help-23

[valid jetty]

alr

@barren bronze

[barren bronze]

Yes

[valid jetty]

how good is ur vectors?

[barren bronze]

Ehm took me like 10 tries to draw what Cnidarian posted

[valid jetty]

oof

alr think of it like this

[barren bronze]

so not so good but i kinda get how they work in this scenario

[valid jetty]

which direction is the arrow facing

right or left

?

[barren bronze]

right

[valid jetty]

so theres a right side component

now is the arrow facing up or down

?

[barren bronze]

down

[valid jetty]

so theres a down component

got it till here?

[barren bronze]

Yes think so

[valid jetty]

now how good is ur trigo?

[barren bronze]

Its Sin 45 and Cos 45

[valid jetty]

which is which?

[barren bronze]

Fv is F sin 45

And Fh is F Cos 45

[valid jetty]

ok good

from question

they gave f =?

[barren bronze]

280 N

[valid jetty]

good

now how much is sin 45?

[barren bronze]

.7 is sin 45

[valid jetty]

uh

alr ig

idk the point values

i just write it as 1/root2

[barren bronze]

I mean 280 x sin 45 = roughly 198

[valid jetty]

so thats ur answer

[barren bronze]

yes 198 N

[valid jetty]

thats the answer for 1

[barren bronze]

Damm so horizontal is just cos 45. ok how about the weight?

[valid jetty]

yes so horizontal component also stays same as cos45=sin45

now coming to weight

they gave u mass as 50kg

now listen here

when u stand on a weighing machine what do u think it measures?

[barren bronze]

my mass

[valid jetty]

no

[barren bronze]

newtons

[valid jetty]

it measures ur weight

[barren bronze]

50 kg x 10 right ?

[valid jetty]

weight is a force

[valid jetty]

yes

[barren bronze]

So 500 N

[valid jetty]

the force
ur mass multiplied by gravity

[valid jetty]

yes

[barren bronze]

do we remove 280 since we are pushing?

[valid jetty]

but it might change if ur book took g = 9.8

[valid jetty]

hmm

one sec

no

if they ask apparent weight then u remove

no in this case u actually add the vertical component force

as weight and force r same direction

[barren bronze]

oh wait its 50 x 9.8 not 10

[valid jetty]

yes

[barren bronze]

490 Newtons is the weight

[valid jetty]

@barren bronze did u get this tho?

[valid jetty]

yes

[barren bronze]

no not really

[valid jetty]

hmm

alr so

[barren bronze]

last one?

[valid jetty]

the weight acts downwards ryt

[barren bronze]

yes

[valid jetty]

no first i gotta clear this concept

[valid jetty]

and a component of the force also acts down right?

[barren bronze]

which force?

you mena Fv

[valid jetty]

the 280 newtons force

[valid jetty]

yes

fv

[barren bronze]

197 N?

[valid jetty]

yes

since both r acting on same direction

u add

[barren bronze]

So 197+490 = 687

[valid jetty]

yes

[barren bronze]

so normal force must be 687

[valid jetty]

that is the total force on the grounf

[valid jetty]

YES

that is normal force

alr now coming to the last one

do u know how friction works?

[barren bronze]

its opposite to the pushing force

[valid jetty]

uhh

no

dont think of it like that

friction is always opposing motion

so if not for friction
normal which direction would this body move?

[barren bronze]

just forward

[valid jetty]

yes

so it moves forward

[valid jetty]

now using this tell me which direction will friction be

[barren bronze]

Backward

[valid jetty]

good

do u know value of friction?

[barren bronze]

not for this question no

[valid jetty]

how its magnitude is calculated

alr listen

[barren bronze]

Should i use this formula

,rotate

[valid jetty]

theres this thing coeffecient of friction which is notioned with mu( the sign)

friction is always equal to mu multiplied by normal force

[barren bronze]

yeah in this case its mu d

[valid jetty]

whats d?

[barren bronze]

dynamic friction

[valid jetty]

oh ok that was prefix

yea

mu d

[barren bronze]

But we dont have mu?

[valid jetty]

that is what they asked us to find

now listen here

the crucial word in the question is

constant velocity

what does that mean?

[barren bronze]

Same speed

[valid jetty]

what it also means is that

the acceleration is zero

[barren bronze]

Yes

[valid jetty]

only then its possible

so we have a condition that acceleration is zero

doesnt that mean net force is also zero?

[barren bronze]

but net force cent be zero if its still moving right?

[valid jetty]

y not?

[barren bronze]

Idk

[valid jetty]

if theres force means its accelerating

means speed is changing

but thats not the case

net force acting on the body is zero

becuz speed is constan

[barren bronze]

we know the friction force?

[valid jetty]

when we equate acceleration to zero we get it

as acceleration is zero

force is zero

as f = ma

then u get this equation

fh - (mu d ) * N = 0

we know fh

we know normal

[barren bronze]

Ya 198

[valid jetty]

cant u find mu d?

so we get

[barren bronze]

198/688

[valid jetty]

fh = ( mu d ) * N

[barren bronze]

fh/N

[valid jetty]

yes

yes

one more interesting thing

[barren bronze]

YES

[valid jetty]

coeffecient of friction is always b/w 0 and 1

its a fractional value

[valid jetty]

did u understand the process?

