help-10

[hearty terrace]

yes sir

if x is a positive number then the absolute value sign doesnt matter

does that make sense

[blazing vale]

so like

so should i divide

72 by 6

and then i get

[hearty terrace]

yea

12

so that's ONE solution because 6*|12|=72

so one solution is $x\ge12$

[warm shale]

Vѳrtєx-

[blazing vale]

-12>|x| > or equal to 12

yea

so i can combine them together

and get

[hearty terrace]

you cant combine them like that

[blazing vale]

oh

[hearty terrace]

because -12>|x| is not correct

[blazing vale]

do i have to use a

and or

[hearty terrace]

you have to write it as two separate inequalities

[blazing vale]

statement

[hearty terrace]

yeah OR

x>12 OR x<-12

[blazing vale]

OKAY

[hearty terrace]

=

[blazing vale]

oops my fault

[hearty terrace]

and <= sorry

[blazing vale]

how do you distinguish

if its a

and

statement

or an OR statement

[hearty terrace]

well let's take this for example

you cant have a number be >4 and <-4

so this is an OR statement

[blazing vale]

yea

OHHH

SO IF

IT WAS

[hearty terrace]

x can either be >4 OR <-4

[blazing vale]

NOT SEPERATED

SAY LIKE

ONE BLUE LINE

[hearty terrace]

yes sir

[blazing vale]

IT WOULD BE A AND????

[hearty terrace]

yea

[blazing vale]

omg

word 2

[hearty terrace]

more or less

[blazing vale]

thanks mayne

that make sso muhc sense

[hearty terrace]

like -4<x<4

x is BETWEEN -4 and 4

in other words

-4<x AND x<4

[obtuse pebble]

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[tranquil sonnet]

Correct

[wooden cipher]

youre on desmos, just write what they give you

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[alpine stump]

Is this legal

[alpine stump]

Jus wanna know

[quaint glen]

Yeah! This is called differentiating under the integral/Leibniz rule. You can read more here https://brilliant.org/wiki/differentiate-through-the-integral/

They have an example thats almost identical to this problem as well

[alpine stump]

Ok thanks

Jus wanted to know if its possible by adding the alpha parameter as an exponent

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[fallen glade]

hello, I'm taking differential equations and it has been a while since I've taken Calc 3 and done stuff with partial derivatives. can someone explain how they arrived at $\frac{\delta}{\delta y}(1-\sin x)...$

[warm shale]

PolloTundra | Aidan B

[stoic yacht]

are you asking why they took the partial derivative?

or why the partial derivative of (1 - ysinx) wrt y is -sinx?

[fallen glade]

why the partial derivative of (1 - ysinx) wrt y is -sinx?

[stoic yacht]

when you take the partial derivative of a multivariable function with respect to just one of the variables, you hold all other variables constant

so in this case, 1 - ysinx is a function of both x and y

and so since you're diff wrt y, you'd consider x to be constant when differentiating

clearly 1 is already constant so its derivative is 0

now when differentiating -ysinx, it looks like something you might want to product rule

but since you're doing it wrt y, x is constant, so sinx is a constant

so technically -ysinx is no different than something like -2y

and you know the derivative of that is just -2

similarly, d/dy (-ysinx) = -sinx

[fallen glade]

gotcha, so why is it the partial derivative wrt y when it's dx and vice versa?

[stoic yacht]

that has more to do with the properties satisfied by exact differential equations

if you have a differential equation

Mdx + Ndy = 0

then it must satisfy d/dy(M) = d/dx(N) to be exact

[fallen glade]

ah ok that makes sense then, thank you!

need to add that to my notes

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[solemn isle]

Im having a stroke reading this question

[solemn isle]

Is it possible you can explain what is being asked

And how a cubic function is associated with a 3D shape

[wooden cipher]

so you start with a square

then you take out square corners, so you end up with a smaller square and flaps

you can fold those flaps up into a box shape without a lid

[solemn isle]

Ok i get that part so far

[wooden cipher]

to make sure the flaps match up, both sides need to be x long

both cuts*

[solemn isle]

So perfect square for each cut out

[wooden cipher]

yes

[solemn isle]

Alr

[wooden cipher]

so then you fold it to be x high

the base lengths you solve for

and you get a cubic

[solemn isle]

How do you get a cubic

Isnt a cubic function not straight

Also wouldn't the answer automatically be A or C since they have the least amount cut out for the corner cubes

I dont get how the cubic function or less than whatever crap relates

Pls ping when you reply, im typing on my phone, so i gotta get back to work

<@&286206848099549185>

Bruh

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[unkempt lotus]

Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.

f(x) = 1/x, 1 < x < 3

[unkempt lotus]

i have no idea how to do this

do i need to get the derivative first?

[keen quarry]

yes

you need to find the critical points

[unkempt lotus]

how do you get the derivative of 1/x?

[keen quarry]

1/x = x^-1

[supple river]

And just use the power rule

[unkempt lotus]

how

ohhh

ha

-x^-2

[keen quarry]

yep

[unkempt lotus]

So do I need to draw the original graph? it looks like you can’t sketch 1/X here

[keen quarry]

you can

not a lot of it, but part of 1/x does fit in that box

[unkempt lotus]

ok so like th eupper half of it then

[unkempt lotus]

but what should i do now , get the critical nimbers by making f'(x) = 0?

