how to make the circle equation
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rain coral
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kindred mauve
can someone explain to me how i'd know where these lines inrersect?
kindred mauve
without using a graphing calculator
mystic scarab
Yep
That's something you should already know
The concept of find the intersection points between two functions
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violet sparrow
can anyone explain this? why it is sin(theta) = sin(180-theta)?
devout snowBOT
can anyone explain this? why it is sin(theta) = sin(180-theta)?
candid maple
have you learnt about the unit circle?
and also how circular/trigonometric functions are represented on the unit circle?
violet sparrow
have you learnt about the unit circle?
i know a thing about unit circle
violet sparrow
and also how circular/trigonometric functions are represented on the unit circle...
ill try
candid maple
I will then assume you know the basics of representing circular functions on the unit circle.
in that case, take a look at this diagram.
does this directly answer your question?
violet sparrow
does this directly answer your question?
why its 180-theta?
candid maple
because an angle of theta is made from the other side (the negative x-axis). what do you recall about two angles that lie on a straight line? what must they sum to?
violet sparrow
because an angle of theta is made from the other side (the negative x-axis). wha...
supplementary angles must sum up to 180
candid maple
exactly so. now, if you make an angle of theta from one side, what must its supplement be?
violet sparrow
exactly so. now, if you make an angle of theta from one side, what must its supp...
if we have one angle that is theta, then the other is maybe 180-theta
candid maple
why is there a 'maybe'? is there any other situation where the linear pair of an angle marked theta is not 180-theta?
violet sparrow
why is there a 'maybe'? is there any other situation where the linear pair of an...
im not sure if its really correct so i mean its possibly correct
candid maple
well then, I will tell you that it is correct (but I am also interested what made you doubt that).
violet sparrow
because you have to minus from 180 to get the other side
violet sparrow
well then, I will tell you that it is correct (but I am also interested what mad...
I cannot just assume that 180-theta is indeed a correct answer, so i say its possible that it is the answer
candid maple
but do you need to assume?
violet sparrow
but do you need to assume?
its necessary for me to say its possible rather than assuming something is that
candid maple
theta and 180-theta form a linear pair, and because they are a linear pair, they are supplementary. it's not a baseless assumption.
violet sparrow
theta and 180-theta form a linear pair, and because they are a linear pair, they...
oh yeah common sense
but have you heard about projectile motion? specifically horizontal formula?
candid maple
there's a difference if you merely guessed the answer, but you even gave me the correct reasoning. what's there to assume if you've substantiated your claim?
candid maple
but have you heard about projectile motion? specifically horizontal formula?
yes. not sure what that has to do with this directly?
violet sparrow
yes. not sure what that has to do with this directly?
why they use 180 degrees?
candid maple
well I will need you to cite that formula then.
violet sparrow
here it is
candid maple
I don't see where the 180 degrees is, though I do see a sin 2theta.
violet sparrow
i was trying to understand this question here:
- A Batang Pinoy athlete from your school throws a javelin, always at the same speed, at four different angles (30, 40, 60, and 80) above the horizontal. Which two throws cause the javelin to land the same distance? A) 30 and 80 B) 40 and 80 C) 30 and 60 D) 40 and 60
the explaination is from ChatGPT
candid maple
I knew the explanation was from ChatGPT (which I highly do not recommend if you have no way to verify its output).
but one way you can think about this problem in particular is to recall that you get maximum range at 45 degrees.
this is because you get the best of both worlds - you get the largest possible air time and horizontal range combination.
violet sparrow
I knew the explanation was from ChatGPT (which I highly do not recommend if you ...
oh its not a math olympiad question but rather a simple one
violet sparrow
but one way you can think about this problem in particular is to recall that you...
yeah i know that
violet sparrow
that advice applies to **any** question.
btw thanks for entertaining my question
candid maple
now, if you move away from 45 degrees, the effects change almost symmetrically (in ideal conditions - in the real world, there's air resistance and all the other good jazz to think about). if you go 5 degrees more than 45 degrees (that is, you aim at 50 degrees), you get slightly more air time, at the cost of slightly less horizontal range, because the vertical component of your initial velocity is now more than the horizontal component (you can check by drawing the triangle formed by the velocity vector and its components).
however, if you dip 5 degrees under (that is, you aim at 40 degrees), your projectile has a faster horizontal speed (because you're aiming at a more horizontal angle), at the cost of less air time.
as mentioned in the first sentence though, these two effects more or less cancel each other out symmetrically.
as such, under ideal conditions, a projectile being thrown at 40 degrees has roughly the same effective range as the same projectile thrown at 50 degrees (of course, assuming same force, etc.).
violet sparrow
as such, under ideal conditions, a projectile being thrown at 40 degrees has rou...
yeah i see
so what does imply?
candid maple
notice the angles add up to 90 degrees. that is not a coincidence.
violet sparrow
notice the angles add up to 90 degrees. that is not a coincidence.
yeah 40 + 50 = 90
candid maple
for any angle theta, throwing the same projectile with the same force at an angle 90-theta yields roughly the same effective range, provided theta is NOT 0 degrees.
if theta is 0 degrees, you still get that relationship only if you interpret throwing at 0 degrees meaning 0 air time and thus 0 range. otherwise, if you give a horizontally-thrown projectile any air time at all, you will definitely get a further range out of that projectile over a projectile thrown 90 degrees; that is, thrown straight up and back down.
violet sparrow
so these two angles produce the same effective range and then also these two add up to 90. So that means any angles that add up 90 (i.e. 60 and 30) then each of those would produce the same effective range?
candid maple
exactly so.
violet sparrow
exactly so.
so it has nothing to do with 180 degrees? and the unit circle?
i also thought of pythgorean theorem (and the right triangles and stuff)
candid maple
it has something to do with the 180 degrees and the unit circle. the unit circle is how you (intuitively) derive the relationship containing the 180 degrees to begin with, and it's the 180 degrees together with the fact that you are taking the sine of 2(theta), NOT just theta, that makes this whole thing work.
violet sparrow
but you just need to know the fact that any two angles that sum up to 90 is the answer?
