It actually said from top to bottom for the D part
#help-23
long oracle
No its fine dont worry! i was confused if i said something wrong, because you described D as being looked out from above
this stuff happens
long oracle
i try to always be careful with these wordings because as i said from the start this is 90% of the trouble
yeah so if we look at this
We get D but pointing outwards right?
fathom jewel
,,\Phi_B + \Phi_R = -\Phi_D
flat frigateBOT
𝔸dωn𝓲²s
long oracle
But R+B is 45
we want it the other direction
so we take into account a minus sign
Yes!
fathom jewel
I understand that part now LOL
but the terminologies are still confusing to me
what's a boundary surface
fathom jewel
yea like D and R are surfaces
long oracle
The reason i was confused about that is say if we already had a surface in 3D
then what is the boundry of that?
long oracle
If we use a topological notion of boundry, then this is just the same object, but when discussing surfaces and such, or in more general a manifold as its called, i think thats even the german word for it lmao you take the boundry curve
Sure
fathom jewel
For a cube this would hold $$\iint _S \vec{v} = \iiint_B \text{div} \vec{v} $$ where $S$ surface and $B$ body.
bruh doesnt exist but $\oint$ exists
long oracle
**𝔸dωn𝓲²s**
ikr lmao
fathom jewel
So the boundary surface(s) would be the side areas of that cube
at least that's how I understand it from the script
long oracle
so when you use area, do you mean area as in the number, or more the word surface?
long oracle
**𝔸dωn𝓲²s**
Also if the terminology of boundary is confusing, you can avoid talking about integrals all together
long oracle
the surface of a cube has 6 areas
Lmao
i guess you mean it has 6 faces then
lets forgot about the intergrals for a momnet, i dont think they will help here
long oracle
Yea its just notation for boundrary
long oracle
so is the word area here confusing for you?
long oracle
usually in English that means the measurement of a surface
fathom jewel
basically the cone is not a closed surface
fathom jewel
so that's not 100 % right
they mentioned afterwards to consider the circle
which would make it closed
long oracle
The disk maybe? circle is the curve
fathom jewel
yea
fathom jewel
yes
long oracle
also the reason for this is because a boundary surface which is oriented MUST be closed
it doesnt otherwise make sense to talk about a boundry surface with respect to some body or even orientation depending on what sort of orientation we mean
this becomes apparent even from Greens theorem
fathom jewel
also the reason for this is because a boundary surface which is oriented MUST be...
what does it mean "which is oriented"
what are non oriented surfaces
long oracle
The Möbius strip cant be assigned an orientation
fathom jewel
I only know it has one direction
long oracle
I think its a bit messy also for surfaces in 3D because as far as im aware theres more than one type of orientation we can talk about
fathom jewel
ah i just read it means you cannot distinguish between bottom and top or whats inwards/outwards
basically it has "one side"
hmm nah
long oracle
Yeah and what's interesting is that even for this shape, one can talk about an orientation that doesnt make sense, but does for closed surfaces
fathom jewel
So if I had a normalvector on the Möbius stripe
it would unnaturally move downwards at some point
I remember from differential geometry a video
where the normal vector acted unnaturally
like had a sudden turn
long oracle
Well for a Möbius strip i think its still continuous but the crux is that theres only one side so to speak
Oh sorry
Yeah if you move to the edge
long oracle
This is the thing i tried avoiding, but yeah you can speak of orientations in which the normal has to move continuously
long oracle
I cant read German so i wont either lmao
long oracle
let me try lmao
fathom jewel
This is the closed surface which means ground surface + lateral surface
long oracle
Well yeah so if i understand right
fathom jewel
but how do get 0
long oracle
they have to take into account that the surface is not closed
so they close it
and then remove the thing that they added
but they have to know by how much, so they calculate the thing they added
which is a much more direct calculation
in this case it seems they closed the surface off with a disk
long oracle
but how do get 0
the div was 0?
fathom jewel
yea but logically I would expect 4pi
long oracle
Deduced from what?
fathom jewel
because 👌🏻K is closed surface
fathom jewel
$$\iint _{\partial K} \vec{v} = \iint _{\text{ground surface}} \vec{v} + \iint _{\text{lateral surface}} \vec{v} = \iiint_K \text{div} (\vec{v})$$
flat frigateBOT
𝔸dωn𝓲²s
long oracle
Your vector field might be flowing in such a way that the flux is as positive as negative
so they cancel out eachother
fathom jewel
this is what I mean
long oracle
Yeah
fathom jewel
So we would have the equation
$$0 + 4\pi = \iiint_K \text{div}{(\vec{v})} $$
flat frigateBOT
𝔸dωn𝓲²s
long oracle
Since the ground surface was surgically added in on purpose
we remove it
to get the original integral
fathom jewel
but then it's not closed anymore to use Gaus, no?
long oracle
Yeah but this is not Gauss anymore
we closed it in order to use Gauss
but now we have a value
so we move on to use the properties of an integral to separate them (disjunct sets)
fathom jewel
wait is a regular surface also always a closed surface?
long oracle
what is a regular surface?
