trellis]

so y=sin2x is growing twice as fast as y=sinx?

[shy shoal]

It's changing twice as fast

[quasi falcon]

"growing"

[shy shoal]

Not growing

[shy shoal]

Yeah lol

[quasi falcon]

this is f(x) = x^2

You can see that visually speaking, the blue version is like a squished version of it

And, yes, its growing faster

Specifically, 4 times faster, because 2^2 = 4

If you imagine that you "go through" functions, by doubling the input, you "go through" at twice the rate.

It so happens that sinx doesnt really grow

[odd edge]

@shrewd trellis Has your question been resolved?

[odd edge]

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[odd edge]

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[grim cliff]

Can you help me please ?

[fair forum]

hi, I think you deleted the first message

[jade turtle]

For which?

[fair forum]

you'll need to reopen another channel, as this one is closed

[grim cliff]

The images

12

Number 12

[unkempt lichen]

!msgdel

[odd edge]

The original post of this help channel has been deleted, and it will abruptly close and possibly lock. (This is irreversible.) Please claim a new channel, and don't delete the first message of any future channel you claim.

[grim cliff]

It’s solving polynomials

I’ve done everything I know and it’s still wrong

[unkempt lichen]

yes, but the channel has been closed

please reshare your problem in another available channel!

[grim cliff]

Ok

[jade turtle]

I think I see where you went wrong

It should be √26 not 26

[odd edge]

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[main grotto]

I dont know how to do this at all (the black print is teacher)

[odd edge]

Please don't occupy multiple help channels.

[fathom vortex]

i can help you

basically you need to find the line and then if you know the line you can give the answer and then you can label it

[main grotto]

what line (btw this is last years test so i dont know how to do any of this)

this tips is so confusing

[fathom vortex]

well that is the tricky part you must find the line and then from that find the circumference of the circle that is inside of the line

i am joking i cannot do this math

but i know someone who can

@fiery granite will hel pyou because he is smart and he my brother

[main grotto]

his pfp 😭

[fathom vortex]

dotn worry he is taking multi variable calculus

and calc 2

and ap clac

and ap statistics

and ap physic

[main grotto]

are u a golem player?

[fathom vortex]

no i like skele barrel

[main grotto]

bru thats stacked

[main grotto]

logbait?

[fathom vortex]

ye or beatdown decks

[main grotto]

How many trophies

[fathom vortex]

8.5k

[main grotto]

Nice

Your getting there

[fathom vortex]

i will get to legendary arena believe trust

everyone else using no skill decks like lumber loon

[main grotto]

Its really easy yea I believe in you

[fathom vortex]

but i will use the random decks like rune giant skele barrel

to reach it

[main grotto]

Trust lumber loon skill 🤓

[main grotto]

dawg rune giant in the big 25?

[fathom vortex]

no it was a joke i never use it i said random because rune giant is random

i just use random cards liike yesterday i used wallbreakers lumberjack and i somehow went on a winstreak

it is a goiod combo actualyl

[main grotto]

Is all these cards maxed out or smth

cause u must be really underlevelled

if u using random decks

[fathom vortex]

correct

like very underleved..

[main grotto]

Gl against level 15 evo mk

[fathom vortex]

he is in my bio for a reason

and so is inferno dragon

i hope that card rots in hell

what a NO skill card

[main grotto]

Inferno dragons is not bad bro

[fathom vortex]

it is so no skill its not even funny

if i dont have a spell in cycle it will literally just one shot everything

[main grotto]

bru thats like saying all cards are no skill u place them down to do something 😭

[fathom vortex]

IT LITERALLY can do like 1k damage a second

[main grotto]

dawg how 😭

[main grotto]

it has to chargeee

[fathom vortex]

evo inferno dragon keeps most the damage

it doesnt reset everytime it kills or swithces a target

like normal inferno dragon

i cant kill it

[jade turtle]

Im currently pushing ult champ using boss bandit cycle with evo ghost

lmao

[fathom vortex]

get out of my face you little p2w

15$ for evo ghost

you little pay 2 win

[main grotto]

bro i know interactions its just that evo idrag resets its charge after like 5-8 sec or somth to 0 charge

[main grotto]

bro is just playing bridge spam

[fathom vortex]

i justhate the card the only dragon that is skill is electro dragon and lots of people think its no skill but it loks cool so it is skill

[jade turtle]

call me daddy

[fathom vortex]

no

[jade turtle]

mommy?

[fathom vortex]

no

[jade turtle]

unc?

[fathom vortex]

ye

unc

[main grotto]

bro no card have skill ure just placing them down, its the decks thatm atter

[unkempt lichen]

!redir

[odd edge]

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

[fathom vortex]

bro did you see we wer ehaving a conversation here

why would you ever do that i will fight you now

[main grotto]

Can u help me with my math 💀

[fathom vortex]

i know you use goblin giant pekka

i can feel it

[main grotto]

me?

[fathom vortex]

i know you do @unkempt lichen STOP DENYING IT

we all know you got ult champ with that deck

now stop lying

[main grotto]

crazy diss

[night rose]

when is sinx=0?

and do you know what HA stands for?

[unkempt lichen]

<@&268886789983436800> think this is leading nowhere, and tbh I'm too lazy to continue convincing you

[main grotto]

HA is horizontal Asymptote

idk about the first part

this was last years test

and the tips is confusing as hell

[fathom vortex]

nothing is leading no where it was very rude to warn us for having a very chill conversation that no one else was taking part off. you could have just said something instead of insta warn. and now you are snitching to mods because i have cursed your deck in clash royale. please be A bit more respectful when you warn someone next time.

