help-33

[marsh citrus]

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[oak oxide]

Also a root

[sick walrus]

if you used OP's method, only x = a is correct.

[oak oxide]

.wut

[sick walrus]

I admit, I should have checked the method first.

[oak oxide]

Why its has my name

[sick walrus]

and you have opened a channel.

[halcyon elk]

yeah but, what are the other answers

like is there any other solution

[sick walrus]

||x = b + c - a|| is the other answer.

[oak oxide]

Oops

[oak oxide]

Yes

[halcyon elk]

oh!

[halcyon elk]

how come

or... how did you get to that

[sick walrus]

I don't think I should spoil the answer for OP.

well, the original OP.

[halcyon elk]

its fiona's channel 😔

[oak oxide]

Dw I'll close this one after this argument:3

[sick walrus]

if you wish, dm me.

[halcyon elk]

great!

[spark otter]

you have a quadratic. two solutions. x = a is an obvious solution. To find the other, you can use ||sum of roots for example||

[oak oxide]

You know me

[oak oxide]

Yeah that's what i used technically

Alright I'll close this:3

.close

[marsh citrus]

Channel closed

Closed by @oak oxide

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[marsh citrus]

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[tardy meteor]

anybody knows how to do b?

[sacred idol]

wait nvm hold on

[verbal loom]

Yeah — you need to find the local maximum on 0 to 4. Take your original equation and subtract 645t since you are losing 645 gallons an hour — then determine the maximum over the interval

[patent sandal]

no

you subtract 645

cus E(t) is the rate of sewage per hour

[sacred idol]

whats up with that em dash

[novel juniper]

wth is a em dash

[sacred idol]

were you able to find the net rate of sewage

[verbal loom]

I always use em dashes — don’t know why

[sacred idol]

creepy

[verbal loom]

And you subtract 645t — 645 is the amount of sewage being removed per hour — 645 isn’t some static amount being removed regardless of time passing

[patent sandal]

subtracting 645t from E(t) would imply that the rate at which the sewage is taken out is increasing with time t

[marsh citrus]

@tardy meteor Has your question been resolved?

[verbal loom]

E(t) is the amount of sewage entering over time t. 645t is the amount exiting over time t. E(t) is not a rate is the amount entering.

• it is the amount of sewage entering

[patent sandal]

Question literally says E(t) us the rate at which raw sewage enters a treatment tank

[verbal loom]

Yeah — you’re right — I dont know why I misread so many times

[lethal bridge]

whats up with the em dash

[paper fiber]

who let the chatbot in here

[paper fiber]

like where in the world did this dot point come from

[devout mauve]

discord automatically formats messages like that if they for example start with a minus or something (iirc)

[paper fiber]

• hello

I see, but I'm suspicious

[lavish pelican]

Hii

[pallid jolt]

Let's get back to helping OP

[tardy meteor]

wait what?

[verbal loom]

Regardless — since the quantity of sewage is the integral, subtract 645 and then integrate, then find the maximum over the interval - or subtract 645 after you integrate (since it’s basically the same thing)

[sacred idol]

they are not the same.

[verbal loom]

Subtract 645t I meant

[sacred idol]

yeah

thats identical

[verbal loom]

Also not a bot — just tired and use a lot of em dashes

[sacred idol]

goofy

[tardy meteor]

.reopen

[sacred idol]

still open

yeah so net rate of change of sewage = $\frac{dV}{dt} = (E(t) - 645)$ gallons/hour

[sacred idol]

V is the sewage volume

recall that for a continuous function extrema occur at the points where rate of change is 0

here, volume changes with respect to time

[lethal bridge]

creepy

[tardy meteor]

ok and ?

[sacred idol]

...

can you figure out the equivalent mathematical equation for this

[marsh citrus]

@tardy meteor Has your question been resolved?

[tardy meteor]

.close

[marsh citrus]

Channel closed

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[brittle herald]

What have you tried so far, Erode?

[vital yew]

Ehhh

Nothing really

I don't get it

there's like 2 variables

when I was solving

aND ONLt 1 eq

wait Imma send where I got stuck

[brittle herald]

Good

What is Rs. ?

[vital yew]

Oh shit

I got it

I mean I found 2 eq.

[brittle herald]

Good

[vital yew]

Is it correct/

I think answer is 8

Based on these 2 eq.

[brittle herald]

I don't understand what Rs. means in the problem so I'll let someone else answer

[lament umbra]

okay so let the initial price of the chocolates be x rs/piece
reduction in price = 25% = 0.25x
final price per peice = 0.75 x
NOW
LET INITIALLY buyer was able to buy y chocolates for 96rs
and now he will be able to buy y + 48for 96
so
96/x was the amount of chocolates that could be bought befoer
and
96/0.75x = 128/x is hthe final
now
128/x - 96/x = 48

calculate now

[vital yew]

Some currency

[paper fiber]

think that's just a currency

[vital yew]

Like dollar

[brittle herald]

Oh

[lament umbra]

x = 2/3

for 12
it's 8

[vital yew]

Yooo

Great :)))

[brittle herald]

Then yeah soft is correct I'll just let soft guide you

[paper fiber]

!nosols

[marsh citrus]

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

[vital yew]

I already said the answer.......

[lament umbra]

oops 😅 new here

[paper fiber]

yep but they were typing that up long before you sent that

[paper fiber]

all good, just keep it in mind for the future

[lament umbra]

hope you got it @vital yew

[vital yew]

that's true..

[vital yew]

yeah

[lament umbra]

🙂

[vital yew]

Now someone pls explain this one

Btw is there a command to pin this too

[indigo nest]

,rccw

[vital yew]

Lmao is Xavier the Admin ?

.close

[marsh citrus]

Channel closed

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[marsh citrus]

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[visual lion]

Given 9^n = 43046721
Is there a quick way of figuring out what n is without computing 9 x 9 x 9 x ... until I get 43046721?

[main idol]

use log base 9

[visual lion]

Ah okay. I haven't learned that yet.

[amber birch]

if you can choose any positive integer instead of just 43046721

n will be almost always an irrational number

so that means that you can assume the solution is at worst, n = k + 1/2, where k is a positive integer (since 9^0.5 = 3), if it's not actually n = k

[visual lion]

n = k + 1/2
k = 9?

