help-0

[buoyant saddle]

my fault gang

[proven leaf]

so we must look at both the positive and negative cases :)

[azure axle]

oh

[proven leaf]

nah nah ur good lol

[azure axle]

i didn’t understand we were actually plugging it into y

[buoyant saddle]

another way to think of it

y^2 - 1 = 0

then factor

difference of two squares

[azure axle]

well thank you . just got back in math

.close

[lone heart]

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[fast nest]

hi i was wondering if i could get help w these 2 tables. i thought i knew what i was doing but what i ended up with wasn’t right so if i could just get some help that’d be great

[proven leaf]

the definition of a linear function is that the slope remains the same

for the first table

can you find $\frac{Q_2-Q_1}{t_2-t_1}$?

[ocean seal]

PajamaMamaLlama

[fast nest]

90.02-95.01/2-1?

[lone heart]

@fast nest Has your question been resolved?

[proven leaf]

right

,calc (90.02-95.01)

[ocean seal]

Result:

-4.99

[proven leaf]

so that is -4.99

now find $\frac{Q_3-Q_2}{t_3-t_2}$

[ocean seal]

PajamaMamaLlama

[fast nest]

oh yeah it’s -4.99 as well

sorry i’m not getting notifs

[lone heart]

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[fast nest]

.reopen

[lone heart]

✅

[proven leaf]

lol no worries

[fast nest]

i believe this one is linear but i don’t know how to set up the problem to convert it to the equation

[lone heart]

@fast nest Has your question been resolved?

[lone heart]

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[ember pelican]

im just confused on how to find the r(x) i know that 1/x^5 is the h(x)

[lone heart]

@ember pelican Has your question been resolved?

[ember pelican]

<@&286206848099549185>

[arctic river]

what is formula for the individual slices of the cylinder

@ember pelican

[ember pelican]

This is using the shell method tho right?

Not the disk method

[arctic river]

formula for cylndrical shells is what

w.r.t x

[ember pelican]

Can't figure how what r(x) would be

[arctic river]

ok what is r(x) and h(x) in this case

u draw a diagram

what variable is responsible as the slice gets closer and closer to b

[arctic river]

this isnt really representitive of the area that u need to find

its underneath 1/x^5

above y=0

between x values

[arctic river]

ok r(x) is x

[ember pelican]

it worked ;-;

honestly i probably shoulda tried that first since thats what it was for two other problems

thx

.close

[lone heart]

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[raw jetty]

find the ammount of odd coefficients of the expansion of (x+y)^1000

[raw jetty]

this is scary honeslty

[woeful pulsar]

Do you know how to find the coefficients?

[raw jetty]

yeah im p sure?

depends on weather i know the method your talking about or not

[woeful pulsar]

Do you have any useful formulas for the coefficients?

Formulas that maybe make it easier to check whether it is odd or even?

[raw jetty]

uh maybe the choose function one?

honestly no

[woeful pulsar]

how about one involving factorials

checking whether a product is even might be much easier

[raw jetty]

dont know any

[sour mica]

yes, thats the one, Binomial Expansion

[raw jetty]

ok

[woeful pulsar]

do you know how the choose function can be written with factorials?

[raw jetty]

uh its like n!/k!(n-k)!

[woeful pulsar]

yeah, so you do know how to write it with factorials, so the problem basically becomes determining whether this is even or odd

[raw jetty]

iirc n is the row and k is the thing from left to right or something

[raw jetty]

how can i do that?

[woeful pulsar]

I'll introduce this idea of a valuation, $\nu_2:\mathbb{Q}\to\mathbb{Z}$, it basically counts ``how many times 2 divides this rational number''

[ocean seal]

Element118

[woeful pulsar]

The relevant relations are $\nu_2\left(\frac{a}{b}\right)=0$ if both $a,b$ are odd, and $\nu_2(2k)=\nu_2(k)+1$.

(Exercise: Show that this is a well-defined function.)

[ocean seal]

Element118

[raw jetty]

can i ask what you mean with "times 2 divides" here means?

nmv im dumb lmao

[mortal trellis]

if you had natural numbers, how often is 2 in the prime factorization

[woeful pulsar]

how many times (2 divides) this rational number

[woeful pulsar]

yeah that also is another way to view it

should have gone for an easier approach tbh, this valuation thing might be overkill

[raw jetty]

uh what can i do with it?

[lone heart]

@raw jetty Has your question been resolved?

[woeful pulsar]

basically, for an integer, if $\nu_2=0$, then the number is odd, and if $\nu_2>0$, the number is even.

[ocean seal]

Element118

[woeful pulsar]

The alternative is to look at the power of 2 in the prime factorisation, and that approach is equivalent

so do whatever's more comfortable

[raw jetty]

hm yea this makes sense but im just not sure what to do with it

[woeful pulsar]

yeah maybe it's easier to work with prime factorisation

since the approach is easier

and equivalent anyway when you unpack it

you want to find the number of powers of 2 in the prime factorisation of $\frac{n!}{k!(n-k)!}$

[ocean seal]

Element118

[mortal trellis]

I mean its not that easy with binomial coefficients. the answer is given by lucas theorem but I am not sure how that can be motivated well

and the proof does something completely different

[raw jetty]

uhhh sorry again 😭 but how do i find that?

[woeful pulsar]

so the subproblem is to find the number of powers of 2 in a factorial

if we can solve that for a factorial, we just need to combine the result for the numerator and the denominator to get the entire thing

[raw jetty]

its like floor(n/2)+floor(n/4)+.... right?

