help-17

[spare portal]

just making sure

[hard atlas]

not sure what you mean

can you give an example?

[spare portal]

lol

yo?

.close

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[median crane]

What do you mean

[vocal sleet]

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[haughty rover]

Sam has two identical red tokens. How many identical black tokens must be added to his collection so that, when all the tokens are used, 55 different arrangements in a row are possible?

[haughty rover]

how do i solve this

Have I written the equation correctly

Or nah

[median harbor]

cancel terms out

notice what happens when you cancel (2+x)! and x!

[haughty rover]

would i like expand x!

so it would be

x(x+1)(x+2)!

then i can cancel out the (x+2)!

[jolly epoch]

no

[haughty rover]

wait no

yeah that wouldnt work

[jolly epoch]

you want to cancel x!

[median harbor]

yes

[jolly epoch]

not (x+2)!

[median harbor]

so what can you do to make (x+2)! x!

[haughty rover]

oh

ohhh

yes

(x+2)(x+1)x!

[median harbor]

noce

noce

nice

now finish from there

[haughty rover]

(x+2)(x+1) = 105

expand and simplify?

[median harbor]

105?

[haughty rover]

gosh

silly me

110

[median harbor]

yes

now finish from there but before you close

you can recognise how n choose 2 = triangle numbers

and 55 is the 10th triabgle number

[haughty rover]

what

[median harbor]

do you lmow what a triangle number is

[haughty rover]

no

x^2 + 3x + 2 = 110

would i uh

use quadratic formula

ofc after i take 110 to the other side

[median harbor]

yeah sure do that

[haughty rover]

and then use the suitable answer by ruling out with common sense

alr

one sec

[median harbor]

altervative you can just factorise 110 and see which pair of factors are one apart

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[undone pier]

I understand that you can find maximums, minimums, and stationary points in a 2 variable function that graphs out a 3d surface

[undone pier]

I was taught that you can find whether the critical points in this 3d surface are maximums, minimums, or stationaries by putting the double derivatives at this critical point into something called a hessian matrix

if the determinant of this matrix is positive, then its a max if the double derivative in the x direction is below 0

if the determinant of the matrix is positive, then its a min if the double derivative in the x direction is above 0

and if the determinant is negative, then its just a stationary point

I was told this by my teacher

I'm just here to ask if its possible to find the max, min and stationaries in the following way and if this is valid too

Max --> Double derivative at the point in both x and y direction are negative

Min --> Double derivative at the point in both x and y direction are positive

Stationary --> Double derivative at the point in x direction is negative, and in y direction is positive OR double derivative at the point in x direction is positive, and negative in the y direction

[magic wasp]

A stationary point is the same as a critical point to me

It's just when the derivative (gradient) is 0

[undone pier]

yes this is what i meant

I guess the third option would look like this

saddle point

[magic wasp]

Yes

It's not the only option though

You can also have a positive double derivative in x and a zero double derivative in y

[undone pier]

and a neg double derivative in x and a zero double in y

and vice versa

yes sure

[magic wasp]

Yes

[undone pier]

i forgot all cases for the saddle point

my question basically just centers around the following

can i just do that instead of bothering with the determinant of the hessian matrix?

[magic wasp]

I mean if I remember correctly the Hessian is basically just a way to find what you're asking about

[undone pier]

yes they give the same result they're just two different methods

[undone pier]

i'm basically just asking this because i'm sitting in class right now and was wondering how to interpret these stationary points and i sort of intuited this in my head

and i want to know if my understanding of this is correct

even though the hessian gives the same result

[magic wasp]

For min and max, I believe so

For other critical points, well there are more options in 2D

[undone pier]

so if we include all combinations of double derivatives being having different signs or just being 0 (+ or -) then its correct, yes?

if we define other nonmax or nonmin critican points that way

[magic wasp]

I think so

In 1D a critical point is either a min, a max, or just a point with horizontal tangent that is neither min or max

In 2D, the determinant of the Hessian gives you the Gaussian curvature

(imagine a sphere that "touches" the function at that point, its radius is the inverse of the Gaussian curvature)

[undone pier]

a sphere in 2d?

[magic wasp]

In 3D, when you graph the 2D function

[undone pier]

oh okay

[magic wasp]

Just another way to visualize it

We use the determinant of the Hessian because it's the product of the eigenvalues, so if it's positive then the eigenvalues are either both positive or both negative, and you have a min or a max

Hopefully you have some experience with eigenvalues

[undone pier]

I do not, the teacher just shoehorned in the idea of a hessian matrix so we could find maxes and mins

[magic wasp]

Hm sounds like a physics lesson

[undone pier]

believe it or not its a math lesson

my understanding of linear algebra isnt the best so I was just coming here to find out if I conceptually understood how to find maxes and mins and stationaries in 3d space

in terms of just looking at double derivatives

because i dont conceptually understand how the hessian matrix is supposed to give us these maxes and mins because i dont know what an eigenvalue is

[magic wasp]

Yes I think your conceptual understanding is good, you just have to be careful and account for all possibilities

[undone pier]

okay, thank you

i'll continue to just use the hessian without knowing what it is at this point just to get the answers right, but its comforting to know that I'm not completely misunderstanding everything

thank you

[magic wasp]

You're welcome
I'm trying to find an analogy for the eigenvalues but it's so abstract...

