Oh so 2 seperate functions
#help-39
pearl pondBOT
@graceful garden Has your question been resolved?
graceful garden
You can copy paste that into Desmos
Thanks
pearl pondBOT
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crimson nebula
pearl pondBOT
crimson nebula
The velocity is expressed in polar coordinates. But I didn't understood the last line .
How can we express the basis vectors r hat and θ hat as a function of time ?
unborn abyss
well, they rotate around the origin along with the wheel, right?
rhat is the direction the spoke is pointing, and thetahat is 90º ccw from that
crimson nebula
My question is
See the fourth last line
They have expressed velocity in terms of r hat and θ hat
And they are saying , we need to know how r hat and θ hat are changing
My question is .... How can I do it
How can I write r hat and θ hat as function of time
unborn abyss
coming up with an equation might be kind of annoying but you need:
rhat(0) = (1, 0) [pointing to the right]
rhat(w / (π/2) ) = (0, 1) [pointing up]
rhat(w / π) = (-1, 0) [pointing left]
and rhat(w / 2π) = (1, 0) [pointing to the right again]
like it needs to be a unit vector rotating in a circle
there's probably a clean way to do it
crimson nebula
I respect your effort , but I think u r not getting my question
chilly trellis
Do you know about polar unit vectors and how to express them as Cartesian unit vectors? This could make explaining the translation easier but if not I don’t want to add confusion
crimson nebula
Yes i know
Basis transformation
But ....
I want answer in purely polar coordinates
Expressing unit vectors as a function of time
But Not in terms of î and j
sharp ravine
rhat is the direction the spoke is pointing, and thetahat is 90º ccw from that
isnt that basically the answer? r_hat = ( cos(wt), sin(wt) ), theta_hat = ( -sin(wt), cos(wt) )
unborn abyss
oh yeah that's a good idea
crimson nebula
isnt that basically the answer? r_hat = ( cos(wt), sin(wt) ), theta_hat = ( ...
U have expressed r hat and θ hat in i and j
.close
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royal galleon
He defines delta k / delta t as F * delta x/ delta t. But then we he multiples be delta t he has dx and dy. How did he just miswrite ?
prime stream
He defines delta k / delta t as F * delta x/ delta t. But then we he multiples b...
again, better add the link to the video and the timestamp
royal galleon
again, better add the link to the video and the timestamp
https://youtu.be/R74EhX8O0Nw he gets the last line at around 23 30
prime stream
https://youtu.be/R74EhX8O0Nw he gets the last line at around 23 30
Listen carefully at 20:10 and 20:34. He switches from 1D to 2D
royal galleon
Yes I see that delta x is just for one dimension. But he can have delta x and delta y represent the x and y components in 2 dimensions. What I don't understand how we get dx and dy since those are instanaous
prime stream
Yes I see that delta x is just for one dimension. But he can have delta x and de...
ah, yes, delta x and dx sometimes are used interchangeably. Especially in physics
when they are not the part of the $\dfrac{df}{dx}$ symbol, then they are different things
jolly parrotBOT
Dedekind
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left anchor
How do I get the eigenvectors of a 3x3
pearl pondBOT
How do I get the eigenvectors of a 3x3
left anchor
I only found methods for 2x2 so far if I’m not mistaken
They were talking about characteristic polynomials and means and other stuff
I’m not really familiar with any computations around this topic but I have a matrix and I’m supposed to find both the eigenvalues and -vectors
toxic lichen
I only found methods for 2x2 so far if I’m not mistaken
They were talking about...
all of that charpoly shit is applicable for any size of matrix
left anchor
I think I understand the meaning of
Matrixvector = scalarvector
Which I wrote as
M x = lambda x
If you rewrite it you get to
(M-lambda*identity matrix) x = 0
Right?
left anchor
all of that charpoly shit is applicable for any size of matrix
Mind going through an example with me?
toxic lichen
I think I understand the meaning of
Matrix*vector = scalar*vector
Which I wro...
yes and this works perfectly fine for 3 dimensions as well
the roadblock is whether you know how to calculate 3x3 dets
left anchor
All i managed so far was writing down rje same matrix but with -lambda on the diagonals which isn’t really helping
toxic lichen
Mind going through an example with me?
sure, perhaps you could have already posted it
toxic lichen
1 1 1
0 1 1
0 0 1
this is your original matrix?
the third -λ is on the wrong entry.
toxic lichen
what you should have is $$\det \bmqty{1-\lambda &1&1 \ 0&1-\lambda&1 \ 0&0&1-\lambda} = 0$$
jolly parrotBOT
Ann
toxic lichen
So it’s (1-lambda)^3
yup. that's your det.
should be fairly easy to tell the eigenvalues here.
toxic lichen
Doesn’t change my result does it
i mean, it does -- your charpoly is degree 3 as it should be. your wrong matrix would give the wrong charpoly.
anyway yeah the only eigenvalue you got is λ=1. with algebraic multiplicity 3.
left anchor
Im looking for the eigenvalues/vectors only
Does the degree have any effect on those?
left anchor
anyway yeah the only eigenvalue you got is λ=1. with _algebraic_ multiplicity 3.
What’s the algebraic multiplicity
toxic lichen
multiplicity as a root of the charpoly.
the degree is a sanity check: for any n by n matrix its charpoly has degree n.
left anchor
I see
So I get degree n
See that I didn’t make a mistake
Get lambda=1
And have the only eigenvalue?
toxic lichen
yes. λ=1 is your one and only eigenvalue here.
left anchor
Can I already decide whether I could diagonalize this matrix?
My guess would’ve been no because there’s one eigenvalue and 3 columns
toxic lichen
so like
generally when you have less eigenvalues than the size of the matrix, you're right to SUSPECT that the matrix could fail to diagonalize.
and in fact this matrix is non-diagonalizable,
but it is not always so in general.
left anchor
Is
100
010
001
A possibility?
toxic lichen
also if you're going to type out matrices the dumb way then please at least put spaces between the entries.
there's no rule of matrix algebra that says a matrix entry can only be one single digit.
left anchor
there's no rule of matrix algebra that says a matrix entry can only be one singl...
We did already agree it’s a 3x3 no?
