help-23

[median vigil]

you can use the piecewise definition of absolute value

[lean otter]

so from negative infinity to 0 the exp0nent section is positive?

like this?

[median vigil]

yes

[lean otter]

ok ty ❤️

!close

.close

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[median vigil]

now that i look at it, i'm not sure how you brought the square down, though

[safe radish]

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[mossy vale]

(Sub rule)

What is a quick way to know what I should sub for "u" when trying to evaluate these integrals

[burnt notch]

Well, if the numerator is the derivative of the denominator the u-sub is of course u = denom.

[mossy vale]

ohh

now that you mention it, i didnt realize the numerator was the derivative

that helps a lot

Pretty sure that's all the help i needed, thanks

.close

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[meager raft]

Could anyone help me w q number 2

What did I do wrong asweel

Nvm I think its just best to expand the denominator

Then realise it and mod arg it

Only easy way I can think of

[dapper star]

,rotate

[safe radish]

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[meager raft]

😔

Someone hlp

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[lucid trout]

the answer is no?

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[urban eagle]

Does n! Grow faster then a^n as n tends to infinity ((a^n)/(n!) = 0)

[peak estuary]

a^n grows with a factor of a at each step

n! grows with a factor of n at each step

at some point n will be bigger than a

try making that precise

[edgy isle]

$\frac{a \cdot a \cdot ... \cdot a}{1\cdot 2\cdot ... \cdot n}$

[flat frigate]

rbit

[edgy isle]

especially the point when n > a will be interesting

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[rapid kestrel]

techniques of integration, regarding even powers of sin and cos

[rapid kestrel]

for example, consider the integral of sinx^4cosx^4, i.e. the fourth power of cos and not x^4.

[split moss]

hi

[rapid kestrel]

now, there is a technique where one replaces each of sinx^2 and cosx^2 with some cos2x +- 1 /2, but this is infeasible for very large even powers, like 10

so the method i am reading about says to use a substitution of tan x = t

this gives a dx of 1/(1+t^2)

[split moss]

$\int \cos^4 x \sin^4 x dx?$

[flat frigate]

mohammed

[split moss]

you mean this?

[rapid kestrel]

and cosx^2 = 1/(1+t^2), and sinx^2 = t^2/(1+t^2)

[supple shore]

seems like bioche's rules

[rapid kestrel]

yes thats correct, sorry i dont know latex yet

[supple shore]

you should look at it

[rapid kestrel]

i see

[split moss]

i know

[rapid kestrel]

i'll have a look at it but i'll finish my full thing

[supple shore]

it's for fractional polynomials of cos and sin

[rapid kestrel]

so this gives an integral of t^4/(1+t^2)^9 dt

[split moss]

https://en.wikipedia.org/wiki/Tangent_half-angle_substitution

[rapid kestrel]

which doesnt seem very easy to integrate

[split moss]

look what i sent

[rapid kestrel]

yeah this is a bit different from tan(x/2) = t, i think?

just directly tanx = t

i shouldve just uploaded this img 😭

anyways, if m = 4 and n = 4 i get

[rapid kestrel]

this

which doesnt seem easy to integrate

i will have a look at this bioche rule though

[rapid kestrel]

i think this is the correct section, with both p and q even, i dont really know what linearization is? i suppose i need to understand that first, then?

[supple shore]

ye it's good to know how to do it to calculate some integrals

like with euler formula, and some basic trigs formula like cos(2x) = .. sin 2x = .. and others

[rapid kestrel]

i see

i'll give it a look ty

without using this, is there any way to integrate my example function except repeated expansion of cos(2nx) + 1 type terms?

[supple shore]

ye maybe like cos(x)*sinx = sin 2x) / 2 multiply by a power 4 and you get only a sin 2x to the power 4 with that you should have it

also knowing that sin 2x is the imaginary part of exp(2ix)

[rapid kestrel]

yeahh i wasnt considering this solution because i gave a bad example, or one that was unrepresentative of my real problem

i guess its like p,q both even and p != q , p,q >= 4 is my actual thoughts

[supple shore]

basically u always want to use bioche's rule and some trigs formulas when you calculate integrals with sin and cos

[rapid kestrel]

okok

ty

i'll learn this stuff

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[next perch]

How do I find the area of the shaded region?

[lean otter]

Find the area of the white area, and then subtract this value from the area of the rectangle, which will give you the area coloured in black.

[safe radish]

@next perch Has your question been resolved?

[next perch]

How do I find the white area?

[main mural]

it is two parallelograms with some rectangular intersection

recall the formula for the area of a parallelogram

[next perch]

base * height

[main mural]

yep

[next perch]

I can't think of what to do after that

[main mural]

base and height

[eternal carbon]

it's not that simple

there's a crucial portion of the problem that has not been addressed

[main mural]

which one

[eternal carbon]

what you're going to do about the white rectangle in the middle

[main mural]

it's a rectangle with sides 2 and 3

[eternal carbon]

not true

it's slanted

[main mural]

looks just rotated to me

[eternal carbon]

?

the side lengths are less than 2 and 3

[main mural]

no? the widths of the strips are 2 and 3 everywhere

oh

i see it

[eternal carbon]

...

