the horizontal side of triangles drawn inside the circle (on the x-axis) has length cos(theta)
#help-36
pale oracle
the vertical side of triangles drawn inside the circle (parallel to the y-axis) has length sin(theta)
because the radius of the circle is 1, the hypotenuse of the triangle is also 1, and we are guaranteed that the coordinates of points on the circle, which are also the tips of these right triangles, have coordinates (cos(theta), sin(theta))
where theta is the angle swept out by the radius
hybrid prawn
Alright, I tried my crack at the problem and I got it correct because it was undefined, but I'm still not super clear on why I got that correct.
this at least means that the denominator was equal to 0
oh I'm trying a similar problem based on the same subject.
So these are the coordinates
and then I just work from there
er wait wrong, hold on
hybrid prawn
.close
final saddleBOT
Available help channel!
real quarry
Hello, guys! Could any of you please help me this one-sided limit? Most of the examples I saw on class were modular functions and stuff like that, so I'm not sure even where to start
final saddleBOT
Hello, guys! Could any of you please help me this one-sided limit? Most of the e...
real quarry
I mean, I know a little bit where to start, but I get stuck in x/x-1
<@&286206848099549185>
quiet zinc
can you tell me what will be f(x) here
we are aproaching the limit from?
@real quarry here?
real quarry
we are aproaching the limit from?
ok, so f = (x + 1 -1)/ x - 1 = x/x-1
quiet zinc
if f(x) = x+1
f(1) =?
real quarry
if f(x) = x+1
f(1) =?
2
quiet zinc
ok, so f = (x + 1 -1)/ x - 1 = x/x-1
you wrote f(1) as 1
final saddleBOT
final saddleBOT
Available help channel!
thorn harness
About to ask sth
final saddleBOT
About to ask sth
thorn harness
It says “Prove that set has a limit”
I assume that the limit in question is supposed to be 1 but a4 is bigger than 1??
I checked and the calculations are correct, any idea?
cursive spruce
at first I'd have thought that it diverges to infinity
because $\sum_{i=0}^{\inf} (1/i)$ diverges to infinity
soft zealotBOT
TRAMPELTIER
thorn harness
at first I'd have thought that it diverges to infinity
So the limit is infinity?
cursive spruce
Maybe not
thorn harness
(I don’t know how some of that terminology is called in English lol)
cursive spruce
because the terms are smaller in your series maybe its not diverging
$a_n = \sum_{i=1}^{n} (\frac{1}{n+i})$
soft zealotBOT
TRAMPELTIER
thorn harness
what does that mean 😭
cursive spruce
thats meaning summation of the terms 1/(n+i), from i=1 to i=n
thorn harness
sorry but we’re still on elementary calculus + I think some of the notation in my country is different(like we haven’t used the sigma sign)
thorn harness
then maybe we would use that at some point, no idea
thorn harness
then I don't have to write that all out
yeah I got what you meant
maybe a stupid question but “infinity as a limit”=“no limit”?
cursive spruce
yes because infinity is no number
we also say "it diverges to infinity"
$\sum_{i=1}^{n} (\frac{1}{n+i}) \geq \sum_{i=1}^{n} (\frac{1}{n+n})$
thorn harness
ohh I see
soft zealotBOT
TRAMPELTIER
thorn harness
also another question
when they ask you to prove that a set has some number as a limit (for example a growing set), can that be proven with mathematical induction?
like for a growing set, proving with mmi that it’s monotonously growing and that every number in it is smaller than the limit
soft zealotBOT
TRAMPELTIER
thorn harness
But what about an>an+1, for every n?
In the case of a monotonously growing smaller
cursive spruce
I mean, it could also converge to another number smaller than a
you'd have to rather prove that for every small $\epsilon$ your $a_n$ eventually satisfies $|a_n - a| \leq \epsilon$
soft zealotBOT
TRAMPELTIER
cursive spruce
that is the meaning of convergence. It means that no matter how small a number $\epsilon$ you pick, there should be a point $N$ where all entries in $a_n$ with $n>N$ will be within epsilon of $a$.
soft zealotBOT
TRAMPELTIER
lyric summit
It says “Prove that set has a limit”
I don't want to disturb , but to show that the sequence converges, it is enough to prove that it is monotone and bounded, this is an existential theorem in terms of the existence of a limit
cursive spruce
Yes, but it does not give you the point of the limit
thorn harness
I don't want to disturb , but to show that the sequence converges, it is enough ...
Yeah that’s what I meant
One part of the proof is the monotony, the other the existence of a bound
lyric summit
One part of the proof is the monotony, the other the existence of a bound
precisely, and i suggest you to start with monotonicity
cursive spruce
Okay I misunderstood and thought you'd wanted to prove also that a specific point is the limit
lyric summit
its ok)
thorn harness
Okay I misunderstood and thought you'd wanted to prove also that a specific poin...