I don’t have the time to learn latex notation
toxic lichen
We did already agree it’s a 3x3 no?
completely orthogonal to my remark
toxic lichen
I don’t have the time to learn latex notation
i am not asking you to learn latex notation. it's a bit clunky. but you can at least put spaces between entries as i already said verbatim
left anchor
completely orthogonal to my remark
There’s three numbers and you know there’s three entries. Where’s the problem?
toxic lichen
the problem is that it looks like ass
anyway we can keep arguing about typography and defense of bad habits, or we could go back to the math
left anchor
1 0 0
0 1 0
0 0 1
Happy?
toxic lichen
yes that's better.
toxic lichen
and in fact this matrix _is_ non-diagonalizable,
but it is not always so _in gen...
to check for diagonalizability, you need to actually find the eigenvectors.
left anchor
Is this approach right for that matter?
toxic lichen
no
first, put lambda=1.
second, just write it as an ordinary algebraic linear system.
left anchor
left anchor
I added a third row with all zeros
I just forgot it on the picture
Checks out with my visual solution for the question
I assumed it was only the x axes that was holding the eigenvectors
Since we have a specific line of eigenvectors
Can we conclude there’s only one such line and say it can’t be diagonalized?
toxic lichen
we have only one eigenvector up to scaling: [1; 0; 0]
so yes, that's not enough
therefore not diagonalizable
left anchor
I was supposed to find all eigenvalues, -vectors and decide whether it’s diagonalizable
Ig I’m done then
.close
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frozen lynx
<@&268886789983436800>
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vestal thistle
Hello, could someone check if this proof looks good please?
\begin{Definition}[Subset of a set]
Suppose $A$ and $B$ are sets. If every element in $A$ is also an element of $B$, then $A$ is a \emph{subset} of $B$, which is denoted $A \subseteq B$.
\end{Definition}
\begin{Definition}[Substraction of sets]
The \emph{substraction} of $B$ from $A$ is the set $A \setminus B = \{ x : x \in A \text{ and } x \notin B\}$.
\end{Definition}
% --------------------------------------------------------------------------------
\begin{Theorem}
Assume $A$, $B$ and $C$ are sets.
If $A \subseteq B$, then $A \setminus C \subseteq B \setminus C$.
\end{Theorem}
\begin{proof}
Let $A$, $B$ and $C$ be sets.
Either $A$ is empty or non-empty.
First, suppose $A$ is empty.
Then, $A \setminus C = \emptyset \subseteq B \setminus C$.
Now suppose $A$ is non-empty.
Then, either $C$ is empty or non-empty.
Suppose $C$ is empty.
Then $A \setminus C = A \subseteq B = B \setminus C$, which is what we wanted to prove.
Now suppose $C$ is non-empty.
Let $x \in A$.
There are two cases for this: either $x \in C$ or not.
\begin{case}
Suppose $x \in C$.
Then, by the definition of the substraction, $x \notin A \setminus C$.
Since $x$ was chose arbitrarily, it holds for any $x$ in $A$.
Thus, $A \setminus C = \emptyset \subseteq B \setminus C$.
\end{case}
\begin{case}
Suppose $x \notin C$.
Then $x \in A \setminus C$.
Because $A \subseteq B$, we have that $x \in B$, so $x \in B \setminus C$.
Since $x \in A \setminus C$ implies $x \in B \setminus C$, by the definition of the subset,
we have that $A \setminus C \subseteq B \setminus C$.
\end{case}
Therefore, in any case, if $A \subseteq B$, then $A \setminus C \subseteq B \setminus C$.
\end{proof}
x in C does not mean A \ C = emptyset
vestal thistle
Allow me to elaborate
jolly parrotBOT
Mor Bras
vestal thistle
x in C does not mean A \ C = emptyset
It should be better now
glass meadow
Did you only change "Case 1"? I'm not reading all that again
vestal thistle
Did you only change "Case 1"? I'm not reading all that again
I only changed case 1
vestal thistle
<@&268886789983436800>
vestal thistle
I know what you're trying to do but that just doesn't work
Every element in A is in C, then if A \ C is every element in A not in C, then there's none in A \ C
glass meadow
That premise is only a special case though
You seem to only consider the cases where all elements of A are in C or none of the elements of A are in C
In general, A can contain some elements that are also in C and some that aren't
Just take A = {1, 2} and C = {2}
Your case 1 would only apply to the element 2, since it's the only element in both A and C
Yet, A \ C is not empty
vestal thistle
I see
glass meadow
Then your case 2 would apply to the element 1, and there it actually works but without case 1 working, your proof is wrong
I'd say you're just going about it wrong
You want to show that A \ C included in B \ C
So pick any element of A \ C, and show that it is also in B \ C
(it's actually just what your case 2 does, so in reality your case 1 is useless, but you need to tidy that up)
vestal thistle
\begin{proof}
Let $A$, $B$ and $C$ be sets.
Either $A$ is empty or non-empty.
First, suppose $A$ is empty.
Then, $A \setminus C = \emptyset \subseteq B \setminus C$.
Now suppose $A$ is non-empty.
Then, either $C$ is empty or non-empty.
\begin{case}
Suppose $C$ is empty.
Then $A \setminus C = A \subseteq B = B \setminus C$, which is what we wanted to prove.
\end{case}
\begin{case}
Now suppose $C$ is non-empty.
Let $x \in A \setminus C$.
Then, by the definition of substraction, $x \in A$ and $x \notin C$.
Because $A \subseteq B$, we have that $x \in B$, so $x \in B \setminus C$.
Since $x \in A \setminus C$ implies $x \in B \setminus C$, by the definition of the subset,
we have that $A \setminus C \subseteq B \setminus C$.
\end{case}
Therefore, in any case, if $A \subseteq B$, then $A \setminus C \subseteq B \setminus C$.
\end{proof}
Mor Bras
glass meadow
Maybe make it clearer that you have x in B and x not in C, so x in B\C
I know you said x not in C in the previous sentence but it reads like you're deducing x in B\C from just A included in B
Also, since you care to special case the case of an empty A, you need to care about the case of an empty A\C
(example, A = C = {1, 2}, then neither A nor C are empty yet A\C is)
pearl pondBOT
@vestal thistle Has your question been resolved?
vestal thistle
Here is version 1.2:
\begin{proof}
Let $A$, $B$ and $C$ be sets.
Either $A$ is empty or non-empty.