candid maple
intuitively you can reason that way, but using that property of sine is the more rigorous method.
and it's not that any two angles sum to 90, it's that any two distinct angles at which a projectile achieves the same effective range sum to 90.
violet sparrow
intuitively you can reason that way, but using that property of sine is the more...
the background explaination is complex
candid maple
then you may need to revisit the unit circle and how the circular formulas are derived from it.
anyway, if you think about it, it kinda makes sense why you would need that 180. the angle of projection is theta, which ranges from 0 to 90, but you are dealing with the sine of 2(theta), so the range of the argument of that sine is 2(0) to 2(90) = 0 to 180. and that's why the relationship is with 180, not 90.
violet sparrow
then you may need to revisit the unit circle and how the circular formulas are d...
i was studying physics but to understand i need a mathematical explaination why 30 and 60 works
candid maple
what is physics if not models created using math to describe reality?
violet sparrow
what is physics if not models created using math to describe reality?
i agree
but you said ChatGPT is not recommended
what can you recommend for me?
i mainly use ChatGPT for studying basic math problems, studying biological stuffs and physics and chemistry
candid maple
human tutors.
or books.
the problem with LLMs is that they sound confident, even when they are wrong, and some mistakes are so subtle you won't even notice it unless you are purposely out to catch it making that mistake. as a learner, though, are you?
plus, I find that LLMs cannot adapt to human learning styles at the speed a good human tutor can. a good human tutor can adapt to your learning and emotional needs. not an LLM. they are designed to feed you wall after wall of text regardless of whether you're ready, and they cannot quickly mix intuition and formalism together.
violet sparrow
or books.
but what if i find solutions that I dont understand?
like I say I have difficulties in understanding
candid maple
that's why communities like these exist, no?
what do you think we are doing at the moment?
violet sparrow
that's why communities like these exist, no?
yeah
but its anxious
maybe you dont get replies
depends on the question
candid maple
I mean, I don't know about you. but if I ask a question and get no replies, I'd probably just bookmark that question and ask again later. or I'd go find a personal tutor to ask them. or I'd ask my teacher. or if I am really that desperate I'd email a teacher, tutor or professor (I haven't had to do this though).
violet sparrow
I mean, I don't know about you. but if I ask a question and get no replies, I'd ...
thx for sharing that
candid maple
unless I am very confident I can verify the LLM's output and that I am prepared to work through what it says, I would probably be cautious about using an LLM.
I mean, your confusion right now? it's caused by the explanation from an LLM!
violet sparrow
unless I am *very confident* I can verify the LLM's output and that I am prepare...
What I trust with these LLMs is that when I see a difficult passage or a stanza in a poem, I would paste it to them and say "provide me a word by word definition"
violet sparrow
I mean, your confusion right now? it's caused by the explanation from an LLM!
they sometimes inaccurate
In fact most of our school repotings/presentations are based on AI research/explaination. Thats why I dont trust with the reportings and stuffs
candid maple
What I trust with these LLMs is that when I see a difficult passage or a stanza ...
so you trust large language models with language, which is exactly what they are designed to do, and thus I do not see a problem with that because that's what they're good at. they're not perfect, but they get the job done in language and linguistics far better than anything requiring reasoning (i.e., STEM).
of course, that is not to say current LLM models are bad - the latest thinking models are honestly scary in their abilities, but at their core they are still kinda language-based. also, note that a lot of the time, if it starts thinking, the answer that comes out is less likely to be explained unless you ask it to, but if you ask it to and it doesn't think, then it may introduce mistakes where it didn't originally. I have been hit by that twice during GPT-5, thankfully both times with people who can catch the output.
violet sparrow
so you trust large **language** models with **language**, which is exactly what ...
yeah, I see
I appreciate your knowledge about AIs and LLMs
candid maple
they sometimes inaccurate
and that is exactly what I mean! they're not perfect. neither are humans, but humans (at least good human teachers and tutors) would be more willing to reassess their stance given evidence and you can assess their confidence in their answers by their language, tone, and body expressions. not so for an LLM - they sound confident no matter what random (and I'm sorry for swearing here, but) bullshit they happen to be peddling at you at that time.
violet sparrow
and that is exactly what I mean! they're not perfect. neither are humans, but hu...
they say when you have problems dealing with mental health or you want to rant, going to ChatGPT is bad because they lack emotional capabilities and semantics, how would you say that?
candid maple
nowadays I really only use GPT as a glorified screen reader because some people send problems that I cannot read with my poor eyesight.
violet sparrow
and that is exactly what I mean! they're not perfect. neither are humans, but hu...
its okay atleast you just demonstatrated your point
candid maple
they say when you have problems dealing with mental health or you want to rant, ...
that would probably be veering very much off topic for this help channel, so if this discussion is to continue I'd very much rather it continue somewhere in a quieter discussion channel rather than here. I don't want to run afoul of the moderators here.
violet sparrow
that would probably be veering very much off topic for this help channel, so if ...
yeah im sorry with that
im just curious
but yeah thanks for answering my math/physics question
candid maple
it's alright to be curious. but right place, right time, right topic.
violet sparrow
it's alright to be curious. but right place, right time, right topic.
yeah anyways have a good day
candid maple
you too! you can .close the channel if you're done with it, or send your next question if you have another one.
candid maple
for most purposes I do not prefer to DM, unless you state up front what you want to talk to me about (which again, is not a topic for this channel).
if you want to DM me, perhaps close this channel first, notify me in #serious-discussion about what you want to discuss, then I'll let you know if I am comfortable discussing with you.
violet sparrow
.close
viral lynx
pearl grove
am I doing something wrong?