It doesnt have to be
But the boundry of a body
always has to be
The problems is they dont teach you about manifolds
and the intuitive reasoning you may use (that comes more from topology)
fathom jewel
I will try again my confusioin
long oracle
Since my trouble was even with Greens theorem
but it all makes sense if you know a little bit about manifolds
flat frigateBOT
𝔸dωn𝓲²s
long oracle
Okay, i was thinking a rather dirty sketch of just the closed surface, but pretty!
Yes!
Green and blue surfaces
fathom jewel
Since the surface double integral is 4pi
long oracle
Which?
fathom jewel
fathom jewel
this then doesn make sense to me
long oracle
Okay so only the surface integral of G is -4pi
fathom jewel
yea
fathom jewel
B?
fathom jewel
the cone
long oracle
Okay
and you want to find the surface integral over K ?
thats the goal i guess right?
fathom jewel
the calculated the flux over the lateral surface K
long oracle
I see
Okay, so if the closed boundry is G+K
and this surface integral was 0
and we know that int over G is -4pi
long oracle
the calculated the flux over the lateral surface K
What must this be in such case?
fathom jewel
well it's 0 here
long oracle
Okay im really confused, then youre done no? i thought K was the cone which was open and not the closed one
What is the question?
fathom jewel
i cant word my question smh
long oracle
Okay but if you dont even understand the goal/premise of the question how do you expect me to help here?
Unless that's the thing you want help with?
fathom jewel
Yeah I am sorry
long oracle
I see
fathom jewel
but it's not closed
long oracle
Okay i see
fathom jewel
Ok
long oracle
for Pine
fathom jewel
ok
long oracle
Okay, so i'll asume the orientation and normal vectors and stuff is something youll know how deal with; so should i avoid talking about that, since it seems you already undestand that here?
fathom jewel
,,P : \begin{pmatrix} x \ y \ -\sqrt{x^2+y^2} \end{pmatrix}
flat frigateBOT
𝔸dωn𝓲²s
fathom jewel
Okay, so i'll asume the orientation and normal vectors and stuff is something yo...
yea
long oracle
Okay, then let me just give the overall idea and see if that is enoghe.
So as you know P is not closed
fathom jewel
yes
long oracle
and you want to use Gauss
long oracle
so then lets close it by some surface G, as long as this surface closes the surface with P we can use Gauss
Yes youll eventually do that
long oracle
since we add this G in
then that changes the integral
we want to know the anwer to the integral without G
so we substract G, once we've used Gauss
Makes sense?
long oracle
A suitable choice for G is the disk
it could of actually been any other surface as long as it closed it P for Gauss
but a disk is probably the best here
fathom jewel
I see no other possibility other than this disk
long oracle
We could have had patricks ceiling from before
mostly any surface that connects with P would work
long oracle
Yeah
fathom jewel
wtf
fathom jewel
yeah but how is that possible
long oracle
How is it not?
fathom jewel
wouldnt the bounds change?
long oracle
Yeah sure
fathom jewel
ok trivially
div(v) = 0
so the bounds dont matter
but imagine div(v) was not 0
or does div(v) = 0 imply not closed surface
long oracle
It says nothing
fathom jewel
ok
long oracle
Well
it does say something
but not about our surfaces
directly
its more about the vector field
we could have a closed surface in which div(v) is not 0
so it does not tell us about closedness
fathom jewel
okokoko
long oracle
that's more of a happy accident
long oracle
yeah
since we'll have to calculate this one separately later on
So we find via Gauss that the integral with them combined (P+G) is zero
We can now solve for the integral over P
What would that look like?
fathom jewel
P = -G + 0
long oracle
Yes in essence!
So to find the value for the integral over P
we have to calculate the one for G
fathom jewel
they described it that way in the script
fathom jewel
almost
long oracle
Alright great!
fathom jewel
it's fucking clear dude thanks
fathom jewel
i think the confusion came because i thought of things differently
fathom jewel
this would help a lot more in the script if they added this
long oracle
I see
Yeah drawing are really imporatant imo
i think it helped us even before too
with this
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steep magnet
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steep magnet
Not sure what my mistake is?
hoary wind
steep magnet
Oh lol
gusty trench
but if f(x) = sin(x^2 * x^(1/7)), your derivative would be correct
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steep magnet
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steep magnet
Which sign is wrong??? I dont see haha
low vector
idk, second answer looks right
faint seal
No, it’s wrong
median vigil
quotient rule not applied correctly
faint seal
Both terms must be negative
faint seal
quotient rule not applied correctly
steep magnet
(g(x) h'(x) - h(x) g'(x))/(g(x))^2
low vector
ohh right
faint seal
(g(x) h'(x) - h(x) g'(x))/(g(x))^2
You didn’t use this
You used the negative of this
Or something idek what you did exactly
You managed to mess up only one of the signs
low vector
$\left[\frac{f(x)}{g(x)}\right]'=\frac{g(x)f'(x)-f(x)g'(x)}{f(x)^2}$
flat frigateBOT
Flappie
steep magnet
Ahhh wait I think I see
Yeyeye I know what I did
Ok thanks everyone!