[night rose]

do you know how to use the unit circle? and when a graph is a HA generally

[fathom vortex]

And now i will lave because i have homework

[main grotto]

bro i didnt learn any of ts

[main grotto]

😭

[night rose]

so you just drew horizontal asymptotes randomly? do you know any trig?

[main grotto]

we just didnt do angles

I dont think we are there yet

idk

Sorry i need to use the washrrooom rq

[night rose]

hm, have you seen the unit circle before in any context?

[unkempt lichen]

so I see that you are to graph 1/[sin(x)] and label asymptotes

think you did the graph fine thus far, so I guess asymptotes are your remaining issue?

[main grotto]

never

[main grotto]

I didnt do the graph

It was someone from last years test who did it

and i have no idea how u do the graph

[unkempt lichen]

first of all, sin(x) oscillates between -1 and 1. i think you agree?

[main grotto]

im just rechecking the graph rn

yep

[unkempt lichen]

cool

so at most points in the sine graph, notice that the value of sin is smaller than 1 (and by that I mean you will have +/-0.xxx)

what happens when you divide by a number smaller than 1?

(because that's what's happening with 1/sin(x))

[main grotto]

it increases?

[unkempt lichen]

mhm

so the smallest 1/(sin(x)) can ever get is when sin(x) = 1

this gives you 1/1 = 1

at all other points, 1/(sin(x)) increases quite quickly (draw a graph of 1/x to see for yourself)

so that gives you a general idea now of what the graph should look like - it will touch sin(x) at -1 and 1, and curve away from it at all other points

[main grotto]

What im confused is why they didnt do x, 1/y for the vertex since thats what we learned to do the reciprocal which is usually 1/something

[main grotto]

I see what your saying yea

[unkempt lichen]

I'm sorry, I don't get this question

wdym x, 1/y for the vertex?

[main grotto]

wait im gonna find it rq

[unkempt lichen]

aight

[main grotto]

u see how the reciprocal

is where its 1/y

but x stays same

[unkempt lichen]

i'm not sure exactly what I'm supposed to be looking at in this graph tbh

yes, I see the center part is a rational function (the reciprocal of a quadratic)

[main grotto]

its a quadratic

and the reciprocal of it

[main grotto]

do u get what im saying

where

it became 1/y

[unkempt lichen]

unfortunately, I don't think I follow. do you have the text accompanying this graph, or a written explanation?

[main grotto]

Well yk for this graph it has a vertex of 1,-16

[unkempt lichen]

ok, yeah

[main grotto]

well for the reciprocal part of it became 1, -1/16

and there are other reciprocals here too but im focusing on the one between the 2 asymptotes

@unkempt lichen idk if ure still here 😭

[unkempt lichen]

a moment

ok I see what you mean

but for 1/sin(x), our vertices are at 1 and -1

so flipping 1/1 gives us... well, 1/1

[main grotto]

OH WAIT

im stupid

im so stupid

[unkempt lichen]

yes, the story is as lame as it sounds

[main grotto]

ok um for key points and asymptotes

[unkempt lichen]

so now then, let's do that

first, notice that sin(x) does cross 0

several times in fact

now, we all know that 1/0 is undefined

so that gives you one of the two kinds of asymptotes

but, notice that sin(x) doesn't "bounce" off the x-axis, meaning that when it crosses the x-axis, it goes from positive to negative, or vice-versa

we are graphing 1/(sin(x)), and the numerator is positive 1, so every time sin(x) changes signs, the whole fraction will too

that's why you see 1/sin(x) appearing first on top, then on the bottom, and then back to the top, etc.

that should give you an idea how to graph 1/sin(x) and what the key points are

[unkempt lichen]

as for HAs, remember this line

[main grotto]

Wait so HA will be 1 and -1 right

[unkempt lichen]

absolutely

[main grotto]

Ayyy thank u bru

Ts so light thx to u

[unkempt lichen]

glad to have helped, esp after... that show

ok wait I just checked my notes

[main grotto]

Well it was a decnet conversation while i was waiting for someone to help 😭

[unkempt lichen]

I stand corrected on the HA

[main grotto]

so y can not equal 1 and -1

[unkempt lichen]

since 1/sin(x) does not stablize on a value as x -> inf, there are actually no HAs

[main grotto]

Wait what

[unkempt lichen]

y can equal 1 and -1

but not any value strictly in between 1 and -1

[main grotto]

So that means there is no HA?

[unkempt lichen]

correct. because of the oscillation here, there is no HA

[main grotto]

👍

also

for this question

i know its either a or d

[warped grove]

(Hanako is correct, please do not have off-topic conversations in help channels. There are other channels for that.)

[unkempt lichen]

narrowing it down to these two options is correct

[main grotto]

um

How will u know which option it is

[unkempt lichen]

so the difference between those two options is in the range

[main grotto]

yep

[unkempt lichen]

now tell me this: what does the range in A mean? what about the range in D?

[main grotto]

y is above 0 for a and y can not equal 0 for d

[unkempt lichen]

focus on D. y not being 0 means that y can be negative

can the fraction be negative?

[main grotto]

Wait it cant cause of the ^2

[unkempt lichen]

mhm

[main grotto]

Thats why its above 0

Ohh

[unkempt lichen]

the range in D is too broad

[main grotto]

Yea 😂

Thanks bro

[foggy lake]

ok

[unkempt lichen]

nps

[main grotto]

again for the help

[foggy lake]

i have a question

or actually we

[unkempt lichen]

!occupied, sorry

[odd edge]

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).