[hexed junco]

Counting the number of digits should work for smaller numbers

[amber birch]

if you know your powers of 3, then you'll be able to figure out that 2187 = 729 * 3 = 9^3 * 3 = 3^6 * 3 = 3^7

so 43 046 721 is around 40 million = 10 * (2000)^2
so you can make a guess that that 43046721 is around 9 * (3^7)^2 = 9^1 * 9^7 = 9^8

indeed, n = 8

[amber birch]

see what I wrote below

[amber birch]

(you can assume that 43 046 721 is an integer power of 3

otherwise you literally won't be able to figure out the answer (without logarithms)

[visual lion]

I think I sorta get it, but I'd still have to know 3^7 etc.

[amber birch]

it's not that hard to repeatedly multiply

[visual lion]

Okay, guess I'll go look up logarithms

[amber birch]

I think the question wants you to figure it out without logarithms anyway

[visual lion]

I know, but I was hoping for a quicker way to do it.

[amber birch]

logarithms are just a way to type it into your calculator

that's the quickest, but probably not what the question wants from you in terms of improving your problem-solving ability

[visual lion]

Okay, thanks 🙂

[amber birch]

no worries!

[visual lion]

I'll do it as asked right now, then worry about logarithms next time

.closed

[amber birch]

.close

[marsh citrus]

Channel closed

Closed by @amber birch

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[humble osprey]

,tex
When we have a clock with 12 hours on it, is it based on $\mathbb{Z} _{12} $ ?

[past thicket]

Put curly brackets {} around the subscript

[elfin berry]

fijokazż

[elfin cairn]

yeah you can think of it that way

[winged delta]

No since you don't have a "0"

so it's rather $\mathbb{Z}_{11}$, with 12 serving as a 0

[elfin berry]

Médicis

[humble osprey]

wait but 0 is every multiple of 12, no?

[elfin cairn]

there are 12 hours so Z12

[winged delta]

yeah nevermind

[humble osprey]

so is there no restriction when we talk about the standard clock? or are there certain properties that have to be ignored?

(i just learnt abt that stuff so idk if my questions make much sense)

[jagged relic]

What's "the standard clock"?

[humble osprey]

the one most people use. with the 12 hours

idk why I said standard but yeah

[jagged relic]

I mean... on an analog clock, the hour hand can go in-between integer hours; you can also have a minute hand and a second hand...

Not sure what you mean by properties

[humble osprey]

the properties that define Z_12, or Z_m in general

[humble osprey]

yeah, i kinda meant it in the situation where we only care about the integer parts of the time on the clock

[humble osprey]

also, im not rlly sure what these properties are, i dont remember them, but im sure they exist lol

[jagged relic]

Z_m is the ring of integers modulo m, so it has all the properties of a ring

It's also the set of congruence classes modulo m, maybe that's something you want to look into

[humble osprey]

From what our prof told us, for Z_m to be a ring, it must be endowed with addition and multiplication. So, Z_m alone, I would suppose, isn't a ring by itself, right?

[humble osprey]

yeah thats what our professor defined it as

[jagged relic]

I mean sure, but if you consider Z a ring, Z_m is automatically a subring

[humble osprey]

Would we need Z_m to be a ring, in order to represent a clock for m=12? Or can we ignore the ring's extra properties?

[jagged relic]

Would be kinda hard to add or subtract hours, for example to get durations, without an addition operator

[humble osprey]

true true

okay then, thank you for helping with this example

its a really weird concept to me

.solved

[marsh citrus]

Channel closed

Closed by @humble osprey

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[jagged relic]

If you don't want those ring properties, I guess you could consider it a set, namely the set of integers form 0 to 11, but what's the point? Z_m is just a notation used for modular arithmetic, so of course we want addition and multiplication (and congruence classes)

[humble osprey]

.reopen

[marsh citrus]

✅ Original question: #help-33 message

[jagged relic]

Z^+ denotes the set of positive integers, and that's a similar notation but with no mention of modular arithmetic

[humble osprey]

if we defined it as just a set, then we also wouldnt be able to circle back while going forward in time, tho, right? at least the way i think about it is: if time continues forward, then the clock, if based on the integers 0 thru 11, would turn back without "counting" the day, but this intuition might be way off

[jagged relic]

What would you define as going forward in time that doesn't require addition?

[humble osprey]

yeah I guess addition must be embedded. so thats another reason why Z_12 should be a ring, right?

[jagged relic]

The reason is just that this is what we want to work with

I wouldn't be very surprised if someone used Z_12 to denote the set for solving a problem that never uses modular arithmetic

But you're asking in the context of modular arithmetic

[humble osprey]

am I reading too much into this? our prof made it a whole thing, basing Z_m to equivalence relations, then classes, then addition properties and all that. she really confused me and now im desperate tryna find an example to make it easier to understand

[jagged relic]

You might be yeah

[humble osprey]

what do you mean by that? Isnt Z_m based on mod m?

[jagged relic]

I think your prof was just giving definitions, which requires some rigor

[humble osprey]

yeah she went really fast and threw me way off without using any examples

[jagged relic]

Just as a shorthand for {0, 1, 2, ... 11} but without ever using modulo

[humble osprey]

ohhh didn't know that

[jagged relic]

I'm not actually 100% sure it would be correct; but it's just notation after all, so surely you can use that shorthand

Though I can't really think of any situation you would need it

(apart from mod)

[humble osprey]

notation is always contextual

anyways, thank you again. ill think of a better example and be back for more brain rot

[jagged relic]

Well, we generally want notation to be agreed upon across the world and across time, so it's a bit of a touchy subject

Like we don't use a giant + as a summation symbol, we use Sigma instead

And everybody knows what Sigma means, how it's used, and how to manipulate it

[humble osprey]

yeah tru. usually they use the same notation for different things only if theyre also in a different field of maths. so it would be weird for two things in algebra to have the same notation

[jagged relic]

Yea

The higher you go in maths, the more specialized the subject matter, and the more obscure the notation

Z_m is far from obscure

[humble osprey]

I'll never get over the fact that ~ is an actual thing

anyway anyway

[jagged relic]

In what context?