[barren bronze]

ya once i had the formula

[valid jetty]

alr thats good

[barren bronze]

so i draw the free body diagram first?

[valid jetty]

yes

split everything into x and y components

and then find normal and based on question react

[barren bronze]

And this was right?

[barren bronze]

Ya makes sense i take it step by step

[valid jetty]

which direction is fh and fv?

[valid jetty]

yes

[barren bronze]

Fh is right and Fv is down

[valid jetty]

well u got it but u gotta put arrows on fbd or u get confused

[barren bronze]

Aight imma have to go but yeah it makes more sense now imma have to practise more though

[valid jetty]

alr

glad i can be of help

[barren bronze]

thanks for your patience man ,ur a real one🫡

can i add u bro

[valid jetty]

clearing those concepts is imp once they clear the chapter gets easy
so ye i had to be patient

[valid jetty]

ye sure mate

gl

[barren bronze]

anyways thx man

Have a good day mate

[valid jetty]

anytime

[valid jetty]

u too

@barren bronze do close the channel

.close

[safe radish]

@barren bronze Has your question been resolved?

[safe radish]

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[dire storm]

how can i graph a complex number in a 3d euclidian space

[pale ether]

3d desmos grapher

3d calculator geogebra

[dire storm]

by myself

[upbeat ridge]

You can graph complex numbers in desmos?

[main mural]

what exactly do you mean

if you have a function $f: \bC \to \bC$ then you can't plot it in 3d regularly

[flat frigate]

artemetra

[main mural]

you usually plot, for example, the absolute value of the result, which makes it a $\bC \to \bR$ function

[flat frigate]

artemetra

[main mural]

or the argument

but to plot a C → C function you'd need 4 dimensions

two for input and two for output

[dire storm]

so two real axes and an imaginary axis

[dire storm]

exciting

[main mural]

huh?

oh

[dire storm]

for the 3d

[main mural]

yes, correct

[dire storm]

what are your sources for this information?

[safe radish]

@dire storm Has your question been resolved?

[dire storm]

@main mural

[main mural]

for what information exactly

[dire storm]

i would need 4 dimensions to graph a complex function

is this common knowledge when studying complex analysis

[main mural]

sorta

you need two dimensions to represent 1 complex number

so a map from one complex number to another needs to be 2+2=4 dimensions

[dire storm]

thank you

.close

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[safe radish]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[primal crescent]

I have completed the problem and don't need help anymore. Thank you.

[red delta]

Use . close

[safe radish]

@primal crescent Has your question been resolved?

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[primal crescent]

wait

wrong one

<@&286206848099549185>

[lilac crane]

bro how long have you been doing this

i remember doing this hours ago

[primal crescent]

IDK

3 hours

[daring crescent]

@primal crescent Can you plug 2 and/or 10 into f(x)?

[primal crescent]

x=2 is zero and x=10 is Remove discontinuty

[daring crescent]

yep

@primal crescent Did you have another question or?

[primal crescent]

no

not now

use.close

[daring crescent]

type just ".close"

[safe radish]

@primal crescent Has your question been resolved?

[wind lion]

https://tenor.com/view/soviet-union-propaganda-stalinium-gif-16289670004460099406

[safe radish]

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[stark vortex]

( Law of Sines and Cosines ) - Trig
Does anyone know where to start for this problem?

[woven hound]

What are you being asked to solve for?

All missing sides/angles?

[stark vortex]

Yes both sides and angles that are missing

[woven hound]

Okay well right off the bat we don't need any trig to find angle A right

[stark vortex]

Ohh yea I forgot
So it's 85 + 47 = 132

Then 132 - 180 = 48

?

[woven hound]

yes

[stark vortex]

Ok

Then do I apply the law of sines?

[woven hound]

exactly

[stark vortex]

Oo ok

That's all I needed, thank you!!

.close

[safe radish]

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[safe radish]

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[upbeat ginkgo]

Don't know how to begin

[sleek plank]

do you know the formula for a tangent line for normal curves?

[upbeat ginkgo]

uhh

can you remind me?

[sleek plank]

y-f(x1) = f'(x1)(x-x1)

[upbeat ginkgo]

ok ok

I do remember that

[sleek plank]

f'(x1) = dy/dx as well

do you know how to find x and y from r?

[upbeat ginkgo]

not quite sure

but isn't it doing substitution?

idk what to substitute for them though

[sleek plank]

x=r * cos(theta)
y=r * sin(theta)

[upbeat ginkgo]

right

but since its 2 theta, would that work?

[sleek plank]

yea, it'll just be cos(2theta) * cos(theta)

[upbeat ginkgo]

lemme write this down rq

wait

i'm not understanding properly

so what exactly are we exchanging to put in cos(theta)?

[sleek plank]

we aren’t really

think about a right triangle

r is the hypotenuse

if we want the x coordinate of r, we can multiply it by cos theta

since cos theta is defined as adjacent(x-value)/hypotenuse

multiplying by r will just give the x value

[safe radish]

@upbeat ginkgo Has your question been resolved?

[upbeat ginkgo]

sorry

got distracted with something

[sleek plank]

that's ok

[upbeat ginkgo]

ok

so then the equation would be r = cos(2theta) * cos(theta)

?

@sleek plank

[sleek plank]

x equals, not r equals

[sleek plank]

my bad dawg

[upbeat ginkgo]

all good

so now I convert the cos theta to x/r?

