Available help channel!
#help-36
final saddleBOT
Available help channel!
novel nexus
how do i calculate the number of terms here
final saddleBOT
how do i calculate the number of terms here
novel nexus
just take the last term minus the last term
this only works if step is 1
i could do it if the sum was iterating by n or n^2 but not this type of problem
vital crag
i see
if n is even and p=1 and the index is k, then the first term is k=1 and the last term is k=n/2. try finding what the last index is for p=2
then continue for arbitrary natural p
novel nexus
one of the sub problems is p=80
vital crag
sure once you get p=2, p=80 shouldn't be too much harder
you can be rigorous and use induction i suppose
final saddleBOT
@novel nexus Has your question been resolved?
barren agate
hello
hi guys, i just downloaded discord so im trying to work my way around it
i know this is a math channel, but i need help with finance. anyone know any servers or is able to help?
civic cloak
finance, believe it or not, is a key part of math
novel nexus
finance, believe it or not, is a key part of math
seems more like other way around to me
final saddleBOT
@novel nexus Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
whole halo
final saddleBOT
whole halo
have you solved the problem yet?
well I think you should try solving it first just to be sure youve accounted for everything necessary for the proof
rocky tusk
bro are you real
hasty mist
well I think you should try solving it first just to be sure youve accounted for...
To be fair, I think you should still show your work.
rocky tusk
tough bro
whole halo
this isnt as easy as it looks, for some reason I figured you could do T^2 - U^2 = (T + U)(T - U) but that doesnt really pan out
in general I need to spend time more productively on other things I do know about
so no, I dont have any work, and I do not have the time to create any - Ive spent it all
if on the other hand you have a proof, Im looking for a hint or a trick that can solve this
rocky tusk
đ¤Ą
hasty mist
Every positive semidefinite operator A has a unique positive semidefinite square root A^{1/2}.
whole halo
yes we need to prove uniqueness
plucky rover
If you know what cooly just said, all you now have to do is show that the square of a psd operator is psd
whole halo
you do need to rule out though that there isnt another square root that would be diagonal in a second basis but wouldnt be diagonal in the original basis
hasty mist
If A is positive (self-adjoint) and T, U are positive operators with T^2 = U^2 = A, then both T and U must be diagonalisable in the same orthonormal basis as A, and on each eigenspace of A they act as multiplication by sqrt{lambda}.
final saddleBOT
@whole halo Has your question been resolved?
hasty mist
This is literally how help channels work, so I donât know why youâre being monkeyâed. 
whole halo
zedias its understandable where youre coming from, but this is not an easy question
a majority of the difficulty Im seeing here is in proving this thing which by all accounts has no reason to be false
so Im asking you to try proving it first, so that if you lead me, I know youve gotten to the end already
Ive made the mistake of trying to help someone in a problem I had no lead in, which ultimately wasted a lot of time for the both of us
you can be specific by which of these so-called dudes youre talking about
hasty mist
That is understandable. However, it is generally recommended to provide any work you have completed when asking a question. If someone requests to see your work, the appropriate response is either to share what you have or to indicate that you have none. Doing so helps not only the person requesting your work but also anyone attempting to assist you.
whole halo
this isnt as easy as it looks, for some reason I figured you could do T^2 - U^2 ...
are none of you even reading that I dont have any work?
Im running out of time you know
hasty mist
have you solved the problem yet?
Iâm referring to this response.
hasty mist
you do need to rule out though that there isnt another square root that would be...
Do you have a question in regards to this? It should clear your concern.
whole halo
...have any of you even tried this problem yet?
hasty mist
If A is positive (self-adjoint) and T, U are positive operators with T^2 = U^2 =...
I meant that this should clear your concern.
whole halo
let me reiterate what the original problem is asking
I was not given that there is exactly one positive semidefinite S such that S^2 = A
I need to prove that that is the case
I cannot use things that are outside of the course, which would include a fact like that
so far we only have seen that one exists
your proof literally states it as a given
look at it
hasty mist
If A is positive (self-adjoint) and T, U are positive operators with T^2 = U^2 =...
This should serve as a hint as to why the second basis idea fails.
whole halo
calling it a "basic fact" does not work, I still need to prove the fact
if T^2 is diagonalizable in A, then you still need to prove that T can only ever be diagonalizable in a basis that is a permutation of A
Im trying to get around this thing you keep leaving out by defining the bases to be sorted by eigenvalues ascending, then seeing if I can worm my way into stating theyre the same
as I have said before, the problem is not conceptual, it is getting the damn words to connect right
you cant appeal to common sense because I am in a proofs class, common sense only allows this problem to be the first one listed on the homework but no more beyond that
theres gotta be something simple though
that sorting idea better be the last thing I need to do
whole halo
if T^2 is diagonalizable in A, then you still need to prove that T can only ever...
ok well this is not correct, I could scalar-multiply them by -1 for instance
I was thinking maybe I rewrite these to be in terms of the eigenspaces instead which dont care about order, I show those sets partition the same then that would be it? but Ive never done something like that before
opaque ember
based on this and your previous posts i strongly suspect AI output. never pass off AI output as your own work. see you in a year
whole halo
oh wow, that was sudden
pure kraken
Hi
hasty mist
yes we need to prove uniqueness
Suppose B is another positive operator with B^2 = A. Show B commutes with A. Conclude B preserves each Eigenspace E_lambda. Analyze B on one eigenspace E_lambda. Use positivity of B. Compare with A^1/2.
whole halo
I dont have a good feeling about this
I did try to show TU = UT much earlier on but that didnt lead much of anywhere, it must ultimately be true but I couldnt get it to work
hasty mist
You said youâve no workâŚ
whole halo
well yea, put it this way
I dont have anything that works
I can show you this
does it give you any new ideas? no
top half is just what youve said before except I wrote it before you got here
bottom part tried to use the inner product definition but got nowhere
I just dont see why you needed something like this
its also very disorganized, the basis beta in the middle of the work is not at all the same beta as at the top
iirc thats like the top half but I tried doing it with one basis instead of two
this also leaves out all of the work I deleted btw, which wouldnt have worked
whole halo
.close found a cheatsheet
blazing lily
Desmos project for Algebra 2 Honors Conic Sections
My teacher added a requirement where we have to put the conic sections in the way shown in the first photo. It does not work when I try to do the F=[âŚ.. but only works when I write 0=[âŚ.
