and then Δy = tanφ Δx
#help-39
meager scroll
you get Δx = 2uo /g (-cos(2φ)sin(2φ) - sin^2 ( 2φ) tanφ)
and Δx/ΔS = cosφ
oh i found the mistake
ΔS = Δx/cosφ right?
i think i did ΔS = Δx*cosφ
uneven cypress
Listen, I am not a machine to follow blunt mathematical steps. I would ideally like you to reason it so I can verify it. Otherwise, wait some time for someone to help you out. I do not even know what Delta S is. Youre going a bit fast.
meager scroll
Listen, I am not a machine to follow blunt mathematical steps. I would ideally l...
i am reasoning everything
uneven cypress
This would require me to work through the problem myself which I am obv lazy to do
uneven cypress
i am reasoning everything
Not really
uneven cypress
Again, I encourage you to try a transformation of axes. It simplifies your mathematics dealing with a flat plane
meager scroll
Again, I encourage you to try a transformation of axes. It simplifies your mathe...
but it has accelartion on both axis
uneven cypress
but it has accelartion on both axis
Yeah
uneven cypress
Why 2 phi?
uneven cypress
idk how its simpler?
Because then you start and end at y' = 0
And you have to find the x' displacement
meager scroll
ah yea
uneven cypress
Also i dont think it should be 2 phi. Please explain
meager scroll
it shouldnt nvm
uneven cypress
How did you come to it then?
meager scroll
the new axis you mean the one that splits the angle 2φ of height and the new velocity direction
uneven cypress
Why 2phi?
meager scroll
2φ is the angle between velocity and height vector
i found 8h sinφ
finally ffs
anyways thanks
.close
pearl pondBOT
abstract pagoda
Bro i was think that the case n=0 is enough
Since our goal is finding some n in N
Showing that H=0Z
Is enough i think
In logic
Forget that
I just didn't sleep much
pearl pondBOT
Available help channel!
wispy bobcat
isnt the formula just base x h. and the base of a hexagon is 3ba (base and apothem)
toxic lichen
that's not a regular hexagon, is it?
based on the picture, i'd imagine the hexagonal base of this prism as being two trapezoids stuck together, with long base 6m, short base 2m, and height 1m for both.
though it is kinda confusing the way they've drawn it, imo.
drifting vale
isnt the formula just base x h. and the base of a hexagon is 3ba (base and apoth...
The base consists of 2 trapezoids of area 4 m²
wispy bobcat
does that mean its ireegular?
just to confirm the trapezoid formula is its too (paralell sides added together, multiplied by its height)/2?
drifting vale
Yes
drifting vale
does that mean its ireegular?
It's not exactly irregular in this case. It has parallel opposite sides which results in convenient symmetry
Technically not regular
These are called parallelogons
pearl pondBOT
wispy bobcat
.reopen
pearl pondBOT
✅ Original question: #help-39 message
pearl pondBOT
pearl pondBOT
Available help channel!
low matrix
pearl pondBOT
low matrix
The ones with the (-1) are always gonna be diverging right
cuz that will make it an alternating sequence
sharp smelt
The ones with the (-1) are always gonna be diverging right
no
low matrix
but it is an alternting seq
sharp smelt
consider $a_n= \frac{ (-1)^n}{n}$
drifting vale
Use the n-th term test for divergence
jolly parrotBOT
wai
sharp smelt
There's a theorm : A seqeunce converges to 0 iff it absolutely converges to 0
low matrix
There's a theorm : A seqeunce converges to 0 iff it absolutely converges to 0
well this one does absolutely converge
low matrix
not to 0 though
1/ inf is zero
sharp smelt
oh seqeunce
low matrix
we hav to apply the lim right
sharp smelt
my bad 😭
sage frigate
The sequence 1/n converges
and that bounds above (-1)^n/n
so this definitely converges to zero
low matrix
Well if I get the limit of the underlying sequence = 0, can I say that the alternating sequence also converges?
sharp smelt
The series however
here it's just the sequence
sage frigate
but not without
naive zinc
There is a thing called alternating series test
low matrix
also converges because of the alternator
did you apply the AST on this
low matrix
There is a thing called alternating series test
yup
low matrix
So it converges by alternating series test
but the questions is about sequence
not series
sage frigate
did you apply the AST on this
I probably should have but I just remember that series converging tbh because it is one of the classic example of the impact of an alternating term.
But I mean it is easy to see that the sequence is monotonic, decreasing, and positive without the alternator
naive zinc
Oh. I thought Σ
Anyway the sequence definitely ->0
sage frigate
Oh. I thought Σ
Anyway the sequence definitely ->0
going to zero as a sequence isnt enough
as 1/n diverges
as a series
but converges as a sequence
naive zinc
He meant the (-1)^n / n itself
sharp smelt
but the questions is about sequence
Okay, 2 theorems you should know
- A seqeunce converges to 0 iff it absolutely converges to 0
- A seqeunce converges to l iff every subseqeunce converges to l
sage frigate
Okay, 2 theorems you should know
1) A seqeunce converges to 0 iff it absolutely...
a sequences converges iff it's cauchy
Which I am pretty sure (without having sat down and thought about it too much) that subsequences pov is equivilent but if you really want to show a sequence converges it is easier to show its cauchy
well
I guess I should be careful and say this only works if you're in R but the sequence in question is in R so it's fine
That is, the cauchy criterion is a necessary and sufficient condition to be a convergent sequence of real numbers.
drifting vale
but it is an alternting seq
In (c) it converges to 0 cuz of the 1/n term approaching 0. But the other one is diverging
low matrix
Okay, 2 theorems you should know
1) A seqeunce converges to 0 iff it absolutely...
idk what subseq are 
we didnt cover that topic in class
Although the first one you stated will be of use
ty
.cloe
low matrix
In (c) it converges to 0 cuz of the 1/n term approaching 0. But the other one is...
yeah
drifting vale
a) converges, b) diverges, c) converges, d) diverges
pearl pondBOT
sharp smelt
idk what subseq are <:sad:532803514892812288>
shoudl I explain real quick?
low matrix
shoudl I explain real quick?
