Are you done with this?
#help-23
inner nest
inner nest
Oh yeah sure
join the VC
bold orbit
I can't vc rn
inner nest
Ok
Now here is a critical detail
As the question mentions that it touches the t' axis
or x axis in common sense
so it has a concavity at that point
which mean?
you know....
the derivative of that point is zero
@bold orbit
bold orbit
Okay yeah
inner nest
Oh yeah sure
so, y'=0
bold orbit
But that still leaves the whole constant problem
bold orbit
What
inner nest
yep
bold orbit
What are the equations in the system?
inner nest
inner nest
😑
inner nest
It is that easy
🤣
bold orbit
Lemme just verify this gives the same answer
inner nest
bruh
inner nest
bruh
bold orbit
.close
halcyon dagger
i get that both numbers will be even so the gcd cant be 1, and its highly unlikely that A is the answer, but idk where to go after that
safe radishBOT
i get that both numbers will be even so the gcd cant be 1, and its highly unlike...
vast obsidian
Is the LHS also divisable by 3?
bold orbit
Yes
halcyon dagger
well yk what maybe, cause it seems that for any even power of two plus and odd power of two it results in a value divisible by 3
but then 6 or 12 is the question
how do i determine that
bold orbit
It's 12
halcyon dagger
why
bold orbit
From the first part
bold orbit
Imma explain ofc
nova creek
Doesn't matter
vast obsidian
how do i determine that
You need to check if the RHS is divisable by 4
bold orbit
From the first part 2^2021 + 2^2022 = 2^2021(1+2)=3*2^2021
Do the same for the rhs
And see which are the common factors
halcyon dagger
3^2021 + 3^2022 = 3^2021(1+2) = 3*3^2021??
i still dont get it
how do i determine the common factors after that
halcyon dagger
oh yeah mb
bold orbit
So it's 4*3^2021
And the other one was 3*2^2021
So you have two 2s and a 3 as the common factors
2*2*3=12
formal quarry
How would I solve part D for this? I don’t know where to start
safe radishBOT
How would I solve part D for this? I don’t know where to start
lean otter
well do you know some very basic polynomials which have a 0 integral?
formal quarry
x?
formal quarry
thats only at 0 right
x raised to an odd power would have an integral of 0 from -1 to 1?
lean otter
yeah x³ is possible too, but think of how you could add a constant to x² to make it 0
formal quarry
x^2-1?
unique bison
Or set the integral of a generic cubic from -1 to 1, to zero
formal quarry
Ok so how do i use that to find the basis?
lean otter
you have
x
x²-1/3
x³
and all those polynomials have a zero integral, now what happens if you add them together or multiply by a scalar, is the integral still 0?
unique bison
you have
x
x²-1/3
x³
and all those polynomials have a zero integral, now what ha...
x^2 - 2/3*
lean otter
,w integral x²-1/3 from -1 to 1
formal quarry
you have
x
x²-1/3
x³
and all those polynomials have a zero integral, now what ha...
it should be right
how do i define that as a basis
lean otter
(x, x²-1/3, x³)
well you still have to prove that there are no more polynomials that are linearly independent from these
or do you need to prove things?
safe radishBOT
@formal quarry Has your question been resolved?
formal quarry
.close
orchid sparrow
I want to find two boolean functions such that f(x)*g(y) = 1 iff x and y = 0 and otherwise f(x)*g(y) = -1
And also that f(x)*g(y) = g(y)*f(x)
safe radishBOT
I want to find two boolean functions such that f(x)*g(y) = 1 iff x and y = 0 and...
lean otter
why f(x)*g(y) = g(y)*f(x)?
orchid sparrow
Because I want them to
lean otter
isnt that trivially true?
orchid sparrow
Oh
I didn't explicitly say that they give off complex numbers
So I had to say that
But assume they do
lean otter
so you want
f(0) g(0) = 1
f(0) g(1) = -1
f(1) g(0) = -1
f(1) g(1) = -1
right? in that case you can derive a contradiction
lean otter
f(0) (g(0) + g(1)) = 0
f(1) (g(0) + g(1)) = -2
=> f(0) = 0
but
f(0) g(0) ≠0
orchid sparrow
Yeah rip
Thanks though
I would have to think of something different to solve what I was hoping this was true for
🥺
swift fjord
f(0) (g(0) + g(1)) = 0
f(1) (g(0) + g(1)) = -2
=> f(0) = 0
but
f(0) g(0) ≠0
idk how you noticed that but thats cool
lean otter
adding the equations together, hoping to do something with the 0
safe radishBOT
@orchid sparrow Has your question been resolved?