[main mural]

hm right

[eternal carbon]

the way you find the lengths of the white rectangle in the middle is by applying pythagorean theorem to find the long slanted lengths of the two white parallelograms, then applying the parallelogram area formula

[main mural]

my bad

[next perch]

alrighty thanks i'll try to figure it out now

.close

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[covert talon]

How do i do this

[covert talon]

The answer is x=6 but i somehow got x=7?

[thin bridge]

can you show your work?
btw ||both values you've provided are wrong||

[covert talon]

I did 4x+5+4x-2+x+4+4x+5

[thin bridge]

how is that leading to getting x=7

[covert talon]

oops

i didnt do the other 4x+5

i did 4x+5 + 4x-2+x+4

so i did 4x-4x + x = 5+2

[thin bridge]

where's that coming from

[lean otter]

Hint: The opposite sides of a rectangle are equal.

[covert talon]

so do i do 4x-2 = x+4

[lean otter]

Yes

[covert talon]

so x=2?

bc thts what i got

[thin bridge]

yes, who said that x was 6

[covert talon]

the answer sheet..

yelp

[main mural]

6 is the side length

i think it's a typo

[covert talon]

Ohh

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[mild bridge]

can someone help with this question?

[queen saddle]

make correspondence b <->c. And then use a standard proof that [a,b) and [a,b] have the same cardinality. Namely, first transform it into [0,1] and [0,1) by one to one map x->(x-a)/(b-a). And then just make correspondence 1/n -> 1/(n+1) for n>=1.

[obsidian oracle]

The idea is basically, for a map I -> I', to map a to c, (a+b)/2 to a, (a+3b)/4 to (a+b)/2, etc... and the other points mapped to themselves

more generally, create a sequence (x_n) valued in [a,b] that never repeats

and map x_0 to c

and map x_(n+1) to x_n

since there's an infinity of x_n, we never run out of x_n to fill by the next one

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[dense tinsel]

derive

im not sure about my answer

[mortal sandal]

is the ^2 on the outside of the log?

[dense tinsel]

yes

[mortal sandal]

it looks like you treated it like it wasn't

,w derive log_3((3x^5+5)^2)

[flat frigate]

Results provided by WolframAlpha

Click here to refine your query online
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Requested by a_mog

[dense tinsel]

wait what

[mortal sandal]

this is equivalent to your answer wolfram is just pepega with simplifying

[dense tinsel]

oh xd

wait so i did it right?

[mortal sandal]

not if the exponent is supposed to be outside the log

[dense tinsel]

tbh im not evne sure if tahts how the teacher wrote it

is there a difference?

[mortal sandal]

yes

[dense tinsel]

would u use u'/u . lna

or whole different thing

[mortal sandal]

log(x^2) is equal to 2log(x), not (log(x))^2

it would be entirely different

[dense tinsel]

oh

ig my teacher made a mistake

how would u solve it then ?

[mortal sandal]

by using ambiguous notation?

[dense tinsel]

yes she always does that

lol

but

considering she only gave us u'/u.lna

[mortal sandal]

solve with chain rule still

[dense tinsel]

wait so how would it look like if thats what we are supposed to use?

the question

[mortal sandal]

i dont have the time to work out the whole problem atm

[dense tinsel]

nono i mean not the answer

like how would the notation be

if i had to use u'/lna.u

[mortal sandal]

do you know what chain rule is?

[dense tinsel]

yes

[mortal sandal]

is this like a very specific restricted version of chain rule that only works when the outside function is a log

[dense tinsel]

uhh

its supposde to be like

log base a then u

ill show wait

[mortal sandal]

ok i give up

[dense tinsel]

lol

[dense tinsel]

this is supposed to be loga u = u'/lna.u

[mortal sandal]

you "use" that in both cases

[dense tinsel]

oh

[mortal sandal]

you just use it at the beginning in the first case and not at the beginning in the other case

[dense tinsel]

so my answer was right the whole time 😭 ?

i just neede to simplify

[mortal sandal]

how should i know

[dense tinsel]

oh u got confused by the notation

so the notation is wrong for the answer i gave ?

[mortal sandal]

yeah it's ambiguous

[dense tinsel]

ah

[mortal sandal]

the answer is fine, aside from the fact that there's a factor you should cancel out from both sides

there's just a 50-50 chance that it's the answer to the wrong question

[dense tinsel]

well

that is the only way she taught us to solve so im assuming thats the way to do it

but

the way she gave the questio is wrong

[safe radish]

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[whole compass]

Can someone help me please

[whole compass]

Let me put my work as well

Stuck here

Don't even know if this is right

[rigid inlet]

denominator should have an (n+1) in there but otherwise fine

[whole compass]

Oh okay

I see the denominator is technically has a higher power than the numerator so the lim goes to zero?

Also don't quite understand part a

[rigid inlet]

part a is precisely what you've just done

[whole compass]

Ohh silly me

😂

I see it now

Thank you

How about this?