[sleek plank]

no, now that we have x and y, we can find dy/dx

[upbeat ginkgo]

wait what

[sleek plank]

if x = r cos(theta)

and y = r sin(theta)

we can differentiate with respect to theta

and get dx/dtheta and dy/dtheta

we can then divide dy/dtheta by dx/dtheta

to get dy/dx

[upbeat ginkgo]

and is r considered a constant when differentiating?

[sleek plank]

is what?

[upbeat ginkgo]

oh whoops

r

[sleek plank]

no, since r is defined in terms of theta

so we can sub in the definition of r from the question

and then we can differentiate

[upbeat ginkgo]

I think it's clicking

let me write it all out and post it here

so that you can check if i'm getting it

@sleek plank

like this?

[sleek plank]

yep pretty much

then you divide dy by dx

and plug in the designated theta value from the problem

which is believe is pi/6

[upbeat ginkgo]

how about the 1/2?

[sleek plank]

i believe the 1/2 is the r-value

i don’t know why the ordered pair was ordered that way, with r first and then theta, but it works out mathematically so

[upbeat ginkgo]

alr, gimme a sec

cause I gotta show my work for this and it has to make sense to be correct

thanks for your patience

[sleek plank]

np man

[upbeat ginkgo]

@sleek plank looks good?

idk why the highlighter blocks the slope

but it's sqrt3/7

[inner parrot]

looks about right

,w slope of tangent line r=cos(2\theta) at (\pi/6, 1/2)

[upbeat ginkgo]

LOL

[inner parrot]

what the hell

[upbeat ginkgo]

https://tenor.com/view/boy-what-the-hell-boy-hell-talking-tom-gif-21980337

[inner parrot]

there we go

wait whys the coordinate system wrong

[upbeat ginkgo]

oh

[inner parrot]

bruh

[flat frigate]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by doctormysterio9

[upbeat ginkgo]

the answers like 0.247...

when I put it into Desmos

[inner parrot]

mm

it should be -sqrt(3) apparently

[upbeat ginkgo]

hmm

[inner parrot]

makes sense that its negative just by looking at the graph

[upbeat ginkgo]

now what have I done wrong?

[inner parrot]

give me a second to look at it

[upbeat ginkgo]

ok, thanks!

[inner parrot]

oh wait wolfram didnt intepret the problem as polar

your answer should be correct

i did it myself and got the same thing

[upbeat ginkgo]

https://tenor.com/view/yippee-happy-celebration-joy-confetti-gif-25557730

yippie!

thanks again!

[inner parrot]

yw!

[upbeat ginkgo]

@sleek plank you too bro!

.close

[safe radish]

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[clever ridge]

how do you find concavity using derivatives I’m having trouble testing and knowing what intervals to use

[clever ridge]

Do I test in between zeros and critical points, to end points

.close

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[dusk cove]

hello can someone help me with this? it's a weighing and scale problem

[west hedge]

A+B+C=D+D+D+D+D

D+D+A=B+B+C

A+A=?

A= 5D-B-C

A=2B+C-2D

A+A=3D+B

E is an option

[dusk cove]

omg thank u I was confused when I made the replacement 😢

[west hedge]

you're just new at these operation

just hone your skill and you will be fine

take more practice

[dusk cove]

thank u buddy, i will

[west hedge]

no problem

[dusk cove]

.close

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[frail beacon]

What are consecutive sides?

[frail beacon]

Apparently they’re congruent???

[quick crater]

huh

[safe radish]

@frail beacon Has your question been resolved?

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[proud veldt]

can someone help? i have tried so many times and always get the same answer which is wrong

I figured it out

.close

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[lean otter]

Help

[light shoal]

maybe try proving the contrapositive

if X is not compact, then either it is not closed or it is not bounded

in each of those two situations, try to construct a continuous function on X that is not bounded

[lean otter]

The contrapositive is true for subsets of "R"

[light shoal]

should also be true for R^2 i believe

[safe radish]

@lean otter Has your question been resolved?

[safe radish]

@lean otter Has your question been resolved?

[safe radish]

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[upbeat ginkgo]

I don't quite understnad what these problems are asking for

I'm expecting to see this type of problem on my exam, so I want to properly understand how to do it

[old chasm]

I am not sure about the absolute sign, but I think they are asking for that term to be less than 10^-6

Hint: ln(a^b) = bln(a)

[upbeat ginkgo]

wdym?

like the term of n to be less than 10^-6?

[old chasm]

$\left(\frac{6}{7}\right)^n < 10^{-6}$

[flat frigate]

#damiforthepresident

[old chasm]

and for $n > 0$, we can see that $\left(\frac{6}{7}\right)^n$ is decreasing

[flat frigate]

#damiforthepresident

[upbeat ginkgo]

cause the denominator is always bigger than the numerator right?

[old chasm]

yes. but at what n it is getting so small?

that is the question.

[upbeat ginkgo]

do we solve this for n?