[keen quarry]

yep

[unkempt lotus]

so $-x^{-2} = 0$ ?

[warm shale]

wakamole

[keen quarry]

yes

[unkempt lotus]

well thats just 0

[keen quarry]

no, cuz x^-2 = 1/x^2

[unkempt lotus]

so it's -1

waity

so it's $-1/x^2 = 0$

[warm shale]

wakamole

[keen quarry]

yes

[unkempt lotus]

then

$-1 = x^2$

[warm shale]

wakamole

[keen quarry]

no, cuz anything times 0 = 0

[unkempt lotus]

wiatwut

[keen quarry]

this equation has no solution

which in turn means that 1/x has no maximums or minimums

[unkempt lotus]

wait so to solve the equation wouldnt it be $sqrt(-1) = x$?

[warm shale]

wakamole

[unkempt lotus]

which isd imaginary

[keen quarry]

no, cuz in order to get to that step, you have to multiply 0 by x^2

[unkempt lotus]

ohhhh

[keen quarry]

which wouldn’t give x^2

[unkempt lotus]

wait but my book is saying theres an answer for a similar problem though

[keen quarry]

which one?

[unkempt lotus]

17

[keen quarry]

what did it give as the answer

[keen quarry]

ohhhh wait im stupid

it’s on a closed interval

so, we didn’t find any critical points, but we still have to check the endpoints

[unkempt lotus]

oh so since 17 is closed then it has ac answer

and since mine is open it doesnt?

[keen quarry]

no yours is also closed

[unkempt lotus]

well 18 is open

[keen quarry]

i simply forgot

[unkempt lotus]

no

18 is open

mine is open

1 < x < 3

not 1 <= x <= 3

[keen quarry]

no i meant like an unbounded interval

yours is bounded

[unkempt lotus]

huh

what u mean

[keen quarry]

cuz it’s not just -inf to inf

but you are right about it not having a solution cuz it’s open

[unkempt lotus]

it looks like there is none still

ok ya

ha

if it was closed then it would have absolute i guess

even though i thought it would have local

i thought closed have local not absolute

[keen quarry]

absolute on the interval

[unkempt lotus]

but the answer to the last one has absolute

absolute on the interval as opposed to something like [-1, 2] ?

which would have a local?

[keen quarry]

no, the absolute max/min is just the highest/lowest point on an interval

if ur not given an interval, then you assume the interval is the domain of the function

[unkempt lotus]

yea but it doesnt include endpoints

or does it

[keen quarry]

it has to

[unkempt lotus]

im getting confused if abs max or local max includes endpoints

ohhh right

absolute includes endpoints local doesnt?

[keen quarry]

no, both can include endpoints

absolute or local is only decided by if it’s the largest one or not

[unkempt lotus]

ok

so

so why is 17 = -1?

why is question 17's answer -1?

[keen quarry]

it’s 1 not negative 1

and that’s cuz the endpoint ended up being the highest point on the interval

plugging in the endpoint into 1/x gives you 1/1 = 1

[unkempt lotus]

ohh

wow ok

i hope i dont fail my exam

seems like idk wtf im talking abt

[keen quarry]

trust you got it

[unkempt lotus]

yeah thanks

[keen quarry]

you have a decent idea, you just need more help putting the pieces together

[unkempt lotus]

for the answers to this i can write no abs/loca; max/min.. right?

yeah

and my TA's arent helping me at all

[keen quarry]

really depends on ur prof

i always like to go extra just to make sure i get the point

[unkempt lotus]

i go to tuoring services 1ce a week plus TA hours

i just wana make sure i got the point to this

[keen quarry]

what unit are you guys doing

[unkempt lotus]

since we derived f(x) and tried to make it equal to 0 we found it is undef.

and since the interval is closed then it means there is no local or absolute min/max

[keen quarry]

i like to write the equation, then x = DNE, then “no max/min”

[unkempt lotus]

i have to fill this out

so DNE for all of them then..

[keen quarry]

yep

[unkempt lotus]

great ty

only 5 more pages of this.....

[keen quarry]

make sure to include the empty circles on the endpoints

for the open interval

[unkempt lotus]

i thought it goes on forever

[unkempt lotus]

ohhh

true

thx

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[chilly basin]

Very simple question but one I was wondering about. If a process is working at 150% efficiency and it takes 1s for the process to complete, 1/1.5=0.6666666666666667 (I usually write this as ≈0.66), what unit would you write that as? 666ms? 0.66s? 0.66th of a second? Or would you simply just put ≈0.66 without any unit?

[eager niche]

This is entirely context dependent. If this is industry/academia try to be consistent with other reporting, and there’s probably a firm/institution/journal etc. style guide. Also you might want to look into sig figs

And if it’s for a class ask your instructor

[chilly basin]

Okay so, didn't know what sig figs were until now :o

What exactly am I suppose to do the significant figure of 0.6666666666666667 (16 significant figures)?

Like, I don't know what to do with the information :o

[obtuse pebble]

@chilly basin Has your question been resolved?

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[obtuse thistle]

Doing systems of linear equations and am so confused on how I should start this problem. I dont know how to set up the equations for this.

