0 if n is odd 1/n otherwise
#help-49
spring latch
past aurora
true
visual tiger
0 if n is odd 1/n otherwise
1/(n/2 + 1)
spring latch
Thats all u need for limit
spring latch
1/(n/2 + 1)
I mean you can use this
past aurora
?
spring latch
But you dont need to
worthy wing
It was a correction
spring latch
For limit
hexed birch
take the limit for both
visual tiger
can someone explain
anyways, have you seen this theorem? https://proofwiki.org/wiki/Convergence_of_Odd_and_Even_Subsequences_to_Same_Limit
hexed birch
if they have the same limit the limit exists
spring latch
You can take the limit of the subsequences and see if they match
If they do thats the limit
past aurora
anyways, have you seen this theorem? https://proofwiki.org/wiki/Convergence_of_O...
did you googled my sequence by any chance
spring latch
In this case they do
junior flower
that theorem seems a little overkill in this case
spring latch
Its a pretty simple sequence to solve
spring latch
I heard there is a sequence encyclopedia
Oeis
past aurora
mmmmmmm
past aurora
You can take the limit of the subsequences and see if they match
how to do that
how to crack this ?
hexed birch
0 if n is odd 1/n otherwise
these are the subsequences
past aurora
oh
hexed birch
and 1/(n/2+1) not 1/n ofc
spring latch
Yes but
past aurora
why 1/(n/2+1)
hexed birch
the answer is the same
past aurora
why then?
spring latch
why then?
Bc it goes 1/2, 1/3, etc
At even numbers
So theres a +1 in denominator
And over 2 bc only for even numbers
The limit is the same tho
spring latch
This formula is for even numbers
hexed birch
but n is even
past aurora
how to calculate the limit of this $\lim_{n \to \infty} \frac{1}{\frac{n}{2} +1}$
grand pondBOT
milanesa de pollo
past aurora
its zero
spring latch
Approaches 0
past aurora
I mean I wanted the limit
spring latch
Bc it approaches 0
past aurora
but it was due to a bigger question
spring latch
ah okay
past aurora
the original question was if the sequence converges or diverges
midnight plankBOT
midnight plankBOT
Available help channel!
forest coyote
How exactly would one go about solving for B?
Also I would like help with the second bullet point
midnight plankBOT
How exactly would one go about solving for B?
Also I would like help with the se...
@forest coyote Has your question been resolved?
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@forest coyote Has your question been resolved?
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@forest coyote Has your question been resolved?
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golden cloak
<@&286206848099549185>
slow wagon
PURE PAIN
slow wagon
wth do i do after i draw diagram
midnight plankBOT
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slow wagon
is it projection vector>??
latent elk
is it projection vector>??
Lowkey yes
slow wagon
i forgot is vector projection a scalar quantity?
latent elk
vector projection is not a scalar quantity itself, but a scalar value obtained from projecting one vector onto another
slow wagon
once ive foudn the projection vector i need to find the distance between the pole and the cat?
i got an answer but its wrong
midnight plankBOT
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midnight plankBOT
latent elk
Modify it then
You know how to do it for 6 sides, then u should know how to do for 8 sides
Probably not to give u the answer directly
Rather teaching u the method
midnight plankBOT
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midnight plankBOT
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pastel tree
"Determine the largest and smallest value of the function"
midnight plankBOT
"Determine the largest and smallest value of the function"
pastel tree
I have determined that the tangentplane originates from the origin with the tangentplane [0,0] which would be the minimum value
However, i'm not sure how i'm supposed to determine the largest and smallest value of the function with those conditions
I would assume that the conditions is what limits the function?
$y <= 10 -x\
y >= 21/x\
y >= 0 \
x >= 0$
grand pondBOT
Merineth
midnight plankBOT
@pastel tree Has your question been resolved?
rose trout
The extreme value theorem tells us that if the domain is closed and the function is continuous on it, then it has to have an absolute minimum and an absolute maximum on it.
The thing is, by having this boundary, we might be artificially introducing extrema that aren't captured by the gradient. (Think of how you have to check the endpoints on a closed interval in 1D).
So, you have to do two things :
- find the points where the gradient is 0 inside the domain
- find the points where the function is optimized on the boundary (that is, when the inequalities are equalities)
For the boundary part, there are a multitude of tools you can use. Sometimes (here it works), you can use the equalities from the boundary to get an expression of y in terms of x, and you can plug that inside the original function. That yields a function of only one variable; you can optimize that fairly easily
You also need to check for cusps in the domain, here there is two.
Once you have all those points (usually not that many), you can evaluate the function there and pick the greatest to be the maximum and the smallest to be the minimum.
pastel tree
Oh wowie that's a lot 
1) find the points where the gradient is 0 inside the domain
I would assume that when i made the partial derivative with respect to x and y which gave me the gradient [2x,8y] It showed me that the only point where the global maximum is when x = y = 0
Would that be correct so far?
rose trout
Is that in the domain though?
pastel tree
I'm honestly not sure
well
x and y is equal or greater than 0
however xy >= 21 does not hold
so i would assume it does not?
rose trout
Indeed!
So that means that on the interior of the domain, there is no place where the gradient is 0.
rose trout
Now the boundary remains. You can separate it into parts.
If we take the part of the boundary that satisfies x+y = 10, can you use that to make f into a function of x only?
rose trout
y = 10 - x
Good, so now sub that in, and find the extrema of that just in terms of x using 1D calc.
rose trout
Using the same principle for xy = 21 where y = 21/x ?