[woeful pulsar]

yeah, you know the answer

but it's not very easy to work with now

perhaps if we can write it as $n-$something we can make it easier

[ocean seal]

Element118

[cursive badger]

But I don't think he'll know about that...

It's not so common to use.

[raw jetty]

how do i do that?

find the "something" i mean

[woeful pulsar]

n=n/2+n/4+...+...

something=n-(floor(n/2)+floor(n/4)+...)
=(n/2-floor(n/2)+n/4-floor(n/4)+n/8-floor(n/8)+...)
={n/2}+{n/4}+...

might still be unwieldy, my next thought would be perhaps looking at the numbers in binary, there's a good chance that would help

since we have a lot of floor(n/2) and that's naturally shifting the integer in binary

[raw jetty]

what does {} mean?

[mortal trellis]

pretty sure base 2 will help

[woeful pulsar]

fractional part

[cursive badger]

yep

[mortal trellis]

the correct answer is expressed using the digits of base 2

[woeful pulsar]

we can always look at small cases to try checking for a pattern

[cursive badger]

well there is another way to write this... I am not sure if you want me to tell you that...

It's like not so common...

[raw jetty]

wdym?

[mortal trellis]

write n in base 2

[woeful pulsar]

writing n in base 2, and k in base 2, then you may be able to see when n! has the same number of powers of 2 as k!(n-k)!

maybe look at doing k + (n-k) = n and think what happens if there are no base-2 carries in the addition

[raw jetty]

wtff

yea i actually dont understand a thing of what your talking about sorry 😭

[woeful pulsar]

Show that when you write $n=\sum_{i=0}^ka_i2^i$ in base 2 where $a_i\in{0,1}$, we have
[\sum_{i=1}^\infty\left\lfloor\frac{n}{2^i}\right\rfloor=\sum_{i=0}^ka_i(2^i-1)]

[ocean seal]

Element118

[woeful pulsar]

(think about how the left and right hand side look like in binary)

[raw jetty]

sorry had s9mething to do

the rhs would like flip all yhe previous digits

the lhs would be uh

would it not be the binary representation of n?

wait no

ok wth im not seeing it

let n=15
7+3+1
111+11+1
1011

a_i from i 1 to 4 would be
1110
checking
1+3+7+0

huh how do i generalize it :(

[lone heart]

@raw jetty Has your question been resolved?

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[heavy prairie]

z^2 = 6i

[heavy prairie]

I want all the z's

I assume there will be two since they are the roots

I forgot how to continue

sqrt both sides might be a good start

but after that it is a blur

z = sqrt(6i)

[uneven isle]

rewrite 6i as [(sqrt6)(cis(pi/2 + 2k pi))]^2

cis = cos + isin

its informal notation but used widely enough so i will just keep referring to it as cis

then naturally square root both sides to find the roots

then apply de movire's theorem and it should be complete

[heavy prairie]

alright I will try to do that

[uneven isle]

k is just some integer

for convenience we will let it take on 0 and 1

[heavy prairie]

Is that the same thing?

The cis thing confused me ngl

I also called k n instead

[uneven isle]

yeah but just make sure ur sqrt 6 is also squared

[heavy prairie]

oh yeah

[uneven isle]

make sure u define n tho as an integer

[heavy prairie]

i see that now

[uneven isle]

just out of habit

[heavy prairie]

ok so hold on

[uneven isle]

for more formal exams or writings it should be expected for you to define n, the +2pi n is there for periodicity

[heavy prairie]

ye i get i get it

let me correct it

or attempt to

If that is correct I can determine the value of cos and sin

because I know the angle

but how did you determine the angle was 90 degrees?

or pi/2?

[uneven isle]

out of practice ig?

[heavy prairie]

Is it just because we say the complex axis is perpendicular to the real axis?

[uneven isle]

u can just evaluate it to show that it is i

or u can say complex axis is perpendicular like u suggested

[heavy prairie]

so is the angle always pi/2?

[uneven isle]

for i?

yes

for other values? no

[heavy prairie]

im not sure i understand

[uneven isle]

so for normal roots of unity

liket say z^n=1

ur angle for ur RHS expression would be just 0+2npi

the angle depends on what u are trying to find roots for

https://en.wikipedia.org/wiki/Root_of_unity

this might help if u want to have a read through

[heavy prairie]

I am not getting the right answer

if x is a member of Z

sry

I mean if n or k is a member of Z

[uneven isle]

dont have to apologise

[heavy prairie]

then cos(pi/2 + 2pi n) should be 0?

[uneven isle]

yes that is right

[heavy prairie]

and sin(pi/2 + 2pi n) would be 1

[uneven isle]

yep

[heavy prairie]

yeah then i have 0 + i

[heavy prairie]

from this expression

wait

i^2

so -1

but uhh

that is not the right answer

[uneven isle]

oh i monkey brained

the trig expression isnt meant to be squared

just only the sqrt(6) is meant to be squared

should be
z^2 = (sqrt6)^2 * cis(pi/2 + 2kpi)

[heavy prairie]

okay

i will try to write that down

isn't (sqrt6)^2 just 6?

[uneven isle]

yeah it is but it is just for showing working out that u are setting up

[heavy prairie]

ok

is 6 just c or something

in the formula?

what is the formula?