[loud osprey]

Index shift

Should start at 0

[whole compass]

Why should it start at 0?

[loud osprey]

Idk the reason why I just know it does

[whole compass]

Ahh okay. How would I change my work to reflect that? as far as the sum of the following series?

[loud osprey]

Well if u shift downwards u must change the exponent to make up for it

So if we go down from 1 to 0

U would get (3/11)^n+1

[whole compass]

Ohh I could probably use the ratio test instead of this hmm?

[loud osprey]

No u can still use geometric thingy

[whole compass]

I was following this here at first

Ohh ok

[loud osprey]

This is equal to 3/11 * (3/11)^n

Oh wait

It seems ur work already accounted for that

[whole compass]

Maybe I messed up on the arithmetic somewhere

[loud osprey]

7*2 is 14

Should be 17/8

[whole compass]

Yep I see it now

Idk why I would multiple 7 by 8 here lol

[whole compass]

Thank you

[loud osprey]

My bad for adding extra complications I didn’t see ur work already accounted for it 😭

[whole compass]

All good

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[cunning roost]

#help-29 message
Something like that

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[cunning roost]

.reopen

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✅

[obsidian oracle]

there are hypotheses to such "local limitedness theorem"?

such as the type of function f is?

because if I set f(x) = 1/x if x not equal to 0 and f(0) = 0, good luck

[brave wolf]

What theorem are you even referring to?

[obsidian oracle]

I think he's talking about "that one theorem some random person on the internet was talking about"

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[gritty lance]

Imma translate eveything

[gritty lance]

Ik sum is equal to 10 and that we need to use the numbers 1 and 4

Just lost

Translated: a region is bounded by the x-axis, the lines x = 1 and x = 4 and the straight line y = kx where k > 0

determine the direction coefficient k algebraically so that the area of the region is exactly 10 area units

[icy lance]

do you know integration or are you doing it geometrically

[gritty lance]

no intergration

[icy lance]

alright, you need the y values at 1 and 4
then you can find the area as a trapezium or a rectangle and triangle in terms of k

[gritty lance]

i think y = x so x = 1 y = 1 and so on with 4

[icy lance]

there is no guessing

if x=1 then y=k

if x=4 then y=4k

[gritty lance]

why is it 4 times k

[icy lance]

because y=kx?

[gritty lance]

ohh ye

what we do now?

can we solve it as a rectangle

we dont know the base

[icy lance]

i told you

[icy lance]

i mean, you can find that quite easily

think about it

[gritty lance]

the base is 3=

3?

@icy lance

so we do 3 x k + 3k x 3/2 = 10?

[icy lance]

hm, no

check your triangle area

oh wait no

sorry youre alright

[gritty lance]

isnt the height going 3k?

ohh okej

imma solve it

got it thank you

but im so confused on this part particulary

i know y = kx in and out

yet im still confused on why it becomes 3k is 3 the x? or wha is it y = 3k im just confused on that part espcesially

@icy lance

[icy lance]

because the base of the triangle lies at y=k and the top is at y=4k so the height of it is 4k-k=3k

[gritty lance]

i get that

ohhh 4 is x

4k

so its the same as k4

ohhhhh

i get it

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[steep magnet]

I dont see what I did wrong

u = 3x-2

du = 3 dx

Umm

Youre giving me a hint but im not getting it 😅

[junior smelt]

You’re forgetting a dx

[steep magnet]

Ahh yeyeye

fixed?

Its been a while since I did these hehe

But oke

(u+2)/3 = x

So sub all that into the original

(u+2)(u^6)/3

Is it wrong? 👀

It IS wrong or thumbs up cause im not wrong? 😅

Im confusedddd 😭

[severe pond]

chartbit is better with the reacts

you forgot du = 3dx

[steep magnet]

Ohhhhhh

Ohhhhhhhhhh

Ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

I see now

[junior smelt]

Yea was gonna say that you need the integral at some point (in case you were getting to it)

[steep magnet]

So its all (u+2)(u^6)/9 ?

[junior smelt]

Yep (but remember you’re integrating that wrt u as well)

[steep magnet]

Ahh yeyeye

I think I can solve it now tho hehe

Thank you so much everyone!!

❤️

.close

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[silent scarab]

I have to find the flux of a sphere with a very ugly vector field and was wondering if there is short cut to find the Jacobian instead of trying to solve it by hand

[empty gyro]

There's divergence theorem

[silent scarab]

oh my gosh you're right

actually the other part of the question is there is another surface in it and I'm not quite sure how to approach that with the divergence theorm

oh do i just adjust my domain

actually I can't do that wither becuase I've defined one surface in cylindrical coordinates and the other in spherical

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[silent scarab]

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.close

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[ruby kestrel]

Hi

[eager plover]

whats ur question?

[broken isle]

!nohelp

!noask

[austere cypress]

!dontasktoask

forgot the command

[broken isle]

!dontasktoask

[austere cypress]

!da2a

[safe radish]

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

[austere cypress]

yay

[ruby kestrel]

Hi

[safe radish]

@ruby kestrel Has your question been resolved?