Yes, but not simultaneously. You'll need to solve for one at a time
pastel tree
Ok!
Well if i had the function y = 10 - x
The extreme point would be when y = -1 Since i'm deriving with respect to x
rose trout
You have to sub it into f. We're trying to find the values of f(x) = x^2 + 4y^2 on that part of the boundary
grand pondBOT
Merineth
pastel tree
$f(x,y) = x^2 + 400-80x+4x^2$
grand pondBOT
Merineth
pastel tree
$f(x,y) = 5x^2-80x+400$
grand pondBOT
Merineth
pastel tree
So this equation here would account for the boundary x + y >= 10
If i'm thinking correct ofc :p
rose trout
Yes. For any given x, this gives you the corresponding value of the function on the boundary x+y=10
So you want to take the derivative of that and get the critical points
pastel tree
$f'(x,y) = [10x-80, 0]$
grand pondBOT
Merineth
pastel tree
It does have an extreme when x = 8 and y = 0
rose trout
Also at this point you can think of it just as a function of x, no need to keep up with the y part.
X=8 is right, but we know that x+y = 10 on this part, so what is y?
pastel tree
y = 10x-80 ?
rose trout
If x=8, 8 +y = 10
rose trout
Yep
pastel tree
So we figured out that an extreme point exists within the boundary x+y <= 10.
Which is when x = 8 and y = 2?
rose trout
So we've got one possible point on the boundary, but we still have to make sure that the point is actually on the boundary of D (here we found an extremum on the whole line x+y=10)
The Line and the hyperbola meet at (3,7) and (7,3), so x=8 is too far. That means we won't be considering this point at the very end.
rose trout
So we figured out that an extreme point exists within the boundary x+y <= 10.
Wh...
We're just going to make a list of points to consider, and at the end we'll see which is the highest and lowest
pastel tree
Oh alright
So we were cosindering (8,2) as a point
Is it because the point xy >= 21 does not satify it?
rose trout
Yep! We're just narrowing down the possible extreme points to a few, and then we can just evaluate the function at those points to retrieve the maximum/minimum
rose trout
Is it because the point xy >= 21 does not satify it?
Yes essentially
pastel tree
Okay nice, just making sure i'm hanging in there :p
rose trout
Okay! One left to go.
We're now considering f when restricted to the hyperbola xy=21
pastel tree
Right so
y = 21/x would make the function :
$f(x) = x^2+4(21/x)^2 \implies f'(x) = 2x- \frac{3528}{x^3}$
grand pondBOT
Merineth
pastel tree
And now we find = 0 ot that derivative
rose trout
Yeah you can just set it =0 and multiply both sides by x^3
The big number you can keep as 42^2, it'll be 4th rooted in a minute
grand pondBOT
Merineth
pastel tree
$\sqrt{42}$
grand pondBOT
Merineth
pastel tree
Would be the answer
pm ofc
And following the same method we did before
$y = \frac{21}{\sqrt{42}}$
grand pondBOT
Merineth
pastel tree
This does however satisfy the condition earlier where x + y <= 10
rose trout
Yep! So this one we keep!
pastel tree
Nice!
rose trout
Now the last last thing is to check the points we found. The thing is, even those points don't take into account the "corners" of the domain, so we're also going to be checking (3,7) and (7,3)
(and we also found only one, so unless the function is constant, it wouldn't make sense for that point to be both the max and min)
pastel tree
How did you get (3,7) and (7,3) ?
rose trout
How did you get (3,7) and (7,3) ?
By solving for the intersection of the line and the hyperbola
rose trout
x+ y = 10 ; xy=21
pastel tree
I do know it looks something like this (for the whole function not including the boundaries)
$y = 10 - x \implies y = 10 - 21/y$
grand pondBOT
Merineth
pastel tree
Is that what you did?
rose trout
Tbf I just tried the couple 3,7 because 3*7 =21 and then 3+7=10
But yeah that's how you'd solve it
You'd multiply through by y and it gives a quadratic
pastel tree
Yea but it'd end up with sqrt(-11)?
That would result in a imaginary value
NVM
$y^2 = 10y- 21 \ implies y^2-10y + 21$
grand pondBOT
Merineth
rose trout
Yep
pastel tree
Well sqrt of 42 is roughly 6,48
Which is my x value
well pm 6,48
Meaning the only value that holds true for that x value would be the positive one
$x = \sqrt{42}$
grand pondBOT
Merineth
pastel tree
i.e the only value that x can be is when the root is positive
rose trout
Yes. So now you can take the points (3,7) , (7,3) and (sqrt(42), 21/sqrt(42)) and evaluate the function there
pastel tree
same goes with y since y=21/42 which is 21/sqrt(42) almost equal to 3,24
So isn't the only extreme point :
(sqrt(42), 21/sqrt(42)
Where both of the sqrt has to be positive for it to be within the (3,7) and (7,3) line
$f(x,y) = \sqrt{42} + 4(\frac{21}{\sqrt{42}})^2}$
rose trout
We're checking the corners as well, since the derivatives we took don't take them into account
grand pondBOT
Merineth
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
pastel tree
I'm not sure i understand why we are checking the corners
We found out that an extreme exists that fulfills the conditions set by our function
Isn't that the only one that applies then?
rose trout
It's kind of the same reason we have to check the boundary of D in the first place.