I put the exercise on desmos and only the roots are correct
but the rest of the graph just isn't
are the channels not working
devout snowBOT
are the channels not working
pearl grove
anyone knows if nel is online
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pearl grove
No
tired surge
who’s nel
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lilac vortex
am I doing something wrong?
regardless of chats working or not... what actually is the question
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mellow willow
Can anybody help me with this one I’m stuck on what to do next
devout snowBOT
Can anybody help me with this one I’m stuck on what to do next
mellow willow
And for this one I don’t even know how to start
tired surge
allie is back
mellow willow
lol
mellow willow
that’s good then
tired surge
And for this one I don’t even know how to start
factor out sin x
both terms have sin x
$$\sec x \cdot \sin x + \sin x = 0$$
woven radishBOT
Ezra
tired surge
$$\sin x(\sec x + 1) = 0$$
woven radishBOT
Ezra
tired surge
then set each factor equal to 0
can you do that?
or do you need more breaking down and babysitting with this
mellow willow
How would I find secx=-1? I think it’s going to be one of those four points on the unit circle but the fact that it is sec and not cos is screwing me up
woven radishBOT
Ezra
mellow willow
Yea irs radius / x
woven radishBOT
Ezra
tired surge
then
mellow willow
So if I flipped the fraction -1/1 would it just be 1/-1 so -1
tired surge
$$\cos x = -1$$
woven radishBOT
Ezra
tired surge
So if I flipped the fraction -1/1 would it just be 1/-1 so -1
yes exactly
mellow willow
I don’t think I wrote my answers down correctly
upper schooner
I don’t think I wrote my answers down correctly
(you are missing some solutions, at the very least 
)
tired surge
you wrote both 0 + 2pik and pi +2pik
mellow willow
Wait how
tired surge
but pik already covers both
upper schooner
Wait how
Actually never mind, you actually aren't 
I read your working out a bit differently
mellow willow
Ah ok that’s good
upper schooner
But note that sin(x) = 0 gives you both of these, whereas cos(x) = -1 gives you the bottom one 
mellow willow
Kk
tired surge
so final answer
mellow willow
isn’t that the final answer?
tired surge
clean it up clean it up
it's correct i guess
no need to go further if your professor isn't nitpicky about it
upper schooner
(yea depending on how you get marked, and if there's "give as condensed as possible" instructions given to you 
)
upper schooner
but pik already covers both
That is fair, I do know of some precalc exercises where they have been though, but anyways, the only additional work you'd need in that case is this part here
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upper schooner
Can you believe it, I'm asking a question 
[Don't mind me, just trying to clear the channel naming 
]
devout snowBOT
Can you believe it, I'm asking a question <:SCshocked:1119493876495953921>
[Don'...
upper schooner
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arctic field
upper schooner
I tried nothing, just give me the answer 
arctic field
uncut crow
lusty sapphire
I tried nothing, just give me the answer <:sky_kekegirlOwO:818700500345290793>
!status
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I tried nothing, just give me the answer <:sky_kekegirlOwO:818700500345290793>
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1. I don't know where to begin.
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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
.Why do I not have the "this is irreversible" sticker 
arctic field
you are simply reversible
cold haven
upper schooner
.istg if the bot closes the channel but keeps it with my name instead, imma scream 
lusty sapphire
https://cdn.discordapp.com/attachments/150423182111801344/1359860883920126032/IM...
Saving this
rare kernel
Can you believe it, I'm asking a question <:SCshocked:1119493876495953921>
[Don'...
Yo uh what's ur question
!xy
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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.
rare kernel
!da2a
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Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
upper schooner
Yo uh what's ur question
.My question is, uh, why is it taking so long for this to refresh and become available again
upper schooner
.One of these days, probably sooner rather than later, I'll actually ask a question 
upper schooner
I blew the <@805217272087838750> up <a:kannafire:1143430218216919090>
.explains why @devout snow seemed to be dead earlier 
heavy current
sorry, I'll give it a headpat instead of blowing it up next time 
rare kernel
Charbit r u a boy or a girl
upper schooner
I am also a bot 
heavy current
I am also a bot <:bot:1040133333591539712>
the cat bit is mathcord's 0th AI. created by the founders of #cat-bit-takes-over-the-world, it was launched on May 24th, 2021. 
upper schooner
.it don't work that way, we aren't the same type of bot
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Available help channel!
severe surge
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woven vale
this x is different to f(x), it's just representing an unknown quantity which you need to solve for
you can write $f(-3)=y$ or $f(-3)=a$ or $f(-3)=\blacksquare$ if that helps you
woven radishBOT
حسیب ♥
versed juniper
that is definitely an odd way to write it lol
severe surge
this x is different to f(x), it's just representing an unknown quantity which yo...
but
doesnt f(-3) = -2
woven vale
doesnt f(-3) = -2
it does, so i think they want you to substitute that into the equation and get -2 = x
personally, why they want to do it that way makes no sense and this seems needlessly complicated. if you can read the value off the graph or off the equation, you are set
severe surge
personally, why they want to do it that way makes no sense and this seems needle...
THIS IS WHAT IM SAYING
EVERYONE
IN THIS MANS CLAS
SAYS
he overcomplicates stuff
severe surge
it does, so i think they want you to substitute that into the equation and get -...
bro
what is the point of that
woven vale
no idea! 
woven vale
i mean, if you don't know what f is exactly, then you work with f(-3) = x as an unknown symbol. so maybe he's trying to get you used to this
woven vale
it would be wrong
unfortunately i'm not your teacher so i don't know what he would mark it as
it's probably best to ask him directly
severe surge
also what does it mean "determine the value of x such that f(x) = 2"
woven vale
here you are given the y-value of the co-ordinate, i.e. y=2
so when the function has a y-value of 2, what's the corresponding x-value?
severe surge
so when the function has a y-value of 2, what's the corresponding x-value?
confusing because we dont get the x value no?
woven vale
well we have the graph
woven vale
we can go to a specific y-value, like y=2, find when the function is exactly that high, and then read off the related x-value
severe surge
because
x = 4
hits y =2
okok
im good now thanks he writes these weird 😭
takes a bit of time to get
woven vale
yeah, i would say "the function hits the line y=2 when x=4"
woven vale
im good now thanks he writes these weird 😭
solving things like f(x)=3 is harder than just f(3), so it may take a second. but well done 
!done
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severe surge
solving things like f(x)=3 is harder than just f(3), so it may take a second. bu...
thank you good sir
.close
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severe surge
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severe surge
I’m confused on A and B I don’t know where to start
supple knot
,rotate
woven radishBOT
severe surge
Domain and range without a graph sucks for me for some reason
autumn girder
Look at a, does there exist any value of x for which you can't evaluate f(x)?