[humble osprey]

for equivalence relations

[jagged relic]

Oh right

[humble osprey]

it feels really weird to use

.close

[marsh citrus]

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[serene falcon]

i need help

[past thicket]

Send it over

[serene falcon]

what’s the easiest way of doing 0.75 times 95? on a piece of paper

[stoic saddle]

3*95/4

[proud peak]

Uh offtopic can anyone help at #help-31

[past thicket]

!noad

[marsh citrus]

Please do not advertise your help channel or thread in other parts of the server. There are many people who need help, so advertising can quickly turn into spam.

[proud peak]

Thanks

Nvm

[serene falcon]

uhm i need help??

[serene falcon]

⬆️

[past thicket]

Be patient, someone will help you eventually. We are busy with this person right now.

[proud peak]

Okok

[serene falcon]

helloooo

[stoic saddle]

i literally told you

3*95/4

0.75 is the same thing as 3/4, so multiply by 3 then divide by 4

[serene falcon]

yes that’s not explained well though i said on paper that’s so confusing

[stoic saddle]

can you do 95*3 on paper

and then can you do long division on paper

[serene falcon]

you know when people do it like
0.75
x95

how do u do like that

[stoic saddle]

do 75*95, then shift the decimal point two positions left.

(if you wanna do it like this at all)

however you're asking for two different and conflicting things

[past thicket]

Long Multiplication

[serene falcon]

but i don’t get what to do with the decimal

[stoic saddle]

didn't i just tell you?

do 75*95, then shift the decimal point two positions left.

[serene falcon]

but i don’t understand

[stoic saddle]

ok

do 75*95
do you understand this bit

[serene falcon]

i don get how to actually figure that out tho like this

it just so confusing

sorry i’m like kinda stupid

[stoic saddle]

do you know how to do multiplication like this in general? yes or no.

[past thicket]

You can watch the video I sent, or listen to Ann.

[serene falcon]

yes but i’m super slow so i don’t understand

[serene falcon]

not really

[stoic saddle]

aight then go watch that vid.

that's your problem really

you need to (re)learn how to do long multiplication

[serene falcon]

ok

[stoic saddle]

just with whole numbers

when you are ok with those, THEN move on to decimals (and not earlier)

[serene falcon]

okay

[marsh citrus]

@serene falcon Has your question been resolved?

[lethal zinc]

You can close the channel

[marsh citrus]

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[marsh citrus]

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[compact moth]

How do I Evaluate integrals using ti-84 plus CE calculator?

[ionic ferry]

I'm not so sure about the calculator, although I can help you on how to solve this

or both of them actually

[compact moth]

just the first one is fine thank you.

[ionic ferry]

Alright

So take x^4 as t

[compact moth]

ok

[ionic ferry]

now differentiate both sides, what do you get

[compact moth]

4x^3 sinx^4dx

[ionic ferry]

No no

See you have t = x^4

Now just differentiate them both

[compact moth]

yeahhhhh. hey man thank you but im going to try and figure the calculator thing out.

.close

[marsh citrus]

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[ionic ferry]

Sure mate, I'm not sure if it'll work though, good luck with it

@compact moth I checked and it says that it only solves and gives an estimate value of definite integrals, it won't algebraically solve it, and that won't be helpful at all for indefinite integrals

[marsh citrus]

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[reef forge]

how are the points supposed to placed I keep getting it wrong

[jaunty blade]

Are you asking where the points are if y=2x+2?

[reef forge]

yea like where do I place em

[jaunty blade]

You know how the x axis and y axis works? Or do you also need help with that?

[reef forge]

x is horizontal and y is vertical yet when I pick those points I get it wrong

[jaunty blade]

Are you plotting x axis first then y axis?

[reef forge]

yes

[jaunty blade]

So, the point (3,8) is wrong? Or did you not plot that one

[reef forge]

I can only plot 2 points, so I have make sure the line is going across both points

[jaunty blade]

Okay, so where are you going wrong? Can I see how you do it so I can have an idea on how to help?

[paper raptor]

the point plotted at (3,8) is right, but why have you plotted another at (0,1), that wasn't on your line

[reef forge]

well I was going wrong

on the point where

actually I got

thank yall for yall help

[marsh citrus]

@reef forge Has your question been resolved?

[marsh citrus]

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[brazen idol]

x2 + y2 = 2ax, a is the parameter.

[brazen idol]

i want to find the orthogonal trajectory of this curve

ill send the solution

my internet is kinda slow

so sorry for the delay

[lament umbra]

to find the orthogonal trajectory
differentiate wrt x and then replace dy/dx with -dx/dy then integrate

cuz dy/dx is the slope of this curve and the slope of the curve orthogonal to it will follow
m1m2 = -1
so dy/dxm2 = -1
hence m2 = -dy/dx

then simply integrate

[brazen idol]

yeah the image isnt sending for some reason

im more confused with the integration per say

i got

[lament umbra]

you saying integration is troubling ?

[brazen idol]

(x^2-y^2)/(2xy) = dy/dx

i just have to put y=vx and solve for this right?

[lament umbra]

yes

as it is homogenous

[brazen idol]

ok nice then i suppose the solution they gave might be wrong

thank you for the help :)

.close

[marsh citrus]

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[lament umbra]

🙂 sure

[marsh citrus]

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[elfin cairn]

y=(x-2)^2 defines are curve (a parabola)
at every point there is a tangent line to the curve, passing through that point. The question asks where is this tangent line perpendicular to the given line

You want to start by understanding:
1 what does it mean for two lines to be perpendicular
2 how to find the tangent linr at a point on the curve

What tools do you have for this?

||derivatives||

One of the geometric interpertations of derivatives is that they give you the slope of the tangent line

Well you should start by taking the derivative

Of y

Yeah the derivative of this

This would be the slope of the tangent to the graph of the function y=(x-2)^2

This is where you should think again what it means for lines to be perpendicular

Yes

You need to understand when are these lines perpendicular

Which wouldn't be the intersection

Also note, 2x-4 is a number which is the slope of the tangent line!

Do you know what it means that two lines are penpendicualr?

This is right yes

But you can't really work with that here

Do you know what it means about their slopes?

The product of the slopes would be -1

Does this ring a bell?

Hmmm.. to explain why is it true I think we will need to get to what are angles

Which is a pain...

||for me||

Explain what?

[elfin cairn]

Continuing from this, the slope of the tangent is 2x-4, the slope of the othet line is 2
So you want
2(2x-4)=-1

This gives x=7/4

And now you need to find the slope of the tangent line (which you already know) and the y-intercept, which can be found using the fact that the tangent line goes through (7/4, (7/4 - 2)^2)

Of the tangent line

This is the slope!