First, suppose $A$ is empty.
Then, $A \setminus C = \emptyset \subseteq B \setminus C$.
Now suppose $A$ is non-empty.
Then, either $C$ is empty or non-empty.
Suppose $C$ is empty.
Then $A \setminus C = A \subseteq B = B \setminus C$, which is what we wanted to prove.
Now suppose $C$ is non-empty. Either $A = C$ or not.
\begin{case}
If $A = C$, then $A \setminus C = \emptyset \subseteq B \setminus C$.
\end{case}
\begin{case}
Otherwise, suppose $A \neq C$.
Then, there exist $x \in A$ that is not in $C$.
Let $x \in A \setminus C$.
Then, by the definition of substraction, $x \in A$ and $x \notin C$.
Because $A \subseteq B$, we have that $x \in B$.
Since $x \notin C$, by the definition of substraction, we have that $x \in B \setminus C$.
Since $x \in A \setminus C$ implies $x \in B \setminus C$, by the definition of the subset,
we have that $A \setminus C \subseteq B \setminus C$.
\end{case}
Therefore, in any case, if $A \subseteq B$, then $A \setminus C \subseteq B \setminus C$.
\end{proof}
Mor Bras
vestal thistle
<@&286206848099549185> Hello, could someone check if this proof looks good please?
shut elm
Looks good
You can skip the C empty/nonempty part though
i.e.
\begin{proof}
Let $A$, $B$ and $C$ be sets.
Either $A$ is empty or non-empty.
First, suppose $A$ is empty.
Then, $A \setminus C = \emptyset \subseteq B \setminus C$.
Now suppose $A$ is non-empty.
Either $A = C$ or not.
\begin{case}
If $A = C$, then $A \setminus C = \emptyset \subseteq B \setminus C$.
\end{case}
\begin{case}
Otherwise, suppose $A \neq C$.
Then, there exist $x \in A$ that is not in $C$.
Let $x \in A \setminus C$.
Then, by the definition of substraction, $x \in A$ and $x \notin C$.
Because $A \subseteq B$, we have that $x \in B$.
Since $x \notin C$, by the definition of substraction, we have that $x \in B \setminus C$.
Since $x \in A \setminus C$ implies $x \in B \setminus C$, by the definition of the subset,
we have that $A \setminus C \subseteq B \setminus C$.
\end{case}
Therefore, in any case, if $A \subseteq B$, then $A \setminus C \subseteq B \setminus C$.
\end{proof}
Coolempire93
shut elm
The A =/= C case covers C empty because we never assume there exists an element in C
vestal thistle
The A =/= C case covers C empty because we never assume there exists an element ...
I see
shut elm
Looks good
Actually, A =/= C could mean there is x in C not in A, which is a case you haven't considered it appears
i.e. if C is a subset of A
Is a case not included in your original
shut elm
Does it matter for this proof?
Well, not covering a case makes the proof not rigourous
Let me reread and make sure
vestal thistle
if x in C not in A, then x not in A \ C
glass meadow
Well, not covering a case makes the proof not rigourous
We only care about elements in A\C, not those in C\A
shut elm
Consider A = {1}, B = {1,2,3}, C = {1,2}
It's down to Case 2 since A is nonempty, C is nonempty and A =/= C
But we go by supposing there exists x in A not in C
This is not true here
glass meadow
Also, since you care to special case the case of an empty A, you need to care ab...
.
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plucky wagon
.
pearl pondBOT
pearl pondBOT
eager jewel
what did bro do
slow oak
spam bot
unborn abyss
what did bro do
divided by 0
slow oak
the mark of the mr beast
eager jewel
o lol
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crimson nebula
In the proof , why do we require |xn|< 1+ |x|
Can't we just write , M to be Supremum of {|x1|,....,|xn| }
Since if, M is the Supremum of absolute value
All |xn| ≤ M
crimson nebula
xn represent the nth term of sequence X
crimson nebula
Well after all this is a sequence
worldly jewel
but what do you mean when you say "M to be Supremum of {|x1|,....,|xn| }"
crimson nebula
A set
worldly jewel
How do you know the supremum exists?
crimson nebula
Okay ....
I understood
We had to take some number that's greater than all of the terms of sequence...?
worldly jewel
The proof strategy is essentially this. You want to pick some M so that |x_n| < M for all n, right?
crimson nebula
Yes
worldly jewel
Now you know a finite set of points {x1, x2, ..., xk} is always bounded because you can simply take the maximum
It is the infinite tail that needs to be taken care of
worldly jewel
Well ok, so by the convergence property you can pick a number K so that x-1 < x_k < x+1 for all k >= K, right?
crimson nebula
Yeaa
worldly jewel
This takes care of the tail, but you still have the finitely many terms in front, ie. x1, x2, ..., x_(k-1)
To take care of the front, you can take the maximum of x1, x2, ..., x_(k-1)
and to take care of the tail you can use |x|+1
crimson nebula
I m not familiar with this infinite tail concept
worldly jewel
By tail I just means the terms that come after x_K
crimson nebula
Okay
crimson nebula
Can't we just write M= 1+|x|
Still not understood
worldly jewel
If you write M = 1+|x| you only bound the terms that come after x_K
the terms x1, x2, ..., x_(K-1) can fall outside this bound
that's why you need to take the maximum {|x1|, |x2|, ..., |x_(K-1)|, 1+|x|}
crimson nebula
Okhayyy I understood
Because in convergence, we consider a k, above which they are within a certain limit
worldly jewel
Yes
crimson nebula
And we have shown that this 1+|x| using that convergence idea
worldly jewel
Mhm
crimson nebula
These proofs seems so easy
But little am I aware of the in depth idea hidden
.close
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cerulean ledge
hmm how do you see if the function is strictly monotone ascending?
weary ledge
is it like this y = 2x ?
no
the y mentioned in the above context is chosen from the domain of f
it is not the same y = f(x) that we normally take
weary ledge
y is just another value like x
cerulean ledge
but how do they see if y > x here?
weary ledge
but how do they see if y > x here?