I'm good, I think the first theorem is sufficient to solve the problems in my course
naive zinc
Though b requires you to use what wai said
low matrix
Okay, 2 theorems you should know
1) A seqeunce converges to 0 iff it absolutely...
btw what if it absolutely converges to 1/2 or 2 or 4
naive zinc
(Two subsequences having two different limits)
sharp smelt
btw what if it absolutely converges to 1/2 or 2 or 4
.reopen
pearl pondBOT
✅ Original question: #help-39 message
sharp smelt
btw what if it absolutely converges to 1/2 or 2 or 4
That's where you use subseqeunces
noble path
$\text{sequence: } a_n = \frac{n}{2n+1} \
\lim_{n \to \infty} \frac{n}{2n+1} \cdot \frac{1/n}{1/n} = \lim_{n \to \infty} \frac{1}{2 + \frac{1}{n}} \
\frac{1}{2 + 0} = \frac{1}{2} \implies \text{converges}$
naive zinc
Anyway a sequence is viewed as a function N->R. A subsequence just means f restricted on a infinite subset of N
low matrix
b
noble path
$\text{sequence: } a_n = (-1)^{n+1} \frac{n}{2n+1} \
\lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2} \
\text{subsequences: } a_{2k} \to -\frac{1}{2}, \quad a_{2k-1} \to \frac{1}{2} \
-\frac{1}{2} \neq \frac{1}{2} \implies \text{diverges (oscillates)}$
jolly parrotBOT
notasnugglebugz
noble path
ig?
naive zinc
Yeah, what wai said. a sequence converges to L iff any subsequence of it converges to L
low matrix
cant i just say thatthe magnitude of the terms approaches $\frac{1}{2}$, the terms of the sequence will oscillate back and forth, getting closer and closer to $\frac{1}{2}$ and $-\frac{1}{2}$
jolly parrotBOT
Sean [Ping On Reply Please!]
naive zinc
Doesn’t sound mathematical
low matrix
is this enough proof for divergence
noble path
ye
because the sequence has two different subsequential limits (in this case, 1/2 and -1/2)
naive zinc
I would either use that result I said, or by definition. (|a_2k+1-a_2k|>1/2, so can’t converge
noble path
well we hv this theorem in our book
but 1/2 is not equal to -1/2
naive zinc
Then use it
low matrix
but 1/2 is not equal to -1/2
thats why it diverges
noble path
thats why it diverges
wdym
low matrix
Then use it
well for even n, L = 0.5
for odd n , L = -0.5
noble path
because the 2 subsequential limits are different
noble path
I would either use that result I said, or by definition. (|a_2k+1-a_2k|>1/2, so ...
should be 1
noble path
why?
naive zinc
I am out
noble path
ok
low matrix
.close
delicate abyss
Is there an easy way to proof this ?
pearl pondBOT
Is there an easy way to proof this ?
delicate abyss
i tried using eulers Form but it gets way to messsy
compact ridge
using the formula for tanh(a + b)
toxic lichen
i tried using eulers Form but it gets way to messsy
euler's form?
delicate abyss
using the formula for tanh(a + b)
I prevent to use it since it will be something i have to remember for an exam but i think its the best way
agile hamlet
.help
pearl pondBOT
Commands:
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.close,.reopen - factoids:
.tag - help:
.help - pins:
.pin,.unpin - reminders:
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Type .help <command name> for more info on a command.
pearl pondBOT
Need help with the approach to this question
agile hamlet
not how this command is used btw
sorry, its like the first time im using this for help
toxic lichen
opening a channel is done by just sending any message in it
except if it begins with a . then that prevents the channel from getting opened
.reopen is for when the channel already had your name on it and you .closed it but want to have it again
agile hamlet
okay got it
toxic lichen
anyway, the fraction appears to be equal to $\frac{g'(\tfrac{2n+1}{2})}{g(\tfrac{2n+1}{2})} - \frac{g'(\tfrac12)}{g(\tfrac12)}$
jolly parrotBOT
Ann
agile hamlet
yeah
toxic lichen
g'/g is the derivative of ln(g), which is called f in your question
agile hamlet
the g(x+1) = xg(x) is like x!g(0) right
toxic lichen
it does mean g(x) = (x-1)! * g(0) for integer x, but you can't do much more with that.
agile hamlet
i changed it to this kind of form
i'll take a pic and send what i did
heady herald
i wants to be pro at doing math
toxic lichen
can anyone help me?
hello and welcome to the server. you've stumbled into an occupied help channel.
if you have a math question to ask, please open your own help channel -- see #❓how-to-get-help for instructions on how to do that.
if you just want to chat, please go to #discussion or #chill.
heady herald
ok
agile hamlet
hello and welcome to the server. you've stumbled into an occupied help channel.
