like, the best you can do to reduce ugliness is to split the region in two
#help-23
quasi bison
slow fern
If I'm asked the question the same thing by some others, I feel I should know the answer
quasi bison
like this
the red region will then be $2 - \sqrt{1 - (x-2)^2} \leq y \leq x$ for $2 - \frac{1}{\sqrt{2}} \leq x \leq 2 + \frac{1}{\sqrt{2}}$, while the green region will be $2 - \sqrt{1 - (x-2)^2} \leq y \leq 2 + \sqrt{1 - (x-2)^2}$ for $2 + \frac{1}{\sqrt{2}} \leq x \leq 3$.
flat frigateBOT
Ann
slow fern
very ugly
quasi bison
you got what you asked for.
slow fern
thank you
merry sleet
quasi bison
i think a big part of math is knowing when not to bash
flat frigateBOT
QuantumBee
slow fern
$P(Y\leq X-4)=\int_{0}^{10} \int_{0}^{x-4} \frac{1}{50} \dd{y} \dd{x}$
flat frigateBOT
QuantumBee
slow fern
I did this, the answer came out to be 0.2, 1-that would be 0.8, which was wrong
I feel I'm always messing up the integrals
@merry sleet @quasi bison
slow fern
Ok
fair bloom
ok
slow fern
int_**4**^10 on the outer one
I'm getting 0.36, it's still wrong
slow fern
no
quasi bison
you've calculated P(Y ≤ X - 4) correctly from what i can tell, but now you need to take the complement.
quasi bison
**QuantumBee**
as per this
slow fern
$P(X+Y \leq \frac{1}{3})= \frac{\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{3}}{\frac{1}{2} \cdot 1 \cdot 1}=\frac{1}{9}$
flat frigateBOT
QuantumBee
slow fern
Similarly, $P(X+Y \leq \frac{1}{5}) = \frac{1}{25}$
flat frigateBOT
QuantumBee
safe radishBOT
@slow fern Has your question been resolved?
glad sedge
I did this question and got 102C2 = 5151 using 2 separators and the 100 lost marks, which was correct according to the answer key.
My question is: Why is 202C2 wrong? I know I'm missing something very obvious, so can someone point me in the right direction?
safe radishBOT
I did this question and got 102C2 = 5151 using 2 separators and the 100 lost mar...
@glad sedge Has your question been resolved?
glad sedge
<@&286206848099549185>
delicate jasper
I did this question and got 102C2 = 5151 using 2 separators and the 100 lost mar...
because you break the constraint of max marks 100 in many combinations
glad sedge
OHHHH
delicate jasper
something like 197 + 2 + 1 might be chosen but it's invalid
delicate jasper
np
glad sedge
no idea how that slipped my mind tbh
slow fern
you can .close if you dont have any more doubts
slow fern
<@&286206848099549185>
safe radishBOT
@slow fern Has your question been resolved?
safe radishBOT
@slow fern Has your question been resolved?
lean otter
is there a concrete method for working this out? by hand
safe radishBOT
is there a concrete method for working this out? by hand
compact ferry
they probably just guessed that 2 would work
and it did
so they factored out (x-2) using polynomial long division
lean otter
i see
flat frigateBOT
toxic stratus
no guessing or polynomial division needed
safe radishBOT
@lean otter Has your question been resolved?
opaque sorrel
@toxic stratus hi
safe radishBOT
<@118663865264242688> hi
toxic stratus
hi
opaque sorrel
pls
toxic stratus
kinda preoccupied in help-25 rn lol
opaque sorrel
hi
i think someone told me this bfore
but i kinda forgot/didnt quite get the theorem
peak estuary
what is the definition of linear independence
opaque sorrel
linear independence means that no vector in the set can be expressed in the form of each other
peak estuary
do you have a more formal definition
opaque sorrel
hmm like textbook formal?
peak estuary
and maybe one which treats all vectors equally?
opaque sorrel
uhh
opaque sorrel
and maybe one which treats all vectors equally?
erm
do u want me to find it in my textbook or smth
peak estuary
well that would help
you need to know definitions
otherwise how do you want to talk about things when you don't know how they are defined
opaque sorrel
alright
opaque sorrel
well that would help
erm i dont quite have a wordy definition tho
would this suffice
peak estuary
this is perfect
peak estuary
so $v_1, \ldots, v_k$ are linearly independent if $c_1v_1+\ldots+c_kv_k=0$ implies that $c_1=\ldots=c_k=0$
flat frigateBOT
Denascite
peak estuary
that we can work with
peak estuary
what does the exercise say. what do we have to show
opaque sorrel
wait but someone told me before that there was like a direct proof for this or something
opaque sorrel
ohh
so we need to show that
if Pu_1, Pu_2, ... Pu_k are LI, u_1,u_2...u_k are LI
so i can also say that
$c_1Pu_1 + c_2Pu_2 + ... = 0$
flat frigateBOT
HellO
opaque sorrel
i think
opaque sorrel
oki
quasi bison
carefully separate assumptions from goals.