No! This is not the equation of the line

This is the value of the derivative at x

Well by definition its the unique line with slope equal to the derivative, that passes through the point on the curve

We know it has slope 2x-4

And it passes through (7/4, (7/4-2)^2)

[marsh citrus]

@still temple Has your question been resolved?

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[native tapir]

how do I do 9c

[ionic ravine]

hello can i ask something except doubt

[stoic saddle]

go to #discussion or #chill

[stoic saddle]

do you have any progress so far or would you like to start from scratch

[ionic ravine]

i want to send a picture

[stoic saddle]

well then you have to take your own help channel

see #❓how-to-get-help for how to do that

[marsh citrus]

@native tapir Has your question been resolved?

[stoic saddle]

op gone again? 💀

[marsh citrus]

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[long prawn]

Could i ask for some practical advice?

[long prawn]

Im doing the Cambridge/Imperial college uni entrance test for engineering and sciences tmr

i know the content well and have done many practice papers but i dont know if its because of the stress of the exam coming up or because of sleep (which i feel like i havent missed that much of) but my basic arithmetic is in shambles at the moment

and sometimes my mind will just go blank because of it which is a problem because the test the 'ESAT' is incredibly time pressured and thats the main difficulty of the exam

is it worth spending the night revising and relying on caffeine and adhd medication to make up for the lack of sleep if i think its beneficial?

especially considering how bad my arithmatic has suddenly become

[elfin cairn]

Go to sleep early and come to exam with good spirit

You can't improve that much overnight

Especially without sleep

[long prawn]

im aware but im just thinking about an extra 2 hours to compliment the entire summer ive spent preparing

[elfin cairn]

You can do a quick review sure

Just make sure to sleep well

[long prawn]

is 6 hours like significantly different to 8

[elfin cairn]

yes

[long prawn]

alright

[ripe coral]

I'm cooked guys i graduated high school while only knowing basic math knowledge

[long prawn]

do you think this is normal? as in my arithmatic getting worse right before the exam

[long prawn]

what course are you gonna do

[ripe coral]

aircraft maintenance or med

[neat elk]

if you let the stress mess with ur head then yes, it’s normal

you shouldn’t be doing that tho

[long prawn]

i cant tell if its me or said sress

stress*

[neat elk]

if you’ve been doing better before and have been doing well during ur review/previous practice

it’s the stress

[long prawn]

alright cheers

[marsh citrus]

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[solar nest]

The question is in turkish
Translation : On the picture below is an archaic lever system to pull water from a well while one of the lever is 30 units above ground the bucket at the other end is at ground level if the bottom of the bucket touches the waters surface zeynel lifts the lever a bit more how many units does does the distance between the tip and the ground increase ?
(the rope is streched in all cases)

[solar nest]

my answer is 45 i will post how i solved it here aswell but im not quite sure if im correct

can anyone please look into it to see if im wrong or not ?

how i solved this

[marsh citrus]

@solar nest Has your question been resolved?

[marsh citrus]

@solar nest Has your question been resolved?

[lament umbra]

it would havebeen better if it was english
🫠

[solar nest]

i literally translated the question

[solar nest]

.

[lament umbra]

oops my bad
🫠
didn't check

[solar nest]

i asked bc im not sure if the actual answer is 45 and i dont have an answer key

[marsh citrus]

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[quaint arrow]

first i found the second order derivative as shown below, am i on the right track?

[spark otter]

I don't know how you got this "w" factor in your first order

[quaint arrow]

the equation is g(w) = a +b^tw + w^tCw

[spark otter]

yep

[quaint arrow]

that's where i got it from. im taking the derivative with respect to w

[spark otter]

then you didn't do it properly

what's the derivative of b^tw with respect to w

[quaint arrow]

omg im dumb, sorry that would be b

so the second order is simply 2C?

ok thanks. So now do i just have to show that 2C is always positive?

wait im slightly confused, in the previous question it asked:

im just confused on how to use the method i used in part a

[spark otter]

you want to show 2C is "symmetric semi-definite"

or whatever you called this

that's the requirement for convexity, for second-order definition

[quaint arrow]

i'm just confused because the second order derivative is 2C, which is not in the form of

[spark otter]

second order derivative is D = 2C

so you'll just show $z^\top Dz \geq 0$ instead

[elfin berry]

Raphaelisius Maximus MMIII

[spark otter]

which is the same thing as this because factor 2

[quaint arrow]

so just to make sure, i am proving that w^T [[2, 2], [2, 2]] w >= 0, right?

[spark otter]

I don't think using w is wise

Let $f(\mathbf w)$ be my function and $D(\mathbf w)$ be my second order derivative. I need $D(\mathbf w)$ to be semi-definite, for all $\mathbf w$

[elfin berry]

Raphaelisius Maximus MMIII

[spark otter]

So I need $\mathbf z^\top D(\mathbf w)\mathbf z \geq 0$, for all $\mathbf z$, for all $\mathbf w$

[elfin berry]

Raphaelisius Maximus MMIII

[spark otter]

w is the point around which I want to know about convexity

z is ANY direction, having started from point w

so "$\mathbf z^\top D(\mathbf w)\mathbf z \geq 0$" means "Starting from point $\mathbf w$, and going in direction $\mathbf z$, I'm never decelerating"

[elfin berry]

Raphaelisius Maximus MMIII

[spark otter]

If this is true for every starting point, and for every direction, then your function is convex

[quaint arrow]

wait is it possible to calculate the eigenvalues of of C, and if they are non-negative, that proves that z^tCz >=0?

[spark otter]

that's exactly what you proved in (a)

so it would indeed be wise to use it

moreover, $D(\mathbf w) = 2C$ for all $\mathbf w$

[elfin berry]

Raphaelisius Maximus MMIII

[quaint arrow]

oh so im calculating the eigenvalues of 2C

[spark otter]

2C or C, whichever. There's a positive multiplicative coefficient between them

[quaint arrow]

ok i see, thank you so much

[spark otter]

if L is an eigenvalue of C, then it corresponds to 2L as eigenvalue of 2C

and vice versa

[quaint arrow]

.close

[marsh citrus]

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[still temple]

Hiii

[still temple]

I needed someone to proof read my work

[past thicket]

Just for future reference, do you mind posting the image/question as your first message so the bot doesn’t pin greetings? This way, if the conversation continues for long, helpers don’t have to dig.

