help-19

[mystic saffron]

alr so the magnitude would be different

[umbral epoch]

yeah

[mystic saffron]

itd be root 2/12

[umbral epoch]

so

sqrt(1/6) ?

or

[mystic saffron]

no no

[umbral epoch]

sqrt(2)/12

[mystic saffron]

this

[umbral epoch]

ok

yeah it was right

how did you do it

[mystic saffron]

idk how ot explain the derivtive of arctan tbh

other than that

u take the magnitude for the demoniator

so thats just sqrt(72)

which is 6sqrt(2)

and the top would play out to be 1

[umbral epoch]

id say this is the most vital part

that i dont understand

i can do the magnitude

[mystic saffron]

is the derivative

u want me to try to explain it

[umbral epoch]

yeah

[mystic saffron]

ok ok ill try alr

so what is u in this fucntion

function

[umbral epoch]

youre referring to u sub or??

[mystic saffron]

alr so derivative of arctan(u) = u'/sqrt(1 + u^2)

yes?

[umbral epoch]

uh

is it not 1/(1+(x)^2) ?

for just arctanx

[mystic saffron]

is that not the same thing

1 is x'

[umbral epoch]

what is q

oh

okay yeah

[mystic saffron]

same thing right

[umbral epoch]

no

[mystic saffron]

so if we set u = (y/x)

[umbral epoch]

where is the sqrt coming from

[mystic saffron]

typo sorry

its just

[umbral epoch]

then yeah

[mystic saffron]

alr so derivative of arctan(u) = u'/(1 + u^2)

[umbral epoch]

i agree

[mystic saffron]

yea

so if we set u = (y/x)

u following

[umbral epoch]

yes

[mystic saffron]

what do we get for fx of this

[umbral epoch]

this

lol

[mystic saffron]

well that was wrong wasnt it

also y would be negative

cuz of the quotient rule

[umbral epoch]

wait

OHH

okay hang on

[mystic saffron]

i can show u my work if u like

[umbral epoch]

nah thats okay, i understand now

[mystic saffron]

got it?

[umbral epoch]

damn

yeah

[mystic saffron]

silly huh 😭

[umbral epoch]

i got it

yeah i did x->y/1 instead -y/x^2

brueh

[mystic saffron]

yea

silly

[umbral epoch]

for this one

its what you were saying

[mystic saffron]

yea its the same exact thing

[umbral epoch]

for max roc is just fx and fy

[mystic saffron]

just worded differently

[umbral epoch]

and then mag

[mystic saffron]

same exact thing

[umbral epoch]

ok lemme see

[mystic saffron]

idk why they put direction after mag tho

considering u use direction to find mag

-pi/8 isnt fun either

but i suppose times two y makes it -pi/4

[umbral epoch]

the direction is the mag right

[mystic saffron]

no

direction is fx and fy

[umbral epoch]

<fx, fy> ?

[mystic saffron]

at the point

[umbral epoch]

oh

[mystic saffron]

so u plug int he point into fx and fy

[umbral epoch]

okay

[mystic saffron]

so i got (-3,-2)/sqrt(13)

[umbral epoch]

how

i did not get that

[mystic saffron]

whatd u get

[umbral epoch]

i got
fx = -3sqrt(2)/2
fy = -sqrt(2)

[mystic saffron]

so yea

can taht not simploify to

(-3,-2)

it does

[umbral epoch]

how do those simplify to -3 and 02

-2*

[mystic saffron]

multiply both by 2

so then u have

-3sqr(2)

and -2sqrt(2)

and then divide by sqrt(2)

so u have

[umbral epoch]

i have -sqrt2

[mystic saffron]

-3 and -2

[umbral epoch]

not -2sqrt2

[mystic saffron]

yea cuz i multiplied by 2

[umbral epoch]

oh

[mystic saffron]

to get rid of the denominator on ur fx term

[umbral epoch]

yeah i see now

ok

[mystic saffron]

yea

algebra

[umbral epoch]

😒

okay and the sqrt13

from where

[mystic saffron]

well what point does ur (fx,fy) give u

(-3,-2)

so then the magnitude of that

sqrt(9+4)

= sqrt(13)

[umbral epoch]

ok

[mystic saffron]

make sense?

[umbral epoch]

yeah

[mystic saffron]

u wanna find the max rate by urself?

[umbral epoch]

how do i even find it

[mystic saffron]

its just the magnitude

[umbral epoch]

oh

[mystic saffron]

so u take ur

buttttt

[umbral epoch]

yeah i go tit

[mystic saffron]

whatd u get

let me make sure

[umbral epoch]

is it not just sqrt13

[mystic saffron]

no

[umbral epoch]

how

[mystic saffron]

cuz that was the simplification

cuz direction is scalable

magnitude is not

[umbral epoch]

oh then sqrt13/2

[mystic saffron]

so we use these

[mystic saffron]

no

sqrt(26)/2

[umbral epoch]

okay makes sense

[mystic saffron]

cuz we factored out the sqrt2/2

so we ultiply that times sqrt(13)

[umbral epoch]

yeah ik

i se

[mystic saffron]

so we get sqrt(26)/2

yes

whats next

[umbral epoch]

thats all

[mystic saffron]

oh alr

u should try to get some practice

until its not so shaky

what is this

multi?