opaque sorrel
what do we start with
we know that u_1, u_2, ... u_k are vectors of n dimensions, and P is a n * n matrix
then suppose Pu_1, Pu_2, ... Pu_k are linearly independent
opaque sorrel
carefully separate **assumptions** from **goals**.
oh so what i just wrote in latex is an assumption?
peak estuary
well it's a random equation out of context
quasi bison
^
quasi bison
hit enter too soon.
opaque sorrel
quasi bison
i would phrase it as this:
== ASSUMPTIONS ==
* P is an n × n matrix
* u_1, u_2, ..., u_n are vectors of size n
* The set {Pu_1, Pu_2, ..., Pu_n} is linearly independent
== GOALS ==
* The set {u_1, u_2, ..., u_n} is linearly independent
you wish to show that {u_1, ..., u_n} is linearly independent.
to this end, you introduce n arbitrary real-number constants c_1, ..., c_n and assume c_1 u_1 + ... + c_n u_n = 0, and set out to prove that for each i from 1 to n, c_i = 0.
== ASSUMPTIONS ==
* P is an n × n matrix
* u_1, u_2, ..., u_n are vectors of size n
* The set {Pu_1, Pu_2, ..., Pu_n} is linearly independent
* c_1, c_2, ..., c_n are real numbers
* c_1 u_1 + c_2 u_2 + ... + c_n u_n = 0
== GOALS ==
* c_i = 0 for all i
thereby transforming your problem into this
lean otter
Hi
lean otter
I don't get why 480 is divided by 10 in the first step. Can anyone help me out?
<@&286206848099549185>
safe radishBOT
@lean otter Has your question been resolved?
pastel briar
Quick question about a basis for a topology. Can there be multiple basis for a topology on a set X?
safe radishBOT
Quick question about a basis for a topology. Can there be multiple basis for a t...
pastel briar
Nvm should have googled. It says "There can be many basis for the same topology (but a basis generates a unique topology)". Just to confirm. Is that true?
I assume it is ofc
safe radishBOT
@pastel briar Has your question been resolved?
safe radishBOT
@pastel briar Has your question been resolved?
honest cairn
<E = <C?
cedar rover
What do you think?
cedar rover
When are two triangles congruent?
honest cairn
when sides congruent
cedar rover
Nice
So you have two triangles that have two sides that are congruent
can you say something about the third ones?
honest cairn
they're also congruent
cedar rover
Yes
so now you have two triangles that are congruent, don't you?
What happens to their angles when they are congruent?
honest cairn
They are also congruent
cedar rover
Yup
thin bridge
just two sides (SS) is insufficient
cedar rover
just two sides (SS) is insufficient
Oh you are right, my bad, from the drawing I mistakenly assumed those were both right triangles
I'm sorry
potent needle
Find the maximum distance a car will drive if:
- The fuel is kept in 3 barrels, 100 litres each
- The car can carry 1 barrel at a time
- The car fuel tank has capacity of 20 litres
- The fuel consumption is 10 litres/100km
My professor got 1690, I either get 3000 if the car can go back to change barrels or 1000 if it can't
safe radishBOT
Find the maximum distance a car will drive if:
1. The fuel is kept in 3 barrels,...
@potent needle Has your question been resolved?
safe radishBOT
@potent needle Has your question been resolved?
safe radishBOT
crisp linden
Yeah, \limits isn't required here
lean otter
Let f(x)=1, and consider the limit x to 0
crisp linden
you require epsilon to be > 0
Not necessarily. You already say that "for every epsilon > 0"
and as you just discovered, the f(x) - L can be zero.
lean otter
if we require epsilon to be 0, then most (maybe all?) discontinuous functions won't satisfy it
eg $f(x)=\begin{cases}x&\text{if }x\neq0,\1&\text{if }x=0\end{cases}$
flat frigateBOT
Toby
lean otter
f(x) is never 0, but the limit x to 0 is 0
yup
basically, a limit looks at the points around the limit, but doesn't care about the actual definition of f at that point
cedar sapphire
can anyone help me with b?
safe radishBOT
can anyone help me with b?
grizzled fossil
can anyone help me with b?