[unkempt lichen]

please get another channel, like #help-49

[unkempt lichen]

glad to have helped

[main grotto]

❤️

.close

[odd edge]

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[odd edge]

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[outer wadi]

<k> is the cyclic group generated by k, but what group is this refering to?

[outer wadi]

Z_n?

wait

k is a divisor n

so <k> refers to orders of k up to n?

when we have something like <3> its always referencing to some other group right?

yeah

3 is an element from another group

so here k is an element from another group

and thats Zn?

is there a different notation for this

let me look it up

<k> seems to be the standard notation

honestly im just confused on the question

I mean I think it makes since but this <k> is tripping me up lol

your basically taking a divisor of n and splitting up Zn into n/|k| pieces and mapping each of those piece to Zk

wait hmm ok lets say I have Z8 and k = 2

$Z_8 = {0,1,2,3,4,5,6,7}$ and $\langle 2 \rangle = {0, 2 ,4, 6}$

[clever fjord]

Branshi

[outer wadi]

so the index of the factor group is 2

which matches to Z_2

ok let me come up with a mapping

the mappig a + <k> = a + k should work I think

because

wait thats just f(a + <k>) = a mod k

and this works because

wait I need to use the first isomorphism theorem

.solved

[odd edge]

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[shrewd trellis]

Im struggling to understand why we need to divide by r?

[unkempt lichen]

!original

[odd edge]

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.

[unkempt lichen]

like, full context

[shrewd trellis]

I was just trying to understand radians

ill come back to it

.clsoe

.close

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[odd edge]

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[fringe shore]

.close

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[obsidian robin]

How i can find the maximum value and minimum?

[vernal yacht]

Can you first find the maximum value of $\sin\theta$?

[clever fjord]

e4 e5 Qh5 Nc6 Bc4 Nf6 Qxf7#

[obsidian robin]

ye

[obsidian robin]

hey i can divide by 2?

[vernal yacht]

wym

[onyx tusk]

derivatives

[obsidian robin]

what dou yo mean ?

[obsidian robin]

idk , because weather divide by 2 i get pi/6

[vernal yacht]

where?

[obsidian robin]

maximum and minimum

[vernal yacht]

okay so, you don't get pi/6 by dividing anything

[vernal yacht]

Like I said, find the maximum of this function

and infer the domain of $\sin(x)$

[clever fjord]

e4 e5 Qh5 Nc6 Bc4 Nf6 Qxf7#

[odd edge]

@obsidian robin Has your question been resolved?

[terse pagoda]

hello everyone

im in new to this community

[unkempt lichen]

hi, welcome! please head over to #discussion or #chill to chat

[odd edge]

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[odd edge]

@covert lotus Has your question been resolved?

[odd edge]

@covert lotus Has your question been resolved?

[odd edge]

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[eager tinsel]

how can I find two consecutive terms for this succession: $$a_n = n^2 - 6n$$ in a way that $$a_{n+1} - a_n > 0$$?

[clever fjord]

Gonçalo Gonçalves

[wooden python]

in English we say sequence not succession

[eager tinsel]

sorry

[wooden python]

anyway, work out $a_{n+1}-a_n$ in terms of $n$ first.

[clever fjord]

Ann

[eager tinsel]

I did, I did $a_{n+1}$, which gave me $n^2 - 6n - 5$

[clever fjord]

Gonçalo Gonçalves

[eager tinsel]

after that I did the subtraction between the two

[wooden python]

that looks strange

[eager tinsel]

but it didn't give me right

[keen elm]

hi

[wooden python]

can you show how you worked out a(n+1) as that

[eager tinsel]

really?

[keen elm]

im new

[eager tinsel]

alright

let me take a photo of the notebook

[wooden python]

hello and welcome to the server. head to #discussion to chat

[keen elm]

thanks

[onyx tusk]

plugin n and n+1 to find n's range at first

[wooden python]

im having them do exactly this

[eager tinsel]

can you see it

[wooden python]

(n+1)^2 isn't equal to n^2+1

[eager tinsel]

ohhhh

yeahhh

binomials

forgot about that...

[wooden python]

you forgor 💀

[eager tinsel]

yeahhh

;_;

alright

let me see if that fixes it

[wooden python]

!nosols

[odd edge]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[onyx tusk]

Ohh sorry sorrry

should i delete

[wooden python]

yes

[onyx tusk]

@wooden python how do i understand if its okay to send answer's snap

[wooden python]

basically never

[onyx tusk]

like, there should be a warning or something

[wooden python]

this !nosols factoid is the warning

[onyx tusk]

some people actually asked me for a pic

[onyx tusk]

got it, thanks

[wooden python]

you are not supposed to give out answers nor ask for them

[onyx tusk]

im new so...

[eager tinsel]

damn it

it gives examples

hmm

I'll send a photo

it gave me

$n > \frac{5/2}$

[clever fjord]

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

[wooden python]

\frac{5}{2}

[onyx tusk]

thats what i got man

[eager tinsel]

yeah I know... dummy mistake

[wooden python]

did you fix your solution anyway

[onyx tusk]

so, u can just plugin n=3,4,..