[forest sinew]

so, you might have something like this

lets work on club A

lets say that the fraction of students in A next year is like this

$A_{n+1} = F(A_n, B_n, C_n)$

[warm shale]

jan Niku

[forest sinew]

where F is linear

can you find F from the given information?

[obtuse thistle]

ngl i havent seen that format before. Our teacher has only gone over like simple equations that we will turn into matrixes and solve

[forest sinew]

can you turn this into a matrix?

the given information i mean

[obtuse thistle]

like the problem itself or the equations u just showed

[forest sinew]

what you posted

i forget the name for these matrices

[obtuse thistle]

ya if you turn it into equations, i just dont know how for this one

[obtuse thistle]

augmented?

[forest sinew]

they describe flow on a directed graph

maybe its markov

anyways

lets say $D = \mqty( A \ B \ C)$

[warm shale]

jan Niku

[obtuse thistle]

let me show you what i did for a past problem so you can see how she wants us to do it

[forest sinew]

so the fraction for each

you could say if theres no one changes clubs

[obtuse thistle]

i didnt post the actual question but this is the general way she wants us to approach these problems

[forest sinew]

sure

well

you have a 3 by 3 system here

[obtuse thistle]

ya

[forest sinew]

so a 3x3 matrix

[obtuse thistle]

yes but i need three equations for that, which i dont know

[forest sinew]

so think of writing $MD_n = D_{n+1}$

[warm shale]

jan Niku

[forest sinew]

were M is a 3x3 matrix

say M was the identity

what would happen

how would D change from n to n+1

[obtuse thistle]

bro ngl i dont even know what u mean by identity, neither have i ever seen that format u just showed

[forest sinew]

okay its like this

lets say theres 2 clubs a and b

[obtuse thistle]

ok

[forest sinew]

half of the people in a go to b

half stay

and no one in b moves

im gonna use d = (a b)

a is proportion of ppl in a

blah blah

seem okay?

[obtuse thistle]

i see

[forest sinew]

so i can write this out

[obtuse thistle]

so this means half of a went to b

[forest sinew]

this is just a vector

here gimme a seci

im on mobine

mobile

lol

[obtuse thistle]

gotchu take ur time

[forest sinew]

$\mqty( 0.5 & 0 \ 0.5 & 1) \mqty(a_n \ b_n) = \mqty( a_{n+1} \ b_{n+1})$

[warm shale]

jan Niku

[forest sinew]

this equation describes this flow

i want you to do the matrix multiplication

write a n+1 in terms of an and bn

[obtuse thistle]

so...i dont think we have done matrix multiplication, idk if that is later but our class started like a week ago

if u could show me that would be great

[forest sinew]

oh man its gonna be hard to type all this

[obtuse thistle]

wait wait

before you do

do you think u could try and do the problem like this

or really just step 1

because thats all i need

[forest sinew]

its like this

An+1 = 0.4An + 0.2Bn + 0.2Cn

n+1 and n mean step

this is just the first eqn

do you see how i got those numbers?

[obtuse thistle]

yes thank you

i was just confused on the initial step

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[fallen inlet]

Hello

[idle quartz]

I'm confused about what this whole page of notes is explaining, but especially part of Descartes rule of signs. When it says "then it differs by a multiple of two". Like why is it by a multiple of two? I can try to get a better picture of my notes if that would be helpful

[fallen inlet]

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[unreal peak]

how odds work?

Take this as an example:
I bet 10 bucks on something. My friend does the same. If I win, I get mine 10 bucks + his 10 bucks.

Total win: 20 bucks.

If I say I want to bet 10 bucks with 3:1 odds what does it mean?

[unreal peak]

I understand that if I win, I will get 3x the value I originally bet, is that correct? If so, how much did my friend bet? It wouldnt be fair if we agreed on 10 bucks but when I win he has to pay me 3x more.

I dont understand how this works

[obtuse pebble]

@unreal peak Has your question been resolved?

[unreal peak]

I understand that the winner will take home 3x his original bet value, right?
But if only two people are playing this, from where will come the extra money?

Look: If 2 people are betting 10 bucks each. The prize pool will be 20, and the winner will get 2:1 thats understandable. But where will the extra money come from if they agree on 10 bucks initially but with 3:1 odds?

You may say the other party will pay the extra 10 bucks, but by doing so, wouldnt it be the same as agreeing on betting more money initially?

Also, wouldnt it be silly to bet on a promise of payment where you dont see all the value but rely on the other party to pay you later a bigger amount?

[vague dune]

ok so 3:1 odds

however, for the other person, it's 1:3 odds

if it helps, think of it like this

the total is 4, which is the combined value of the odds (3+1)
converted into percentage, each number is worth 100/4 = 25
so 3:1 odds means 75:25 - you have a 75% chance of winning

for the other person, they have a 25:75 or 25% chance of winning

if you both bet $10, the winner (regardless of who it is) will walk away with $20, and the loser will walk away with $0

Now, if someone says they'll give 3:1 odds, that means the person with the 3:1 odd is willing to pay 3 times more because the chance is 3 times less likely
it is not like the $10 and $10 example - the people agree on $30 to $10 at the beginning

does this make sense?