@umbral epoch

[umbral epoch]

yeah

my exam is next thurs

[mystic saffron]

want some practice problems

[umbral epoch]

shore

ill do them after my 2nd hw

[mystic saffron]

what is that 2nd hw for

[umbral epoch]

the same stuff

a cosmic joke

[mystic saffron]

want me to help

ive got time

[umbral epoch]

yeah lemme see which ones i cant do

[mystic saffron]

alr alr

[umbral epoch]

are you a math major

[mystic saffron]

well long story actually

[umbral epoch]

go ahead

[mystic saffron]

umm ho wold r u

how old r u

[stone blade]

Wait so are you a math major?

[mystic saffron]

who're u 😭

[stone blade]

Im just interested lmao

A passerby

[mystic saffron]

im 14 lol

got moved up a shit ton of grades in math at my private school growing up

[stone blade]

Oh awesome sauce

[mystic saffron]

yea

im more than qualiffied for multi tho lol

[umbral epoch]

must be nice to be a prodigy

[stone blade]

Multivariable calc?

[mystic saffron]

yea

[stone blade]

Oh fun stuff

[mystic saffron]

took that like 5 yrs ago 😭

so @umbral epoch

[stone blade]

Do you do lots of set theory?

Intersection man

[mystic saffron]

who me?

oh nah intersection man 😭

[stone blade]

LOL

[mystic saffron]

thats kinda funny icl

😭

[umbral epoch]

no ironically

im more into group and number theory

[stone blade]

Man I need help in Group Theory

[umbral epoch]

it's pretty cool

and surprisingly

i has helped me in my own major

[mystic saffron]

oh so u do abstract algebra as well?

[umbral epoch]

i did not see that coming but pleasant surprise

[umbral epoch]

not really

just specifically group and number

and not crazily

[mystic saffron]

riing fields and vector spaces lol

[stone blade]

You should look into more Abstract Algebra

Yeah find some fun stuff like Rings, Modules

[mystic saffron]

abstract algebra is fun

[short terrace]

Feel free to claim an available channel to ask questions!

[stone blade]

Then you can give Algebraic Number Theory a shot

[stone blade]

haha I just need to do some more reading thanks though :)

[short terrace]

Sure sure

[mystic saffron]

@umbral epoch so back to ur homework

[short terrace]

That's more linear algebra I'd argue

[short terrace]

Will leave you two to it, sorry for the intrusion

[mystic saffron]

ummm i feel like rings wouldnt appear in linear algebra till pretty far on

no?

[stone blade]

Definately, advanced linear algebra is fun

[mystic saffron]

i mean id say theyre much more prevalent in abstract algebra

[stone blade]

Yeah, basic ring theory; but Vector Spaces in particular are def more linear

[short terrace]

I meant vector spaces, should have specified

[mystic saffron]

wwell ofc

[stone blade]

Alright good luck on your homework

[mystic saffron]

if hed answer lol

[umbral epoch]

this is it

im working on it rn

[mystic saffron]

alr u getting it?

[umbral epoch]

breuh fuck

no i plugged in the vector

hang on lol

i think so tho

[stone blade]

Do you know how to find the directional derivative?

[umbral epoch]

yeah

i got it

[stone blade]

Like a general process

[umbral epoch]

yeah

[stone blade]

Okay good

So what did you get for your gradient

[mystic saffron]

@stone blade i got it or do u wanna help along

[stone blade]

yes sorry

[mystic saffron]

nah nah

@stone blade

stay 😭

[stone blade]

Alright hey hey

I think you got is

I gotta stop procrastinating

[mystic saffron]

ummm more the merrier

whatre u taking

this semester in college

[stone blade]

Abstract Algebra

first year

[mystic saffron]

is that how college works?

oh nice nice

pure math major?

[stone blade]

yeah

[mystic saffron]

thats what i wanna minor in

[umbral epoch]

<4/133,8/133,50/133>

right?

[mystic saffron]

yes

[umbral epoch]

oki

[mystic saffron]

no do that times u

[umbral epoch]

i got -92/247

[mystic saffron]

yea

good

[umbral epoch]

hooray

[mystic saffron]

i had

so -644/1729

but that simplifies yes

good job

hmu if u need anythign else alr

[umbral epoch]

ookay

thanks

[mystic saffron]

are u a first yr in college as well?

[umbral epoch]

no

[mystic saffron]

second?