[eager tinsel]

I don't know, the book gives the "example" of 8 and 9

there are more...

so maybe

do you want to see my reasoning to be sure..

or do we close this?

yeahh

it's probably correct

let's just close it I guess.

sorrry I saw you were gonna type something

no?

alright I'll close

.close

[odd edge]

Channel closed

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[obsidian frost]

can someone help me out

[obsidian frost]

i understand from 1-12 but 13-17 is where im genuinly lost

[slender geyser]

Bro these are easy

[obsidian frost]

for you i guess

[slender geyser]

Wait ill say

[obsidian frost]

can we start with 11

[slender geyser]

K

Wait

[obsidian frost]

take ur time

[slender geyser]

The pink ones are actually false they just said you to solve and prove it

[obsidian frost]

the pink ones are the answers

[slender geyser]

The answer

Nope

I solved one but it doesn't match

[obsidian frost]

wait are u serious

which one

[slender geyser]

The 13

Just solve it and you won't get the answers for some questions

Which is why they said you to express

Well some pink ones are correct

[obsidian frost]

ohh

alo

also

do u know what they mean

by

assume no denominator=0

[slender geyser]

That's what I thought at the start

But how

[obsidian frost]

its fine just ignore it

[slender geyser]

K

[obsidian frost]

so besides 13 which pink ones are wrong

[slender geyser]

Wait I just solved these questions mind

[obsidian frost]

but hows it wrong it all looks correct

[obsidian frost]

hmm take ur time and do it on a piece of paper

[obsidian frost]

(q13)

[slender geyser]

Only that

[obsidian frost]

its the only incorrect one?

[slender geyser]

No

Wait ill solve

[obsidian frost]

alright

[slender geyser]

Wait a minute

I do get the correct answer but wait how do the power gets both multiplied and powered to the numerator and the denominator

Hey bro you there

?

[obsidian frost]

cause

its all raised

to the power of 2

distributive property

[slender geyser]

But how

Can ya explain me

[obsidian frost]

just take that 2 and multiply it to everything in the brackets

cause we cant have power of power

[slender geyser]

Oh

Ok

Now I get the answer

Sorry I was wrong

I just did everything in mind

[obsidian frost]

so is it correct

like the pink answer

[slender geyser]

Yeah

Now I solved everything

K bro should go to sleep bye

[obsidian frost]

wai

wait

can u send me

ur solving

.cloes

.close

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[steady cargo]

I'm currently working with transformations. We are told to graph an asymptote and I am having a little trouble understanding how to graph it. I understand that an asymptote is a line that gets very close to a certain point but never touches it. I also understand that to get the x-axis that it never touches you look at the fraction and pick a number that would be the denominator 0. I am just confused on how to find out the y-axis point that it never touches.

[quasi sparrow]

can you just show the original question

[steady cargo]

I have to graph f(x) = 1/x-2. I understand that x cannot be 2 since that would make the denominator 0, but how can I figure out y?

[quasi sparrow]

graph 1/x first

then shift

[golden juniper]

there is no y-intc

[steady cargo]

so y cannot be 0?

[golden juniper]

wait im slow

😭

[golden juniper]

no no there is a y-intc

my apologizes

y-intc is when x=0 so solve that plug that value into the given equation and find it out

[steady cargo]

okay thanks

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[obtuse aurora]

i know that x = -4 + or - root 17
but how do we get the area?
​

[weary pelican]

What are the formulas for area of a triangle you know

[odd edge]

@obtuse aurora Has your question been resolved?

[onyx tusk]

u need to use determinants

1/2 det(x1,y1,1 (1st row) x2,y2.1 (2nd row) x3,y3,1 (3rd row))

otherwise u can find equations of lines joins these points, and find the height and base

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[quiet swan]

Can you please explain why this equation should be broken apart like that?

[quiet swan]

For example how do I know -6x^3 should not be 10x^3 - 16x^3 or something like that

[lofty spade]

does it successfully solve the equation?

[quiet swan]

Yes

[honest turtle]

So you have a common term x-2

[quiet swan]

Yes but I will only find that out after some time

Is there any way to

Know it before tryin out

[lofty spade]

you just have to be really lucky. I'm guessing this is from some software. In that case, the software probably used some other algorithm, and then worked backwards to figure this out

[lofty spade]

No, not as far as I know

[lofty spade]

especially for a fourth-degree equation. Who gave you this?

[quiet swan]

My teacher

[quiet swan]

Damn if thats really the case

That’s all I wanted to know

Thx

[lofty spade]

then there's probably some other trick

like extra information given to you that can help you solve it

if the teacher actually wanted you to solve it

and wasn't just giving you the problem to keep you quiet

[quiet swan]

there are but its even more nerve wracking than this

[lofty spade]

oh, what is it?

[quiet swan]

Polynomial division

[lofty spade]

oh

can you send the entire problem?

[quiet swan]

This is the entire problem

If I were to solve it using polynomial division, I’m gonna have to try every divider of 8 and its negative counterpart

In this case it’s probably better maybe

[abstract fulcrum]

yeah probably you have to find atleast one root

[quiet swan]

Sorry, but why do I need root?

[abstract fulcrum]

and then you can factorize

https://youtu.be/WRkgMDVIETE?si=MpOgfRIAZMqOshXF

[quiet swan]

Ah thanks

[abstract fulcrum]

you can refer to this

its more convenient and easier

just in case you have to be lucky

using counter tricks with constant part

[lofty spade]

oh, so the piece of extra information is that it has some rational roots. In that case, yes, you can just try every integer divisor of 8

[abstract fulcrum]

yeah

its 2

[abstract fulcrum]

apply for 2 you'll get it factorized

[quiet swan]

thanks got it

[abstract fulcrum]

can you close it

[quiet swan]

how do I close?