But how can you prove that a limit exists without figuring out exactly what the limit in question is?
lyric summit
btw: the limit si equal to ln 2, but it is other story
thorn harness
Like for example I might see the members of the set, think that it looks like the limit could be 1, for example, and try to prove that
lyric summit
<@1035542975503138836> why is the limit ln 2?
i have computed it 500 time sduirng my lectues, but there are two ways: sandwich thoerem and seocnd way , Riemann sums
cursive spruce
wait
What about this:
$\sum_{i=1}^{\infty} (\frac{1}{n+i}) \geq \sum_{i=1}^{\infty} (\frac{1}{n+n}) = \frac{1}{2}\sum_{i=1}^{\infty} (\frac{1}{n}) = \infty$
soft zealotBOT
TRAMPELTIER
cursive spruce
wait something is wrong I'm doning something stupid cause n is not defined
lyric summit
that shud be written in other way
lyric summit
Like for example I might see the members of the set, think that it looks like th...
bu tyou have to start with monotonicty
thorn harness
bu tyou have to start with monotonicty
Why?
And do I do that with mathematical induction?
lyric summit
And do I do that with mathematical induction?
no math induction here
thorn harness
no math induction here
What other method of proof?
thorn harness
monotonicty is the first thuign you must dientify in your case,
Okay sure
But why can’t I prove said momotonicity by mmi?
lyric summit
What other method of proof?
th eother method is to find the limit, then you also show convergence, but it is hardder
lyric summit
Okay sure
you need to remind how we show that seqence is increasing or decreasing i writ efist lien fo royu :
thorn harness
you need to remind how we show that seqence is increasing or decreasing i writ e...
Huh?
lyric summit
Huh?
if this differwence
wil be positive
then yoru seqeuence is increasing
if negative then decreasing ok ?
no math induction here at all
and additioanally, if you show yoru given sequence is increasins then you onyl study boundedness from the top
and if yoru given seq is decreasing then you find bound from the bottom
thorn harness
no math induction here at all
Oh so expressing the difference with n is enough to tell us if it’s positive or negative?
cursive spruce
smart
lyric summit
Oh so expressing the difference with n is enough to tell us if it’s positive or ...
yes but you shud say: the sign of the received difference gives you monotonicity
cursive spruce
so if it holds for all n>N would be enough right
lyric summit
yes
cursive spruce
I mean if the sign is positive
thorn harness
But what about the bound part of the proof? What method is there to be used instead of mmi?
lyric summit
But what about the bound part of the proof? What method is there to be used inst...
as you noticed, the monotonicity helps you from what side you have to search the bound
thorn harness
Yeah
cursive spruce
If it increases, we upper bound. if it decreases, we lower bound a_n
lyric summit
exactly 🙂
cursive spruce
I'm so smart like joanna
thorn harness
But how can we prove that the number in question is the bound for sure
cursive spruce
do you have to finde the number?
lyric summit
But how can we prove that the number in question is the bound for sure
you must only find if limit exists or not
thorn harness
For example the set 1/2, 1/4, 1/8… has 0 as a bound, which we can see. But how can we logically prove that for sure?
lyric summit
For example the set 1/2, 1/4, 1/8… has 0 as a bound, which we can see. But how c...
wait
lyric summit
For example the set 1/2, 1/4, 1/8… has 0 as a bound, which we can see. But how c...
the set of limits in yoru words, contains zero or 1 element
limit exists
or not
thorn harness
Ohhh
lyric summit
there are no two limits
thorn harness
So to prove that the set is convergent, you just need to find the limit, without having to prove how said limit was found and whether we’re sure that it is the correct one?
lyric summit
So to prove that the set is convergent, you just need to find the limit, without...
i change your words:
thorn harness
- having first proven monotonousity?
lyric summit
the sequence is convergent if its finite limit exists
lyric summit
+ having first proven monotonousity?
yess, first monotonicity and next boundedness from the appropriate side
thorn harness
yess, first monotonicity and next boundedness from the appropriate side
Okayyy
And to prove boundness from appropriate side, you just need to figure out the number by Common Sense?
Like in 1/2, 1/4, 1/8… I can tell that the number is 0 without doing any proofs
lyric summit
And to prove boundness from appropriate side, you just need to figure out the nu...
look Juno, if you maybe notice but i showed something fwe liens a**ove, soem photo for TRAMPELTIER, and there i shwoed
cursive spruce
you can find for any epsilon a number 1/n that is smaller than epsilon
thorn harness
Ohhhh
lyric summit
so if you shwo sequence is increasing, then you get the bootm boudn for free
thorn harness
Nvm I think I mixed up limits and bounds
lyric summit
i thikn so )
thorn harness
Because I wasn’t sure how they were called in english
lyric summit
ok eng is not my native either
thorn harness
Like 1/2 is the bound in this case
cursive spruce
hablas espanol
thorn harness
What I meant was that 0 is the limit
And I can tell that it’s the limit without doing proofs
Is that+the monotony proof first enough?
cursive spruce
I mean 1/n > 0 thats your bound
lyric summit
What I meant was that 0 is the limit
0 can be also your bound but it is not the limit
cursive spruce
but it is the limit
thorn harness
Don’t they all infinitely decrease and approach 0 but never reach it ?
Thus making 0 the limit
cursive spruce
yes
lyric summit
Don’t they all infinitely decrease and approach 0 but never reach it ?
i do not discuss 1/n now, i tlak about yoru main example
$a_{n}=\sum_{k=1}^{n}\frac{1}{n+k}$
soft zealotBOT
Joanna Angel
thorn harness
i do not discuss 1/n now, i tlak about yoru main example
Ignore the main one
thorn harness
I’m asking about the simpler one because it’s easier to understand through it
cursive spruce
Juno trust me
thorn harness
So in the simple one the limit is 0?
cursive spruce
SEE JUNO
lyric summit
or - 10000 too
cursive spruce
joanna knows it
thorn harness
So to prove that a set has a limit, we prove monotony, find the direction and then eyeball a number that should be the limit, and the existence of that number leads to the conclusion that the set has a limit?