[umbral epoch]

yeah

[mystic saffron]

alr alr

so this problem i assume u made ur i hat j hat k hat

[umbral epoch]

PQ YEAH

yeah*

sorry

[mystic saffron]

yea

[umbral epoch]

lol

[mystic saffron]

so its the same as last problem

pocess wise

[umbral epoch]

well i did something wrong

[mystic saffron]

u can plug in either point

well yes

i assume u got a new problem tho

u wanna just start fresh on that

makes it easier

[mystic saffron]

alr alr

so pq is. <-3, 4, 1>

[umbral epoch]

for
fx = -y/(x+z)^2

and same for fz

[mystic saffron]

yes

and for y

x+z

[umbral epoch]

and fy as 1/x+z

[mystic saffron]

'. sorry yes

that

1/x+z

[umbral epoch]

maybe i did wrong pq

[mystic saffron]

so now u can plug either p or q in

to find ur gradient

[umbral epoch]

<5,-1,5>

?

[mystic saffron]

which point did u use

[umbral epoch]

q

[mystic saffron]

yup

so we have

that dot ur vector

[umbral epoch]

-21

[mystic saffron]

PQ

[umbral epoch]

-24

[mystic saffron]

pq was <-3,4,1>

so -15 -4 +5

yes

so -14 no?

[umbral epoch]

damn

yeah

[mystic saffron]

wai tlet me solve it our rq

[umbral epoch]

yeah i keep getting it wrong

[mystic saffron]

wait one sec sorry

alr so the formula

is delF(Q) dot pq

divided by mag pq

mg pq is sqrt 26

[umbral epoch]

yeah im taking a break

[mystic saffron]

one sex

[umbral epoch]

thank you for your help, i appreciate your time

[mystic saffron]

im there

one sec

[umbral epoch]

for why

there is no point this hw is annoying me lol

[mystic saffron]

alr i got

6/sqrt(26)

check that rq

just to make sure

[umbral epoch]

if i check it then i wont do it

[mystic saffron]

why

[umbral epoch]

i need to be able to do it

[mystic saffron]

alr umm ho wlong u gonna breka for

[umbral epoch]

probs until tomorrow lol

[mystic saffron]

lets finish this first alr?

[umbral epoch]

after my class

[mystic saffron]

then ill let u break

ok?

[umbral epoch]

i suppose

so what do i do

so i js divide it all by the mag

[mystic saffron]

calm down calm down

alr alr

so lets start from the top

we computed one of our vectors incorrectly

thats my fault alr

@umbral epoch

[umbral epoch]

which on

e

[mystic saffron]

lets plug in point P into our gradient

so plug in (2,1,-1)

so x+z = 1

what wuld u get

[umbral epoch]

these are the pts now btw

[mystic saffron]

yeayea

[umbral epoch]

i got <-1,1,-1>

[mystic saffron]

yup

[umbral epoch]

w P

[mystic saffron]

so now we dot that

with out direction vector

which was

<-3,2,-1>

is htis a new problem btw

[umbral epoch]

yeah

ik

i got 6

for dot

[mystic saffron]

yup

and the magnitude of the direction

[umbral epoch]

sqrt15

[mystic saffron]

u sure

9+4+1

[umbral epoch]

14

sqrt14

[mystic saffron]

and r u sure abt 6

[umbral epoch]

3+2+1??

[mystic saffron]

its 3 +2 -1 no?

or am i tweaking

[umbral epoch]

<-1,1,-1> dot <-3,2,-1>

so +1

[mystic saffron]

its dotted by <-3,2,1>

so -1

cuz z goes from -1 to 0

so +1 not -1

[umbral epoch]

oh yeah

okay

so 4

[mystic saffron]

so 4/sqrt(14)

tell me if its correct

[umbral epoch]

thats the ans?

k

yeah

its right

thank you

[mystic saffron]

u get the concept behind it?

[umbral epoch]

nope

[mystic saffron]

i shoudl be better... im stupidly rusty

[mystic saffron]

alr so whatre we looking for

the directional derivative

yes?

[umbral epoch]

yeah

[mystic saffron]

so what is the direction

????

[umbral epoch]

pq?

[mystic saffron]

YUP

and we found the derivative of f(x,y,z)

for teh point P

and by dotting them we find the directional derivative

so why wasnt it working before...

any guesses?

[umbral epoch]

bc we used q

[mystic saffron]

we used q

[umbral epoch]

yeah

[mystic saffron]

and put it in the direct

p1

pq

so it doesnt work like that ofc

if we did q

and put it in direction qp

yk what youd get?