Also donât know how to plot complex number points either
final saddleBOT
Desmos project for Algebra 2 Honors Conic Sections
My teacher added a requiremen...
blazing lily
<@&286206848099549185>
vital crag
Just google desmos tutorials
blazing lily
But I did exactly what my teacher did
And its not showing
And I donât really know what to google
final saddleBOT
Available help channel!
old quarry
final saddleBOT
sonic crystal
what you think
runic needle
is it homogenous?
idts
old quarry
final saddleBOT
old quarry
dont know where to start
i tried taking x^2 out from both brackets
but nothing was simplifying
glossy zephyr
you probably should try moving around the parts of the equation
mostly because, this DE doesnt include y
old quarry
you probably should try moving around the parts of the equation
ye ok i can do variable separation but after that?
glossy zephyr
Try to write it as dy/dx = f(x)
runic needle
integrate
glossy zephyr
dy/dx = (2x^2+2x+3)/(x^4+2x^3+3x^2+2x+2)
i hope you good with dividing polynomials
runic needle
i tried taking x^2 out from both brackets
i doubt
old quarry
i hope you good with dividing polynomials
i know how to..i divide the denominator with the numerator?
how do i know that will work
runic needle
no
old quarry
then what
runic needle
maybe like factorize or smth
old quarry
how to factorize that
runic needle
idk
old quarry
;/
glossy zephyr
i know how to..i divide the denominator with the numerator?
youre basically in for some work doing polynomial factorization and division, youll eventually get pretty well known integrable polynomial fractions
glossy zephyr
Tbh there just might be some other way to solve it, but im not precisely sure about it
runic needle
but u could complete the square in the denom or smth
old quarry
youre basically in for some work doing polynomial factorization and division, yo...
how to factorize the polynomial?
runic needle
quartic formula
old quarry
do you _really_ need to solve this question?
its practice
candid pulsar
how to factorize the polynomial?
partial fractions
yeah the denominator is unfortunately factorizable
but not in terms of the numerator directly
old quarry
yeah the denominator is unfortunately factorizable
what are the factors bruh
old quarry
how did u find that
candid pulsar
search
glossy zephyr
classic w
old quarry
no manually if u have to do then
candid pulsar
yeah this is weird
old quarry
goofy ahh question
glossy zephyr
im trying to do some trickery integrating with y and x through different variables and doing some curve fitting lmao
old quarry
yeah i works
ye cause u know the factor is x^2+1
candid pulsar
ye cause u know the factor is x^2+1
yeah maybe ill consider this from now onwards
old quarry
yeah maybe ill consider this from now onwards
but they wont give the exact same denom đ
back to the question..is there any practical way to do it?
or did they just give this so noone gets full?
candid pulsar
i guess you just resort to partial fractions for denominators with a larger degree than numerators?
like consider $\int \frac{x^4+1}{x^6+1} dx$
soft zealotBOT
candid pulsar
i mean the question setter probably just wanted you to check with i and -i
old quarry
:( who tf checks i and -i bro đ
candid pulsar
out of the box thinking
old quarry
ok ig i can whine but thats not going to help
ill not waste more time on this question
thanks bro
.close
crimson ermine
final saddleBOT
crimson ermine
for this proof of bezouts lemma why is gcd(a,b) considered to be the smallest possible element of n when ax+by=n when in this example of a linear combination 2x+4y where the gcd is 2, one is the smallest number
hearty socket
2+2 is 5
final saddleBOT
hearty socket
hi
tired walrus
hi
hello and welcome to the server. if you have a question of your own then please post in an available help channel.
!help
final saddleBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #âhow-to-get-help for instructions.
hearty socket
i dont need help
tired walrus
then #discussion
final saddleBOT
Available help channel!
worthy radish
Can anyone explain each of these signs and give an example of EACH?
final saddleBOT
Can anyone explain each of these signs and give an example of EACH?
dense coral
Can anyone explain each of these signs and give an example of EACH?
from right to left: âis an element ofâ, âis not an element ofâ, âis a subset ofâ, âis not a subset ofâ, âempty setâ, âempty setâ
worthy radish
Can you give example for ø?
worthy radish
Can you give example for ø?
dense coral
could you elaborate on what you want? 
the empty set isnât something for which there are several examples
worthy radish
So if A is 1
and B is 2,3,4
Then
A ø {B}?
dense coral
ah, ø isnât an symbol meant to designate a relationship
willow tiger
Nooo
worthy radish
Then what is it
dense coral
itâs a standalone symbol meant to signify the empty set itself
willow tiger
A = {} = enpty symbol
dense coral
ø = {}
scarlet sequoia
You want to say their intersection is empty
scarlet sequoia
So if A is 1
and B is 2,3,4
Then
A ø {B}?
In your example $A \cap B = \emptyset = {}$
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
**Raphaelisius Maximus MMIII**
And which is empty A or B?
scarlet sequoia
Coming back to your example, neither
A = {1}
B = {2,3,4}
Neither is empty
But the intersection (the set that contains elements that are in both at the same time)
The intersection is empty
worthy radish
The intersection is empty
Huh?
scarlet sequoia
That's what the symbol $\cap$ means, "intersection"
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
What is that symbol
scarlet sequoia
What is that symbol
Like the elements in common
If A = {apple, tree}
B = {tree, branch}
"Tree" is the only element in common
So $A \cap B = {tree}$
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
$\in$?
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
Yes
scarlet sequoia
$a\in A$ means "object $a$ is in the set $A$"
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
So, for example
livid plover
Hello can someone help in this problem !?
scarlet sequoia
In the set $A = {apple, tree}$
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
Hello can someone help in this problem !?
!occupied
final saddleBOT
Hello can someone help in this problem !?