At the very base of it, the reason is that the derivative gives us information on open sets, and not on their boundaries.
So when we solve for the critical points on x+y=10, nowhere in there is there information that the line is between (3,7) and (7,3). The line itself we can think of as a closed domain and we have to check the endpoints.
The graph of XY=21 is closed as well, so we have to check the endpoints.
Moreover, if that value was the only extreme value, it means it's both the absolute min and max, and the only way that's possible is if f is constant on D, but that's not true.
So in general : check the interior of the domain, check the boundary of the domain, check any cusps in the boundary
pastel tree
Hmm
I'm honestly not exactly sure of what you mean
I went and did some research about it but
This guy never checks the boundaries that you meantion with (7,3) and (3,7)
He did exactly like we did to find out the extreme points
But then it stops?
Aren't we essentially done?
We found the extremevalue that is withing the boundary of our function
x = sqrt(42)
y= 21/sqrt(42)
Where the sqrt are both positive
rose trout
That's because the boundary he checks has no cusps
rose trout
We found the extremevalue that is withing the boundary of our function
The thing is, the domain is closed, so there has to be an absolute max and an absolute min
pastel tree
That's because the boundary he checks has no cusps
I sadly don't know what a cusp is
I tried googling it and get something about teeth? lmao
rose trout
A pointy part if you'd rather.
What's creating this is our boundary is made of two curves, no just a circle like in his case.
rose trout
That's the boundary. Just like we checked the boundary of D when finding critical points on it, we need to check the boundary of the lines to find the critical points on them, as you did.
But the boundary of the red line is just the two points (3,7) and (7,3).
Same with the blue graph
pastel tree
I'm so sorry haha but i have no idea what we are doing
We had a function f(x,y) x^2+4y^2
we found out that there is an extremepoint within this function with the set boundaries of x+y <=10 and xy >= 21
Is the problem that we don't know if this is a max or min?
rose trout
We found out there is a .possible. extreme point.
And again, unless the function is constant on that domain D, which it isn't, you'll at least get two points to check.
pastel tree
We proved that it is an extreme point because the x and y fit the equations of our boundaries?
rose trout
We proved that it is an extreme point because the x and y fit the equations of o...
We proved it's a critical point of the function on the boundary.
We want critical points inside, given by the gradient, and on the boundary.
For the boundary, we split it into 2 parts and found a single critical point on there, but the derivatives we took to do that don't take into account that the lines that make up the boundary are cut off.
Just like if we wanted to find the extrema of x^2 on [1,4], we have to check the endpoints.
Taking the derivative gives 2x, but we can't stop and say that the function has no min/max because 0 is not in the interval. Clearly, 1 is a min and 16 is a max.
pastel tree
I've never encountered this before
I sadly don't understand
Is it possible to visualize this somehow?
Seems impossible to learn just by looking at it like this
rose trout
I think just the thumbnail of the video you sent helps a bit.
Can you see how the function itself, the blue paraboloid, doesn't have any maximum without the domain specified?
rose trout
Right, so adding the domain did something, and that something is that the boundary introduced some extreme value on it.
But if you take the gradient, you wouldn't be able to find this max
pastel tree
But we found the critical points of the function which was at 0,0 which was not within the boundaries, hence we disregarded it.
Then we found the critical points of the boundaries and found that one was x=sqrt(42) and y=21/sqrt(42)
We found that this critical point is indeed inside the boundary and inside the function.
I don't understand but aren't we done here?
We just have to identify if it's a max or min
rose trout
We want to find all critical points on the boundary.
In some sense, we could say that a critical point arises when the derivative is 0 (we did that) or when it doesn't exist. The derivative doesn't exist on those endpoints to the domain.
pastel tree
But we found all the critical points?
By deriving a function and setting it = 0, and solve it we find them?
rose trout
We found the points where the derivative is 0 on the boundary
pastel tree
Isn't it essentially the same in one variable calc+
rose trout
Not when it doesn't exkst
rose trout
Isn't it essentially the same in one variable calc+
Think of how you would do this on, say, y=|x|
The derivative isn't defined at 0, but it's still a critical point since it doesn't exist
pastel tree
Since it doesn't exist? It clearly does?
pastel tree
Haha i'm sorry but this is very very very confusing xD
I was under the assumption that all critical points are found when we derive and set the derived function = 0
Is that not how it's done?
rose trout
No. Critical points include points where the derivative isn't defined as well.
pastel tree
Alright no wonder i have no idea what we are doing since i had no idea that was even a thing
I'm totally lost now, what are we trying to figure out?
We found that a critical point exist on the boundary which is inside the function
Isn't that what the question asked for?
rose trout
But unless we consider all critical points, we can't know if it's a max, a min or neither
pastel tree
Ok so we want to find imaginary critical points to determine if it's a max, min or neither
rose trout
Sort of yeah. The derivative is oblivious to the boundary of D and the bojdnarh of the curves that make up D, so we have to check manually
tribal temple
wait where does the z = 84 come from again 
pastel tree
i'm not sure lmao
I have no idea anymore what we were originally doing
I just found a critical point at (sqrt(42), 21/sqrt(42))
which is on the boundary and inside the function
tribal temple
Oh so you put that into the function right?
pastel tree
Yea
tribal temple
Fairs, well, a better picture would be (it does work out to 84 for sure)
it doesn't seem to be working out the way I wanted it to 
Oh no I'm just dumb 
pastel tree
Then what does that make me?...
literally might as well remove my brain since i'm not using it
tribal temple
Anyways, as per here, you can see where you have the y = 10 - x, the intersection of that and the surface f is basically like "a bit of a parabola", and that if you go along that intersection, that the maximum is "at the top"
tribal temple
literally might as well remove my brain since i'm not using it
I mean, you haven't reached "didn't even adjust the graph scale" level yet, so there's that 
pastel tree
I've already done that mistake like 5 times
tribal temple
An easy one to make tbf 
tribal temple
Anyways, as per here, you can see where you have the y = 10 - x, the intersectio...
