f''(x) ≤ 0 for all x?
#help-23
finite yacht
wind laurel
i guess yea
i say i guess cause i never got a formal introduction to concavity, or really any of this so i'm not sure if im using the wrong definitions
finite yacht
There is one related to the line between any two points on the graph and the graph of f being above that line
finite yacht
i say i guess cause i never got a formal introduction to concavity, or really an...
But then why do you have to prove that?
wind laurel
But then why do you have to prove that?
Its not an exercise given by my teacher or anything
just something im doing in my free time
to learn latex and proof writing
we barely even started trigonometry at school ahahah
I know about the defintion for convex/concave angles
finite yacht
There is one related to the line between any two points on the graph and the gra...
Then you should have a look at this definition before, I believe you have to use it to prove this
finite yacht
we barely even started trigonometry at school ahahah
Oh
wind laurel
in the sense that any 2 points in a convex angle can be connected by a line which is fully inside the angle
finite yacht
wind laurel
concave being the contrapositive
finite yacht
in the sense that any 2 points in a convex angle can be connected by a line whic...
Well in the case of functions it's related to this
A function is convex when the area over the graph of f is convex with respect to that definition
wind laurel
so basically:
convex: the segmented created by connecting two points on a curve is above that curve between those 2 points
concave: ... below ...
finite yacht
in the sense that any 2 points in a convex angle can be connected by a line whic...
Because of course this definition does not require the region to be an angle
finite yacht
so basically:
convex: the segmented created by connecting two points on a curve ...
Yes
wind laurel
would a straight line be both concave and convex or neither?
hearty egret
A straight line is convex
but not strictly
wind laurel
wait so even if a function is not C^2 i can still say something about its concavity?
wind laurel
ohh yea
finite yacht
This definition is a lot more convenient and less artificial than the second derivative being negative
wind laurel
So just to make sure, what changes do i need to make to the proposition?
for any -> for all
finite yacht
So just to make sure, what changes do i need to make to the proposition?
for any...
I would only change that
wind laurel
Okay!
And does my original approach where i basically show that: for any convex f there exists a "less" convex g s.t. g > f is false so f must be concave
still work
Intuitively it does but rigurously should i preffer moving to something closer to the defintion of a concave/convex function and maybe find a contradiction there
finite yacht
Intuitively it does but rigurously should i preffer moving to something closer t...
I would do this too
I can't think of a way so that the other proof technique works, which doesn't mean that it doesn't, but it looks harder and less probable
wind laurel
something like:
since g > f
any segment connected by 2 points on g must never intersect f
if both f and g are arbitrarily convex this isnt necessarily true (here is where i add the intuitive argument)
g is convex => f is concave
finite yacht
if both f and g are arbitrarily convex this isn't necessarily true
Why not?
And what does arbitrarily convex mean
Or concave
Also I think you need more than one g(x)
And more than one t
Because of the counterexample
wind laurel
And what does arbitrarily convex mean
By arbitrarily i mean that i can chose its curvature to be as big or small as i desire (aslong as its positive in this case).
as stated in the proposition: g''(t) <= epsilon for any epsilon > 0
finite yacht
something like:
since g > f
any segment connected by 2 points on g must never i...
But the two first lines are true nevertheless
wind laurel
Because of the counterexample
oh yea
wind laurel
But the two first lines are true nevertheless
Yea i just noticed that aswell
i mean they are logically equivalent so its unneccesary to add the step
finite yacht
@wind laurel I think I came up with a rather convoluted proof which involves the concept of compactness of a set
Do you know about that?
wind laurel
Perfect timing!!
im reading a book right now in measure theory ahahah
compactness meaning a set is closed and bounded right?
finite yacht
Well
Not exactly, but yes
In the real numbers, it's equivalent
But this is like convexity, there is a more general definition for other contexts
In this case it is not so intuitive unfortunately
finite yacht
Yes, but it can be something different, not just a cube
wind laurel
or in general a covering i guess
finite yacht
A sphere for example
https://en.m.wikipedia.org/wiki/Sierpiński_carpet or this thing
wind laurel
A sphere for example
In that case it would be the ball no? but i guess we could always approximate the sphere with its covering
finite yacht
In that case it would be the ball no? but i guess we could always approximate th...
Both the whole closed ball and its sphere (or border) are compact
wind laurel
okay! I just finished the section on exterior measure and now its starting to talk about lesbegue measures
anyways, how does compactness help with the proof?
finite yacht
okay! I just finished the section on exterior measure and now its starting to ta...