Post it!

[still temple]

Sorryyy

[past thicket]

it’s okay.

[still temple]

YAYYY

[marsh citrus]

@still temple Has your question been resolved?

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

i need a helper 😭

<@&286206848099549185>

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

😔

.close

[marsh citrus]

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[craggy pelican]

what is the answer for this?

[marsh citrus]

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[marsh citrus]

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[hollow crane]

Can someone just explain how could I rewrite the equation to represent all points of x that would be a solution for this equation?

[wary kite]

parameterize it is what i assume they want

[devout mauve]

well the best way to represent the solutions would be to draw the line I would say

[hollow crane]

alr

[wheat ridge]

i think you make one of the two variable in term of the second one and then write all the solution as pairs (x,y) and sub y or x from the given formula

[hollow crane]

so how would that go? could ya just give an example

[wary kite]

solve for y

it’s how i said

parameterize

[wary kite]

^

[wheat ridge]

for example if you given the equation 3x+7y=3 and want to represent all of the solutions to this equation
first you solve for x or y it is up to you lets solve for x you get x=(3-7y)/3--()
then you write the solutions as pairs (x,y) and sub the variable you solved for, here i solved for x
finally from () you get ((3-7y)/3,y) is the pairs of all solution
take y=0 for example
you get the pair (1,0) is a solution
take y=3 --> (-6,3) is another solution and take more value of y you get more solution...
solve your problem like this...

[marsh citrus]

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[gilded surge]

For a function ( f(x, y) ) developed around ( (2,2) ), the following holds:

$$
p_4(h, k) - p_3(h, k) = \frac{4h^4 - 32h^3k + 102h^2k^2 - 140hk^3 + 61k^4}{3}
$$

and

$$
f(3,3) - p_4(1,1) = \frac{17}{3} + 0.03002\ldots
$$

Which function is ( f(x, y) )?

[elfin berry]

Skiii

[gilded surge]

Any idea where to start with this one?

[empty raft]

guys I need help with a question: two taps are filling up a countainer. One of them takes x hours to fill it up, the other one takes y hours. Which expression would represent the time they would take to fill up the container together?

[marsh citrus]

@gilded surge Has your question been resolved?

[rancid galleon]

Hello

[marsh citrus]

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[gilded surge]

For a function ( f(x, y) ) developed around ( (2,2) ), the following holds:

$$
p_4(h, k) - p_3(h, k) = \frac{4h^4 - 32h^3k + 102h^2k^2 - 140hk^3 + 61k^4}{3}
$$

and

$$
f(3,3) - p_4(1,1) = \frac{17}{3} + 0.03002\ldots
$$

Which function is ( f(x, y) )?

[elfin berry]

Skiii

[gilded surge]

Any idea where to start?

[charred egret]

Many function shares the same 4th order Taylor expansion at a point, I don't think you can uniquely determine the whole function

[gilded surge]

Thats what i though, but does that implies with the taylor error also?

Otherwise, i think im supposed to give an example of a function rather than ”solving it”

[marsh citrus]

@gilded surge Has your question been resolved?

[marsh citrus]

@gilded surge Has your question been resolved?

[quaint badge]

@still temple justice por el xd

[still temple]

What

@quaint badge

[marsh citrus]

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[lofty blade]

How do I turn this trinomial into a binomial? 6x^2 + 3xy + 8y^2

[lofty blade]

,tex 6x^2 + 3xy + 8y^2

[elfin berry]

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

[slim shard]

factorise it

[lofty blade]

Yeah how do you factorize?

[slim shard]

well people have different methods for it

[lofty blade]

So, how would you do it?

[slim shard]

what method were you taught to factorise a quadratic? it may be extended for this

[lofty blade]

i think it was like adding the first and last co-efficients or multiplying them

[slim shard]

wait are you sure you didnt make a typo with this?

try to send the original question, maybe with an image

[lofty blade]

oh wrong question lol

its this (3x+4y)(6x+9y)

[slim shard]

okay

you can expand that

[lofty blade]

my teachers said to use foil

,tex \displaystyle{\displaylines{(3x+4y)(6x+9y)=}}

[elfin berry]

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

[slim shard]

idk what that is lol. over here they teach us to use the 'rainbow method' (which is shown in the image)

in any case, just apply foil here to expand it.

[lofty blade]

oh ok

so like

[lofty blade]

so like this?

,tex \displaystyle{\displaylines{9x+(3x)(9y)+(4y)(6x)+13y}}

[elfin berry]

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

[slim shard]

no

[lofty blade]

are the variables gonna combine

like xy

[slim shard]

youre getting 9x by adding 3x and 6x right? and 13y by adding 4y and 9y?

that's wrong, you multiply 3x and 6x; and multiply 4y and 9y

[lofty blade]

OHHH its multplying

im dumb

so is it this

,tex \displaystyle{\displaylines{18x^2+27xy + 24xy+36y^2}}

[elfin berry]

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

[slim shard]

looks right

[lofty blade]

ok, so i add 27+24

[slim shard]

yup

btw

[lofty blade]

then i get this

[slim shard]

use $ $ for latex math mode

[lofty blade]

ohh ok

,tex $\displaystyle{\displaylines{18x^2 + 51xy + 36y^2}}$

[elfin berry]

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

[slim shard]

$18x^2 + 27xy + 24xy + 36y^2$ for example. put non-math stuff outside the dollar signs

[lofty blade]

,tex $18x^2 + 51xy + 36y^2$

[elfin berry]

QwertyDev

[lofty blade]

ohhh ok

[slim shard]

yeah

[lofty blade]

ty

[slim shard]

that's correct

[lofty blade]

ok ty

so

yeah

.close

[marsh citrus]

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[ocean patio]

im like super confused on the content of this page

[marsh citrus]

@ocean patio Has your question been resolved?

[marsh citrus]

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[oak oxide]

I'm seeing Geo, seems like the paper ask about special line segments

[marsh citrus]

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[pliant coral]

how is the horizontel shift of this equation 8 and not 56? and why is it to the left instead of right?

[pliant coral]

isn't horizontel shift meant to be h?