[umbral epoch]

nope

lol

[mystic saffron]

think abt it like a curve

if u parameterize y = x to go from infinity to -infinity

what woudl the slope be

[umbral epoch]

i wanna say +1 lol

[mystic saffron]

well ur moving backwardds

bad example

but every 1 u move to the left... so in the direction set

so + 1

youd go down 1

so -1/1

so slope woudl appear to be -1

same concept applies to f(x,y,z)

[umbral epoch]

oh cuz inf to -inf

[mystic saffron]

yea

flipped direction

its becoem y = -x

[umbral epoch]

yeah gotcha

i didnt see inf to -inf

[mystic saffron]

so wopudl this be

starting with pointq

[umbral epoch]

ive no clue

[mystic saffron]

well think abt it

its intuitive

if start from p in direction pq

was

4/sqrt(14)

if we flipped the direction

[umbral epoch]

..

sqrt14 / 4

?

i really dont know

[mystic saffron]

why would u flip it

[umbral epoch]

bro idk

[mystic saffron]

ok so when we went opposite direction on y = x what happened

[umbral epoch]

-x

[mystic saffron]

what happened to the slope

thats waht we're concerned with

since we're workign with the derivative

[umbral epoch]

i am going to explode

[mystic saffron]

x to -x

what happened

[mystic saffron]

ur overthinking it 😭

[umbral epoch]

idk i am just very confused

[mystic saffron]

it became negative

if we use poitn q

in direction

qp

itd be

-4/sqrt(14)

bro left 😭

[umbral epoch]

no im still here

[mystic saffron]

does that make sense tho

intuitivelty

[umbral epoch]

mkaes sense

yes

it make snse

[mystic saffron]

u can verify it if u want 😭

[chilly ridge]

i disagree

[mystic saffron]

well fuck 😭

[mystic saffron]

hwo come

[chilly ridge]

nevermind actually i think ur right

[mystic saffron]

yea

😭 🤯

well thats fun

@umbral epoch would u say u get it now

[odd edge]

@umbral epoch Has your question been resolved?

[odd edge]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[glass dew]

Hey, working on graphing here, how do I know what the table of contents is going to be?

[glass dew]

Like for 31 it just gives y=2x, how does that help me at all?

[austere jolt]

btw, it's called a table of values, not table of contents

good thing is that you're given the specific x-values you need

[short terrace]

(not for 31)

[austere jolt]

so if y = 2x, at x = 0, what will y be?

[glass dew]

Oh alright!

[austere jolt]

oh I just saw 31, mb

if for 31, even better!

[glass dew]

Yeah!!

[austere jolt]

you get to choose what points you want to use!

[glass dew]

You do…?

[austere jolt]

but of course, don't choose like x = 100000 or something

yes

[glass dew]

Right LOL

So if I get to choose my points, what am I even finding for the work?

[austere jolt]

dump some x-values into the formula to get corresponding values of y

plot those ordered pairs (x, y) on the graph and join them

that's all

[glass dew]

Really?

[austere jolt]

mhm

[glass dew]

That’s great!!

Ok so one more thing, if it’s y= 2x I’ll just add 2, right?

And then if it’s 2x-1 I’m not really sure what the idea is for that

[austere jolt]

same idea

[austere jolt]

wait, no

2x is 2 times x

not 2 plus x

2 plus x is just 2 + x

[glass dew]

Ohhhh

So 2x-1 you’d multiply by 2 and then sub by 1?

[austere jolt]

yes, in that order.

[glass dew]

Ok, great!!!

Thank you so much mate your saving my grade🙏

[austere jolt]

no problem, glad to help!

anything else?

[glass dew]

Just a couple more, is that alright?

[austere jolt]

why not?

[glass dew]

So what would I do if it’s x= instead of y? Do I just make up for y instead?

[austere jolt]

ok for this I will need to check the question

[glass dew]

And what does x-y entail for a problem?

[austere jolt]

in case the question asks for something else

so, original question please.

[austere jolt]

oh ok

so for these, same strategy. choose for x, solve for y

(you may want to rearrange some of these equations if it helps.)

[glass dew]

Oh, so there’s no change in how I approach them?

[austere jolt]

nope

[odd edge]

@glass dew Has your question been resolved?

Channel closed

Closed by @glass dew

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[polar cairn]

I'm stuck on this problem and i'm not sure where to even begin. I tried finding f(0) but couldnt rigorously prove it.

[polar cairn]

I think after proving f(0)=0, we can derive relations like f(x)=f(f(x)) or similar stuff

[quasi sparrow]

can you show you proved f(0) = 0

[undone palm]

ive never really saw anything like that before but the lower equation is relating each component

[polar cairn]

i didnt prove it yet

im not sure how to

[odd veldt]

If f(f(0))=f(0), can you say anything about f(f(f(0)))?

[polar cairn]

i derived f(3/2f(0))=2f(0)

f(f(f(0)))=f(0)

and so on

i also derived f(2f(0))=4f(0)

i tried proving f is injective but couldnt do it

i def think proving f(0)=0 is the first step. If we establish this we can cancel out so many things

[odd edge]

@polar cairn Has your question been resolved?

[polar cairn]

Help?

<@&286206848099549185>

[odd edge]

@polar cairn Has your question been resolved?