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severe surge
Look at a, does there exist any value of x for which you can't evaluate f(x)?
uh no
cant you use any value
for x
couldnt i do 4
4wai
wait
seven is the vertex
right @autumn girder
so i just
draw the vertex
and see where it hits
on the y axis
and the vertex is h+k
so
it would be 2,7
so domain is {xer} and range is {yer | y >= 7}
okok
a makes sense
b
.close
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pale pecan
I need a function with the following properties(for personal curiosity on desmos graphing calculator):
- Variables "x" and "n"
- I guess function f(x,n)?
- Domain: [0,100]
- as n -> infinity, f(100,n) -> infinity
- as x -> 0, f(x,n) -> 0
- as n -> Infinity, median of y values -> 0
- The mean of y values should remain consistent regardless of the value n, and preferably be able to be controlled by a variable if possible?
devout snowBOT
I need a function with the following properties(for personal curiosity on desmos...
@pale pecan Has your question been resolved?
pale pecan
<@&286206848099549185>
solar goblet
I need a function with the following properties(for personal curiosity on desmos...
i think i have a rough idea what the graph could look like, but i don't have an exact formulation yet
pale pecan
oh I totally overlooked the last point mb
Its ok if the very very last bit is ignored, but the mean does need to stay consistent with N increasing
pale pecan
i think i have a rough idea what the graph could look like, but i don't have an ...
No worries !!!! It's a silly question for a silly use anyways
solar goblet
maybe two normal distributions could do?
make this into a piecewise, tweak the domains and stuff
this is just a rough sketch on the graph shape
pale pecan
maybe two normal distributions could do?
What is the red line in this case?
Also yeah normal distributions make perfect sense here haha
I'm not super math-y I just got this idea for a silly dice game 😭
solar goblet
the blue graph is just a horizontal shift of the red graph
solar goblet
maybe two normal distributions could do?
what i was thinking is actually the upside down of this
as n goes to up, the (for lack of better word) base of the graph goes up
pale pecan
Mhmmm...
Sorta?
Like the idea is uhh
A dice game, where you can pick a die of any number of faces
Dice with less faces have a lower maximum payout
But the same average as a die with more faces
The idea is the payout is never lower than 0 either*?*
If you don't have an answer that's fine it's a really random question
Thank you for your effort either way
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@pale pecan Has your question been resolved?
craggy dagger
Dice with less faces have a lower maximum payout
Minimum number of faces >= mean
pale pecan
Minimum number of faces >= mean
?
craggy dagger
if the mean is 5.5, the die cannot have less than 6 faces
pale pecan
I need a function with the following properties(for personal curiosity on desmos...
This is the original question if it helps at all !!!
craggy dagger
Other than that there is a lot of flexibility in the problem
pale pecan
if the mean is 5.5, the die cannot have less than 6 faces
Oh, how come?
pale pecan
Other than that there is a lot of flexibility in the problem
I see I see
craggy dagger
If all possibilities are < 5.5 you can never get a mean of 5.5
pale pecan
Ohhh
The mean in this hypothetical function isn't correlated to the actual value assigned to the faces
pale pecan
I need a function with the following properties(for personal curiosity on desmos...
The idea is to assign a second value to the faces, that would be proportional in some way creating the properties I described here
Thank you for your help !!!!
As in the function would create coordinates that would describe the percent out of the whole of the faces of dice
And what value to assign them
To make the dice have the same average value when rolled while making them varied in median
craggy dagger
So how likely each option is and what value each option has
Not necessaily 1 to 5
If the mean is x, the median can't be larger than 2x (assuming all options are nonnegative)
pale pecan
I think we're both confused😭
It's not about how likely each option is I'm just tryna use this graph in a weird way
craggy dagger
If the mean is 5.5, the median can't be bigger than 11?
You want to produce physical dice?
pale pecan
It's not about how likely each option is I'm just tryna use this graph in a weir...
So like if the die has 6 sides (100,y) represents the ig "points" value the dice's face of 6 is
pale pecan
You want to produce physical dice?
Nah it's uhh
The idea is like plinko kinda?
Like a function to make a graph, 1-100
The dice analogy doesn't really matter 😭✌🏽
Its hard to explain
pale pecan
I need a function with the following properties(for personal curiosity on desmos...
This is what I need I'm sorry for confusing </3
craggy dagger
I need a function with the following properties(for personal curiosity on desmos...
Issues so far
- function has 2 variables but domain is given as a set of reals. Is that domain for x?
- As n -> infinity, "median of y values" would require some sort of probability distribution (uniform?) So median of f(y, n) induced by uniform distribution of y in [0, 100]?
- mean also requires the same treatnent
pale pecan
Issues so far
- function has 2 variables but domain is given as a set of reals. ...
Yeah the domain is for x
Which I suppose could be done as a domain restriction/peice wise
And the other 2 things I don't understand, as I said I'm not very math-y I'm only in precalc 🥲
craggy dagger
And the other 2 things I don't understand, as I said I'm not very math-y I'm onl...
To talk about a median, you need a random variable, sometimes given via a probability distribution
Is the intention that y is uniformly distributed to form a random variable f(y, n)?