...
ann help pls
toxic lichen
your handwriting is hard to read & i can't tell if that letter is x, n, u or something else
agile hamlet
its all x
toxic lichen
doesn't look convincingly like x
you might want to work on your handwriting in the meantime
but it looks like you got to g'(x+1)/g(x+1) - g'(x)/g(x) = 1/x
or to put it in more compact terms, f'(x+1) - f'(x) = 1/x
now you want to sum that for x going from 1/2 to (2n-1)/2 in steps of 1
and that'll get you your target expression
pearl pondBOT
pearl pondBOT
Available help channel!
low matrix
pearl pondBOT
low matrix
Does exact values of first four partial sum mean that I have to add the first four values?
tulip marsh
fairly certain it means to add the first four terms. so, for a), it would be adding 2, 2/5, and the next 2 terms...
modern talon
Does _exact values of first four partial sum_ mean that I have to add the first ...
first four partial sums means to calculate s_1 = a_1, s_2 = a_1 + a_2, and so on until s_4
low matrix
fairly certain it means to add the first four terms. so, for a), it would be add...
For (c), what does closed form mean?
Do I have to find partial fractions of the general term, and then add all the terms upto n to get an expression for the sum?
tulip marsh
'A closed form is an expression that can be computed by applying a fixed number of familiar operations to the arguments.' Is the definition I seem to get. So, yes, you need to find a general formula for S_n
managed to find the formula for a)?
c) has a little trick you'll need to observe... about fractions of that form
low matrix
c) has a little trick you'll need to observe... about fractions of that form
yes
I did it, sum is 0.5
tulip marsh
Maybe. I thought you'll say the formula for S_n first
low matrix
Maybe. I thought you'll say the formula for S_n first
it is 1/k+1 - 1/k+2
tulip marsh
I did it, sum is 0.5
You're correct, ig. the sum seems to converge to 1/2
low matrix
ty!
low matrix
You're welcome!
btw can i separate the sum into TWO series and then proceed by inndividually finding the two sums?
Like if i have 1/k-1 and 1/k+2 then i can find their sum separately
low matrix
it is 1/k+1 - 1/k+2
and then subtract them to get this
tulip marsh
Like if i have 1/k-1 and 1/k+2 then i can find their sum separately
most terms should cancel out, if we talk about c)
you should be left with just 2 of them
low matrix
you should be left with just 2 of them
i'm just asking if i can do it the other way
low matrix
btw can i separate the sum into TWO series and then proceed by inndividually fin...
this way
tulip marsh
If both converge, yes.
Otherwise, definitely not.
Oh, you mean for the general formulas...sure. You can group the terms however you want
my previous comment is only about proving it converges and to what it converges
sharp smelt
I don't get the part after we define the common refinement
pearl pondBOT
I don't get the part after we define the common refinement
sharp smelt
oh, we're just adding the first eqn to the additive inverse of the 2nd
got it
.clos
.close
pearl pondBOT
pearl pondBOT
Available help channel!
simple elbow
pearl pondBOT
patent blade
mhm, and where are you stuck?
cause you have the formula for the function
2x meaning for 1 x to the right[or left] it goes 2 up
and the -8 decides your starting point
from there forwards I believe you can solve this no?
@simple elbow
untold valve
Hey simon your a mod right
patent blade
no?
simple elbow
The whole thing
untold valve
i mean a helper
patent blade
ah ye
simple elbow
I’m not too familiar with graphical method
patent blade
mhm
simple elbow
It’s as if I never learned it before
patent blade
and where does it go wrong?
simple elbow
If you get what I mean
patent blade
you don't know how to visualise the function?
untold valve
okay it seems like you got this but @simple elbow what are you going wrong with and you can ask @patent blade that bye!
patent blade
Thank you Pianoman[wonderful song btw]
Alright so @simple elbow do you know who a function f(x) = x looks like?
simple elbow
Alright so <@851436668002369568> do you know who a function f(x) = x looks like?
Hmmm
simple elbow
Alright so <@851436668002369568> do you know who a function f(x) = x looks like?
Kind a
y=x simplified am I right?
patent blade
I, guess. f(x) and y are the exact same. But in some sense, yeah, simplified
simple elbow
Sooo it’s kinda a slanted line across the graph
patent blade
yes
and 9/10 when you need to draw it
y=x is 45degrees from y or x
exactly like this
simple elbow
Yes I understand that part
patent blade
So what do you think happens when you multiply that x by 2
proper nova
so $\abs{2x - 8}$ is just basically $y$ in your graph
jolly parrotBOT
1 divided by 0 equals Infinity
patent blade
yup
proper nova
you are trying to find the $x$ values such that it's above the $y = 4$ line
jolly parrotBOT
1 divided by 0 equals Infinity
simple elbow
you are trying to find the $x$ values such that it's above the $y = 4$ line
Uhhh sorry I’m tryna understand how graphical method works can you not like get straight to the answer? Sorry bout that 😅
proper nova
Uhhh sorry I’m tryna understand how graphical method works can you not like get ...
that's the dang method
simple elbow
Please be patient with me I’m kind of a handful
proper nova
Please be patient with me I’m kind of a handful
np
simple elbow
Thanks
proper nova
draw a dotted line at $y = 4$
jolly parrotBOT
1 divided by 0 equals Infinity
simple elbow
that's the dang method
Dang method
proper nova
Dang method
...
proper nova
Dang method
hold up Dang is not the name
patent blade
@proper nova I gotta go so I'm gonna let you handle it :D
proper nova
Why? As if it’s the asymptote?
borderline
simple elbow
Okok
proper nova
it's more like a borderline drawn like an asymptote
simple elbow
Yes
proper nova
so the part above the line is when $y > 4$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
and the part below the line is when $y < 4$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
some part intersects the line is when $y = 4$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
What’s next
and since your $\abs{2x - 8} > 4$, the solution is basically all values of $x$ coordinates above the $y = 4$ line
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
Bruh
you would've figured by now if you don't rely on my explanation to copy your answer
simple elbow
I just draw the curve
simple elbow
you would've figured by now if you don't rely on my explanation to copy your ans...