❤️
.close
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rain dagger
in the line where the arrow is at, do I really have to divide "exists c"?
rain dagger
I mean if it's in the leftmost side it'll try to make the overall statement True, so it doesn't really matter right?
like in step before
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@rain dagger Has your question been resolved?
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lyric salmon
hello i am back this one may be even more difficult than the other question i sent earlier
safe radishBOT
hello i am back this one may be even more difficult than the other question i se...
lyric salmon
i tried multiplication and saw no correlation very confused
lyric salmon
thank uuu
lyric salmon
sure
lyric salmon
how did u do it?
solemn terrace
So basically the next number in the sequence is the sum of the previous two numbers
Do you understand what Im saying
solemn terrace
Ok
lyric salmon
theres no way i can find out the answer using a formula right??
because u basically eyeballed it right
solemn terrace
No not really
No not really
Actually you can think of this as a formula, the fact that you have to add the previous two numbers to get the next consecutive number in the sequence
lyric salmon
yea but the thing is
solemn terrace
So no, not really eyeballing it would help you solve it that easily
lyric salmon
this uses a different formula
and if i keep getting questions like these im just going to be lost
solemn terrace
yea, well just remember for every sequence, there may be a different pattern
Give me a min for this one
lyric salmon
okay
solemn terrace
Ok got it
lyric salmon
??
lyric salmon
what was the pattern?
solemn terrace
first, you have to find the difference between each number
I mean each set of two numbers
solemn terrace
And then we actually notice that each subsequent difference is double that of the previous difference
So if we keep performing that way, we get 407
Do you understand
lyric salmon
25-13 is 12
solemn terrace
Yea keep doing it that way, subtracting sets of numbers until you find the pattern I just told you
lyric salmon
i asked someone ab this question earlier so i know the answer but the thing is
how do i find the answer in the first place
solemn terrace
I just told you right
lyric salmon
since theyre 2 completely diff ptterns
solemn terrace
This whole time
lyric salmon
?
solemn terrace
Well i told you the methof\d
lyric salmon
jut keep subtracting?
solemn terrace
By subtracting sets of two numbers, finding their difference and doubling the difference together to get the next set of numbers' difference
Thats the method
low vector
Can you share the original question again?
lyric salmon
like someone helped me, but its just a matter of understanding
how to get the answer
like finding a mthod tht applies to all sequence questions
low vector
Usually you look at the differences
3 1 4 5 9 14
-2 3 1 4 5
5 -2 3 1
-7 5 -2
12 -7
-19
It's a bit hard to spot, but there's a pattern in here
low vector
So the next item should be 23
lyric salmon
u added the prev 2 numbers right?
low vector
Yeah
Cuz if you continue the pattern
It becomes like this:
3 1 4 5 9 14 23
-2 3 1 4 5 9
5 -2 3 1 4
-7 5 -2 3
12 -7 5
-19 12
31
You can see that the row below, is the row above but shifted by 1
lyric salmon
oh wow i see
and if i do that for every sequence question ill be able to find the answer like that?
do u remember the other question i had earlier
can i do that process but with the first question
lyric salmon
this uses a different formula
e
lyric salmon
ok ill lt u know
low vector
13 25 51 101 203
13 25 51 101 203
12 26 50 102
14 24 52
10 28
18
Not quite
But if you go the other way you might see it
lyric salmon
idk if i did my math wrong but i got different nubmbers
Yea i did get it wrong srry
lyric salmon
But if you go the other way you might see it
wym?
low vector
wym?
Now we are going "down"
We are getting the differences from our sequence
But what if our sequence is the difference of another sequence?
lyric salmon
28??
next as in next to each other??
im so sorry for the dumb questions but i rlly appreciate ur help i rlly do
low vector
I just picked a number
lyric salmon
oh oh
low vector
That looked nice to start from
lyric salmon
ok and how did u get the 13?
low vector
We can also start at 12
low vector
ok and how did u get the 13?
That's the first item in our sequence
lyric salmon
so u just picked a random number
lyric salmon
but why
low vector
When we went down
We took the differences of the itmes in our sequence
Now, if we go up, our sequence is the differences of the sequence above
lyric salmon
hmm
low vector
ok so u added it?
Indeed
lyric salmon
ok im going to try it
low vector
I'm on mobile so I can't make a nice diagram out of it
low vector
Yep
low vector
You had the first term right, and then messed up all the rest of the terms
lyric salmon
WHAT
low vector
38, 26 difference is -12, which is not 25
lyric salmon
But u said to add all the. Numbers no?
low vector
Here, look at this one
Imagine the top row doesn't exist
How would you generate it, given that it starts at 13 and the differences are 12, 26, 51, 102
lyric salmon
i would add the number from the row?
low vector
Yeah
First term is, 13, difference to the next term is 12, so the second term is 25
The second term is 25, the difference to the next term is 26, so the third term is 51, etc
And that's how you generate our initial sequence
So, given our original sequence, start with some number, how would the rest of that sequence be constructed?
flat frigateBOT
The current time for flappie. is 03:46 AM (CEST) on Sun, 30/06/2024.
lyric salmon
i think i should let u go its late for u
low vector
Nah
lyric salmon
im sorry to keep u awake
low vector
Look at the differences in the sequence you just made
Is that our original sequence?