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[faint coral]

I just need a quick clarification on dominated convergence theorem

[faint coral]

my textbook states it as

is the convergence of $E{X_n}$ to $E{X}$ pointwise?

[flat frigate]

00100000

[faint coral]

and if so, is it fair to say that if DCT implies to a series of r.v. $X_n$, $Y_n$ converge to $X$ and $Y$ respectively, then $E{X_n}+E{Y_n}\rightarrow E{X}+E{Y}$?

oops

[flat frigate]

00100000

[faint coral]

or like, some form of using absolute value to shuffle it around?

[midnight delta]

i studied this theorem for the lebesgue convergence of a sequence in which case the convergence is pointwise, so i assume that also in this case the convergence is pointwise

[faint coral]

but this notion of convergence is weaker than pointwise

the notion of convergence for the random variables

so I'm just not entirely sure

oh wait

I don't know what lebesgue convergence is

so I'm not sure if that's a strictly weaker or equally strong criterion

every probability resource just says that it converges

and not pointwise/in probability/a.e.

so I'm just confused :(

[midnight delta]

basically, it converges pointwise for not all x's in the domain, so is a little weaker than a normal pointwise convergence

[faint coral]

so, then is it fine for me to just add the convergence of the expectations?

i.e. $E{X_n}+E{Y_n}\rightarrow E{X}+E{Y}$

oops

[flat frigate]

00100000

[faint coral]

I honestly totally forgot if this is a property of pointwise convergence 😅

[raven heart]

the E(X_n) are just numbers right, there's not a zillion ways for sequences of real numbers to converge

[faint coral]

yes

[faint coral]

oh also, sorry to bother you so much 😅 , but to apply DCT to a r.v. $X_n$ that already is shown to converge to $X$, I can just say that $|X_n|\leq |X|+1$ a.s. because otherwise there are an infinite number of terms for which it's a certain distance away from the convergence point, so it wouldn't converge, right?

[flat frigate]

00100000

[midnight delta]

you just need to find a variable that "dominates" over you X_n, it can be anything, now i don't specifically know which proprieties can have a r.v. but for reference, in the lebesgue case, the function that dominates it's defined only on positive numbers

[faint coral]

a random variable is real-valued

is that enough information to tell if it's dominated by $|X|+1$? 😅

[flat frigate]

00100000

[midnight delta]

however, if you define $Z_n = X_n + Y_n$, for which i know that $|X_n| \leq X and |Y_n| \leq Y, then |Z_n| \leq X + Y$ that we call $Z$. So i think that $E{X_n}+E{Y_n}\rightarrow E{X}+E{Y}$(?)

[flat frigate]

mercidy

[faint coral]

hmmm....

[midnight delta]

This is valid because of the linearity of the expected value?

I'm trying to justify the thing i'm not totally sure about it ahahah

[faint coral]

I'm not sure what this is meant to show tbh

the connection between $Z$ and the expectations is unclear to me

[flat frigate]

00100000

[faint coral]

it's probably just me being bad tho

[midnight delta]

are we trying to understand if this is possible right?

[faint coral]

ohhh, no

I'm trying to prove that it's true

[midnight delta]

yes absolutely

[faint coral]

oh ok

thank you

I'm glad that I'm not going insane :))))

[midnight delta]

dw i went insane for this theorem about a week ago 😄

[faint coral]

lmaooo

[midnight delta]

oh okai

[faint coral]

but im pretty sure if you're right abt the convergence being pointwise, it's true

take https://math.stackexchange.com/questions/1214225/my-proof-that-sum-of-convergent-sequences-converges-to-sum-of-limits for instance

the convergence here is pointwise

[safe radish]

@faint coral Has your question been resolved?

[midnight delta]

okai, sorry if i'm late, i searched what a.e. means xD

I'm sure that the almost everywhere convergence implies the pointwise convergence

you are basically saying that it converge pointwise for every point that is not in the set of measure zero

[safe radish]

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[lean otter]

I’m confused and the answer sheet gave a different answer than a scan app and I got a diff answer.

I’m also new to this

[light shoal]

what did the answer sheet say?

[lean otter]

To make it 0 they would have to be negatives?

X

-5 and -10

[light shoal]

i guess you're asking, are the solutions to (x+10)(x+5) = 0
-5 and -10

yes they are, you can see by plugging in those values for x

[lean otter]

Let me send another pick of a different problem

[light shoal]

to see it from the equation, just recognize that if the product of two numbers is zero, then one of the numbers has to be zero

[lean otter]

That gave a weird answer

[light shoal]

so either x+10 = 0 or x+5 = 0

equivalently, either x = -10 or x = -5

[light shoal]

ok,any thoughts?

[lean otter]

But the answer sheet says 2 and -7

[light shoal]

answer sheet is wrong

[lean otter]

I also just talked to someone who already pointed out a typo in my book

[light shoal]

plug in x=2 to the original equation, it doesn't hold

your answer is correct

[lean otter]

But I wasn’t sure which way is actually right because this is my first time doing this

[lean otter]

Okay thanks. I just wasn’t sure if the answer was the opposite or not cause this book is backwards.