[undone pier]

i wont burden you with teaching me about eigenvalues, i'll close the channel now so others can use it, thank you again

.close

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[rose shard]

b)

This is what I thought

So, I tried to do something with complementary

Like saying 1 - Probability there are no balls that has same number as round picked

Which must be the probability that at least one satifies

And I tried to do something with like the binomial saying it had 0 successes

but not quite the right result -< left side is the right result, and right side is what i got

[dark kiln]

google derangement

[rose shard]

Oh shit

That's literally the formula

or almost

the one he has

In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.

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@rose shard Has your question been resolved?

[cyan shadow]

(holy hell!)

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[hard atlas]

<@&268886789983436800>

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[slender depot]

Need help finding all functions f(n) : N -> N. Satisfying f(mn) = f(m)f(n), and f(n) < f(n + 1) for all natural numbers m, n.

[slender depot]

I’ve proven that f(n^k) = f(n)^k

and trivially f(1) = 1

[sudden compass]

Didnt you post this problem, like quite a while ago?

[slender depot]

I have not

posted this problem

Atleast I think I haven’t

[sudden compass]

Hmm okay

[slender depot]

I posted another functional equation

a couple months back

[sudden compass]

https://math.stackexchange.com/questions/1703615/the-functional-equation-fmn-fmfn-where-f-mathbbn-rightarrow-mathbb
this might help

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[long jackal]

What should I do from here?

[vocal sleet]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[long jackal]

🥸

<@&286206848099549185>

[proud vessel]

Misri?

[long jackal]

yes

[proud vessel]

Wait whats a dhil3?

[long jackal]

side

[proud vessel]

Oh

Ight

[vocal sleet]

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[mortal cloud]

hwo do u differentiate this

[cyan shadow]

have you heard of the chain rule

[mortal cloud]

yh

managed to complete it

[cyan shadow]

[mortal cloud]

.close

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[viscid pumice]

latex:
$$w_0+f/I_0=w_1\times R-f\times R/I_0$$

Could you clarify what these values are? Is everything here a real number or is there more going on?

[merry python]

yes you have written this very badly wrong

[twin meteor]

nHail

[viscid pumice]

if you recognize this formula I'll let you handle this

[merry python]

this seems to be a question from rotational mechanics with torque stuff

please post the original question

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[vagrant rose]

hi need help with understanding how to do this

[vagrant rose]

dont know how to start

[sudden compass]

I would start by writing out a few terms, see if theres a pattern

[vagrant rose]

lowkey just did ratio test but idk how that helps me

oh

[sudden compass]

also how are we defining n choose k when n is rational?

[sudden compass]

hmm wdym? what did u get from ratio test?

[vagrant rose]

i dont actually understand what n choose k is, teacher just told us thats sorta the form of what it looks like

should just be 1.7^n on the right hand side there in that denominator

[sudden compass]

how did u go from line 2 to 3

[vagrant rose]

everything cancels except for 1/5-n on the numerator and n+1 on the denominator

[sudden compass]

oh I got it

[vagrant rose]

and also the 1.7 yeah

[sudden compass]

yeah just got it sorry

[sudden compass]

well doesnt this approach 0

[vagrant rose]

doesnt it converge anything less than 1 for ratio test

[sudden compass]

u did this wrong

u forgot abs value!

[vagrant rose]

fax

thats true

so it just diverges?

[sudden compass]

Its 1.7 with abs value i think

[vagrant rose]

yeah

this was the answer choices

[sudden compass]

Let me check on wolfarmalpha but I believe that is the case

[vagrant rose]

alr

that would make sense

especially since we didnt get taught anything that would help us know what exactly it converges to

[sudden compass]

it converges...

[vagrant rose]

oh

how do i do it then

[sudden compass]

this is it right (1/5 = 0.2)?

[vagrant rose]

i dont know if thats the correct form

[sudden compass]

I wrote $\choose{n}{k} = \frac{n!}{k!(n-k)!}$

[twin meteor]

ItzKraken

[sudden compass]

fudge

[vagrant rose]

this how i have it written out

from this

[sudden compass]

yeah it should be the same

[vagrant rose]

alr

[sudden compass]

wait no its the opposite

n! should be on top

[vagrant rose]

wtf

[sudden compass]

its $n\choosek = $ $\frac{n!}{k!(n-k)!}$

[vagrant rose]

so my prof just taught us opposite or?