38-23=15( not in our sequence)
76-38=38 (not in our sequence)
152-76=76 (not in our sequence)
So you have not generated the correct sequence
lyric salmon
so i just have to keep guessing numbers?
low vector
No
low vector
First term is, 13, difference to the next term is 12, so the second term is 25
What happened to this?
Did you read this?
low vector
Nevermjind
I'll give you the sequence
Starting from 10, we get
10 23 48 99 200 403
If we start from 11 we get
11 24 49 100 201 404
From 12
12 25 50 101 202 405
lyric salmon
how did u get 48?
lyric salmon
hmm
ok
so howcome u also started with 11 and 12?? were u able to find the pattern?
should i continue the sequence for those 2?
this is all a little confusing for me
its not u i [promise]
low vector
so howcome u also started with 11 and 12?? were u able to find the pattern?
The first term I can decide
Do you notice that how the differences of the items in my sequence is exactly our initial sequence?
lyric salmon
yes
low vector
So, I can start with any number
low vector
Do you notice that how the differences of the items in my sequence is exactly ou...
And still create a sequence that satisfies this condition
Since it would just be shifted
lyric salmon
soo... how would i find the next number
would i
add 203 and 304?
no
101 and 304?
low vector
The problem is, I still don't see a pattern in this sequence
lyric salmon
😭
low vector
*in terms of what I have not already found
low vector
The problem is, you can think of an infinite amount of ways to complete the sequence
That's the problem with these types of questions
It either has to be almost blindingly obvious, otherwise you can think of almost anything
lyric salmon
hmm
because i have less than a minute to answer each question
so what do u think will be my best bet
low vector
13, 25, 51, 101, 203
low vector
because i have less than a minute to answer each question
If it's not blindly obvious, guess
Idk
low vector
13, 25, 51, 101, 203
Like, the pattern here is previous term * 2 + 1 or -1
lyric salmon
yea
low vector
Yeah
lyric salmon
i just wish there was a formula or something to solv this without guessing
low vector
There is something
lyric salmon
??
lyric salmon
i hope u find it 🙏
low vector
Nvm, can't find it
lyric salmon
is it something w arithmetic?
low vector
But It gave some weird results anyway
lyric salmon
hmm
okay
guess ill get the questions wrong ty though!! : )
have a nice rest of ur night
.close
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iron onyx
$(0.2)^{3x}<5\cdot(0.2)^{x+1}$
safe radishBOT
$(0.2)^{3x}<5\cdot(0.2)^{x+1}$
flat frigateBOT
roi
iron onyx
why is ans $x>0$? When does the inequality sign flip?
flat frigateBOT
roi
iron onyx
i know how to get to $(0.2)^{3x}<(0.2)^x$ but i dont understand why the sign flips when equalling exponents
flat frigateBOT
roi
royal kiln
Try dividing both sides by one of the exponential factors
iron onyx
by (0.2)?
iron onyx
ohhh
royal kiln
If you do that, and divide out the constant 5, you get
1/5 < 0.2^{(x+1) - 3x}
0.2 < 0.2^{1-2x}
What could you do then?
iron onyx
If you do that, and divide out the constant 5, you get
1/5 < 0.2^{(x+1) - 3x}
0...
i think that's not correct man
the x shouldnt be dividing, it should be substracting
also 0.2= 1/5 so it simplifies the 5
royal kiln
i think that's not correct man
Shit you are right
white umbra
It's because 0.2^x is a decreasing function
If it helps, try rewriting it as 5^(-x)
So,
[5^{-3x} < 5^{-x}]
flat frigateBOT
kalman_filtERIC
white umbra
Whenever you have a decreasing function involved, like multiplication by a negative number, or 0.2 to the something, you have to flip the inequality
Yeah
iron onyx
okok thank you
flat frigateBOT
white umbra
hopefully helpful visual ^ note the value of the function goes down as x goes up
iron onyx
yeah, thx
safe radishBOT
white umbra
no worries
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mystic vale
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mystic vale
i'm trying to follow this solution, but i don't understand what they mean by =! than pi/2 , and why it's a contradiction.
solving the 2 partials i find a couple of coordinates but not all of them.
i find
(+-pi/2, +-pi/2)
and
a = -b/2
b = -a/2
which i don't really know what to do with them
i can put them into system but i get nonsense
b = -(b/2)/2
b = b/4
or i should put them equal to each other?
a = b
-b/2 = -a/2
so yeah a= b
wind stream
i'm trying to follow this solution, but i don't understand what they mean by =! ...