orchid sparrow
@lean otter
How about this:
Is it possible to do
f(x)*g(x) =
1 if x isn't y
-1 if x=y=0
i if x=y=1
If so then I can still solve my problem
lean otter
so
f(0)*g(1)=1
f(1)*g(0)=1
f(0)*g(0)=-1
f(1)*g(1)=i
lean otter
g(0) (f(0)+f(1)) = 0 right?
but g(0) cant be 0, so f(0) = -f(1)
but that implies -f(1) g(1) = 1
orchid sparrow
so
f(0)*g(1)=1
f(1)*g(0)=-1
f(0)*g(0)=-1
f(1)*g(1)=i
What if this is the case
this one is my last attempt at success in this thing
Then I'll simply give up on that idea
You may be thinking wtf am I trying to accomplish
But trust me
I need it for things
Ohhhh I think i got it
Ok ok so so like
orchid sparrow
so
f(0)\*g(1)=1
f(1)\*g(0)=-1
f(0)\*g(0)=-1
f(1)\*g(1)=i
Is it possible?
@lean otter
I think I can't give a counter example
I think it's just a matter of figuring what they are
orchid sparrow
No solution!?
safe radishBOT
@orchid sparrow Has your question been resolved?
orchid sparrow
Oh my fucking god I'm a total idiot I've got it finally omggggg
safe radishBOT
Oh my fucking god I'm a total idiot I've got it finally omggggg
gray plaza
Can someone please explain to me the theory behind this code?
safe radishBOT
Can someone please explain to me the theory behind this code?
gray plaza
def gram_schmidt(X):
O = np.zeros(X.shape) for i in range(X.shape[1]):
# orthogonalization
vector = X[:, i]
space = O[:, :i]
projection = vector @ space
vector = vector - np.sum(projection * space, axis=1) # normalization
norm = np.sqrt(vector @ vector)
vector /= abs(norm) < 1e-8 and 1 or norm
O[:, i] = vector return O
chrome summit
If you have a list of linearly independent vectors, they may not be orthonormal. E.g. consider (2,0) and (1,1). They are neither normal nor orthogonal, but they are a basis.
Gram schmidt makes them orthonormal by basically going one vector at a time and fixing the problems
so (2,0) isn't normal, but it is when you scale it by 1/2, so do that
Then make (1,1) orthogonal by projecting it onto the line spanned by (2,0) and then subtracting that projection from (1,1). Basically, remove the part of (1,1) that goes in the same direction as (2,0)
There's a gif here: https://en.wikipedia.org/wiki/Gram–Schmidt_process
chrome summit
orthogonal to what? It's not orthogonal to (2,0). The angle between them is 45 degrees
gray plaza
Ahh! Yes
Sorry
So, in these first two lines it will take the columns of X and A.
Then it will mutiply them. Right? 😀
chrome summit
well it makes no sense to multiply vectors
there is no multiplication
you can take a dot product, which I imagine is what the code is doing (because that's what Gram schmidt is)
(u dot v) v is the same thing as projecting u onto the line spanned by v
lean otter
This is about factoring the differences of perfect cubes, using the two cubes identity formula. I don't understand why on the left, he simplified b^2 (3^2) to 9, but on the right, he did not simplify b^2 (5^2) to 25 ?
safe radishBOT
This is about factoring the differences of perfect cubes, using the two cubes id...
halcyon carbon
It doesn’t matter; he just wants to show the formula
worthy hemlock
This is about factoring the differences of perfect cubes, using the two cubes id...
3^2 is 9
You can write it both ways
Showing it as 5^2 was a method to help you see what B is easier
lean otter
Is the correct way to always simplify it down, so 3^2 to 9, and 5^2 to 25, or it doens't matter and I am okay with either?
worthy hemlock
Doesn't matter
safe radishBOT
safe radishBOT
Available help channel!
balmy folio
hi
balmy folio
this is partt of a homework question, but essentially it's a minimization problem and im not sure what im supposed to simplify the expression to
and i guess while im here im not sure how this is wrong
sudden birch
When u make a cube root x into x^(1/3), you remove the root symbol. You forgot to do that in your first wrong answer
sudden birch
How do you even find the length of the window? Idk
balmy folio
How do you even find the length of the window? Idk
the width, x?
it's a calculus problem using minimization
but i must have messed it up
balmy folio
When u make a cube root x into x^(1/3), you remove the root symbol. You forgot t...
also i simplified it like that and got something weird
so im still not sure for that one
sudden birch
the width, x?