(oh wait, took the wrong one 
not that one-)
Better 
anyways notice how you like effectively again have that "bit of parabola" and that it has that minimum point at the one you found
Which corresponds to what we'd expect hopefully, that the point we find would be something like that
pastel tree
ah right
Yeah that is the point that we found
If i make the x and y negative then i get another point on the opposite side aswell
tribal temple
If i make the x and y negative then i get another point on the opposite side asw...
yep
Of course along that constraint there isn't really any maximum you can reach, nothing "holds you back" from raising 
tribal temple
Yep, corresponding to being on the boundary xy = 21
There's also that other boundary to consider (which I should scroll up for 
)
pastel tree
He explained very well
i just got lost on the last part
I was sure that was it
chartbit :c
am i not done? :C
tribal temple
i just got lost on the last part
The bit about not being differentiable?
tribal temple
chartbit :c
Sorry I was getting food 
tribal temple
Some burritos 🌯
pastel tree
yum C:
tribal temple
And oh yea, so, like, you know if, say, I told you to find the maximum and minimum value of a single variable [continuous] function on some closed interval, right?
And that in addition to checking the points where you have derivative zero (and also points where the derivative doesn't exist, as per |x|) you need to check the endpoints of the closed interval too?
pastel tree
I didn't know i had to check the endpoints
if i were given such a question i would find all the max and min by deriving and setting = 0
and then i just exclude all of the points that aren't in the intervall
tribal temple
Alright, well, for an explicit example, let's say-
We want the maximum and minimum values of x^3 - x on the interval [-1, 2]
The maximum value you can get on that interval is actually 6, which you get from x = 2
(that's also the difference between "local" maximums and "global" ones, the turning points you'll find are only "local" because when you move far away enough, you can exceed/go below said max/min)
pastel tree
$f(x) = x^3 - x \
f'(x) = 3x^2 - 1 \
3x^2 = 1 \implies x^2 = 1/3 \implies x = \sqrt{1/3}$
grand pondBOT
Merineth
pastel tree
That would give me 2 possible critical points
then i'd just check if either of these are withing the interval -1 and 2
Isn't that essentially what we did?
tribal temple
Yes, but you also need to check the endpoints: where's the largest value you can get here?
It isn't the one corresponding to -1/sqrt{3}
pastel tree
Oh so the endpoints and startpoints count as derivative = 0?
incline/decline is zeor?
tribal temple
They don't, but you still need to check them: the derivative at the endpoints here isn't zero (you can see you're still increasing at both)
pastel tree
So what are we checking them for?
tribal temple
Yes, but you also need to check the endpoints: where's the largest value you can...
Because you might have that they're the largest point: as before, see the picture, the largest output is 6, isn't it?
pastel tree
ooooooooooh
we are going for largest point
not the actual derivative = 0
the derivative = 0 only finds all the points where it isn't increasing or declining not if it's the actual highest or lowest value
tribal temple
More that the derivative being zero will tell you about "local" max/mins, but you don't know if they're global, because you might end up increasing at some point again
as per the picture, there's that turning point at -1/sqrt{3}, but eventually we go above it, in some examples you won't, but in this one you do
grand pondBOT
Merineth
pastel tree
So we take our boundaries and figure out from where it starts to ends?
So when we input (7,3) we get 85
meanwhile (3,7) we get 205
So what does this point mean? xD
seems wrong?
tribal temple
Make sure to also include that x + y = 10 I think
pastel tree
ooh this is the point where the green penetrates?
or dissapates rather
So our global maximum is (3,7)
global minimum is sqrt(42), 21/sqrt(42)
and local min is (7,3)
tribal temple
probably worth adding the z value too
tribal temple
A point is also that it may be worth checking some of the other points you find are also within the domain but "global" would be that of all the points you've tried, that it's the largest/smallest
pastel tree
So what does this point mean? xD
Yeah but i mean for example the blue point here
Is essentially this point here
I see that it's a local min because of the graph, but how do i know algebra?
tribal temple
Well you could e.g. use the second derivative to see it's a local one if you wanted, sure
pastel tree
I've never taken second derivative of a two variable before
tribal temple
[though considering we're only caring about the "global" ones, you don't need to]
pastel tree
do i just derive the vector?
tribal temple
Well, take e.g. the partial wrt x and differentiate that wrt both x and y, same for the partial wrt y
See e.g. Hessian matrices for how to determine whether it's a max/min/saddle point etc
pastel tree
$x^2+4y^2$
grand pondBOT
Merineth
pastel tree
$[2x, 8y]$
grand pondBOT
Merineth
pastel tree
$[2, 8]$
grand pondBOT
Merineth
pastel tree
Not like this?
tribal temple
Not exactly like that, and the second derivatives will get you a 2 x 2 matrix
Differentiate 2x wrt x, and wrt y, and then also differentiate 8y wrt x, and wrt y
tribal temple
Then there's something that tells you about the nature of the point you found, which I don't remember 
pastel tree
Well im finally done now
Honestly i'll have to wife you if i manage to pass multivariable calc
rose trout
I'm so sorry something came up
pastel tree
Quite literally holding my hand like a toddler trying to make me pass calc 3
Don't worry about it!