Notice that when alpha = pi/2, f_y=0. Hence, they are interested in solutions where beta is not also pi/2 such that f_x = 0
can you find any beta in [0, pi/2] in such case?
mystic vale
i've tried to plug pi/2 into b and i get
b = -pi/4
wind stream
right, forget about the general solutions that you found when solving sin(2alpha+beta)=0 and sin(alpha+2beta)=0. I'm talking about why alpha=pi/2 and beta!=alpha/2 is a contradiction
anyway, as you can see, in the case alpha=pi/2 and beta != pi/2. f_alpha=0 iif sin(2alpha + beta)=0
wind stream
anyway, as you can see, in the case alpha=pi/2 and beta != pi/2. f_alpha=0 iif s...
do you follow why this is?
wind stream
alright sure, take your time
mystic vale
what i see is that if i plug pi/2 in f i always get zero, cause one of the terms will go to zero
so we want values where f is not zero. yes.
But i dont really understand how to see other values a part from what i've found with the partials
wind stream
alright, lets just start from the bit where they found f_alpha and f_beta. Do you follow up until that bit?
mystic vale
yes
wind stream
right, so do you agree that $f_\beta(\frac{\pi}{2},\beta) = 0$ no matter the value of $\beta$?
flat frigateBOT
waler
wind stream
oops sorry, typo'ed
mystic vale
yes, cos(pi/2)= 0 so yes it always going to be zero. that's one of the solutions i found indeed
wind stream
yes, however it is not the case that $f_\alpha(\frac{\pi}{2},\beta) = 0$ for any value $\beta\neq\frac{\pi}{2}$
flat frigateBOT
waler
mystic vale
yes
wind stream
so we need to find $\beta\in[0,\frac{\pi}{2}]$ such that $f_\alpha = 0$
flat frigateBOT
waler
wind stream
can you find any such value for $\beta$?
flat frigateBOT
waler
mystic vale
i found b = -a/2
wind stream
eh not really
mystic vale
so what's inside the sin goes to zero
actually i found before
(pi k - a)/2
but i've scrapped it cause we're looking only 0,pi/2
wind stream
right, let's jsut take it slow here
we are looking for solutions of $\beta\neq\frac{\pi}{2}$ such that $f_\alpha(\frac{\pi}{2},\beta)=0$
flat frigateBOT
waler
wind stream
this is what we are currently doing yes?
mystic vale
ok
wind stream
but since $f_\alpha(\frac{\pi}{2},\beta)=\cos(\beta)\sin(\pi+\beta)$, do you argee that $\sin(\pi+\beta)=0$?
flat frigateBOT
waler
wind stream
right, so that implies $\beta=-kpi$ for integer k
flat frigateBOT
waler
mystic vale
yes
wind stream
which means that there does not exist any $\beta\in[0,\frac{\pi}{2}]$
flat frigateBOT
waler
wind stream
can you see that this is a contradiction then?
mystic vale
mmm ok. and k starts from 1
mystic vale
can you see that this is a contradiction then?
looking into it hold on
yeah. it's a contradiction ok. cause before we tought b = pi/2 was a good choice
ok but now we've put it into a system so not all solutions match. so all good right?
cause i want to find where both partials =0 are true at the same time
wind stream
right, similar argument is made for the case $\alpha\neq\frac{\pi}{2}$ and $\beta=\frac{\pi}{2}$
flat frigateBOT
waler
wind stream
so that's the 'contradiction' bit out of the way
wind stream
i find
(+-pi/2, +-pi/2)
and
a = -b/2
b = -a/2
which i don't really know what t...
now, to this bit. The general solution you found is correct, everything you did afterwards to get b=-b/4 is also correct
can you solve for the value of beta in this case?
mystic vale
it can only be zero i think
mystic vale
yes cause similar argument can be made for alpha
wind stream
yes, or rather by the fact that alpha=-2beta
so you have found that alpha=beta=0 is another solution
mystic vale
ok yes
but (0,0) is where f is maximum so that is also not the point 'im looking for
wind stream
can you post the original question? From the snippet you sent I assumed that the problem asked for critical points
mystic vale
ok but the short version is i need to find the critical point with lower value (should be negative)
maybe you need also the problem3 he's mentioning
i'm trying to solve question a)
wind is w = (1,0) and i need to find the angle of 2 vectors that indicates the direction of the sail and of the boat.
wind stream
right so basically you want to find the minimum
mystic vale
so i need to find the best angle to have the boat to go contrary to the wind
yes
there's a missing piece in the solution, was in the other page.
wind stream
right let's return to $\sin(2\alpha + \beta)=0$ and $\sin(\alpha+2\beta)=0$
flat frigateBOT
waler
wind stream
and let's just solve this for all real alpha and beta
we should have that $2\alpha + \beta = k_1\pi$ and $\alpha + 2\beta = k_2\pi$ for $k\in\mathbb{Z}$
flat frigateBOT
waler
wind stream
do you follow up to this?