its just a comparison; a general case
compact ridge
yeah they're trying to show that if $s < t, f(s) < f(t)$ for monotone increasing
jolly parrotBOT
south
compact ridge
and $s < t, f(s) > f(t)$ for monotone decreasing
jolly parrotBOT
south
weary ledge
they say one thing for y>x and another for y<x
compact ridge
for all s, t in the domain
cerulean ledge
they say one thing for y>x and another for y<x
so the way you show it is by showing that if there was an y that was x < y in this case then 2x < 2y then it would also be bigger than x in this case?
second case if x < y and -x^2 > -y^2
yeah i don't even see a number where this is possible?
must be a negative number
-5 < -1 and -(-5)^2 > -(-1)^2
so this is true for negative numbers?
weary ledge
so the way you show it is by showing that if there was an y that was x < y in t...
yes, since this function is relatively easy to prove
weary ledge
so this is true for negative numbers?
why do you think it only applies to negative nubers?
sorry for being afk for a while btw
compact ridge
-5 < -1 and -(-5)^2 > -(-1)^2
oh that's not what's being shown, be careful
the domain is when x, y are now nonnegative
so from x < y you can directly conclude x^2 < y^2
then -x^2 > -y^2
pearl pondBOT
@cerulean ledge Has your question been resolved?
cerulean ledge
yeah i feel like some parts of math are confusing because of definitions like this
like most of the concepts are very easy to understand anyway they usually just present it in a complex way
compact ridge
so from x < y you can directly conclude x^2 < y^2
(for a, b, c, d positive, if a < b and c < d, then ac < bd)
that's one of the field axioms
compact ridge
like most of the concepts are very easy to understand anyway they usually just p...
it also helps knowing 1) what the axioms are and 2) how to derive it from those axioms
it's so easy to take mental shortcuts which will hurt later in mathematics
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junior anchor
Do variables in LP always have to be non negative
pearl pondBOT
Do variables in LP always have to be non negative
junior anchor
LP = Linear Programming
bitter herald
Do variables in LP always have to be non negative
LP being?
toxic lichen
technically no. but if you do have one that is allowed all real values, and has a nonzero coeff in the cost function, then p sure you'll end up with unbounded optimum.
bitter herald
No they dont
junior anchor
technically no. but if you _do_ have one that is allowed all real values, _and_ ...
Ah I see
Is it just a "pretty sure"
Or a guarantee of any soet
Sort
toxic lichen
can probably cook up some proof there
i am reasonably certain of my assertion in the case of linear programs
bitter herald
most algorithms like the simplex method require the problem to be converted into a 'standard form' anyways where all variables are nonnegative
junior anchor
Yeah , so does it work to cut up a variable x which can be any real value into
x- and x+ of sort
x+ being (-1) times x-
And both forced to be non negative
Idk I'm like very new to lp
Just started learning about it like 10 minutes ago
toxic lichen
you can write $x = x_+ - x_-$ with $x_+ = \max(x,0)$ and $x_- = \max(-x,0)$ and then $x_+, x_- \geq 0$
jolly parrotBOT
Ann
toxic lichen
in this way you can reduce an LP with an unconstrained variable into one with ≥0 constraints but +1 variable count
note btw that even if some x is not itself constrained, other constraints may imply it can't actually vary over R
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unborn charm
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unborn charm
idk how to start, I can just say if x = 0 then I must have a_n = 0
worldly jewel
Hint: base-p expansion
autumn fossil
is it really that obvious that we can express every number in base-p expansion?
there is definitely something to prove
unborn charm
no but if I am using that to begin with, I already have a sequence
or are you implying I prove that
i guess yeah that would make sense
my bad lol
autumn fossil
yep, im implying exactly that. The statement is equivalent to "every number in 0,1 can be represented in base p expansion"
but its still something not entirely obvious and it does deserve a proof
keen star
can u construct a sequence like that works for all p
worldly jewel
You can give an algorithm which outputs a1, a2, ... given an x, and then prove that the corresponding sum converges to x
unborn charm
hmm yeah
I guess we are to use the division algorithm since a_n is strictly less than p
cursive wraith
I'm pretty sure you'll have to do a separate case for x = 1, but apart from that the "algorithm" should work
unborn charm
yeah I would need a starting point
unborn charm
should it really include x = 1?
pearl pondBOT
@unborn charm Has your question been resolved?
worldly jewel
if they always pick .999... then I guess it should work for 1
junior anchor
Given two random non equivalent Boolean functions, what is the chance that they agree for >p% inputs
pearl pondBOT
Given two random non equivalent Boolean functions, what is the chance that they ...
@junior anchor Has your question been resolved?
azure kite
can you give us some of your inputs on the problem
junior anchor
Inputs as in?
What I have worked out?
It started with this
He seems to have said "B1 B2 may disagree rarely"
As a way to discard that algorithm
But if the B1 B2 disagreeing rarely case itself is rare
Then I would reconsider discarding it
That's why I was wondering how to start working out the chances
I have no idea of how to begin tackling this
<@&268886789983436800>
pearl pondBOT
@junior anchor Has your question been resolved?
pearl pondBOT
@junior anchor Has your question been resolved?
empty oriole
Show that the set of differentiable real-valued functions 𝑓 on the interval
(−4, 4) such that 𝑓′(−1) = 3 𝑓 (2) is a subspace of 𝐑^(−4, 4)
empty oriole
I don't know how a set* of functions works as vector spaces
rough stream
Vector spaces are anything that can be added or scalar multiplied
(Where adding/scalar multiplication are defined by the 10 axioms for a vector space)
f(x) + g(x) is a new function
a(f(x)) is a new function
So functions can be added and scalar multiplied
toxic lichen
I don't know how a set* of functions works as **vector spaces**
functions are things that can be added together and scaled by numbers
that's (kinda) all you need for a vector space
empty oriole
Yeah
But, for proving the existence of a zero vector, I need to prove the existence of the zero function
I still don't get how there exists a zero function for any x on the differentiable real set of numbers (-4, 4)
rough stream
f(x) = 0 is the zero function
toxic lichen
I still don't get how there exists a zero function for any x on the *differentia...
you're really mixing up words here
quite badly
empty oriole
How do I know that f(x) = 0 holds true for any x over (-4, 4)?
toxic lichen
"differentiable" is a property of functions, not numbers, not sets of numbers.
rough stream
How do I know that f(x) = 0 holds true for any x over (-4, 4)?