What I was getting at is that: is eyeballing enough
cursive spruce
you need to also bound
thorn harness
you need to also bound
In the simple case that’s 1/2?
cursive spruce
eyeballing can help you find the bound
lyric summit
So to prove that a set has a limit, we prove monotony, find the direction and th...
So to prove that a set has a limit, we prove monotony, find the direction = to prove that a sequence has a limit, we prove monotonicity and boundednss from appropriate side, but we do not ned to fidn the limit, jsut to know its existence
thorn harness
I mean, in the simple set, isn’t the lower bound 1/2 and the upper infinity?
cursive spruce
1/2 is a lower bound. but that only helps us if the series decreases
thorn harness
So to prove that a set has a limit, we prove monotony, find the direction = to ...
So me eyeballing that 0 is enough as long as I was first sure of monotony?
lyric summit
So me eyeballing that 0 is enough as long as I was first sure of monotony?
sometiems you can "eyebal
but very rare
cursive spruce
So me eyeballing that 0 is enough as long as I was first sure of monotony?
monotonically decreasing frmo somethiing above 0
thorn harness
but very rare
Is that enough for elementary calculus?
lyric summit
Is that enough for elementary calculus?
i m afraid to say,but rather not, unelss you have examples like 1/n
cursive spruce
what's mmi
thorn harness
Like for example in the simple example, proving through it that every single member an<0 for every n
And that can be done via mathematical induction?
thorn harness
what's mmi
Method of Mathematical Induction
Idk if it’s often called this in English tho
cursive spruce
But if an>0 for all n, that does not tell us much about convergence
Maybe you can prove it with induction
But still also you need to prove monotonicity
Maybe you can also prove that with induction
You can try
usually many way to prove things
lyric summit
Method of Mathematical Induction
to use the principle of mathematical induction you must have some equality or inequality theorem to prove, or some other fact true for natural numbers
thorn harness
Isn’t an<0 an inequality?
lyric summit
Isn’t an<0 an inequality?
but if you discuss existence of the limit of the given sequence, you do not need math induciton
lyric summit
But then what other method?
the other methods, you can guess the limit and prove it using the definition of the limit of the sequence
for example
lim 1/n = 0
then you can sue the defintion
to rpove it
but in yoru main exampole as yu remind
you have no chance
to predicit
the reesult like
ln 2
lyric summit
But then what other method?
the final method is just findign the requyired limit if it is yoru case, then there is a wide range of variours methods to obtain such the liimt depends on the character of the given sequence
lyric summit
But then what other method?
anywway, once again, your teacher at school wants you to investiagte the monotonicity and boundedness of your main sequence, and if you show those two condtions satisfied, then you write finally that your main sequence is convergent
thorn harness
Convergent = has limit, right?
lyric summit
Convergent = has limit, right?
correct
thorn harness
Okayy
So proving boundedness and monoticity = a limit exists = the set is convergent?
cursive spruce
kinda, but in math language also people say "limit at infinity" or "limit at negative infinity" if the series diverges and is not convergent.
thorn harness
you wrote all well , except one word, write sequence, not set
Ohh yeah my bad
lyric summit
and yes, limit must be finite
thorn harness
So proving boundedness and monoticity = a limit exists = the set is convergent?
So this proves that a FINITE limit exists?
thorn harness
Okayy
So in the simple sequence
1/2 is the bound
And we can prove that it’s monotonously growing smaller by subtracting an+1 - an and getting a negative value
Is that so?
lyric summit
And we can prove that it’s monotonously growing smaller by subtracting an+1 - an...
yes, and is it possibe for you to shwo it maybe here?
thorn harness
Okay
lyric summit
cool )
cursive spruce
I'm cheering for you
cursive spruce
looks correct
lyric summit
but it is your main first one sequence?
thorn harness
but it is your main first one sequence?
The more complex one?
Should I test it on it too?
lyric summit
The more complex one?
but the main oen shows compelxity )
thorn harness
Do we have to prove boundedness or is saying “[number] is the bound” enough?
lyric summit
if your sequence is so simple liek yours 1/2n, then it is enough to say or formally write in a follwiong way: look :
thorn harness
Like for example in the complex sequence, a1=1/2 and it’s monotonously increasing so that must mean that a1 is the bound?
lyric summit
Like for example in the complex sequence, a1=1/2 and it’s monotonously increasin...
that is formal definition that your sequence 1/2n is bounded from bottom by the M = 0
thorn harness
Isn’t it bounded by 1/2?
lyric summit
no need to justify it in this case
thorn harness
Like every an>=1/2
lyric summit
Like every an>=1/2
yes it is
because
if sequence
is boudned form the bootm
then by th emany numbers
0, -`1, -100
1/2
is the largest one
but 1/2
si the boudn fromt he top
so you can write:
thorn harness
What was the difference between bottom and top bound?