Negative numbers I just make it positive
x=256 I don’t know what went wrong
proper nova
this part is wrong
proper nova
x=256 I don’t know what went wrong
a better approach is to turn your $4\log_2 x^2$ into $\log_2 (x^2)^4$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
and THEN apply the rule
proper nova
this part
in your draft
leads to the whole answer being wrong
$(a + b)^2 = a^2 + 2ab + b^2$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
however you only had $ab$ in your expansion
jolly parrotBOT
1 divided by 0 equals Infinity
pearl pondBOT
@simple elbow Has your question been resolved?
simple elbow
pearl pondBOT
simple elbow
I can’t seem to solve the angle part
uneven cypress
The derivative of a function gives the tangent of the angle it makes with the x axis counter clockwise @simple elbow
pearl pondBOT
@simple elbow Has your question been resolved?
simple elbow
The derivative of a function gives the tangent of the angle it makes with the x ...
Ok
simple elbow
How to intergrate natural log?
pearl pondBOT
How to intergrate natural log?
simple elbow
<@&286206848099549185>
crystal dew
!15m please.
pearl pondBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
shut elm
In case no one comes by, this is something there's a plethora of youtube videos on 👍
brave sluice
you can write ln x = 1 * ln x, and do integration by parts
blazing garden
How to intergrate natural log?
Following what Axe was saying, then you would chose:
u = ln(x)
dv = 1dx
And then go from there. Good luck
blazing garden
We treat ln x as a constant?
No. Do you know what integration by parts is?
simple elbow
No
I only know the power number add one then divide it
Then divide the differentiation
😅
blazing garden
I only know the power number add one then divide it
Ok, the inverse power rule. Do you know the product rule for derivatives?
blazing garden
Yes
So, much like derivatives, integrals are not separable by multiplication.
Integration by parts is kind of like the product rule of integration
toxic lichen
No
then you have to go learn integration by parts, bc it is necessary to integrate ln(x)
blazing garden
Integration by parts is kind of like the product rule of integration
It comes from actually integrating the product rule
simple elbow
then you have to go learn integration by parts, bc it is necessary to integrate ...
Aight
simple elbow
It comes from actually integrating the product rule
Okok
blazing garden
Okok
Do you have any other questions before I go to sleep? You too @compact ridge, since I saw you reacted to my message
compact ridge
No I'm just lurking to see if I could help
blazing garden
No I'm just lurking to see if I could help
I love helping too. I started doing Calc in Grade 7, and tutoring uni students since grade 8. That’s when I noticed my love of teaching others.
pearl pondBOT
@simple elbow Has your question been resolved?
crimson nebula
How to prove,
Every finite set has a maximum element.
here elements are real numbers
pearl pondBOT
How to prove,
Every finite set has a maximum element.
here elements are real...
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
Don't know where to begin.
I had an idea :
to define a set φ such that
I will fill elements from Original set to this ,
Comparing each time with previous element of φ
sullen cedar
Induct on the number of elements in the set.
toxic lichen
How to prove,
Every finite set has a maximum element.
here elements are real...
did you mean every finite nonempty set has a maximum element?
crimson nebula
did you mean every finite **nonempty** set has a maximum element?
Yes
crimson nebula
Induct on the number of elements in the set.
Can I do it by creating some other set
Somehow filling elements in it , in such a way , That finding maximum in this new set is easy
toxic lichen
you dont need to create any other set
induction is basically your only option here, one way or another
crimson nebula
I know there's no need
I was just asking , of something like this can be done or not
dusky widget
Try it and we will see!
crimson nebula
I some sense
Couldn't Precisely define it mathematically
My idea was to first define a function φ
Such that φ (1)= first element of original set
Than I went from one element of original set to next and compare with φ(1) ,
If it's larger , define φ(2)= that element
Ana so on , till I have reached end of original set
pearl pondBOT
@crimson nebula Has your question been resolved?
autumn fossil
Is there a continuous function f : R^2 -> R, s.t. when we graph it and intersect it with planes x = c, we can get every 2D (R -> R) continuous function as a result?
autumn fossil
it would be trivial if it didn't need to be continuous. There are 2^|N| continuous functions R -> R and 2^|N| intersecting planes, so we could just form a bijection between the 2 and construct some messy function
cinder flower
fascinating question
autumn fossil
indeed, its been bothering me for few days at this point
ivory swallow
I would say no but all I can offer is vibes
Um you don't really have an order on function space
What makes two functions close?
Close together? As in is there a natural order?
autumn fossil
Does that matter? I dont see how closeness is related
ivory swallow
Closeness matters for f being continuous
autumn fossil
okay i think i see your point
sup(|g(x) - h(x)|) could work, but it allows infinite distance
and its probably unnecessarily strong too
ivory swallow
So like let's say you get some function at the cut x = c. What happens at x = c +- δ?
autumn fossil
So like let's say you get some function at the cut x = c. What happens at x = c ...
I actually think we cant say much, can we? I don't think it'd need to be "uniformly" continuous, if you're implying that the function would differ only by some epsilon
autumn fossil
I don't really get what you mean, sorry
ivory swallow
Yeah let's make this more concrete.
Lets say that at x = c you get some function g. Then at say, x = c + δ, you should have a small change in the outcome space. But what is "g + ε"?
autumn fossil
What matters is the continuity of f, which should be well defined (simply continuity of multivar function), i dont see how being able to define "g + epsilon" is required for existence of f
if we were able to define it, it would probably make the construction easier, but i dont see how not being able to define it shows that f cant exist
ivory swallow
Which is exactly the issue.
z = f(x,y), and you want g(y) = f(c,y).
So as an example:
z = xy. Fixing x, we get every linear function of z = cy passing through the origin.