[uneven isle]

there isn't really a formula , at least i was never taught one

i mean, u could generalise it to taking nth root as

z^n = r(e^(ix)), where x is some real such that r(e^(ix)) will represent ur RHS expression

[heavy prairie]

i am running into the same problem

where this isn't the answer

[uneven isle]

show ur working out

[heavy prairie]

first term equals 0

second is i

then when i square z

[uneven isle]

yes and ur lhs is still z^2

[heavy prairie]

i get sqrt(i)

that's not the answer

[uneven isle]

u need to apply De moivres theorem or use euler's form

and sqrt6* sqrt(i)is a root

[heavy prairie]

Okay I am still confused

z^2 = 6i

z^2 = 0 + 6i

where a = 0

and b = 6

then I get the modulus of z

which is the same as r?

I think

again I am confused so I don't know

[uneven isle]

z^2=6i = (sqrt6)^2 * (cis(pi/2 + 2kpi))
=>
z= sqrt6 * [cis(pi/2+2kpi)]^1/2
=>
z=sqrt6[ (cis(pi/4 + kpi)]

[heavy prairie]

That doesn't clear it up for me

[uneven isle]

dont use x+iy form

it isn't helpful here

[heavy prairie]

I am turning it from rectangular to polar

You can do it in your head

I can't

|z| = r = sqrt(0 + 6)

so now that explains to me how you arrived at sqrt6

Assuming that is right

[uneven isle]

mhm ok lets just start over, i will try my best to show not skip any steps

z^2 = 6i = 6 [ cis(pi/2 + 2kpi) ]

[heavy prairie]

De Moivres says z^n = r^n (cos(n theta) + isin(n theta))

[uneven isle]

if z = r cis theta yes

[heavy prairie]

bro this cis stuff

is just confusing to me

[uneven isle]

ok i will stop using it

[heavy prairie]

I understand it is just the shorthand

for (cos(n theta) + isin(n theta))

well I say I understand

but I don't know

I assume that is what cis means

but I don't know

[uneven isle]

cis theta is just cos(theta) + i sin(theta)

regardless, i wont use the shorthand

[heavy prairie]

going from here

|z| = r = sqrt(0 + 6)

is that true?

is |z| = r

[uneven isle]

yes that is true

[heavy prairie]

ok

so I put my sqrt6 into De Moivres

all my givens that I can figure out

z^2 = (sqrt6)^2 (cos(2 theta) + isin(2 theta))

then you said the angle was pi/2

I have no clue how you got there

[uneven isle]

i would avoid working out in your current order but i will xplain why in a few steps

for now, just to clarify

what demovire's is saying is just (cosx+ i(sinx))^n = cos(nx) + isin(nx)

[heavy prairie]

Can you summarise the order

in words

Your recommended order

[uneven isle]

let me wirte down on paper and i will just send an image, easier htis way

[heavy prairie]

Given a complex number in rectangular form

no wait in this case it isnt a complex number, it is a squared complex number

this is so confusing to me

[uneven isle]

@heavy prairie my handwriting isn't the neatest in the world so lmk if anything is too difficult to read

[heavy prairie]

i will read this in a sec

Arg(0 + 6i) is the angle right?

[lone heart]

@heavy prairie Has your question been resolved?

[lone heart]

@heavy prairie Has your question been resolved?

[heavy prairie]

<@&286206848099549185>

z^2 = 6i

goal is to find roots

[alpine sable]

ok

[heavy prairie]

I rewrite it as z^2 = 6i + 0

[alpine sable]

i know

[heavy prairie]

Brother

Actually

Nah nvm

[alpine sable]

use z = a+bi

[heavy prairie]

yes Daniel yes

[heavy prairie]

<@&286206848099549185>

$z^2 = 6i = 0 + 6i = a + bi$\\De Moivre's Theorem tells me that $z^n = r^n(cos({\theta}n) + isin({\theta}n)$\\I don't know Theta or r, but to find r I can:\$z^2 = 0 + 6i\Longrightarrow z = \sqrt{0 + 6i} \Longrightarrow |z| = r = |0 + 6i| = \sqrt{0^2 + 6^2} = 6$\\I'm not sure how to find Theta.

[ocean seal]

Neo

[warm musk]

You need to find the theta in the geometrical form of $z^2$

[ocean seal]

Lilly (NasaExploration)

[warm musk]

Aka you need to normalize the vector and identify your sinus and cosinus then solve

So when you normalize you have $i$ because the norm of $z^2$ is $6$

[ocean seal]

Lilly (NasaExploration)

[warm musk]

And then you will have $sin(2*\theta)=1$

[ocean seal]

Lilly (NasaExploration)

[warm musk]

Then solve

[heavy prairie]

I'm sorry but I can't follow that: I feel like I am lacking some really essential stuff, that, or people are just on a higher level of maths than I am at the moment.

I also feel like the language barrier is a problem because I had my whole math education in Swedish

I don't know what a geometrical form is

Should I just envision a square?

[warm musk]

Okay

So let’s start from there

[heavy prairie]

Yeah, would that be okay?