wind stream
distinguishing between k_1 and k_2 so as to help not miss any solutions
anyway, you can do a lot of thigns to solve for alpha and beta. But here, the quickest way is to just do row operations on the equations
solving this should give you $\alpha=\frac{2k_1+k_2}{3}\pi$
do you follow up to this?
mystic vale
i'm trying
wind stream
sure, just take your time
flat frigateBOT
waler
wind stream
oops sorry, a little typo, should be 2k_1
mystic vale
yes got the same yes
wind stream
yeah, and similarly $\beta=\frac{k_1+2k_2}{3}\pi$
flat frigateBOT
waler
mystic vale
yes
wind stream
so since we want solutions in [0, pi/2], we just need to restrict 2k_1+k_2 and k_1+2k_2 to [0,3/2]
mystic vale
ohhhh
wind stream
right, so since k_1 and k_2 are integers, there should be finite solutions to this system of inequalities
can you take it from here?
mystic vale
i know the sum of k1 and k2 can be only 0 or 1 to stay in the 3/2 limit
so i can have the combinations
0, 0
0, 1
1, 0
1, 1
wind stream
well usually, you would want to conider the equality case first
mystic vale
so since we want solutions in [0, pi/2], we just need to restrict 2k_1+k_2 and k...
oh no i wrote this wrong
wind stream
which of those did you write it wrong?
mystic vale
i'm trying to understand the k1_ + 2k_2 (i didnt wrote the 2)
do i need to find all combinations of k_1 and k_2 now?
wind stream
kind of, yes, but you should rather graph it out.
At the same time, you should also notice that you can simplify the inequality.
Since k_1 and k_2 are integers, it is not possible for any linear combination of k_1 and k_2 to be 3/2, in fact, any values between 1. and 3/2
so you can now simplify the inequalities down to:
0 <= 2k_1 + k_2 <= 1
and
0<= k_1 + 2k_2 <= 1
mystic vale
ok this is what i was doing before mentally. ok.
wind stream
right, so now you should solve for the intersections between the four equalities
that is (2k_1 + k_2 = 0 and 1) and (k_1+2k_2 = 0 and 1)
this should restrict the bounds of k_1 and k_2 further after which you would consider combinations of k_1 and k_2 satisfying the mentioned inequalities
mystic vale
so those are other 4 systems of equations to solve? or i should just test all the values? and find where those are equals?
wind stream
you would solve them as if they are real, so you can find the upper bound and lower bound for k_1 and k_2
to save time, I'm just going to continue, you can check the answers after you have worked it out.
Solving the intersections should give you (k_1, k_2) in
||{(-1/3, 2/3), (0,0), (1/3,1/3), (2/3,-1/3)}||
which gives you k_1 and k_2 in ||[-1/3, 2/3]||
and after this just check for integers values inside this bound to see which values satisfies the inequalities
wind stream
**waler**
Oops, I made a big mistake solving for $\alpha$ and $\beta$ here. It should rather be:
[\alpha = \frac{2k_1-k_2}{3}\pi, \beta=\frac{2k_2-k_1}{3}\pi]
flat frigateBOT
waler
wind stream
really sorry for that, the method still is the same however. The solution should instead be:
||{(1/3, 2/3), (0,0), (1,1), (2/3,1/3)}|| which implies that k_1 and k_2 is in ||[0,1]||
right, so only values of 0 and 1 are possible for k_1 and k_2, and you should see that k_1=0 and k_2=1 does not satisfies the inequalities.
This also implies that k_1=1 and k_2=0 does not satisfies the inequalities as well since the inequalities are symmetric
So you can easily see that only k_1=k_2=0 and k_1=k_2=1 satisfies
mystic vale
so i can find beta for those values
wind stream
yes
in the end, you should get that $\alpha=\beta=0$ and $\alpha=\beta=\frac{\pi}{3}$
mystic vale
and alpha. ok
flat frigateBOT
waler
mystic vale
ok I'll try to solve the inequalities.
but do i need to treat the k1 and k2 differntly if they are only integers? i mean . should i restrict the operations i do to them ?
i guess i need to work it out a bit more. for some reason i'm struggling a lot in some passages
wind stream
you do not, ideally when solving for system of inequalities for discrete variables, you want to solve for the boundaries so that you can test out some possible combinations
if you try to think 2k_1 - k_2 as a whole, you can think of a lot of combinations of k_1 and k_2 to test out.