So you run into the issue of closeness and enumeration. How can you parameterize all of function space? What is your path?
autumn fossil
idk, thats what makes the question difficult for me
i cant see a path, but i dont see why it cant exist
oh wait
z = f(x,y), and you want g(y) = f(c,y).
g(y) = f(y, y) + 1
diagonalization
this function surely cant be there
but thats weird, it didnt even require continuity
oh continuity is required to make g continuous
what if we focused only on some compact domain?
e.g. closed interval
[0, 1] lets say
[0, 1] x [0, 1] -> [0, 1]
then everything would be uniformly continuous
ivory swallow
I don't think that changes the problem. At least to me it seems to be one of enumeration. Like a space filling curve. R is self similar so it might kind of work, but R^n has no order except for the partial order, so you'll have discontinuous lexicographic jumps. So I'd say you'd need stronger constraint than uniform continuity or compact domain. You need a way to collapse dimension into a single scalar. Which only seems natural and obvious for any family of functions with only a single parameter.
autumn fossil
well, space filling curves are continuous and exist for R^n
but it's probably gonna be similar, it shouldnt change the problem that much
it just feels like there are "less" functions on the compact space, so it might be possible. And the diagonalization argument wouldnt work either
worldly jewel
If such an f exists, you have a continuous surjection f(–, y) from R to C(R, R) with the pointwise convergence topology. I think there should be a few ways to see such a surjection cannot exist
autumn fossil
If such an f exists, you have a continuous surjection f(–, y) from R to C(R, R) ...
I see, and would it exist if R was replaced with [0, 1]?
rustic gate
[0,1] -> won't work because the target isn't compact
autumn fossil
[0,1] -> won't work because the target isn't compact
As in C([0, 1], [0, 1]) isn't compact?
I'll probably need to learn more topology for this
rustic gate
yeah
feral sedge
it's a metric space
so it's sufficient to find a sequence that doesn't have a convergent subsequence, ex, sin(nx)/2 + 1/2 I think
rustic gate
x^n
rustic gate
feral sedge
But obviously a limit of continuous functions would have to be continuous
autumn fossil
x^n
So one (actually more than one)of those functions can't be in f, because they dont converge uniformly?
It kinda makes sense
rustic gate
no C([0,1], [0,1]) just isn't compact
feral sedge
thats probably true too though
feral sedge
It needs a compact image, which means it can't contain that infinite sequence of functions
so some x^n must not be representable
autumn fossil
no C([0,1], [0,1]) just isn't compact
oh right and its not compact because its not closed
makes sense
rustic gate
oh wait hold on
bolzano weierstrass probably forces this
some x^n won't be there yeah
mfw compact subsets that aren't closed are a thing
autumn fossil
mfw compact subsets that aren't closed are a thing
There are compact sets which arent closed?
feral sedge
In non-hausdorff topologies, yes
autumn fossil
Sounds weird
feral sedge
Ofc, C[0,1] is very hausdorff
autumn fossil
If such an f exists, you have a continuous surjection f(–, y) from R to C(R, R) ...
Btw is there something I could replace R with to make it true?
Something non-trivial
rustic gate
which R
autumn fossil
any R
rustic gate
like X -> C(R, R)?
rustic gate
that one won't work either
rustic gate
autumn fossil
Replace first R with C(R,R)
okay that's what I meant by trivial lol
rustic gate
R^n -> C(R, R) won't work for any finite n
autumn fossil
so ig it won't work in most of the non-trivial case
ivory swallow
Thanks everyone <:catlove:920698762932920370>
Bring more problems like this, that was fun
pearl pondBOT
Bring more problems like this, that was fun
pearl pondBOT
pearl pondBOT
Available help channel!
low matrix
pearl pondBOT
jolly parrotBOT
latent quail
What’s the problem?
latent quail
No, I mean which part do you need help with
Differentiation, identification or others
low matrix
Differentiation, identification or others
diff
latent quail
kk, do you know the formula?
low matrix
yea quotient rue
feral olive
that is a VERY long exercise
lilac ocean
good luck
feral olive
yeah just differentiate the functions from inside to outside and seam it all together ig
latent quail
low matrix
yeah just differentiate the functions from inside to outside and seam it all tog...
did i do it write
doesnt seem to agree with wolfram 
low matrix
why is there a cube here
lilac ocean
why is there a cube here
did you forget to square your denominator by chance
low matrix
did you forget to square your denominator by chance
yea i did
but how does that change the numerator?
i got a 5 in there
low matrix
damn dude i dont feel like doin this
lilac ocean
finish the question and then take a break maybe
also remember that you dont have to do every single question from a set
low matrix
also remember that you dont have to do every single question from a set
that means i dont have to do this one lmao
i did all the last ones btw
lilac ocean
in theory yeah but you should at least finish what you started
i recommend doing a couple of questions of every different type at the very least
low matrix
,w 2nd derivative of x^(4/3) - x^(1/3)
jolly parrotBOT
lilac ocean
like 15, 16, 18, 19, 20, 24, 26 are all baically the same so i dont think there is a point in doing all of them
low matrix
part 24
i get the 1st derivative for this as (4x-1 )/ x^2/3
should i use quotient rule or separate the fractions (for computing 2nd derivative)
low matrix
just separate
that gives 4/x^(2/3) + 2/x^(5/3)
low matrix
that gives 4/x^(2/3) + 2/x^(5/3)
How do I find the root of this?
lilac ocean
factor
low matrix
the "rule" is just take whatever makes it look easiest i guess
im asked to find minimum and maximum using ANY method
is there a way to not use calc in this
low matrix
do you know what log graph looks like
well it starts from 0
i mean undef at 0
then it grows without bound
low matrix
so is 0 a minima
lilac ocean
there is actually no minima or maxima since it is unbounded in both directions
lilac ocean
does it?