I thought that for example bottom bound only existed in infinitely increasing sequences
lyric summit
0 si bound from the "down" = bottom, and 1/2 from the top
thorn harness
And you only had both top and bottom if it was a closed sequence
Like for example {2;4;6;8} or something
lyric summit
And you only had both top and bottom if it was a closed sequence
but if we tlak ab the limits then sequence is not finite, we discuss infinite sequences
thorn harness
Okay
But if, for example, we have a monotonously growing infinite sequence
Like 1/2, 1/4, 1/8…
lyric summit
the sequence 1/2n is decreasing so we do not need to know the upper bound, it is enough for convergence to state that we have the lower bound by 0
thorn harness
Wouldn’t the bottom bound be 1/2
Because that’s the number all values are bigger than
Or equal to
lyric summit
your sequence 1/2n is: 1/2, 1./4, 1/6, 1/8 , ....
right ?
so 1/2 bounds from the top
not from the botom agree?
every term of your sequence is equal or less than 1/2
hence 1/2 is the upper bound
but not lower bound
lower boudn is zero
or -1
or -1234556
etc
thorn harness
not from the botom agree?
So “all members are equal or bigger than X”= “X is the top bound”?
OHH
My bad I mixed up the directions
So if all numbers are equal or smaller than 1/2, then 1/2 is top bound?
lyric summit
So “all members are equal or bigger than X”= “X is the top bound”?
you should write, : if all terms ( not members) of the given sequence are equal or less than M, then we call M as the bound from the top = upper bound
thorn harness
oh yeah my bad with the terminology lol
lyric summit
oh yeah my bad with the terminology lol
it is also important to be undeRstood )
thorn harness
you should write, : if all terms ( not members) of the given sequence are eq...
Okayy
So is proving that the sequence is monotonously growing smaller, then find out that there exists a number M(in this case =1/2) which is the upper bound, this means that a limit exists and the sequence is convergent?
lyric summit
So is proving that the sequence is monotonously growing smaller, then find out t...
you forgot some sentences of mien i wrote 15 minutes ago )) let me write it again
:
if you prove that the given sequence is decreasimg, then you need to find out whether there exists a number m whcih is the lower bound, then you are alllowed to state that the sequence is convergent
ok ?
thorn harness
Okay
lyric summit
decreasing and bounded from the bottom is convergent
or
increasing and boudned from the top is convergent
thorn harness
And proving that it’s monotonously decreasing is through the an+1 - a method, right?
lyric summit
you must memorise it
lyric summit
And proving that it’s monotonously decreasing is through the an+1 - a method, ri...
right
thorn harness
Alright
thorn harness
decreasing and bounded from the bottom is convergent
But wait, in the 1/2n sequence, the bound 1/2 is top but the sequence is decreasing?
lyric summit
But wait, in the 1/2n sequence, the bound 1/2 is top but the sequence is decreas...
it is true, but i wrote to you that you did not need 1/2
you rather need zero from teh bottom
i write it for you
wait 2 minutes
thorn harness
you rather need zero from teh bottom
Ohhh
But wasn’t 0 the limit?
Or in this specific case limit=bound?
lyric summit
But wasn’t 0 the limit?
in this simple case, the bound we showed, is the limit too, that happens often, when we cope with such simple sequences,
final saddleBOT
@thorn harness Has your question been resolved?
thorn harness
Okay sorry for dipping i had to get something done lol
thorn harness
Ohhh
So 1/2n>0 for any n means that 0 is one of the bounds, which means that the sequence is bounded, which plus the monotonous decreasing makes it convergent?
lyric summit
So 1/2n>0 for any n means that 0 is one of the bounds, which means that the sequ...
very well )
thorn harness
Okay let me try a harder one
soft zealotBOT
Joanna Angel
thorn harness
Okay but there’s a problem where they do ask what the exact lim is
I suppose at this point i can (and they expect me to) eyeball it?
lyric summit
I suppose at this point i can (and they expect me to) eyeball it?
no, you cant take it form black magic hat haha ))
there are precise methods
to find it
thorn harness
Okayyy
For example there’s another problem where an=(n-1)/n
How do i go about finding the limit?
thorn harness
Yeah
cursive spruce
plug in n-> infinity
precise method is show for a small epsilon that there's an N such that |a_n-a| <epsilon
for all n>N
lyric summit
Because you said eyeballing didn’t work
1/n, goes to zero, you know, , 1 goes to 1, since it is the constant seqeunce, so ?
thorn harness
What do you mean by “goes to”
I do know that n=/=0 because it’s in the denominator
lyric summit
What do you mean by “goes to”
=reaches its limit
lyric summit
I do know that n=/=0 because it’s in the denominator
1/n ---> 0
means, when n approaches +infinity then 1/n reaches zero
lyric summit
$\lim_{n \to \infty } \frac{1}{n}=0$
soft zealotBOT
Joanna Angel
thorn harness
That makes sense
thorn harness
How are we sure that it’s zero that it’s approaching?