[obtuse pebble]

@unreal peak Has your question been resolved?

[obtuse pebble]

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[random grotto]

Hi I need help with a matrix transformation question

[random grotto]

Ill share the set up of the question, its from a university resource I found online

I know that if a matrix A is the standard matrix of T, its size has to be 3 x 2 (am I right)?

[brave bramble]

Height 3, width 2, yes

[random grotto]

But how how would I find what matrix A is?

[brave bramble]

You have an easy shortcut in this case:
I suggest first getting T(1,0)

See how it's done in this case?

[random grotto]

Not exactly

Also, why would we want T(1, 0)?

[brave bramble]

That's okay. Note they give
T(1,-1) and T(0,1)

If we add them:
T(1,-1) + T(0,1)
= T([1,-1] + [0,1])
= T(1,0)

[random grotto]

Ah I see

[brave bramble]

If you have the transformations on the standard basis, then the matrix is just given by the outputs stacked into a matrix

[random grotto]

Okay so that works luckily in this case then, but if it were something like T(-1, 4) and T(1, -1) that they have given us, how would we proceed then?

Would we do scalar multiple multiplications or similar?

[stoic yacht]

if you want to repeat this approach, you could always find what linear combinations of (-1,4) and (1,-1) you'd need to take to get each of the standard basis vectors

[random grotto]

right, so in the case I just created how exactly would that work - would we still be working to find T(0, 1) and T(1, 0)?

[random grotto]

wait sorry I did not understand this

[stoic yacht]

to get the matrix representing the transformation

see what it does to the standard basis vectors

and write the transformed versions of those as the columns

you don't know what T does to (1,0) or (0,1), but if you can take some linear combination of T(-1,4) and T(1,-1), maybe you can find T(1,0)

you're allowed to take linear combinations of these since they're linear transformations

[random grotto]

So if we want to T(1, 0), we could do T(-1, 4) + 4T(1, -1), which gives T(3, 0), and then divide by 3

Is that the right idea?

[stoic yacht]

sure

that works

[random grotto]

But how would I write something like that out, because going back to the original question, the solution did something like this

Would the divide by 3 be outside the whole thing?

[stoic yacht]

yes

technically the dividing by 3 is the same thing as 1/3 T(-1,4) + 4/3 T(1, -1)

which is the same as T(-1/3, 4/3) + T(4/3, -4/3)

which is the same as T(1, 0)

[random grotto]

Would you mind terribly if I asked if you could write it out in a similar format and share a picture - I'm slightly confused on how to write it out, once I have T(1, 0) I can do T(0, 1)

[stoic yacht]

write what out?

do you agree the matrix representing T here has T(1,0) and T(0,1) as its columns?

[random grotto]

The thing that you are explaining, but in a similar format to how the picture shows it

[random grotto]

Yup

[stoic yacht]

so we want to find what those are

let me try to latex it

[random grotto]

That works, I appreciate it

[stoic yacht]

you used T(-1,4) and T(1,-1) as examples but never said what they actually were

let's say you knew T(-1,4) = a and T(1,-1) = b for some vectors a and b

[stoic yacht]

then you know that T(1,0) = 1/3 T(-1/4) + 4/3 T(1,-1) as per this reasoning here

[random grotto]

this is better, because I made up the -1 and 4 part to explore different version

[stoic yacht]

then you know that T(1,0) = 1/3 a + 4/3 b

of course if you had the vectors a and b, you could write this as a single vector

which would form the first column of your matrix representing T

[random grotto]

I see, let me try it out for T(0, 1) too

Okay I think I solved it

There was a question after this that asked to find a vector in R^2 whose image under T is a vector they gave

And I used the A constructed from (T[1, 0] T[0, 1]), and augmented it with the 3rd vector they gave. I then turned into RREF, got x1 and x2 as basic and x3 as free, and x1 and x2 ultimately turned out to be 1 and 0 respectively, so the vector in R^2 was x = [1 0]

Thanks for all your help

If i have more questions ill come back

.close

[obtuse pebble]

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[vernal acorn]

would this be 7 over tan 55 degrees?

[tawdry osprey]

Yep

[vernal acorn]

these are the other options

i’m mainly confused between a and d

[sage bloom]

why?

tan(55°)=7/BC

BCtan(55°)=7
BC=7/tan(55°)

[obtuse pebble]

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[random grotto]

if a matrix has Nul(A) = 0 then is it guaranteed the linear transformation x to Ax is 1:1?

[random grotto]

i think yes because it means all col are pivot columns which is a req for 1:1

[obtuse pebble]

@random grotto Has your question been resolved?

[random grotto]

<@&286206848099549185>

[shut lagoon]

Yes. The kernel is trivial iff the transformation (or its associated matrix) is injective,

[random grotto]

thanks

.close

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[fallen inlet]

Are -1/3 , 1/-3, and -(1/3) the same slope??

[fallen inlet]

Like in the end my line will be the same right

[gilded needle]

they are all the same number so yes

[timid silo]

a way u can check is to test them with real coordinates to see if they are the same slope

and do rise over run to find if they are the same 🙂

[fallen inlet]

Ohh ok

[nova bane]

-1/3 = 1/-3 = -(1/3)

[timid silo]

yup!

