help-49

[runic thunder]

oh yeah..

💀

[cinder fox]

LOL

[runic thunder]

do u think i should add 69 and 4 or just leave it as fraction

[cinder fox]

leave as frac

[runic thunder]

cause it said add x and y right

[cinder fox]

yeah

[runic thunder]

yeahh

ok ok

thanks bro

could u help me with 2 or 3 more or na

[cinder fox]

npnp

naa watching mushoku tensei atm

gl tho

[runic thunder]

alright thanks

.close

[midnight plank]

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[last slate]

Can someone help me use FPI to guarantee convergence for the backwared Euler method? PLS

[last slate]

<@&286206848099549185>

[midnight plank]

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.reopen

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✅

[last slate]

@stiff heart

<@&286206848099549185>

[cinder fox]

i got no scoobs what year is this?

[last slate]

wym

im iin university

numerical analysis

[cinder fox]

yeee im 2nd year

that shi sounds hard

glgl

[last slate]

lolllll

[midnight plank]

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@last slate Has your question been resolved?

[last slate]

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[mint oriole]

I have no idea how to start this one

[mint oriole]

I've tried approaching it two ways

both which took

a lot of time

and i got lost with

I let X' = (1, D_i, W_i', D_i (W_i - E[W_i])'

And computing E[XX']^{-1}E[XY]

and i got pretty far

i managed to invert E[XX']

but there was anothe rmatrix i had to invert

and i had such a complicated form for beta

and i had no clue how to use conditional unconfoundedness to simplify it

i then tried doing frisch waugh lovell, but i got stuck in the same way

[midnight plank]

@mint oriole Has your question been resolved?

[mint oriole]

<@&286206848099549185>

[midnight plank]

@mint oriole Has your question been resolved?

[midnight plank]

@mint oriole Has your question been resolved?

[midnight plank]

@mint oriole Has your question been resolved?

[midnight plank]

@mint oriole Has your question been resolved?

[midnight plank]

@mint oriole Has your question been resolved?

[midnight plank]

@mint oriole Has your question been resolved?

[last slate]

What topic is this?

[mint oriole]

statistics

[eternal frost]

Should i skip Algerbra and go strait to AP Geometry

Actuall my asian dad said i should go to AP Physiscs any htoughts

[mystic mason]

do them all

Algebra, then take calculus and physics at the same time

why calc AND physics?

because a LOT of calculus is used in physics. they go hand in hand. even if you take a ap physics without calculus applications, calculus is still there because a ton of formulas are connected and if you learn calculus youll see how their connected

and to do calclulus, u should know basic algebra 😭 at least

for geometry... uhh... idk

[crisp oxide]

It looks like you need to use conditional unconfoundedness to show both the (Wi - E[Wi]) and the Wi component are uncorrelated with the treatment, which means they have zero effect on beta when you calculate E[e’e]

[midnight plank]

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[wide dagger]

Hi, im looking for someone whos familiar with Calculus Taylor Series questions

[wide dagger]

Im studying for an Exam and need recommendations on who to look for online who can teach me how to solve questions like the following:

[hard shard]

if need help with a range of topics, try khan academy

[wide dagger]

Is that free

[hard shard]

yes

[wide dagger]

I just need a dumb down version of how to solve problems like these

[stable owl]

Khan academy is a great resource! Also organic chemistry tutor is pretty good for computational problems.

[wide dagger]

any other individuals on yt ?

[stable owl]

Are you looking for a more intuitive/geometric intuition? Or a more computational approach?

[wide dagger]

Im sorry those words a too big for me lol, I have an exam tmrw I just need to note down step by step how to approach solving those style taylor series problems

[stable owl]

Organic chemistry tutor is the way to go!

It offers easy, comprehensive explanations that are geared towards exam-like questions, instead of geometric intuitions.

Hope you ace your exams!

[wide dagger]

Thank you

[midnight plank]

@wide dagger Has your question been resolved?

[wide dagger]

How do you get the 4x^2 from this ?

I understand the x^4 and 4 but 4x^2 ?

[stable owl]

Is it a volume of revolution question?

[wide dagger]

Its Find volume of solid generated by revolving the region about x axis

[wide dagger]

This is the only thing im confused about

from (2-x^2)^2 how do you get the 4x^2

[stable owl]

Foiling

Try expanding (2-x^2)(2-x^2)

[wide dagger]

ohhhh

lol

ok now I get it

[stable owl]

Great!

[wide dagger]

thank you

[stable owl]

Np.

[wide dagger]

So I don't have to do distribution of the ^2 I just do (2-x^2)(2-x^2) to get all 3 of those x^4 - 4x^2 + 4 ?

[stable owl]

So you just foil out term by term.