[twin meteor]

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

[worthy citrus]

$\binom{a}{b}$ if you want that syntax btw

[twin meteor]

ΣΑCu

[sudden compass]

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

[twin meteor]

ItzKraken

[sudden compass]

OH THANK GOD

yeah

[vagrant rose]

ok so my prof taught us that n! is denominator

i have so many notes written out that way

still dont know how to find what it converges to

[sudden compass]

ur prof has a skill issue

[sudden compass]

well first apply ratio test again

Lemme see what I can come up with

[vagrant rose]

ngl bro

this def looks like it diverges

[worthy citrus]

The n is in the bottom in their question

[lapis marten]

https://en.m.wikipedia.org/wiki/Binomial_theorem#Newton.27s_generalised_binomial_theorem

[worthy citrus]

Fractional binomial have different definition

[lapis marten]

Look under generalized

[vagrant rose]

yeah that looks like what i learned

so then ratio test says it diverges then

[sudden compass]

still his denom had $n!$ but it should be $n!*(\frac{1}{5} -n)!$

[twin meteor]

ItzKraken

[worthy citrus]

Not for Generalised coefficients

See jelles link

[vagrant rose]

i got answer

.close

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[sage cargo]

How can o solve this integral:

[sage cargo]

$\oint_{\gamma}^{}\frac{e^{iz}}{z^2}dz$

[twin meteor]

Olá, me chamam de Misas

[sage cargo]

$\gamma=e^{it},0\leq t\leq 2$

[twin meteor]

Olá, me chamam de Misas

[sage cargo]

I've tried Cauchy's integral theorem but doesn't worked well

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@sage cargo Has your question been resolved?

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@sage cargo Has your question been resolved?

[sage cargo]

.close

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[knotty vigil]

hey yall dont know where to start with this one

[eager lichen]

So

I think

Since it is a right triangle, thus the hypotenuse will be 13, since two legs are 12 and 5

And for cosine, it will be adjacent/hypotenuse

Thus cosx=12/13

[knotty vigil]

damn iam just supposed to adjust the formulas on right to somehow get cos x :D

ill send another example

[eager lichen]

Well then

No clue actually 😭

[knotty vigil]

its alright ill try to do it have an idea might work :D

[eager lichen]

I’m back

I think you can do

[knotty vigil]

iam still clueless

[eager lichen]

Sinx/cosx =5/12 first

Then you do cross multiply

You got 5cosx=12sinx

Then you square both side, you got 25cos^2x=144sin^2x

You can change it to 25cos^2x-144sin^2x=0

Then you know cos^2x+sin^2x=1, you multiply this by 25, and subtract it from the first equation

Then you got -169sin^2x=-25

Then you got sinx=5/13

Then you got cosx=12/13 by using cos^2x+sin^2x=1

[knotty vigil]

give me a minute i think iam on the right path

[eager lichen]

Ok

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[ocean anvil]

obviously the matrix is symmetric

so we have $M=P^{-1}DP$ where $D$ is diagonal and $P$ is reversible.

[twin meteor]

michelson prime

[ocean anvil]

i don’t know where to start

[naive herald]

,tex say $t=\alpha/n$. Then you have $\lambda_1 = \cos t +\sin t$ with $v_1 = (1,1)$ and $\lambda_2 = \cos t - \sin t$ with $v_2 = (1,-1)$.

Thus [P = \begin{bmatrix}1&1\1&-1\end{bmatrix}] and [D=\begin{bmatrix} \cos t +\sin t&0\0&\cos t - \sin t\end{bmatrix}]

[ocean anvil]

did you have a logic way to function lambda1 and lambda2

[twin meteor]

RaD0N

[ocean anvil]

like how did you find them ?

[naive herald]

1 moment. Let me fix that

[twin meteor]

RaD0N

[limber wedge]

My 2 cents on identity matrix of dimension 2x2

[naive herald]

basically, I chosed the vector (1,1) and done the multiplication and saw that it is an eigen vector. Then I tried (1,-1) and done the same thing.

[ocean anvil]

but how did you know 1,1 would work

[naive herald]

because the columns are almost identical, it's only interchanging the rows

[ocean anvil]

oh ok

but if i didn’t know that

[naive herald]

but u also could solve but brute force

[ocean anvil]

how can i find them with eigen values ?

i don’t see obvious eigen values

[naive herald]

solve $\det(A-\lambda I_2)=0$ for $\lambda$.

[twin meteor]

RaD0N

[ocean anvil]

and i tried X.In - M

[ocean anvil]

may i know what you call this ?

characteristic polynom?

[split heart]

yes, that's the characteristic equation for a matrix

[ocean anvil]

ok

ill try that thanks a lot all of you

[naive herald]

and then [\cos t +\sin t = \sqrt{2}\sin(t+\pi/4)]
with
[\cos t - \sin t = \sqrt{2}\cos(t+\pi/4)]

[twin meteor]

RaD0N

[split heart]

don't think that's necessary. just need to take limit as t->0, no?

[naive herald]

not that fast

[vocal sleet]

@ocean anvil Has your question been resolved?

[naive herald]

I guess that matrix could approach 0

,tex $\lim_{n\to\infty} \sqrt{2}\sin^n(\alpha/n+\pi/4)=\sqrt{2}\left(1/\sqrt{2}\right)^{+\infty}=0.$

[twin meteor]

RaD0N

[naive herald]

Hence the limit is the zero matrix

@ocean anvil

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[tardy mountain]

is this not right?

[rough zenith]

should be correct

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@tardy mountain Has your question been resolved?