[fallen inlet]

Asking because the answer to my problem says -1/3 and -3/1 but I got -1/3 and 3/-1

[fallen inlet]

Ohhh

Yeah

[nova bane]

-3/1 = 3/-1

[fallen inlet]

So it’s same thing right

Yes

Ok

[nova bane]

-3/1 = (-3/1) * (-1/-1) = 3/-1

since (-1/-1) is 1

[obtuse pebble]

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[bitter terrace]

How would you tell this is a perfect square?

[meager glade]

all the coefficients are squares

2^2=4
6^2=36
9^2=81

[nocturne bronze]

(a+b)^2 = a^2+b^2+2ab

[upper sorrel]

the middle should be 2ab tho, so 2(2)(9) instead

[upper sorrel]

yeah this one

[nova bane]

its just a coincidence that 36 is also a perfect square

[nocturne bronze]

yea

[meager glade]

i misunderstood the question

mb

[nocturne bronze]

like (x+2)^2

[upper sorrel]

np

[bitter terrace]

I’m still kind of confused, what steps would you take to figuring out the perfect square

[prisma hatch]

see you factorise it

[nova bane]

check if the first and last terms are perfect square

[prisma hatch]

you can identify if it's in form of a^2 + 2ab + b^2 or a^2 -2ab +b^2

then you can be sure that it's a perfect square

[bitter terrace]

Ok I think I see now

Thank you

[obtuse pebble]

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[obtuse pebble]

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[bronze shoal]

if I want to understand easily about polynomial equations better, what kind of math topic should I review?

[sage dagger]

polynomials

[bronze shoal]

its really confusing to factoring those numbers and had to group them

[timid silo]

You should learn some basic algebra topics
Such as polynomials (monomial, binomials trinomials), linear equations in 1 variable 2 variable 3 variables, system of linear equations,
quadratic equations
And in addition you can learn to draw graphs of linear equations

[obtuse pebble]

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[radiant marsh]

can someone please explain when exactly I use the => arrows in math? I know that it means "X implies Y" and I know that <=> means "X implies Y AND Y implies X" but I don't know when not to use them

[radiant marsh]

like for example when I have to use the quadratic formula for an equation, does it matter if I use the arrows there?

[prisma hatch]

x = 238
y = x +2
=> y = 240

[radiant marsh]

that I understand

[tender tusk]

its not necessary to use implies for a quadratic equation

[radiant marsh]

I'm not sure when for example I got something like this

[tender tusk]

its not an incorrect usage of it, but its not necessary

[radiant marsh]

when is it necessary? because my exam deducts points for not using them when needed

or if you don't use them you have to write down a check for if your answer is correct where you plug in x that you found

so far I know that I have to use them when applying algebra and that I have to copy the whole equation everytime I do it

so for example
x + 2 = 0
=> x = -2
and not
x + 2 = 0
=> -2

but that's about it when it comes to my understanding of it

.close

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[solemn compass]

Hello

[solemn compass]

I need help with one radical

[prisma hatch]

[meager glade]

!da2a

[obtuse pebble]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[nocturne bronze]

where you learn these commands/?

[prisma hatch]

just by being active

[kind hawk]

#helpers-info has a list of commands

[obtuse pebble]

@solemn compass Has your question been resolved?

[restive gorge]

Thats the exact same question like yesterday lol

[nocturne bronze]

!close

.close

!forceclose

!timeout

!intolerance

!lag

[gloomy vector]

tf

[obtuse pebble]

@solemn compass Has your question been resolved?

[nocturne bronze]

!forceout

won't this chat ever close

[meager glade]

v has to close it or you have to wait for it to timeout

[nocturne bronze]

'@solemn compass '
ok

[obtuse pebble]

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[round orbit]

Someone help me with b?

[prisma hatch]

find it's circumcenter

[round orbit]

How exactly?

[prisma hatch]

by drawing perpendicular bisectors

and where they meet is known as the circumcenter

[round orbit]

So i should draw a perpendicular bisector on all A, B and C?

[prisma hatch]

no any two sides would be enough

[round orbit]

I see

[round orbit]

Once I find the circumcenter do I just draw a circle that touches lines A, B and C or just make it big that it passes through them?

[prisma hatch]

no it should just pass through ABC

Steps:

Construct the perpendicular bisector of one side of triangle
Construct the perpendicular bisector of another side
Where they cross is the center of the Circumscribed circle
Place compass on the center point, adjust its length to reach any corner of the triangle, and draw your Circumscribed circle!

[round orbit]

Would this be correct?...

[prisma hatch]

show

[round orbit]

Is the Circle too big?

[prisma hatch]

no no no not like this

like this

[round orbit]

Ohhh

Ahhh

I get it now

Wow thank you so much for your time and teaching

Didn't know it was that easy

.close

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[topaz sluice]

can i please have some help?

[pseudo raft]

ask

dont ask to ask tho

just ask

[topaz sluice]

can someone please tell me how to complete the square?