Can you write an equation with that?
cedar sapphire
Can you write an equation with that?
yea thats where im stuck on
grizzled fossil
Say if p=10 could you calculate the change, q then?
cedar sapphire
im just looking for the formula on how to do it
i can solve it in my head
oh wait
nvm
lemme retry
yea no im still confused
cedar sapphire
So how do u calculate it?
that dosnt really help me
cedar sapphire
i have to write the formula
cedar sapphire
no my teacher wants the formula
cedar sapphire
oh yea
i got mixed up
i though it was divison instead of subtraction for some reason
cedar sapphire
So seems like we do 50-3p to get q
wait can i write my answer as q=(3p-50)
or would that be formatted wrong
grizzled fossil
50-3p not 3p-50
sick light
hi
sick light
the formula in question is the geometric series formula
but can we use that formula to prove this?
in the geometric series we assume a constant raised to a power
flat frigateBOT
Paschalis Chinos
peak estuary
they indexed the sum wrong. should be k=0 to N-1. w=e^(2pi i/N)
sick light
yeah but w is a function of the N parameter
peak estuary
N is fixed
sick light
Do we agree that the geometric series can be proven by induction?
peak estuary
<@&268886789983436800>
peak estuary
Do we agree that the geometric series can be proven by induction?
yes
wild cape
ty
sick light
how does it work with $\sum^{n=0}^{N-1} w(N)^n$
flat frigateBOT
Paschalis Chinos
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
sick light
oops
peak estuary
N is fixed
it's just a fixed number
like it could be $\sum_{n=0}^3 e^{2\pi i n/4}$ or something
flat frigateBOT
Denascite
peak estuary
w is constant with respect to n
peak estuary
first prove $\sum_{n=0}^k w(N) = \frac{w^{k+1}-1}{w-1}$ by induction on $k$
flat frigateBOT
Denascite
peak estuary
then set k=N-1
peak estuary
ehh dont worry, it is confusing. it could definitely be meant like you thought
sick light
I mean it doesn't matter does it
it's the same however it was meant
if it works for k it works for N-1
peak estuary
you'd have a division by 0
blazing dew
How come this graph is considered hamiltonian but is not complete?
blazing dew
safe radishBOT
@blazing dew Has your question been resolved?
hazy sentinel
vertex
buoyant shadow
it's not clear what you mean
a graph doesn't need to be complete to be hamiltonian
blazing dew
a graph doesn't need to be complete to be hamiltonian
then what about the travelling salesman problem which aims to find the hamiltonian cycle of minimum cost
one of its assumptions is that the graph is complete, right?
buoyant shadow
i suppose
blazing dew
then since hamiltonian cycles come from hamiltonian graphs
and considering the constraint, it has to be complete
buoyant shadow
sorry i don't know
lament creek
Traveling salesman is for (directed) weighted graphs
Completely unrelated
Do you know the definition of Hamiltonian?
blazing dew
yes
lament creek
What is the definition, as you understand it?
blazing dew
What is the definition, as you understand it?
its when you can visit each vertex exactly once
lament creek
Yes, and you should be able to do that here.
blazing dew
Yes, and you should be able to do that here.
but one of the things that TSP assumes is that the graphs are complete
also, I just googled and it says that TSP can be modelled as an undirected weighted graph
lament creek
but one of the things that TSP assumes is that the graphs are complete
I don't know why your so fixated on that
blazing dew
I don't know why your so fixated on that
my book says that
lament creek
TSP is about Hamiltonian paths on a weighted (possibly complete) graph
TSP does not define a Hamiltonian path
blazing dew
i see
blazing dew
TSP is *about Hamiltonian paths on a weighted (possibly complete) graph*
why possibly?
lament creek
For a weighted graph is doesn't really matter if it's complete or not
blazing dew
why not
lament creek
Since you can assume edges that aren't there are there and just have weight zero.
blazing dew
Since you can assume edges that aren't there are there and just have weight zero...
is a complete graph a graph where a vertex is connected to every other vertex by a single edge, or when every pair of vertices are connected by a unique edge?
i have no idea if these are the same or not
lament creek
depends
generally I'd say these are equivalent because multigraphs are not graphs
but these sorts of definitions are going to vary depending on the text you're reading
blazing dew
i got another question
lament creek
ofc weighted graphs by default also refer to real weighted graphs, which differ from multigraphs, although you can represent any multigraph as an integer weighted graph
blazing dew
why are TSP graphs often complete?
lament creek
it's just easier to assume that all possible edges are present than it is to keep track of a list of edges
blazing dew
it's just easier to assume that all possible edges are present than it is to kee...
so you can make up connections
peak estuary
you can just give those edges a weight of like 25 million or some other high number so any algorithm won't take it
blazing dew
you can just give those edges a weight of like 25 million or some other high nu...
but you can make them up right
for the sake of simplicity
peak estuary
yes
blazing dew
it's just easier to assume that all possible edges are present than it is to kee...
this may sound dumb, but what do you mean by 'keep track of a list of edges'
blazing dew
Since you can assume edges that aren't there are there and just have weight zero...
but for weighted graphs, do I still need to establish that an edge exists even though it has a weight of 0?
safe radishBOT
@blazing dew Has your question been resolved?