[warm musk]

Actually let me pull up a whole talk I did about that form

[warm musk]

Yeah of course

[warm musk]

Firstly we will aim to explain what the trigonometric form of a complex number is. We will built it from the algebraic form you know

As you mean know, complex number are a thing (the only building element you need to know is that i^2=-1 which is weird but not too much if you think about it) and under their algebraic representation it looks like

z=a+ib

With a the real part and b the imaginary part. Noted :

Re(z)=a | Im(z)=b

You can also calculate the module of it :

|z|=sqrt(a^2+b^2)

This is a positive number and it would represent the length of the segment drawn between the origin of the complex plane and the actual point. You can easily realize that :

-|z|<=Re(z)<=|z| | -|z|<=Im(z)<=|z|

And therefore :

-1<=(Re(z)/|z|)<=1 | -1<=(Im(z)/|z|)<=1

What handy tool do we have to describe number between -1 and 1 ? We have trigonometric functions to do so. Therefore you can prove (here proving it is useless and likely out of your reach) that it exist a unique number theta in between 0 and 2 pi such as :

Re(z)/|z|=cos(theta) | Im(z)/|z|=sin(theta)

And this angle theta represent the angle between the segment drawn between the origin of the complex plane and the point AND the real axis (pointed toward the positive increment). Here I can’t really explain why as you don’t have the proof of it, it does explain a lot.

So you can write :

z/|z| = Re(z)/|z| + Im(z)/|z|i= cos(theta) + sin(theta)i

And us mathematicians can realize that writing e the complex exponential function such as :

z/|z| = e^(i times theta)

Makes a lot of sense as you can realize that it works nearly the same way as the exponential function. Every single property of it applies to this view of complex numbers (except a few exceptions). Here I won’t go into much detail as it’s just a bunch of trigonometric properties you need to demonstrate in a very painful way.

So you have to differentiate the classic and the complex exponential function. It’s for their relative closeness that we kept the same notation.

So now it’s as trivial as multiplying by |z| to find the trigonometric form of complex numbers. It’s called trigonometric cuz at its core there is trigonometric functions. So :

z = |z|e^(i times theta)

Please note that this definition don’t work if z=0 as you would multiply (and divide when building it) by 0 and therefore don’t guarantee the uniqueness of theta. So the trigonometric form of 0 is just 0.

There a whole talk about it

[heavy prairie]

Is that euler's formula?

At the end

z = |z|e^(i times theta)

[warm musk]

We call it trigonometric form here because it uses trigonometric function

[heavy prairie]

Okay I was confused about that

But I could replace trigonometric

with geometric

here?

So I get the geometric form of my complex number by rewriting it using euler's formula?

My complex number being 0 + 6i

[warm musk]

Yeah and you aim to find theta

[heavy prairie]

ln(z) = iTheta * ln(|z|)

ln(z)/(i*ln(|z|)) = Theta

[warm musk]

No no don’t use ln

[heavy prairie]

no

okay

[warm musk]

You can’t use it there

[heavy prairie]

yes ok i uhh dont know how to do this then

[warm musk]

You have to develop it

With the sin and cos

Okay so like

[heavy prairie]

develop it ehhrrr

yeah explain if you can

[warm musk]

In here you found r right ?

[heavy prairie]

Assuming I did everything correctly I would say so

Looking at it, it still looks correct

So I would say I found r

Yes

[warm musk]

What value did you found ?

[heavy prairie]

6

oh yeah

[warm musk]

Okay nearly there

[heavy prairie]

and r is |z|

i forgor

[warm musk]

That r^2

So you need to square it

[heavy prairie]

no way

[warm musk]

I mean root square it

[heavy prairie]

6 isn't r?

[warm musk]

In your formula you have r^n which is r^2 in your case

[heavy prairie]

I never ended up using that formula

I just want to get to use that formula

Oh wait

Oh you mean

Now I use that formula

and insert 6 into it

r is 6

but r^2 = 36

typo

[warm musk]

Okay @heavy prairie I am sorry but I am not really free right now

Do you think we could talk about it in like 1 to 2 hours ?

I will be able to help you more cuz I would want to talk about it from scratch instead of losing you like that

Either here of DM I don’t mind both

[heavy prairie]

I don't mind either way and yeah it can wait

Don't feel forced to help though, only when you have time and want to

I will close this in the meantime

.close

[lone heart]

Channel closed

Closed by @heavy prairie

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[lone heart]

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[keen glade]

How to solve for L ⁉️

[cursive badger]

It means, solve for l, in terms of w and P

[keen glade]

Say what now

[cursive badger]

I mean, try to bring l on one side of the equation

[keen glade]

Okay so now I got -2W +P = 2L
———————
2. 2

Would I make the 2 cancel out the 2W?

Or just leave as -2W + P/2

[cursive badger]

Yes, divide both sides by 2. Because we only want the right hand side to be in terms of l only.

[keen glade]

Okay so thoughts on -W + P = L

[cursive badger]

How?

[keen glade]

What

[cursive badger]

I mean, if you are dividing both sides by 2, then the equation should be:
$-W+\frac{P}{2}=L$

[ocean seal]

moaforlife

[cursive badger]

You have to divide P by 2 too

[keen glade]

Oh

Could you help with another one 😔

[cursive badger]

Yeah

Well that's what I am here for...

OK, WAIT!

[cursive badger]

LET ME GIVE YOU THE INSIGHT

DO YOU KNOW WHAT EQUATION MEAN?

[keen glade]

Uh probably

It's when it has =

[cursive badger]

It means that there are 2 things which have the same value...

[keen glade]

And has a variable

[cursive badger]

And when we do any operation, we have to apply it to the whole...

Forexample

If we divided the right side by 2

then we have to divide the whole left side by 2

[keen glade]

I know but I forgot 😔

[cursive badger]

So that it remains an equation...