[pseudo raft]

ask.-.

post the question

[high lily]

,tex .cts

[warm shale]

ℝαμΩℕωⅤ

[topaz sluice]

i dont have a question im asking how to complete the square because idk how

[hoary kindle]

let's use the image above

[topaz sluice]

i dont really get the image

[hoary kindle]

what do you not understand

[topaz sluice]

where the c came from

[hoary kindle]

ok we'll ignore that for now

let's take this example

x^2 + 14x = -24

first step of completing the square

we always divide the 'b' term by 2

in this case we divide 14 by 2

so that gives us 7

now we square it

so we get 49

now whatever we do on the left side

we do it on the right side

so we add 49 to -24

so it'll look liike this x^2 + 14x + 49 = -24 + 49

now can you tell me what the LHS simplifies to?

[topaz sluice]

idk is it x^2 + x +49 = 25/14?

[hoary kindle]

nope

basically i want to know the factored version of the perfect square trinomial

so x^2 + 14x + 49 is what we call a perfect square

and it's a trinomial because it has 3 terms

the x^2

the 14x

and the 49

do you know how to factorise?

[topaz sluice]

kinda ive done it before so i kinda remember but im not 100% sure

[hoary kindle]

Okay do you want to try factorise the perfect square?

and tell me what you get?

[topaz sluice]

ok

is it (x+7)^2?

[hoary kindle]

yep

that's why it's a perfect square

i can't explain what a perfect square is perfectly

so I would search it up

so when we complete the square we're just trying to make the LHS a perfect square

[topaz sluice]

ohh ok

[hoary kindle]

so now (x+7)^2 = 25

now we square root both sides

so we get x+7 = plus minus 5

[timid silo]

x = -2

[hoary kindle]

yes or x = -12

[topaz sluice]

wait what

how

[hoary kindle]

plus minus

[topaz sluice]

surely its (x+7) = 5??

[timid silo]

square root gives pluse/mines 5 for 25

[hoary kindle]

yes

because if u think abt

it

waht does -12 + 7 give u

i['ll just answer it it gives u -5

and -5 squared is 25

now do the same for -2

-2 + 7 = 5, 5 squared is 25

therefore x can accept both values of -2 and -12

hence we get plus minus 5

[topaz sluice]

im confused where -5 came from and where -12 and -2 came from still

[hoary kindle]

ok so let's go from (x+7)^2 = 25

if we square root the value

we're left with x+7 = plus or minus 5

reason being is because either plus or minus 5 works if we square it

cause it's going to give us 25 either way right?

[topaz sluice]

so surely its either x= -2 or x=12?

[hoary kindle]

yes

-12*

[topaz sluice]

hows it -12?

[hoary kindle]

-5 - 7

[topaz sluice]

not js 12

[hoary kindle]

essentially plus minus 5 is like having two cases

[topaz sluice]

but surely plus or minus so 5 + 7 = 12 and 5-7 = -2

[hoary kindle]

plus 5 (positive 5 basically) is one case

nono lemme explain

[topaz sluice]

ok

[hoary kindle]

and minus 5 (negative 5 basically) is another case

if we take case 1

we completely ignore the minus

we assume 5 is just positive

so when we move the 7 to the other side

we get 5- 7 = -2

now scrap the first case and focus on the second case

assume 5 is negative and not positive

so x+7 = -5

x = -5 -7 = -12

do you understand?

[topaz sluice]

ohh ok yeah i get it thanks so much for ur help

[hoary kindle]

you're welcome

[topaz sluice]

ive got a exam in a few days and i didnt rly understand how to do that

[hoary kindle]

it's okay

just practice

[topaz sluice]

will do

[hoary kindle]

if you have a notebook or like a spare piece of paper, write this down

[topaz sluice]

yeah im also taking screenshots

[hoary kindle]

alr good

good luck for the exam then

[topaz sluice]

thanks

[obtuse pebble]

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[warm galleon]

not sure how to start?

[meager glade]

what is h_3?

[warm galleon]

huh

[meager glade]

h1=100

[warm galleon]

ye

[meager glade]

h2=100*0.9

then what is h3?

[warm galleon]

r=0.9

[warm galleon]

100 * 0.9 * 0.9

[meager glade]

exactly

and then h4?

[warm galleon]

100 * 0.9 * 0.9 * 0.9

[meager glade]

perfect

now, how would you write $h_n$?

[warm shale]

Flappie

[warm galleon]

uh

idk

[meager glade]

$h_1=100\h_2=1000.9\h_3=1000.9*0.9$

[warm galleon]

ye

[warm shale]

Flappie

[warm galleon]

yh

[tender tusk]

do u not notice anything

[meager glade]

how would you write this in terms of n

so that we can find the height of any arbitrary bounce

[warm galleon]

uh

hn = 100 * 0.9n?

[meager glade]

close

[warm galleon]

0.9 times of n

[meager glade]

how would you write 0.9 times itself, n times?

[warm galleon]

0.9^n

[meager glade]

exactly

so then what is h_n?

[warm galleon]

100*0.9^n

[meager glade]

exactly!

we hae the height of the ball

after n bounces

[warm galleon]

ye

[meager glade]

we want the height of the ball

to be less than 25 cm

after how many bounces does it achieve that

[warm galleon]

100*0.9^n<25 ?

[meager glade]

perfect

now, solve that inequality

[warm galleon]

aight

how do i solve 0.9^n < 0.25

💀 i forgot

[meager glade]

take the logarithm

[warm galleon]

so log10 0.9^n < log10 0.25 right?