[abstract fulcrum]

.close

[quiet swan]

.close

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[blissful vigil]

I got 1.755. But its none of the answers of (0.671, 2, 0.287,-1/2,and 9). I don't know what to do now

[wicked kestrel]

where's the 2.2 bit at the bottom coming from?

[odd edge]

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[woven stag]

Hello. I would like some help with the Derivatives problem, "If f(1) = 6 and f'(1) = -4, find d/dx (f(x)/x) at x = 1.

More specifically, I'm just looking to ask if there's some way to find the function that produced these outputs, from the outputs themselves.

[sharp oak]

You can find the smallest polynomial, actually. That's called Lagrange interpolation

But there's infinitely many functions that can fit this

Actually, I'm just now realizing there's only two points, so Lagrange interpolation will give you f(x) = -4x + 10

A much more boring answer than in other cases

[woven stag]

Hold on.

The question says, find "find d/dx (f(x)/x) ".

[sharp oak]

You can (and should) imagine that the function may not be a line

[woven stag]

So wouldn't it have to be something like, -4x +10 / x ?

Sorry, I'm assuming that there has to be two terms (I think that's the right word) in this thingymabob.

Because it's f(x), {term 1} over x {term 2}.

[sharp oak]

I was answering your second question "What function may produce these outputs?"

I will now restart completely to answer your first question, "what's d/dx (f(x)/x)?"

Note we don't need any form for f(x) to answer this question

[woven stag]

Sorry. But thanks! 🙂

[sharp oak]

You should have a rule for differentiating a quotient. Know it?

[woven stag]

If I may interject, wouldn't it just be f' * g - g' * f / g^2 ?

[sharp oak]

Except g(x) = x here

[woven stag]

Wrong sign.

[woven stag]

Gng are WE DEADASS!??! 😭

[sharp oak]

You are differentiating the quotient between f(x) and x.

Don't let f(x) trip you up. It's a function, much like sin(x) is. It has a derivative, f'(x)

[woven stag]

I thought this whole time that the function which produced 6, and its derivative, producing -4, was supposed to be some sort of "f(x)/x".

[sharp oak]

With all that in mind, can you simplify d/dx (f(x)/x)?

[woven stag]

I assume I can just plug in the x that the question gave us?

That being, 1.

[sharp oak]

You had it exactly right here, you gave me d/dx( f(x)/g(x) )

Now, small change to turn that into
d/dx( f(x)/x )

We just let g(x) = x

[woven stag]

Alright. 👍

[woven stag]

When doing the quotient rule, do I keep the x as "x" or should I replace it with 1? The way we do it in class, we only plug in numbers once we finish deriving something.

[sharp oak]

That's a good strategy! Don't plug in the numbers yet

We can find d/dx(f(x)/x) for all functions f this way

[woven stag]

Alright. I got -4 * x - -4x +10 * 1 / x^2 => 10 / x^2.

[sharp oak]

I expected to see f(x) and f'(x) in your answer

Remember, f(x) is not -4x + 10. Was answering a different question at that point

[woven stag]

Oh, okay.

Well then, it'd be f' * x - x' * f / x^2.

[sharp oak]

Bingo! You made it look easy

That's the derivative of f(x)/x, no matter what f is

We can plug in the numbers now

[woven stag]

Ah, this makes more sense now.

You don't have to find the original function after all.

You can just replace f and f' with the given numbers.

[sharp oak]

Oh wait, x' is just 1 here

[woven stag]

Yup.

Thank you so much for your help.

I wish you a nice, mathematical day.

[sharp oak]

You too! Have a good one

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[molten knot]

hello i would like some help on understanding this process

[molten knot]

so i understand how 75 turns into 3*25

but idk where the denominator came from

[abstract fulcrum]

5^4 = 5^2 . 5^2

5^2=25

[molten knot]

okay wait i understand it now

[molten knot]

i thought u were saying 5 to the power of 2.5 and i was confused lol

thanks

.close

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[proven hill]

Im having trouble setting up this double integral (specifically the bounds). Any help would be awesome! I don't really need help evaluating.

[olive needle]

Don't concern yourself with the value of z when determining the region when setting up the bounds. What does the projection of the space look like on the xy-plane?

[proven hill]

the bottom bound for y is -10 but it seems to be type 1 since the lines are being bounded vertically (?) (f(x,y)dydx)

[olive needle]

What is the upper bound of y?

[proven hill]

would the top bound of y be the y=4-x^2 ?

like the parabola that sits on the xy plane that its making

[olive needle]

Yes.

[proven hill]

would the bottom and upper bound of x be -2 and 2

[olive needle]

z = 4 - x^2 - y

If z = 0 and y = -10, what is x?

[proven hill]

plus or minis sqrt(14)?

[olive needle]

Yes.

[proven hill]

Thank you for helping with the bounds! I should be good to evaluate now.

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[thin dagger]

Wdym by congruency in probability context

<@&286206848099549185>

[merry finch]

It’s more in terms of definition

Like that’s just how the symbols are defined

[thin dagger]

Oh

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[thin dagger]

.close

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[onyx spoke]

im stuck here, i can't find another way of isolating x, i also tried taking log base 2 but i haven;t gotten anywhere. I'm not looking for an answer, a hint about what to do will be appreciated

[pastel orbit]

try applying 2^x to both sides

[modern sundial]

They have mixed bases

[onyx spoke]

which step

[pastel orbit]

eh? not at the end though

[modern sundial]

Oh, I didn't read that far because I would have done that at the second step. Sorry

[pastel orbit]

I'll admit, I didn't really read all the steps and only read the part that OP was stuck at

[onyx spoke]

like this?