[cursive badger]

Oh ok ok

[keen glade]

So on the coffee question I have T = 5.50P +7F

What's it mean solve for F

And what would the constraints be

[cursive badger]

Solve for f, it again means, bring f to one side of the equation...

Like f=smt

And you have to manipulate what that smt is gonna be

[keen glade]

Yeah but i would have T -5.50P = 7F

[cursive badger]

Yeah..

THEN?

you know what to do next...

[keen glade]

Divide

[cursive badger]

Yep

[keen glade]

But like

[cursive badger]

How?/

[cursive badger]

what?

[keen glade]

Divide by 7 on both sides

[cursive badger]

yep

[keen glade]

When it asks me to solve for F does it want me to have the $40 still

From a

[cursive badger]

Nope

Otherwise, they would have mentioned it

[keen glade]

Oh alright I just thought it would use more numbers

And then for the constraints. Does it just mean that like he has to choose what he wants more of?

[cursive badger]

rule: If they have not mentioned anything, you should never assume smt on own.

[keen glade]

Also thank you for the help

[cursive badger]

Yeah no problem

[cursive badger]

wait..

So, by constraints, probably they mean, what can be the maximum and minimum possible value of p and f, when T=40.

[keen glade]

Oh so like
"The max amount of coffee they can get is _"

[cursive badger]

Actually not really...

[keen glade]

Is there any special way to solve the maximum value of P and F or do I just gotta mental math it kinda

[cursive badger]

If you read the question again, p is the cost of plain coffee and f is the cost of flavoured coffee

So, kind of:

"the maximum and the minimum value of both coffees that she/he can afford"

[keen glade]

So she can afford 5 flavored or like 7 plain

Like that?

Alright I gotta go, thanks again

.close

[lone heart]

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[cursive badger]

Yeah you can simply make inequality by putting one 0.

My network is so bad... sorry I couldn't reply before...

[lone heart]

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[alpine sorrel]

can sm1 solve this

[alpine sorrel]

<@&286206848099549185>

[long axle]

why dont u just do it

[alpine sorrel]

im stuck while adding the equations

[long axle]

show ur work

[alpine sorrel]

im in laptop i cant show

let me explain wait

[alpine sorrel]

first i squared the 1ST equation then did the same with second

subtracted both

after that i got 4tanAcosA = m^2 - n^2

now i cant proceed

please help

[gritty bramble]

thats value of m^2 - n^2

now plug values of m and n in the rhs part (4sqrt(mn))

you'll get lhs = rhs

[cursive badger]

yep

[long axle]

m^2 - n^2 = (tanA + sinA + tanA - sinA)(tanA + sinA - (tanA - sinA)) = (2tanA)(2sinA) = 4tanAsinA

[long axle]

u did something wrong in ur algebra

[gritty bramble]

also yes, it should be sinA not cosA

[alpine sorrel]

yea sorry my bad

wait let me do that as you told

yea i got it

ty @gritty bramble@long axle

[gritty bramble]

[alpine sorrel]

.close

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[lost abyss]

How do i simplify this summation?

[thorn monolith]

Hmm i dont see how to simplify it but seems like you can split it to two sigma sums

[alpine sable]

2 power n is summation of ncr coeff you can use that

and simplify

i did it

[alpine sable]

here you go

tell me if you dont understand anything

[lone heart]

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[lethal glacier]

Is this english or math 😕

Doesnt matter, put 77 outside

[gritty bramble]

how are you "helper" (your role)

[lone heart]

@full badge Has your question been resolved?

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[austere shell]

help zero

wow

[austere shell]

think i got part of it now sigh.

I was confused on this for a very long time regarding how this is equal, my new understanding is that no it isnt equal.

but that c/a is still apart of the equation even if it is removed(negative)

and that because of that when inputted into the quadratic formula it would undo the negative and give the result for the normal equation.

is this correct?

[fresh wedge]

Normal quadratic formula is
ax²+bx+c=0
Now if you divide everything by a you will have
x²+(b/a)x+c/a=0
Subtracting c/a both side will give you
x²+(b/a)x=-(c/a)

Where are you confused?

[austere shell]

they are deriving it btw

[austere shell]

nowhere shown

well sort of

i understand how they did this not how its possible

people tell me this is somehow equal to the original polynomial

but if polynomials are a bunch of components that make up the behavior of a parabola

wouldnt removing c/a alter that behavior

yes -c/a is still there on the other side but its negative

would that not then alter the behavior causing the parabola to be more flat or go in a negative direction since negative value was subtracted from the answer?

or by equal do they just mean when you solve it out via the quadratic formula it would become equal afterwords because the formula would undo it and generate the original quadratics answer?

[fresh wedge]

Ummm why are you jumping in between parabola and quadratic equations?

Yes if you alter anything in parabola equation

It will change the graphs

[austere shell]

are they not the same?

[fresh wedge]

But solving a quadratic equation is a different topic than seeing parabola, graphs finding major minor axes etc

[fresh wedge]

You can't say exactly parabola and quadratic equation is same. Quadratic equation is a type of equation, parabola, hyperbola, circle are well...

[austere shell]

what are major and minor and how are they different dont you need the zeroes to find a parabola?

[fresh wedge]

Idk the exact term to define them

The general formula for a parabola is y = a(x-h)² + k or x = a(y-k)² + h, where (h,k) represents the vertex

ax²+bx+c=0```

If you consider y=0, then parabola equation will change into a quadratic equation yes

Or x=0 for 2nd parabola formula

[austere shell]

define Y

also define H and K

(their representations and meanings)

when graphing this what values would be assigned to X and what to Y (axis)?