A rectangle has a width and length. The window rectangle has width x. But then what’s length? Also gotta go and sleep so good night sleep tight don’t let the bed bugs bite
glossy cypress
Question: where did the (cos^2(A) + Sin^2(A)) go? I just want to know why it got replaced...
safe radishBOT
Question: where did the (cos^2(A) + Sin^2(A)) go? I just want to know why it got...
glossy cypress
great i havent learned that yet
is there a simple way you could explain it?
like maybe an example
obtuse plover
There isn’t a rlly simpler way of explaining it
glossy cypress
fair
i did kind of want to go into this while understanding the proof but i guess identities are for another time
obtuse plover
Eh they’re not too bad, im sure u can catch it on quickly
glossy cypress
I have no clue why the lesson's already using identities before Im even told how they work but
thats alr ig
thanks
.close
wary wave
if im integrating acelleration
safe radishBOT
if im integrating acelleration
wary wave
which is 10e^-0.2t
and acceleration at the start is 10
then to get speed
its 10 + the integral of acceleration
between 0 and t
is
10 - 50(e(^0.02t)+1)
right?
nova creek
No
wary wave
wait yea i see
nova creek
The constant after integrating acceleration isn't the initial acceleration
Also idk how you calculated that integral
nova creek
How are you getting that
wary wave
between 0 and t
wary wave
why not?
oh wait
yea i mean if its from rest
it would be from 0
but in the question it says
nova creek
First off, what are you trying to do?
wary wave
a vehicle accelerates from 10ms
trying to determine the power required that 1300kg vehicle to accelerate from 10ms at an acceleration 10e^.2t takes
and P = m * u(acc) * u(speed)
nova creek
What's u(acc) and u(speed)?
nova creek
Also, 10 m/s is velocity, not acceleration
wary wave
Also, 10 m/s is velocity, not acceleration
correct
nova creek
So u(acc) = u''(t) and u(speed) = u'(t)?
wary wave
and speed = intial plus integral of acceleration between 0 and t
wary wave
So u(acc) = u''(t) and u(speed) = u'(t)?
yea
nova creek
Typically, when you have an acceleration function and you want velocity, you find the indefinite integral. At least, I do
Then you can find the constant since you know u'(0) = 10
wary wave
yea i just wanted to know how i got the integral wrong
when integrating
10e^(-0.2t)
cause u divide 10/-.2
which is 50
so u get 50(e^(-0.2t))
right
nova creek
- C, yes
wary wave
yea but imma do through 0 and t
so then
when t = 0
it goes to 1
and when t =t
its just the e^(0.2t)
nova creek
It's kinda bad practice to use the same variable in the integral and in the bound, but that's just a notational thing
wary wave
yea i have a number for t
but i was just figuring out where i went wrong
for future reference
nova creek
Then everything seems right now
You do add 10, if 10 is the initial velocity rather than acceleration
Which it seems like it is
wary wave
Then everything seems right now
yep sorry i just used wrong term ahah a
it is inition vel
how can i simplif
(1300)(5e^(-0.2t))(35-25e^(-0.2t)
nova creek
How'd you get 35 - 25e^(-0.2t)?
safe radishBOT
@wary wave Has your question been resolved?
safe radishBOT
@wary wave Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
south skiff
safe radishBOT
south skiff
I just need help with part B
slow fern
,rcw
south skiff
I can’t wrap my head around how I’m going to deal with the no zero
flat frigateBOT
south skiff
Would any 1 be able to help ?
safe radishBOT
@south skiff Has your question been resolved?
south skiff
safe radishBOT
south skiff
I need a bit of help for part B
safe radishBOT
@south skiff Has your question been resolved?
restive palm
my answer was 78.75, right?
safe radishBOT
my answer was 78.75, right?
restive palm
h = 78.75
split ether
It's 80
restive palm
It's 80
h = -5t^2 + 30t + 35
h = -5(t^2-6t-7)
h=-5(t-7)(t+2)
t= 0.5(7-2), 2.5
h = -5 (2.5)^2 + 30(2.5) + 35
h = 78.75
that's what I did
what's wrong here
split ether
(t - 7)(t + 2) =/= t^2 - 6t - 7
split ether
The factoring it wrong
restive palm
the roots are incorrect I guess?
split ether
ax^2 + bx + c expression reaches its extrema when x = -b/2a
So in this case -5t^2 + 30t + 35 is maximal at t = 3
So just plug in t = 3
split ether
30/2*(-5) = 3
restive palm
in my book, it was like 0.5(root1+root2)
but I think they're the same mainly, but I didn't take x=-b/2a thing
safe radishBOT
@restive palm Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
safe radishBOT
blazing pagoda
Well
Take 2 common
That way, 2(1+1)
=> 2(2) = 2×2 = 4
Therefore, 4 is our answer.