[tardy mountain]

I'm getting a 6/9 🤦‍♂️

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[unreal sparrow]

I’m trying to find f’(x) of the integral on the right (equal to f(x)), but the textbook says the answer is cosx(sqrt(sinx))

[unreal sparrow]

.close

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[trail sinew]

Let $Q:\mathbb{C} \rightarrow \hat{\mathbb{C}}$ be a rational function with no real poles such that $\lim_{|z|\rightarrow \infty} Q(z)=0$.\
Show that:\
$\lim_{R\rightarrow +\infty} \int_{-R}^R Q(x)e^{ix}dx = 2\pi i \sum_{\substack{z_0\ pole\ of\ Q\ Im(z_0)>0}}Res(Q(z)e^{iz},z_0)$.

[twin meteor]

Casiel368

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[river hedge]

hello

[river hedge]

so far I have that
$S={s_1,s_2,...,s_n}, V={a_1s_1,a_2s_2,...,a_ns_n:a_1,a_2,...,a_n\in\mathbb{F}}$

[twin meteor]

george clooney real account

[river hedge]

i understand the intuition for this pretty well

but don't know how to prove formally

but basically T fulfils equation $b_1t_1+b_2t_2+...+b_mt_m=0, m\leq n$

[twin meteor]

george clooney real account

[river hedge]

if all the b_i are equal to zero then i've proved linear independence and if not all of them are then i need to show that there is a smaller subset that spans it

how...?

[cyan talon]

well yeah we can go by contradiction

suppose not all of the b_i are 0

what would that mean ?

@river hedge

[river hedge]

that some of them "cancel out"

but idk how to show that really?

that one of them is a multiple of the other i suppose but that's only true when there's 2

like if there's three base vectors going at 0, 2pi/3, 4pi/3 radians from the origin then three of them would cancel out

but i find it hard to state that mathematically

[cyan talon]

that you have a nontrivial lin combo which gives you 0 yeah

[river hedge]

but where do i go from there?

[cyan talon]

so say we have $c_1 t_{i_1} + \cdots + c_k t_{i_k} = 0$, with all the $c_i \neq 0$, and the $i_k$ select some of the indexes $1\cdots m$

[river hedge]

indexes ?

like the letters at the bottom ?

[twin meteor]

aPlatypus

[cyan talon]

yeah

I'm selecting the vectors who are part of the nontrivial lin combo

[river hedge]

ah okay

[cyan talon]

well then

[river hedge]

well you can substitute one out entirely

[cyan talon]

yep

[river hedge]

so the list is now one element smaller

and repeat if there are still nontrivial combos?

or just

by contradiction tbh

"not smallest"

[cyan talon]

well you've already shown the contradiction yes

[river hedge]

okay thank you so much

[cyan talon]

so all the b_i are 0

and we won

[river hedge]

man linear algebra is really interesting but

the proofs are weird

i like analysis more

ok tysm for the help

[cyan talon]

welcome to math

[river hedge]

yeah its my first year of undergrad

very fun and i love it

but

omg dude

see u :)

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[forest haven]

How shool I solve this?

[vocal sleet]

@forest haven Has your question been resolved?

[vocal sleet]

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[prime umbra]

stuck on this problem here is what i have tried so far

[pale perch]

[late ingot]

implicit diff the equation and plug in 3 and 1 and for x and y and solve for y'?

[prime umbra]

do you know where i messed up?

[pale perch]

oh wait was that file what youve done so far?

[prime umbra]

yea

[pale perch]

i was waiting for a screenshot or something

[prime umbra]

yea the file is a picture of my paper work

[pale perch]

id prefer not to download anything

[prime umbra]

alright hold up

is that better?

[pale perch]

ah nice, yeah

so, your second line seems okay
what were you doing after?

[prime umbra]

what i got was a slope of 9/130 +103/130

but that does not sound right

what do you mean doing after?

[pale perch]

just looks very jumbled
were you trying to get dy/dx=f(x,y) or were you just subbing in (3,1) then solving after

[prime umbra]

first i tried to get the derevative which i thougt was 48x/(4x^2+4y^2)(52y)

then i plugged in 3

i think i might have messed up on the derevative

sorry for the jumbled work

[pale perch]

your second line after differentiating was fine
this isnt though

if you were going to rearrange for dy/dx you would have to expand all the brackets first

it would be easier to just plug (3,1) into your second line

[hoary blaze]

chain rule lol i'm blind

[prime umbra]

chain rule

yea my bad

rearrange

ok hold on let me process

[pale perch]

youd have to do this, its a bit lengthy

[prime umbra]

wow

thanks

[pale perch]

never closed my square bracket, oops

[prime umbra]

that is wayyyyy yclearer

wait it is wrong?

[pale perch]

no its fine

[prime umbra]

ok so why could i not have divided part of it like the way i did

it

instead of combining

[pale perch]

what exactly were you actually dividing by?

[late ingot]

that denominator doesnt look right to me

[prime umbra]

mine or his

or hers

[late ingot]

the picture

[prime umbra]

idk

[late ingot]

i know you divided by 2 but the powers of the vars in the denominator arent similar

[prime umbra]

wait ok i am confused now is -9/13 wrong?