modern lynx
how do I integrate this? I cant use partial fraction decomposition and not regular devision either since the numenator is a power less then the denominator so I'm kinda stuck
safe radishBOT
how do I integrate this? I cant use partial fraction decomposition and not regul...
nova creek
Completing the square and trig sub
modern lynx
how do I complete the square?
stray bluff
how do I integrate this? I cant use partial fraction decomposition and not regul...
you can take out -2 as a constant
nova creek
modern lynx
oh thanks, I tried looking at it in my book but didnt understand but I'll try the video
nova creek
https://youtu.be/C206SNAXDGE
Only difference is you're doing it to an expression, not an equation. Instead of, say, adding a constant to both sides, you'd add and then subtract the constant
nova creek
lemme write this down i swear u dont have to use trig sub
The answer has a trig function so...
nova creek
Well I don't have the inverse trig integrals memorized so for me thats just a trig sub in disguise 
modern lynx
Only difference is you're doing it to an expression, not an equation. Instead of...
is there a straightforward way to find out which constant? Or is it just understanding things in which case I just gotta watch the whole of that video
nova creek
Completing the square is pretty intuitive imo
You just want to add something to get something in the form a² + 2ab + b²
For instance, what could you add to x² + 2x to make it a perfect square?
modern lynx
1?
nova creek
Yep
So we can write x² + 2x + 2 as x² + 2x + 1 - 1 + 2
Which is (x + 1)² + 1
Square completed
modern lynx
ohhh alright it was just the phrasing that I couldnt make sense of
thanks for the help
.close
dire spoke
Why is n+nZ not in Z/nZ
safe radishBOT
Why is n+nZ not in Z/nZ
dire spoke
Since Z/nZ = { a + nZ | a in Z }
I think n+nZ is equivalent to 0+nZ but i don't understand why
In my mind, it seems 0 + nZ includes the zero element
but 5+nZ does not
light shoal
Why is n+nZ not in Z/nZ
of course it is, it equals 0 + nZ as you noted
light shoal
but 5+nZ does not
depends, what is n?
if n is 5 then 5 + nZ = 5 + 5Z = 0 + 5Z = 5Z, which certainly contains zero
safe radishBOT
@dire spoke Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
delicate cape
Hello
delicate cape
I need to find theoretical and actual slope
But they told me to add all 5 points for one of them
safe radishBOT
@delicate cape Has your question been resolved?
safe radishBOT
@delicate cape Has your question been resolved?
misty bay
you need to give the entire problem here
what is hanging
is anything attached to the thing that's hanging
delicate cape
Oh Dw I figured it out!
safe radishBOT
Available help channel!
severe pebble
Let f(x) = $\tan(\ln(x)) \cdot \tan\left(\ln\frac{x}{2}\right)\cdot \tan(\ln2)$ \
$x \in [1, e^{\pi}] \setminus {e^{\frac{\pi}{2}}, 2e^{\frac{\pi}{2}} }$\
Then which of the following is equal to f(x)\
A. $\tan(\ln(x)) + \tan\left(\ln\frac{x}{2}\right) + \tan(\ln2)$\
B. $\tan(\ln(x)) - \tan\left(\ln\frac{x}{2}\right) + \tan(\ln2)$\
C. $\tan(\ln(x)) + \tan\left(\ln\frac{x}{2}\right) - \tan(\ln2)$\
D. $\tan(\ln(x)) - \tan\left(\ln\frac{x}{2}\right) - \tan(\ln2)$
safe radishBOT
Let f(x) = $\tan(\ln(x)) \cdot \tan\left(\ln\frac{x}{2}\right)\cdot \tan(\ln2)$ ...
severe pebble
is that right?
safe radishBOT
@severe pebble Has your question been resolved?
severe pebble
<@&286206848099549185>
safe radishBOT
@severe pebble Has your question been resolved?
safe radishBOT
@severe pebble Has your question been resolved?
plain lion
severe pebble
alright don't give me that shit, the fact I'm waiting hours for this problem means i got the answer key with me already, don't ping me for such subtleties, get me a genuine solution of your own and ping me all you want
Thank you
@plain lion
or you can point my error as well, that also is commendable
stoic owl
note that ln(x/2) = ln(x) - ln(2)
and tan(a - b) = (tan(a) - tan(b))/(1+tan(a)*tan(b))
after that it is just a pretty simple comparison
ping me if it's not clear
safe radishBOT
@severe pebble Has your question been resolved?
severe pebble
it is, thanks for your help ayayaclap, appreciate it
lean otter
Write an equation perpendicular to y = -2x - 7 that passes through the point (4,3)
safe radishBOT
Write an equation perpendicular to y = -2x - 7 that passes through the point (4,...