[old chasm]

yeah

[upbeat ginkgo]

oh

alr

gimme a sec

i'm gonna try to do it and if you could possibly check my work, that'd be great

[upbeat ginkgo]

NOW I see what you meant by this

[old chasm]

and remember to check the sign

[upbeat ginkgo]

?

wym

[old chasm]

you will see

[upbeat ginkgo]

ok ok

[old chasm]

yeah that is pretty much it

[upbeat ginkgo]

ln 6/7 is negative it seems

wait

[old chasm]

6/7 < 1

so yes, it will be negative

[upbeat ginkgo]

mhm

now I calculate the number

and will the greater than sign affect my answer?

cause it's asking for N, not n

so it's tripping me up now

[old chasm]

so you got n > something and your N is the minimum n that is the whole number

[upbeat ginkgo]

n > 89.6234163446

[old chasm]

ok.

[upbeat ginkgo]

so then it'd be 89?

[old chasm]

no

because n must be at least that much

but N also satisfy the condition

I mean, if not then I can pick negative infinity and I will be fine

[upbeat ginkgo]

ok

let me make sure i'm understanding

this confuses me, so bare with me

89.6234163446 is the minimum amount you need

and N is the minimum whole number that you need

so then it'd be 90?

[old chasm]

yes

[upbeat ginkgo]

ah!

ok

so if the number came to be 88.1234

the minimum whole number would be 89

[old chasm]

yes

[upbeat ginkgo]

ahhh

ok ok ok

cool cool

thank you sir!

[old chasm]

no problem

[upbeat ginkgo]

.close

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[lean otter]

Hi, I was struggling to understand how to do this question as there is limited information and the working out provided doesn't make sense.

[south lotus]

Do you know how to find the mean of binomial dist.

And the variance

[inner parrot]

essentially solving a system of equations here

[lean otter]

ik variance, idk binomial dist

[lean otter]

how do you do that with probability?

[inner parrot]

you have to use the formula for mean and variance of a binomial distribution

do you know them?

[south lotus]

Okay, what do you know about variance

[weary hull]

Recall that the mean of a binomial random variable is μ=np where n is the number of trials and p is the probability of a given trial’s success. Furthermore var(X)=np(1-p)=μ(1-p). Since μ=12 and var(X)=3, the argument from here is simply arithmetic

I’ll leave the answer below with a spoiler, but I implore you to try and solve the question from here before revealing the answer. If there’s anymore confusion, feel free to ping me :)

||p=3/4||

[safe radish]

@lean otter Has your question been resolved?

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[safe parrot]

Find all the solutions $u(x,y)$ to the partial differential equation \$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = x^2+y^2$ which depends of $r=\sqrt{x^2+y^2}$, i.e is of the form $u(x,y)=f(r)$.

[flat frigate]

Merineth

[safe parrot]

Are there actual people who understand these type of questions from the book and know how to solve them?

How do i actually learn when i'm unable to solve the questions?

[split fulcrum]

@junior smelt '

[safe radish]

@safe parrot Has your question been resolved?

[safe parrot]

i wish i died

[dry wolf]

haha, did you try to view $u(x,y)=f(\sqrt{x^2+y^2})$ and plug that into the equation?

[flat frigate]

NulledOutChicken

[safe parrot]

no, i have literally no idea what i'm supposed to do

[dry wolf]

haha

[safe parrot]

Like i can't even interpret the question right

[dry wolf]

Okay I just gave you a hint

Use that expression, just plug it in and try to simplyfy for now

[safe parrot]

Yeah but i want to understand what i'm doing

I need to be able to solve these

[dry wolf]

The understanding usually comes AFTER you solved it

do you know what $\frac{\partial f}{\partial x}$ means?

[flat frigate]

NulledOutChicken

[safe parrot]

yes, the partial derivative of f wrt x

[dry wolf]

Do you know how to compute it?

[safe parrot]

Yes.

[dry wolf]

And do you also know how to use the chain rule?

[safe parrot]

When it comes to one variable, yes very. Multivariable i'm not sure

[dry wolf]

if it is a partial derivative, the chain rule works like with one variable

[safe parrot]

Ok but i dont understand the question that i'm given.\ \

It asks me to find all the solutions to: $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = x^2+y^2$, right?

[flat frigate]

Merineth

[safe parrot]

and it gives me a condition r?

[dry wolf]

Okay, you are thinking the wrong way around

The first thing you should think about is how to simplify the question

Try to use the info you have got

My first instinct would be to replace u(x,y) with its condition

[safe parrot]

1. FInd all the solutions to the function u(x,y)
2. add the condition r

Does this simplify it?

[dry wolf]

Try it

[safe parrot]

$u(x,y) = \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=x^2+y^2$

Am i thinking correct?

[dry wolf]

Hmm

[flat frigate]

Merineth

[dry wolf]

you already have $u=f(r)$

[flat frigate]

NulledOutChicken

[safe parrot]

i dont know what u=f(r) means

[dry wolf]

Maybe start by just plugging that in first?

[safe parrot]

\o/

i dont know`?

[dry wolf]

Lets start simple. Try to expand $\frac{\partial f(r)}{\partial x}$

[flat frigate]

NulledOutChicken

[safe parrot]

I have no idea what that means'

Like i dont think you understand, i do not know what to do

i literally dont understand the question

This isn't simple to me

How do i read the question such that i understand what i need to do

i need to be able to solve these questions on my own

and that requires me to understand

[dry wolf]

Okay they ask you to find this derivative $\frac{\partial f(\sqrt(x^2+y^2)}{\partial x}$

[safe parrot]

what they are asking for

[flat frigate]

NulledOutChicken

[safe parrot]

and how do you know this?