[split vine]

yeah

but there's a -h

so it's the negation of whatever's there

[pliant coral]

I'm not quite sure what you mean

is it just meant to be 7h = 56?

[split vine]

no the 7 only applies to the x

[pliant coral]

Mmmm

okay

[split vine]

so these equations are usually written with the -h right

[split vine]

but here there's a 56

the only way it can go from a - to postiive

is if there was a double negative

in other words if there was x - -56

[pliant coral]

okay?

[split vine]

so if you compare x - -56 to x - h

h has to be -56

[pliant coral]

yeah

so far so good

[split vine]

and then a negative shift will move it to the left

because the left contains the negative numbers

like if you move to the left you move to the more negative end of the x axis

[pliant coral]

okay I understand why its to the left now

but I still don't understand why its 8 and not 56

[split vine]

sorry i misread this before yeah so it would be 7h = -56

so then h = -8

[pliant coral]

but

it my ciriculum is saying h = 8?

oh maybe it just doesn't care

[split vine]

ngl it's been almost four years since i've taken algebra/geometry so idk if im the best personn to help sorry

[pliant coral]

because saying to the left is the same as saying its negative

its all good

[split vine]

ohh i see

oh yeah so if you specify left yeah you don't have to say -8

but if you said right ig you'd have to say -8

[pliant coral]

you still awnsered my question

thank you

[split vine]

ofc

[pliant coral]

.close

[marsh citrus]

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[quaint arrow]

i just proved B but now im having trouble proving C

[quaint arrow]

in part b i did this;

so for part c i have

am i allowed to factor this the way i did? @chilly valley

[chilly valley]

One minute, I am busy.<

Ok, so...

[quaint arrow]

is this factorization correct, or is not possible lambda I is being added to each outer-product

[chilly valley]

Yes, that’s fine, you can factor it that way. The expression still makes sense mathematically because it’s a scalar (a quadratic form). It’s equivalent.

[marsh citrus]

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[naive moat]

Hi, I am following the book Linear Algebra done right by Sheldon Axler to teach myself linear algebra, but I am finding it very hard to understand some concepts. Can you recommend some additional resources that I should use with the book?

[naive moat]

<@&286206848099549185>

.close

[marsh citrus]

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[naive moat]

.reopen

[marsh citrus]

✅ Original question: #help-33 message

[indigo nest]

You could just ask here about the concepts you're finding difficult

As for another reference, my professor likes Hoffman and Kunze for his introductory course

[naive moat]

I am having trouble in understanding vector space and subspace

I mean, I just got started with the text, and I am unable to understand them

.close

[marsh citrus]

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[odd agate]

is this your first rodeo in linalg?

(reopen if interested in discussing.)

[novel juniper]

.reopen

[marsh citrus]

✅ Original question: #help-33 message

[novel juniper]

@naive moat

are you still here

[naive moat]

Hi

[novel juniper]

what are you finding hard in LADR

do you know how to write proofs?

[odd agate]

if I may say, do consider this question first

[naive moat]

I have never done proofs before

[naive moat]

yes

[odd agate]

because this is quite some important context considering the target audience of LADR

unfortunately, LADR is written as a second exposure to linear algebra

it's not for newcomers as far as I know

[naive moat]

Understanding new concepts, Vectorspace subspace and proving them as well

[novel juniper]

IT can work for them

but proof writing is essential

[naive moat]

I see

[novel juniper]

without knowing how to write proofs LADR isn't doable

[karmic widget]

how does linear algebra work?

[odd agate]

good question, but unfortunately I'm not sure if OP wants to discuss that

if you want to discuss what linalg is, do open your own help channel please

[naive moat]

So, we have linear algebra as one of our course in uni but the thing is, they have greatly reduced the syllabus of linear algebra, so I wanted to learn all there is to learn in an introductory linear algebra, so I started exploring,

[karmic widget]

thank you for the info, see you next time.

[cobalt sedge]

you can follow video lectures too, there are plenty available on yt

[novel juniper]

for that use strang's lectures

https://youtu.be/J7DzL2_Na80?si=xJoc83hh-GLdohjr

start with this

[naive moat]

Alright

[novel juniper]

And please learn proof writing

[naive moat]

I will, can you guide me how should I start with that?

[novel juniper]

There's a book

https://richardhammack.github.io/BookOfProof/Main.pdf

[naive moat]

Thank you so much

[elfin cairn]

Practice is the real way you improvr with proofs

[naive moat]

Got it

[elfin cairn]

Make sure to solve exercises, and maybe get some feedback here

[novel juniper]

But knowing the 4 main techniques is key

[naive moat]

Will do

[novel juniper]

esp contrapositive and induction

[naive moat]

i see

Thank you for the help @novel juniper @elfin cairn @odd agate @cobalt sedge .

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[odd agate]

I didn't help much, but yw

[marsh citrus]

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[hexed pike]

I don't know how to do part c

I'm pretty sure that the answer is similar to the function in part a. but I don't know how do I get to it

Like I can't just assume X(x) = e^kx something like that

I don't know what to do next

Pls help

[marsh citrus]

@hexed pike Has your question been resolved?

[marsh citrus]

@hexed pike Has your question been resolved?

[gray spire]

d²X/dx² = -λ²X and d²Y/dy² = -λ²Y
notice that the second derivative of X(x) and Y(y) is proportional to X(x) and Y(y) itself, if you remember your derivative formulas, you know d/dx(e^ax) = ae^ax, so if you differentiate that again:
d/dx(ae^ax) = a²e^ax
so X(x) = e^-λx and Y(y) = e^-λy

pretty sure the cosine terms from question a come from complex exponents

[hexed pike]

Thank you for answering but the problem is that, I know the answer, but I don't know how to proceed to the answer.
Im pretty sure that X(x) = e^kx OR e^-kx and Y(y) = cos(ky) OR sin(ky) because these satisfied the differential equations and the Laplace's equation as well(actually same as part a).
But how do I prove that these are the general form instead of random answers that satisfy the equations

Because I realized that when I saw part a

And the funny part is that I've never learnt about ordinary differential equations and how to solve them

[gray spire]

i'm pretty sure you get e^kx by solving with integrating factor and cos(ky) with separation of variables

you can use integrating factor since laplace's equation is linear btw

[hexed pike]

Hmmm how to do that. Sorry this is my first time handling it lol

Like I don't know what is the X(x)

[gray spire]

when you have an ODE with the form:
dy/dx + P(x)y = f(x)
find e^∫P(x)dx and let u(x) be equal to it
d/dx[u(x)*y] = u(x)f(x)

then you can just integrate

[hexed pike]

OK I see, but I don't understand how do I change when there's a second derivative

Idk if it would still work

[chilly valley]

the integrating factor method only applies to first-order linear ODEs.
If there’s a second derivative, the equation becomes a second-order ODE, so you’ll need different techniques, like finding the characteristic equation or using reduction of order, depending on its form.