[lone heart]

@austere shell Has your question been resolved?

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[tight thunder]

Guys this matrix would be

2-parameter solution?

i often read 1 parameter solution

2 parameter solution

but dont know what mean

it means this?

because of the 4th coloumn having unknowns which are not on the main diagnal?

.close

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[alpine sable]

Functions f and g are defined from A to A, where A is a non-empty set.
If f(g(x) function is one-to-one which of the following is absolutely correct?
A)f is surjective B)g is surjective C)f is one-to-one D)g is one-to-one E)g(f(x)) is one-to-one

A video solution says:
If f(g(x1))≠f(g(x2)) then g(x1)≠g(x2) so g is one-to-one
But I didn't understand how other options can be false

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

<@&286206848099549185>

[valid trout]

Based on the fact that it's just one-to-one you can't conclude it's surjective

Because it doesn't necessarily mean that every element of A is covered

So that's A and B gone

I am not sure I fully agree with the solutions

We can say that g(x1) = y1 and g(x2) = y2 and assuming that f(y1) is different from f(y2) it means that y1 is different from y2

Meaning that f is one-to-one

However with this information we don't know anything about g and thus we can't conclude E is true either

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

But f(g(x)) is one-to-one from A to A, so shouldn't it be surjective?

[lone heart]

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[woven plaza]

Is this proof correct?

[lone heart]

@woven plaza Has your question been resolved?

[woven plaza]

<@&286206848099549185>

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@woven plaza Has your question been resolved?

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[waxen turtle]

idk what classifies as each

someone plz tell

[lone heart]

@waxen turtle Has your question been resolved?

[waxen turtle]

<@&286206848099549185>

[potent owl]

Is it 3-6 or all problems

[waxen turtle]

prob all idk i haven tooled at the others

[potent owl]

Do you understand the difference between jump and infinite

[waxen turtle]

idek wat jump is

[solar narwhal]

,w plot floor(x)

[ocean seal]

Results provided by WolframAlpha

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Requested by rishabh_nirwan

[solar narwhal]

This is a example where it jumps

You can see it's constant for some time then it suddenly jumps upto 1 at 1

[waxen turtle]

Ohhh

idek how to find that

is that like piecewise

[solar narwhal]

Do you know about limits?

[waxen turtle]

yes

[solar narwhal]

Then take the Right hand limit and the left hand limit

If they're different then there exists a jump on that point

[waxen turtle]

wat is infinite

and wat is removable and non removable discontinouty

i forgor

[solar narwhal]

Infinity is when the function shoots upto infinity

,w plot absolute value of 1/x

Here at x=0

Function shoots upto infinity

[ocean seal]

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[waxen turtle]

am I doing like limits or sum idk it doesn't say lim

[solar narwhal]

A removable discontinuity is when the RHL and LHL are equal and there's no jump

BUT

The value of the function is not defined or different than limit

[waxen turtle]

wat does that mean

wats a rhl

and lhl

[solar narwhal]

Right hand limit

[solar narwhal]

Left hand

[waxen turtle]

o

ok the first one

i have no idea

do I test points around 3

idk

[solar narwhal]

I'll help 1st one then you try others

[solar narwhal]

.

[waxen turtle]

1/(x-3)^2

idk

[solar narwhal]

Yeah, since we see that the denominator can be zero we check there

[waxen turtle]

x=3 is prob important

[solar narwhal]

Take the RHL and LHL at 3

[waxen turtle]

right hand limit it goes to infinity

[solar narwhal]

Yeah

[waxen turtle]

si does left

so

[solar narwhal]

It's infinite

Do you understand why?

[waxen turtle]

Ya

wat if one goes to neg infinity

or both

wat would it be theb

[solar narwhal]

That'll be infinite I guess

[solar narwhal]

Then the limit doesn't exist

There's a jump or more accurately a asymptote

[waxen turtle]

do I need anything else for 1

or do I just write infinity

[solar narwhal]

Write infinite at x = 3

[waxen turtle]

is that it

[solar narwhal]

Yeah

[waxen turtle]

is it a removable discontinuing or no

cuz both go to inf

same

[solar narwhal]

I think it's neither

[waxen turtle]

wait wat is a normal disconiouty

idrk anything Abt continuity tbh

[solar narwhal]

Removable would mean that we can define the function to equal the RHL and LHL so that it becomes continuous

[solar narwhal]

It means that there are no abrupt value changes

Or there are no 'holes'

[waxen turtle]

is this continuous

[solar narwhal]

No this isn't

It's a example of discontinuity at every integer

[waxen turtle]

okkkkkk

[solar narwhal]

Can you try the second one?

[waxen turtle]

okk

remov disc on 4 right

[solar narwhal]

Lemme see

[solar narwhal]

Yeah

[solar narwhal]

No

No

It's continuous on

4

[waxen turtle]

ruh roh

i think i sold on my qui

quiz

on this

today

💀

i def said something like that was a remov disc

[solar narwhal]

I think you'll be fine

[waxen turtle]

😭

i took it today

earlier

anyways

[solar narwhal]

Actually it maybe

I don't think my brain is functioning properly

I just woke up

[solar narwhal]

You'll be fine

Anyways

There's another point you'll have to check

[waxen turtle]

is it a remov disc or wat

[solar narwhal]

Idk gimme a sec so you find about the other point in this

There are 2 points

[waxen turtle]

heudhshsh

[solar narwhal]

Where we check

[waxen turtle]

5

[solar narwhal]

I think it's remov discontinuity

[solar narwhal]

Yeha

[solar narwhal]

You were right..