[dry wolf]

Thats what it basically boils down to

[safe parrot]

Why isn't that mentioned in the question?

[dry wolf]

That is not something that you just know

In máthematics the burden of proof is on you

[safe parrot]

Yeah but if i sit down on the exam, i'm supposed to understand this

I can't have someone add additional information that is neeeded

for me to even understand what they are asking for

[dry wolf]

And you will if you stop whining, man up and take that damn derivative

[safe parrot]

whining

i'm litearlly fucking suicidal

since

i dont understand

whjat i'm supposed to do

[dry wolf]

Try to answer that for now

And calm down men

Really chill

[safe parrot]

$\frac{1}{2\sqrt{x^2+y^2}}$

[flat frigate]

Merineth

[dry wolf]

haha okay lemme do it for you

$\partial_x f(\sqrt{x^2+y^2})= (\frac{x}{r})\partial_r f(r)$

[flat frigate]

NulledOutChicken

[dry wolf]

Okay I know a way to explain what they want from you

They basically wanna you to rewrite $\partial_x^2 f(r) +\partial_y^2 f(r) like \partial_r^2 g(r)$

[flat frigate]

NulledOutChicken

[safe parrot]

Is this correct?

[dry wolf]

Not at all

The chain rule definetly does not work that way

Also, you used the absolute chain rule for partial derivaives

and you should expand partial x of f(r)

not the other way around

You should be using it like that

$\partial_x f(r)=(\partial_x r)\partial_r f(r)$

[flat frigate]

NulledOutChicken

[dry wolf]

Also, do not set everything equal to u, you already have u in the equation, use something more appropriate like g(x,y). I also think your approach si wrong, since you should definetly simplify the partial derivatives first

So your mission is to find anotehr expression for $\partial_x^2 f(\sqrt{x^2+y^2})$

[flat frigate]

NulledOutChicken

[dry wolf]

you think you can do that?

[safe radish]

@safe parrot Has your question been resolved?

[dry wolf]

:=(

But I did...

:(((

[safe parrot]

I'm very sorry but i just dont understand what you mean

[safe radish]

@safe parrot Has your question been resolved?

[safe parrot]

.close

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[deft vigil]

if given
$$S = {x \in \mathbf{Q} | x} S \subset \mathbf{R}$$

[lean otter]

use \{

[deft vigil]

sorry

[flat frigate]

Mephisto

[deft vigil]

that's better

does that mean that Q are the accumulation points of this set?

and that there is no interior because of the fact that between each rational number, there is a irrational one?

I'm also assuming IR are the boundry points of S, because boundry points don't have to be a part of the set

meaning it's not a closed set, but also not an open one?

is this possible?

[quiet plume]

What exactly is the set S you're defining? The formatting makes it hard to understand what you mean.

[deft vigil]

I was trying to use the notation from my book, would be wayy easier to just say all rational numbers as a subset of all irrational ones

so just a set $\mathbb{Q} \subset \mathbb{R}$

[quiet plume]

So just a subset of Q or Q itself?

[flat frigate]

Mephisto

[deft vigil]

that's better

the set of all rational numbers as a subset of all real numbers

it has to be specified, otherwise if we consider all rational numbers independent of all real numbers, the set of boundry points would be an empty set

[quiet plume]

Yeah I see what you mean, it wasn't exactly clear. So Q inside Euclidean space.

Indeed, the interior of that subset is empty.
Its boundary is just R, which includes irrationals, thus it's not closed.

So it's neither closed nor open.

[deft vigil]

oh alright, so it is possible?

[quiet plume]

Yes, sets are not doors, they can be open and closed simultaneuously and neither as well.

[deft vigil]

it just seemed like a contradiction at first, but it might be because of the words used to define the terms

[quiet plume]

It's because closed sets are not defined as sets that are not open. Rather, they are sets whose complement is open.

[deft vigil]

ohhh okay

so the complement of Q sub R, would be R\Q which is both closed and open?

can you clarify?

[quiet plume]

No. If Q is neither open nor closed, then its complement is also neither open nor closed.

[deft vigil]

so if I'm following correctly, an interior is a property of any subset? is that what you mean?

[deft vigil]

ohhh okay I see it now

Q sub R, is not closed and not open its complement must be, not open because Q is not closed, and not closed because Q is not open

what were you thinking of?

alr guys, thanks for your time, I think I got it, closing this one

[empty gyro]

If X is not open, then R\X is not closed

[deft vigil]

ye, I was a bit confused at the start but I figured it out

.close

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[worthy frost]

The task is to Using a double integral, calculate the area of the region D bounded by curves.

[worthy frost]

I don't understand how to get the upper limit for x

the curve for the upper limit is y=lnx

how do i get x?

In case my drawing isn't readable

like that?

<@&286206848099549185>

I got this, but I don't think it's right

[stoic dune]

Yeah that looks good

[worthy frost]

oh nvm

I had the answer given like this

why would you do that

lol

my teacher is trolling me

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[narrow venture]

hey i have an uncanny question

[narrow venture]

how would i graph this graph in desmos?

im trying to make a semi realistic sailing game

and im struggling with the math

i have already caluculated the boats angle to the wind and the wind speed

or get the formula for a graph like this

thats the main issuelol

or really

whhat would a formula for this be?

what does that eben look like

is it poossible?

<@&286206848099549185>

could i acomplish it with a gausian function?