[pale perch]

hm one momento

[late ingot]

that is correct for slope of the tangent

[pale perch]

yeah youre right

[prime umbra]

ok

i got it then

i think

[pale perch]

my 4y should be 4y^3

[late ingot]

show us the equation for the tangent

[prime umbra]

yea algith let me caclulate

[late ingot]

the calculation is done, slope was all that we needed calculation for

the equation should just be plug and chug now

[pale perch]

fixed

[prime umbra]

so i got -9/13 +40/13

is that good

[pale perch]

where is the 40/13 coming from

[prime umbra]

?

[late ingot]

we need the equation of the tangent

hint: the tangent is a linear function

[prime umbra]

i plugged in 3 and then -27/13 add to 1 or 13/13

and then that would be equal to b right?

[late ingot]

you are overthinking it

do you remember the point-slope intercept formula?

[prime umbra]

so -9/13x +40/13

yea ok it is coorrect

i just put it into the thing

i am good thank you for the hlep

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[sullen geode]

Could someone help me with part c? I am getting the initial velocity to be zero, but I am not sure if its right

[sullen geode]

This is my attempt

Im assuming its right? Because if the door is slammed shut (y=0) position is zero and velocity would be 0

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@sullen geode Has your question been resolved?

[vocal sleet]

@sullen geode Has your question been resolved?

[vocal sleet]

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[marble anchor]

I evaluated it as 124-(-17) / 6-(-3). Which resulted in 141/9, but the answer was still wrong.

please tag me when a helper is here

[drifting jackal]

How did you get -17 for f(-3)?

[marble anchor]

Let me redo it

oh for fucks sake

Got it

Help with this one please @drifting jackal

Let me go back and do the problem

The mean theorem I got 364/14 which equals 26

The deriv is 6x^2-18x-60

Which I factored out to get x^2-3-10

Would I ahve to do quadratic formula?

[drifting jackal]

Yes but you're missing the part where it says f'(c) is equal to the mean slope

If the mean slope is 26

That means f'(c) = 26

[drifting jackal]

So this is equal to 26

[marble anchor]

Ah yes yes forgot about that

So I set the deriv equal to 26

I believe thats what we did in class at least

so x^2-3-10 = 26?

[drifting jackal]

No

You factored out a 6

So it's 6(x^2-3x-10) = 26

[marble anchor]

So would I divide 26 by 6

I’m sorry i’m not sure what step i’d take after

[drifting jackal]

You need to have it equal to 0

Then use the quadratic formula

Also I suggest doing 6x^2-18x-60 = 26

Instead of dividing by 6

So then you don't have to deal with a fraction

[marble anchor]

You just said two different things so

Should i bring 26 over to the left

and then do quadratic?

[drifting jackal]

Yes, move the 26 over first

[marble anchor]

Which makes it 6x^2-18x-86

And then do the quadratic formula from there

?

[drifting jackal]

Yes

[marble anchor]

Okay I got it thank you

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[vast shale]

What’s the conjugate of i square root 2?

[vast shale]

i √2

What’s the conjugate of that

[cyan shadow]

$-i\sqrt{2}$

[vast shale]

Ok

[cyan shadow]

assuming you just mean the complex conjugate

[vast shale]

Would -i √2(i √2) become -i^2+2?

If I multiplied them together

[cyan shadow]

... no

[vast shale]

Wait what

[cyan shadow]

1. if you know what a complex conjugate is, you should be able to do very basic multiplication of complex numbers to see what it results in
2. ... what do you think the point of a complex conjugate is

look up "how to multiply numbers" and us the property that $i^2 = -1$ to solve this yourself

[vast shale]

So the answer would be 3

[cyan shadow]

????

please show your work

[vast shale]

-i^2 +2

[cyan shadow]

first of all, what you wrote isnt equal to 3, but also

where did you get addition from?

why are you adding anything? your question is pure multiplication

[vast shale]

Square root 2 Times Square root two is 2

[cyan shadow]

okay yes good work

so now we have $-i\cdot i\cdot2$

[vast shale]

Oh shit mb

I was supposed to mult the two

2

[cyan shadow]

a very convenient property of complex numebrs and their conjugates is that when we multiply them together we get a real number

also a "complex conjugate" is just the same as the first number, but the imaginary part has the opposite sign

[vast shale]

Ok so my final answer is x^3-3x^2+2x-6

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[cyan shadow]

im not convinced i wasnt being trolled

[vast shale]

Original problem included complex numbers but it required roots

[cyan shadow]

(mostly cuz of the random final answer at the end thats unrelated to everything else)

okay fine sure

glad i could help

[vast shale]

I just needed help with the first part of the question

[vocal sleet]

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[faint wing]

for #37 idk where to even start

[faint wing]

i think you gotta change it to spherical coordinates

[weary sentinel]

Yup

It makes the question trivial

[faint wing]

so sqrt(x^2 + y^2 + z^2) = p right?

[weary sentinel]

Yup

[faint wing]

pe^p?

ep^(p^2)

[weary sentinel]

… no

[faint wing]

no no

[weary sentinel]

This but no sqrt in exponent

[faint wing]

pe^(p^2)

[weary sentinel]

Yeah that’s right

[faint wing]

im confused on the bounds of integration now

[weary sentinel]

What about them?

[faint wing]

so x^2 + y^2 + z^2 = a^2 is one half of the sphere?