.close

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[silk talon]

Hello, I want help with a concept, if I have a sphere and I slice it in two, each half contains half the surface area of the sphere. By the way Im not counting the new surface area created by cutting the sphere, it's not relevant to my stuff, only focused on the outer curved bit.
Ok so if I cut the sphere at 33% of it's diameter, would the smaller 33% chunk's surface area be 33% of the surface area of the sphere, and the larger chunk be 66%? That's how I assume it works, but is there some weird shenanigans due to it being curved?

[silk talon]

Does it exponentially becomes less surface area as you go farther out or is it just linear?

[safe radish]

@silk talon Has your question been resolved?

[silk talon]

<@&286206848099549185>

[safe radish]

@silk talon Has your question been resolved?

[safe radish]

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[rancid sierra]

Hi

[rancid sierra]

I was wondering if anyone could explain limits in two variables in brief

I have an exam on it today

Thanks!

<@&286206848099549185>

[plucky elk]

https://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx

[safe radish]

@rancid sierra Has your question been resolved?

[rancid sierra]

I want theory bro

Idk what it really means

That's what I'm tryna understand

[plucky elk]

Wut

What are you actually asking for

[plucky elk]

That isn't explained here

[rancid sierra]

Like in normal limits they explain stuff using graphs right?

How does that work for two variables

I don't understand it

They do do it

But I kinda need help

[plucky elk]

You want someone to draw pictures for you?

This is a good tool
https://www.desmos.com/3d

[rancid sierra]

Aight thanks

.close

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[steep magnet]

I dont really know how to get things to cancel out here

u = sqrt(x) + 1, du = 1/2sqrt(x) dx

How do I get rid of the x in the du?

<@&286206848099549185>

[severe pond]

well this is just the wrong u sub

[steep magnet]

Oh yeah oops

[severe pond]

consider u = sqrt(x)

[steep magnet]

Ok...

[severe pond]

you could’ve done $dx = 2(u-1)du$ if you wanted to though

[flat frigate]

knief

[severe pond]

$\int \frac{2u-2}{u^4} du$

[flat frigate]

knief

[severe pond]

then split it

both work

how’d you get the 1 though

[steep magnet]

Its inside the brackets

[severe pond]

no lol i meant like

as an answer

you said you didn’t know how to proceed

oh i’m blind

😭😭

nevermind

before i zoomed in i thought you typed 1

[steep magnet]

Oooooh hahahaha

But oke I thiiiiiink I know how to do it now

Thank you so much for your help!!

❤️

.close

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[severe pond]

you’re welcome

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[unborn wigeon]

If a series contains a $\bar z$ term, can it converge to an analytic function? Moreover, if a power series contains a $\bar z$ term, can it also converge to an analytic function?

[flat frigate]

Kakaka

[unborn wigeon]

series of functions that is

[safe radish]

@unborn wigeon Has your question been resolved?

[empty gyro]

Wdym a $\overline{z}$?

[flat frigate]

SWR

[unborn wigeon]

complex conjugate

[safe radish]

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[quiet smelt]

which of the of the expressions in the square can be written as the following.

[quiet smelt]

I need help with understanding how to use double angle to solve them

[terse lichen]

state the double angle identity

[quiet smelt]

for sinus its sin2x = 2 sinxcosx

[terse lichen]

ok good

now for a its 4sin(2x)cos(2x)

any ideas

[quiet smelt]

nope

[terse lichen]

well you see its 2x inside sin and cos

[quiet smelt]

yes

[terse lichen]

what if you put x = 2x in the identity

[quiet smelt]

2sin(2x)cos(2x)?

[terse lichen]

yep

and whats the left side

[quiet smelt]

2sin4x?

[terse lichen]

no

just sin(4x)

[quiet smelt]

a

[terse lichen]

so you got the sin(2x)cos(2x) down

but you got a 4 in the front

how do you get a 2

[quiet smelt]

idk

[terse lichen]

whats the relationship between 2 and 4

[quiet smelt]

half

[terse lichen]

so 4 = 2*2

[quiet smelt]

yes

[terse lichen]

so thats the same as 2 * 2sin(2x)cos(2x)

and what was 2sin(2x)cos(2x)

[quiet smelt]

2sin4x so twice
sin4x

[terse lichen]

2sin(2x)cos(2x) = sin(4x)

[quiet smelt]

yes

[terse lichen]

so you just get 2sin(4x)

and thats a

same operations with b and c

[quiet smelt]

ah, i get it

thanks

.close

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[jaunty steeple]

how do i evaluate this integral as a power series

[safe radish]

@jaunty steeple Has your question been resolved?

[safe radish]

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[quiet smelt]

so like sin(2x)=1/2 * 2 * sinx aka sin(2x)=sinx is it solvable or does it require more information?

[quiet smelt]

nvm it was not

.close

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[night seal]

The nth statement in a list of 100 statements is “Exactly
n of the statements in this list are false.”
a) What conclusions can you draw from these state-
ments?
b) Answer part (a) if the nth statement is “At least n of
the statements in this list are false.”
c) Answer part (b) assuming that the list contains 99
statements.
//
I am stuck on this proposition question. Cannot get through it. I need help.
<@&286206848099549185>

[safe radish]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[devout scroll]

which part?