[midnight plank]

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[wide dagger]

t

[wide dagger]

How did he get the (3-x^2)^2 - (x^2)^2 those values in the brackets

[tiny drum]

does f(x) = 3 - x^2 and g(x) = x^2

[wide dagger]

yes

thats the formula for it

[tiny drum]

looks like they just used the formula to the right then

[wide dagger]

exactly what was done

my question is how'd get the 3-x^2 and x^2 inside the brackets

[tiny drum]

they just substituted f(x) and g(x) into that formula

[wide dagger]

wait nvm sorry

I found it

[midnight plank]

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[midnight plank]

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[wide dagger]

.reopen

[midnight plank]

✅

[wide dagger]

howd they get the 6x^3/3

[dawn dagger]

It's the power rule simply

If you differentiate 6x³/3 you end up with 6x²

[wide dagger]

Will you always make it _x^3/3 ? for any problem ?

[dawn dagger]

if you have x^2 yes

[wide dagger]

got it

let me show you an example real quick just to make sure

[dawn dagger]

,,\int x^n : \dd x = \frac{x^{n+1}}{n+1} + C \quad n \neq -1

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

So for example

x^4 integrated would be x^5/5

[dawn dagger]

it's basically the power rule from differentiation but reversed

[dawn dagger]

yes

[wide dagger]

This is another problem, the last part would that be right?

ok awesome

thank you

[wide dagger]

in the next step how did they get 1/2 ?

and where did the 6x^3/3 go

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

They applied this

[wide dagger]

ok got it

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

if it was like the problem in my notebook ill send pic again how would that look like ?

[dawn dagger]

You mean the bounds?

You do upper bound plugged in minus lower bound plugged in

[wide dagger]

not sure the terminology but like I know for as

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

Does the above make sense?

Your F(x) is here 4x - 4x³/3

[wide dagger]

Sorry im writing all these notations down real quick before we continue on that example sorry, the prior example howd the get this second part

why is there no 1/2 for that one

[dawn dagger]

I explained it before.

[dawn dagger]

This

You can rewrite the expression either way

[wide dagger]

ok got it, im gonna write all these notes down, is it ok if I tag you once I note down all this if I have further questions about this

[dawn dagger]

sure

[wide dagger]

thx

@dawn dagger is the 2 underlined from 6/3 ?

[dawn dagger]

yes

6/3 = 2

[wide dagger]

ok and you said I could rewrite the expression eitherway for the -2(3/2)3/2 so does that mean it could also be -2(3/2)1/2 ?

this portion

[dawn dagger]

no

you left out the 3

why

[wide dagger]

oh

so just 1/2 * 3 ?

[dawn dagger]

,,\sqrt[a]{x^b} = \left ( \sqrt[a]{x} \right )^b = x^{\frac{b}{a}}

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

You can apply these power rules

[wide dagger]

still little confused so what would you plug into that ?

like how would it look with everything plugged in

[dawn dagger]

what do you want to do here?

[wide dagger]

im just trying to better understand how the (3/2)^3/2 you get

I know its (3/2)^1/2 when you plugin that formula you gave me

but how does the ^1/2 turn into 3/2

[dawn dagger]

,,\left ( \sqrt{\frac{3}{2}} \right )^3 = \left ( \left (\frac{3}{2} \right )^{\frac{1}{2}} \right )^3 = \left (\frac{3}{2} \right )^{\frac{1}{2} \cdot 3}

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

so what I said here was correct ?

You just have to do 3 * the 1/2

[dawn dagger]

yes

[wide dagger]

ok

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

where is there no fraction for this portion ?

if its negative theres no 1/2 ?

[dawn dagger]

i dont understand your question

[wide dagger]

where is there no ^1/2 or ^3/2 on the otherside ?

[dawn dagger]

they rewrote it

and dragged the minus out

[dawn dagger]

it's this

[wide dagger]

still confused, why is there no ^1/2 or 3/2 anymore if its that

[dawn dagger]

there is

I see a 3/2

and 1/2

[wide dagger]

on the right side I dont see it ?

the red underlined part

[dawn dagger]

because they wrote it with the root

expression

,,\sqrt{}

[wide dagger]

oh

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

so you can do it either way ?

[dawn dagger]

yea

[wide dagger]

oh im sorry its the step below that it shows it

my bad

ok now I understand this form of solving the problem

@dawn dagger for this style of problem am I doing that part correct ?

I feel like i am missing something

step 5

[dawn dagger]

uhm you missed the 2nd term

[wide dagger]

4/3 doesnt go into itself so how would I write that

[dawn dagger]

,,\pi \left [ 4(1) - \frac{4(1)^3}{3} - \left ( 4(-1) - \frac{4(-1)^3}{3} \right ) \right ]

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

got it

thanks

[dawn dagger]

dont forget brackets

[wide dagger]

yep

[wide dagger]

question for the next step

does it always go to 2 pie ?

is the 2 pie represent the other half that was in the previous step ?

[dawn dagger]

dont see things that way

use logic

the 2 comes from simplifying the other stuff

This simplifies to

,,2 \cdot \left ( 9 \left( \frac{3}{2} \right )^{\frac{1}{2}} - 2 \left( \frac{3}{2} \right )^{\frac{3}{2}} \right )

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

it's like having

[wide dagger]

got it

[dawn dagger]

here it's like a + a which is also the same as 2a

and you can pull out 2

[wide dagger]

how do they get this portion and then the solution

whenever I input it into the calculator it gives me a decimal answer

[dawn dagger]

what common factor can you see

[wide dagger]

3/2 ?