Oh hmm looks like at some point i confused x for y
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lapis raft
z = x + iy
where x = a
a=5
(\omega ) = ln(z)
is this plot of the line z = 5 + iy correct?
woven radishBOT
idk
lapis raft
where the red lines are ln(z)
devout snowBOT
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lapis raft
no
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lapis raft
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✅ Original question: #help-27 message
@lapis raft Has your question been resolved?
lost laurel
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lost laurel
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✅ Original question: #help-27 message
lost laurel
lost laurel
I feel like I'm missing something here
the cdf is t^{n+1}; 0<t<1
so surely it converges to 0
lost laurel
the +C is because t isn't defnite?
lost laurel
each Z_n’s cdf looks like t^{n+1} + C for some C
then it converges to C
but how would we know what C is
lost laurel
C is just 0 here anyway
won't limiting the cdf give us 0 thne
uncut crow
0 for what?
lost laurel
$\lim_{n \to \infty} t^{n+1}; 0<t<1$
woven radishBOT
wai
uncut crow
i don’t see what the problem is
lost laurel
I have to show {Z_n} converges in dist to Z=1
uncut crow
yep
uncut crow
yea i agree now. maybe this question is sus. Z’s cdf is the identity function (well on [0,1])
lost laurel
yea
okay, so I guess it;s wrong
wait, lemme see if there;s a soln
@uncut crow
this is what the book says
uncut crow
lost laurel
<:ducky_skull:1121451613224173689>
this is wrong is it not
uncut crow
nothing they wrote there is wrong, but concluding Z_n converges to Z from that is. hence the
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drowsy thicket
Can someone explain the alternating series test to me?
devout snowBOT
Can someone explain the alternating series test to me?
pseudo basin
are you looking for an explanation of why it's true or just how to apply it?
drowsy thicket
are you looking for an explanation of _why it's true_ or just _how to apply it_?
Both if possible, but mainly how to apply it, it is confusing me for some reason
pseudo basin
ok, do you have a particular series you're struggling to apply it on
drowsy thicket
ok, do you have a particular series you're struggling to apply it on
Not particularly, I just don't really understand it with the formal definitions online
pseudo basin
ok, do you know what an alternating series is in the first place
drowsy thicket
ok, do you know what an _alternating series_ is in the first place
Yeah like (-1)^n where the sign alternates, right?
pseudo basin
the (-1)^n encodes the sign alternation yes
but if you just write $\sum (-1)^n$ you'll end up adding and subtracting specifically 1's
woven radishBOT
Ann
woven radishBOT
Ann
pseudo basin
to apply the alternating series test you:
- check whether the sequence {a_n} (without the (-1)^n) is monotone decreasing, or at least eventually so
- work out the limit of a_n, expecting to get 0
if both of these checks pass then the series is convergent
thats the "execute application of AST" part
maybe you could link one of those formal things you saw online but could not understand?
drowsy thicket
if both of these checks pass then the series is convergent
so you can disregard the alternating part for those two?
pseudo basin
i... would not say you are "disregarding the alternating part" exactly
rather you are putting it out of the way
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silver breach
uhmm i forgot what my prof said in class about the havel-hakimi theorm... how do i draw a graph from it xD I got my numbers but yeah- this is the only example i got
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uhmm i forgot what my prof said in class about the havel-hakimi theorm... how do...
@silver breach Has your question been resolved?
kindred mauve
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kindred mauve
how do we know what the values in yellow are?
scarlet sequoia
These are well-known formulas, they can be derived, but also simply memorized
Or do you mean which function is f and g?
kindred mauve
Or do you mean which function is f and g?
yes
scarlet sequoia
for 1 < x < 2 you have 2x + 3 > -x, so f(x) = 2x + 3 ("higher one") and g(x) = - x ("lower one")
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kindred mauve
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✅ Original question: #help-27 message
kindred mauve
is f(x) always the higher one?
lilac patio
is f(x) always the higher one?
i have little knowledge of physics but all of these integrals are analogous to the area integral so yes
potent yew
I need help for 14iii
devout snowBOT
I need help for 14iii
potent yew
dry oak
Alr
fair juniper
you already did a major portion of the progress in part ii
dry oak
What did you do in the question before?
potent yew
you already did a major portion of the progress in part ii
Ye what do i do next
potent yew
What did you do in the question before?
A proof
dry oak
Ye what do i do next
You can use the previous result
fair juniper
because they say give your answer to 3 decimal places, you are expected to use a calculator
youre not done when you solve for theta
remember, we want x
so you plug theta back in, and your multiple values disappear
potent yew
Yeah do i only write the calculator solution for theta?
fair juniper
your infinite values*
fair juniper
Yeah do i only write the calculator solution for theta?
asin(3/4) is not a special value, so they expect you to use a calculator
dry oak
How many values for theta did you write?
potent yew
1
dry oak
You can write more
fair juniper
dont forget to plug the value back in to be sure you got a good value
dry oak
Remember the unit circle
potent yew
But they equal the same thing
fair juniper
sin(3theta) equals the same thing for all of them, but sin(theta) is different
dry oak
How am i supposed to know how many values of theta to wrote
Generally if you expect there to be 3 solotiouns write 3
dry oak
Some angles add to 180 so they are equal
Was this before or after you divided by 3?
potent yew
After
fair juniper
solving for sin(3theta)=3/4 looks like this:
3theta = asin(3/4) + 2*k*pi
theta = asin(3/4) + (2/3)*k*pi
k is an integer
notice how when you pick two k's which differ by 3, you get theta which differs by 2pi
potent yew
fair juniper
those are the same (coterminal) angles
potent yew
Can i just say
fair juniper
oh lol i forgot about the second solution
potent yew
That a polynomial degree of 3 means it has 3 roots so i test until i find 3 roots?
fair juniper
once you find one, you can use polynomial division to get a quadratic, and quadratics are easy
potent yew
O
dry oak
once you find one, you can use polynomial division to get a quadratic, and quadr...
I dont think thats wise cause hell get soke quadratic with a bunch of decimals
dry oak
Bit yeah this is good just check they satisfy the original equation
fair juniper
i wasnt sure if the sin gives all solutions, so thats what i would go with
potent yew
Is the working fine
dry oak
Is the working fine
Ye its fine
dry oak
i wasnt sure if the sin gives all solutions, so thats what i would go with
Hmm it should give all the solotiouns
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lost laurel
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woven radishBOT
wai
pseudo basin
you don't rly need the marginal as such no
or are you forbidden from using the law of the unconscious statistician?
lost laurel
well , I can
pseudo basin
$E[g(X,Y)] = \sum_{(x,y)} g(x,y) p_{X,Y}(x,y)$
woven radishBOT
Ann
lost laurel
I can, but If I need E(X) , I'll want to restrict it no?
pseudo basin
this law works perfectly fine with g(X,Y) = X
lost laurel
okay, so just $\sum_{(x,y)} x^2 p_{X,Y}(x,y)$
pseudo basin
lowercase p and also you're gonna sum over the entire support of p_X,Y
putting down $\sum_{(x)}$ will make no sense
lost laurel
lowercase p and also you're gonna sum over the entire support of p_X,Y
entire support for a given x , obviously NVM, this would be for marginals
pseudo basin
no
lost laurel
so $4 \cdot \left [ p(2,10)+p(2,12)+p(2,14) \right] +49 \cdot \left [p(-7,10)+p(-7,12)+p(-7,14) \right]$
woven radishBOT
wai
pseudo basin
so it seems, yes
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grizzled iris
Question about translation and translation vector
devout snowBOT
Question about translation and translation vector
grizzled iris
When applied to a equation y = f(x)
the vector (a b) would change the equation like y = f(x -a) + b or y = f(x+a) + b?