[woven hound]

This is definitely polar

[safe radish]

@narrow venture Has your question been resolved?

[narrow venture]

any chance you can make it for me?

i havent covered it in school yet

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[jolly ledge]

How do I do (iii)? Tried multiplying sum of square and sum of cube up but to no avail.

[jolly ledge]

Especially after multiplying sum of square and sum of cube up I don't know what to do with the shitload of strange variable multiplies

[eternal carbon]

would consider Newton's Sums here

[jolly ledge]

Ok so it is something my lecturer haven't taught yet

I'll skip it for now

Thanks!

.close

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[brittle rivet]

okay so

you know the properties of exponentiaton?

a^2 * a^3 = a^2+3

alright, so using the same property

we can say

srory

there

okay, so using that

[shadow sparrow]

() on exponent

[brittle rivet]

we can do x^2 * x^(-2) = x^(2-2) = x^0 = 1

therefore x^2 must be multiplying 1/x^2

for it to be 0

1*

in general, negative exponents indicate that the the expression is inverted

and the reason is what I just mentioned

no problem

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[willow nexus]

Question:
How do I "expand" values as they approach 1? Red line is the input, blue line is the desired output. The the value of the blue curve at 1 would ideally be controllable with a variable, I don't want it to necessarily approach infinite at 1.

Bonus:
Could this be done in a way where the "sharpness" of the curve can be controlled with another variable? See the light blue lines.

What I tried:
The answer ChatGPT gave me, e^(k*(x-1)), seemed to do the opposite (affecting only values below 1), and I couldn't quite figure out how to flip it around. Also, it doesn't give me control over the curve tightness. I think it might just have been too difficult for me to explain the problem to it.

Context:
I'm writing a shader for a video game where this came up. Normally I can google my way out of math problems like this, but I just can't wrap my head around this one.

Any help or just hints would be appreciated.

[austere brook]

what is this

you trying to do graphwars?

oh this aint it

[willow nexus]

I'm writing a shader. Looked up graphwars, and it looks like fun if you're not as bad as me with math.

[austere brook]

nah its just some graph you need to memorize

like the abs(x+5)-abs(x-2) to shift the graph

for 1

dont you do like x+e^(x-k)

k is the adjustable value

itll look smth like this

e^(k*(x-1)) + x

actually that seems right

-1 is how you adjust where the curve "spikes"

[willow nexus]

That looks right, let me give that a try real quick

[austere brook]

and k is the amount you want it to increase before it goes to infinity

add +x to that

okay here we go

g is the initial gradient

k is how fast you want the graph to "spike"

[willow nexus]

Maybe I'm just implementing it wrong. I'm still getting the other way around, it's lifting the lowest values instead of the heighest.

[austere brook]

and -1 is the position of where you want it to spike

[austere brook]

wdym by this?

https://www.desmos.com/calculator/f3h2eqqbrh

try this

play around with the values until you get your desired result

[willow nexus]

Does this look right? Maybe I'm making a mistake when converting things into nodes. I'll try writing it out as a script instead

ooh Desmos is very helpful. Thanks for the tip.

[safe radish]

@willow nexus Has your question been resolved?

[willow nexus]

I haven't figured it out yet, but I'm going to close the room while I fiddle. @austere brook Thank you

.close

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[plucky swan]

Could someone help out? I proved the first part, but what does it mean by the second part?

[plucky swan]

Hence find an expression for I(n) in terms of n for n> 0

[forest spruce]

It means you need to express I_n in explicit forms of n, instead of I_{n-1}

[plucky swan]

so i'm assuming that'd mean find a way to integrate the I(n-1) term

so it's in terms of N throughout and not recursive...?

would you hve any idea how I could go about doing that?

[safe radish]

@plucky swan Has your question been resolved?

[obsidian oracle]

You have a recursive formula about In

So start from I0

How do you get I1?

You do 3/5 I0

How do you get I2

You do 6/8I1

Which is 18/40*I0

And so on

Find a way to generalize this

[plucky swan]

Okay

[plucky swan]

Would I form a series of some kind?

I haven't done anything like this before, would you mind sharing the answer

@obsidian oracle

[odd crest]

factorials!

[plucky swan]

Still, it doesn't end up being true for n = 0 with factiorials

(3n! / (3n! + 2)) * 1/2

Still only true for n > 1

I really have no clue where to go I've tried everything I know

And I need to know how to do this

[forest spruce]

I tried to work on this but can't find the answer. If you somehow get the answer without the proving step, you can prove it by induction.

[plucky swan]

Could you show me how it could be proved via induction?

[forest spruce]

for example we know that $S_n=\sum_{i=1}^n i=\frac{n(n+1)}{2}$ and $S_n=S_{n-1}+n$
then we have $S_1=\frac{1(2)}2=1$, then assume that this hold for n=k, we can prove the case for n=k+1 by the recursive relation
$S_n=S_{k+1}= S_k+k+1= \frac{k(k+1)}2+k+1= \frac{k^2+3k+2}2= \frac{(k+1)(k+1+1)}2= \frac{n(n+1)}{2}$

[flat frigate]

Micni9

[odd crest]

can you write your answer in terms of the product symbol?

[forest spruce]

normally I assume the answer need to be in closed form

[safe radish]

@plucky swan Has your question been resolved?