Chartbit to the rescue
rose trout
As was foretold
pastel tree
Also i usually tend to take a long while to understand and learn from what i'm doing so it's understandable
But now i'm finally done 
HAHAHAHHAA
I didn't mean to send that but i love it
thanks for the help 🫶 both of you
See you tomorrow
.close
midnight plankBOT
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Available help channel!
eternal pine
i need to solve at least 2 more of these which ones are easy to find the limits for
midnight plankBOT
i need to solve at least 2 more of these which ones are easy to find the limits ...
zenith stone
i need to solve at least 2 more of these which ones are easy to find the limits ...
4 and 5 ig
eternal pine
4 and 5 ig
can 5 be solved quickly by using l'hospital
eternal pine
ah nice 👍
midnight plankBOT
@eternal pine Has your question been resolved?
eternal pine
oh wait 5
is 4 doable as well?
i dunno how to use l'hopital on that one
someone solved a similar one like this
i don't get the double fraction
midnight plankBOT
@eternal pine Has your question been resolved?
eternal pine
/solved
zenith stone
do .solved
midnight plankBOT
Available help channel!
zenith tree
This is an accurate description of the accelerometer in Android phones. However, Apple decided to negate the vector. i.e placing an iPhone flat will read (0,0,-1) g instead of (0,0,1) g .
midnight plankBOT
This is an accurate description of the accelerometer in Android phones. However,...
zenith tree
Whats a good description for the way Apple does it? This will help me to avoid confusion
I'm thinking "the acceleration of an object in free fall relative to the device"
latent elk
Probably its just how Apple set it
Mathematically speaking, u can set the whatever vector to prepresent "flat" position
zenith tree
Mathematically speaking, u can set the whatever vector to prepresent "flat" posi...
Can you expand on that? Acceleration is the 2nd derivative w.r.t time, and Apple is clear about the coordinate system of the device.
To say something like "this vector is the acceleration of the device" seems plain wrong to me.
latent elk
Your question seems like a question of convention
zenith tree
Yes. I'm not saying either is wrong, just that one is clearly described
Apples docs don't cover this
midnight plankBOT
@zenith tree Has your question been resolved?
midnight plankBOT
@zenith tree Has your question been resolved?
zenith tree
zenith tree
I'm thinking "the acceleration of an object in free fall relative to the device"
I'll just roll with this ig
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alpine hound
midnight plankBOT
alpine hound
i am having problems with taking the limit of negative infinity, when i compute i get negative infinity as my answer but still like photo math says it is not defniedn
yet when i plug the equation into desmos, it is defined and continios
ie:
why?
am i missing something?
ruby imp
What’s your working out
ruby imp
Do you know about indeterminant forms and lhopitals rule
alpine hound
no we havent been taught that....
we are doing curve skethching rn i think thats our next unit tho
rose trout
,rccw
grand pondBOT
alpine hound
that is a cool ass tool never seen that
is my computation correct?
other classmates have gotten that aswell
rose trout
Both go to -infinity. Your working is fine. If x is big, -x^2/3 dominates and that goes to -infinity. If x is -big, both terms are negative and grow, so it goes to -infinity
alpine hound
so then why do online tools say that it is undefined?
rose trout
Sometimes, programs will say a limit doesn't exist if it diverges.
alpine hound
can i just assume that it approaches negative infinity then?
it was multiple programs that said it was undefined thats why i was confused
photomath aswell
shut canyon
I'm trying to show that a string of composable arrows of any length has one unambiguous composite. How to approach this when I struggle to begin?
midnight plankBOT
I'm trying to show that a string of composable arrows of *any* length has one un...
shut canyon
Does it have more possible composites? Or is there always one
"one unambiguous"
midnight plankBOT
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lilac nebula
I'm trying to show that a string of composable arrows of *any* length has one un...
Can you guess why there might be multiple possible compositions?
A priori, at least, why there is something that we have to prove
midnight plankBOT
@shut canyon Has your question been resolved?
shut canyon
Can you guess why there might be multiple possible compositions?
If there is a path from a to b, and a path fromb to c, then there is a path from a to c. If these places are connected, then they are connected.
If the characters of the alphabet are connected, then we can get from a to z
shut canyon
A priori, at least, why there is something that we have to prove
Perhaps, for example, so that we don't have to draw infinitely many arrows, or visualise arrows in many dimensions. If we know the regularity, and that it works in the world we are navigating, then we can deduce truths about stuff that would be too complex to think about otherwise.
midnight plankBOT
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copper ginkgo
midnight plankBOT
copper ginkgo
does AB have change in KE?
ik that OA has change in KE cause it starts from rest and travel up to A
.close
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hearty inlet
hi does anyone know what substitution i could make to solve this. i've tried u = 1/x and ended up with (-1/u)(sin(1/u)/(sin(1/u)+sinu) which doesnt seem to help
midnight plankBOT
hi does anyone know what substitution i could make to solve this. i've tried u ...
twilit field
are you sure it's integrable in terms of elementary functions?
midnight plankBOT
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hearty inlet
are you sure it's integrable in terms of elementary functions?
i have it as a question in my tuition's booklet. there's also boundaries 0.5 --> 2. all it says is to make a suitable substitution to solve. also idk what elementarry functions are sorry lol
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balmy zodiac
Is this correct?
midnight plankBOT
Is this correct?
dawn dagger
Is this correct?