[modern sundial]

When they say apply 2^x, they mean write 2^x on both sides, but replace the "x" with the entire left (right) side

So for example, if you have 5x = sin(x), you would write 2^(5x) = 2^(sin(x))

[onyx spoke]

ah i see

okay

so now i have this

[modern sundial]

I really shouldn't be speaking for someone else because this wasn't my vision, but they meant the last line of the second image, not the first

[onyx spoke]

ohh

😭

okay lemme try that

[modern sundial]

Right, and now you just use properties of exponents and the fact that $2^{\log_2(x)} = x$ to simplify (x>0).

[clever fjord]

JessicaK

[onyx spoke]

im not sure waht to do for the bottom fraction

[modern sundial]

Well, you would use 2^(ab) = (2^a)^b, but this is not turning out as nice as I would have expected

[onyx spoke]

yeah i get this

i have to go eat, let me try this after

thanks for your help tho

i might be able to get it

[pastel orbit]

@onyx spoke I hate to say it, but I don't think the algebra works out nicely and I'm afraid I may have lead you down the wrong path

[onyx spoke]

Its okay, i can try something else

This isn’t that important anyways, we’re learning logs and found this on google and decided to try it

[pastel orbit]

this log equation might not be solvable via your usual algebraic techniques

the solution is clearly x = 8, but I cannot seem to get it with exponent laws

[onyx spoke]

That’s interesting

I spent like 30min on this 😭

It’s okay

Thanks for your help guys

[pastel orbit]

I apologize for the misguidance

[onyx spoke]

No worries

[pastel orbit]

if there is a way to solve it algebraically, I fail to see it

[onyx spoke]

Though im interested how u would solve this then

[pastel orbit]

but the idea to raise everything to (base)^x when you have log_base in your equation as the sole logarithm is usually helpful

[onyx spoke]

Alright

I checked photomath, it only shows graph

So yeah u were right

What was the command to close

.close

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[modern mortar]

Struggling with understanding #30

[modern mortar]

First time using this server so if I’m doing it wrong lmk

[weak token]

It’s asking you sin of what degree is equal to -sqrt2/2 and it might help to look at a unit circle

[modern mortar]

I mean I understand what’s it’s asking for I just don’t really know how to start it

Ykwim

[brazen hare]

@modern mortar were you taught the basic angle + ASTC method?

[modern mortar]

Yes

[brazen hare]

so what's the basic angle?

[modern mortar]

I don’t understand 😥

[brazen hare]

what is $\sin^{-1}\left(\frac{\sqrt 2}{2}\right)$

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[unborn obsidian]

is it sin 45?

[nocturne belfry]

is what?

[toxic bear]

root 2 will cut the 2 and then make 1/root 2 and then the value of theta will come is sin 45

[toxic bear]

yes

[fair forum]

quick note: the value of theta is just 45 degrees, not sin 45

but also, do you have another question?

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@unborn obsidian Has your question been resolved?

[lone solar]

-sqrt(2)/2 = -1/sqtr(2)
Which is just sin 135
cos60 = 1/2

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[frank tide]

two forces, r and f, acting on lengths AC and AB, if f is perfectly resembled on AC (It's magnitude is equal to AC's length), what is the magnitude of torque around B

answer choices;
1- magn(f) * magn(r)
2- 2magn(r) * magn(f)
3- magn(r)/magn(f)
4- 2 * the area of the triangle

[frank tide]

Im not even sure if this is justified or not but... I said "it's gonna equal to the torque of r around b + torque of f around b" and since b is on the line of work of r then we neglect the torque of r

now we want torque of f

so I drew a perpendicular line on F to say F * BC

now im kinda stuck because they didnt even mention that r is perfectly resembled on BA

ok yeah i figured it out

for anyone that was wondering,

split the f into sin(x)f and cos(x)f where x is the angle BAC

and- it will be r * f * sin(theta) which is 2 * area IF r perfectly resembles BA

eh ig so bc i dont thyink this owuldve been solved otherwise

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[vernal yacht]

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[full lark]

How is the answer 2.5 km?
A hiker travels a distance of 5 km from point 𝑃 to point 𝑄 on a bearing of 030∘. She then travels from point 𝑄 to point 𝑅 on a bearing of 330∘ for 10 km. The distance west of 𝑅 from 𝑃, in kilometres, is given by:

[full lark]

I am not sure how this visually looks

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[small cloak]

it's multiple choice I know but I read it as >=

so I thought it was |w| >= 2000, so I imagined at least 2001 states so therefore 2000 states to accept, 1 state for the initial state to move on from (2001)

[woeful briar]

Try

You have nothing to lose

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[small cloak]

.close

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[cyan chasm]

Hello , I have been struggling to understand differentiability and critical points

Could a point be critical if the function was differentiable at it

MUST a point be critical if the function wasn't differentiable at it ( In all functions excluding piecewise because I sorta understand it)

[cyan chasm]

Oh and one more thing ,
I haven't faced any questions related to differentiablity "specifically" (except in piecewise functions) , so I really want to ensure that I am also capable of knowing whether a point is differentiable or not too

[mystic saffron]

to ensure a point is differentiable u may use LHD,RHD IG

[cyan chasm]