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #âhow-to-get-help for instructions).
scarlet sequoia
In the set $A = \{apple, tree\}$
For example, in the set A there is "apple"
So we have $apple\in A$
worthy radish
So
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
A ⏠Apple, tree
severe canyon
???
scarlet sequoia
A ⏠Apple, tree
It's kinda the opposite
worthy radish
So itâs used when one element is in a set
scarlet sequoia
Think of sets as boxes
And objects are things you can put in boxes
"$a\in A$" means "object $a$ is in my box $A$"
soft zealotBOT
Raphaelisius Maximus MMIII
worldly mesa
$\notin$ means that
scarlet sequoia
"$a\notin A$" means "object $a$ is not in my box $A$"
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
So
scarlet sequoia
What is that
So, we can compare boxes between each other
worldly mesa
a set A is called a subset of a set B, notated $A \subseteq B$ if $\forall x. x \in A \implies x \in B$
soft zealotBOT
ExpertEsquieESQUIE
worthy radish
So, we can compare boxes between each other
So thereâs no one object but two sets?
scarlet sequoia
So thereâs no one object but two sets?
Yes, it's between sets, so two boxes $A$ and $B$
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
So like
scarlet sequoia
If everything that's in my box A is also in my box B
worthy radish
{A} = 5,6
{B} = 5,6,7
{A} c {B}?
scarlet sequoia
We write $A \subset B$
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
{A} = 5,6
{B} = 5,6,7
{A} c {B}?
You're putting the brackets on the wrong side but it's fine
worldly mesa
**Raphaelisius Maximus MMIII**
I prefer \subseteq
and \subsetneq for strict inclusion
scarlet sequoia
I prefer \subseteq
Different countries/regions, different notations
scarlet sequoia
and \subsetneq for strict inclusion
In France for example, \subset means \subseteq
worthy radish
But what if
A = {5,6,7}
B = {5,6,7}
Is it still
{A} c {B}
scarlet sequoia
But what if
A = {5,6,7}
B = {5,6,7}
Is it still
{A} c {B}
{5,6,6} is in fact the same as {5,6}
Copies of an object only matter once
worldly mesa
sets don't care about the order of their elements nor about duplicates
worthy radish
{5,6,6} is in fact the same as {5,6}
I mean {5,6,7}
scarlet sequoia
I mean {5,6,7}
It depends on what your book intended when they wrote $\subset$
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
If they haven't written $\subseteq$
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
But what if
A = {5,6,7}
B = {5,6,7}
Is it still
{A} c {B}
Then my bet is that A c B here
worthy radish
Okok
scarlet sequoia
$A\subset B$ means "everything that's in my box A is also in my box B"
And so $A\not \subset B$ means "not everything that's in my box A is also in my box B"
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
So ${5,8} \not \subset {5,6,7}$
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
Is 0 a natural number
scarlet sequoia
That's a different topic
And the answer: it depends on who you ask
And which conventions you took
worthy radish
Yeah but they also teach about these in this chapter
scarlet sequoia
Some anglophone systems use the convention that the set of natural numbers is $\bN = {1,2,3,...}$ and thus they write $\bN_0 = {0,1,2,3,...}$ the set of non negative integers
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
Some other anglophone systems and, for example the french system, use the convention that the set of natural numbers is $\bN = {0,1,2,3,...}$ and thus they write $\bN^* = {1,2,3,...}$ the set of positive integers
soft zealotBOT
Raphaelisius Maximus MMIII
scarlet sequoia
But 0 isnât a negative number
For some people, natural = positive integer
worthy radish
For some people, natural = positive integer
And zero is a positive number?
As the result wonât result in a minus
scarlet sequoia
And zero is a positive number?
Zero is not positive
willow tiger
For some people, natural = positive integer
Yea those people are not getting educated right
scarlet sequoia
Yea those people are not getting educated right
Let's not judge "righteousness" based on conventions
I like the convention that the naturals start at 0, but we have to accept the fact that not everyone thinks like that
We still do the same maths and end up with the same results eventually
scarlet sequoia
Why isnât i) true?
Because $\sqrt 9 = 3$
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
But dude
worthy radish
Q are all numbers in sqrt and pi and powers
scarlet sequoia
Q are all numbers in sqrt and pi and powers
Huh?
worthy radish
scarlet sequoia
Q are all numbers in sqrt and pi and powers
Q is all rational numbers, you're confusing with R, the set of all real numbers
scarlet sequoia
https://i0.wp.com/mathema.me/wp-content/uploads/2023/07/znimok-ekrana-2023-07-26...
Q is "Liczby wymierne"
scarlet sequoia
Why isnât i) true?
To come back to this, not everything that can be written as $\sqrt{something}$ is nothing but a real number
soft zealotBOT
Raphaelisius Maximus MMIII
worthy radish
Yeah so sqrt 9 is Q
scarlet sequoia
https://i0.wp.com/mathema.me/wp-content/uploads/2023/07/znimok-ekrana-2023-07-26...
It's in every "region" here
scarlet sequoia
Yep
Because sqrt9 = 3 which is in Z
Also I see that in your convention
N = {1,2,3,4...}
So 0 is not a natural integer according to the conventions of your book/source
worthy radish
Ok whatever
k) is fake?
It should be {-1, 0} c Z
Oh wait
Itâs /c/
So itâs true
Lmao this notebook considered 0 as a N number
.close
final saddleBOT
final saddleBOT
Available help channel!
fluid rain
I am completely lost on this task, this feels way out of our scope and I need someone who can ELI5 and run me through this please
final saddleBOT
I am completely lost on this task, this feels way out of our scope and I need so...
fluid rain
the numbers after letters are small index numbers or "subscript" if thats what you call it
see for reference the original (in german)
final saddleBOT
@fluid rain Has your question been resolved?
cursive bough
a set of vectors is linearly independent if $k_1 a + k_2 b_2 + k_3 b_3 + k_4 b_4 = 0$ only if all of the $k_i$'s are 0
soft zealotBOT
Doaby
cursive bough
so if you can prove that all or the k_i's are 0, you are done
fluid rain
so if you can prove that all or the k_i's are 0, you are done
this sounds easier said than done (I think?), also you need to ELI5 more, "linearly independent" is something ive not heard before
cursive bough
this sounds easier said than done (I think?), also you need to ELI5 more, "linea...
oh, that is the definition of linearly independent. another way to think about it:
vectors are linearly dependent if you can combine some of them to get another. for example, the vectors [1, 1] and [2, 2] are linearly dependent because you can scale the first one to get the second
and the vectors [1, 1], [2, 3], and [3,4] are linearly dependent since you can add the first two to get the third
so linear dependence means that isn't possible, and that boils down to the equation i wrote
why did they give you that if you haven't heard of linear independence? that's like one of the first things you cover in linear algebra
final saddleBOT
@fluid rain Has your question been resolved?
fluid rain
oh, that is the definition of linearly independent. another way to think about i...