@lean otter
upbeat swan
<@&268886789983436800>
blazing pagoda
What happened master
upbeat swan
my friend got banned from asking this kind of question
so i thought it was against the rules
but since no mod is respoding i guess it isnt
upbeat swan
ok
harsh shoal
Yeah you joined literally just to post this. F off
safe radishBOT
harsh shoal
Ty
upbeat swan
np
safe radishBOT
Available help channel!
uncut cloak
This is not making much sense to me
safe radishBOT
This is not making much sense to me
halcyon carbon
First part?
uncut cloak
Yeah
halcyon carbon
Do you know the group axioms
uncut cloak
I guess
halcyon carbon
halcyon carbon
For closure
Firstly reals are closed under addition and multiplication, so we only need to concern ourselves with x≠1 condition
uncut cloak
K
uncut cloak
Then it’s not close bruh
halcyon carbon
No I meant suppose for contradiction
halcyon carbon
Where did you even get that
halcyon carbon
uncut cloak
What now
halcyon carbon
Factorise x+y+xy+1
uncut cloak
Idk bruh how
uncut cloak
Oh okay so x=-1 or y=-1
halcyon carbon
Yes so x*y = -1 iff x=-1 or y=-1
In other words x*y≠1 iff x≠-1, y≠1
So our group is close under * because -1 is not in the set
Okay?
uncut cloak
No
uncut cloak
Because how do you know -1 in not in the set
halcyon carbon
Read the question again
uncut cloak
Okay my bad
Okay so it’s closed under star
Now associativity I’ve done
So identify
.close
safe radishBOT
uncut cloak
.reopen
safe radishBOT
✅
uncut cloak
uncut cloak
K
So
x+x^-1+xx^-1=0
x^-1 =(-x)/(1+x)
This?
That’s correct I think
I got -11/18 for the next part
marble yarrow
Help
marble yarrow
I found DE = 5/2
And AE 3/2
We can know that BF is also equal to 3/2
If we make a CFB a 90 degree angle, would CBF be 30 degrees and BCF 60
Is the DC part 13
<@&286206848099549185>
deft shadow
If we make a CFB a 90 degree angle, would CBF be 30 degrees and BCF 60
Yes
Ur DE is not correct
marble yarrow
Why
deft shadow
Try to count again
marble yarrow
We have to convert it to standard form
Of a 30-60-90
5/2)^2 + 3/2)^2 = 3^2
9 = 9
deft shadow
5/2 is which one
deft shadow
Yes
deft shadow
Wait let me count
marble yarrow
Ok
deft shadow
$\frac{3\sqrt{3}}{2}$
flat frigateBOT
Noone
marble yarrow
This is an example from book, and the way i did it, the book got the same answer
deft shadow
marble yarrow
So based on that
deft shadow
That is the rules
marble yarrow
Yeah
flat frigateBOT
Noone
deft shadow
What u get?
deft shadow
3/2
deft shadow
So FC is?
marble yarrow
And is CF also 2.6
flat frigateBOT
Noone
marble yarrow
Its 3
deft shadow
Not
deft shadow
Use this
marble yarrow
Ye so 3/2 is a times root 3
marble yarrow
3/2root3
deft shadow
Ok
safe radishBOT
@marble yarrow Has your question been resolved?
lean otter
I already know that the sum of all numbers from 1 to n is n*(n+1)/2
But is there a formula for the sum of the sum of all the numbers up to all the numbers up to n
For example for 4 it would be
1+1+2+1+2+3+1+2+3+4
lean otter
Sorry if I didn’t explain it well
lean otter
I already know that the sum of all numbers from 1 to n is n*(n+1)/2
But is ther...
I’ve tried stuff like n*(n+1)/4*n
<@&286206848099549185>
cosmic grove
I already know that the sum of all numbers from 1 to n is n*(n+1)/2
But is ther...
you mean this :
$\sum_{k=1}^n \left( \sum_{k=1}^{n} k \right)$
flat frigateBOT
Ｈｅｒｅｌｓ
cosmic grove
?
lean otter
Yes ig
cosmic grove
well we know that :
$\sum_{k=1}^n k = \frac{n(n+1)}{2}$
lean otter
Yes
flat frigateBOT
Ｈｅｒｅｌｓ
cosmic grove
so you just calculate :
$\sum_{k=1}^n \frac{n(n+1)}{2}$
flat frigateBOT
Ｈｅｒｅｌｓ
cosmic grove
thats easy tho
lean otter
Is there no formula do I have to calculate it?