[polar cairn]

.close

[odd edge]

Channel closed

Closed by @polar cairn

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wide drift]

This is the question and the proof they wanted. Would I lose any marks for the proof I gave? Was anything important missing that was necessary to gain full marks in your opinion?

Also is there anything else you would change in my answer to improve it?

[mental lotus]

well, if you claim x is the largest in S, it makes no sense for y to exist in S

you claim to be able to pick a y in S which is greater that x

but you cant 'pick' a number larger than the largest number in set.

by pick it means that such a number already exists, which is same as assuming the theorem is already true

[wide drift]

Isnt that what they did in their solution though?

[mental lotus]

well, they calculated a number and showed that it should also exist in the set

which is different than picking it up from the set

[low locust]

why is (x+1)/2 <1? or >x?

you would lose points for that

[mental lotus]

picking it out means you already know its existence (which means you knew x was not the greatest one in the set aka you lied)

[wide drift]

Alright I think I see your points

[wide drift]

So I would need to go into more detail about this calculation?

[low locust]

yes

[wide drift]

and I would need to start with an element in y and then turn it into an element thats still in y but its greater than x?

.close

[odd edge]

Channel closed

Closed by @wide drift

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wide drift]

This is the question and the proof they wanted. Would I lose any marks for the proof I gave? Was anything important missing that was necessary to gain full marks in your opinion?
Also is there anything else you would change in my answer to improve it?

[rose wigeon]

if you wanna be super strict and rigourous aroung the

\in naturals

thereis not parentese

sorry i sent the start of my answer on accident im going to keep reading

instead of on you can say in

[wide drift]

Ahh ok good to know

Other than that would I have gotten full marks in your opinion?

Anything else you would change or add or remove?

[rose wigeon]

mhm I can tell you are somewhat new to proofs but the ideas are good

just the wording mostly

but that comes with time

you can use words like

Thus, then

for logical implications

but take and such that is well used

and let

in terms of markings if the class is a beginner proofs class id give full marks ?? depends how strict they are but i suppose you could remove points for the wording im really not sure

oh ur first implicatoin statement is redudant a little bit

like u just said it exists

you could say take ym = z, this when y is subtituted by xn gives ...

makes sens ?

[wide drift]

Yeye 😇

[rose wigeon]

but like i said its all like my critiques are mostly redaction quality

anyways have fun in ur proofs class

[wide drift]

Yeye these are the little things I want to improve on haha

Thank you so much for the analisis! That answers my question perfectly

❤️

.close

[odd edge]

Channel closed

Closed by @wide drift

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[gloomy mason]

can someone tell me how diagonal matrix works ?

[low locust]

in what sense

its a matrix where all entries that are not on the diagonal are zero

so the only place where nonzero entries could appear are on the diagonal

[gloomy mason]

but we have 0 on corners

[low locust]

"the diagonal" in this context means exactly those entries where the matrix has 3,5,1

the entries A_ii if you are familiar with that notation

so only the one diagonal from top left to bottom right

[gloomy mason]

whats double i

[low locust]

no other diagonal

[gloomy mason]

i onnly know i and j

[low locust]

j=i

[gloomy mason]

wha

[low locust]

A_11, A_22

same index

[gloomy mason]

oh

[wooden python]

the entries in row 1 col 1 or row 2 col 2 or row 42069 col 42069 (if your matrix is big enough)

"the diagonal" of a matrix is only the entries whose row and column index are the same

[gloomy mason]

oh

alr

ty

.close

[odd edge]

Channel closed

Closed by @gloomy mason

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wanton galleon]

im out of my wits with this question :3 someone help?

[woeful briar]

notice this has a special form

if f(x)=x^2-3x+1

then the equation is f(f(x))=x

[pastel steeple]

let $y = x^2 - 3x + 1$, then $y^2 - 3y + 1 = x$

[clever fjord]

spoopy

[wanton galleon]

mb if I completely misunderstood but is this what u guys meant

idk where to proceed from here either though ;(

nvm gimme a sec

[woeful briar]

for example any solution of f(x)=x will be a solution to f(f(x))=x

[wanton galleon]

THATS WHAT MY FRIEND SAID but i didn’t get how that works

could you explain that

[woeful briar]

f(f(x))=f(x)=x

if f(x)=x

[wanton galleon]

oh

omgg

i see it 😭

THATS SO SMART HOW DID YOU NOTICE THAT

[acoustic nexus]

but that doesn't include every possible solutions right?

[woeful briar]

yeah it only yields one solution

its very likely to have more

[acoustic nexus]

I wonder how we can do the rest w/o dealing with quartic

[woeful briar]

it gives two solutions

not one

and actually this means we can reduce to a quadratic

long division by x^2-4x+1

[wanton galleon]

i ended up doing this :p

[woeful briar]

looks correct

[wanton galleon]

i hope it is i dont have the answers 😔

tysm for the help!