[weary sentinel]

A full sphere

[faint wing]

and x^2 + y^2 + z^2 = b^2 is the other half?

oh

[weary sentinel]

they’re two spheres total

One of radius a

One of radius b

so instead of integrating p from 0 to R

you integrate from radius a to radius b

[faint wing]

ah

you want to find the area between sphere a and b

[weary sentinel]

Not area per say

[faint wing]

sphere a is inside sphere b

[weary sentinel]

You’re integrating a function

[faint wing]

yeah eayh

with the bounds being sphere a inside of sphere b?

[weary sentinel]

Uh between the two no?

[faint wing]

ah

[weary sentinel]

Integrating across the volume between surface a and surface b

Those surfaces are spheres

[faint wing]

i see

[weary sentinel]

I gtg don’t forget the jacobian !!!!

[faint wing]

yup

thank you youre fucking awesome

[weary sentinel]

It was my pleasure

[faint wing]

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[flint breach]

how do i do this?

and this

for 9 i tried setting up the integral

$$\pi\int_{0}^{2}(y^2)^2-(2y)^2)dy$$

[twin meteor]

Arctic

[sudden compass]

the shaded area is what we are rotating, yes?

[flint breach]

i did that

😭

im so stupid im sorry

2y should have gone before y^2

but 13 still doesnt make sense

[sudden compass]

lemme see

okay

[flint breach]

soooo

[sudden compass]

lemme paint it

this is what hes rotating

[flint breach]

ohh

i thought it was like

inbetween...

[sudden compass]

no cus that wouldnt make sense

so to find orange you do $\int_{-1.047}^{1.047} 3 dy - \int_{-1.047}^{1.047} \sec x + 1$

[twin meteor]

ItzKraken

[sudden compass]

(1.047 is approximately the intersection point of sec x + 1 and y =3)

[flint breach]

well isnt this washer method

ohh i see

okay but what about once we rotate it

[sudden compass]

well let orange area = O

then $\int_{0}^{2\pi} O dx$, i think

[twin meteor]

ItzKraken

[flint breach]

shouldnt it be pi times it

[sudden compass]

oh yes 🤦‍♂️

i missed the square too

$\pi \cdot \int_{-1.047}^{1.047} O^2 dx$

[twin meteor]

ItzKraken

[sudden compass]

this should be it

@flint breach

[flint breach]

hmm

it isnt

this is the answer

[sudden compass]

[vocal sleet]

@flint breach Has your question been resolved?

[flint breach]

<@&286206848099549185>

[sudden compass]

oh wait am dumb

[sudden compass]

we dont need to subtract

[flint breach]

sooo

what do we do

[sudden compass]

am close but I dont have the exact answer (I am off by a few decimal places)

[flint breach]

(bounds are +-pi/3)

[sudden compass]

thx

BRUH AM STILL OFF BY 0.1 TF

okay i gotta do this by hand

[flint breach]

did you get it @sudden compass

[sudden compass]

no 😭

still thinking

I cant get it

<@&286206848099549185>

[vocal sleet]

@flint breach Has your question been resolved?

[vocal sleet]

@flint breach Has your question been resolved?

[hoary arch]

YEAH IT IS INBETWEEN

GENERALLY, ITS:
[VOLUME OF HIGHER FUNCTION] - [VOLUME OF LOWER FUNCTION]

THE Y=3 LINE IS HIGHER THAN Y=1+ SEC(X) FOR THE REGION OF INTEREST

[vocal sleet]

@flint breach Has your question been resolved?

[vocal sleet]

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[vast shale]

2n^+n``` how can I solve this?

[night herald]

2n^2 + n or 2n^n

[river minnow]

Solve what?

[outer warren]

the expression you typed is also questionable

[little portal]

i think they meant

= instead of +

and 2^n

actually no i have no idea 💀

[vocal sleet]

@vast shale Has your question been resolved?

[vast shale]

can I say it is 3n^2?

[onyx fog]

Hi

Oh

Hmm

Try to make tree numbers and resolve

[outer warren]

no idea what your expression is supposed to be

[little portal]

what do you want to solve for though?

[outer warren]

do you have a pic of the original problem

[vast shale]

can 3n^2 equal to 2n^2 + n

[outer warren]

they're not equivalent, but can have the same value depending on what n is

[little portal]

if n is 1

then yeah

or 0

[vast shale]

what if n is > 1000?

[outer warren]

no

[vast shale]

so they cannot be equal if n > 1000

[outer warren]

you can determine when they're equal by solving the equation
3n^2 = 2n^2 + n

[vast shale]

thank you

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[upper bluff]

which ones are bipartite? i thought B,C and D but its not correct

[hard atlas]

when is a graph bipartite?

[vocal sleet]

@upper bluff Has your question been resolved?

[upper bluff]

when the edges go across ???

but cant you redraw some of the ones i sent so the edges dont go across?

[hard atlas]

are you confusing it with planar?

bipartite means that you can put the vertices into two sets, such that all the edges are from one of the sets to the other

[upper bluff]

so like a graph that is k2,4 ?

2 at bottom that connect to 4 at the top?

[hard atlas]

for example, yes

although in that case all of them are connected

which doesnt have to be the case in general

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@upper bluff Has your question been resolved?