[waxen turtle]

light work

i never lose

[solar narwhal]

Check on 5

[waxen turtle]

ok from pos direction it's inf

neg inf from neg

[solar narwhal]

Yeah

So it's non removable discontinuity since

They're different

[waxen turtle]

okii

[solar narwhal]

Try third

[waxen turtle]

OK

idk

[solar narwhal]

Try to make it a piece wise function

[waxen turtle]

idk how

like

idk

idek how to solve absolute value functions

I was never taught

[solar narwhal]

Look at the zeros of the function inside Abs Value

Those are the points we care

Recall

That Abs Value of x is x itself for x>0

And -x for x<0

[waxen turtle]

help

[solar narwhal]

So solve for when the function is positive or negative

[solar narwhal]

?

[waxen turtle]

so like

-x-2

idk

[solar narwhal]

Abs(x) = { x for x>=0,
-x for x<0

So

[waxen turtle]

wat does abs mean

[solar narwhal]

Abs( x+ 2 ) = { x+2 for x+2 >=0
-(x+2) for x+2<0

[solar narwhal]

Absolute value

[waxen turtle]

do I make a version where x is neg

like -x+2

[solar narwhal]

..

[waxen turtle]

oh the whole thing

[solar narwhal]

Solve the inequality in there

[waxen turtle]

do I include the bottom

[solar narwhal]

That'll be the condition on x for the function to be x+2 or -(x+2)

[waxen turtle]

or liek forget it

[solar narwhal]

Yeah include it?

[waxen turtle]

like

nvm

[waxen turtle]

uhhh

so 4 and -4

idk

[solar narwhal]

Ask

[solar narwhal]

No

[waxen turtle]

😮‍💨

[solar narwhal]

The inequality is x+2<0

[waxen turtle]

OH

[solar narwhal]

Subtract 2 from both sdes

[waxen turtle]

WAIT

i thought it was

[solar narwhal]

Do the other one similarly

[waxen turtle]

x+2>=0

[solar narwhal]

Yeah that's the other one

Both are pieces of the same function

[waxen turtle]

I'm uhh

i got

x >= 2

and x>2

😭

[solar narwhal]

-2

[solar narwhal]

-2

[waxen turtle]

oh

[solar narwhal]

You subtract 2 from both sides

So -2

[waxen turtle]

-x-2<x

-x<2

x>-2

[solar narwhal]

Yeah

[waxen turtle]

but they r like the same thing

[solar narwhal]

Do the other one too

[solar narwhal]

No?

[waxen turtle]

wdym i just did both

[solar narwhal]

.

[solar narwhal]

Wait how'd you get this

[waxen turtle]

the second line

[solar narwhal]

No

The inequalities are

x+2 >=0

And

x+2<0

[waxen turtle]

oh

okkk

[solar narwhal]

Solve those

[waxen turtle]

x<-2

so like that includes everything

Fr

[solar narwhal]

Yeah

[solar narwhal]

Everything less than -2

[waxen turtle]

plus the other thingy

it's everything fr

[solar narwhal]

Yeah

[waxen turtle]

oki

now wat

[solar narwhal]

Now write the function as piecewise

[waxen turtle]

idk how

[solar narwhal]

For x less than -2

Abs(x+2) = -(x+2)

So open the initial function like this

[waxen turtle]

IDK

[solar narwhal]

What's the function

[waxen turtle]

-x-2 for x<-2

[solar narwhal]

the initial function

[waxen turtle]

and x+2 for x≥2

is that the piecewise

[solar narwhal]

That's piecewise but we want the piece wise of the initial function

So I ask you again

What's the initial function

[waxen turtle]

omg

do I divide both of those by x+2

[solar narwhal]

Yeah

[waxen turtle]

EZ

[solar narwhal]

Now what's the resulting piecewise function

[waxen turtle]

OK

1 for x ≥ -2

-1 for x < -2

[solar narwhal]

Yeah

Take limit at 2

[waxen turtle]

1

[solar narwhal]

RHL and LHL

[waxen turtle]

don't u just plug it in

[solar narwhal]

No

[waxen turtle]

it's greater than -2

so it's 1

[solar narwhal]

Take limit

[waxen turtle]

how

[solar narwhal]

RHL and LHL

[waxen turtle]

it's a straight line

at 2

[solar narwhal]

How have you been taking limits till now?

[waxen turtle]

liek a month

[solar narwhal]

Bro just take the RHL and LHL they're different

[waxen turtle]

ISNT IT A STRAIGHT LINE???

[solar narwhal]

IT'S NOT

[waxen turtle]

at 2.1 it's 1

and 1.9 it's 1

???

[solar narwhal]

-1

[waxen turtle]

how i

it's not less than -2

so it doesn't count for that one

[solar narwhal]

Ohh

-2 please

[waxen turtle]

.

ok

it's a jump

[solar narwhal]

.

[solar narwhal]

Yeah

[waxen turtle]

jump at x=-2

right

[solar narwhal]

Yeah

[waxen turtle]

it's non remov right

idk

[solar narwhal]

Try the next one

[solar narwhal]

Yeah

[waxen turtle]

i have to go tho

sooo

rip

😔

.close

[lone heart]

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[trail imp]

I wanted to know which one of the steps is wrong here. The correct answer is 1/6.