Like no quick way to get it like with n*(n+1)/2
cosmic grove
there isnt a formula for this, thats something you can calculate yourself
cosmic grove
Like no quick way to get it like with n*(n+1)/2
dunno
lean otter
Alright thanks for your help anyways I was just curious
safe radishBOT
@glad onyx Has your question been resolved?
left agate
I think -15 is the correct ans?
safe radishBOT
I think -15 is the correct ans?
lean thistle
I think -15 is the correct ans?
it is
left agate
Thank u!
safe radishBOT
@left agate Has your question been resolved?
spring horizon
a passenger trains leaves the train depot 2 hours after a freight train left the same depot. the freight trin is traveling 20 km/h slower than the passenger train. Find the rat eof each train, if the passenger train overtakes the freight train in three hours
safe radishBOT
a passenger trains leaves the train depot 2 hours after a freight train left the...
woven swan
a passenger trains leaves the train depot 2 hours after a freight train left the...
what have you tried? what are you stuck on
spring horizon
what have you tried? what are you stuck on
the whole question
im not very good at word problems
woven swan
the whole question
ok so for word problems i think u should start by making variables
let’s say
p = speed of passenger train
f = speed of freight train
we know the freight train is 20km/h slower than the passenger train so f = p-20
you can determine the duration for each of the trains
we know speed * duration = distance
the distance traveled by both trains is equal, so you’ll have an equation you can solve
spring horizon
ooohh ok
safe radishBOT
@spring horizon Has your question been resolved?
lusty bronze
If 5^x = 3^y = 45^z, then prove that 1/z = 1/x +2/y
safe radishBOT
If 5^x = 3^y = 45^z, then prove that 1/z = 1/x +2/y
lusty bronze
How tho
glass carbon
for example
flat frigateBOT
Modus
stray socket
It's just log properties
lusty bronze
It's just log properties
And I don't know about log properties
halcyon carbon
If you haven’t done log. Consider $5=3^{\frac{y}{x}}$ and $3^y = 3^{2z} \times 5^z$
flat frigateBOT
Pure
stray socket
You haven't done any logarithms at all?
lusty bronze
You haven't done any logarithms at all?
I'm just a 8th grader
Anyways thanks
I got it
.close
lean otter
what "x" equals
safe radishBOT
what "x" equals
lean otter
it goes to infinite but i cant understand how i should solve it
swift fjord
yea are you sure theres a patter
lean otter
i think we should give a name for pattern than equal it
but cant find pattern
yes i am sure
swift fjord
yea first you have to figure out the pattern
but its like
i dont get why theres 48/2
intead of 48/x+something
lean otter
i think it can be like x+24 than x+12
swift fjord
yea whats the answer
lean otter
it is 8
swift fjord
what "x" equals
yea ok so
you know the denominator comes to 12
and for that the smaller denomator comes to 12
ok i get the sequence
lean otter
do you find pattern
swift fjord
theres like a chain of 12 denomators
the pattern is that basically
but i dont get why it ends on
48/2
it should end on 48/12 or something
swift fjord
basically i think the
denomitaor repeaats
x + 48/(x+48/(x+48/12)
every denomitaor should be
8+4
and then
48/12
continues
8+4
x is 8
8+4 is 12
48/12 is 4
which keeps it going
lean otter
how were we going to do it without knowing that the answer was 8
shouldn't we name the pattern part
like "a" or something
swift fjord
uh idk what you mean by that
what i did
was
ok obviously subtract 5 from both sides
then divide by 4 and cross multiply
you know the sequence is = 12
actually yea i dont know how you would solve with guess and checking 8 from there
swift fjord
yes
lean otter
<@&286206848099549185>
safe radishBOT
@lean otter Has your question been resolved?
foggy salmon
safe radishBOT
foggy salmon
does this hold for m=0? we cant use geometric series if m=0 right? also what if m>0, does this still hold? or is there anything we can use?
arctic schooner
привет может кто-нибудь помочь мне
привет может кто-нибудь помочь мне
привет может кто-нибудь помочь мне
привет может кто-нибудь помочь мне
привет может кто-нибудь помочь мне
arctic schooner
does this hold for m=0? we cant use geometric series if m=0 right? also what if ...
привет ты можешь мне помочь
compact wraith
привет может кто-нибудь помочь мне
<@&268886789983436800>
edgy cloud
привет может кто-нибудь помочь мне
Read #❓how-to-get-help, don't hijack already occupied channels, and then ask in English.
arctic schooner
Read <#488120190538743810>, don't hijack already occupied channels, and then ask...
no entendí
edgy cloud
does this hold for m=0? we cant use geometric series if m=0 right? also what if ...
For m=0 you just have O(1) -- but that still allows the denominator 1+O(1) to get arbitrarily close to 0.
arctic schooner
OK
safe radishBOT
@foggy salmon Has your question been resolved?
edgy cloud
does this hold for m=0? we cant use geometric series if m=0 right? also what if ...