[night seal]

b part

[devout scroll]

so assume k statements are true

than what do you get?

[night seal]

In question b par a i mean. Sorry for not giving any clarification.

*b part a

[devout scroll]

wdym?

i dont get you?

[night seal]

b) Answer part (a) if the nth statement is “At least n of
the statements in this list are false.”

[weary obsidian]

the binary property always help, what if this statement is correct, what if it is not

sry is boolean lol

[devout scroll]

so, that means atmost 100-n are correct?

assume k are correct

so that means 100-k are correct?

[night seal]

100-k are false

[devout scroll]

yes

and how many statements are true if 100-k statements are false?

statement 1,2,3,4,...,100-k are correct, right?

[night seal]

1 to k?

1,2,3,.....,k are correct, right?

[devout scroll]

1 to 100-k

it says atleasy n are false

[night seal]

I didn’t get that. Like when i say in statement 1: at least 1 statement is false.
Statement 2: at least 2 are false.
.
.
.
In 99 : at least 99 are false.
In 100: at least 100 are false. Which isn’t possible because if 100th is true then 100 statement can't be false.
Then how do i determine which is true.

[devout scroll]

so if 100-k statements are false

let us say k = 40 for sake of simplicity

than atleast 1 statement is fasle

atleast 2 satemnts are false

atleast 3 statements are false

...

atleast 100-40=60 satements are false

so thes statements are trye

so 100-k statements are true

[night seal]

Sorry i don’t have much time. I have to go. Thanks for trying to help me. I really appreciate it.

.close

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[lusty hornet]

Does set of exponential functions over R form a ring?

[peak estuary]

do you mean the functions of form f(x)=e^(ax) ?

under which operations?

[lusty hornet]

Yea, + and *

[peak estuary]

what are your thoughts

[lusty hornet]

Idk what axioms it may hold or not to be a ring or not

Wait

No?

As in, there is no 0

In set of the exponential functions

[peak estuary]

yes, that is one of the problems

you could fix that by allowing all functions of the form f(x)=c*e^(ax)

does that make it a ring?

[lusty hornet]

c can be 0?

What's 'a' here

[peak estuary]

the same as before

some parameter

some real number

if you allow c=0 then you would at least have a zero element in your set

so then its closer to a ring

[lusty hornet]

Yup

[peak estuary]

but its still not a ring

thats what I meant

[lusty hornet]

Which axiom

Unsatisfied

[peak estuary]

for example e^x + e^(2x) is not again in the set

[lusty hornet]

Thus?

[lusty hornet]

Why?

Shouldnt the set of ce^x (+,*) be a ring?

[peak estuary]

ok so now you only allow one exponent?

but then what about e^x * e^x

[lusty hornet]

Yea you are right

[lusty hornet]

Why won't ce^ax(+,*) be a ring?

[peak estuary]

.

[lusty hornet]

Why it aint

[peak estuary]

its a bit ugly to show

not sure what the nicest way is

you can pick three points, eg x=0,1,2 and then show that no function c e^ax goes through the relevant points

or another option is to differentiate and see that the derivative is not the a multiple of the original function

thats actually not that bad

[lusty hornet]

So closure under addition isn't satisfied?

[peak estuary]

yes

[lusty hornet]

But e^x+e^2x is convergent and the new function is also an exponential one

[peak estuary]

what does convergence have to do with this

what do you even mean with the function being convergent

convergent to what and in what sense

do you mean continuous?

e^x + e^2x is not of the form c e^ax

and those were the functions we allowed

[lusty hornet]

Wait

[lusty hornet]

Acc. To this yea, e^x + e^2x isn't the form of f(x)

[lusty hornet]

But it's an exponential function still ain't it?

c(e^x+e^2x)

I.e., Set of exponential functions of the form ce^ax form a ring

Yo?

[peak estuary]

no

e^x+e^2x does not have the form c*e^ax

two terms are not one term

just because it still has exponentials in it does not mean it has the form c e^ax

[lusty hornet]

I get what you mean

For it to satisfy the closure under addition,
e^x+e^2x have to return element of form ce^ax?

[peak estuary]

for the set to be closed under addition the sum of two elements has to again be in the set

but the only things in the set are functions of the form c e^ax

so the sum again needs to have that form

[safe radish]

@lusty hornet Has your question been resolved?

[lusty hornet]

I fully understand your argument, but I'm confused over the fact that any exponential function of the form ce^ax can be mapped to a number in R, and so can be the sum of such functions.
How do we establish that the axiom doesn't hold

[peak estuary]

what does that have to do with it

what do you mean with "can be mapped to a number in R"

[lusty hornet]

As of the function has a number as it's image,

[peak estuary]

do you mean that the set of functions of the form c e^ax has the same cardinality as R?

a function has a set as its image

[lusty hornet]

No, as of that image belongs to set R

[peak estuary]

and for a function of the form c e^ax the image is always one of (-infty,0) or {0} or (0,infty)

depending on c<0, c=0 or c>0

[lusty hornet]

Yea

What if c belongs to set of integers

[peak estuary]

doesnt change anything

[lusty hornet]

I get it

Thanks for your persistent effort

I really do appreciate it

.close

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[fathom jewel]

Is it true that $(\mathbb{Z}{10}, +) \cong (\mathbb{Z}{11}^*, \cdot )$?