Im still there btw Im just taking this all in rn and doing some looking into previous lectures
terse dagger
You write this down
and then find k_{1, 2, 3, 4}
if they are all 0 you've proven it
remember that all the original b_{1, 2, 3, 4} are linearly independent so if you have an expression with them in it they have 0s as coeficients
final saddleBOT
@fluid rain Has your question been resolved?
fluid rain
remember that all the original b_{1, 2, 3, 4} are linearly independent so if you...
I literally know close to nothing about this topic, could you simplify, also..
how would I go about "finding" k_{1, 2, 3, 4}, idek what finding means, it might be a language barrier
terse dagger
You have $k_1 a + k_2 b_2 + k_3 b_3 + k_4 b_4 = 0$
soft zealotBOT
Katharine
terse dagger
This is an equation that you need to figure out if the set {a, b_2, b_3, b_4} is linearly independent
fluid rain
right
I kinda see what theyre asking, I dont know how to go about it is the issue
if you have the god given talent to explain that to a simpleton, go ahead
I got 3 days for this task so it should be doable incase I need to set some foundation first
fluid rain
alright, you werent up to the task but our beloved artificial friend was
thanks to everyone nonetheless
.close
final saddleBOT
final saddleBOT
Available help channel!
left trail
If the change in kinetic energy is equal to the derative of work why does my simplification not yield that. The issue I see is that the bounds cause there to be extra stuff. But you are finding the work from a start position to an end position so there must be bounds
final saddleBOT
If the change in kinetic energy is equal to the derative of work why does my sim...
left trail
left trail
so you *plug in* t = a and t = b to v, not multiply it
But our integral starts off like this
Where a and b are positions
How could they turn into time?
left trail
position varies with time
Yes but isn't but I have force set up as a function of time?
left trail
and so does velocity
Are you saying that I need to convert the position bounds into time bounds where the time is when it would be at that position?
formal trail
whenever you change the variable of integration in an integral you must change the bounds to match
left trail
I see how that is done with u sub but here there is nothing with x. Only dx
final saddleBOT
@left trail Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
stone path
Test questions:
Confused on the circled questions as i got the positive/ negative switched and for others im not completely sure on what i did but i didnt fully understand them
final saddleBOT
Test questions:
Confused on the circled questions as i got the positive/ negativ...
tired walrus
#3, you have a negative number under the square root
these are not real and you should defo be treating them as nonexistent for now
the others we'd need to see your work to see where the errors happened (incl. sign errors)
there is also the issue of your curly braces having an... interesting shape, and the notation {NS} seems a bit nonstandard too.
stone path
the others we'd need to see your work to see where the errors happened (incl. si...
i figured out where i went wrong with the others, my roommate helped
stone path
there is also the issue of your curly braces having an... _interesting_ shape, a...
just how answered were formatted for test was a blank box with curly brackets on both sides
stone path
#3, you have a negative number under the square root
that makes a lot of sense thank you
tired walrus
fair, it's just that the solution set for an eq that doesn't have any would usually be written either {}, or â (but with no {} around that symbol)
stone path
.close
final saddleBOT
final saddleBOT
Available help channel!
candid pulsar
would the degree of $\sin(\frac{dy}{dx})+x=2y$ be 1 or not defined?
final saddleBOT
would the degree of $\sin(\frac{dy}{dx})+x=2y$ be 1 or not defined?
soft zealotBOT
frosty sigil
would the degree of $\sin(\frac{dy}{dx})+x=2y$ be 1 or not defined?
I believe this eq cannot be written as a polynomial in terms of its derivatives.
It should be not defined
candid pulsar
would $\frac{dy}{dx} = arcsin(2y-x)$ be legal is my actual question
soft zealotBOT
candid pulsar
since some other questions similar to this used the same justification to have an answer for the degree
frosty sigil
Right
final saddleBOT
@candid pulsar Has your question been resolved?
candid pulsar
waiting for 15 min
final saddleBOT
final saddleBOT
Available help channel!
stray notch
final saddleBOT
stray notch
So... i tried to find the range of f(x,y) in 1b. but i got Rf = Real number and when i checked in wolfam for some reason it said that -pi/2 <= Rf <= pi/2.
and im wondering what did i do wrong
my logic is that since arctan(u) basically give you an angle (lets name it v angle) that makes tan(v) = u.
and as far as i know there isnt a restriction on angle thats why i think Rf = real number
glossy zephyr
Do you know how arctan looks?
glossy zephyr
do you know what arctan x is with respect to tan x?
stray notch
i did try plot it on a graphing calaculator tho
stray notch
do you know what arctan x is with respect to tan x?
inverse of tan(x)?
glossy zephyr
more specifically, inverse of the main branch of tan x
Since tan x is just a repeating function
stray notch
by main branch u mean the one that intersect (0,0)?
glossy zephyr
tan(x), defined in its whole domain doesnt have an inverse
since it obviously isnt injective
So we actually constrain the domain to define arctan as its inverse.
stray notch
So we actually constrain the domain to define arctan as its inverse.
i think.. i understand
glossy zephyr
Instead of the full domain, we just limit ourselves to this particular stripe.
Mostly cause all the others look the same.
stray notch
so in a nutshell we need to use the main branch to find the inverse because if we don't arctan won't be a function because 1 x value can gives multiple y value right?
static hedge
Instead of the full domain, we just limit ourselves to this particular stripe.