Solving for the intersections helps you exhaust some of the combinations
if you need a visualisation. This is essentially what we are doing when solving for intersections
real values of k_1 and k_2 that satisfies the inequalities are points inside of the parallelogram spanned by the four points
mystic vale
ye i was using geogebra to visualize it and i've figured i had to find that
but hard time putting the pieces together with math itself
but those axes are alpha and beta right?
wind stream
eh no, k_1 and k_2 rather
mystic vale
yea sorry my bad, i was thinking something else
i'll keep trying retracing allthe steps today
you've been super helpful, thanks a lot
wind stream
no problem, glad I could help
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vale oriole
show that $\int_2^4 \frac{x}{ \sqrt{6x-8-x^2}}dx = 3 \pi$
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show that $\int_2^4 \frac{x}{ \sqrt{6x-8-x^2}}dx = 3 \pi$
vale oriole
bruh
flat frigateBOT
Galaxy
vale oriole
aight thx
im not sure where i went wrong here
i first simplified under the root to 1-(x-3)^2
then i tried letting x=sinu + 3
which means dx=cosu du
and bounds become from 3pi/2 to pi/2
which means i now have
$\int_ \frac{3 \pi }{2} ^ \frac{ \pi }{2} \frac{sin(u)+3}{ \sqrt{1-sin^2(u)}} * cos(u) du$
flat frigateBOT
Galaxy
vale oriole
denominator simplifies to cosu which cancels with the other cosu
so its just sinu+3
which integrates to -cosu+3u
but when i sub in bounds i get -3pi instead of 3pi
and idk where i went wrong
quasi sail
but when i sub in bounds i get -3pi instead of 3pi
Note that you always have to calculate Upper Bound Minus Lower Bound.
Since $\pi/2$ is less than $3\pi/2$. The roles of the bounds change.
flat frigateBOT
226PHIL
quasi sail
yes
vale oriole
are u sure?
u can calculate integrals from 1 to 0 and it works fine
in fact i integrate from v to 0 in a lot of my mechanics questions
where v>0
quasi sail
are u sure?
I am pretty sure you have to do it only when substituting. Hold on I am going to try to find out more
ok nvm you dont have to do it yeah. I rememberd that wrong (and it wouldnt change anything because you would change sign anyway)
quasi sail
denominator simplifies to cosu which cancels with the other cosu
I think the problem is here
vale oriole
right
quasi sail
denominator simplifies to cosu which cancels with the other cosu
the correct symplification should be$\sqrt{\cos^2(x)} = |cos(x)|$
flat frigateBOT
226PHIL
vale oriole
hmm that looks correct
quasi sail
and now using the fact that $\cos(x) \leq 0$ for $x \in [\pi/2,3\pi/2]$ you can further simplify to $|cos(x)|= -cos(x)$
flat frigateBOT
226PHIL
quasi sail
and now it cancels but you keep the -
vale oriole
right thx man
quasi sail
Sorry for my confusion earlier
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lone spruce
Can someone explain me why how z transform ROC works ?
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Can someone explain me why how z transform ROC works ?
lone spruce
<@&286206848099549185>
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<@&286206848099549185>
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quartz wren
Is this an differential equation? At first glance I would have said no but the result sinh and the fact that x and y are parametrized make me wonder
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Is this an differential equation? At first glance I would have said no but the r...
quartz wren
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kheerii
peak estuary
I mean I would just call it an integral
faint seal
all indefinite integrals are differential equations in disguise
quartz wren
Ok thanks, from my understanding a differential equation is a equation that contains the function itself as well as the derivative of the function. Here I just see the derivative but not the function itself. Does that have to do with it beeing paramterized?
faint seal
it doesn't necessarily have to contain the function itself
i mean, that just becomes an integral then i guess
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quartz wren
So is it a real differential equation or a kind of differential equation?
glad grail
guys i need help in this equation: 4x²- 12x + m-2 = 0
and find m
and delta need to equals 0
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guys i need help in this equation: 4x²- 12x + m-2 = 0
and find m
and delta need...
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glad grail
.close
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fathom adder
guys i need help in this equation: 4x²- 12x + m-2 = 0
and find m
and delta need...
144-16(m-2) = 0
and you solve
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uneven wren
I am trying to get 1/14 of 30, so the result is subtracted from 30
What I did first was to multiply 1/14 * 30 (fraction multiplication) = 15/7
then 30 - 15/7 (fraction subtraction) which I got 195/7 or the mixed fraction (27 6/7), the thing is that the solution on the book is 28. Maybe I am doing something wrong, or I have to round it up?
safe radishBOT
I am trying to get 1/14 of 30, so the result is subtracted from 30
What I did fi...
glass carbon
Make sure you rewrote it well
It seems they want 14/15 or 30 minus 1/15 of it
(or post the exact question)
uneven wren
(or post the exact question)
if the result of 30 - (1/14 of 30) is 28 or is it 27 6/7
glass carbon
27 6/7
uneven wren
27 6/7
gotcha. maybe I should reread the problem because the solution is 28
Thanks
.close
wise hound
What the best free resources are to learn math? And also what tools I should get?
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What the best free resources are to learn math? And also what tools I should get...
empty gyro
What the best free resources are to learn math? And also what tools I should get...
Khan Academy
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left python
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left python
folks what is the solution for this?
safe radishBOT
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 dont know where to begin
mortal arrow
Factorize the radicals
white umbra
I dont know where to begin
Do you know how to factorize a number?
left python
Do you know how to factorize a number?
wait lemme search it on youtube
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left python
Do you know how to factorize a number?
so i tried to factorize 1183 using the factor tree method but to no avail
white umbra
ah okay, what factors did you try dividing 1183 by?