lilac ocean
yeah
low matrix
nice
lilac ocean
?
did you not mean the minimum value
or were you looking for the x value for which it is minimal
you were a bit unclear so i kinda presumed you just meant minimum value
low matrix
you were a bit unclear so i kinda presumed you just meant minimum value
well the minimum is -1
lilac ocean
what is the smallest value |x| can take
low matrix
what is the smallest value |x| can take
0
lilac ocean
right, at which x value
lilac ocean
yeah exactly
now, 1+cbrt(x) is surjective, meaning it can take any value
so for |1+cbrt(x)| to be minimal, what does 1 + cbrt(x) need to be?
low matrix
0
lilac ocean
and what x value does that occur for
low matrix
-1
lilac ocean
:)
low matrix
wat
lilac ocean
that is all
low matrix
what motivates professors to give this godforsaken concoction in exams
lilac ocean
seeing whether you understand the content
in general, if they dont want you to use calculus, the function is probably not as hard as it seems; just think about the bigger picture and surjectivity
low matrix
cool
now doing these
@lilac ocean Please give me 3-4 questions so that I get all the flavors
lilac ocean
hmmm
lowkey they are all basically the same
so maybe do just like
51, 54, 55, 57, 59
low matrix
alright
low matrix
doing 66
lilac ocean
cos(x) = cos(2pi-x)
unkempt yacht
I get cosx = 1/2, by solving for critical pts
how did you solve for x here?
low matrix
how did you solve for x here?
i set 2cosx - 1 (numerator of f') = 0
low matrix
i mean that once you got cos(x) = 1/2, how do you proceed?
isnt 60 the only ans
unkempt yacht
well your desmos says otherwise
also please use radian, we are working in radians here
low matrix
next repetition would be at pi/3 + 2pi
unkempt yacht
would it?
lilac ocean
lilac ocean
cos(x) = cos(2pi-x)
^ yes
frank violet
<@&268886789983436800>
pearl pondBOT
bronze heath
mr beas
pearl pondBOT
Available help channel!
wary vault
hi there. i tried proving that sup(A+B) = supA + supB. my approach is contradicting the fact that sup(A+B) < supA + supB. Is what ive done correct?
autumn fossil
hi there. i tried proving that sup(A+B) = supA + supB. my approach is contradict...
looks correct, but could probably use some polishing
spiral pivot
For the record, you don't need to do a proof by contradiction, but I don't see an error.
wary vault
For the record, you don't need to do a proof by contradiction, but I don't see a...
oh i see. ill look the proof up now that i did it
thank you
.close
pearl pondBOT
twin portal
What is A?
waxen agate
good job fijo
wary vault
.reopen
pearl pondBOT
✅ Original question: #help-39 message
wary vault
What is A?
its a set
waxen agate
your proof is still very symbol heavy
wary vault
good job fijo
thank you cooly
waxen agate
how stylistic
twin portal
What is A + B?
spiral pivot
If sup(A) is within epsilon/2 of any member of A and similar with B then sup(A+B) is within epsilon
twin portal
A union B?
wary vault
your proof is still very symbol heavy
a prof once told me if you wanna prove something rigourously you use only maths and nothing else
autumn fossil
What is A + B?
its defined within the text, A + B = {a + b | a in A, b in B}
waxen agate
wary vault
A union B?
no, i defined it, look inside the brackets
twin portal
My bad.
wary vault
If sup(A) is within epsilon/2 of any member of A and similar with B then sup(A+B...
oh and thats enough?
autumn fossil
a prof once told me if you wanna prove something rigourously you use only maths ...
only maths or only symbols?
wary vault
only maths or only symbols?
the symbols are maths. theyre a language everyone who does maths understands
waxen agate
a prof once told me if you wanna prove something rigourously you use only maths ...
Did he also mention that it might make others not read your proofs?
autumn fossil
If u look at research papers, >90% of it is written in words (obviously). And that includes the proofs
wary vault
Did he also mention that it might make others not read your proofs?
yeah i try to use language when needing validation lol
wary vault
If u look at research papers, >90% of it is written in words (obviously). And th...
i didnt know that. well im a lazy person and having learnt that i can just use symbols to describe everything ill always tend towards that
waxen agate
indeed I do see some words here which is an improvement
spiral pivot
oh and thats enough?
That proves that sup(A+B) is ≥ sup(A) + sup(B), to prove it is exactly equal consider some value sup(A+B) + δ with δ > 0
wary vault
That proves that sup(A+B) is ≥ sup(A) + sup(B), to prove it is exactly equal con...
we can just use the fact that supA + supB is an upper bound
spiral pivot
Or that
pearl pondBOT
pearl pondBOT
Available help channel!
crimson nebula
My proof for 2.4.5 corollary:-
since t>0 so 1/t >0 , By Archimedean property there exist n ∈N such that 1/t< n which implies
0< 1/n < t
Is it correct?
toxic lichen
seems okay
cursive wraith
My proof for 2.4.5 corollary:-
since t>0 so 1/t >0 , By Archimedean property th...
yeah this is literally what they did with epsilon in the corollary proof above
it looks good
pearl pondBOT
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Available help channel!
untold valve
Who pinged me?
pearl pondBOT
Who pinged me?
bronze heath
Who pinged me?
No one
untold valve
i Got ping here
robust lynx
Who pinged me?
Hi! You have the “Helpers” role, which pings you for outstanding questions that haven’t been answered. You can remove this role by typing ,iamnot helpers.