[flat frigate]

𝔸dωn𝓲²s

[obtuse jackal]

they are indeed both cyclic and of 10 elements

(because 11 is prime)

[fathom jewel]

So that makes them isomorphic?

I wrote down the group tables and was asked what I could notice

And I thought they should be isomorphic

[obtuse jackal]

prereq: prove that there is only one cyclic group of size n, up to isomorphism

often writen Cn

[obtuse jackal]

the hard part is showing the second group is cyclic
There's a theorem saying Zp* is always a cyclic group, but you don't know that, so you just need to find a generator (try it by hand, there's no great method)

[fathom jewel]

Looking at my table I noticed the generator should be 2 so Z_11* = <2> which would make it cyclic

So since both groups are cyclic and have the same cardinality that makes the isomorphic

[obtuse jackal]

1 -> 2 -> 4 -> 8 -> 5 -> 10 -> 9 -> 7 -> 3 -> 6 -> 1 is indeed a cycle of length 10

[obtuse jackal]

hopefully you've proven that before
Otherwise I recommend you make sure that's obvious

[obtuse jackal]

cause you didn't seem to

[thorn wagon]

i am not even going to try to understand 💀✨

[fathom jewel]

The task was just about writing down both tables and then write down what I could notice about both groups

It seemed to me conclusive that it's about them being isomorphic since that's also something new we recently studied

The generator of Z_10 being 1 is obvious, and for Z_11 * I wrote down the remainder of 2^n mod 11

[thorn wagon]

what does isomorphic mean

[obtuse jackal]

step 1: study intro group theory

[fathom jewel]

In my understanding, imagine you had a big cube from a set A and 4 small cubes from another set B, that together resemble the big cube.
Even though one is just the big cube and the other "big cube" is made up from the 4 smaller cubes, they are still the same, since they both represent that big cube. So the big cube is isomorphic to the 4 small cubes.

They basically have the same structure

[safe radish]

@fathom jewel Has your question been resolved?

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[earnest rain]

(The solution on the book)

Did i do smth wrong or is the books solution just missing a piece?

[safe radish]

@earnest rain Has your question been resolved?

[earnest rain]

<@&286206848099549185>

<@&286206848099549185>

[plucky elk]

hmmm welps

[earnest rain]

Wdym?

[plucky elk]

can't read your image since it's rotated

and can't use the bot since it only rotates the most recently sent image

[earnest rain]

Ah wait

,rccw

[earnest rain]

@plucky elk there

[plucky elk]

yea unclear what the question is

[earnest rain]

It asks to calculate the job of the vector field

With that vector field in the curve written on the 2nd line

[plucky elk]

can't read this

F = (-x, x) ?

what does L represent

[earnest rain]

-y,x

[earnest rain]

The job, the question of the edercise which is integral of F(r(t))r'(t)

[earnest rain]

Røcosø Røsinø

[plucky elk]

L = job?

[earnest rain]

Yes its italian

Job=lavoro

[plucky elk]

,w int 0 to 4pi of x [-sin(x) + cos(x)] + x^2 dx

[flat frigate]

Results provided by WolframAlpha

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

Requested by rie.mann

[plucky elk]

yea maybe there's a part of the question you're missing

that cancels the 4pi R^2 part

[earnest rain]

It just asks the job

Of the spiral p=Rø

And that vector field

What can I be missing?

[plucky elk]

¯_(ツ)_/¯

wording of problems matter and only you know the original wording

[safe radish]

@earnest rain Has your question been resolved?

[safe radish]

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[covert yoke]

@twin owl based on the size of this video, it seems that I will need to upload it here.

[covert yoke]

actually, it's too big even for that.

.close

[safe radish]

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[safe radish]

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[tired python]

How could this be?

I thought WXQ and QZY are equal

[magic junco]

What are you looking for?

[tired python]

how can XWQ and ZYQ be equal?

i'm just used to these two being equal

[magic junco]

It’s a statement

[tired python]

which is why the question confused me

[magic junco]

W isn’t guaranteed equal to Z, because XW and YZ isn’t parallel

[tired python]

if we remove the A would these two be equal?

[magic junco]

[tired python]

so the ones furthest to the left and right have to be parallel?

[magic junco]

Here’s an exaggerated example

[magic junco]

Yes

[tired python]

i see, that makes sense

how could I apply that rule in general?

as in how do I know which lines have to remain parallel

[magic junco]

The one on the furthest left and the one on the furthest right

[tired python]

no but like if it's in a different scenario / different orientation

[magic junco]

Can you demonstrate it?