Wow, you find/make illustrations related to questions really quick
glossy zephyr
Knowing that we use this specific restriction of the domain to define the inverse of tan(x)
You should easily see why the range of arctan is limited
stray notch
Instead of the full domain, we just limit ourselves to this particular stripe.
ohhh so since inverse is swapping x and y the range of arctan is [-pi/2,pi/2] right?
glossy zephyr
really slight consideration:
(a,b), not [a,b]
since tan x itself isnt defined at -pi/2 and pi/2
stray notch
since cos(pi/2) is 0 right?
alr
ye, i think i understand now
dont think i can find the answer to this alone lol
glossy zephyr
If arctan(x) is limited to -pi/2 and pi/2
stray notch
even with u explaining, it took me a while to comprehend
glossy zephyr
then no matter what you input with arctan(y/x), it cant be greater than that
stray notch
mm i see
alr, thx a lot
appreciate it
prob took me a lot longer if not thx to u
.close
tranquil pine
Prove that any convex polygon with vertices n>=3 can be triangulated with (n-2) triangles
final saddleBOT
Prove that any convex polygon with vertices n>=3 can be triangulated with (n-2) ...
onyx peak
Prove that any convex polygon with vertices n>=3 can be triangulated with (n-2) ...
Try induction
tranquil pine
Try induction
Hmm how in this case, i did try
onyx peak
Hmm how in this case, i did try
Well, what would the base case be?
tranquil pine
Well, what would the base case be?
It'd be a triangle
Which is already triangulated with 1 triangle
So the base case is tru
onyx peak
Alright, now imagine I give u some convex polygon, maybe sth like this. If u wanted to triangulate this, what would be the first line you'd draw?
tranquil pine
Alright, now imagine I give u some convex polygon, maybe sth like this. If u wan...
Join two vertices, i guess that have 1 vertex inbetween
onyx peak
Join two vertices, i guess that have 1 vertex inbetween
Alright, sure
now what shapes are left for us to triangulate?
static hedge
Prove that any convex polygon with vertices n>=3 can be triangulated with (n-2) ...
Can you elaborate the question?
onyx peak
Alright, now imagine I give u some convex polygon, maybe sth like this. If u wan...
Suppose this is a general convex n-gon
static hedge
Like triangulate means whaat?
tranquil pine
Suppose this is a general convex n-gon
So the polygon left is n-1 gon
onyx peak
Like triangulate means whaat?
divide it into triangles basically
onyx peak
So the polygon left is n-1 gon
Yep, and that's how u apply the induction
tranquil pine
Yep, and that's how u apply the induction
Oh ohk i get it
static hedge
ok
onyx peak
by induction, that n-1-gon can already be triangulated in (n-1)-2 triangles
tranquil pine
Thanks
onyx peak
There are some details we need to mention btw
tranquil pine
There are some details we need to mention btw
Hmm like what
onyx peak
why do we require the polygon to be convex? What step of this proof would fail if it wasnt convex?
tranquil pine
why do we require the polygon to be convex? What step of this proof would fail i...
Hmm
onyx peak
here is a hint (the black shape is the polygon)
tranquil pine
here is a hint (the black shape is the polygon)
In this case joining that top vertex with the concave vertex does triangulate it thoo
4 vertex, triangulated into two triangles, so this one case does workđ
onyx peak
In non-convex polygons, the diagonal doesn't necessarily lie inside the polygon.
It is actually provable that every non-convex polygon can be triangulated as well, but we would need to prove that we can pick a diagonal which lies inside the polygon
tranquil pine
In non-convex polygons, the diagonal doesn't necessarily lie inside the polygon....
Hmm
onyx peak
In this case joining that top vertex with the concave vertex does triangulate it...
Indeed it does, but our proof doesn't explain it. We would have to explain why we can always pick a diagonal which lies inside the polygon, even in non-convex polygons
tranquil pine
Indeed it does, but our proof doesn't explain it. We would have to explain why w...
Hmm
onyx peak
here is a hint (the black shape is the polygon)
Here u only told the reader to join two vertices with 1 vertex inbetween. The reader, trying this on a non-convex polygon could draw this red diagonal and then claim that the proof doesn't work. You'd have to explain to the reader, which diagonal should he pick (one that lies inside the polygon) and you'd also have to explain why he could always pick one (i.e. prove that every non-convex polygon has diagonal which lies in its interior)
Does that make at least a bit of sense
final saddleBOT
final saddleBOT
Available help channel!
pearl egret
Guys I messed up. Drop year me abhi tak makkari kiya and ab kuch bhi syllabus complete nahi hai. 11th and 12th me guidance nahi mila to jee preparation nahi kiya boards pe focus kiya and 87% mila. Ab jee ke liye prepare karna hai. What do i do? Physics I am trying to cover from Eduniti sir ke 1700 question challenge. Maths me mathematically inclined se karunga. Chemistry zara sa bhi samajh nhi aati. Kuch bhi nahi hua hai chemistry me. Physical and inorganic thoda sa padhne se ho bhi jaega, organic nahi ho raha. I need help and guidance
final saddleBOT
Guys I messed up. Drop year me abhi tak makkari kiya and ab kuch bhi syllabus co...
candid pulsar
hinglish đ¤Š
candid pulsar
Guys I messed up. Drop year me abhi tak makkari kiya and ab kuch bhi syllabus co...
practical chem is v easy though
i think you would understand question pattern for most chapters if you went through ncert and then pyqs
final saddleBOT
@pearl egret Has your question been resolved?
carmine plaza
i dont understand how they did the end
final saddleBOT
i dont understand how they did the end
carmine plaza
how did they figure out that it was f(0.01)
trail mango
1 - 0.99 = 0.01
carmine plaza
yeah but why did they do 1-.99
trail mango
look at fâs definition, and look at the value they ask you to estimate
if you want them to match, what should x be?
pliant shore
yeah but why did they do 1-.99
they didn't do 1 - 0.99
carmine plaza
oh i see
1+x would make the bottom 1.01
not sure i undrestand what this one is asking
carmine plaza
derivative?
pliant shore
yep!
and when you take d/dx on both sides
you need the chain rule, so for example d/dx (3y) = 3 dy/dx
carmine plaza
but how do you find the point that has a slop equal to two
pliant shore
but how do you find the point that has a slop equal to two
find dy/dx first and then we'll talk
you'll get an equation when you set dy/dx = 2
pliant shore
howd they get 8.5 here?
do you understand v(t) = 3/2 t^2 - 7t + C?