Yeah I think I used to do that in piecewise because LHD and RHD were clear as they were written , but idk I just froze when I saw a function other than piecewise lol

I think they are the same function on both sides in any function excluding piecewise

[merry finch]

What is a critical point

[cyan chasm]

A point where its derivative is zero

Or 1/0 "undefined"
I think

[merry finch]

I’ve seen when it’s 0 but I’ve not heard a point without a defined derivative be called critical

[merry finch]

Is that what you’ve been told

[cyan chasm]

Yes

[merry finch]

Well then clearly the second one is true

If f is not differntiable at a point, by definition the derivative doesn’t exist

[cyan chasm]

I was told , all the points making the derivative of the function zero or undefined is considered a critical

[merry finch]

And if that means it’s a critical point then it is easy to see that non-differentiability implies critical

[cyan chasm]

I guess that's true , however I think I have heard one of my friends talk about this specifically to a teacher , he said that it is not necessary for a point which is not be differentiable to BE critical

Which was really confusing so that's why I asked here haha

[merry finch]

So your first point

[cyan chasm]

I am guessing that one is right , just a hunch tho

[merry finch]

If you have a derivative of 0, does this mean you are not differentiable

[cyan chasm]

Well my friends told me "it depends" and showed me an example of the piecewise function

[merry finch]

No no

Think about this again

If f’(a) = 0, is f differentiable at a?

[cyan chasm]

Sorry battery ran out , umm I think yeah it is

[merry finch]

Why

[cyan chasm]

Well if we are talking about all functions excluding the piecewise I guess the LHD is same as RHD

Piecewise is unknown

At x =a ofc

[cyan chasm]

If he states in the question the function is not differntiable at this point specifically then that's another case
I would guess that this point doesn't exist in the domain or sth

Maybe a vertical tangent passes by it
Or it's a sharp/corner point

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[spring pond]

Can I get some help with Algebra? In my online school they're solving this problem by adding the power to the parentheses first, but isn't that wrong because of the order of operations requiring parentheses to be done first? If the values are first summed inside of the parentheses and then the outside exponent gets placed into all of the values inside then wouldn't the answer be different?

[strange aspen]

You will get the same result either way.
You could start from the inside and then taking the result expression to the power of 4 and you will still have the same expression, even if you started with the power of 4 then simplifying

[spring pond]

Thanks a ton, that's really interesting! ❤️

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[warped glacier]

if you think about what's happening to just the exponents

track the exponent of x for example

in your online school, they did exponent = 2 * 4 - 3 * 4

but you can just do (2 - 3) * 4

and that's the distributive law!

2 * 4 - 3 * 4 = (2 - 3) * 4

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[sour jolt]

Hey, I wanna start learning geometry and trigonometry all over again but I'm lost, Could I get some help

[warped glacier]

you should ask this in #study-discussion

[sour jolt]

Ohh ok thanks

[warped glacier]

nw!

[sour jolt]

.close

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[merry birch]

somebody explain this to me without using the formula

[merry birch]

I know that you have to multiply 5 by 4/3 but idk how to get 4/3

i got 5 from the rise from Ax to Bx

[quasi sparrow]

what formula are you against using

[merry birch]

the dividing a line segment

im not against it our teacher just wants to make us use a different way

to understand it better

[quasi sparrow]

show the formula

[merry birch]

idek if u use it in this question

[merry birch]

b and a are the ratio

eg 4:1 a is 4 b is 1

[quasi sparrow]

where are you getting 4/3 from then

[merry birch]

mathgpt

[quasi sparrow]

you should do your own work

[merry birch]

Ya but its hard when idek what to do

[quasi sparrow]

yea but learning from something that's wrong a lot is even worse

[quasi sparrow]

especially when you make it sound like you "know" with certainty 4/3 is needed for some reason

[merry birch]

well i thought i know cuz i checked my answer and it was right

[quasi sparrow]

[quasi sparrow]

then you should know the answer to this question

[merry birch]

Idk bru

This too confusing for me idk what im do9ing

[quasi sparrow]

yes which is why ^

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[lapis hazel]

guys I need help with a question

[lapis hazel]

lemme send a picture of it one sec

if z = x + yi, with x,y E R, a complex number that satisfies that satisfies the system below is: a) 3+or-4i ; b) 3 +16i ; c) 4+5i ; d) 4+or-3i ; e) 3-16i

[sullen ferry]

do you know what the conjugate means? (the bar above the z)

[lapis hazel]

yes

[sullen ferry]

well if you know that z=x+iy then you know what the conjugate of z is, have you any work on this question? :3

[lapis hazel]

wait let me try to do it again so I can show you where I stopped

[sullen ferry]

,rccw

[lapis hazel]

this is the best I could do

yea mb about that I should’ve turned it before sending

[lapis hazel]

ok sure

[sullen ferry]

i would make sure to write parentheses for the first one, the way you have it written a bit ambiguous and confusing

[lapis hazel]

I just learned complex numbers and I had to do it by a youtube video since my teacher didnt show us that, thats why Im kinda lost atm cus Idk exactly what to do

[lapis hazel]

ok sure

[sullen ferry]

well each number has a 'real part' and an imaginary part, think of them like coordinates

so a+bi=(a,b)

[lapis hazel]

ok I think I got it

[sullen ferry]

everything with an 'i' is a y coordinate, and i^2=-1

[lapis hazel]

ok got it

[sullen ferry]

do you see how you can simplify the second equation?