It'S in German
balmy zodiac
danke schön
What about this?
i am a bit confused since the betrag can be minus and plus and technically you can add Flächen if you use the Betrag?
dawn dagger
yes correct
balmy zodiac
the first one is incorrect since one of the grau markierten Bereiche is minus on one side?
dawn dagger
If you wanna do it correctly dann betrachtest du die Flächen separat
grand pondBOT
𝔸dωn𝓲²s
dawn dagger
Die 2. Ankreuzmöglichkeit ist das was dabei herauskommt, wenn man den Betrag richtig einsetzt
Die 1. Antwortmöglichkeit ist richtig
Die Funktion is punktsymmtrisch
balmy zodiac
**𝔸dωn𝓲²s**
this looks so confusing
dawn dagger
thanks
balmy zodiac
xd
dawn dagger
2 mal die Fläche von 0 bis 4 entspricht der 2. antwortmöglichkeit
Betrag heißt nur
wenn der innere Term negativ wird
dann kannste die Striche wegmachen
machst aber halt ein Minus davor
wenns sonst positiv ist, lässt du die Striche ganz weg
ganz einfache Geschichte
balmy zodiac
**𝔸dωn𝓲²s**
why is there an extra minus before intagral -4 to 0?
dawn dagger
I just explained why
Die Fläche von -4 bis 0 ist negativ
das Innere ist also negativ
um die Betragsstriche loszuwerden
machen wir ein - davor
grand pondBOT
𝔸dωn𝓲²s
balmy zodiac
okay so basically i explained it already:
If we put away the Striche then we get a minus in front of our integral which is the negativ Bereich
dawn dagger
yea (kinda)
balmy zodiac
although i thought we use betrag in order to add two Bereiche regardless if one is negativ
so that we get an addition without loosing the value of the one in a negative zone?
dawn dagger
No we do it if we are interested in only positive values
balmy zodiac
i think we are on the same page just with a different explaination?
dawn dagger
i am not even in the book
dawn dagger
yes i know
balmy zodiac
is that sometimes
dawn dagger
my humor confuses you
balmy zodiac
yeah
that too
i will just t pose like this so that the teachers let me pass
okay so
dawn dagger
nah you good
balmy zodiac
sometimes we subtract the negative Bereiche from the positive ones
and sometimes we are interested in the whole Breiche regardless if they are negative
why?
In real life there is no negative Bereiche so sometimes you do have to count everything together
or why?
do my questions make sense?
sorry if not
dawn dagger
Bilanz
Imagine you have a Konto
Last month -200 €
this month 400 €
Insgesamt sind es 600 € vom Betrag her
aber die Bilanz wäre 200 €
und je nach Kontext interessiert man sich für die Bilanz oder den Absolutwert
Bilanz hier, um zusehen wie viel du übrig hsat
In anderem Kontext kann es sein du hast eine Funktion die dir angibt wie viel Fläche du brauchst um zum Beispiel eine Dose zu produzieren, da kann es vorkommen, dass du negative Fläche ausgerechnet hast, aber den absoluten Betrag nimmst
balmy zodiac
so you were in -200 but since you payed 600 and your current standing is 400, the whole money you payed is still 600?
I feel like this is very simple but I keep overthinking it
balmy zodiac
okay so we add minus and plus when we are interested in the Flächenbilanz
balmy zodiac
so you were in -200 but since you payed 600 and your current standing is 400, th...
okay so we add minus and plus when we are interested in the Flächenbilanz
And then there is the Absolutwert
which is only the plus without the negative space?
balmy zodiac
ah i see and when it comes to bilanz, for example in economics we are only interested in the money that we have zur Verfügung?
dawn dagger
we are interested in the difference basically
yes
are you really from Germany
Ah ok (basically the same)
No
You asked if I am from Austria
I said no
And you concluded I am not from Germany
Yea
Germany and Austria are alike
balmy zodiac
okay, now i get it
dawn dagger
ok
balmy zodiac
.close
midnight plankBOT
dawn dagger
that was too much german experience huh
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shell axle
Hey can someone help w this i dokt get anything
midnight plankBOT
Hey can someone help w this i dokt get anything
midnight plankBOT
@shell axle Has your question been resolved?
pallid juniper
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radiant mesa
Hello guys, I'm struggling with combinatorics. My question: (1) The only thing you remember about your classmate's phone number is that it is nine digits long, starts with a three, contains no two
the same digits and is divisible by twenty-five. Determine how many phone numbers are involved.
midnight plankBOT
Hello guys, I'm struggling with combinatorics. My question: (1) The only thing y...
main current
"no two of the same digits"?
radiant mesa
yeah, it doesn't contain 2 same digits
15120
correct answer
yet im unable to solve this
midnight plankBOT
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bleak wren
If sinxcosx = 1/6
What is cos4x?