More concretely, note that $e^{-z}-1$ is $O(z^m)$ for all $m\ge 0$, so the left-hand side might be $\dfrac{1}{1+(e^{-z}-1)} = e^z$.
flat frigateBOT
Troposphere
foggy salmon
For m=0 you just have O(1) -- but that still allows the denominator 1+O(1) to ge...
arbitrarily close to 0 or infinity right?
edgy cloud
If the denominator goes arbitrarily close to 0, the whole fraction gets close to infinity, yes.
foggy salmon
note that $e^{-z}-1$ is $O(z^m)$ for all $m\ge 0$
flat frigateBOT
flat frigateBOT
foggy salmon
More concretely, note that $e^{-z}-1$ is $O(z^m)$ for all $m\ge 0$, so the left-...
does this work if we replace it with any other function that is O(z^m), m >=0? what would the general form be like
changing it from e^(-z) to e^(z) feels drastically different
edgy cloud
The usual meaning of an equation with O(...) on both sides is that for every way to replace each O(...) on the left with a concrete function that satisfies the growth rate, there will be a way to replace all O(...) on the right that makes the whole equation true.
So just e^{-z}-1 is a counterexample to the claim.
foggy salmon
ahh so O(f(z)) = O(g(z)) means asymtotically equal?
edgy cloud
Not quite.
foggy salmon
More concretely, note that $e^{-z}-1$ is $O(z^m)$ for all $m\ge 0$, so the left-...
wait actually sorry, this doesnt work as a counter example, e^-z - 1 is not O(z^m), m>=0. consider m=0, O(1).
oh wait actually
i forgot to specify, complex domain
so infinity means complex infinity from any direction
edgy cloud
ahh so O(f(z)) = O(g(z)) means asymtotically equal?
It means that every function that is O(f(z)) is also O(g(z)) -- not necessarily the other way around.
foggy salmon
it was ambiguous earlier
edgy cloud
so infinity means complex infinity from any direction
Whoops. Then everything I've said is probably rubbish.
edgy cloud
Hmm. I think in that case m >= 0 would still allow O(z^m) to be close to -1 arbitrarily far from the origin. My concrete example of e^{-z}-1 won't work now, though.
foggy salmon
Hmm. I think in that case m >= 0 would still allow O(z^m) to be close to -1 arbi...
oh i think whats being applied is geometric series with 1/(1-r), |r| < 1 is the condition. thus we cant have m>0 as then z^m >1
im not too sure how it works on the boundary
or 1/(1+ O(1))
hm actually
i think its just 1+O(1)
or
O(1)
yea um
i think m>0 is trivially false
take a polynomial (only entire functions bound by z^m)
trivially just take z^m
1/(1+z^m) goes to 0 as z goes to inf
1/(1+O(z^m))
while 1+O(z^m) goes to inf
edgy cloud
I'm slightly unsure here, but I don't think there's any implicit assumption that the function O(z^m) stands for is even differentiable.
foggy salmon
i guess?
foggy salmon
I'm slightly unsure here, but I don't think there's any implicit assumption that...
hm for what? geometric series?
wait nvm geometric doesnt use differentiation idk im confused
edgy cloud
I don't understand why you keep speaking about geometric series.
foggy salmon
oh yea
this thing said geometric series
so i kinda just took it as a geometric series plugged it and it works i think
when m<0
edgy cloud
Huh.
I'd be wary of trusting that source, unless there's a proof that makes it clearer why the heck it thinks it would be true for m=0.
There doesn't seem to be anything "geometric series" about the actual claim.
flat frigateBOT
foggy salmon
wait
edgy cloud
For which limit of z?
foggy salmon
this feel weird
inf
ok this fails hm
oh wait
no it doesnt fail
cuz m<0
LOL
oops
edgy cloud
Let's write the claim out in full without the "=" abbreviation:
$$\forall f:\bC\to\bC: \Bigl[\begin{array}{c} \exists g\in O(z^m)\bigl[ f(z)=\frac{1}{1+g(z)}\bigr] \\implies \exists h\in O(z^m) \bigl[ f(z) = 1+h(z) \bigr] \end{array}\Bigr]$$
flat frigateBOT
Troposphere
edgy cloud
m < 0 seems to be fairly easy. For large enough |z|, we have |1+g(z)| > 1/2, so everything happens in an area where 1/(1+z) is Lipschitz, and so the bound on g translates to a bound on h.
foggy salmon
$\lim_{z \to \infty} \frac{|g(z)|}{|z^m|} \le K$
flat frigateBOT
edgy cloud
A strong form of continuity: https://en.wikipedia.org/wiki/Lipschitz_continuity
foggy salmon
oh cool
edgy cloud
It basically means that the difference in h(z) will be at most a constant times the difference in g(z) whenever |1+g(z)| > ½.