What do you mean by "holds true"?
toxic lichen
How do I know that f(x) = 0 holds true for any x over (-4, 4)?
that's like asking "how do i know the number 33333 consists of nothing but 3's"?
empty oriole
What do you mean by "holds true"?
I don't know what f(x) is, really. So I am unsure how I know it's for, let's say, x = 2
toxic lichen
what you're examining is the space of all possible differentiable functions from (-4,4) to R,
rough stream
f(2) = 0
toxic lichen
in particular kaynex is directing your attention right now to the zero function in that space
and the zero function's whole thing is that, viewed as a function, it maps all x's to 0
R^(-4,4) is a huge space. one single function is just one point in that space.
empty oriole
What do you mean by "holds true"?
How do we prove that the zero function is an element of the set mentioned above?
empty oriole
in particular kaynex is directing your attention right now to the *zero function...
This set
toxic lichen
How do we prove that the zero function is an element of the set mentioned above?
by observing that it satisfies the equation f'(-1)=3f(2)
not sure what else there is to say though like... hm
have you seen other vector spaces before this
empty oriole
Sure
toxic lichen
R^3 for example
empty oriole
Yeah
empty oriole
P_5(R) or something
I don't think so
toxic lichen
mmm
right
yeah this sounds like a case of abstraction related motion sickness
but im not sure how to treat it
tiny nexus
Yo guys
Checkout ts question
Funner than whatever was here before
Quit this derivative stuff
empty oriole
but im not sure how to treat it
I get how R^3 maps to R
rough stream
Like, I think you're maybe asking how to prove f(x) = 0 is indeed the zero vector? This kind of thing is usually taken as "known" and isn't in every proof
toxic lichen
Funner than whatever was here before
!occupied
pearl pondBOT
Funner than whatever was here before
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
empty oriole
And I understand it as a prototype of R^{1,2,3}
toxic lichen
Wrong grammar
this channel is occupied. please move.
tiny nexus
Ann ur buns at English
empty oriole
?
rough stream
<@&268886789983436800>
toxic lichen
gtfo of this channel that's what
mortal seal
tiny nexus
What does gtfo mean
toxic lichen
What does gtfo mean
= "get the fuck out"
toxic lichen
don't tell me you don't know this old internet slang
dim linden
Quit this derivative stuff
this channel is occupied. please claim your own channel and stop derailing this channel
empty oriole
toxic lichen
Wrong grammar
nice ragebait btw 3.5/5 stars
toxic lichen
Take a day off from the trolling please
bro got noped
mortal seal
Let’s not get mad at each other and go back to helping andre, you shouldn’t give attention to trollers, that only encourages them
empty oriole
You drew this that fast? wow
empty oriole
lol
toxic lichen
whio can hielp me wioth 1plius 1 i need heolp
<@&268886789983436800> obvious troll is obvious
plain wing
pliease i need help
radiant terrace
Don't troll here
plain wing
yiou doniunbt beliuve me
empty oriole
Does this happen often? lol
plain wing
sioryuy i need help
radiant terrace
I muted them and they left
rough stream
Clearly an alt?
empty oriole
So, how do we define a function to map to the same result?
That is what's confusing me
Because I had the intuition that the elements of a set don't repeat
is f(2) a different element than f(-1)?
rough stream
These functions are in R^(-4,4).
That is, they take elements from the real interval (-4,4) and return back real numbers.
The only other rule we have is that this function must be differentiable.
And that's it. f(2) can be the same real number as f(1) if they want. Can even be the same as f(1.2).
empty oriole
Certainly, every real number is differentiable, right? (off-topic real analysis question)
Eh
doesn't matter
Thank you for your help!
How do I give points?
.close
pearl pondBOT
rough stream
Functions are differentiable, not sets. Maybe I misunderstood you?
empty oriole
Functions are differentiable, not sets. Maybe I misunderstood you?
No, it was my fault. I meant to ask what functions that map to a real number are differentiable
Forget it. It's a silly question (I can just google it.)
Thanks
.close
pearl pondBOT
Available help channel!
royal galleon
For 24 how could you pour water in if the vertex is up?
pearl pondBOT
For 24 how could you pour water in if the vertex is up?
sharp vigil
assume the cone has a base
royal galleon
assume the cone has a base
Yes I did. I can understand if the vertex is down and we are pouring water through the base but if the vertex is up and we pour through the base wouldn't our rate change since gravity is opposing it and if not wouldn't it just be an identical problem?
sharp vigil
where exactly you pour water in from doesn't affect things really
what does matter is that the shape of the tank is different
royal galleon
I don't see how. It is just upside
sharp vigil
yes
royal galleon
It is still a right circular cone though
grave mirage
For 24 how could you pour water in if the vertex is up?
Assume like an infinitely small hole attached to an infinitely narrow tap
sharp vigil
royal galleon
Oh ok so it is filling up a different shape which will affect how the height is changing.
Thanks. .close
.close
pearl pondBOT
pearl pondBOT
Available help channel!
amber pagoda
hi
amber pagoda
i asked lot of ai like chat gpt gemini copilot and grok all are giving me different anwers
sly jacinth
ok so y and z have to be equal right?
amber pagoda
ok so y and z have to be equal right?
yeah x too ig
sly jacinth
do you knowwhat the sum of all the interior angles of a triangle should be
amber pagoda
180
amber pagoda
oh wow ur right
amber pagoda
so if you were to have all the angles equal to 50
then whats the 50 there for
sly jacinth
it helps you determine the angle measures of y and z
since this is an isosceles triangle y = z
50 + z + y = 180
amber pagoda
ah
sly jacinth
you can set y and z to the same variable
amber pagoda
i asked lot of ai like chat gpt gemini copilot and grok all are giving me differ...
so there are 2 interior angles in this pic
warped violet
i asked lot of ai like chat gpt gemini copilot and grok all are giving me differ...
!noai
pearl pondBOT
i asked lot of ai like chat gpt gemini copilot and grok all are giving me differ...