But this?
cursive spruce
cuz for any epsilon, you will find an N such that |1/n - 0| = 1/n is smaller than epsilon
this is true for $n\geq N > 1/epsilon$
soft zealotBOT
TRAMPELTIER
cursive spruce
e.g. epsilon = 0.0001, then for all n > 10000, the difference between 1/n and 0 is less then epsilon
thats the formal way of limits
but in school you probably didn't learn this epsilon stuff
btw, Here's a bit thinking
$a_{n}=\sum_{k=1}^{n}\frac{1}{n+k} =\sum_{k=n+1}^{2n}\frac{1}{k} = \sum_{k=1}^{2n}\frac{1}{k} - \sum_{k=1}^{n}\frac{1}{k}$\
$a_{n+1} - a_n = \sum_{k=1}^{2n+2}\frac{1}{k} - \sum_{k=1}^{n+1}\frac{1}{k} - \sum_{k=1}^{2n}\frac{1}{k} + \sum_{k=1}^{n}\frac{1}{k} =
\frac{1}{2n+2} + \frac{1}{2n+1} - \frac{1}{n+1}$\
and then we need to check if $\frac{1}{2n+2} + \frac{1}{2n+1} < \frac{1}{n+1}$
soft zealotBOT
TRAMPELTIER
cursive spruce
so and if it is true that this decreases, then we know that it has a limit because we showed earlier 1/2 is a lower bound
thorn harness
but in school you probably didn't learn this epsilon stuff
We have mentioned it but honestly i find it a little difficult to grasp
lyric summit
so 1/2 is the boudn from the bottom but we need from the top
yes
that is not hard to find
cursive spruce
ehm, isnt it diverging to infinity?
lyric summit
no ))
i said i did it
m500 time sin my life at the uni with my students
that is oen of very nice example
to sjhow
famous example
thorn harness
so 1/2 is the boudn from the bottom but we need from the top
So the top bound in an increasing sequence is the limit?
cursive spruce
ah we can bound by harmonic number
lyric summit
So the top bound in an increasing sequence is the limit?
we need to find upper bound = from the top, but not necesairory its limit
thorn harness
But in the problem they’re asking us to find the limit
lyric summit
So the top bound in an increasing sequence is the limit?
we jsut need find any uppper bound
lyric summit
to infinity
i said before it slimit is ln 2
cursive spruce
prove
thorn harness
we jsut need find any uppper bound
So any upper bound is the limit?
cursive spruce
no
lyric summit
So any upper bound is the limit?
no, you shud say, that if sequence is bounded from the top then there are many bounds , upper ones, greater or equatl to supremum, and in this acse
limit wil be supremum
but now i shwoed just oen fo easier upper bounds
thorn harness
Okay it’s kinda late here and I’m tired and starting to get kinda confused, can i ping you again tomorrow to ask you about all that? 😭
cursive spruce
We have an upper bound and the series is monotonically increasing (no fluctuations) => it can not get any bigger than the upper bound => either upper bound is the limit or some number below that.
Joanna you convinced me your proof is nice
lyric summit
Joanna you convinced me your proof is nice
when i was a student i was doing it for free )) smiles )
cursive spruce
I always thought about bounding by harmonic sum
but that's not a good boudn
letting the n stay there is a good thing cuz it makes the sum small
cursive spruce
Btw Joanna what are your habits of becoming a math genius
I need to adopt your lifestyle
cursive spruce
but hows that true
cursive spruce
I dont see that
lyric summit
and htat comes form Bernoullis inequality
lyric summit
what ?
cursive spruce
You're not justifying
lyric summit
You're not justifying
i jsut give you a hint
if you want more, coem to mys chool
pay fo rit
and hten i prove it
cursive spruce
damn
lyric summit
the other hitn is :
that can be also taken from Lagrang;e theorem
though ineng literature it can have other name
i think they call Mean Value Theorem
but we call it Lagrange Theorem in my country
cursive spruce
Lagging range ?
lyric summit
Okay it’s kinda late here and I’m tired and starting to get kinda confused, can ...
yes ofc
lyric summit
Lagging range ?
i am writing quickly i do parallel thing
cursive spruce
huh
lyric summit
and the other cute methosd is:
cursive spruce
drum roll
lyric summit
lyric summit
I dont' see where the second equation holds
it requiers knowledge about concept of definitie integrals and to be eprcise Riemann sums
so i agree, but i glaldy explian it
cursive spruce
ok
lyric summit
durign fwe motnhs of lessons every day
cursive spruce
what class is that
lyric summit
what class is that
wilbe indiivudal class for oyu )
lyric summit
what class is that
indiivdual classes aer the best 🙂 smiles
cursive spruce
what 😄
lyric summit
what 😄
Zoom or Teams or other way, though i prefer classes face to face, but i asusme you do not live in my country so i can teahc you onyl online haha )
lyric summit
Are you a real prof
if you pay me fo rht elesosn then ic an even show you my diploma :))
cursive spruce
nice, but no prof
lyric summit
but you must be a student of mine
lyric summit
nice, but no prof
i pass you contact mail box on DM if you like )
lyric summit
do you know fourier analysis and signal processing?
ofc i can teach it as well my researchers are about; compelx analysis and partial diffeertnual equations, but hamornic analysis is also in my interestes
lyric summit
do you know fourier analysis and signal processing?
but classe ab Forier cost more than about limits liek here )))
cursive spruce
:d
lyric summit
do you know fourier analysis and signal processing?
i hope you are rich )
lyric summit
you're too expensive
parents of yours are, let them cal me
cursive spruce
Nah my parents are way too poor
lyric summit
Nah my parents are way too poor
soem students tel it the same and enxt they coem to my hosue on Mercedes
so i do not believe ))
cursive spruce
I come to your house with a broken bike and almost die riding it
lyric summit
I come to your house with a broken bike and almost die riding it
ok )) there is a palce fo rbikes as well
lyric summit
I come to your house with a broken bike and almost die riding it
ok anywya thank sa lto for cute conversaiton ands ee you on lessons with me )) wish you al the best i ned to go offine )
cursive spruce
alright thanks joanna for the wisdom
final saddleBOT
@thorn harness Has your question been resolved?