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lone spruce
how the z transfomr of the first expression is z/z-1/4 while i have found 1/1-4z ?
safe radishBOT
how the z transfomr of the first expression is z/z-1/4 while i have found 1/1-4z...
lone spruce
<@&286206848099549185>
lone spruce
<@&286206848099549185>
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vague urchin
@lone spruce the z transform of (1/4)^n u(n) would be the sum from n=0 to infinity of (1/4)^n z^(-n), which is just a geometric series with common ration (1/4z). so the sum is 1/(1-(1/4z)), which simplifies to z/(z-1/4)
lone spruce
<@1211587354234458172> the z transform of (1/4)^n u(n) would be the sum from n=0...
yes if yo simplify it further its 4z-1/4z
lean otter
Can we use (y+c)/dy instead of integral?
Cus /(integral) d/dx y dx = y+c = (y+c)/dy d/dx y dx
safe radishBOT
Can we use (y+c)/dy instead of integral?
Cus /(integral) d/dx y dx = y+c = (y+c)...
lean otter
<@&286206848099549185>
lean otter
Can you post this in latex or as an image
There is no image because i didnt read it on anywhere, I found it when thinking
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compact wraith
There is no image because i didnt read it on anywhere, I found it when thinking
I mean can you draw it?
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lean otter
.reopen
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compact wraith
@lean otter can you please write it out on paper, take a picture and send it because I'm struggling to see what exactly you mean
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pastel plover
Hello! Only missing one spot but My answers arent working
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Hello! Only missing one spot but My answers arent working
thin bridge
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odd crest
Hello! Only missing one spot but My answers arent working
its just the volume of parabolid - volume of cone no?
so 1/2 pi r^2 h - 1/3 pi r^2 h = 1/6 pi r ^2 h
r = x coordinate where y=2x and y =x^2meets
and h = y coordinate
x=2 y=4
1/6 * pi *16 = 8pi/3
pastel plover
I am trying here sorry for taking so long.
pastel plover
First I got 64/15pi and then I simplify. V=π(15/160−15/96)𝑉=𝜋(6415)V=π(15/64)
thin bridge
show full work
start with how you set up your integral
and what you did to get those values
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upbeat ginkgo
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vapid pebble
Huh
Never seen this type of problem
Prohably wants you to use the limit definition of integrals
upbeat ginkgo
ah
vapid pebble
You're giving a series, after all ig
white umbra
What do you get if you plug in that first equation into the integral?
white umbra
Prohably wants you to use the limit definition of integrals
I don't think that's related
vapid pebble
Never seen this type of problem
.
upbeat ginkgo
What do you get if you plug in that first equation into the integral?
haven't tried that
vapid pebble
Was just guessin
white umbra
haven't tried that
Ah yeah, try it out
upbeat ginkgo
I was going off this:
white umbra
You want to do an approximation though, so the trick is to figure out how many terms from the first equation you need
Yeah
That's a very similar problem
upbeat ginkgo
I just don't know how to turn it into a series
white umbra
the first equation you're given is already a series
oh, do you mean how to write it in terms of the Sigma-notation
like (\sum)
flat frigateBOT
kalman_filtERIC
upbeat ginkgo
yeahhh
white umbra
Okay, so the first step would be to put that first equation you're given into that notation
How might you go about doing that
white umbra
just try to express the first equation as a sum first
without worrying about the rest of the problem
upbeat ginkgo
like this?
white umbra
no, I thought you wanted to write it using the (\sum) notation?
flat frigateBOT
kalman_filtERIC
white umbra
I'm just saying, write the first equation using the (\sum) notation
flat frigateBOT
kalman_filtERIC
upbeat ginkgo
oh
white umbra
btw the first equation is centered at u=0, you can tell because it's powers of u, and not powers of (u-0.1) or something
upbeat ginkgo
so the sigma n = 0 to infinity of tan^-1(x^2)
white umbra
No
white umbra
We're gonna have (\tan^{-1}(u) = \sum_{n=0}^\infty \dots)
I'm just asking you to fill in what the dot dot dots are
flat frigateBOT
kalman_filtERIC
white umbra
we haven't started with anything, we're just rewriting the first equation they give you with the sum notation
white umbra
no, (\sum_{n=0}^\infty \tan^{-1}(n)) would mean (\tan^{-1}(0) + \tan^{-1}(1) + \tan^{-1}(2) + \dots)
flat frigateBOT
kalman_filtERIC
white umbra
no, that's not what we want
white umbra
What we want is (u - \frac{u^3}3 + \frac{u^5}5 - \frac{u^7}7 + \dots)
flat frigateBOT
kalman_filtERIC
white umbra
I guess I should've asked first, do you know what the (\sum) symbol means?
flat frigateBOT
kalman_filtERIC
upbeat ginkgo
yeah, but I guess I don't have a proper grasp on it
white umbra
okay let's start with a simpler example then
can you tell me how you would write (1 + 2 + 3 + \dots + 100) using the sum symbol?
flat frigateBOT
kalman_filtERIC