Because in this lesson I was doing exercises but I messed up the order, but it marked it as correct, but the theory states otherwise, what is the correct way?
crystal dawn
gods why are they bringing in vectors right now
grizzled iris
Because they literally give me this in the theory lesson
OH wait
YOu aren't blaming me
I thought you are
crystal dawn
I have no incentive to blame a learner over a syllabus
grizzled iris
Ok so what are they trying to do here, because they are contradicting themselves
It's IB material by the way, official one
crystal dawn
the super simplified gist of it is that if a function is translated by (a, b), its graph is shifted by (a, b) (where positive x and y represent right and up respectively), and the translated function is represented by f(x - a) + b
these two are two of the four transformations I mentioned in our DMs
grizzled iris
Ok I understand this
But How do I apply vector transformation to cordinates?
lul
crystal dawn
if a point (a, b) is translated by (c, d), then the new coordinates are simply (a+c, b+d)
grizzled iris
Why is it inverted here for x
crystal dawn
translation is literally just adding coordinates (or subtracting if you have a negative)
grizzled iris
I know
crystal dawn
Why is it inverted here for x
because of how function inputs work
crystal dawn
suppose I have f(x) = something
let's notate that something as a
so f(x) = a
eh wait sorry
f(0) = a
but the general formula is f(x)
now, suppose I rewrite f(x) as f(x - 3)
grizzled iris
Then it moves to the right
crystal dawn
in the original function, substituting x = 0 gives us f(0), which is a
but now, if we substitute x = 0 into our new function, we find that we get f(0 - 3) = f(-3)
so to compensate, we have to increase the value of x, aka moving right
grizzled iris
Holy quacks what
crystal dawn
in this case, right by 3 gives us x = 3, and indeed, f(3 - 3) = f(0) = a
grizzled iris
Uh, Me is confused
crystal dawn
which part
grizzled iris
but now, if we substitute x = 0 into our new function, we find that we get f(0 -...
.
crystal dawn
so now suppose we want to know which value of f(x) gives a
grizzled iris
inverse function?
crystal dawn
we know that f(0) = a; keep this in mind
we don't care about inverses since we know key values already
so given the original function f(x), what must x be for f(x) to equal a?
grizzled iris
I don't know I am lost
crystal dawn
I will remind you, f(0) = a.
grizzled iris
x =0
grizzled iris
3
crystal dawn
good
now, originally, the point (x, a) was at (0, a).
but after this translation we find that the point has moved to (3, a) (assuming we want to find a point with the same y-coordinate, namely a)
this is a movement to the right by 3 units
grizzled iris
so to have a still in the output, we had to add 3 units to the x axis?
crystal dawn
yes.
grizzled iris
Ok Im still confused
crystal dawn
alternatively you can also directly see the point here
grizzled iris
Ok so why it's inversed?
crystal dawn
say you have a fuel tank in a car that's good for some number of miles when full
now, I empty out, say, 40% of the tank. I now ask you this: what is the amount you need to add or remove back for the tank to achieve the same number of miles as a full tank?
grizzled iris
didn't know 60% + 60% = 100%
I am math
crystal dawn
now notice that if the input was decreased, you will need to increase it back to reachieve a given output
that's the key behind why this whole "direction reversing" thing happens
grizzled iris
It still doesn't make sense
crystal dawn
similarly, if I overfilled the tank by 10% and asked you the same question, your answer would have been to throw the extra 10% away
crystal dawn
It still doesn't make sense
why not then?
grizzled iris
I don't know
crystal dawn
or maybe a symbolic approach would help you here
well I don't quite appreciate random gifs in the middle of a help session
tell me when you're ready to proceed
grizzled iris
I am
crystal dawn
cool
grizzled iris
Im sorry im barely holding my anger
crystal dawn
then consider taking a break
crystal dawn
how sure are you you can get it with your current mindset? I'm not judging you or anything but I will note I have experience with learning while angry and teaching students who do that
and I will say neither are pretty
grizzled iris
Ok so I will get back in 5 minutes
grizzled iris
No problem
crystal dawn
or if someone else gets to this channel before she does then listen to them Ig
grizzled iris
I should be beaten physically because of the amount of help I am gettgin here from random people
crystal dawn
there's nothing wrong with that and we would really rather not have too much self-depreciation when learning
candid maple
I'll park myself here for when you're ready then.
candid maple
so as mentioned by Hanako, we can try a symbolic approach.
do you agree that for any point with an x-coordinate of x, its y-coordinate is f(x)?
grizzled iris
Yes
candid maple
so keep the point (x_0, f(x_0)) in mind.
now, consider a function g(x), defined as g(x) = f(x - a).
we know that (x_0, f(x_0)) lies on the graph of f. but this point doesn't lie on the graph of g because g has been shifted relative to f. at this point, we don't care how it has been shifted (though we will find out soon how), we just know it has been shifted.
suppose, then, we want to find the point on g such that g(x) = f(x_0).
substituting x = x_0 into g yields g(x_0) = f(x_0 - a), which gives the point (x_0, f(x_0 - a)). this is of not much use to us. but since the function f has been shifted by -a to get g, we know that we can remove this from consideration by instead substituting x = (x_0 + a). why? because then:
g(x_0 + a) = f([x_0 + a] - a) = f(x_0)
so we find that the corresponding point on g(x) is (x_0 + a, f(x_0)), and that's why the shift in input has to be opposite the direction that a is in.