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wanton spindle
what does this have anything to do with the largest odd divisors
final saddleBOT
what does this have anything to do with the largest odd divisors
scarlet sequoia
the point is that once you look at a without its odd divisors, then it's just a power of 2.
So if two numbers share the same odd divisors, then they're bound to divide one another (because you always have 2^m|2^n or 2^n|2^m)
wanton spindle
cud u explain the second method
scarlet sequoia
well the second method is just trying to pair numbers together to use pigeonhole principle
you have a total of 15 elements
you want to prove 9 is enough
to find 8 pigeonholes, 7 pairs and one standalone element should do the trick
wanton spindle
so If I make 8 pigeonholes it automatically means 9 is enough
scarlet sequoia
yes
wanton spindle
bet
ionic swift
yo
I have a question
Name each polynomial by its degree and number of terms
3gn⁷ - 8d⁶ - 7y⁴s³ - 6z⁷k³ + 9d⁵y4
That's the question
final saddleBOT
@wanton spindle Has your question been resolved?
next pagoda
idle fractal
ok Imma leave this channel
final saddleBOT
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scarlet sequoia
<@&268886789983436800>
brisk charm
yup
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indigo cobalt
Im stuck on the last problem on what I need to do, I keep looking and I havent found anything
final saddleBOT
Im stuck on the last problem on what I need to do, I keep looking and I havent f...
indigo cobalt
I tried uusing a line and picked two points for slope to find the answer but it didnt help
<@&286206848099549185>
lapis warren
Sorry, could you give me a clearer image of the question or type it? i dont have my glasses with me
lapis warren
that’s perfect, ill be back with an answer soon
digital steeple
having only some points of the graph does not allow to calculate the instantaneous velocity at a given point. you could only estimate it - with different methods and therefore different results.
lapis warren
you got it right before i did haha
indigo cobalt
what are some methods? to estimater
digital steeple
what are some methods? to estimater
you could take the average velocity in [2,4] as estimater. you could fit a (maybe) qudratic funtion to the given points and then calculate it.
indigo cobalt
omg
the answer was right infront of me
i was overcomplicating it for myselfg
tysm
digital steeple
the answer was right infront of me
and the answer was?
digital steeple
ok, so you had to use the average velocity for the instananeous valocity.
pure cypress
final saddleBOT
pure cypress
need help with these 3 questions
rapid sky
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
rapid sky
OK, so the function is called f.
So, you have f(x) = 5/(x - 2).
Does that make sense how I got that from the question?
pure cypress
yes
rapid sky
OK, do you know how to find f(12) from there?
pure cypress
you plug 12 into x
rapid sky
Right, so what do you get for it?
pure cypress
f(12)=5/10
rapid sky
OK, now reduce the fraction.
pure cypress
1/2
pure cypress
no.
rapid sky
OK, so try to find f(2).
rapid sky
No.
pure cypress
so undefined
rapid sky
It's just undefined. If it has a number for it, that means it's defined.
Since it's undefined, it can't have a number for it, so it can't be 0.
Does that make sense?
pure cypress
yes
rapid sky
OK, so why is it undefined?
pure cypress
because theres no number for it
rapid sky
Right, but why is there no number for it?
pure cypress
because 5/0 is undefined
rapid sky
OK, but why?
pure cypress
because anything divided by 0 is nothing
rapid sky
Right, so the answer to part b is that it's undefined because you can't divide by zero.
Does that make sense?
pure cypress
yes
rapid sky
OK, now do you know what a domain is?
pure cypress
no
rapid sky
OK, a domain is all the values that work when you plug them in for x.
Like you plugged in 12, and that gave you 1/2.
So, 12 worked.
But 2 didn't because it was undefined.
So, the domain will have 12 in it, but it won't have 2 in it.
Does that make sense?
pure cypress
so if the number you plug in isnt undefined it can be put in the domain
after reducing it
pure cypress
3
pure cypress
17
rapid sky
OK, yes, 17 would work because f(17) = 1/3.
But there are a lot of numbers to check if you do them one by one.
So, what you want to do is to look at how 5/(x - 2) can be undefined.
We saw that it's undefined for 2, right?
Is there anything else it's undefined for?
pure cypress
no
rapid sky
Why not?
pure cypress
nothing else minus 2 equals 0 besides 2
pure cypress
yes
rapid sky
So, the domain is every real number except 2.
rapid sky
You're welcome.
pure cypress
.close
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fiery bronze
• On what circumstances am I required to 'flip' the denominator and numerator of the sine law formula?
• to find angle, do I use the sinA/a formula or the a/sinA formula?
• vice versa. To find side, do I use the sinA/a or the a/sinA?
final saddleBOT
• On what circumstances am I required to 'flip' the denominator and numerator of...
hybrid heath
• On what circumstances am I required to 'flip' the denominator and numerator of...
The ratio by itself does little to help you solve anything. You need the rest of the equation
fiery bronze
The ratio by itself does little to help you solve anything. You need the rest of...
I know, it's just an example of the entire formula. My questions also applied to the rest of the formula (sinB/b and sinC/c)
So
• On what circumstances am I required to 'flip' the denominator and numerator of the sine law formula?