[vocal sleet]

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[zealous pewter]

∫dx/(2cos^2(x)+3)

[urban valley]

multiply numerator and denominator by 1/cos^2theta

[hearty oyster]

Guys

I need help

[mental falcon]

!occupied

[vocal sleet]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[zealous pewter]

?

[mental falcon]

what is happening there, why is there dx/ right after the integral? 🤔

$\int\frac{dx}{2cos^2(x)+3}$

this is your problem?

[marble schooner]

dx/y = (1/y)dx isn't it?

[mental falcon]

oh nm i was confused

[urban valley]

convert sec in denominator in tan terms

the substitite tan

[twin meteor]

Soosh

[urban valley]

@zealous pewter

did u get?

[zealous pewter]

I need to solve it (I don't know English very well)

They asked at school

[urban valley]

$\int\frac{sec^2(x)dx}{2+3(1+tan^2x}$

[twin meteor]

control

[urban valley]

now substitute tan^2x

I mean not now

first solve denominator

little bit

$\int\frac{sec^2(x)dx}{5+3tan^2x}$

[twin meteor]

control

[urban valley]

$\int\frac{1/5 sec^2(x)dx}{1+3/5tan^2x}$

$\int 1/5 \frac{sec^2(x)dx}{1+3/5tan^2x}$

$1/5\int\frac{sec^2(x)dx}{1+3/5tan^2x}$

$3/5\int\frac{sec^2(x)dx}{5/3 +tan^2x}$

$1/3\int\frac{ sec^2(x)dx}{5/3+5/3tan^2x}$

oof finally

no

shit

$1/3\int\frac{ sec^2(x)dx}{5/3+tan^2x}$

[twin meteor]

control

[urban valley]

oof finally

@zealous pewter

now substitute tan x = t

[zealous pewter]

$1/3\int\frac{ 3dx}{5+3t^2}$

$1/3\int\frac{ sec^2(x)dt}{5/3+tan^2x}$

$1/3\int\frac{ 3dt}{5+3t^2}$

[twin meteor]

HolyDevil

[zealous pewter]

$\int\frac{ dt}{5+3t^2}$

[twin meteor]

HolyDevil

[zealous pewter]

Right or wrong?

[copper anchor]

find <jkm

[zealous pewter]

<@&286206848099549185>

[dawn kestrel]

yes its right

[zealous pewter]

.close

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[mellow geode]

helo

[mellow geode]

i need to find the eigenvalues

[vocal sleet]

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

[mellow geode]

1

[dawn kestrel]

first, do you know what are eigenvalues?

[mellow geode]

yes

[ionic pecan]

Hello guys, please assist me. in this tutorial the result with my ? mark how come he equaled- 180x10 with 5 exponentiation equals to 1.8x10 with 7 exponentiation? as first will be 180,000,000, and second 18,000,000,000

[dawn kestrel]

sorry this help channel is being used rn

[cursive comet]

How did you get 18 000 000 000

[crimson jetty]

This is not their channel, please do not derail.

[cursive comet]

Sorry i just opened it without looking

[dawn kestrel]

are you familiar with the equation Av = λv, where λ is our eigenvalue

[ionic pecan]

please suggest right channel to ask about that

[dawn kestrel]

an available math help channel

[mellow geode]

no

[dawn kestrel]

ok so basically, an eigenvalue is some constant λ such that the equation Av = λv is satisfied, where v is the corresponding eigenvector (which is non-zero)

we find the eigenvalue by solving the equation det(A - λI) = 0

[mellow geode]

ok

but i cannot find the solutions

[dawn kestrel]

do you know what a determinant is?

[mellow geode]

yes

here it would be the det of a 2x2 matrix

[dawn kestrel]

do you know how to find the determinant of a 2x2 matrix?

yes

exactly

[mellow geode]

but how do i simplify the determinant here?

[dawn kestrel]

(t+4-λ)(4t+1-λ) - 4(1-t)^2 = 0, and then solve for λ

[mellow geode]

and how do i solve it?

[dawn kestrel]

just expand

or if ur lazy put into wolfram alpha 🙂

[mellow geode]

and where is the 1/5?

[dawn kestrel]

multiply both sides by 5 to get rid of it

[mellow geode]

?

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[barren falcon]

Is the set of polynomials with integer coefficients with the polynomial composition operation set on it a monoid? I think yes, but i want someone to confirm

[barren falcon]

Also I think it's not a group, just monoid

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[tardy jetty]

can some1 please explain this. I have an exam tomorrow and cant figure this shit out

[tardy jetty]

the solution:

[dawn kestrel]

Multiply the first fraction by z-1 on both the numerator and denominator

Multiply the second fraction by x-1 on both top and bottom

Same with third fraction multiply by (y-1) on both top and bottom

Now the denominator of the three fractions is all the same and you can combine the numerator

[tardy jetty]

but how do i know to do that? is it just the regular procedure?

[half imp]

that's how you add any fractions together

It's least common denominator

[tardy jetty]

ahhhh yea now i get it i was thinking too far ahead

thx guys

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[polar pumice]

How to prove that $\sum_{n=1}^{\infinity} \frac{3^n}{n^3}$ is divergent? I have tried with comparison test, but I cannot find a divergent sequence that is smaller than this sequence.