carmine plaza
yes
carmine plaza
oh they set it equal to 3
pliant shore
yeah
carmine plaza
wait how did they get that from the graph
pliant shore
the line through (0, -7) and (3, 2)
carmine plaza
oh it says it in the question
pliant shore
the line through (0, -7) and (3, 2)
you can figure it out from this info
the person used (0, -7) and (1, -4) which is easier
carmine plaza
the slope of the line =3 but why does that mean v(1) = 3?
pliant shore
the slope of the line =3 but why does that mean v(1) = 3?
hold on
carmine plaza
they use the same thing here, does it just work for all the intervals?
pliant shore
the slope of the line =3 but why does that mean v(1) = 3?
v(1) can be anything
because of the +C when you integrate acceleration
carmine plaza
yeah
carmine plaza
yeah but it applies to all v(t) im not sure why
pliant shore
yeah but it applies to all v(t) im not sure why
what applies to all v(t)?
pliant shore
yeah but it applies to all v(t) im not sure why
that doesn't apply to all v(t)
carmine plaza
they use the same thing here, does it just work for all the intervals?
whyd they apply it in this part then
pliant shore
they use the same thing here, does it just work for all the intervals?
what's the full question of this?
what are parts a and b
pliant shore
oh right, yeah a(t) really is 2 from t = 3 to t = 6
then you can just do v(3) = 2 * 3 + A = 1, so A = -5
v(3) = 1.5 * 3^2 - 7 * 3 + 8.5 is incorrect
the velocity function changes so it's not the same one as for 0 < t < 3
carmine plaza
v(3) = 1.5 * 3^2 - 7 * 3 + 8.5 is incorrect
well no they just put that there to show what they did in part a
and apply it to part b
that being v(3)=1
carmine plaza
then you can just do v(3) = 2 * 3 + A = 1, so A = -5
im not sure why you can do this
pliant shore
im not sure why you can do this
it's the same logic as in part a
pliant shore
i mean the slope is zero
yes, a(t) = 2
if you integrate both sides what do you get?
carmine plaza
v(t) = 2t
carmine plaza
yep
pliant shore
v(t) = 2t
also it's v(t) = 2t + c
you always need a constant of integration
2t + 5 has derivative 2
so does 2t - 67
so does 2t + 69420
pliant shore
oh right
carmine plaza
yeah..
carmine plaza
well they just brought over from part a for some reason
pliant shore
they're using the velocity at t = 3 because that's where the function from part a still applies
exactly
yeah so it works
carmine plaza
i see
pliant shore
yes then you get v(3) = 1 omg
carmine plaza
i understand this
idk the corollary
ik rolles theorem is just case of mvt
when mvt = 0 like f(b)-f(a)=0
pliant shore
when mvt = 0 like f(b)-f(a)=0
I have to go sorry
carmine plaza
ok
pliant shore
if you want, you can close and reopen this channel so that it moves to the top
pliant shore
no worries!
carmine plaza
part b i dont understand
faint locust
Did you have 'the corollary' before this?
@carmine plaza
Something like "If f is differentiable on (a, b) and f' has no roots, then f has at most one root"? Have you had that as a corollary somewhere?
carmine plaza
we've learned about it at some point
faint locust
Ok yeah that's good then, essentially you just use that
g''(x) has no roots since it's > 0
so g' has at most one root
carmine plaza
but doesnt g'''(x) also >0
faint locust
Yes, sure, but why does that matter?
carmine plaza
why does g'' have no roots
faint locust
The idea is that g^(n) has no roots => g^(n - 1) has at most 1 root => g^(n - 2) has at most 2 roots => ...
faint locust
why does g'' have no roots
Because g''(x) = 2 + 1/25 e^(2x) and you know that 2 is positive and e^(2x) is positive
carmine plaza
ok do you stop only when you know for sure that its positve
that all the terms are positive
like why dont you stop at g'(x) did you test it in your head that i would be negative at some point
faint locust
Something like "If f is differentiable on (a, b) and f' has no roots, then f has...
To make this argument? Yes, you want either strictly positive or either strictly negative
faint locust
like why dont you stop at g'(x) did you test it in your head that i would be neg...
Yep. You can quickly check that for x = 0, it's negative but for x -> infinity it goes to +infinity so somewhere there is a zero
faint locust
The idea is that g^(n) has no roots => g^(n - 1) has at most 1 root => g^(n - 2)...
Anyways after that, it's just applying this corollary
That you have somewhere in your notes, probably
carmine plaza
this is aids
my exam is gonna be very time constrained
is the only way going to be to use two subs here
i guess it wouldnt take that long just seems a little concerning
faint locust
Yeah, I'd u-sub, then that sin and cos should cancel out and it gets nicer too
faint locust
i guess it wouldnt take that long just seems a little concerning
It's surprising how much simpler it gets after the u-sub so yeah, just start
carmine plaza
Do you have any advice for related rates problems
faint locust
Do you have any advice for related rates problems
Someone asked the same a while ago and I found this page some prof wrote up:
hushed bluff
Please donât let this be on the test
carmine plaza
shoot my exam is pretty soon i dont think i have time to read it
hushed bluff
carmine plaza
Please donât let this be on the test<:blankstare:1438715334390583406>
it always is man
faint locust
Please donât let this be on the test<:blankstare:1438715334390583406>
It simplifies a lot after the u-sub
carmine plaza
yeah it's not bad
would it be bad for me to ask you to summarize that page
im pretty much looking at that and riemman sums before i head out
faint locust
would it be bad for me to ask you to summarize that page
I mean, I think this part on the page is the main strategy
Everything else is just examples
carmine plaza
Oh ok cool
So there's right riemman sum and left riemman sum I'm not sure what the difference is in writing like I know how it looks on a graph
Midpoint is easier because it's just always the same
faint locust
So there's right riemman sum and left riemman sum I'm not sure what the differen...