• to find angle, do I use the sinX/x formula or the x/sinX formula?
• vice versa. To find side, do I use the sinX/x or the x/sinX?
hybrid heath
At the core, you have $a/b=c/d$. You need to know how to solve for any of the four variables.
soft zealotBOT
SWR
fiery bronze
At the core, you have $a/b=c/d$. You need to know how to solve for any of the f...
Okay
So in this case, I have to find the angle of A first before doing the side length of c.
So do I use sinX/x or x/sinX?
which order do I have to use to find angle and side?
Correction : turns out I don't have to find c. And I just need to find angle A. My question remains though. Do I use sinX/x or x/sinX?
final saddleBOT
@fiery bronze Has your question been resolved?
final saddleBOT
@fiery bronze Has your question been resolved?
limber isle
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limber isle
why is this correct?
fallen comet
yes
ur not expanding x and 4 only
its (x+4)(x+4)
its not just x^2+4^2
u have to multiply 4 by x, and x by 4 as well
distribute across every term
lean magnet
remember
fallen comet
did they ever tell you FOIL
lean magnet
(a+b)^2 = a^2 + 2ab + b^2
fallen comet
first, outer, inner, last
final saddleBOT
@limber isle Has your question been resolved?
limber isle
<@&286206848099549185>
hasty pollen
Well it can’t be A because it has roots
And it definitely can’t be C because that’s not how parabolas work
limber isle
i actually figured that one out lol
now im stuck on this one
bc tbh idk what a perfect square trinomial
is
hasty pollen
now im stuck on this one
alrifht
Ohhhh
Uh
No idea actually
What this means
Probably just means perfect square
feral sphinx
now im stuck on this one
A trinomial is where there are 3 non zero terms present
final saddleBOT
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ruby lake
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ruby lake
i know that the normal vector of s surface for cartesian coordinates is found this way
but how did we even get this formula
i know that N(x,y) is -dg/dx, -dg/dz, 1
but still i don't understand how we got this
soft zealotBOT
spq_64_t
wise moon
help
hollow wolf
final saddleBOT
warm python
IBP probably
hollow wolf
IBP? What is that?
warm python
Integration by parts
warm python
Alternatively first simplify the expression
hollow wolf
warm python
,rotate
soft zealotBOT
soft zealotBOT
Why am. I here
warm python
in the denominator
hollow wolf
warm python
wait, not $x^2cos^2(x)$ just $xcos^2(x)$
soft zealotBOT
Why am. I here
hollow wolf
I see
warm python
You'll then obtain $\int \frac{x}{xcos^2(x)+2}$
soft zealotBOT
Why am. I here
warm python
I doubt it, that would be really nasty
Or you know what
it may work
,w differentiate xsec(x)+cos(x)
soft zealotBOT
warm python
hmm , so a direct u sub isn't of much use
try x=arccos(y)
ok, ykw, I'm not too sure of how to solve this
sorry
are you sure the problem is right
integral calculator.net says there's no elemenatry solution for this
cold gorge
,w d/dx(tan(x) - xsec(x)/(xsin(x) + cos(x))
soft zealotBOT
cold gorge
It's not the same, is it?
warm python
@hollow wolf
warm python
, w integrate (x/(xsec(x)+cos(x)))^2
soft zealotBOT
Requested by physicsrocks
final saddleBOT
@hollow wolf Has your question been resolved?
hollow wolf
cold gorge
<@816521583564030013> Has your question been resolved?
Would you like your work checked or something?
cold gorge
yeah
ruby lake
final saddleBOT
ruby lake
Where did u get that from ??
delta = e^(x-y),e^(x-y) -ye^(x-y)
smoky plover
Oh u mean the gradient ?
final saddleBOT
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dusty mesa
I am trying to understand this better. I think I understand the concepts behind it but I'm unsure as to why we need subsequences to negate the statement
final saddleBOT
I am trying to understand this better. I think I understand the concepts behind ...
dusty mesa
the negation going from there exists to for all N makes sense, and filling the =< epislon to > epsilon makes sense. I know that if for all N there is n>= N such that |x_n - x| > epsilon and then from that all the subsequences of x_n also have |x_n_k - x| > epsilon. my question is. why do we NEED this property to negate x_n -> x. just seems like we have done this without introducing subsequences
is it as simple as just making it easier for further analysis or is there something im missing?
as in the two statements are equal but with subsequences future results are easier to find?
soft zealotBOT
spq_64_t
dusty mesa
Im currently trying to understand sequential continuity and epsilon delta continuity definitions being equal and explaining the proof to myself. this part of the proof references the first screenshot i sent. I'm still puzzled for the use of subsequences in the proof. Surely I can just negate the statement, not worry about subsequences and then say " for any delta and any n => 1, i have |x_n - x|< delta BUT |f(x_n)-f(x)| => epsilon which leads to a condradiction?"
also how do you type that format haha
sleek hornet
Hello there
final saddleBOT
Hello there
sleek hornet
I got a question
fallow wren
royal gust
No!
hazy veldt
sleek hornet
Because we are missing something?
hybrid heath
Determine f '(x0) when f(x) = 34 og x0 = –2
og?
royal gust
Yes!
hybrid heath
because there is no specific function?