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[sour violet]

Would the proper way to read this integral be

The integral of x times sine( x) with respect to the derivative of x?

Or like the integral of x times the derivative of sine(x)?

[thin bridge]

neither

[sour violet]

How would it be read then?

[thin bridge]

integral of
x times sin(x) (would be better described as the product of x and sin(x))
with respect to x

[sour violet]

I dont need to specifically state that I’m using the derivative of sin(x)?

[thin bridge]

what derivative of sin(x)

there is no derivative of sin(x) there

[sour violet]

!rccw

Ugh

Ok so im integrating by parts with this one so im just splitting it into x and sinx dx

[sour violet]

But when I do that u=x

And v=sin(x) dx

[thin bridge]

you want
dv = sin(x) dx

[sour violet]

And that leaves me with v=-cos(x)

But how would u call out the problem it’s giving me without explicitly just writing the numbers down

Like if I had to call this out after seeing it what would I call it

My first thought was what I said earlier

So I’m still thinking

[thin bridge]

the initial problem itself was what i described earlier

[sour violet]

Yeah

Just if I were to say this out loud what would I be saying

If it isn’t what I thought it was earlier

[thin bridge]

the initial problem itself?

[sour violet]

Ignore the ex

Yes

[thin bridge]

was what i described earlier

[sour violet]

But how would I be able to tell it’s sin(x)dx

[thin bridge]

wdym

[sour violet]

Like what’s the indicator that it’s an integral or derived with respect to x

[thin bridge]

whether you have an integral sign
or derivative operator

[sour violet]

What if it isn’t written though

[thin bridge]

wdym

[sour violet]

Assume as if I’m writing this down as someone’s telling me

How would by word alone I break this down to write the question

[thin bridge]

integral of
x times sin(x) (would be better described as the product of x and sin(x))
with respect to x

THAT

if you understand the terminology

[sour violet]

So with respect to x just means dx

[thin bridge]

that will give you precisely what you gave me

[lime dust]

dx is referring to the increment

[sour violet]

But what if for some reason I wanted the second derivative of x how does that change the wording?

[thin bridge]

d(what to integrate with respect to) at the end of the integral expression tells you what to integrate

[sour violet]

Ahhh

[thin bridge]

if you wanted the second derivative of someothjing

[sour violet]

But like here’s the work I used

It tells me to integrate x but then after u get like

[thin bridge]

that would literally be
second derivative of (that something)
add with respect to variable of choice if intentions are unclear

[sour violet]

dv=sin(x)dx and just turn it all into v=-cos(x) when I derive it

[thin bridge]

you're getting integrals and derivatives mixed up

[sour violet]

I’m integrating by parts

[thin bridge]

you get
v = -cos(x) when you integrate it

[sour violet]

What

I don’t know like the in between then can you help me out with that

[thin bridge]

starting with
$$\dd{v} = \sin(x) \dd{x}$$
$\textbf{integrating}$ both sides:
$$\red{\int}\dd{v} =\red{\int} \sin(x) \dd{x}$$
gives
$$v = -\cos(x) \gray{+C}$$
(you don't care too much about the C there as you can just assign that somewhere else in the overall integral

[flat frigate]

ℝαμΩℕωⅤ

[sour violet]

But what exactly happens between integrating it and the solution? Like how do you know what it integrates to

[thin bridge]

integration is the reverse of differentiation

[sour violet]

Ahhhh that’s what I’m missing

I forgot to write those formulas down

[thin bridge]

knowing v differentiates to 1,
the antiderivative of 1 is v

knowing the derivative of cos(x) is -sin(x), derivative of -cos(x) will be sin(x)
the antiderivative of sin(x) is -cos(x)

[sour violet]

Ok just taking notes now thanks for the help

.close

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[lean otter]

How do I do this ?

[carmine knoll]

firstly: what does it mean for a matrix to be nilpotent

[lean otter]

There exist a power raised to which the matrix becomes zero

e.g A^n=0. If there exist such a "n" the matrix is called nilpotent.

[carmine knoll]

I'm trying to think of a good way to motive the next part of this proof

basically, we want to factor (I + A^n) into the product of (I+A) and another matrix

you can think of this as polynomials, if it makes it easier

[lean otter]

Ok..

[carmine knoll]

this factors into something like (I + A^n) = (I+A)(I-A+A^2-A^3... + (-1)^{n-1} A^{n-1}) and we get our inverse

maybe an easier way to see it: if B is invertible, then the determinant of B+A is det(B)+det(A)+det(A^-1 B)

here, we have det(I+A) = det(I)+det(A)+ det(IA) = 1 + 0 + 0 = 1

which means it's invertible

[lean otter]

If we can get this part

Do we still need to do the determinant ?

Oh

[carmine knoll]

no, but sometimes the determinant is easier and sometimes it's harder

[carmine knoll]

note this isn't always true actually, nevermind

[lean otter]

If A is nilpotent then A have a inverse. (That is the determinant is not zero)

[carmine knoll]

if the determinant is zero it doesn't have an inverse

[lean otter]

Yes

[mossy fractal]

If A^n = 0 then detA = 0

[carmine knoll]

this doesn't prove the thing we want it to

ooh wait I have a snappier proof

if (I+A) is non-invertible there is a nonzero v such that (I+A)v = 0

this means that v + Av = 0 or Av = -v

let's say that A^m = 0. On the one hand (A^m)v = (-1)^m v, but also (A^m)v = 0v = 0, so we have (-1)^m v = 0 i.e. v = 0

bam, no polynomials needed

[lean otter]

Wait

[lean otter]

(A^m)v = (-1)^m v (How?)