[hexed pike]

Oh thank you. let me look it up

[chilly valley]

Ok!

[hexed pike]

Oh my

Damn I just found a organic chem tutor video about 2nd order ode

And I understand everything now

Damn

Thank you

You guys saved me

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[gray spire]

oh yeah i forgot that

[gray spire]

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[calm harbor]

. @royal chasm post here

[royal chasm]

bruh

[royal chasm]

i dont have any match questions rn

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[earnest belfry]

I was trying to find the interval(s) within [0, 6pi] for the function a(x) for which it is negative so I did the following:

[earnest belfry]

Did I do something wrong? How can I interpret the answer from there?
That's the Ti Nspire
if anyone can help

The full output to solve() is this:

Is that a normal output?

I didn't want the output to have a second veriable in there (n_3) . . .

I'm quite startled as to why the final output added n_3 when I just wanted to find the inequalities of x in terms of numerical boundaries . . .

[marsh citrus]

@earnest belfry Has your question been resolved?

[marsh citrus]

@earnest belfry Has your question been resolved?

[marsh citrus]

@earnest belfry Has your question been resolved?

[marsh citrus]

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[fickle shell]

Is this a valid proof and correct way to write the proof? Im still learning how to write them so any tips to improve it are welcome, no matter how nitpicky haha

[jagged relic]

That first sentence doesn't make much sense

[small berry]

lgtm, like I would give full marks for this. To be a bit nitpicky, No need to write "and there exists y in R" in the first line. You cn go straight to the second line and say y = 4-x (in R), to satisfy the condition that y must be real

[pallid tapir]

yeah you don't attach logical quantifiers to variable proclamations

[viscid pond]

And i would probably add like "4 >= 4 for all x"

[mellow kraken]

For any given x, you need to find this y.

[small berry]

Also no "for all" before x in line 1, just x in R

[pallid tapir]

just "let x in R" is fine. you can't just say "let y in R" however, because you want to define it, not let it be free

[pallid tapir]

not sure if I should say this, but I wouldn't give full marks for this. the language is important at all levels beyond this point!

[small berry]

Retrospectively, I agree once I read it more closely. I would cut a small bit if it's part of a full-on RA proof. If it's a literal "proof" class, this would be borderline at best.

[fickle shell]

https://tenor.com/view/mochi-mochimons-cute-kawaii-kitty-gif-492947389134426846

Alright thank you everyone this is all really useful info!!

❤️

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[pallid tapir]

. you can share any revisions you have for further notes

[marsh citrus]

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[earnest belfry]

.reopen

[fickle shell]

Why can we say X is a subset of Y and Y is a subset of X?

[fickle shell]

This is the question and its solution btw

[jagged relic]

Definition of antisymmetric relation

[fickle shell]

But the solution is just the definition of an antisymmetric relation no?

Whats the actual proof?

[jagged relic]

"if XRY and YRX then X=Y" -> R is antisymmetric

Let X,Y in P(A). Assume XRY and YRX, so X subset of Y and Y subset of X. Then X=Y. So R is antisymmetric.

[fickle shell]

Ok just so that I understand can you explain what 'on P(A)' means again

Like I know that P(A) is the powerset of A is but idk what 'on' means still

Like what the original statement is trying to say

[fervent rampart]

a relation on a set S is a subset of S x S. so it's saying what set the elements being related are from

[jagged relic]

R_8 is a relation on P(A), meaning it's a set of ordered pairs of members of P(A)

And XRY means (X,Y) in R

[fickle shell]

So only the sets in P(A) that contain ordered pairs? No more and no less?

[jagged relic]

No

The elements of the ordered pairs are in P(A)

[fervent rampart]

R is a subset of P(A) x P(A)

[jagged relic]

Not the pairs themselves

[fickle shell]

Ok another question are ordered pairs (1, 2) and (2, 1) both within a power set of {1, 2}

[jagged relic]

I just said the pairs are not in P(A)

[fickle shell]

But they are in the powerset of {1, 2}, no?

[jagged relic]

No

Say A = {1,2} instead so I don't have to write too much...
P(A) = { {}, {1}, {2}, {1,2} }
Those four elements are subsets of A. Those four elements can be chosen in pairs to form 16 different ordered pairs.
For example, ({}, {2}) and ({2}, {1}) are 2 of the 16 pairs.
A relation on P(A) is a set of such pairs. There are 2^16 possible relations since there are 16 pairs.

[fickle shell]

Ok yeah thats what I was wondering

So you cant have {2, 1}

[jagged relic]

R_8 is a "subset" relation, so it contains ({}, {1}) and ({2}, {1,2}) and ({1}, {1}) for example

[jagged relic]

Not sure what you mean by that

[fickle shell]

P(A) = { {}, {1}, {2}, {1,2} }

Theres no {2, 1}

Which was what I was trying to ask

[jagged relic]

{2,1} = {1,2}

It's the same set

[fickle shell]

Oh

But for relations it matters right?

[jagged relic]

Ordered pairs are not sets

As the name implies, order matters for ordered pairs

Not for sets

[fickle shell]

But you get the order pairs from the sets of the power set no?

[jagged relic]

No, you get the elements, and you make the pairs

The set {a,b} has two elements: a and b. You can make 2^2 pairs with them: (a,a), (a,b), (b,a), (b,b)

A relation on {a,b} is a set of such pairs, for example R = {(a,a), (b,a)} is one

R = {} is another

The pairs come from {a,b}^2, and the relations are elements of P({a,b}^2)

[fickle shell]

So the elements of P({a,b}^2) are ((aRa), (aRb), (bRa), (bRb))

[jagged relic]

No

The elements of {a,b}^2 are (a,a), (a,b), (b,a), (b,b)

The elements of P({a,b}^2), you should be able to imagine by taking the power set...