toxic stratus
pret sure i just ignored the case where m=0 when i did the proof for this
foggy salmon
like its the easiest way ik to throw stuff into numerator
foggy salmon
pret sure i just ignored the case where m=0 when i did the proof for this
u didnt do the proof
toxic stratus
i dont think it works
foggy salmon
it doesnt work when denom=0
edgy cloud
However, for the m=0 case, $f(z) = e^{|z|}$ seems to be a counterexample. We then have $g(z) = e^{-|z|} - 1$ which is a perfectly cromulent $O(z^0)$ function, but $h(z)$ becomes $e^{|z|}-1$ which is definitely not $O(z^0)$.
flat frigateBOT
Troposphere
edgy cloud
Ignore "prefectly cromulent" -- that's a meme, not technical content.
edgy cloud
Sorry.
toxic stratus
like
foggy salmon
nani!
flat frigateBOT
foggy salmon
from the first one to geo series
foggy salmon
how wld i use geo series
ah ok
i pulled out a half n manipulated a bit n got this
$\frac12 \sum_{n=0}^{\infty} [\frac{1+O(z^m)}{2}]^n = O(z^m)$
flat frigateBOT
foggy salmon
yay?
foggy salmon
:c
i did
wot
:c
$\frac{1}{1+O(z^m)} = \frac12 \frac{1}{1-(\frac{1+O(z^m)}{2})}$
flat frigateBOT
foggy salmon
you cant do that
y not
toxic stratus
okay sure
foggy salmon
oh
yayy
ok well
im kinda done here for now
thanks so much
trop and baker
💕
Breakthrough
Level up Imperial Sky Layer 5
.close
wintry axle
While we are determining whether they are a proper subset/incomparable, do we count the subsets of the element type subset or we don't? I mean is cardinality important? Cuz for the solution B2, it says that "the elements in the sets which are elements of B" meaning it doesn't care about cardinality, while for B3 it says that A and C are incomparable due to cardinality and B is a subset of A so...
safe radishBOT
While we are determining whether they are a proper subset/incomparable, do we co...
wintry axle
From what I understand, C's cardinality not specified so for k=3 (a subset of k >= 2 condition), it includes all of these 3: {3} {3,6} {3,6,9}
B's cardinality is 3 with conditions of elements of the subsets being divisible by 3 thus it includes, {3,6,9} {6,9,12} {3,6,12}
A's cardinality is 3 without any conditions so : {1,2,3} {2,3,4} {1,2,4} are all valid
Proper Subset: All values of subset is in the superset but superset has at least one more element that is not in the subset
Incomparable: none fully contains the other, they each have something that the other doesn't
!!NONE OF THESE HAS TO BE SEQUENTIAL!!
B2 and B3 differ because C already has every subset (element) B has (element)
nvm solved my own issue lol
.close
next garden
how does telescoping work?
And how is it useful for a series like
1/1(2)(3) + 1/2(3)(4) + 1/3(4)(5).... + 1/(n)(n+1)(n+2) (I was told it can be used there)?
next garden
I was also having a problem with finding the numerator of the same series (partial sum).
quasi bison
,w partial fractions 1/(n(n+1)(n+2))
flat frigateBOT
quasi bison
how does telescoping work?
telescoping is essentially mass cancellation of parts of adjacent terms in your summation
to say it in a somewhat vague way
next garden
Oh alright.
next garden
I am still a bit confused here though, we're finding the partial sums.
Unless I didn't notice.
halcyon carbon
Hm for this question, rewrite -1/(n+1) as -2/(2(n+1)) then group the terms in a way that gives you two telescoping sums to deal with.
next garden
How does that make a difference though?
next garden
,w partial fractions 1/(n(n+1)(n+2))
And hold on for a second, shouldn't partial sums have cross multiplication?
We are only seeing the terms here.
next garden
Sure!
unique basalt
when calc implicit derivative, when do we make dy/dx?
safe radishBOT
when calc implicit derivative, when do we make dy/dx?
stray bluff
derivative of y wrt x is dy/dx
unique basalt
derivative of y wrt x is dy/dx
so d/dx is ilke a fraction?
stray bluff
so d/dx is ilke a fraction?
huh
nova creek
d/dx is not a fraction, no matter what physicists tell you
stray bluff
dy/dx just represents the change in rate of the derivative of y with x changes
lean thorn
so d/dx is ilke a fraction?
it represents a ratio of infinitesimally small change in y over infinitesimally small change in x. There are times you can treat it like a fraction, but it more or less represents a change in y over a change in x.