[lapis hazel]

I dont

can I just combine them to make it disappear?

oh ok

[sullen ferry]

you mean the -yi and yi?

[lapis hazel]

yup

[sullen ferry]

yes

[lapis hazel]

ok so 2x=8

[sullen ferry]

that is exactly what you can do :)

[lapis hazel]

is that right?

[sullen ferry]

indeed

[sullen ferry]

now for the first equation, you know FOIL?

[lapis hazel]

oh I dont think so

actually Im brazilian so I might have learned it with a different name

[sullen ferry]

this look familiar?

[lapis hazel]

oh ok sure I know that

[sullen ferry]

so do that multiplication

[lapis hazel]

ok one sec

[sullen ferry]

you should get something nice at the end :)

[lapis hazel]

x*2-(yi)*2

oh I meant to elevate them

[sullen ferry]

what is i^2?

[lapis hazel]

-1?

[sullen ferry]

indeed

[lapis hazel]

I think Im starting to understand it

[sullen ferry]

so whats the final simplifcation

[lapis hazel]

wait lemme think

[sullen ferry]

is this supposed to be x^2 - (iy)^2?

[lapis hazel]

yea sorry mb

[sullen ferry]

you remember that (a * b)^2=a^2 * b^2 yes?

[lapis hazel]

oh yea sure

sometimes I forget those

[sullen ferry]

so you can write (iy)^2 as (i^2)(y^2) yes?

[lapis hazel]

sure sure done

[sullen ferry]

and you know i^2=-1 so what is the final expression you have

[lapis hazel]

x^2-y^2 = 25

[sullen ferry]

remember you were subtracting before

[lapis hazel]

oh mb

[sullen ferry]

so what should it be instead of subtracting?

[lapis hazel]

x^2 + y^2 = 25

[sullen ferry]

there you go!

[lapis hazel]

now I change the x for 4?

[sullen ferry]

yep!

[lapis hazel]

since 2x = 8

okok

so y=3

[sullen ferry]

technically you have y^2=9

so it can be negative as well

[lapis hazel]

oh ok sure

I always forget that

so letter d

4+-3i

ok I think thats it

I got it

tysm 🙂 @sullen ferry

I really appreciate it

how do I close the chat?

.close

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[normal ferry]

hey so how do i solve this

[normal ferry]

nevermind i figured out it

.close

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[keen tendon]

what even is that lowk

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[mystic saffron]

how do i find the direction with steps?

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@mystic saffron Has your question been resolved?

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[golden wadi]

Hello, I'd just like to check if these are right? I think I'm ok on the word problem, it's the drawing that I'm mostly unsure of. Thank you.

[golden wadi]

Think I screwed up on A and C and just fixed it, I think.

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@golden wadi Has your question been resolved?

[golden wadi]

.close

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[outer wadi]

im having trouble understanding the proof of this corollary

[outer wadi]

We have that p divides m which divides n

but I dont see where the subgroup is coming from and where we are using cyclic properties

what is cyclic here?

or hmm

are they saying

K is cyclic

we can generate a cyclic subgroup from G and thats K

hm but why does K have order p

let me recheck theorem 11.1

let me add that here also

so we can write G as the direct product of cyclic groups of prime power order

and we have a prime number p that divides n

I think im getting somewhere

so the cyclic properties are being applied to the direct product of G

mmm

We can write G as

Z_p + Z_x ...

if we take the subset

(a, 0, ...)

this is a subgroup of Z_p + Z_x ...

the reason its a subgroup is because

because for an group K (Z_p), if H <= Y then H + K <= Y + K

here we have <0> is a subgroup of Z_x

so Z_p + <0> + <0> ... is a subgroup of Z_p + Z_x + Z_x2 ...

now all that is left to prove is that if G is isomorphic to Z_p + Z_x + Z_x2 then if Z_p + Z_x + Z_x2 ... has a subgroup, then G also has a subgroup isomorphic to that subgroup

So I need to prove if $G \cong H$, and $K \leq H$ then there exists a $G_k \leq G$ s.t $G_k \cong K$

[clever fjord]

Branshi

[outer wadi]

hmmmmm

ive proven there exist an ismorphism between two groups before but I've never proven the existence of a group isomorphic to another?

hmm let me try

I think I see

I just need to let the group be the elements that map to K to get the isomorphic subgroup

[echo ginkgo]

what makes you think you actually have a Z_p in G ?

[outer wadi]

since p divides |G|, G can be written as cylic group of prime ordered power so wouldn't be able to write G in the form Z_p + ...

something similar to here I think

or let me find another example, I could be wrong tho

[echo ginkgo]

there's a counterexample right there in your picture

[outer wadi]

yeah thats true

[echo ginkgo]

Z_p^2 + Z_p^2, there's no Z_p in there

[outer wadi]

hmm

then

can I say

for integer i

G can be written as Zp^i + ...

and the other stuff should still follow?

mm wait let me think

if G has some Zp^i in it

we can take the cyclic group of order p which is a subgroup of Zp^i

and so then we have some group of order p <a>, with <a> + <0> + <0>

whic should for a subgroup of Zp^i + Z_x + Z_x2....

and then the order of this group is just p

and so G has a subgroup isomorphic to this group

so G has a subgroup of order p

is that ok?

[echo ginkgo]

can you give such an a with order p in Z_p^i ?

[outer wadi]

ah your right I got that confused

I was thinking since p divides p^i

then since its cyclic it has a subgroup of order p

but it has a subgroup of order p^i/p