.close

[odd edge]

Channel closed

Closed by @wanton galleon

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[vernal yacht]

. @steel terrace post here

[steel terrace]

ok

[vernal yacht]

講中文ㄇ？

[steel terrace]

uh

so my problem here is id wanna make it spread

to actually form a rollercoaster bruh

[proven harbor]

English

[steel terrace]

without doing anything in the letters.

[vernal yacht]

bruh, I thought they speak Mandarin due to their pronouns

[steel terrace]

im literally he/him bruh 😭

[vernal yacht]

Could you demonstrate it on a paper?

[steel terrace]

lemme grab a reference

so my rollercoaster looked like this.

but i want to make it looked like this

here are my equations

[steel terrace]

that turned all to these

[proven harbor]

For that use a square

Like (x-c)²

[steel terrace]

like id just wanna expand these

at some point at both ends are locked.

[proven harbor]

Just shift the roots

[steel terrace]

this is like my 1st time using ts bro 😭

im so cooked.

[proven harbor]

Firstly make it so that from a to k the values increase

So make a -2.1

Then b -1.7

So on

[vernal yacht]

.solved @proven harbor he left

[odd edge]

Channel closed

Closed by @vernal yacht

Use .reopen if this was a mistake.

[proven harbor]

I don't think we should close it tho

[vernal yacht]

He left the server 💀

[proven harbor]

The bot will do it after sufficient time

[vernal yacht]

Not the channel LOL

[proven harbor]

Oh 💀

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[feral dome]

i dont understand the solution from "we need to focus on the k=4 term" and everything onwards.

[wooden python]

(xy+y^2)^k contains only terms of degree 2k

you want the coefficient on x^3 y^5, which is a term of degree 8

so the only way to get that is if 2k=8, thus k=4.

[feral dome]

do they not all have a total exponent of 8, say one term would be xy^7, another might be x^6y^2?

[wooden python]

not what the word "coefficient" means.

[feral dome]

exponent

[wooden python]

do they not all have a total exponent of 8
yes thats the point

thats why you focus only on (xy+y^2)^4

terms from (xy+y^2)^69 would have total exponent 138, and not the 8 that you want

[feral dome]

yeah i agree, im saying that in the binomial expansion of (xy + y^2)^4, all of the terms would have the same total exponent, so how does this zone in on one term?

[wooden python]

you zoom in on it in two steps

the first of which is figuring out that you want (xy+y^2)^4 and not (xy+y^2)^(something else)

[feral dome]

right, so now we fix k = 4

[wooden python]

yes

[feral dome]

so now how is this part formed?

[wooden python]

replace xy+y^2 with Z for the moment

in expanding $(1+Z)^n$, the term with $Z^4$ is $\binom{n}{4}Z^4$

[clever fjord]

Ann

[feral dome]

right yes

so now sub the Z back

[wooden python]

that is where the C(n,4) comes from

[feral dome]

why specifically 4?

[wooden python]

bruh did we not spend like a page discussing why 4

[wooden python]

.

[feral dome]

thats the exponent

[wooden python]

.

[feral dome]

of the binomial expansion

[wooden python]

no, what you've got is a binomial expansion inside of a binomial expansion

[feral dome]

explain

[wooden python]

fuck idk how to explain w/o repeating myself sorry

gonna go back to resting

[feral dome]

oh 💔

[river island]

i think if he use the multinomial expansion it would be easier ...

[odd edge]

@feral dome Has your question been resolved?

Channel closed

Closed by @feral dome

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[river island]

just expand the term (xy+y^2)^4 again so you gonna have an expansion inside expansion
and then take the possibilities of the exponents

[feral dome]

yeah i think i figured it out

only in the expansion of (xy+y^2)^4 do we start to think which term gives us x^3y^5

and this is when the (xy)^3 and (y^2)^1

[river island]

true

[feral dome]

in the outer binomial expansion, we only care about what power the (xy+y^2) is

and that is = 4

once we know its = 4

then we can binomial expand the inner term

thanks for your efforts

[river island]

yes it would be easier to determine the solution

[feral dome]

that sounded sarcastic but i meant it

[river island]

your welcome

[feral dome]

i have another question coming up so 😭

[river island]

its ok just ask🫣

[feral dome]

is this the rule when looking at multinomial expansions?

zoom in on one part then zoom in again

[river island]

this is it
it is generalisation of the binomial expansion

if you have just two term it gonna be same as the binomial expansion

.close

[odd edge]

Channel closed

Closed by @river island

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[winter berry]

Prove or disprove that h(x)=(x+1)/(x^2+1) is uniformly continuous. This is to be done strictly by the definition of uniform continuity, since I haven't seen any theorems related to this yet.