lean otter
i know how to write an equation parallel just not perpendicular
im not sure if theres another formula or what 😭
white hollow
the product of gradients of perpendincular lines is -1
white hollow
lean otter
OMG ok thank u sm
white hollow
this is just a rabndom pic I found online
lean otter
i missed the entire lesson on this topic
white hollow
it's ok you'll catch up
white hollow
yeah
once you get the gradient of the perpendincular line
just substitute htat for m
and then use the coordinate
white hollow
where'd you get 3/4 from
lean otter
umm i
ok let me write this out on a pic hold on
@white hollow ignore the highlight shes making us highlight the equation we finished w
lean otter
anyway 3/4 x -3/4 =/= -1
im confused
white hollow
dude this is not the question you asked at first
lean otter
how did u get that
white hollow
it's just a formula for perpendicular liens
lean otter
ohhhh ok
white hollow
so what's the gradient of the perpendicular
lean otter
im so sorry but do i like plug in the m to the formula u sent
white hollow
-2 x m = -1
m is the gradient of the perpendicular here
lean otter
m = -3?
lean otter
im genuinely so confused
white hollow
ok so we have the line
-2x - 7
we want to find a line perpendicular to that
the gradient of a perpendicular line is the negative reciprocal of the other gradient
so the gradietn we have is -2
-2 x m = -1
reciprocal would give 1
white hollow
that's the formula
white hollow
ok -2 x m is -1
if you want to know why it equals -1 you can derive it from some simple trig
but anyway, m here would be 1/2
so the gradient of the perpendicular line is 1/2
so y = 1/2x + c
we need to find c
we do this by using the point that the perpendicular passes through
lean otter
(4, 3)
white hollow
mhm
lean otter
ok i think i get it
i have to go out somewhere so ill jus finish this later
thank u for helping tho
white hollow
sure no worries
lean otter
.close
safe radishBOT
safe radishBOT
Available help channel!
misty coral
safe radishBOT
misty coral
is this proof by induction?
l <= m meaning the base case then the p(m+1) being the induction
compact ferry
inequality is the wrong way to say anything
trim swan
inequality is the wrong way to say anything
I don't think that's the problem. It's the base case for strong induction
trim swan
It's just that they only give you the base case here
compact ferry
I don't think that's the problem. It's the base case for strong induction
oh shit woops
misty coral
so a. not sure , depends on other answers
b. depends on the carried out proof?
c. no idea how to picture that
d. is the only thing that looks familiar
wait wouldnt c be true maybe
because l <= m < n and since it would prove all things >=m
trim swan
If you were going to use strong induction. You would first prove that P(l) is true for all l<m.
But that's not all you would do, right?
You'd also have to show the induction step
misty coral
hmm
I see
ill read over the chapters and come back to the q in like 5 - 10 mins
.close
lean otter
safe radishBOT
https://math.stackexchange.com/questions/2440047/colour-change-in-drawing-balls-...
safe radishBOT
@lean otter Has your question been resolved?
lean otter
No
safe radishBOT
@lean otter Has your question been resolved?
random rivet
safe radishBOT
random rivet
is this right
safe radishBOT
@random rivet Has your question been resolved?
nimble venture
hi i got the an exercise but its in greek so ill try to translate it perfectly
we got a continuous function f: R -> R with f(3) = 2. If xεR then f(x) * f(f(x)) = 1, find f(2) and f(1).
I found f(2) = 1/2, by rearranging x = 3, but i cant find f(1)
I rearranged again x = 2 so, the function becomes f(2) * f(1/2) = 1, but i dont know how to continue. Any help? I thought of f(1/2) < f(1) < f(2)
nimble venture
<@&286206848099549185> please?
safe radishBOT
@nimble venture Has your question been resolved?
nimble venture
.close
grim tangle
safe radishBOT
grim tangle
I'm wondering if these two curves are orthogonal? My derivatives look like negative reciprocals but they have different variables...
thorn chasm
You need to find the points of intersection
And show that at those points the derivatives are negative reciprocals
safe radishBOT
@grim tangle Has your question been resolved?
tired basin
Can someone help me with Categorical Naive Bayes? I cannot seem to get how the 3 is calculated in the data set? Am i missing something? (its highlighted in yellow)
safe radishBOT
Can someone help me with Categorical Naive Bayes? I cannot seem to get how the 3...