[dawn dagger]

almost

(3/2)^(1/2)

aka

sqrt(3/2)

that's what they pulled out

[wide dagger]

understood, but then what happens to the ^3/2 at the end on the far right

[dawn dagger]

3/2 = 1/2 + 2/2

so you endup with (3/2)^(2/2) = (3/2)^1 = 3/2

[grand pond]

𝔸dωn𝓲²s

[wide dagger]

how would this be shown with the numbers inputted

still confused with how the 3/2 * 3/2 ?

[dawn dagger]

what

[wide dagger]

the formula you gave above, what would that look like with the numbers from that problem implemented. The way you explained it before you sent the formula still confused on

[dawn dagger]

they are rules XD

[wide dagger]

but like didnt the 1/2 go away because it was sqrt

so how would you input the 2(3/2)^3/2 into that formula

[grand pond]

𝔸dωn𝓲²s

[dawn dagger]

That's it

you could have tried it by yourself

to see

x = 3/2
a = 1/2
b = 2/2

[wide dagger]

ok so 3/2^2/2 just equals 3/2

[dawn dagger]

yes

it's like 3/2 to the power of 1

[wide dagger]

right

now once we reach this point, how do you get 12 pie ?

I understand the sqrt 3/2 stays the same

its 2pie * (9-2*3/2) ?

[dawn dagger]

evalutate the things inside paranthesis

[wide dagger]

I just calculated it I got it now

Thank you

[dawn dagger]

haha

[wide dagger]

lol I’m sorry I’m slow

[dawn dagger]

nah it's ok

you should try more and experiment i think

that's all

so you discover stuff and get an aha moment

[wide dagger]

Yeah im running through different problem scenarios for my exam tmrw

[midnight plank]

@wide dagger Has your question been resolved?

[wide dagger]

@dawn dagger are you by chance familiar with taylor series

[dawn dagger]

yes kind of

[wide dagger]

would you by chance know how to find the taylor series for functions like these

[dawn dagger]

well you need to use the taylor series formula

calculate some derivatives at x = 0

and find a pattern that matches to one of the series

[midnight plank]

@wide dagger Has your question been resolved?

[midnight plank]

@wide dagger Has your question been resolved?

[wide dagger]

how does this become sqrt 2 ?

[civic bough]

It is the form of complex numbers with euler's formula
e^i (angle) = cos(angle) + i sin(angle)

[last slate]

First (image)
then, sin(pi/4) and cos(pi/4) = 1/root2
if you multiply and divide by root2 it becomes root 2 / 2

[wide dagger]

thank you

[wide dagger]

how did they get this value underlined ?

[last slate]

the underlined value is integral(vdu)
Is it your question?

[wide dagger]

the underlined red portion the v is not there as its an unknown

underlined part was found prior to V

[last slate]

sorry for the delay
they have made the equation into x^2 . xe^(x^2) in order to integrate dv easily

[wide dagger]

how would I find v

[last slate]

integrate dv which is integrate (xe^(x^2))dx

[lilac nebula]

Wowie that's a lot of independent questions

[midnight plank]

@wide dagger Has your question been resolved?

[midnight plank]

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[neat sandal]

can someone help with this

idrk how to do it

[orchid vault]

By using u = pi - x, what do you get

[neat sandal]

I did it up to this point but idk what to do after

[orchid vault]

Do you realise this is $\int_0^{\pi} \pi f(\sin(u)) \dd u + \int_0^{\pi} -u f(\sin(u)) \dd u$

[grand pond]

! What the hell am I doing here?

[neat sandal]

yes i did that too on another page earlier but then got stuck

[orchid vault]

And also $\int_a^b f(x) \dd x = \int_a^b f(u) \dd u$

[grand pond]

! What the hell am I doing here?

[neat sandal]

oh

[orchid vault]

I mean, the variable doesn't matter as long as it's the same integral, now does it?

[neat sandal]

thats right u and x would be the same either way

oh shit makes sense now

ty ty

.close

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[bleak pier]

lim n -> x [(pi * sum i = 1 to n sin((i*pi)/n))/n]

[honest urchin]

n -> x?

x a natural number?

do you mean to infinity?

@bleak pier

[honest urchin]

yeah, infinity

Looks like a Riemann Sum.

[grand pond]

M8732

[jagged saffron]

its not complex i just an index

[midnight plank]

@bleak pier Has your question been resolved?

[honest urchin]

ohhhhhh 🫣

@bleak pier Yeah forget about the second comment, it still looks like a Riemann sum though.

[midnight plank]

@bleak pier Has your question been resolved?

[midnight plank]

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[midnight linden]

hello! any idea on how to proceed with this?

$\lim_{(x, y)\to(0, 0)} \frac{\sin^2(x + y)}{|x + y|}$

[grand pond]

protsac

[midnight linden]

I know it tends to 0

but I'm a bit at a loss on how to prove it

the denominator makes it so that anything over it will tend to infinity

[hard pewter]

not really familiar with multivariables, but is it against the rules of multivariable calculus to put x+y=t and go t tends to 0 here?