I really liked that theory when I studied it, it's really interesting
finite yacht
anyways, how does compactness help with the proof?
Well
Assume f is not convex
Then there is an interval, [c,d], where the definition fails
So there is a point z\in[c,d] such that f(z) is under the line
From now on, it's not a formal proof but a sketch, or maybe just an idea
For each t\in[c,d], find some g
That gives you infinitely many open intervals (the domains of the g) who cover the compact set [c,d]
By topology, you can find a finite subset of those open intervals which still cover [c,d]
finite yacht
And then you try to go from the borders to z in a finite number of steps to reach a contradiction (?)
finite yacht
any covering has a finite subcovering
Yes!
finite yacht
And then you try to go from the borders to z in a finite number of steps to reac...
I don't know, it's more an idea than anything else, it made more sense in my head
wind laurel
nvm i missread
wind laurel
And then you try to go from the borders to z in a finite number of steps to reac...
what would be the contradiction?
that z cant be below the line
finite yacht
To show that f(z) must be above the line
Maybe by taking limit \varepsilon\to 0
But that changes the number of intervals in the finite subcover, which is bad
wind laurel
how about
finite yacht
With a proper process of refinement this must work
wind laurel
And then you try to go from the borders to z in a finite number of steps to reac...
wait sorry i still dont understand this
what do you mean by going from the borders to z
finite yacht
wait sorry i still dont understand this
So you have a chain of open intervals
And c is in the first and z in the last
wind laurel
So you have a chain of open intervals
each subsets of eachother and all containing z
wind laurel
ohh okay
finite yacht
And in each of those intervals you have a g
I have a better idea
If f is concave in [s_1,s_2] and in [s_2,s_3] then it is concave in [s_1,_s_3]
Use that to find a binary sequence of chained subintervals of [c,d] where f is not concave
Now you have a point and a sequence of intervals which converges to it
This sounds better
wind laurel
So from the beggining and starting from the proposition:
- Assume for contradiction f isnt concave
- Then there is an interval [c,d] where the defintion fails
- So theres a point z in [c,d] s.t. f(z) is underneath the line
- For each t in [c,d] theres a g defined in its open neighborhood (by the proposition)
- The union of these infinitely many open intervals will be a covering of the compact set [c,d]
- By topology there exists a finite subset that still covers [c,d]
wind laurel
If f is concave in [s_1,s_2] and in [s_2,s_3] then it is concave in [s_1,_s_3]
convex* i assume
wind laurel
if you could that would be great cause im not sure i understood
finite yacht
So you have [c,d]
f is NOT concave there
Let y = (c+d)/2
Then f can't be concave in both [c,y] and [y,d]
So choose the interval where it isn't
Iterate
wind laurel
like f isnt concave in both c,y and y,d
like it isnt in both rather than in one yes and one no
finite yacht
f might be concave in one of them
wind laurel
ohhhh yea sorry i was stuck with this notion of not concave = convex but you can also be neither
finite yacht
Yes
wind laurel
what does constructing that sequence bring?
finite yacht
A point
The intersection
But now you have an infinity of intervals where a g could be impossible
wind laurel
could we maybe try an example with sin(x)?
finite yacht
Well
Once you choose [y,d] = [pi, 2 pi] as the second interval it becomes trivial
As it is not concave in any subset
Maybe |x| is better
wind laurel
okay
finite yacht
Oh wait a second
wind laurel
yea no i guess its the same as sin(x)
finite yacht
If f is concave in [s_1,s_2] and in [s_2,s_3] then it is concave in [s_1,_s_3]
This is just the dumbest thing I've said in a while
It's totally false!
finite yacht
|x| is a counterexample
wind laurel
i guess so is this
finite yacht
Yes
wind laurel
could we maybe try the measure theory solution
i think thats very elegant and probably good practice
since thats well what im studying rn ahaha
like self-studying
finite yacht
could we maybe try the measure theory solution
Sure
wind laurel
tbh
someone else in this server asked
lol
and i answered with my intuitive approach
and it was enough for them
but im trying to reach a more rigurous answer
finite yacht
Alright then we have no clue about the difficulty of this question
finite yacht
It might be a hard contest problem
finite yacht
i mean it shouldnt be that hard though no?
It can
wind laurel
i guess we just need to translate the intuitive approach into something rigurous
wind laurel
It can
Does it happen that something has a very simple intuitive answer but it becomes really hard when you try to prove it?
wait
i guess 1+1=2 lol
finite yacht
i guess we just need to translate the intuitive approach into something rigurous
Well, once I was in a differential equations lecture, and the lecturer was continuously assuming a very simple geometric fact about derivatives. It seemed obvious, but I wanted to actually prove it rigorously
It took me a week
wind laurel
Well, once I was in a differential equations lecture, and the lecturer was conti...
oh...
finite yacht
Does it happen that something has a very simple intuitive answer but it becomes ...