@tired basin Has your question been resolved?
safe radishBOT
@tired basin Has your question been resolved?
misty bay
Can someone help me with Categorical Naive Bayes? I cannot seem to get how the 3...
of different classes iirc
tired basin
# of different classes iirc
omg that makes sense now, thank u so much!!!
misty bay
wait it might also be the number of predictors
lemme do a quick derivation of it
it is indeed the number of classes
tired basin
thank u so much!
tired basin
@misty bay I'm trying to apply the equation that I just asked earlier from this channel into this but im stuck on how to classify them into the equation e.i. (P=(Bedroom = 1 | Apartment) and so on. --Train datset is above and test dataset is below which is what im trying to clasify
I've done the classifying in the above table but i'm stuck on how to do this ? or if im doing this correctly?
safe radishBOT
@tired basin Has your question been resolved?
south linden
could anyone explain what i did wrong for this/how to correct
safe radishBOT
could anyone explain what i did wrong for this/how to correct
safe radishBOT
@south linden Has your question been resolved?
radiant jetty
Given:
plane 1 = 2x-4y-z=-1
Express in scalar (normal form) an equation of a second plane that passes through P(2,1,-3), Q(3,2,-2) and is perpindicular to plane 1
So I know the normal of plane 1 is <2,-4,-1>
I can make a line equation which passes through P
line 1 = <2,1,-3> + t<2,-4,-1> where t belongs to real numbers
line 2 = <3,2,-2> + t<2,-4,-1>
These would both be perpindicular since i'm using the normal of plane 1
however i'm not sure how to get a second plane
safe radishBOT
Given:
plane 1 = 2x-4y-z=-1
Express in scalar (normal form) an equation of a se...
@radiant jetty Has your question been resolved?
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@radiant jetty Has your question been resolved?
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glad dragon
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glad dragon
I got 4 points as my answer
But when I plug in two of them back into dy/dx, I don't get the right slope
Also, if I had solved for x instead and did x = y -3, I wouldn't have even gotten those two points in red at all....
Can someone help?
trim swan
Yeah, those are just extraneous solutions @glad dragon
glad dragon
Yeah, those are just extraneous solutions <@696528190516232353>
How would we have known to check for extraneous solutions? There's no absolute value or radical
And if we did x = y - 3 to get the 2 right solutions, how do we know those are the only 2 solutions?
Why were they even extraneous? I didn't do any algebraic manipulations like squaring both sides, doing radicals or absolute value, etc
@trim swan ?
Hm, looks like you are away
trim swan
I'm still here, sorry I'm still thinking about it
glad dragon
Thank you
trim swan
Oh wait
The points you circled are the extraneous ones
The ones you didn't circle are the correct solutions
glad dragon
Yes
I'm just wondering why they are extraneous, and why we needed to check.
If we had done x = y -3 instead, we would have just gotten the two correct solutions with no extraneous
trim swan
Yeah, because if you plug those solutions back into x, the equation you have is linear
it only has one soltution
You should sorta always check for extraneous solutions, unless you're positive there aren't any
When you solve an equation and you go from
equation P
to
equation Q
to
equation R
You're showing that if equation P is true, then equation Q is true, which means equation R is true
But then you solve equation R, and you find what values make it true, that doesn't necessarily mean they make equation P true as well
Even if the logic works forwards, it doesn't necessarily work backwards, so you should always check
glad dragon
Yeah, because if you plug those solutions back into x, the equation you have is ...
Do you mean y = 3 - x part is supposed to have only one solution since it is linear?
balmy sky
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balmy sky
,rotate ac
flat frigateBOT
balmy sky
how do I do question 8?
spiral crescent
how do I do question 8?
What is the question
balmy sky
integrate
balmy sky
How do I integrate question 8?
safe radishBOT
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balmy sky
<@&286206848099549185>
balmy sky
<@&286206848099549185>
jaunty belfry
you can usually put this on integral calculator online, and it will give you the step by step explanation and solutions
jaunty belfry
How do I integrate question 8?
.
muted nest
Graph of which trigonmetric function doesn't meet x axis y axis and origin?
muted nest
Csc right?
thin bridge
yes
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lone grove
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lone grove
,rotate
flat frigateBOT
dusk vapor
lone grove
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zinc dust
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zinc dust
Im looking at a proof of „continuity from below of messures“ and i dont quite get the meaning of the equation in the circle
quasi bison
the arrow means "approaches from below" i think
zinc dust
Can it some how be written in other ways
Like is it lim sup?
Lim sup ( mu(A0\Aj)- mu(A0\A))
And j goes to infinity?
peak estuary
if something converges, then lim=limsup=liminf
zinc dust
Yes
peak estuary
I've seen that notation in two contexts. either x_n <= x for all n or even further x_n is a monotone sequence
where lim x_n = x
peak estuary
measure theory really likes monotone sequences
peak estuary
well mu(...) is a number
zinc dust
The one in blue circle has another arrow
Yes!
So one number converges to another number?? From below??
Or from above???
zinc dust
Then the equation of that in the middle is suppose to be a number?