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
sly jacinth
so there are 2 interior angles in this pic
6, and one of them is given
warped violet
rather, come here 🙂
amber pagoda
if i understand what vertex means
so whats the answer here
sly jacinth
i cant just tell you it
amber pagoda
true
warped violet
This is all you need
sly jacinth
yeah ig so
amber pagoda
yeah ig so
wait
what if i check a yt video rq about this
and come back is that possible
warped violet
what if i check a yt video rq about this
What do you not understand?
sly jacinth
what he said
its not a very complicated concept to grasp
just try to understand what i said
amber pagoda
cause if all angles should be equal to 180
all the answers are more than 180 cobined
warped violet
I didn't mean it that way. We all start from somewhere. I just want to understand where to begin explaining
warped violet
This is all you need
You understand all these 4 segments are same length
amber pagoda
the blue sticks u drew?
sly jacinth
cause if all angles should be equal to 180
noooo
warped violet
Yes
amber pagoda
affirmative
sly jacinth
well my wifi is dying so ill lleave it to him
amber pagoda
yeah
warped violet
What do we know about them
amber pagoda
oh
so there are 2 traingles
so both of the triangless sum shoudl add up to 180
not both
but each
warped violet
"triangles sum" doesn't really make sense
amber pagoda
interior angles sum
amber pagoda
warped violet
z, y and one that measures 50°
warped violet
And their sum must equal to 180°
warped violet
so z and y should be 50 too
Why?
amber pagoda
erm cause the angles arent equal no?
warped violet
erm cause the angles arent equal no?
what?
amber pagoda
forget it cause it doesnr make sense
amber pagoda
exactly
jolly parrotBOT
USS-Enterprise
jolly parrotBOT
USS-Enterprise
warped violet
So the sum of z and y must be 130°
But this yet doesn't tell us enough to figure out what z and y are
amber pagoda
yeah
warped violet
z could be 1 and y 129. z could be 80 and y 50
So we need another piece of information
Look at the triangle
amber pagoda
warped violet
What can you tell me about it
amber pagoda
that y and z should be equal
warped violet
Yes, why?
amber pagoda
cause they look the same
sly jacinth
he aint wrong
amber pagoda
vro why u making it hard
warped violet
what
amber pagoda
the angles side by side are equal
amber pagoda
ture
amber pagoda
equirtaeral traingle
warped violet
What's an equilateral triangle?
amber pagoda
warped violet
Yes
amber pagoda
i see ur tiny sticks here
amber pagoda
exaclty
amber pagoda
2
warped violet
yes
we don't know anything about the length of the segment from angle y to angle z
amber pagoda
we just know that they are same value
warped violet
But we do know the two segments going to the center are the same length
So this is not an equilateral triangle
But it is a different type of triangle
amber pagoda
wat
warped violet
3rd times the charm
amber pagoda
An equilateral triangle has all 3 sides the same length
yeah
warped violet
I just said the segment from angle y to z is not the same length
Therefore this is not an equilateral triangle
amber pagoda
ok
warped violet
Plus, if it were
all 3 interior angles would be 60°
and we clearly see one is 50°
Okay
amber pagoda
true
warped violet
ever heard of an isosceles triangle?
warped violet
Yes
warped violet
2 sides are equal, like here
amber pagoda
TRUE
amber pagoda
YES
warped violet
The base is the one segment which doesn't have a "little stick"
Okay, so
$z + y = 130^{\circ}$
jolly parrotBOT
USS-Enterprise
amber pagoda
where did 130 came from
warped violet
$z + y = 130^{\circ}$
?
warped violet
yes
amber pagoda
why didnt the command work
amber pagoda
$180/2$
jolly parrotBOT
!ᐯᗩᒪIᑌᗰ
warped violet
130, not 180
amber pagoda
i forgor
amber pagoda
65
warped violet
yes
warped violet
Ever heard of vertical angles?
amber pagoda
jolly parrotBOT
USS-Enterprise
amber pagoda
same
warped violet
yes
warped violet
If it were, the right triangle's sum of interior angles would be 150°
That can't be
not because of that
but because the sum of interior angles of a triangle is always 180°
so it can't be 150°
amber pagoda
casue here ost of them are equal so ithought this one would be too
warped violet
$180 \neq 150$
jolly parrotBOT
USS-Enterprise
amber pagoda
true
pearl pondBOT
pearl pondBOT
Available help channel!
empty oriole
Show that the set of differentiable real-valued functions 𝑓 on the interval
(−4, 4) such that 𝑓′ (−1) = 3 𝑓 (2) is a subspace of 𝐑^(−4, 4)
west sapphire
unfortunately the channel auto-closed because you deleted the first message
you'll have to open a new one
plush bramble
Show that the set of differentiable real-valued functions 𝑓 on the interval
(−4...
Open it after you fix the latex
empty oriole
unfortunately the channel auto-closed because you deleted the first message
.reopen
okay
pearl pondBOT
Available help channel!
dense jasper
.close
pearl pondBOT
pearl pondBOT
Available help channel!
crimson hawk
Let P be a point outside circle (O). PA and PB are two tangents to the circle at A and B. A secant line PDC is drawn such that D lies between P and C. A line KL passes through A and is parallel to PB, where K and L lie on BD and BC respectively. Prove that A is the midpoint of KL
dapper kraken
smells symmedian
pearl pondBOT
@crimson hawk Has your question been resolved?
crimson hawk
smells symmedian
Can u explain further pls
dapper kraken
idk i havent got it either lmao
but its construction is like a symmedian construction
CD is the symmedian to ABC
naive sandal
this makes my head hurt bro im working on it tho
crimson hawk
i asked my classmate construct auxiliary lines and try to find similar triangles but i have not found the solution 🙁
crimson hawk
CD is the symmedian to ABC
but CD is not fixed :<
dapper kraken
err
ok i kinda reversed the problem, with a given triangle ABC first
should be identical tho
KDLC is concyclic but dunno how that helps
crimson hawk
KDLC is concyclic but dunno how that helps
no it isnt concyclic
dapper kraken
u sure?
dapper kraken
crimson hawk
sorry im mistaken CKD with AKD 😭
subtle oak
KL and BP are parallel so trying to use equal angles?
crimson hawk
KL and BP are parallel so trying to use equal angles?
ive tried that 🙁 i even used thales thereom but it didnt seem to work
subtle oak
ive tried that 🙁 i even used thales thereom but it didnt seem to work
No? For me it looks right now as if it would just need tangent-secant relation after it but could be mistaken
crimson hawk
No? For me it looks right now as if it would just need tangent-secant relation a...