[winter berry]

I got that
$|h(x)-h(y)|= |x-y|\cdot \frac{|xy+y+x-1|}{(x^2+1)(y^2+1)}$

[clever fjord]

Erk Gah

[winter berry]

So I need to limit that fraction somehow... But I'm out of ideas

[quasi sparrow]

try using triangle inequality on the numerator

[warped glacier]

you are actually trying to apply Lipschitz continuity when you did this

and Lipschitz continuity is a stronger condition than being uniformly continuous

I'm not saying your approach doesn't work, but you can't bound |xy + y + x - 1| above by 100|xy| or 1000|xy| and so on

[winter berry]

Shouldn't I try to make |x-y|≤E to show uniform continuity?

I haven't seen Lipschitz continuity yet

[winter berry]

This is by no means rigorous but the graph suggests I can bound the fraction by 2 (I think), I just dont know how

[warped glacier]

oh that's nice

yeah I didn't think of bounding it by 2

[winter berry]

Oh I made a typo

Nvm I didnt

Its correct

I thought I had put in x^2-1 instead of x^2+1

[warped glacier]

yeah so you can just choose $\delta = \frac{\epsilon}{2}$

[winter berry]

Yeah so I make |x-y| less than epsilon then choose a delta based on that right?

[clever fjord]

south

[warped glacier]

Lipschitz is motivated by choosing delta = epsilon * constant, cause it's neat

[winter berry]

Yes the problem is how do I even get to that conclusion

I just thought of putting it on the graphing calculator but idk how I can rigorously conclude the fraction is bounded by 2 lol

Hold on let me try what that guy suggested I almost forgot

I'll try applying triangle inequality to the numerator

Ok, so I got

$|x-y|\cdot \frac{|xy|+|y+x|+1}{x^2y^2+y^2+x^2+1}$

[clever fjord]

Erk Gah

[winter berry]

I should have something here

Can I just say that the fraction is <1...

No I can't I think

[quasi sparrow]

if you can prove it sure.

show your proof here

probably would help to break it into cases of |x| and |y| <= 1 and > 1

[warped glacier]

I feel this defeats the purpose of RA if it's just inequality training

but I have an idea: notice that your expression is symmetric in x, y, so WLOG you can assume x >= y

$\cdots \le \frac{|(x + 1)^2|}{x^2 + 1} \frac{1}{y^2 + 1} \le \frac{(x + 1)^2}{x^2 + 1} \frac{1}{0^2 + 1}$

[clever fjord]

south

[warped glacier]

so now you can use calculus to find the maximum, which is 2

(or if you're astute, Cauchy-Schwarz gives $(x^2 + 1^2)(1^2 + 1^2) \ge (x \cdot 1 + 1 \cdot 1)^2$)

[clever fjord]

south

[warped glacier]

oh wait there's a problem with |(x+1)^2| but hopefully you can tweak it

[winter berry]

Hmm

[warped glacier]

||it's the same reasoning but with (|x| + 1)^2 instead||

[winter berry]

I was trying to limit the numerator with some other inequalities but I just got some weird stuff

By AM-GM I got that (x^2+y^2)≥2|xy|

Then (|x|+|y|)^2= x^2+y^2+2|xy|≤x^2+y^2+(x^2+y^2)=2(x^2+y^2)

So |x+y|≤|x|+|y|≤√(2(x^2+y^2))

I think i got it figured out

[winter berry]

I applied AM-GM again here
√ab≤(a+b)/2, so I let a=2 and b=x^2+y^2

Then √2(x^2+y^2)≤(2+x^2+y^2)/2= (x^2+y^2)/2+1

[winter berry]

The combining with this i get

|xy|+|x+y|+1≤(x^2+y^2)/2+(x^2+y^2)/2+1+1

=x^2+y^2+2

Also the denominator is ≥x^2+y^2+1

x^2y^2+x^2+y^2+1≥x^2+y^2+1 <=> x^2y^2≥0 which is true

$\frac{|xy|+|x+y|+1}{x^2y^2+y^2+x^2+1} \leq \frac{x^2+y^2+2}{x^2+y^2+1} = 1+\frac{1}{x^2+y^2+1}\leq 2$

[clever fjord]

Erk Gah

[winter berry]

@warped glacier @quasi sparrow sorry for the ping but this should be correct yes?

.close

[odd edge]

Channel closed

Closed by @winter berry

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[plush flicker]

help please idk what to do

[plush flicker]

nvm i got it

[steep mantle]

!done

[odd edge]

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

@plush flicker Has your question been resolved?

Channel closed

Closed by @plush flicker

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[green birch]

help

[green birch]

:D

The composition of the functions () defined by () is the function

is the

b

but, wtf means 2k+1 𝜋/2,0

[green birch]

https://tenor.com/view/cute-kitten-begging-attention-cat-gif-8380127

[manic sleet]

uhhh im gonna bec honest, idk either that is come goofy notation

[green birch]

twins

[manic sleet]

okay well you're right, in that the composition will be sqrt(sin^2(x)-1)

I think I can kinda understand what it means

it saying the domain of the function

BUT pay attention to that sqrt()

what happens if I try to put a negative number inside the sqrt?