Solve this using trignometric identities, and a four function calculator (no cos function).
midnight plankBOT
If sinxcosx = 1/6
What is cos4x?
Solve this using trignometric identities, and...
bleak wren
Only got to sin2x = 1/3, anything I'm trying right now ends back here.
sharp coral
now you should try to find cos(4x) in terms of sin(2x)
bleak wren
Can you please elaborate?
tribal temple
I can see you're somewhat familiar with trig identities: if I gave you something like cos(2u), can you write that differently?
midnight plankBOT
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indigo peak
how to find x in a circle
midnight plankBOT
how to find x in a circle
indigo peak
<@&286206848099549185>
surreal moon
!15m
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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.
indigo peak
???
surreal moon
<@&286206848099549185>
Please wait 15 minutes before pinging helpers if you have received no help
indigo peak
Yea the pic wont upload
how do i do number 21
Can someone help me find the value of x?
<@&286206848099549185> bot isnt helping me can someone help me instead?
small jasper
<@&286206848099549185> bot isnt helping me can someone help me instead?
The bot won't ever help you (it's a bot)
indigo peak
true
small jasper
Assuming that the dot is the centre, you have that the bottom side is a diameter
which implies this is a right angle
small jasper
well you have a triangle and all three angles (though two of them are in terms of x)
can you think of a relationship w/ these three angles
indigo peak
right triangle?
small jasper
yup you can apply that
small jasper
what about the 90 deg angle
small jasper
yeah
small jasper
,w 2x+9+4x-2=91
grand pondBOT
indigo peak
deduce?
small jasper
deduce = determine from previous info
indigo peak
st?
indigo peak
adjacent?
small jasper
I meant its measure
indigo peak
ohh
small jasper
given arc RS
indigo peak
110
small jasper
yeah
so arcs RS and ST are 110 deg
and we need to find RT
Hint: ||what is the measure of the total circumference of a circle||
indigo peak
so 360-110
small jasper
there's two 110 deg arcs
small jasper
?????
small jasper
yeah your teacher is wrong
small jasper
If BA is a diameter, what can you deduce abt the arc measures
just list whatever you can
small jasper
abt the arc measures
indigo peak
oh
ca is 2x+17 and bc is 4x- 5
right?
4x-5+2x+17=360?
or 180
yo
,w 4x-5+2x+17=360
grand pondBOT
grand pondBOT
indigo peak
,w 20x-40=360
grand pondBOT
grand pondBOT
grand pondBOT
indigo peak
,w 8(x+8)=6(12)
grand pondBOT
indigo peak
,w 4(4+x)=8^2
grand pondBOT
midnight plankBOT
indigo peak
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hushed solar
How do I prove that the only way to completely fill rectangles with L tetrominos is when the L tetrominos pairs up to form small rectangles
midnight plankBOT
How do I prove that the only way to completely fill rectangles with L tetrominos...
hushed solar
Is proving this in a black and white chessboard a good idea?
I feel like there's an argument to be made with colors
surreal moon
oh you mean the L-shaped tetrminos?
hushed solar
Yes
hushed solar
I feel like there's an argument to be made with colors
Nvm
This doesn't work
I have a stupid idea. We can easily show that rectangles that are able to be filled with L tetrominos can be split into smaller 4x2 rectangles
And then uhh
surreal moon
That's only showing that 4x2 rectangles is a way that works. You're asking to show that no other way works at all
hushed solar
That's only showing that 4x2 rectangles is a way that works. You're asking to sh...
Yeah I'm thinking of some way of doing that
surreal moon
things to note: a tetrimino has 4 pieces, so you only need to consider rectangles with areas in multiples of 4
hushed solar
That's only showing that 4x2 rectangles is a way that works. You're asking to sh...
With the fact that the L tetrominos must also fill all the smaller rectangles
surreal moon
And then, if you can prove that an odd number of L minos can never fill a rectangle, then you only need to rects with areas in multiples of 8
hushed solar
And then, if you can prove that an odd number of L minos can never fill a rectan...
And then what do I do
surreal moon
Then, if you can prove that no side of the rect can have odd length, then you've reduced the problem to only rectangles where one side's length is a multiple of 2 and the other side length is a multiple of 4
after that is where it gets hard, but you've narrowed it down (assuming you can prove all of those other things)
but I suppose once you get to that stage, you can consider the rectangle as being built from several 2x4 rectangles and that might be like your domino idea
midnight plankBOT
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hushed solar
Is this statement even true?
surreal moon
Is this statement even true?
Unsure. It's your question
midnight plankBOT
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Available help channel!
waxen silo
can someone help explain this question to me?
midnight plankBOT
can someone help explain this question to me?
waxen silo
is it just saying that r is sqrt(3+4i) sand s is sqrt(3-4i)
because the non negative real part is throwing me off
.close
midnight plankBOT
novel herald
are you familiar with central limit theorem
basically when you have IID Xi, their mean tends toward a normal distribution
which is what you have here
each Xi is a bernoulli distribution
iid is independent identically distributed
Xi is $X_i$ a random variable indexed by i
grand pondBOT
chebyshev's infinite pee norm
novel herald
so you have 100 independent identically distributed bernoulli trials
they are basically just saying, let's consider this to be a normal distribution since we have a really high sample size
so 100 coin flips is a binomial distribution but it's really close to a normal distribution
so what will happen when we approximate the probability with a normal distribution?