[midnight linden]

I would suppose so, yes 😦

the most I've seen is try and use polar coordinates, which is really helpful in x^2 + y^2 cases

as with x = r cos theta and y = r sin theta you get the trig identity

and it suddenly becomes single variable

I suspect this would be based on the squeeze theorem, but I don't know how

in fact, I just saw an argument that would allow to make that substitution

but it feels a bit sloppy

but I think I'll go with that, yes 😁

thank you!

.close

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[cedar kraken]

small angle approximation after polar coordinates

[midnight plank]

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[sturdy crystal]

.rotate

/rotate

[hard pewter]

,rotate

[sturdy crystal]

!rotate

thx

can ya see the mcq

abt the domain

the one having a quadratic equation inside the underrrot

[hard pewter]

yeah

[sturdy crystal]

the ans is d

[hard pewter]

you basically want to exclude values where it is undefined

[sturdy crystal]

what does the U mean there

[hard pewter]

union

[dawn tiger]

ok what can you say about the domain of a square root function? and what can you say about the domain of 1/x?

[hard pewter]

it means both the sets are included

in the domain

[sturdy crystal]

do you guys know a trick to guess the domains of these

by looking at the options

[hard pewter]

its not something for which you should rely on tricks, but still if you insist you can always test some extreme values and eliminate the options which have them

for example you know the denom can be zero for x=1 or x=2

so the options including those if any can get eliminated

but this one is so simple that you dont need any elimination at all, just exclude wherever the things become undefined

[sturdy crystal]

but i have to keep track of both the factors of the quadratic

things get tricky there

i can solve other functions which dont have any quadratcics

[hard pewter]

theres the wavy curve method which can be applied to any factorised expression

for sign analysis

[sturdy crystal]

ig ill hav to skip this domain topic

[hard pewter]

why do you want to skip it

[sturdy crystal]

and js pray it doesnt come in my exam tomoro lol

i dont have time to study new methods

[hard pewter]

is solving (x-1)(x-2)>0 bothering you?

[sturdy crystal]

the wavy curve etc

[hard pewter]

its not too complicated tbh

wont take more than 10 mins

[sturdy crystal]

how do i solve it next

[hard pewter]

and for quadratics i wont even call it a 'method'

[sturdy crystal]

muliply it

[sturdy crystal]

what do i search on youtube

[hard pewter]

there are 2 terms multiplied to each other here

the possibilites are:

1. one negative and the other positive

2. both negative

3. both positive

you want cases 2 and 3

[sturdy crystal]

yup

[hard pewter]

meaning x>2 (which automatically makes x>1 )
or x<1 (which automatically makes x<2)

[sturdy crystal]

is there always a union of two sets for domain of quadratics

like this

[hard pewter]

not always no

it all depends on the problem

sometimes the domain can be just one set, or interesection of 2 sets

https://www.khanacademy.org/math/algebra-home/alg-quadratics/alg-quadratic-inequalities/v/quadratic-inequality-example

you can refer this

[sturdy crystal]

.close

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[left pagoda]

Can anyone help? I found out that X_k = 16 for all k>31 but can't seem to do anything with y

[midnight plank]

@left pagoda Has your question been resolved?

[midnight plank]

@left pagoda Has your question been resolved?

[midnight plank]

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[dawn prism]

This matrix should be orthogonal. How do I solve this?

[wary thorn]

Do you know what it means for it to be orthogonal

[dawn prism]

yes

I get 8 quadrics

4 equations in the form a^2 + b^2 = 3/12
and 4 in the form ac + bd = -19/84

g^2 + h^2 = 3/12
e^2 + f^2 = 23/84
c^2 + d^2 = 53/84
ge + hf = 3/12
gc + hd = -3/12
ce + df = -29/84

[midnight plank]

@dawn prism Has your question been resolved?

[wary thorn]

I would just find two other linearly independent vectors from these

Then you know that you have a basis of R4

Which you can reduce to an orthogonal basis using Grahm Schmidt

And then just normalize all your vectors

Might be the most straightforward way

[dawn prism]

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[hard shard]

testing something rq

[hard shard]

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[hard shard]

.reopen

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✅

[hard shard]

oh cool lmao

.close test over

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[dull yoke]

u still have help channel limit role

got*

[hard shard]

yeah

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[wraith flower]

Calculus 1 question about derivatives

[wraith flower]

I have this problem and it's a true or false

it says

If f isn't continuous at x=2, then f isn't differentiable at x=2

I'd say no because for a point to be differentiable, it has to be continuous right

[tired vapor]

you would be correct (assuming you mean that it isn't differentiable).

[pine thicket]

recall the definition of a derivative

[wraith flower]

it's a limit

[pine thicket]

but what is that limit

[frail burrow]

if you're saying that, then you're agreeing with the statement

[olive lintel]

yep

[wraith flower]

true.. I meant true

[olive lintel]

to be differentiable

its derivative should be same approaching from left and right

[wraith flower]

yeah

[olive lintel]

and should have limits left = right

and continious

[wraith flower]

if it is a function like this, with the same derivatives from the sides

but not the same limits

it can't be differentiable

[olive lintel]

yep

not continious + no limit

[stable owl]

Even if it is continuous over R does not mean its differentiable.