grizzled iris
What is x_0?
candid maple
a random specific value of x.
grizzled iris
so x is x or 0
candid maple
x_0 is a single variable.
grizzled iris
Ok
candid maple
you can replace it with m if you want to.
another intuition for the same situation is this.
grizzled iris
substituting x = x_0 into g yields g(x_0) = f(x_0 - a), I lost it here
candid maple
consider a machine that converts water to, say, food. don't care how it's done, just know that this machine does it.
now, you are told that 50 liters of water produces 80 kg of food.
so if 50 liters go into the machine, then you expect to see 80kg of food out.
but now suppose I have a rule that any time you want to put water into the machine, you have to pay a "water tax" - 20 liters will be removed from what you have before it goes into the machine.
how many liters of water do you need to bring to get 80kg of food under this rule?
grizzled iris
70
candid maple
and why do you say so?
grizzled iris
Because it will cost you 50 Liters for the conversion, and 20 Liters for tax
To get 80KG
candid maple
what if the tax was higher, say, 35 liters taken away before it goes into the machine?
grizzled iris
85Liters?
candid maple
absolutely. now, let's get a little general.
if the tax was m liters, how much water do you need to bring, and compared to some specific value of m, do you expect to increase or decrease the amount of water you need to bring if m was increased?
grizzled iris
m+35
candid maple
(assuming you want the same output of food.)
grizzled iris
Tax is now 35, so I need to add 35 liters anywhere if I want to get 80KG of food for 50 liters
candid maple
absolutely. now, let's get a little general.
if the tax was m liters, how much w...
well, the tax is now m liters.
grizzled iris
Then m+50
candid maple
great job. now, suppose m increases. do you then expect to have to bring more or less water for the same amount of food?
candid maple
good.
now, let's formalize this.
suppose the function governing the amount of food for any given amount of water x is f(x) = y.
changing f(x) to f(x - m) is introducing this "water tax". this tax reduces the amount of water going into the machine f(x), so to get back the same amount of food, we need to compensate by adding back m liters of water to the input to counteract the tax.
candid maple
then either consider a longer break, or consulting a personal tutor, or even playing around with some graphs in Geogebra or Desmos.
in the meantime, if you want to verify what you do and/or do not understand with me, I'm all ears.
grizzled iris
Ok im gonna close this ticket, copy the last explanation and sit around with it for a while, if I will have more questions I will reopen
candid maple
sure thing. all the best.
candid maple
you don't have to apologize. you owe us nothing.
I'm the one who should be apologizing for failing to help you out here.
grizzled iris
.close
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lost laurel
Really silly question, but here once I find c=3/16 all I do is (3/16xy^2)/(marginal pdf) right for f_{1|2} and f_{2|1}
devout snowBOT
Really silly question, but here once I find c=3/16 all I do is (3/16xy^2)/(margi...
tired surge
Really silly question, but here once I find c=3/16 all I do is (3/16xy^2)/(margi...
hello
lost laurel
hello
tired surge
not silly
yes you divide joint by marginal
also
$$
f_{1|2}(x_1|x_2)=f_1(x_1), \quad
f_{2|1}(x_2|x_1)=f_2(x_2)
$$
woven radishBOT
Ezra
tired surge
notice they’re independent
lost laurel
notice they’re independent
for conditional expectation I just integrate xf_{1|2} or yf_{2|1} right
tired surge
yesss
lost laurel
cool
tq
lemme find them rq
$\mu_{2 \mid 1}= \int_{0}^{2} y \cdot \frac{3}{8} y^2 dy$
woven radishBOT
wai
lost laurel
which is 3/2
woven radishBOT
wai
lost laurel
Independence comes in clutch here
it allows me to say $E[X_1^nX_2^m]=E[X_1^n]E[X_2^m]$
woven radishBOT
wai
lost laurel
which helps with the last but one problem
tired surge
very good
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grave idol
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grave idol
Is my solution correct? (Do I need to find the limit of everything at the same time or I can do it step-by-step like this)
supple knot
no
you can't take the limit of different parts of the expression differently
you have to take the limit all at the same time together
devout snowBOT
grave idol
.reopen
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✅ Original question: #help-27 message
supple knot
you inputted [ log(1 + 1/n) ]^n to wolfram. the expression in the problem is log[ (1 + 1/n)^n ]
the main thing you need is
$\lim_{n \to \y} (1 + 1/n)^n = e$
woven radishBOT
riemann
supple knot
and lim[ log(stuff) ] = log[ lim (stuff) ]
reef sail
i miss the days of simple lim
native gyro
what?
grave idol
i miss the days of simple lim
So do I, I am about to get clapped by fourier series or smth.
native gyro
ln(x)^y != ln(x^y)
grave idol
ln(x)^y != ln(x^y)
The thing u see in my input in wolfam is wrong. It's suppose to be:
grave idol
can someone give me a hint or 2 on where did i go wrong.
this not what i type
native gyro
The thing u see in my input in wolfam is wrong. It's suppose to be:
yes i know
thats why it shouldnt be ln(e)
or wait
is it (ln(1+1/n))^n or ln((1+1/n)^n)
coz i think its the first one
so ln before ^n
supple knot
so it will become (ln(e)) = 1 right?
yes
grave idol
is it (ln(1+1/n))^n or ln((1+1/n)^n)
i think it's the 2nd
reef sail
its ln am blind
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alpine python
<@&268886789983436800>
wicked rover
looks like the ban didnt delete this
devout snowBOT
ebon flicker
consider a machine that converts water to, say, food. don't care how it's done, ...