[meager glade]

yes

[warm galleon]

got it

but its n<13.157

answer is 14

is something wrong?

[meager glade]

no

you have to round it up

100*0.9^13=25.4

which is just above

so you have to let it bounce one more time

[warm galleon]

i twill become n<13

which is the same

13

[meager glade]

you have to round it up

not down

the threshold is inbetween two whole numbers

if you take the lower number

then it wont have passed that threshold yet

and =thus is not the number we are looking for

so you have to round it up

[warm galleon]

got it

thankss so much

[meager glade]

👍

[warm galleon]

@meager glade For b how do i actually start?

[meager glade]

you need to take the sum

you need the total distance it has travelled

and that is each height

summed

[warm galleon]

Its height is 100cm

[meager glade]

yes

but you want the distance its travelled

[warm galleon]

How do i find the distance travelled

[meager glade]

i just told you

read what i type please

i am not typing just to type

[warm galleon]

.

Wdym is each height

[meager glade]

what is the distance it travels between each bounce?

its goes down

and up

what is that distance

[warm galleon]

Yh

It doesnt say?

[meager glade]

?????

how far does the ball go down

when it bounces

[warm galleon]

Theres no distance in the question

[meager glade]

whats the height?

[warm galleon]

0.9h

[meager glade]

is that not a distance?

[warm galleon]

Oh

I thought u meant the distance between em like left and right thats why i got comfused

[meager glade]

nono

up and down distance

[warm galleon]

100-(100*0.9)

[meager glade]

why a -?

it wants the total distance

[warm galleon]

Yh

[meager glade]

so you take the distance it goes down

and the distance it goes up

[warm galleon]

Oh

So 100+2*[100 * 0.9]

And so on ?

[meager glade]

yep

[warm galleon]

Is this the only way tho

Theres no other methods?

[meager glade]

i dont see how else

you want the total distance

so you have to add up all the up and down bounces

[warm galleon]

Aight thxxx

.close

[obtuse pebble]

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[fathom ridge]

for a group with infinite order, such as 2^n

[fathom ridge]

n=0 is technically the order

so why isn't the order 0, but rather inf?

thats what im asking

he did r-s saying that the only way it can equal the identity is by it's exponent being 0

but, how is that a given

abstract math seems to put exponents into recipe without telling and it's stupid

i

i'm okay with it being exponantial but

[marble ingot]

that's literally the definition of an element having infinite order, if a^n = e implies n = 0

[fathom ridge]

if it has an order, a^0 = e, then it means that ord(a) is zero, not infinite

[fathom ridge]

oooh

so inf order = 0 order?

[marble ingot]

if you want to think about it like that sure

[fathom ridge]

that's stupid, thanks for help

abstract algebra makes less sense the more i learn it

[marble ingot]

think about it like this

a generates some subgroup

the order of a is the number of elements of this subgoup

if the order is 0, this would imply that the subgoup is the empty set

can't be tho

[fathom ridge]

2^n isn't empty tho

[marble ingot]

because it contains at least a

right

that's why it makes sense for the order to be infinite

[fathom ridge]

run a function zero times, and you get the identity

i believe this means this

but if you run a function zero times, also, not run it

you don't get anything

not even the identity

[marble ingot]

there is no esoteric reason why a^0 = e, it's defined to be like that so that the exponent rules make sense and {a^n in G | n in Z} is a subgroup.

it's just a notational device

[fathom ridge]

oh

so we're assuming that the G group is using the exponantial fuction

[marble ingot]

no

if o is the operation a^n = a o a o a ... o a n-times

it's just a matter of definition

and if you have say $a^n$ and $a^{n-1}$ then $a^n * a^{-n} = e$ obviously

[warm shale]

belabutter

[fathom ridge]

multiplication*

i believe, is the thing i should have said

[marble ingot]

so for the exponent rules to make sense we define $a^0 = e$ so that $a^n * a^{-n} = a^{n-n} = a^0 = e$

[warm shale]

belabutter

[marble ingot]

maybe these help

[fathom ridge]

thank

s

[marble ingot]

np

[fathom ridge]

by the definition of order, a^r can't equal e, so why must it be r=0?

im certain that in the definition of order,

it was

0 =< x < n, x != e

i acknowledge that we need to set r to zero, to make a^r equal to e

but doesn't that contradict the definition of order?

that's the rule i believe

does it allow a^0 being e?

[obtuse pebble]

@fathom ridge Has your question been resolved?

[obtuse pebble]

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[wise lance]

why is 24 being divided by 4000?

[wise lance]

integral of cos is sin, I get that

and I get that you can pull the 24 out

[meager glade]

because of the chain rule

$\frac{d}{dt}\sin(4000t)=4000\cos(4000t)$

[warm shale]

Flappie

[meager glade]

so if oyu take the inverse $\int\cos(4000t)dt=\frac{1}{4000}\sin(4000t)+C$

[warm shale]

Flappie

[wise lance]

is it the "t" which makes you require the chain rule in this case?

im looking at pauls online notes and it seems different

cos(x)dx = sin(x) + c

not xsin(x) + c

[meager glade]

what?

where are you getting xsin(x)+c from?