That I don’t understand
peak estuary
your notes should hopefully define this notatin somewhere
peak estuary
well mu(A_0 \ A_j) is a number that depends on j
peak estuary
and mu(A_0 \ A) is a number that is fixed
and this says that the first sequence converges to mu(A_0\A)
from below
peak estuary
we have a sequence $x_j = \mu(A_0\setminus A_j)$
flat frigateBOT
Denascite
zinc dust
Oh wait
peak estuary
with $\lim x_j = x = \mu(A_0\setminus A)$
flat frigateBOT
Denascite
zinc dust
Am I suppose to read the equation this way?
peak estuary
and this limit is "from below"
zinc dust
peak estuary
if that helps you, sure
zinc dust
Oh yeah I get it now
That confused me with how can one convergence be a number
That’s a lot!
.close
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Available help channel!
rain fog
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https://discord.com/channels/268882317391429632/903481105888452609/1034106173907...
rain fog
So
I did these
But I'm not sure if it's correct
They drop medicine to the patient 120ml
A) how much it drops 120ml?
B) drop rate if the dropping time is 1 hour
2nd picture is 150ml medicine if the dropping rate is 30 gtt/min how long it takes? 1ml=20gtt
3rd picture the ingredients is 0, 8% of we make out of 200 mo how much gram it has
rain fog
https://discord.com/channels/268882317391429632/903481105888452609/1034106173907...
click this to see the pictures
safe radishBOT
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rain fog
<@&286206848099549185>
anyone? .7
its very easy math for advanced people
im very bad at maths
rain fog
all of them ?
seems? :
😮
i have an exam this wenday
nice to see another finnish fella ❤️
woven swan
all of them ?
yes but i think ur unit on the last one is wrong
the answer should be in grams
same for the third question
u forgot ur units, and for the step in the middle it should be 1.5ml/min instead of 1.5gtt/min
woven swan
nice to see another finnish fella ❤️
👍
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@rain fog Has your question been resolved?
rain fog
<@&286206848099549185>
woven swan
so to calculate volume
for the first shape which is a rectangular prism, the formula is
$volume = width * length * height$
flat frigateBOT
Intrer
flat frigateBOT
Intrer
flat frigateBOT
Intrer
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lean otter
IS X in 52 65 ?
dusk goblet
- x=90-25
dusk goblet
rectangular property, angles at a corner =90 (given that the shape is not a rhombus)
dusk goblet
rectangular property
lean otter
oh i dint know that
so this ?
so in 53 answer is 60 ?
or wait
30 ?
@dusk goblet
dusk goblet
30 seems more precise
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@lean otter Has your question been resolved?
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@lean otter Has your question been resolved?
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thin bridge
sqrt(x^2) = -x
for negative x
you can if you want, though unnecessary if you apply
sqrt(x^2) = |x| = -x
under these conditions
in the original form, you have inf-inf which is indeterminate
using conjugates is fine
you just need to remember that sqrt(x^2) isn't x
poor implication
$|x| = \begin{cases} x \ &\text{if } x > 0 \\ -x \ &\text{if\ } x\leq 0 \end{cases}$
flat frigateBOT
ℝamonov
thin bridge
it's part of the definition of the absolute value
sqrt(x^2) = |x|
both of these have the same pieceeise definition and are equivalent
since you're considering the lim as x to -inf, you're dealing with negative x
when manipulating your fraction you'd be dividing numerator and denominator by sqrt(x^2)
you can divide the roots like you did earlier,
and then for the numerator, you can use sqrt(x^2) = -x
not sure what you mean
you can factorise sqrt(x^2) if you want
either way you'd apply sqrt (x^2) = -x
at some point
spark steppe
how do you represent a 2x2x2...x2 game as matrix (that can act on vectors) and what are that matrix's properties?
intuitively, I was thinking that you could just represent each player's strategy as a bit, either 0 or 1, where 0 represents strategy 1, and 1 represents strategy 2. From this, we could make a binary string, where player 0's strategy is the least significant bit.
for example, in a 2x2 game, consider the payoff:
$$
\begin{bmatrix}
(3, 3) & (10, 1) \
(1, 10) & (2, 2)
\end{bmatrix}
$$
where $(x, y)$ represents the payoff; $x$ is the payoff to player 0, and $y$ is the payoff to player 1
Now, for each strategy, we can create a matrix:
$$
\begin{bmatrix}
3 & 10 & 1 & 2 \
3 & 1 & 10 & 2
\end{bmatrix}
$$
Column 0 is the strategy profile $00$, column $1$ is the strategy profile $01$, column $2$ is the strategy profile $10$, etc.
We can easily call the vectors $\begin{bmatrix}1 \ 0 \ 0 \ 0\end{bmatrix}$ to get the payoff for profile $00$, and use vector $\begin{bmatrix}0 \ 1 \ 0 \ 0\end{bmatrix}$ to get the payoff for $01$ and so on.