can u illustrate it pls, my brain doesnt function properly rn :0
subtle oak
Wait could be mistaken, gimme a minute
crimson hawk
this makes my head hurt bro im working on it tho
have u found anything yet :<
subtle oak
‘‘‘ok, AK/AL = AD/AC; AB^2 = AD * AC; then you have AK = APAD/AB and AL = AP * AC/AB; multiply these to get AKAL = AP^2 * (AD*AC)/AB^2 now substitute AD * AC = AB^2 and you‘ll end up with AK * AL = AP^2; you know AP = PB; and AKL and APB are similar so AK/AL = AP/PB = 1‘‘‘
crimson hawk
‘‘‘ok, AK/AL = AD/AC; AB^2 = AD * AC; then you have AK = AP*AD/AB and AL = AP * ...
why do we have AK/AL=AD/AC????
subtle oak
KL || PB So AKD and APB should be similar and ALC and APB should be, so AK/AP = AD/AB and AL/AP = AC/AB, I think
dapper kraken
‘‘‘ok, AK/AL = AD/AC; AB^2 = AD * AC; then you have AK = AP*AD/AB and AL = AP * ...
AK/AL is indeed not AD/AC
crimson hawk
ABP isnt equal to AKD or APB?
dapper kraken
AKD and APB isnt simmilar
ALC and APB isnt even close to simmilar??
yeah lowk idk
maybe you meant diffrent triangles?
crimson hawk
maybe you meant diffrent triangles?
if u talk to me then i was mentioning angles ( i js want to clarify :3 )
dapper kraken
nah i was talking to the guy that said that the triangkes were simmilar
yeah im out of ideas
maybe menelaus might work? but i couldnt get it
my last idea is trig bashing but id rather not
subtle oak
You‘re right, made the drawing wrong, then I don‘t know
crimson hawk
*maybe* menelaus might work? but i couldnt get it
i dont really know, i rarely use it 🙁 it seems inefficient and tiring
subtle oak
Sorry
crimson hawk
You‘re right, made the drawing wrong, then I don‘t know
:<
subtle oak
Maybe with coordinates? That would be my last guess how to do it tbh
crimson hawk
Maybe with coordinates? That would be my last guess how to do it tbh
ive not learnt that yet
subtle oak
ive not learnt that yet
ah, yeah, uhm, wait let me try something
naive sandal
pearl pondBOT
@crimson hawk Has your question been resolved?
crimson hawk
u drew it wrong :<
naive sandal
i just realized after i sent it ts has me crying ill try again
subtle oak
You haven‘t learned vectors?
maybe try projecting the line BD and BC onto line KL from center A. AC/ADD = BC/BD. I think that could help coming to a result?
crimson hawk
You haven‘t learned vectors?
yes i have not learnt that yet
subtle oak
maybe try projecting the line BD and BC onto line KL from center A. AC/ADD = BC/...
that seems to could work
crimson hawk
maybe try projecting the line BD and BC onto line KL from center A. AC/ADD = BC/...
idk 🙁 i will go back to class now, good byr
subtle oak
Bye
pearl pondBOT
@crimson hawk Has your question been resolved?
vast berry
can someone please explain the hihglighted part to me
pearl pondBOT
can someone please explain the hihglighted part to me
tropic saddle
you should have seen that same construction previously
G/H
x~y iff x^-1 y in H
the same here with H=stab(s)
vast berry
oh ok
twin portal
pearl pondBOT
twin portal
If D is the midpoint of the side BC of a trianlge ABC and E is the midpoint of AD, find the value of AF/FC.
proper nova
If D is the midpoint of the side BC of a trianlge ABC and E is the midpoint of A...
!status
pearl pondBOT
If D is the midpoint of the side BC of a trianlge ABC and E is the midpoint of A...
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
are vectors allowed or not
twin portal
noup
toxic lichen
twin portal
-# no vectors allowed, it's just a geo problem
Geometry problem doesn't imply vectors aren't allowed.
proper nova
Geometry problem doesn't imply vectors aren't allowed.
i can try to assume that $\frac{AF}{FC} = \frac{1}{2}$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
it's just a diagram assumption
toxic lichen
i can try to assume that $\frac{AF}{FC} = \frac{1}{2}$
bad
assuming the answer equals 1/2 is just a bozo tier move
proper nova
and knowing that, i can try to let $G$ be the midpoint of $FC$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
and prove $AF = FG = GC$
jolly parrotBOT
1 divided by 0 equals Infinity
toxic lichen
guessing the what
thorn agate
ew geo
twin portal
Let area(BED) = area(DEC) = x.
Let area(GEA) = y and area(AEF) = z.
We have area(GEB) = x - y and area(EFC) = x - z
Now area(CEF)/area(AEF) = area(BFC)/area(BAF)
So, (x-z)/z = (3x - z)/(x + z)
x^2 - z^2 = 3xz - z^2
x = 3z
x - z = 2z
(x-z)/z = 2:1
area(FEC)/area(AEF) = 2:1
FC/AF = 2:1
AF/FC = 1/2
Is this correct and good?
proper nova
Let area(BED) = area(DEC) = x.
Let area(GEA) = y and area(AEF) = z.
We have area...
i don't think $S_{GEB} = x - y$
jolly parrotBOT
1 divided by 0 equals Infinity
twin portal
ABC is any triangle.
proper nova
can you explain why?
twin portal
AE = ED, and hence area(ABE) = area(DEB) = x. Now subtract from this area(GEA) = y.
proper nova
let.. is that only true for isoceles triangles
and i was also wondering about this too
proper nova
AE = ED, and hence area(ABE) = area(DEB) = x. Now subtract from this area(GEA) =...
since when did you do that?
since when did you let $S_{ABE} = x$?
jolly parrotBOT
1 divided by 0 equals Infinity
toxic lichen
Is this correct and _good_?
criterion of goodness?
twin portal
I let area(DEB) = x, which is equal to area(DEC). Then I concluded that area(BEA) = x. I realize that I missed a few steps.
thorn agate
I let area(DEB) = x, which is equal to area(DEC). Then I concluded that area(BEA...
why do you think so
twin portal
D is the midpoint of BC. Again, I could have written in the first line that x = area(BED) = area(DEC) = area(BEA).
proper nova
D is the midpoint of BC. Again, I could have written in the first line that x = ...
he means explain why the areas are equal
as in prove why the areas are equal
twin portal
Oh.
thorn agate
D is the midpoint of BC. Again, I could have written in the first line that x = ...
actually it is correct
twin portal
Both have bases of equal magnitude and the corresponding heights are the same as seen by dropping the altitudes.
thorn agate
ED still remains median dividing the triangle into two equal parts
pearl pondBOT
twin portal
Thanks btw.
thorn agate
I guess it's correct. I have another question.
what question
pearl pondBOT
Available help channel!
wary vault
Suppose there are 100 students, and we wanna make a 5-member committee out of them, which will consist of 5 different roles: president, vice president, secretary, cashier, and a prospective member. Suppose, now, that we have 2 students, George and Catherine, both of which are part of the 100 students eligible for a place in the commitee. We wanna calculate the number of ways that { George is elected president or Catherine is elected cashier }.