[wise lance]

4000t = x

right?

err

I haven't actually reached the chain rule for integrals yet in my book so im not sure

only for derivatives

[meager glade]

yes, but $\int \cos(ax)dx=\frac{\sin(ax)}{a}+C$

[median dome]

sin(ax)

[warm shale]

Flappie

[meager glade]

,w int cos(ax)

[warm shale]

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[meager glade]

and since hte constant in the examples is 1 (a=1), you divide by 1, and thus doesnt do anything

[wise lance]

ah so a in this case would be 4000

so you pull it out and divide 24 by it

.close

[obtuse pebble]

Channel closed

Closed by @wise lance

Use .reopen if this was a mistake.

[obtuse pebble]

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[ruby wyvern]

im using the product rule

[meager glade]

!show

[obtuse pebble]

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

[ruby wyvern]

give me a sec

[trail musk]

you can use implicit differentiation as follows:
first divide both sides by the radical, then square both sides, then take the derivative and solve for y'

[meager glade]

almost

you need to multiply the first term with the derivative of whats inside the sqrt

$\frac{d}{dx}\sqrt{f(x)}=\frac{f'(x)}{2\sqrt{f(x)}}$

[warm shale]

Flappie

[timid silo]

can any1 here help me with limits

[ruby wyvern]

open a empty chat

help-13

oh, I was playing with the expresion and I got to the answer

how can I close a chat_

?

[obtuse pebble]

@ruby wyvern Has your question been resolved?

[meager glade]

type .close

[obtuse pebble]

Channel closed

Closed by @ruby wyvern

Use .reopen if this was a mistake.

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Remember:
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• Show your work, and if possible, explain where you are stuck.
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[wicked crown]

is an algorithm that runs with expected runtim O(d * n) still considered an efficient las vegas algorithm if d in N+.
The thing here is that 1/d is the difference of two terms in my analysis and here 1/d is between 0 and 1. Then d can be any positive integer. Which makes this whole runtime weird because it could be arbitrarily large.
Does anyone know how to judge this ?

[brave bramble]

Can n also not be arbitrarily large?

I am maybe not understanding what the issue is. What information are you trying to extract out of the big-O runtime?

[obtuse pebble]

@wicked crown Has your question been resolved?

[wicked crown]

well n is the input

[wicked crown]

I want an efficient probabilistic algorithm for a good independent set

now I have some bound on the independence number

[timid silo]

I'm not certain but

[wicked crown]

and I am trying to derive a las vegas algorithm

[timid silo]

as far as I know regarding big o notation

O(1e8888*n)=O(n)

[wicked crown]

Now usually if the graph has m edges, then the las vegas algorithm needs to run O(m) times to find a good set. But unfortunately my analysis tells me that in some cases there might be a value d that can be arbitrarily large, depending on what the fractions do in my term, but it does not directly depend on the graph size

[wicked crown]

yes but

[timid silo]

as far as I know big O notation doesn't refer to how fast or efficient an algorithm is but how well it scales as n does

[wicked crown]

if we have something like 10^10 * O(n) thats not pracitcal

[wicked crown]

thats the thing

technically its linear in the edges

but by analysis it could have some insanely large constant

[timid silo]

well

[wicked crown]

and I didnt find any way to bound that value

[timid silo]

O(n) is still the worst case scenario

[wicked crown]

I could give details if you want

but its a little bit technical

[timid silo]

what language

[wicked crown]

english

could do german too

[timid silo]

oh it's not a program?

[wicked crown]

lol no its math

[timid silo]

😭😭😭

[wicked crown]

didnt implement it yet

[timid silo]

I'm not quite sure how it would work in math

[wicked crown]

wdym

[timid silo]

idk if I can help

give it a try tho

[wicked crown]

kk

no problem

[timid silo]

next person may find it useful

[wicked crown]

I kind of asked this in here a few times already lol

nobody seemed to want to take a look

I know that the expected size of an independent set is n^2/4m

therefore there is an independent set with size at least n^2/4m

[timid silo]

without looking at it too much wouldn't that be O(n*m)?

What in the computer science is this

[wicked crown]

this is in O(m) expected ?

1/3m+1

is the probability that a randomly constructed independent set has size at least n^2/4m

so geometric distribution has expected value 1/p

here it is >= 3m+1

O(m)

in the case where the ceil(n^2/4m) is in N

the other case is the tricky one

the difference in the numerator can be arbitrarily close to 1

therefore 1-D = d can be arbitrarily small

so the fraction can skyrocket as far as I am concerned

forgot this

where U is an indpeendent set in a graph G, randomly selected by a certain method

and n is at most 2m

[timid silo]

question real quick

do n and m grow at similar rates

[wicked crown]

the only relation we know is that 2m >= n

other than that you could have everything up to m = n choose 2

[timid silo]

so that would be O(n^2/m) then, no?

[wicked crown]

why ?

[timid silo]

cause we have two things that change

[wicked crown]

but by bounding n through m , I get rid of n ?

also we know that m is always in O(n^2)

its at most n choose 2

[timid silo]

u said that n=2m

worst case scenario

[wicked crown]

2m >= n

yes

and m <= n choose 2

thats just by definition of a graph