and those are where the phi(1.1) - phi(-1.1) come from
are you familiar with zscores
yes
it's just the zscore
yes
what is the average number of heads from 100 coin flips
and what is the standard deviation
novel herald
i know i know your name from somewhere
from where!!!
junior flower
somewhere
shut canyon
$2^{40}, 2^{40}+1$ and $2^{40}-1$ are elements of the composite numbers
midnight plankBOT
$2^{40}, 2^{40}+1$ and $2^{40}-1$ are elements of the composite numbers
flat spire
use dollar signs only around the math environment
shut canyon
Okay
flat spire
u dont need to back slash every word
grand pondBOT
شعاع الماء
grand pondBOT
kytsu
midnight plankBOT
@shut canyon Has your question been resolved?
midnight plankBOT
@shut canyon Has your question been resolved?
shut canyon
Just found this
in Daniel J Velleman - How to Prove It
I still struggle to see the generalisation, I guess primes are unwieldy
grand pondBOT
kytsu
shut canyon
$2^{n}=2^{ab}=2^{xy}$
grand pondBOT
kytsu
shut canyon
" Thus, we have shown that 2^n − 1 can be written as the product of two positive integers x and y, both of which are smaller than 2^n − 1, so 2^n − 1 is not prime."
How to read $x = 2^b - 1 < 2^n - 1$?
grand pondBOT
kytsu
shut canyon
x is less than 2^n - 1
Why, it is a jigsaw puzzle!
I'm also trying to read the proof as it was computer code, not line by line, but in a structural way.
this is from The Nuts and Bolts of Proofs by Antonella Cupillari
I also sense that it is the relationships in this proof, rather than the actual powers of 2 that are illuminating
shut canyon
I don;t understand what they do with x and y,
x = 2^b - 1
y = 1 + 2^b + 2^2b + ... + 2^(a-1)b
...
xy = 2^b - 1
midnight plankBOT
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last slate
my book is showcasing a proof of [
\vj A \cross (\vj B \cross \vj C) = \vj B(\vj A \vdot\vj C) - \vj C(\vj A \vd \vj C)
]
midnight plankBOT
my book is showcasing a proof of \[
\vj A \cross (\vj B \cross \vj C) = \vj B(\v...
grand pondBOT
last slate
i dont understand the highlighted part
i cannot rationalise why D + KA_|| makes sense, i mean, they are orthogonal vectors?
i would understand something like D + ka_d instead
btw the notation a_i denotes the unit vector in the i direction
Oh nvm got it
.close
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near flame
<@&286206848099549185>
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near flame
<@&286206848099549185>
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high hill
Claim
worthy kestrel
claimed
high hill
Does anyone know about how to draw this
midnight plankBOT
@high hill Has your question been resolved?
vagrant cairn
hii, so this is my progress so far and like idk how to continue starting from the part with the second term
midnight plankBOT
hii, so this is my progress so far and like idk how to continue starting from th...
rain wasp
grand pondBOT
your local hot fungus
vagrant cairn
so when we simplify the powers does it go like 4 - (-2) + 3?
last slate
so when we simplify the powers does it go like 4 - (-2) + 3?
u messed up that second term
also remember its (-5)^3
there is a minus inside
last slate
it would make it 5^6
no
Just decompose it as (-5)^2 * -5
5^2= (-5)^2
1/ (5)^-2 is 5^2
vagrant cairn
1/ (5)^-2 is 5^2
uh huh... 🤔
vagrant cairn
1=5^0
the '1' over here comes from the 1/(5)^-2, right? (sorry if this question sounds dumb)
last slate
the '1' over here comes from the 1/(5)^-2, right? (sorry if this question sounds...
Yeah
last slate
(5)^0+2= 5^2
5^0 + 2 isnt 5^2
last slate
5^0 + 2 isnt 5^2
Grah I'm addin' the power
Multiply 5^2*5^2=5^4
5^4 × 5^1 × -1= 5^5 × -1= -5^5
vagrant cairn
wait where did the 5^1 come from
last slate
wait where did the 5^1 come from
5^4 ÷ 5^2 × (-5)^2 × -5
,align
125^{\543} \divsymbol \41{25^{-1}} \2 (-5)^3 &= (5^3)^{\543} \divsymbol \41{(5^2)^{-1}} \2 (-5)^3 \ &= 5^4 \2 5^{-2} \2 (-(5^3))\ &= -5^{4 -2 + 3} \ &=-5^5
grand pondBOT
last slate
notice between the first and second line the division symbol gets replaced by multiplication
vagrant cairn
notice between the first and second line the division symbol gets replaced by mu...
ooo yea yea
okay wait so just to make sure I understand everything here, why does the last term get one more bracket when in the second line
last slate
the idea is that (-a)^(odd number) = -(a)^(odd number)
(-a)^(even number) = (a)^(even number)
take the simplest case of (-1)^3 for example
-1 * -1 * -1 = -1 as the first two terms become 1
last slate
1/(5)^-2 should be 5^2
last slate
1/(5)^-2 should be 5^2
i flipped the division sign to multiplication
so the power doesn't change
,, a \divsymbol \41b = a \2 b
grand pondBOT
vagrant cairn
okkk, I finally understand the whole thing ald
thank you so much guysss 🫡🫡
.close
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cosmic zenith
inner wolf
!status
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What step are you on?
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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.
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inner wolf
Did you try converting tan into sin/cos