$\frac{dy}{dx}$ represents that x is our independent variable that we're taking derivative of (hence the term "taking the derivative with respect to x"). y is our dependent variable that depends on x.
flat frigateBOT
MellowDramaLlama
lean thorn
d/dx is not a fraction, no matter what physicists tell you
well... kind of yes and no. I mean we consider the slope of a straight line a fraction, yeah? And the change in slope at a point comes down to a tangential line.
This probably goes beyond the scope of this discussion but it would also be considered a fraction in stuff like DE yeah? That's how we can separate to solve for separable diff equations.
nova creek
well... kind of yes and no. I mean we consider the slope of a straight line a fr...
d/dx is an operator
Imo, the separation found for seperable diff equations is purely a notational hack
lean thorn
sorry I'm talking about dy/dx, not just d/dx (should've read your comment closer haha). d/dx represents taking the derivative, but after we have a dy/dx, which represents the tangental slope of a point
nova creek
dy/dx is a function, and I still wouldn't call it a fraction
It's lim Δx->0 Δy/Δx
So it's the limit of a fraction
nova creek
Doesn't help that people would solve "dy/dx = x" as "multiply both sides by dx then integrate"
Which I think is a notational hack rather than something you can do rigorously. But hey, it works
safe radishBOT
@unique basalt Has your question been resolved?
fervent siren
(A – B) – (B – C) = A – B
safe radishBOT
(A – B) – (B – C) = A – B
fervent siren
i have to prove it and it says using set identities
compact wraith
these are sets and - is the complement?
fervent siren
i think?
lean otter
(A\B)\(B\C) i guess
fervent siren
im so confused with this question lmao
compact wraith
when ur confused with a question like this then draw it , either a venn diagram or make up some sets and check if it's true
patterns then arise and you can sometimes see how it is going to work
fervent siren
im trying to look at the set identities but they dont use - and +
compact wraith
(A\B)\\(B\C) i guess
I think it's this
fervent siren
thats my problem, im not even sure what to draw or how to start this
compact wraith
thats my problem, im not even sure what to draw or how to start this
then start by reading / watching some stuff on set theory basics
then reattempt the question
fervent siren
use an available help channel @opal roost
it looks like 14 24 44 and 48 are open
oh im starting to get it
i have to rewrite it
A - B also means A n notb
torn kestrel
How would I graph 4-cos(x/3) ?
safe radishBOT
How would I graph 4-cos(x/3) ?
compact wraith
what have u tried?
torn kestrel
Well first I wanted to try and identify everything. I think the amplitude is 4 and the period would be 0 I think. I don’t know if this problem has a phase shift though
compact wraith
instead of thinking in trig terms i prefer to think in terms of general functions
first of all identify the common function, that is cosx
it will be useful to graph cosx first
well in this case we dont have cosx we have -cos(x/3)
so the -1, what effect does this have on the function?
torn kestrel
Would it make the whole function negative
Or would you change it to be -3cos(x/3) if you minus the 4-1cos
compact wraith
Would it make the whole function negative
yeah it would
it would reflection the function in the x axis
now we have -cos(x)+4
what does the +4 do to the function?
(dw we'll get to the x/3 next)
compact wraith
yeah so it's a vertical translation
so we have that 4-cos(x) is cosx first flipped in the x axis and then translated upwards by 4
finally the x inside becomes x/3
so we have 4-cos(x/3)
now what is the effect of x/3 (this is the hardest to remember / understand so dw if ur not sure)
torn kestrel
Not fully sure but would 3 be the phase shift in this equation?
compact wraith
well it's a stretch in the x direction by factor 3
let's look at cos(x)
Now cos(x)=0 for pi/2 and 3pi/2
but cos(x/3)=0 for 3pi/2 and 9pi/2
so them 2 points have been increased in size by a factor of 3
and in fact all points have
so whenever you have f(ax) it's a stretch of the function by a factor of 1/a in the x direction
putting it all together: cosx is first reflected in the x axis, it is then translated up by 4 and stretched in the x direction by a factor of 3
with that information you should be able to graph it
torn kestrel
Random question but in this case is the amplitude still 4 or no because we moved it in the equation?
.close
safe radishBOT
safe radishBOT
Available help channel!
solar bramble
safe radishBOT
solar bramble
kinda stuck here trying to figure out how to write sine in terms of cosine so that I can substitute 1/5 in it
maybe im going about this the wrong way
safe radishBOT
@solar bramble Has your question been resolved?
solar bramble
<@&286206848099549185>
sullen parrot
kinda stuck here trying to figure out how to write sine in terms of cosine so th...
sin^2 x+ cos ^2 x=1