[ocean whale]

,rotate

[ocean whale]

Are you using L'hopital?

[trail imp]

Yes, in the first step

[ocean whale]

Your derivative of the numerator looks wrong

[trail imp]

I used sinx/x = 1 to delete the sin from sin(sin x)

[ocean whale]

I'm saying your derivative of cos * sin - cos is not correct

[trail imp]

What would be the correct derivative?

[ocean whale]

Well, you'd use product rule for cos * sin, correct?

[trail imp]

But it is not cos * sin, the angle of cos is sin x

[ocean whale]

Oh my bad

I misread that

[trail imp]

Np

[ocean whale]

That derivative is still incorrect, you'd use chain rule

Wait

I just can't read

[trail imp]

That just made me laugh

This solution was given for this question.

Oh, got it

Sin(sinx)/sinx wouldnt equal to 1 bcoz it is x tending to 0, not sin x tending to 0

But isn't it the same thing?

[lone heart]

@trail imp Has your question been resolved?

[lone heart]

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[unreal pawn]

Who could graph the quadratical function: f(x) = -(x+3)^2+4?

[trail imp]

Do you know what shape do quadratic equations make?

[lone heart]

@unreal pawn Has your question been resolved?

[unreal pawn]

yes, a parabola. Correct?

[trail imp]

Exactly

So, what is the standard equation for a parabola?

[unreal pawn]

f(x) = a(x+p)^2+q

-p I mean

[trail imp]

Yep

So, when it's -a, it just means that it's an upside down parabola

And you can substitute x values to find the corresponding y value to graph the parabola

Tho, there's a really great website for it. Search up desmos graphing calculator

[unreal pawn]

i am asked to graph it, using the given formula. I did use graphing calculators, its just kinda unclear to me why it resulted in a -1 x-intercept

[trail imp]

Oh, you can just substitute the value of y = 0 and find the x intercept

[unreal pawn]

wdm

[trail imp]

Here, put y= 0 as on x-intercept y= 0, right ? So, then you would get a quadratic and then you gotta solve that to find the values of x-intercepts

By y, I mean f(x)

Function of x is 0 on x intercept

I am so used to throwing around y's in place of f(x) to make the question easier, I didn't realize that it's not even a variable here

[lone heart]

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[small badge]

Need Linear Algebra help; I've tried this homework problem a few different ways and still getting it wrong.

[hushed locust]

i would try reversing s and t

[small badge]

wouldn't s represent x3 and t represent x4?

[unreal pawn]

thx alot

[hushed locust]

if we have that r represents x5 then it seems like it would make sense for s to represent x4 and t to represent x3

[small badge]

hmm didn't work but I'll ask my professor tomorrow morning, thank you though!

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[trail imp]

My answer here seems to be wrong. The correct one is 1/6. Where is the mistake in the solution?

[unborn plover]

,rccw

[gritty bramble]

how

[trail imp]

I used sinx/x =1

[modern kernel]

sin(sin x) became sin x

[modern kernel]

In your work

[gritty bramble]

how

you cant do that

[modern kernel]

How is that supposed to work

[trail imp]

why?

[gritty bramble]

i mean

firstly

where did u get sinx/x from

u didnt split the denominator

even if u do

its sin(sinx)

not just sinx

[trail imp]

well, I have sin(sinx). so, I multiplied and divided by sinx and got sinx

[gritty bramble]

what

[modern kernel]

That's not right

[gritty bramble]

in your work, sin(sinx) became sinx out of nowhere

if u diviied by sinx, wheres the sinx in denominator?

also the x you took out of 4x^3, its still 4x^3 instead of 4x^2

in short its all wrong

[trail imp]

no, like sin(sinx)*sinx/ sinx

so, sin(sinx)/sinx would become 1

and sinx would remain

[gritty bramble]

oh

but the problem is

you dont put limit in between like that

its affecting the whole function here

the numerator has another term in addition

you cant do what you are trying to do

[trail imp]

but dont limits have additive propertry?

[gritty bramble]

what

[trail imp]

like if you have lim x tends to 0 (x+y), you can put it as lim x tends to 0 x and lim x tends to 0 y

[gritty bramble]

(x+1)x^2, if i multiply and divide by x, as in x*x/x, then cancel out the x^2 in numerator and denominator, this just changes the whole function

infact all these ^ steps are wrong

[trail imp]

i see. then, do I have to lhopital all the way or is there any other way I can do this question?

[gritty bramble]

except you didnt split or anything

u just multiplied by sinx/sinx (that too wrong)

if u have (x+1) and you multiply by x then the its x(x+1) not x^2 +1

[trail imp]

yeah, i splitted and recombined them.

[gritty bramble]

if u split them already then you dont combine them back

[trail imp]

but why?

[gritty bramble]

because you changed the split part

it just changes the whole function completely

[trail imp]

yeah but the denominator and the limit is still the same

[gritty bramble]

but the function is changed

[trail imp]

yeah, but according to the addtive property why does that matter

if the limits are same, I can add them

my basics do suck real bad so, I really need to clear these doubts

this was the solution provided for the question

[gritty bramble]

yes

because using lhopital it will get complex

since you will have to keep differentiating until the indeterminate form goes away (so 4 times)

[gritty bramble]

ok lets say you are right, see these 2 functions