Is this method well known/what are the other properties of this matrix? Is there any relation to this matrix and the nash equilibria?
safe radishBOT
how do you represent a 2x2x2...x2 game as matrix (that can act on vectors) and w...
flat frigateBOT
StevenJohn
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@spark steppe Has your question been resolved?
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@spark steppe Has your question been resolved?
spark steppe
<@&286206848099549185> can i have help
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unkempt solstice
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Available help channel!
tough obsidian
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cedar saffron
The equality does hold in case 2 right
no, in that case the inequality does hold but not the equality
tough obsidian
Why check for equality
tough obsidian
Why check for equality???
earnest wagon
Why check for equality???
That depends on the question..what is the question?
placid cove
3Fn −Fn−2 = Fn+2,for n ≥3
IS THIS STATEMENT VALID?
safe radishBOT
3Fn −Fn−2 = Fn+2,for n ≥3
IS THIS STATEMENT VALID?
neat kiln
what is Fn?
grizzled shoal
scenic gull
Or Fibonacci number. But doesn't make a whole lot of sense
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lean otter
i dont get it
safe radishBOT
i dont get it
lean otter
hello/?????
i need help with this
is it 6 root 3
??????/
<@&286206848099549185>
<@&286206848099549185>
i dont get it
is it 6 root 3
plees help me
.close
safe radishBOT
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Available help channel!
tiny frost
Hello, im having brain farts trying to understand direct proportionality. y = kx. is K the difference between y and x?
safe radishBOT
Hello, im having brain farts trying to understand direct proportionality. y = kx...
tiny frost
if you have a table say X = hours 1, 2, 3, 4, 5 and distance y, 500, 1000, 1500, 2000. to get direct proportionaly between say x-3 and y-2000, is that what K is?
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@tiny frost Has your question been resolved?
dry hazel
Hello, im having brain farts trying to understand direct proportionality. y = kx...
You make 10 dollars an hour. How much do you make when you work x hours?
($) = ($ / hour) * (hours)
y = 10 * x
dry hazel
for your example with distance, can you figure out the distance travelled per hour?
tiny frost
Yea, x times y, so 1 hour 500 whatever the metric is
dry hazel
yep
so proportionality is basically saying,
I have (this) for each (thing)
okay that is crappy phrasing
tiny frost
Right, so if I have a question like this: "You know that P is proportional against T with the proportionallyconstant K. (im not used to do math in english but I hope you understand what I mean)
so it asks deside P if T is 44, and you know that K = 0,75
you just muiltiply those togheter
but ok Im kinda getting it now 😄
Thanks for the explanation
.close
knotty gorge
Hello
hardy lion
x+1=x+2-1
deep apex
Use $a^x = e^{x \ln{a}}$
flat frigateBOT
Assass!n
knotty gorge
I undestsnd
hardy lion
oh yeah that also works
deep apex
yeah this should do it
safe radishBOT
@knotty gorge Has your question been resolved?
compact egret
Claim
compact egret
What is it then?
nova creek
||cA|| is not √(c(a² + b²))
compact egret
cA • A
c (a,b) * (a,b)
(ca,cb) * (a,b)
ca^2 +cb^2?
nova creek
What about it?
nova creek
Remember ||v||² = v • v
compact egret
Remember \||v||² = v • v
That says nothing about cA
nova creek
That formula is true for all v
compact egret
I want to know cV not V V
nova creek
||A||² = A • A and ||cA||² = (cA) • (cA)
nova creek
Well, if you don't want it squared, then just take the square root
compact egret
Ik but why do you not
nova creek
So I don't have to write the square root symbol
compact egret
It helped tho; thanks
nova creek
Purely out of laziness
compact egret
Remember \||v||² = v • v
However this was terrible advice i would never figure it out
Maybe if you did (V)
Or let V = aB
nova creek
Wdym
compact egret
Nvm
jagged sequoia
1st pic qst 2nd one my work I'm trying to find k such as what it's written, I suspect |(un-vn) /2(un+1 + vn+1 ) | to be k but idk how to start my work
safe radishBOT
1st pic qst 2nd one my work I'm trying to find k such as what it's written, I su...
jagged sequoia
<@&286206848099549185>
arctic locust
Are you from Louis-le-Grand?
Anyways you get $u_{n+1}- v_{n+1} = \frac{u_{n}^{2}+v_{n}^{2}-4u_{n}v_{n}}{2(u_{n}+v_{n})}$
flat frigateBOT
black_couscous
arctic locust
And in terms of absolute values you have that $|u_{n}- v_{n}| \leq u_{n} + v_{n}$ because the two sequences are positive
flat frigateBOT
black_couscous
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@jagged sequoia Has your question been resolved?
jagged sequoia
well I didn't write it correctly but I got the idea thanks man : )
well we get k between 0 and 1/2
right?
jagged sequoia
Are you from Louis-le-Grand?
Nop 🥲