You should have a precise definition somewhere. Basically you partition your big interval [a, b] into subintervals [x_k, x_{k + 1}]. Now you sum up over the rectangles, i.e. length * height = (x_{k + 1} - x_k) * f(something), where something in case of a left Riemann integral is taken as x_k and for right as x_{k + 1}
carmine plaza
for part 2 why is the second term 0 or that rectangle
faint locust
for part 2 why is the second term 0 or that rectangle
Because the partition of [1, 9] is into [1, 3], [3, 5], [5, 7] and [7, 9]. For an underestimate you pick the function value that is the smallest in that subinterval, for x from [1, 3] that is y = 5, for x from [3, 5] that is y = 0 and so on
carmine plaza
its an over estimate though
faint locust
(ii) says underestimate
carmine plaza
oh
right yeha
well i just dont understand why that one is 0
not rectangle is drawn at all
if it was a tiny bit bigger the area under there could you have a rectangle of height 1
hushed bluff
It simplifies *a lot* after the u-sub
the anwer is -2e^(e^1-sin^2(x))|0 -> pi/2 right?
faint locust
well i just dont understand why that one is 0
Because you partition [1, 9] like this and for each subinterval (parts between two red lines) you look at the smallest value the function takes on
And draw the rectangle from there
carmine plaza
ah i get it
faint locust
And between x = 3 and x = 5, what's the smallest function value?
It's 0
So you draw a rectangle with height 0 and length from x = 3 to 5 (so 2)
But its area is 0 * 2 = 0
Same principle as for the other subintervals.
Between x = 1 and x = 3, the smallest value of the function is 5.
carmine plaza
thanks for your help!
faint locust
np
carmine plaza
.close
final saddleBOT
faint locust
the anwer is -2e^(e^1-sin^2(x))|0 -> pi/2 right?
I haven't checked, you can put this into WA to see if that's correct
final saddleBOT
Available help channel!
brazen stump
hello what step did i do wrong im not sure
final saddleBOT
hello what step did i do wrong im not sure
tired walrus
looks like the very end is the problem
hybrid adder
hello what step did i do wrong im not sure
you didn't divide 10 on both sides
tired walrus
10m = p means m : p = 1 : 10
not 10 : 1
m is smaller than p after all
also your 5 looks too much like S. be mindful of that handwriting-wise.
brazen stump
you didn't divide 10 on both sides
then it would be m=p/10
brazen stump
10m = p means m : p = **1 : 10**
i tho we need 10ms to get 1p so the ratio is 10:1
tired walrus
you have it backwards
it takes 10 m's to get one p, that means m is smaller than p
so how can the ratio m:p be 10:1 with m matched to the big number 10
brazen stump
then you'd divide both sides by p to get m/p = 1/10
whats next
tired walrus
10m = p means m : p = **1 : 10**
.
brazen stump
so how can the ratio m:p be 10:1 with m matched to the big number 10
ohh thats a good way to think about it
hybrid adder
ratios are basically the same as fractions in a mathematical sense
brazen stump
kis bb whats ur method
brazen stump
ratios are basically the same as fractions in a mathematical sense
so what do i do as thenext step put the fraction into ratio?
hybrid adder
so to solve m:p, it's basically asking you to find out what m/p as a simplified fraction is
hybrid adder
so what do i do as thenext step put the fraction into ratio?
yes
m/p would be the same as m : p
brazen stump
m/p would be the same as m : p
wait i dont get it m=p/10
so m=p:10
is that waht you want me to do
brazen stump
yh
hybrid adder
so if you divide both sides by 'p' next, what do you get?
brazen stump
?
brazen stump
is just gives m=p/10
brazen stump
10
hybrid adder
it's 1/10
brazen stump
p/p=1?
brazen stump
alr ok
brazen stump
wait
brazen stump
which is the same as m:p = 1:10 in this context
dont u need to corss multiply
hybrid adder
what's the question asking you to do?
brazen stump
m:p
brazen stump
.close
final saddleBOT
final saddleBOT
Available help channel!
plush mason
help with this i don't know how to start
final saddleBOT
help with this i don't know how to start
rugged merlin
help with this i don't know how to start
What i like to do is to convert all of those to explicit functions without piecewise definitions
Using the heaviside and impulse functions
u[n] and delta[n]
Can you do that
plush mason
u[n] and delta[n]
i think u can use only one to convert
plush mason
What'd that be
h = u(n + 2) - u(n - 2)
and x
$x[n] = \frac{1}{3} \times n \times (u(n)-u(n-6))$
soft zealotBOT
Tee
plush mason
$h[n] = u(n + 2) - u(n - 2)$
soft zealotBOT
Tee
final saddleBOT
@plush mason Has your question been resolved?
tranquil pine
i have a question regarding linear algebra
final saddleBOT
i have a question regarding linear algebra
tranquil pine
i wanna understand the intuition behind linear transformations
are they a form of linear combination?
because you can write a linear transformation as Ax = b
scarlet sequoia
Ax = b is more of an equation to solve than a linear transformation
glossy zephyr
Linear Transformation are essentially a special kind of function,
It connects two vectors spaces, and it has to satisfy 2 essential conditions
scarlet sequoia
A linear combination of vectors u,v,... is any sum of scaled versions of u,v,...
So for example 2u -1/4v is a linear combination of u and v
A linear transformation is such that it works well with linear combinations
glossy zephyr
Addition: T(u+v) = T(u) + T(v)
And Scalar Multiplication: T(kv) = kT(v)
tranquil pine
I understand linear combinations but in hopes of understanding linear transformations o thought some things would be similar
I*
scarlet sequoia
I understand linear combinations but in hopes of understanding linear transforma...
If you want the image of a linear combination, like T(2u+3v)
It's the same as computing the linear combination of the images
So T(2u+3v) = 2T(u) + 3T(v)
So if you want to find the image of any vector
Just find the image of each "coordinate"
We love to work with linear transformations because you need so little to actually encode one
scarlet sequoia
Im tryna connect Ax=b to that as well
You have a linear transformation T(x) = Ax
tranquil pine
and what the solution to those planes lines represent the image
scarlet sequoia
And you want to know when you attain a specific value
It's like trying to find the range and inverse images
glossy zephyr
Your prob have seen how a matrix multiplication over an arbitrary set of coordinates (x,y) yields something like a "distorted" plane, right?
tranquil pine
Your prob have seen how a matrix multiplication over an arbitrary set of coordin...
idk about distorted but i know it yields planes
or some subsapce
Subspace(s)
tranquil pine
whatâs happening here
glossy zephyr
this is one example of the (usually) most basic sort of linear transformations
If you just focus in the blue square and red parallelogram
You can see that the i,j vectors get transformed into new i' and j' vectors.