With that aside, do you have a question?
untold valve
No i got ping here but anyways bye
pearl pondBOT
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Available help channel!
low matrix
How to do (2)
pearl pondBOT
How to do (2)
toxic lichen
shells
toxic lichen
no it's not
since when would it be included
you don't integrate from 0 to 2
you integrate from 0 to wherever the circle and line intersect
low matrix
you integrate from 0 to wherever the circle and line intersect
my professor said so
i asked him
toxic lichen
???
low matrix
YES
toxic lichen
what exactly did you ask him and what exactly did he say
low matrix
what exactly did you ask him and what exactly did he say
I precisely asked him if I had to ALSO include that extra area, he said yeah, that one will be included. (He was struggling with finding a solution so he asked me if I had tried it, I showed him my solution WITHOUT that extra area, then he explained his approach to me and said that mine was incorrect.)
Btw the question was not about shaded region, It was about area from 0 to 2
same equations
toxic lichen
...
well damn ig you have to split it up into the sum of two integrals
from 0 to sqrt(2) and from sqrt(2) to 2
autumn fossil
do you? It's gonna be squared anyway (so the sign doesnt matter)
low matrix
from 0 to sqrt(2) and from sqrt(2) to 2
So I integrate $2\pi x(x-\sqrt{4-x^2})$ ?
jolly parrotBOT
Sean [Ping On Reply Please!]
toxic lichen
yes
toxic lichen
do you? It's gonna be squared anyway (so the sign doesnt matter)
no
y2 - y1 isnt squared here
autumn fossil
y2 - y1 isnt squared here
Oh, right, sorry
toxic lichen
2 pi y( y - y^2 + 2) isnt correct smh
can you show prof feedback again
actually im off to sleep sorry
pearl pondBOT
@low matrix Has your question been resolved?
brave sluice
2 pi y( y - y^2 + 2) isnt correct smh
it seems right
but you need to find the bounds
low matrix
it seems right
from -2 to 2?
brave sluice
no, 0 to 2
low matrix
how?
low matrix
no, 0 to 2
what do the bounds represent btw
brave sluice
the lowest and highest points of the shaded region
(given that we're revolving about the x-axis)
if it's the y-axis then the bounds would be the leftmost and rightmost points
low matrix
the lowest and highest points of the shaded region
got it
canu help with this btw
how to express the middle part
im thinking integral of x/3 from 0 to the intersection point, and add that to integral of the curve from 0 to that point
brave sluice
no, because the integral calculates signed area
it should work if you multiply the integral of the curve by -1
low matrix
it should work if you multiply the integral of the curve by -1
does the integral of the curve from 0 to 1.732 give me this area?
brave sluice
it gives you -1 times that area
low matrix
well i always take modulus
@brave sluice for this part, can i do integral of (line - curve) with the given bounds
and then take modulus of course
brave sluice
no, just the integral of the curve
pearl pondBOT
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Available help channel!
latent glen
Given a list of of integers from 1 to n ($2 \le n \le 10^9$). With each turn, we take elements at positions that have a remainder of 2 when divided by 3 (the list is 1-indexed) and create a new list with it. We do this until one number remains. Given n, find the remaining number.
jolly parrotBOT
Thomas
glass meadow
... isn't it just 1?
latent glen
no
cinder flower
1 will get removed
latent glen
For example, with n = 15, we have: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
then 2 5 8 11 14
then 5 14 then only 14 remains
glass meadow
Oh you keep the new list not the old one
brave sluice
try writing the first element of the kth list formed this way
hint || there's a geometric series ||
viscid shale
Also, this should be pretty obvious, but the list is essentially cut into (basically) a third everytime
brave sluice
try writing the first element of the kth list formed this way
in other words what is the sequence 1, 2, 5, 14 ... ?
latent glen
in other words what is the sequence 1, 2, 5, 14 ... ?
where'd you get that list from?
brave sluice
the first element of each list
latent glen
1 + 3^0 = 2, 2 + 3^1 =5, 5 + 3^2 = 14, ...
jolly parrotBOT
Axe
brave sluice
this can be simplified
latent glen
this can be simplified
how?
latent glen
so that gives $\frac{3^{k+1} - 1}{2}$
jolly parrotBOT
Thomas
latent glen
- 1 = $\frac{3^{k+1} + 1}{2}$
jolly parrotBOT
Thomas
brave sluice
yeah 👍
latent glen
but how do we find k tho
brave sluice
the largest k such that ( 3^(k+1)+1 ) / 2 <= n
u can solve for k and take the floor
latent glen
WA half the later tests
brave sluice
WA?
latent glen
wrong answer
brave sluice
probably floating point error
latent glen
how do i fix it then?
latent glen
still got the WAs
brave sluice
but it works for small values?
why is the "while" line different from the "cout" line?
brave sluice
**Axe**
this was kinda wrong because it gives 2 when k=0
so it misses the first 1
u should be able to remove the -1 from the "cout" line
latent glen
why is the "while" line different from the "cout" line?
when we finished adding k, (3^(k+1)+1)/2 > n
so we need to trace back to the last k that can still fit
brave sluice
i don't think so
oh
because of the "while"
i see
does / do floating point arithmetic?
in python there's // for integer division
brave sluice
1 + 3^0 = 2, 2 + 3^1 =5, 5 + 3^2 = 14, ...
you could avoid the formula completely and just implement it like this
latent glen
does `/` do floating point arithmetic?
if you divide int by int then it does integer division
brave sluice
ok
latent glen
or just any part that has int results in int division
brave sluice
long long has 64 bits i think, so there shouldn't be overflow
can you link the website?
latent glen
it's from a different language
brave sluice
oh ok
can you see for which values of n it's giving a wrong answer?
and what is it printing out?
latent glen
sadly no
pearl pondBOT
pearl pondBOT
Available help channel!