140L
You put your first 50l u get 30kg then another 50l wich becomes 60kg then you add only 40l at the end so 20l/20kg get takken out(taxes) and then you are left with 30 + 30 +20 = 80kg
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charred eagle
can someoen explain why the normal vector of a plane is the values before the x + y + z
devout snowBOT
can someoen explain why the normal vector of a plane is the values before the x ...
charred eagle
like lets say your trying to find point of intersection of the line r(t) = 6i -2j + k + t(i + 2j - 4k), t belongs to R, and the plane 4x + y + 2z = 6
how do i do this
supple knot
When F(x,y,z) = ax + by + cz + d, the normal to the plane is simply the gradient and that's the vector (a,b,c)
charred eagle
but wouldnt the gradient be on tjhe plane
charred eagle
When F(x,y,z) = ax + by + cz + d, the normal to the plane is simply the gradient...
likehow does it make sense for a perpendicular line to be the gradient of a plane
supple knot
but wouldnt the gradient be on tjhe plane
the gradient is a vector, not a point
supple knot
likehow does it make sense for a perpendicular line to be the gradient of a plan...
take any vector in the plane and dot it with the normal vector and it'll equal zero. that's the definition of perpendicular
charred eagle
the gradient is a vector, not a point
the gradient is a vector in the plane?
supple knot
nowhere did i say that
charred eagle
nowhere did i say that
Im asking a question
supple knot
if it is, you should try to prove it
charred eagle
no im ngl
i have no idea wahts going on in this planes topic
like im trying to udnerstand
supple knot
scroll to the middle for planes
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.05%3A_Equations_of_Lines_and_Planes_in_Space
queen mason
This is Calculus 1 Optimization, is this the right thought process? From here I would simplify and then derive right?
devout snowBOT
This is Calculus 1 Optimization, is this the right thought process? From here I ...
supple knot
,rccw
woven radishBOT
rich barn
This is Calculus 1 Optimization, is this the right thought process? From here I ...
looks right to me
dry oak
Me aswell
rich barn
just find the global minima for the t(x) you came up with
dry oak
Ye and Id recommend not calling it t(x) but call it f(x) instead cause t is a mid letter
queen mason
Any tips on simplifying
drifting sierra
Ye and Id recommend not calling it t(x) but call it f(x) instead cause t is a mi...
t for time
dry oak
t for time
Huh good catch
dry oak
Any tips on simplifying
But yeah just derive it without simplifying I say
queen mason
Oh I can do that
dry oak
Ye
I mean its already pretty simplified
Just maybe split thr fraction if its easier for you
The second one
queen mason
I only ask because my teacher tried simplifying more and it did not make sense but thanks
.close
restive river
Can someone help me in factoring
devout snowBOT
Can someone help me in factoring
timber pebble
factoring what?
devout snowBOT
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
timber pebble
can you show a problem you're having trouble with?
restive river
It’s many
timber pebble
lets start with one 
restive river
Thank you a lot
I know this is supposed to be easy I’m just very not smart with math 🙁
woven radishBOT
jan Niku
timber pebble
so you know, if we can factor this, it will look like (x-a)(x-b)
for some a and b
usually when they say, factor quadratic, they mean write it like this
restive river
ohhhhh
timber pebble
(x - something) times (x - something else)
timber pebble
this one is one we'd say uhh you'd solve by inspection
restive river
Like the process
woven radishBOT
jan Niku
timber pebble
we want two numbers that multiply to 2, and add to -3
timber pebble
we almost always just try small, whole numbers
1, 2, 3 ,5
things like that
or -1, -2, etc
restive river
1 x -2?
timber pebble
and -1 + -2 = -3
restive river
thank you
timber pebble
do you know how to write it now?
restive river
And then you write it like
timber pebble
x^2 + 2x - 3 =
timber pebble
A for effort but off a little bit
timber pebble
we want two numbers that **multiply** to 2, and **add** to -3
so if we find these two numbers
we have to write it in a very specific way
say the numbers are a and b
we have to write, $(x-a)(x-b)$
woven radishBOT
jan Niku
timber pebble
here your numbers are -1 and -2
that means we have to write $(x - (-1)) (x - (-2))$
woven radishBOT
jan Niku
restive river
I was close
restive river
Yes
timber pebble
**jan Niku**
the negatives cancel out
timber pebble
3=0?
restive river
3 different coefficients
restive river
Why is there even a 0 at the end I don’t get it
timber pebble
this is usually the first thing you try, because its easy if it works
you try it by staring at the quadratic for 15 or 30 seconds and then you give up
and try something more complicated
its usually a good first thing to try no matter what
restive river
Should you find the common factor first tho
timber pebble
its a lot easier to calculate if x^2 has a coefficient of 1
so if you see a problem like, $6x^2 + 10x + 14$
woven radishBOT
jan Niku
timber pebble
you might not waste too much time trying to solve by inspection
but, a problem like $x^2 + x - 2$
woven radishBOT
jan Niku
restive river
you would want to get rid of 6
timber pebble
you would want to get rid of 6
yea, and if we factor out a 6, we create a lot of fractions
fractions are hard to deal with in your head, usually
wanna do your next problem?
restive river
It says in the book the answer is -3 and 1
restive river
no worries lol I feel the same
restive river
?
restive river
How do you even get that answer tho I’m confused
timber pebble
x^2 + 2x - 3
we need numbers that add to 2
and multiply to negative 3
i mixed up add and multiply when i told you to do this initially
and then since i was reading my own post instead of the question i didnt catch your answer being off
you solved it correctly based on what i told you to do, though
restive river
Ohhh
restive river
so it’s 3-1
timber pebble
forgive me for bungling that
woven radishBOT
jan Niku
timber pebble
so we want two numbers that add to 1, and multiply to -2
restive river
1- + 1
timber pebble
but -1 * 1 = 1
lethal zinc
could someone help me in vc im studying calclus 1
timber pebble
could someone help me in vc im studying calclus 1
we dont have vc, you'll have to open your own channel
whats an easy way to make -2 using whole numbers
we could try, -2 * 1, or -1 * 2
theres only really two options
restive river
1- 2
timber pebble
so which one of them adds to 1
yea -1 and 2
since 2 - 1 = 1
and -1 * 2 = -2
so we'd write, $(x- (-1)) ( x-2) = x^2 + x - 2$
woven radishBOT
jan Niku
timber pebble
make sense?
devout snowBOT
@restive river Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
dire hinge
Sketch a graph for 0 ⩽ t ⩽ 60 , labelling the endpoints
devout snowBOT
Sketch a graph for 0 ⩽ t ⩽ 60 , labelling the endpoints