[tired vapor]

generally, since [f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}], if f'(x) does not exist, then the limit for f(x+h) is different, meaning f is not continuous at x, or f(x) does not exist, which also means f is not continuous, or both.

[grand pond]

fish

[wraith flower]

ok

I got it

thanks everyone

a lot

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[dusk rover]

can anyone help me approach this problem

[stable owl]

Does CBA denote the angle between CB and BA?

[dusk rover]

yea

[stable owl]

What are you thoughts on this?

[dusk rover]

i have no idea where to begin

i dont know what formula to use

i dont see how it could help me

its a non-calculator problem

[cinder basin]

you can try the law of sines

[stable owl]

Yea

Do you know the formula?

[cinder basin]

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

[dusk rover]

yeah

i kow the formula

[stable owl]

Try and relate to theta

[dusk rover]

but i dont have 2 angles to use it with

[stable owl]

You can find angle BCA

In terms of theta

[dusk rover]

what liek this ?$$ \frac{\sin\theta}{BC}=\frac{\sin BCA}{2} $$

$$ \frac{\sin\theta}{BC}=\frac{\sin BCA}{2} $$

[grand pond]

Jakey

[stable owl]

Do you know how to find BCA?

The angle

In terms of theta

[dusk rover]

no

2pi-pi/4-theta

i mean pi*\

not 2 pi

[stable owl]

Yes

You can simplify the pi part right?

Leaving us with an expression with only pi and theta

[dusk rover]

so its 3pi/4

[stable owl]

The hint that I would give is: AC/sin(pi/4)=2/sin(BCA)

And just find AC in terms of theta

[dusk rover]

hm ok ill try

like this ? $$ \frac{\sin\frac{\pi}{4}}{AB}=\frac{\sin\left(\frac{3\pi}{4}-\theta\right)}{2} $$

[grand pond]

Jakey

[stable owl]

It would work.

Then cross multiply

[dusk rover]

hm ok

[stable owl]

Find an expression in terms of theta.

I think you will also need double angle?

[dusk rover]

what is that\

[stable owl]

Sorry I mean angle sum

[dusk rover]

this

?

[stable owl]

Yes

[dusk rover]

k ill see what i can get

[stable owl]

So you get something in sin and something in cos

As required in the question

[dusk rover]

$$ \frac{2\sin:\frac{\pi}{4}}{AB}=\sin:\frac{3\pi}{4}\cdot\cos\theta-\cos:\frac{3\pi}{4}\sin\theta $$

[grand pond]

Jakey

[dusk rover]

theres no way thats solvable

[stable owl]

3pi/4 is a special angle.

[dusk rover]

ive never seen it

is it on the unit circle

omg waut

im stupid

i got it lemme try again

ok i got part a

ill attempt part b

[stable owl]

Sure. Do you need any help for part (b)?

[dusk rover]

ill try it and if i cant do it i will ping you

is that ok?

[stable owl]

Sure.

[dusk rover]

k thanks\

ok @stable owl i got this as the derivative

$$ -2\left(\cos:\theta+\sin:\theta\right)\left(-\sin:\theta+\cos\theta\right)=0 $$

[grand pond]

Jakey

[dusk rover]

which means (cos+sin)=0 and -sin+cos=0

so i got the angles

$$ \theta=\frac{\pi}{4}\frac{3\pi}{4}\frac{5\pi}{4}\frac{7\pi.}{4} $$

[grand pond]

Jakey

[stable owl]

I don't think your derivative is correct?

[dusk rover]

ok

im not sure how to find the derivative of this one

omg

wait

[stable owl]

Use quotient rule.

[dusk rover]

dont i just need chain rule

$$ -2\left(\cos:\theta+\sin:\theta\right)^{-2}\left(-\sin:\theta+\cos\theta\right)=0 $$

[grand pond]

Jakey

[stable owl]

Yes.

That is correct.

[dusk rover]

ok so i just forgot to minus the exponent

ugh let me get the angles again

[stable owl]

Sure.

[dusk rover]

ok i expanded to get

$$ \frac{-2\left(-\sin:\theta:+\cos\theta\right)}{\cos^2+cos\theta\sin:\theta+\sin^2}=0 $$

[grand pond]

Jakey

[dusk rover]

would it be ok to completly get rid of the bottom

cuz i can just times both sides by it

[stable owl]

Just set its derivative to 0

[dusk rover]

yes yes

[stable owl]

Also, consider the restriction to the answer.

[dusk rover]

what restriction

[stable owl]

So, the answer above by setting the derivative to 0 gives us infinetely many possible solutions

The derivative has infinite number of roots.

[dusk rover]

im not understanding what your getting at

[stable owl]

However, since we are restricted to a triangle the angle must be less than pi/2.