[carmine knoll]

Av = -v
A(Av) = A(-v) = v
AAAv = Av = -v
etc

each application of A picks up a factor of -1

[lean otter]

ok

[lean otter]

Is this a theorem ?

[carmine knoll]

Yeah, if a matrix isn't invertible it has nontrivial kernel

[lean otter]

If a matrix is not invertible => the determinant is zero => kernel is non trivial

Makes sense

When you looked at the statement for the first time did you know that it is true ?

[mossy fractal]

I think in this case you kinda just intuitively hope it’s true first

Since it’s easier to prove that it’s true than to prove it’s false

At least I think it is

If you really dont get anywhere with trying to prove its true, maybe then you can start doubting and try to prove that its false

Actually tbh if it was false, it may not be hard to prove that the determinant is 0 for example

[lean otter]

Do you have anything to say @carmine knoll

[carmine knoll]

It looked true yeah

Because of the polynomial thing

[safe radish]

@lean otter Has your question been resolved?

[safe radish]

@lean otter Has your question been resolved?

[safe radish]

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[void blaze]

greetings

[void blaze]

if 2 cos 45degrees = 2 * pi/2 and sin 90degrees = pi/2, is 2 cos 45degrees = sin 90degrees?

[prisma shuttle]

sin 90 isnt pi/2

and 2cos45 isnt 2 * pi/2

[void blaze]

oops i apologize

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[swift matrix]

so i was looking at some cambridge interview questions. the first question was to figure out what sin(x)/x would look like. what is the thought process for trying to figure that out

[swift matrix]

what sin(x)/x would look like in a graph

[empty gyro]

Knowing its zeros, critical points, and limit at 0 would be useful

Also knowing its end behavior

[swift matrix]

idk man none were given

[obtuse jackal]

find them

[swift matrix]

none were stated idk what to tell you

[gusty trench]

yeah this would be really easy if they were stated

you're supposed to find them

using algebra/calculus

you can also think intuitively about what its end behaviour would look like at +/- infinity

[obtuse jackal]

then you aren't deserving of Cambridge if you don't even try to find basic information about a function
gotta get to work

[swift matrix]

i thought it was simpler than that i'm sorry😭 I don't know that math yet

[shadow glade]

i'd say think about it when x > 1 or x < -1, then it's basically just sin(x) but the amplitude keeps getting smaller and smaller as you move further and further from 0 in either direction.
when -1 < x < 1 ask yourself how is the amplitude changing?
what is happening at x = 0? obviously the function is undefined, can you find a limit?

consider if anything like even\odd function is relevant here

[obtuse jackal]

well either you're preparing for the test during the same year, or your knowledge is lackluster yeah

[shadow glade]

have you taken calculus yet \ are you familiar with limits?

[obtuse jackal]

if you haven't don't study uni entrance tests

[swift matrix]

no not at all I'm getting to that next year. I didn't know the question involved that

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[obtuse jackal]

that only makes sense when you're already very familiar with all the material

[shadow glade]

not necessarily to get a good idea of what the graph looks like, but it's much more likely you'd have the familiarity needed with functions and the important bits to look out when thinking of graphing and the behavior at x = 0

[obtuse jackal]

also surprised they'd only start calculus during the last year of HS

[shadow glade]

but agreed with what bezier said about focusing on that before entrance exams

[shadow glade]

maybe they're a freshman looking ahead!

[obtuse jackal]

2 years ahead?

[shadow glade]

it's been known to happen i guess if someone is planning for a specific school ahead of time

[obtuse jackal]

if they look 2 years but can't figure out it'd be beneficial to learn the material first idk what to say besides sorry for you

.reopen

[safe radish]

✅

[obtuse jackal]

before it closes

since you're still typing

ok now I look silly

.close

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[swift owl]

can someone explain this one please?

[buoyant epoch]

Are you asking about the f•g notation or something else?

Also to answer the question you need to have other information about f and g

[swift owl]

reallly?

that's all i am given

i am lost at what its asking

[buoyant epoch]

Does it assume f is a polynomial?

[sleek plank]

f and g are inverses

you’re supposed to use that fact to help you figure it out

[buoyant epoch]

Is that the problem? Where does it say that

[swift owl]

nothing in there says about inverses

[swift owl]

how did you come up with that? is it the numbers?

[sleek plank]

g(-4) = -2
f(-2) = -4

[swift owl]

oh

that acutally might make sense

[sleek plank]

wait nvm there are quite a few exceptions here

[jolly kiln]

(fog)(14) isnt 14 tho

[sleek plank]

yea things like that rebuke my theory

[swift owl]

is it possible that the problem is not complete?

i mean that's all im given but still

and I don't see a pattern

[left gyro]

what if you assume that different inputs must lead to different outputs?

for example the topmost answer box cannot be 14, -4, or -2

if you make this assumption, youre assuming that f and g are "injective" which is a normal thing for functions to have

[swift owl]

Hmm would it be 12?

Probably not

[left gyro]

theres no other options

go try 12 in both boxes

then see if it works out

[swift owl]

I have to fill it all out