If you can just define how those two translate into eachother, you have basically described how the whole plane gets transformed
If you give me a quick second i can actually pull off a demo of the concept in desmos for you to play around
drowsy epoch
my brain: covariant and contravariant basis
tranquil pine
If you give me a quick second i can actually pull off a demo of the concept in d...
alright thanks bro
so the point of linear combinations is to transform planes?
like the whole coordinate plane
like whats the intuitive graphical goal
drowsy epoch
i wouldnt reduce the concept of linear combinations to just "transforming planes"
but it is a good intuition to understand that concept geometrically
tranquil pine
i wouldnt reduce the concept of linear combinations to just "transforming planes...
wait no sorry i meant transformations
not combinations
glossy zephyr
alright thanks bro
mb took a solid second
This is actually just a really basic introduction to what a transformation is, tbh
but the intuition of it sort of remains
Just as a quick example, good old derivatives count as linear transformation, where the vectors are actually real valued functions themselves.
final saddleBOT
@tranquil pine Has your question been resolved?
wheat vale
im stuck on these
final saddleBOT
im stuck on these
wheat vale
I thought i could assume K to start, KâK 3,4 Cp?
strange iron
hey
wheat vale
hello
final saddleBOT
@wheat vale Has your question been resolved?
wheat vale
Does it refer to the subsets?
no
icy tulip
ohh
final saddleBOT
@wheat vale Has your question been resolved?
storm haven
Not sure but, isnt K implies K always true?
With that, R is then true
And so is R V M
and then the problem is solved?
wheat vale
i got this but i need another K for KâK CP
wheat vale
<@&286206848099549185>
wheat vale
.close
final saddleBOT
final saddleBOT
Available help channel!
dapper heath
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and I'm really poor in math like I'm bad at all chapters and I have my board exams (one of my most important exams in my life) coming up starting on Feb 9'th and i just finished my Pre Boards and I got 43/80 which is really bad especially for someone like me who aims for 95% above is there someone who would be willing to start teaching me all of this from scratch?
final saddleBOT
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
worldly spruce
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
Was this just math? On the exam?
dapper heath
yeah
worldly spruce
I see
Well, unfortunately I highly doubt anyone will have the time to teach you everything, as I am assuming everything took your professor tens if not hundreds of hours
But, most likely people will recommend very good sources where to learn from
dapper heath
ummm im still in high school so its simple things like intro to trigonometry and stufff
and quadratic equations statistics etc.
worldly spruce
I can't because I am not that familiar with them in English
I see
Yeah just wait a bit
Someone will tell you
tranquil pine
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
Go to local tuition.
Or use online resources.
dapper heath
Someone will tell you
yup i hope someone will but thank you for your time
tranquil pine
Like PW or Next toppers or Magnet Brains.
tranquil pine
i need a lot of practice
Ofc.
dapper heath
anyway i gtg cause llemme go practice maths and chemistry
shell condor
Ahhh another 10th grader, don't worry dude, just solve enough questions and you should be good enough
Even I was scoring 48ish in my Half-yearly's, ended up scoring 96
dusty helm
the boards are really easy bro u dont have to worry
just solve lots of questions
if you study enough the paper doesnt even take 2 hours to finish
work hard and u got this
shell condor
Yeah I mean the 10th guys are too worried, something to do with their first serious exam, but it's nothing like that, dude when you give 12th you'd be so careless
dusty helm
yeah
shell condor
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
BTW how are your other subjects?
final saddleBOT
@dapper heath Has your question been resolved?
icy tulip
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
ok
Iâm in 9th but i can help
boards is basically like school final exams
just donât overthink it
shell condor
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
If you want you can add me, I've got no issues with helping personally here
shell condor
boards is basically like school final exams
I mean there is that psychological factor, you're in a new place in a different environment with all that hype developed, it's less about ignoring that and more about rolling with it, like be confident at your approach
icy tulip
Yeh
and boards actually have more significance than anyone other exam written before so yeah there would be some more stress
icy tulip
hello all I am a student who studies in a CBSE curriculum I'm in 10'th grade and...
you can add me if you want to
plush sparrow
Hi guys need help with an indction proof
final saddleBOT
Hi guys need help with an indction proof
plush sparrow
I really dont understand the last party
I thought the estimation would be smth like this
11/7 < 7/4
plush sparrow
and int he inductions start he randomly started with n=2
i mean n=1 and n=2
so he showed it for all n>2
cann i just start at 1 and show it for all n>1?
final saddleBOT
@plush sparrow Has your question been resolved?
turbid ravine
How large is the clockwise torque if A is the pivot point and F = 100 N?
turbid ravine
I need help understanding how to solve this. I am quite bad at math so explain it like im 10 xD
storm haven
What have you tried?
tired walrus
!status
final saddleBOT
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
okay, so status 1.
gloomy kelp
wait, this is high school level??
storm haven
wait, this is high school level??
Yes
gloomy kelp
hm maybe i should know how to do this then
tired walrus
clearly pre-simplification the denominator is 5!
tired walrus
a favorable outcome is a sequence that goes either down->up or up->down
or it's 12345 or 54321
if it's not one of those then the up->down transition has to occur wherever the 5 is
ditto for down->up and 1
do you understand why this is
ok
gloomy kelp
how do i do spoiler
tired walrus
can you count the up->down sequences of the form *5***
tired walrus
how do i do spoiler
|| like this ||
gloomy kelp
||is it 10 acceptable strings, and 5! combinations||
wait
that was dumb
yes
dnt click
but its wrong anyways
tired walrus
are you going to try or have you just given up immediately
...
A) "i did not understand what you want me to count"
B) "i did not understand your question just now"
C) "something else entirely"
ok
let us try this again and overcome this difficulty in communication
how many ways exist to place the digits 1, 2, 3, 4 (once each) into the asterisks in *5***, so that the part up to & including the 5 is increasing, and the part after the 5 is decreasing
great
so you CAN count these. why say no before.
alright, now do the same for **5**
no
there is also ***5* and then also all these 3 same cases but with 1 instead of 5
and then those two specific ones that dont fit into any of these: 12345 and 54321
final saddleBOT