[dusk rover]

why is that

OH

wait

[stable owl]

nvm

[dusk rover]

waaaait

it can be obtuice

[stable owl]

Yes I do think I can be obtuse

[dusk rover]

but your right it cant be more than pi

[stable owl]

But the restriction would still be pi

Sry about that.

[dusk rover]

its ok dw

so my anlge is pi/4

[stable owl]

yes.

[dusk rover]

omg

thats so good

thank you

[stable owl]

Np.

Do you have other questions?

[dusk rover]

are you willing to help with another problem

[stable owl]

Sure.

[dusk rover]

ok lemme get a screenshot

this one is also non calculator

[stable owl]

Which part are you struggling with?

[dusk rover]

part a, i dont know how im supposed to solve the function when i set it to 0

$$ \sin x=\sqrt3\cos x $$

[grand pond]

Jakey

[dusk rover]

that doesnt seem possible

[stable owl]

Yes it is.

[dusk rover]

unless im missing comething agian

[stable owl]

Notice that, if you divide by cosx you can tanx.

[dusk rover]

hm yes

but still tanx=sqrt3

[stable owl]

Do you know the special triangles?

[dusk rover]

like 30 60 90 and 45 45 90?

[stable owl]

Yes. Which one do we use now?

[dusk rover]

30 60 90

if the adjecent side is sqrt3 then the short side is 1

[stable owl]

Yup. Let's use the convention of radians.

[dusk rover]

i think

[stable owl]

For the sake of this question.

[dusk rover]

ok

[stable owl]

So, let's draw out a triangle such that the tan(of some angle) is sqrt(3). What would it look like? What would be angle be?

[dusk rover]

the opposite would be sqrt 3 and adjacent 1

[stable owl]

Yup.

[dusk rover]

so its the 30* angle

or pi/6

[stable owl]

Not quite.

[dusk rover]

oh

60*

[stable owl]

Yup!

So, pi/3

Notice that, pi/3+2pik, for all positive integers of k, would give us sqrt(3)

[dusk rover]

yes

so pi/3 and 4pi/3

maybe

[stable owl]

Yes.

[dusk rover]

so those are the coordinates

[stable owl]

Yes.

That is the x-coordinate

You have to include y as well

Give it in the form of (x,y)

Instead of just x=...

[dusk rover]

ok thanks

[stable owl]

Do you need help with other parts of the question as well?

[dusk rover]

nah i should be good for this one

[stable owl]

Sure.

Do you have other questions?

[dusk rover]

yeah on another problem im confused a little

but i still need to finish this one

do you want me to ask you my question anyuway?

[stable owl]

Sure, if you want to.

You can keep asking.

[dusk rover]

ok for this one

when i am finding g(x) is it just 2.5 or 10.5

like

$$ 4\sin\frac{\pi}{8}\cdot\left(x-10.5\right)+12 $$

[grand pond]

Jakey

[dusk rover]

i dont know if i am shifting a whole new function or just f(x)

cuz it doesnt say f(x) is shifted right any

i also need help with (c)

cuz i stopped when i was confused there

[stable owl]

Which part are you struggling on?

[dusk rover]

(b) ii

the question is above

what im struggling on

@stable owl

😠

😭

its ok

.close

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[gentle pebble]

Is it wrong ?

[cerulean gust]

X^(1/4)

[cobalt sigil]

No

[cerulean gust]

It’s correct

[midnight plank]

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[normal plover]

what is he doing here

[ember shuttle]

just balancing it

[normal plover]

oh ok

what does he mean 5n down tho

[ember shuttle]

5 newtons

[normal plover]

og

oh

[inner wolf]

You can see it in the diagram

The resultant force is 14-9 which is in downward direction

[normal plover]

ohh ok

[normal plover]

i think of it as the 5n crosses out the 5n on the 10n as an equilibrium thing is that right?

[ember shuttle]

just 10-5

[normal plover]

ohok

[inner wolf]

Kinda, but equilibrium is when forces cancel each other out completely, here u have some left out force

[normal plover]

yes

[inner wolf]

It's called a resultant

[normal plover]

ohh

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[small jasper]

Please use induction to solve this problem, if possible. Prove that k^(2^n) -1 is divisible by 2^(n+2). K is an odd number

[olive shoal]

Start by proving the base case

[small jasper]

wym

[midnight plank]

@small jasper Has your question been resolved?

[tiny drum]

the base case is n=1

[small jasper]

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[uncut echo]

Hi is correct to ask for help here?

[midnight plank]

@uncut echo Has your question been resolved?

[uncut echo]

<@&286206848099549185>

[outer herald]

It's true

@uncut echo It's correct

[uncut echo]

All the questions?

[outer herald]

Wait let me check

[uncut echo]

Ok thanks! I also need some help on some other questions after this one. Thank you so much

[gusty dune]

what?

[uncut echo]

Are my response in the images correct?

[uncut echo]

Here

[gusty dune]

idk

sorry

[uncut echo]

Ah ok, how can I ask for help from people?

[midnight plank]

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[gloomy socket]

Is this correct?

<@&286206848099549185>

[midnight plank]

@gloomy socket Has your question been resolved?