Is that a language thing?
what language is that?
crystal birch
og 🤨
sleek hornet
differential calculus
hybrid heath
differential calculus
What spoken cultural language is this?
sleek hornet
I have no idea what your asking 🤣 haha
hybrid heath
do you speak english normally?
sleek hornet
no
hazy veldt
what does og mean?
hybrid heath
what language do you speak?
sleek hornet
what does og mean?
same
crystal birch
what does og mean?
Old gangster !
hybrid heath
because there is no specific function?
English translation: $f(x)=34$ and $x_0=-2$
soft zealotBOT
SWR
hybrid heath
"og" is Danish for "and"
hazy veldt
oh ok
hybrid heath
@sleek hornet yes this is solvable.
hybrid heath
Do you know the derivative of a constant?
hazy veldt
yes
sleek hornet
But if you have no x value
or
is x0 the x value
sooo
were looking at
f'(x)= -2?
cus if u take this right?
U do 3x*2=6+4 right?
= 10
so f'(x)=10
welp :3
anyone? :3
final saddleBOT
@sleek hornet Has your question been resolved?
gentle aspen
anyone? :3
f' will be 3 at all points differentiation of 3x =3 and 4 is constant so 0.
final saddleBOT
@sleek hornet Has your question been resolved?
sage siren
final saddleBOT
https://cdn.discordapp.com/attachments/486841106298961930/1204896471078674483/IM...
sage siren
right
im going through my old hw and i seem to have gotten a different answer
(my handwriting is bad) but my final answer was 2/5ln(|sec(5x^1/2|)
or is this just a situation of i dont remember half the rules of logarithms
and since sec is 1/cos
you can take the ^-1 and put the - outside
?
idk
im just gonna assume its that
.close
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ember sable
can anyone explain why its -1/2cos2x? where does 1/2 come from
final saddleBOT
can anyone explain why its -1/2cos2x? where does 1/2 come from
sonic crystal
u-sub: u = 2x
ember sable
so then du = 2? but i dont understand the next step
final saddleBOT
@ember sable Has your question been resolved?
final saddleBOT
@ember sable Has your question been resolved?
high laurel
so then du = 2? but i dont understand the next step
close, du = 2 dx
generally, you solve for dx so you can substitute afterwards
$u = 2x$
$du = 2dx$
$\frac{1}{2}du=dx$
soft zealotBOT
gerald
high laurel
then substitute dx into the integral
ember sable
then substitute dx into the integral
i see ty
ember sable
.close
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slender pewter
Beg
cunning abyss
help🙏
slender pewter
LOL
cunning abyss
fuck
slender pewter
go to #❓how-to-get-help
cunning abyss
mf trolled me 🤓
final saddleBOT
slender pewter
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tranquil pine
final saddleBOT
tranquil pine
trying to simplify this a bit but
I think im cnofusing myself
I'm letting sqrt(x)=a
but what would sqrt(x^3) be in respect to a?
warm python
$(x^{1/2})^3$
frigid hawk
$(x^{1/2})^3$
warm ether
use {}
warm ether
$\left(x^{\frac{1}{2}}\right)^3$
soft zealotBOT
AℤØ
tranquil pine
so a^3 right?
warm ether
si
tranquil pine
oh okay, was tripping myself out abit because I kept thinking to myself that the cube was inside the sqrt and that would've meant it's not actually ^3 more than a
lol
thanks for clarifying though
.close
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hollow rune
final saddleBOT
hollow rune
i need help with stat questions, i have the answers but I am not sure how my professor got them
final saddleBOT
@hollow rune Has your question been resolved?
hollow rune
<@&268886789983436800>
brisk charm
Please don't ping Moderators for math help, you can ping Helpers since your question wasn't answered for 15 minutes though
hollow rune
@scenic moth
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gray trench
Not really sure where to start with this one? Do I use the formula for a parametric curve or one for arc length here?
final saddleBOT
Not really sure where to start with this one? Do I use the formula for a paramet...
silver maple
I am finding it hard to read your writing
gray trench
I am finding it hard to read your writing
Sorry, let me try and type it out then
The first one is integral of a to b of the square root of 1 + dy/dx squared
The second one is the integral of t0 to t1 of the square root of dx/dt squared plus dy/dt
silver maple
I am not sure what you are trying to do they are asking for volume. For this question we can use cylindrical shells
gray trench
I am not sure what you are trying to do they are asking for volume. For this que...
So 2pi r h dx?
But what would be the integral then?
gray trench
Um, I don't see an x value listed here- where would I derive it from? Sorry, I find it hard to see these connections.
r is just the distance between something and something, right?
And would the integral be from -2 to 2?
lean magnet
Um, I don't see an x value listed here- where would I derive it from? Sorry, I f...
no you dont need a specific x value
the radius is just x
gray trench
Becaause it's at y=0, right?
lean magnet
yea
generally the radius is always x
the height is the y value
which in this case is just the function
as it is bounded on the bottom by the x axis
gray trench
Alright, I'll be sending my work so far in a sec just to double check my steps
lean magnet
gotchu
gray trench
lean magnet
looks good
gray trench
Ok, lemme try an difnishit it off
wow
My brainpower has officially been all diverted to math
Alright gimme a sec
lean magnet
How does this look
mb i had to go for a while
yea you should be good
unless im missing something
gray trench
unless im missing something
Could you check my set up for this problem?
turbid sable
final saddleBOT
warm ether
3