[green birch]

i

imaginary

[manic sleet]

right, in other words cringeness if we want real numbers

[manic sleet]

so we require sin^2(x)-1>=0

thanks sky

[green birch]

yeah

and

It's not because the sine takes all the real numbers, but the square limits it to not taking the negative numbers?

but what means that

[manic sleet]

sorry had to help my roommate with math too

[green birch]

dont worry

[manic sleet]

not quite what we're looking for here, so we have sin^2(x)-1>=0, right?

that came from the sqrt() always being non-negative

[green birch]

yeah

[manic sleet]

can you simplify that inequality?

[green birch]

i tink sin in both sidesor something like that

[manic sleet]

not quite that much, alas, do you agree that sin^2(x)-1>=0 ==> sin^2(x)>=1?

[green birch]

ah

i agree, i thinked that but after the sin in both sides

[manic sleet]

not quite yet

is it possible for sin to ever be greater than 1?

[green birch]

hmmmm

no

because the range is -1 and 1

[manic sleet]

right!

and so surely sin^2(x) can never be greater than 1, right?

[green birch]

https://tenor.com/view/cool-fun-white-cat-dance-cool-and-fun-times-gif-15078139155535862541

[-1,1]

[manic sleet]

then when we have sin^2(x)>=1, it's really nonsensical to ask which real x sin^2(x)>1, so we only care about the sin^2(x)=1, yes?

[green birch]

yes

[manic sleet]

now, I leave it to you, can you solve that trig equation?

those values of x, will be exactly the ones where sqrt(sin^2(x)-1) will be defined

[green birch]

i dont get it

[odd edge]

@green birch Has your question been resolved?

[green birch]

https://tenor.com/view/hola-grupo-cat-grupo-contesten-busy-gif-10460945

[green birch]

i get until this

[green birch]

.close

[odd edge]

Channel closed

Closed by @green birch

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[spark igloo]

hi! i need some help with my geometry project. we are making a parallel lines project and im having some seconds thoughts on where i should place certain things. here is the attached rubric and the base ive done.

[spark igloo]

is this outline good for the project?

<@&286206848099549185>

??

.close

[odd edge]

Channel closed

Closed by @spark igloo

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sharp geyser]

how can i quickly multiply something like f(x) = (x-r1)(x-r2)(x-r3) is there a method for it?

[sharp geyser]

has to be on paper with no calculator bc its for amc

[graceful viper]

u know the highest order term is x^3 so add up all the coefficients which yield x^3 terms, then do the same for x^2, and so on

for instance, the only possible combination to get x^3 here is x*x*x which gives a coefficient of 1 so your x^3 term will have coefficient 1

[sharp geyser]

ohh i see

thank you

.close

[odd edge]

Channel closed

Closed by @sharp geyser

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[outer wadi]

I was wondering why wouldn't a closed set be bounded

[outer wadi]

looked it up and google gave an example saying integers aren't bounded but are closed

what makes the integers closed?

The definition wiki gave is that a set is closed if its complement is open

but whats the complement of Z?

oh

the complement of Z is R - Z if were dealing with the real numbers

hm

so R - Z is open?

I mean I guess that makes sense but it doesnt really make sense intuitively on how Z is closed still

it stretches from -infinity to infinity

let me retake a look at the definition of an open set

yeah I dont really understand ill ask my prof tmr

.close

[odd edge]

Channel closed

Closed by @outer wadi

Use .reopen if this was a mistake.

[odd edge]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[woven stag]

Why in the heck is f'(x) = -3 (cosx * cosx - sinx * sinx) ??

[woven stag]

Product rule?

[nimble blaze]

yes

[woven stag]

Does the constant, 3, get removed afterwards?

[nimble blaze]

wdym removed

[nimble blaze]

are you trying to find when f'(x) = 0

[woven stag]

Yeah.

[nimble blaze]

well that's a result of basic algebra

(dividing both sides of the equation by -3)

[woven stag]

Alright. I guess that makes things simpler.

How does sin^2x = cos^2x lead to tan^2x = 1?

Wouldn't it be simpler to convert cos^2x - sin^2x into cos2a ?

[short terrace]

What is tan x

[woven stag]

sinx/cosx.

[short terrace]

Good

What is tan²x then

[timber river]

so sorry

[short terrace]

No worries

Feel free to claim an available channel

[woven stag]

sin^2x / cos^2x ?

[short terrace]

Good

And now they're equal so...?

[woven stag]

Sorry, I'm not getting it. I'm trying to check my notes if a similar instance to this problem has occured, and that doesn't appear to be the case.

[short terrace]

sin²x = cos²x

tan²x = sin²x/cos²x

What is tan²x then

[woven stag]

You divide the right side by cos^2x to get tan^2x, then the right side is left as 1?