This is an exercise our prof gave us, and upon solving it, i found the number of ways to be equal to:
2C(99,4) -C(98,3) . The reason for that is because i took the ways George can be president, which equals the ways Catherine can be cashier, and then i subtracted the ways both of them can be in their respective roles. It comes out to 7.376.656 .
The solved exercise says it is equal to 179.777.472 . Like wtf? did i do smth wrong?
wary vault
heres the proof provided in the notes
honestly the solution provided does look correct too. i just dont get what i did wrong
unborn abyss
the 5 roles are all different
so P(99, 4) is the correct thing to use, not C(99, 4)
wary vault
oh shit, using C it chooses them for any position
yeah i forgot abt that
thanks
.close
pearl pondBOT
pearl pondBOT
Available help channel!
sly jacinth
pearl pondBOT
thorn agate
thorn agate
instanttaneous
sly jacinth
LOL
thorn agate
the curves would flip after intersection point
thorn agate
add these for each
thorn agate
yeah
sly jacinth
from the same range i assume?
thorn agate
yes
thorn agate
wait i have a doubt
thorn agate
add these for each
they don't look like this
sly jacinth
yeah
thorn agate
i'm sorry unable to help ping helpers
pearl pondBOT
pearl pondBOT
Available help channel!
eager jewel
pearl pondBOT
eager jewel
ive got the answer in a form but im not able to convert it to the questions form
let me send my working
my t = sqrt(1+x^2) + x
.close
pearl pondBOT
pearl pondBOT
Available help channel!
wary vault
Hi. So, a pizza restaurant advertises that they sell over 500 kinds of pizza. It is given that they have 9 different ingredients that they can put in a pizza. To calculate the amount of different pizzas they can make, the given solution does it with whether one ingredient is used or not every time, and thus the amount of pizzas is 2^9 = 512. That seems correct, but doesnt that assume that we dont concern ourselves with which order each ingredient is put in the pizza? If we did care for that, would the amount of pizzas equal 9!+1 or am I far off?
fast briar
The calculation of (2^{9}=512) is correct for the scenario described, and if order mattered, the calculation would involve a summation of permutations of subsets, not (9!+1).
jolly parrotBOT
uraverageindian
fast briar
thats what i think
wary vault
oh i think i get you
fast briar
(\sum _{k=0}^{9}P(9,k)=\sum _{k=0}^{9}\frac{9!}{(9-k)!})
jolly parrotBOT
uraverageindian
toxic lichen
Hi. So, a pizza restaurant advertises that they sell over 500 kinds of pizza. It...
if you have a pizza how can you tell if the pepperoni was put on it first and then the jalapeños, vs the other way around
fast briar
To calculate all possible unique ordered pizzas (assuming you can choose any number of ingredients from 0 to 9), you would need to sum the permutations of every possible subset of ingredients. The general formula for this would be the sum of permutations for all subset sizes (k) from 0 to 9:
wary vault
yeah 9!+1 would be if we used 9 ingredients for sure
jolly parrotBOT
uraverageindian
wary vault
if you have a pizza how can you tell if the pepperoni was put on it first and th...
sheldon would
fast briar
sheldon would
"according to the bubble and the fact that my left buttcheek is itchy, then that means that the pepporoni was put on first"
pearl pondBOT
fast briar
np
pearl pondBOT
Available help channel!
wary vault
Hi. Suppose we have a roundtable, and 5 people who wanna sit around it. How many ways are there, if 2 specific persons out of the 5, have to sit together? the solution says it is 3!,
. But the way i see it, it equals 2×3! because the two people can exchange seats, no?
pearl pondBOT
wary vault
i scroll down a bit each time to not spoil myself lol
toxic lichen
the columns for the edges in a cycle add up to 0
with a linearly independent set of columns this never happens
the columns for the edges in a cycle add up to 0
this is already a nontrivial linear combination
pearl pondBOT
Available help channel!
warped violet
pearl pondBOT
Available help channel!
celest turret
can someone tell me if this is correct and if not what am i doing wrong
unkempt yacht
can someone tell me if this is correct and if not what am i doing wrong
looks correct
pearl pondBOT
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Available help channel!
fading ledge
pearl pondBOT
fading ledge
i applied ratio test for checking convergence
dusty jungle
x is any positive number? or are they asking you to determine all x for which it converges?
pearl pondBOT
@fading ledge Has your question been resolved?
still hamlet
i applied ratio test for checking convergence
what did you get R as
fading ledge
R is r
pearl pondBOT
@fading ledge Has your question been resolved?
midnight haven
what's lim n-> inf n²/(n+1)²
are u fr asking?
crimson nebula
I have learnt about *m-tail of a sequence".
But what do we mean when we we say "tail of a sequence".
ie "m" is not mentioned
autumn fossil
I have learnt about *m-tail of a sequence".
But what do we mean when we we say "...
Is there any context?
usually it'd just mean some m-tail of a sequence
crimson nebula
Some of the people wrote in their replies
So i thought there might be some concept
autumn fossil
I think it's just m-tail, where you dont want to specify the m
so like "all tails do x" would mean "for all m, m-tail does x"
autumn fossil
"some tail does x" would mean "for some m, m-tail does x"
autumn fossil
So for every m, that holds
whether its every m or some m depends on the context
autumn fossil
yeah, sure
pearl pondBOT
pearl pondBOT
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green aurora
Why is the probability of the empty set 0 if it is in A then P(A)=1/n???
crystal dew
need context
tardy reef
there are 2^n elements in P(A)