[tired python]

hmm cant find a tricky one rn

[tired python]

@magic junco for this one could u pls show how they would be equal

like by drawing the lines

[safe radish]

@tired python Has your question been resolved?

[silent forge]

If AB is parallel to BC then the angles marked in red are equal and the angles marked in blue are equal.

The fact that, in your problem, ∠XWQ=∠QYZ is only true because the problem statement said so.

[tired python]

can you draw it on this image, i still don't get it

[silent forge]

you wanted was ∠XWQ=∠QZY but this is not nessecarilty true, what SkyAndNight was trying to show with that image is that they are visibly not equal hence why they said an "exaggerated" example

so α≠β

[tired python]

is this why they are equal?

[silent forge]

they are not equal

[tired python]

wait why

[silent forge]

the vertical lines you drew are not parallel

[tired python]

in this example they are parallel, no?

[silent forge]

and they are not parallel in the problem either

[tired python]

if we assumed they were parallel

would this be a correct way to deduce the anglews

[silent forge]

if we assume they are parallel then yes, α=β

[tired python]

kk yeah I got it, ty

i'm just so used to these types of questions always being parallel

[silent forge]

the parallel problems are very common, but your original problem doesn't have the lines parallel, you have to find another method

[tired python]

btw if the lines (to the most left and right ) are parallel, then the angles on that the lines are always the same right

[silent forge]

yes, it's the same rule but with a different common line

[tired python]

Thank you

[safe radish]

@tired python Has your question been resolved?

[safe radish]

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[patent musk]

am i right with C here?

[patent musk]

<@&286206848099549185>

[rigid inlet]

!15m

[safe radish]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[patent musk]

oh my bad.

<@&286206848099549185>

[severe pond]

if the sequence decreases with each successive term how is the ratio 3?

[safe radish]

@patent musk Has your question been resolved?

[patent musk]

oh yeahh the common ratio must be less than 1 not greater. so it has to describe a decreasing sequence? in that case its B right

[severe pond]

nope

you just guessed

check that it actually goes through one of the points

like let n = 0 or something

[safe radish]

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[primal pendant]

,rotate

[primal pendant]

,rotate

[primal pendant]

,rotate

[primal pendant]

i need help with 1

[empty gyro]

^Me seeing your assignment spin around

[empty gyro]

The problem is a little tricky because you will need the perpendicular height of the triangles and parallelogram, but they are not given and there is no way to calculate them

However, if you recall the properties of ratios in similar triangles, that may help you

[quick crater]

let the area of ABC be a variable and simmilar triangles bo brrt

[primal pendant]

brrt?

[quick crater]

like you use it alot

[primal pendant]

what does brrt mean?

is bd=de

im so lost

[quick crater]

no

what are the properties of a parallelogram in the context of length of sides?

[primal pendant]

two sets of parallel lines

[quick crater]

yeah do those lines have anyhing in common?

[primal pendant]

.close

[safe radish]

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[patent musk]

am i right with B here?

[safe radish]

Please don't occupy multiple help channels.

[patent musk]

am i correct with this or did i make a mistake

thank you again!!

.close

[safe radish]

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[safe radish]

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[split moss]

Calculate the weighted mean inverse square root of all perfect numbers between (10^{15}) and (10^{18}), multiplied by (\pi^{e^{\pi}})

[flat frigate]

Tazculator

[main mural]

[echo bronze]

😭😭

Keep looking bro

[safe radish]

@split moss Has your question been resolved?

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[lusty pendant]

can smb help with #1? i know it's easy but i can't like.. grasp the concept of it

[finite igloo]

1+2 = 90 deg

2 = 60 from sum of angle = 180

[lusty pendant]

so i need to find w, 1 & 2?

[pale lily]

find the measure

of each

numbered angle

[lusty pendant]

because isn't there supposed to be 3 interior angles inside of each triangle? or is that entire thing counted as one?

[pale lily]

every triangle adds up to 180 right

[lusty pendant]

yeah

[pale lily]

RWS

look at that triangle

what angles are given

[lusty pendant]

<2 and S

[pale lily]

no

whart

what angles

are given

[lusty pendant]

wdym

[pale lily]

in the triangle

[lusty pendant]

wdym

[pale lily]

RWS

do you know what a angle is

[lusty pendant]

bro yes

i'm not slow

[pale lily]

what is it

[lusty pendant]

uhh

[pale lily]

there genuinely is not

a easier way

to ask

the question

I asked you

[lusty pendant]

angle 1 and angle 3 are given

[pale lily]

what angles

are given

the

actual anghles

in degrees

[lusty pendant]

60 deg. and 30 deg.

[pale lily]

in

the triangle

RWS

[lusty pendant]

only 30

s

[pale lily]

what does

that

red thing mean

in the corner

[lusty pendant]

i actually don't know, our teacher didn't explain that

those are right angles basically

[pale lily]

what is a right angle

equal to

[lusty pendant]

90!

were getting somewhere

[pale lily]

so whats angle 2 equal to

[lusty pendant]

90

[pale lily]

no

why do u think its 90

[lusty pendant]

why isn't it