[gloomy socket]

<@&286206848099549185>

[midnight plank]

@gloomy socket Has your question been resolved?

[gloomy socket]

<@&286206848099549185>

[midnight plank]

@gloomy socket Has your question been resolved?

[gloomy socket]

<@&286206848099549185>

[midnight plank]

@gloomy socket Has your question been resolved?

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.close

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[proven ice]

please help me, I am not sure I am correct...

[proven ice]

pretty please,

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[sinful socket]

Hi, I have a problem with this limit

[sinful socket]

Can u help me pls?

[midnight plank]

@sinful socket Has your question been resolved?

[wraith basin]

question so big nobody even trying

[sinful socket]

:(

[wraith basin]

I'm in grade 10 so I won't even try

[sinful socket]

aaah ok, no problem bro

[wraith basin]

Tag the helpers

<@&286206848099549185>

[sinful socket]

<@&286206848099549185>

[midnight plank]

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.close

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[cunning bobcat]

$y=ln(tanx)$ where X $\in ]0,\frac{\pi}{4}]$ then $\frac{dy}{dx}$=

[grand pond]

Joeller

[cunning bobcat]

now I know that the derivative of ln is $\frac{f(x)'}{f(x)}$

[grand pond]

Joeller

[cunning bobcat]

but the answer is a lil different and I wanna know how to get to that answer

[olive matrix]

there are often many ways to express trig things

[cunning bobcat]

the answer is 2csc2x

[olive matrix]

and what do you have?

[cunning bobcat]

$\frac{sec^2}{tan}$

[grand pond]

Joeller

[uncut cloud]

\LaTeX{} tip: there are macros to properly format special functions, like \( \ln(\tan x)) \).

[cunning bobcat]

I tried something but got a different answer

[olive matrix]

well i don't immediately see how to convert one to the other, but let's just check if those are equal

[grand pond]

Invariance

[olive matrix]

,w plot {sec^2 x / tanx, 2 csc(2x) }

[grand pond]

Results provided by WolframAlpha

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Requested by kitten.in.a.teacup

[cunning bobcat]

they are

[olive matrix]

try converting to sin and cos

[cunning bobcat]

I tried something and ended up with sec(x)csc(x)

[olive matrix]

that also seems to be equal

[cunning bobcat]

ye

I think the answer might be related to the double angle rules

[olive matrix]

yep

,tex .double angle

[grand pond]

hayley table

[cunning bobcat]

I'm still a little confused

[normal river]

i can try explaining if youd like

[cunning bobcat]

sure

[normal river]

sec^2 x is equal to 1/cos^2 x

[cunning bobcat]

yes

[normal river]

tan x = sinx/cosx

[cunning bobcat]

yep

[normal river]

so you have 1/cos^2 x * cosx/sinx

bc od dividing fractions

[cunning bobcat]

yeahh

[normal river]

youre left with 1/cosxsinx

[cunning bobcat]

then multiply by 2 ?

[normal river]

then you can multiply by two and divide by two

yeah

to get 2/2sinxcosx

which is 2/sin2x

[cunning bobcat]

yes

[normal river]

which is then 2csc2x

[cunning bobcat]

perfect

thank you sm

[normal river]

yeah np

[cunning bobcat]

.close

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[midnight plank]

@red cave Has your question been resolved?

[cobalt swan]

Because AT is perpendicular to plane of ABF

[midnight plank]

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[wooden knoll]

is (3, 2) the point of inflection?

[digital walrus]

hi pixel

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[safe oriole]

how do i solve b

[worldly peak]

You have to show that Un+1 > Un for all n

[safe oriole]

ik

my question is how to show that

i started by $U_n<U_n+1$

[grand pond]

pirateking0723

[safe oriole]

but now i am thinking about how to show that dividing the rhs of this inequality by 2 wont affect the inequality

idk if this is the correct/best way to approach this problem

so if this isnt the correct approach then correct me please

and if it is correct then help me proceed into the next step

<@&286206848099549185>

[worldly peak]

@safe orioleYou can show that Un+1 - Un > 0 for all n

[safe oriole]

@worldly peak how

[worldly peak]

bro pls

[safe oriole]

did anything annoy you ?

if the ping did then sorry i wont ping again

[safe oriole]

how will i do this

[midnight plank]

@safe oriole Has your question been resolved?

[safe oriole]

<@&286206848099549185>

[safe oriole]

<@&286206848099549185>

[green marlin]

Square both sides

And note that if $0\le U_n \le 1$ then:
[U_n\ge U_n^2]

@safe oriole

[grand pond]

Max

[hollow gate]

or you can just prove the function (sqrt(1-x)/2)-x >0 with all 0<x<1

[safe oriole]

both sides of U_n=sqrt(...)?

[green marlin]

Yea

[safe oriole]

so $U_n\geq\frac{1+U_{n-1}}{2}$

[grand pond]

pirateking0723

[green marlin]

now because $u_{n-1}\le1$ make a lower bound. (Btw the left hand side should be $u_n^2$)

[grand pond]

Max

[safe oriole]

why u_n^2 should be on left hand side i just substituted u_n^2 according to this so on RHS

[safe oriole]

so how can we proceed from here

[midnight plank]

@safe oriole Has your question been resolved?