[devout hill]

its a badly written solution.
they said, basically:
For which X,Y is XR8Y symmetric? it is when XR_8Y and YR_8X. they showed that this happens only when X=Y. So it is not symmetic for all X,Y.

[jagged relic]

{}, {(a,a)}, {(a,b)}, ..., {(b,a), (b,b)}, ..., {(a,a), (a,b), (b,a), (b,b)}

16 total elements

[fickle shell]

Ok I get it

[devout hill]

ignore this im blind

[fickle shell]

So what does it mean by let X and Y exist on P(A)?

Like X and Y are subsets?

[jagged relic]

It never says "exist on"

It says they are elements of P(A), meaning they are subsets of A

[fickle shell]

Sorry yes

Mental typo

[fickle shell]

Ok I see

So X relates to Y iff X is a subset of Y (and X and Y are both subsets of P(A)^2)?

[jagged relic]

... no, subsets of A

[fickle shell]

Wait why A sorry

[jagged relic]

.

[fickle shell]

Ahhhhhhhhh

Ok so X relates to Y iff X is a subset of Y AND X and Y are both subsets of A

[jagged relic]

For R_8, yes

[fickle shell]

Okok

But wait why does the answer say Let X and Y be subsets of P(A)? Werent they only supposed to be subsets of A?

[jagged relic]

Where does it say that?

[fickle shell]

In the proof right?

[jagged relic]

This?

[fickle shell]

Yeye

[jagged relic]

Can you show me the symbol for "subset of"?

[fickle shell]

I thought you said this ^^

Oh wait

Nevermind

I get it

[queen sun]

If you see “in P(A)” you can think of “subset of A” vice versa

[fickle shell]

On P(A) is the same thing?

[queen sun]

On?

Do you mean forall a in A?

[fickle shell]

X is a subset of Y on P(A)

[queen sun]

X is a subset of Y isn’t that it, what’s the on P(A)?

[fickle shell]

Thats what it says in the statement

[queen sun]

Oh I see

[jagged relic]

No that's not a thing, you are mixing up terminology

[queen sun]

X and Y are in P(A)

“On” in this context is a part of the semantic definition of a relation

[jagged relic]

A relation on a set
A subset of a set
An element of a set
"x in S" means x is an element of S
"R on S" means R is a relation between elements of S

[fickle shell]

Okkkk

I think I understand it all now hahaha

thank you so much!

❤️

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[finite forum]

is this a decent graph for part d

except from stating the coordinates, i will do that soon

[fervent rampart]

if it's in freefall then the acceleration (slope of v-t graph) should be constant (with value -g)

[finite forum]

yeah this is what i though

but

i feel like the longer that it is in freefall, the faster the velocity will be

and it will lurk at 0 for a bit

[spark otter]

you're mixing up position and velocity

the stone's position indeed lurks around its maximal height for a bit

but the acceleration is constant

[finite forum]

velocity will be decently close to zero at the blue point?

[spark otter]

and so the velocity is...

[spark otter]

not nearly as much as you think

[finite forum]

hmm

[spark otter]

if the acceleration is constant

then by integrating

what type of function is the velocity?

[finite forum]

linear?

[spark otter]

yes

so v(t) = at+b

[finite forum]

okok

thanks

guys

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[elfin badger]

gauss jordan elimination is making a upper triangle with the leading entires = 1

so for this question id use these rules

so are we trying to get this

[wispy stump]

What exactly is the problem?

[elfin badger]

part c

[fervent rampart]

that's the form it should be in if we have a unique solution, yes

[elfin badger]

right okay

is there a technique to use to get it into that form

or do we just test it out

[queen sun]

For reduced row echelon form you need it to be in row echelon form and then the missing numbers in the 3x3 need to all be 0’s

So in this case it needs to look like I_3

[fervent rampart]

the technique is called gauss jordan elimination

gauss jordan provides you a specific set of steps to follow to get a matrix to reduced row echelon

[elfin badger]

it involves Adding or multipling the rows to get it into that form

[queen sun]

Yeah

[fervent rampart]

yes, those are the basic operations you can do

gauss jordan tells you which operations you should perform and in which order

[queen sun]

Since you know the answer to b you can say what the other 3 numbers are

[elfin badger]

i thought we had to determine this our selves

[fervent rampart]

you can use your judgement and perform any sequence you want

but gauss-jordan elimination does give you a set sequence to follow, so if there isn't a compelling reason to do it another way you can just follow those steps

[elfin badger]

and the steps are?

[fervent rampart]

your class notes or textbook will probably have the steps of gauss-jordan elimination listed out somewhere

[queen sun]

Are we talking about other problems that might come up in general here because this one requires no row operations

[fervent rampart]

from the note below, i'm pretty sure the question wants you to solve it by gauss-jordan and check afterward that the solutions match with (b)

[elfin badger]

i know the answers for b

[fervent rampart]

for reduced row echelon form, the columns should be all 0 above the pivot

[queen sun]

So it should look like $I_{3}$

[elfin berry]

BBMaths

[queen sun]

Can I make texit images smaller? I don’t like how big they are sometimes

[elfin badger]

this is I3

[queen sun]

Just in this case, as in general you might have zero rows

[fervent rampart]

,tc alwayswide off

[queen sun]

I_3 should be in here no fractions

[elfin badger]

the answer to be was fractions

[queen sun]

That’s not where they go

[fervent rampart]

in this instance we expect that at the end of gauss-jordan elimination, the only fractions will be in the augmented column

[queen sun]

They will go in the vector on the right

[elfin badger]

and why do they go there?

[queen sun]

Imagine the written down answer to b is actually 3 linear equations 1x+0y+0z=X 0x+1y+0z=Y 0x+0y+1z=Z

[elfin badger]

oh

[queen sun]

Where X, Y and Z are the answers

[elfin badger]

so the advantage is is that you can just read off the answers/

[queen sun]

Yeah

[elfin badger]

thanks

[queen sun]

For REF you need to back substitute it but I think it might be faster to compute that way

Not sure

Might be same complexity actually

I have no idea

[elfin badger]

so here we have to back sub

the yellow bit

[queen sun]

You can also back substitute the previous one

As long as it’s upper triangular

[elfin badger]

hmmm

[queen sun]

Scaling won’t affect that

[elfin badger]

so what would it look like if i did back sub