[twin meteor]

abbiefung

How to prove that $\sum_{n=1}^{\infinity} \frac{3^n}{n^3}$ is divergent? I have tried with comparison test, but I cannot find a divergent sequence that is smaller than this sequence.
```Compilation error:```! Undefined control sequence.
l.57 How to prove that $\sum_{n=1}^{\infinity
                                             } \frac{3^n}{n^3}$ is divergent...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]```

[lapis marten]

What is the limit of 3^n/n^3 as n goes to infinity?

[polar pumice]

Infinity

[lapis marten]

yeah, so would it be possible for the sum to converge?

[polar pumice]

No, but how can I prove it?

[lapis marten]

An infinite sum is the limit of a finite sum as the number terms goes to infinity

But the finite sum is always greater than its last term

and the last term already diverges, so there's no way the sum also converges

[obsidian stream]

I think it's easier to use something like the nth root test or ratio test than construct a comparison here

[polar pumice]

Oh! I overthought it. Thanks very much

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[spring sigil]

Have no idea how to do this

[flat whale]

read https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/05%3A_Integration/5.03%3A_Riemann_Sums

[tidal dock]

omg are these your sums? /j

[spring sigil]

For Riemann sum all I remember is x_k=a+k(delta x) and delta x=(b-a)/n

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[flat whale]

Read the link to learn more than just what you remember

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[unreal sparrow]

How to complete this if I don’t know what f(t) is equal to?

[vale nexus]

F(x) is the area under the curve from 3 to x

the problems are set up so you wont need to know f exactly

remember that integrals consider area under the x-axis as negative

[unreal sparrow]

So is F(7) also 0?

Bc its symmetrical and cancels out?

[vale nexus]

yeah

[unreal sparrow]

For part c, F(0) and F(5) are positive, and F(8) is negative?

[vale nexus]

no yes yes

but dont proceed until you understand why

eep sorry

[unreal sparrow]

Oh its bc 0 is less than 3?

[vale nexus]

yes the integral from 3 to 0 is the opposite of the integral from 0 to 3

[unreal sparrow]

Ok thanks

[unreal sparrow]

Finished the rest, this correct?

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[crimson grove]

Could anyone help me understand how to solve this problem? I am being asked the following:For the function, let g(x,y,z) = xy^(2)+2z^(2). Compute the directional derivative at the point (1,1,0) for <u_1,u_2,u_3>.

[bold hound]

just note that the x they use there is a vector

it is what you call (x,y,z)

v would be yourr u

[crimson grove]

right, but shouldn't I have been given values in that vector?

[bold hound]

you have values

the definition right there gives you the directional derivative of f(x,y,z) at point (x,y,z) in the direction v

so instead of the x, you put in your r

afaik it should look like this

[crimson grove]

I'm a little lost....I haven't seen that formula before.

[bold hound]

which formula did you use before?

[crimson grove]

Well, I know my first step is to find the partial derivatives with respect to x, y, and z. Doing so gets me y^2, x2y, and 4z, in which case I plug in my values for each one afterwards. Doing so, I get my gradient vector of <1,2,0>.

That's how I've always done it.

[graceful surge]

this is also right

[ruby vapor]

I get the same for the gradient at that point. Since they don't give you the actual values for that vector I think you'd just give an answer in terms of u_1, u_2, and u_3

[bold hound]

the thing i posted was the definition we used in class
what you do will lead you to the jacobian matrix where you use total differentiability

[crimson grove]

Yeah I’ve never used this or heard of it

[bold hound]

it has some nice positive side effects, meaning you will only have to calculate the jacobian once if you need multiple directional derivatives

[crimson grove]

Where do I go from here tho?

[graceful surge]

directional derivative is the dot product of the gradient and the vector

[bold hound]

this is what you want to do

you already know what u is

and Dg(r) is your jacobian, which is this thing

if you deal with vector valued function, this jacobian will become a matrix

here the D1g(r) is the partial derivative of g(x) with respect to x1 at r

like this

so the whole Dg(r) will be a horizontal vector with all your partial derivatives at the point r=(1,1,0)

[crimson grove]

So just apply the dot product on the gradient <1,2,0> and the vector given <u1,u2,u3>?

[bold hound]

yes

[graceful surge]

[crimson grove]

For directional derivative problems, shouldn’t we be calculating magnitude of gradient vector at some point?

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[crimson grove]

Could I tell you answer?

[bold hound]

sure

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[vast shale]

How does the ASTC method in trig work, im strugglying to use it in equations

[vast shale]

.close

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[covert salmon]

how was the second line achieved?

[mental falcon]

multiplying both sides by (d-1)^2 and dividing both sides by 10, its valid as long as d isnt one

you would end up with $\frac{1}{10} > (d-1)^2$ but then they switched the inequality to keep the d's on the left i guess

[twin meteor]

Soosh

[covert salmon]

thanks

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[river kettle]

yo wait my brain is lagging how did they isolate the x

[ivory yarrow]

its a propety of logarithms that log(x^y) = ylogx

[river kettle]

Yeah I know that property