[green marlin]

Okay so squaring both sides gives:
[ U_n^2\geq\frac{1+U_{n-1}}{2}]

Then using that $0\leq U_n\leq 1$ we have:
[ U_n^2\geq\frac{1+U_{n-1}}{2}\geq \frac{U_{n-1}+U_{n-1}}{2}=U_{n-1}]

In short
[U_n^2\geq U_{n-1}]

The crux is realising that because $0\leq U_n\leq 1$ then $U_n\geq U_n^2$

[U_n\geq U_n^2\geq U_{n-1}]

[grand pond]

Max

[green marlin]

@safe oriole hope that helps

[safe oriole]

tysm

.close

[midnight plank]

Channel closed

Closed by @safe oriole

Use .reopen if this was a mistake.

[midnight plank]

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[twilit field]

Let $x_1, x_2..$ be non negative real numbers such that $\sum_{i=1}^{2024}x_i=1$

[grand pond]

ƒ(Why am. I here)=I don't Know

[twilit field]

Find with proof

the maximum and minimumvalues of the expression

$\sum_{i=1}^{1012}x_i+\sum_{i=1013}^{2024}x_i^2$

[grand pond]

ƒ(Why am. I here)=I don't Know

[twilit field]

where do I even start

source UGB 2024

is this number theory ?

[junior flower]

you can probably do this with langrange multipliers

it’s more like optimization

[twilit field]

This is suppsoed to be solvable by highschoolers

[junior flower]

high schoolers better start learning about lagrange multipliers then

[frail burrow]

💀

[sage helm]

This is probably just playing arounf and estabilishing boundaries around square terms and stuff

[twilit field]

mind elaborating ?

[sage helm]

complete the square on the square terms or something

I'm grasping at straws

[frail burrow]

I think the maximum is pretty easy to do

First find a bound, and then show it is satisfied by some specific example

[sage helm]

Maybe a 2024 dimensional unit vector

[junior flower]

yea the max is ez

min not clear

[twilit field]

how would I do that

Intutively, it's 1

[frail burrow]

Yeah

Look at the squared terms

[dull path]

When you square a number it will get smaller

[frail burrow]

What happens when you square a number that's between 0 and 1?

[twilit field]

it gets smaller

[dull path]

So the total can't possibly be bigger then 1 using squares

[twilit field]

hmm

oh yeah

[frail burrow]

I think for the min it's the same trick

[twilit field]

makes sense

how would I prove that though

[frail burrow]

Write the inequality

What is x_i^2 bounded by?

[dull path]

||For min you would put all the sum in the squared terms||

[frail burrow]

Maybe spoiler that

[twilit field]

between 0 and 1, right?

[frail burrow]

Yes, but there's something better

Not a constant

As we said, squaring a number thats between 0 and 1 makes it smaller

[twilit field]

let me TeX it

just a minute

$\sum_{i=1013}^{2024}i^2<\sum_{i=1}^{1012}x_i<1$

[grand pond]

ƒ(Why am. I here)=I don't Know

[junior flower]

this line of thinking is not very good me thinks… if you were trying to minimize x + y^2 subject to x + y = 1, x,y >= 0, you wouldn’t put everything into y, would you?

[last slate]

Can we solve it with variance and mean ?

Just asking

[frail burrow]

Why not

A

Wait

[junior flower]

that would make x + y^2 equal 1

but you can get smaller than that

[frail burrow]

Ok that's not what I did

[junior flower]

with some weight on x

[frail burrow]

It's not similar mb

||x + y^2 >= x^2 + y^2 >= 1/2||

I meant its similar as in the idea of how you use the inequality

[twilit field]

none of this makes much sense to me tbh

[frail burrow]

Ok so,

You have the number 1

You want to split it into 2024 parts, and then take the second half of the values and square them

To maximize the expression, it's best to give the parts you DON'T square more weight

That's because squaring makes the number smaller

If you want to write it down using variables, this is basically just:

[twilit field]

hmm, so this is sort of like statistics

[frail burrow]

x_i^2 <= x_i for all 1 <= i <= 2024

So:

$\sum_{i = 1}^{1012} x_i + \sum_{i = 1013}^{2024} x_i^2 \geq \sum_{i = 1}^{1012} x_i + \sum_{i = 1013}^{2024} x_i$

[grand pond]

RedstonePlayz09

[frail burrow]

Literally just replacing all squared terms with x_i

And what do you get on the right hand side?

[twilit field]

1

[frail burrow]

Right

Now to make sure it's actually the maximum value and not just an upper bound

[junior flower]

i don’t think this is tight enough. the actual min of x + y^2 there is 3/4

[frail burrow]

What is a simple example that actually gives 1 as the value?

[frail burrow]

Is it not?

Can you give an example where it's smaller

Wait what

[twilit field]

all numbers in the sequence but one are 0

[frail burrow]

You are saying it's 3/4

Hmm

[junior flower]

well it should be on you to give me an example where x + y^2 is 1/2