wary vault
hi there. is this proof that 1/2n converges to 0 fine? our prof taught us to do it without giving any value to n_0
wary vault
hi car
autumn fossil
this chain of inequalities doesnt look right
autumn fossil
if n = 5, 1/100000000000000 will surely be < 1/(2*5)
autumn fossil
I'd omit this condition, it seems unnecessary
and if you insisted on that, it would make the next claim valid only for eps < 1
wary vault
I'd omit this condition, it seems unnecessary
true, it just comes from the archimidean thingy and i thought id write it
autumn fossil
Your archimedean property is stated with that condition? Ok then ig
wary vault
and if you insisted on that, it would make the next claim valid only for eps < 1
ohhhh ur right
no i see your point
well ig theres no harm in taking epsilon<1
autumn fossil
But it looks right
pearl pondBOT
waxen agate
you're welcome
atomic ermine
hi
pearl pondBOT
Available help channel!
quasi estuary
f(100) = 200/101
pearl pondBOT
f(100) = 200/101
frank violet
,rccw
jolly parrotBOT
toxic lichen
f(0)=0, f(1)=1, f(2)=4/3, f(3)=6/4, f(4)=8/5, ...?
what even is the pattern to these numbers?
quasi estuary
2n/(n+1)
quasi estuary
yes
pearl pondBOT
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
I am stuck after finding the value of constant
f(x)(x+1) - 2x = k(x)(x-1)(x-2).....(x-100)
I got k as -2/101!
toxic lichen
that sounds about right
quasi estuary
What should I do after that
toxic lichen
well then you know that (x+1)f(x) - 2x = -2/101! * x(x-1)(x-2)...(x-100)
i think this should let you find f(x) in an explicit enough form to find f(101) and f(-1)
idk why the question also asks for f(1) when f(1) is explicitly given
quasi estuary
That was a mistak
toxic lichen
f(-1) might be a little harder cause you may need to do some derivative-adjacent bullshit
quasi estuary
f(-1) might be a little harder cause you may need to do some derivative-adjacent...
Can u tell me how to do that
That's the part I am stuck on
I alr calculated the value of f(101)
toxic lichen
f(x) = (2x - 2/101! x(x-1)...(x-100))/(x+1) right
quasi estuary
f(x) = (2x - 2/101! x(x-1)...(x-100))/(x+1) right
yes
toxic lichen
at x=-1, the value of this thing will equal the derivative of the numerator
quasi estuary
at x=-1, the value of this thing will equal the derivative of the numerator
Why is that ??
toxic lichen
laziest possible explanation is l'hôpital
quasi estuary
Shouldnt it be of the form 0/0 but in this case its finite quantity/0
plucky oriole
Shouldnt it be of the form 0/0 but in this case its finite quantity/0
its 0/0 here as well
blazing garden
<@&286206848099549185>
What’s the question? I tried scrolling back, and couldn’t see that
quasi estuary
.
quasi estuary
And I’m assuming you’re trying to find the function?
f(-1) is what I need but yeah
blazing garden
f(-1) is what I need but yeah
Shit. I gtg, hopefully someone doesn’t leave you here alone. gl 😭
pearl pondBOT
@quasi estuary Has your question been resolved?
quasi estuary
<@&286206848099549185>
midnight haven
<@&286206848099549185>
How can I help you
quasi estuary
see the ques
pearl pondBOT
hollow cosmos
<@&286206848099549185>
i haven't done the lesson yet, but interesting
is the quest use trial number right?
quasi estuary
is the quest use trial number right?
quest use trial number ?
what;s that
*what's
hollow cosmos
that's abt polynom
quasi estuary
is the quest use trial number right?
I dont rly know abt that
hollow cosmos
quest use trial number ?
*question
quasi estuary
I dont know what trial number is
hollow cosmos
i think the main problem is a polynom bcs i don't really see a limit here
hollow cosmos
I dont know what trial number is
i've learned abt polynom. which mean the trial number helps solve that's problem to find approximate result
quasi estuary
i think the main problem is a polynom bcs i don't really see a limit here
I dont think that's true cuz the poly. part is easy but even thinking about using limits was harder
quasi estuary
i've learned abt polynom. which mean the trial number helps solve that's problem...
U dont have to find the approx ans we just needed to get the final ans using sigma
pearl pondBOT
Available help channel!
hollow cosmos
U dont have to find the approx ans we just needed to get the final ans using sig...
aaahhh i haven't done the lesson yet is true. turns out limit is not only used to find the approx ans. i'm really sorryy bcs i cannot helps that probb
plush bramble
aaahhh i haven't done the lesson yet is true. turns out limit is not only used t...
did you have a question
hollow cosmos
did you have a question
i think sooo
bronze heath
i think sooo
What is it?
quasi estuary
aaahhh i haven't done the lesson yet is true. turns out limit is not only used t...
No worries man dont sweat it
hollow cosmos
What is it?
why the limit is not only an approx ans? so far i learned it was like that. i never to finds are an fix ans. if i find it, i only used l'hopital
warm current
why the limit is not only an approx ans? so far i learned it was like that. i ne...
Wdyk by approximate answer?
plush bramble
why the limit is not only an approx ans? so far i learned it was like that. i ne...
Show the actual problem and the two answers and approximate answers
pearl pondBOT
@hollow cosmos Has your question been resolved?
rotund nymph
why the limit is not only an approx ans? so far i learned it was like that. i ne...
theres supposed to be an exact limit; thats the definition of a limit
fluid oar
Yes
azure ether
pearl pondBOT
azure ether
Is f’ defined at -6, -2 and 2?
Or they are all undefined because they are sharp points?
proven sleet
what direction is this facing?
toxic lichen
what direction is this facing?
!occupied
pearl pondBOT
what direction is this facing?
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