It happens most in number theory, second in combinatorics and third in elementary calculus, I believe
wind laurel
It might be a hard contest problem
i dont think so, i talked quite a bit with the person that asked the question and answered also some other questions which seemed not too experty
finite yacht
There are no extremely hard problems in calculus but they can indeed be convoluted
finite yacht
i dont think so, i talked quite a bit with the person that asked the question an...
Alright, most problems are doable anyway, so it's worth trying
wind laurel
should we go with a whole new approach, the measure theory one, or a translation of my intuitive proof into something rigurous
finite yacht
I don't really know hahahah
wind laurel
Should i maybe reexplain the approach i took
finite yacht
No, I think I understood it
But it seems to fail with functions like the Weierstraß' one, doesn't it?
wind laurel
the problem with weierstrass was that a g doesnt exist for every t and every epsilon
if im not mistaken
finite yacht
Yes, and your argument tries to show that f is concave locally with just one t
wind laurel
hm right
we could try expanding the idea for multiple t's
in the case of a convexity, let t to be the local minimum around that convex part of the function and show that for a small enough epsilon g < f
basically how here you said to set t to be the local minimum next to t
finite yacht
in the case of a convexity, let t to be the local minimum around that convex par...
You mean a non-concavity set (?)
wind laurel
sorry maybe adding a comma would make it clearer
finite yacht
Yes but
wind laurel
edited it
finite yacht
Not concave =/=> convex
wind laurel
hm yea
wait but if something is not concave and not convex is it necessarily a case line sin(x) where at some intervals its concave and at others its convex or can it be not convex and not concave in every interval in the domain
finite yacht
It can be not concave and not convex ar each interval
wind laurel
just looked online and... wierstrass lol
finite yacht
Yes
wind laurel
So from the beggining and starting from the proposition:
1) Assume for contradic...
Could we maybe try and look at the measure theory part a bit more
finite yacht
And c is in the first and z in the last
Maybe reverse it?
You start at z
It's easier if we write f(c)=f(d)=0
We can do that without loss of generalisation
wind laurel
- We go from z to borders through a chain of open intervals:
- (z, z1), (z, z2), ... (z, d)
is this what you mean?
wind laurel
ohhh yea
finite yacht
(r0,r1), (r1,r2), ..., (r_{n-1},rn)
And z\in(r0,r1), d\in the last one
Oh but wait
The intervals must intersect themselves
(a0,b0),...,(an,bn)
But I don't see it
finite yacht
would these be the sets we are creating?
Yes, but in the line
Maybe with ordinals there is a proof
But that's just too convoluted
wind laurel
But that's just too convoluted
yea most likely
wind laurel
okay sorry i have to leave this as im going to dinner, do you think its best we continue talking here or moving to dms?
finite yacht
okay sorry i have to leave this as im going to dinner, do you think its best we ...
Maybe moving to dm will be better
wind laurel
Okk! Byee ttyl
finite yacht
Bye!
safe radishBOT
@wind laurel Has your question been resolved?
safe radishBOT
@wind laurel Has your question been resolved?
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@wind laurel Has your question been resolved?
wind laurel
I'll close the channel as to leave it open for others, If anyone is interested in helping out feel free to dm me
crisp mural
A and B are playing a game in which whoever gets to 11 first, wins. They can choose upto 3 consecutive numbers and they take alternating turns. How many games can be played?
safe radishBOT
A and B are playing a game in which whoever gets to 11 first, wins. They can cho...
brave wolf
A and B are playing a game in which whoever gets to 11 first, wins. They can cho...
I don't understand the game
can you give an example?
or even better, send the OG question?
crisp mural
for example a game that got finished within 4 moves would be something like:
A=(1,2,3)
B=(4,5,6)
A(7,8,9)
B=(10,11)
end
crisp mural
or even better, send the OG question?
uhhhh i made this question :D
brave wolf
for example a game that got finished within 4 moves would be something like:
A=(...
once some numbers are chosen, they can't be chosen again?
and there are numbers from 1 to 11?
crisp mural
yeah, whoever starts first starts at 1, can take upto 3 CONSECUTIVE numbers at a time and they can only be chosen once for that specific game
desert swallow
This seems equivalent to A and B getting to add numbers 1, 2 or 3 to an accumulator until 11
brave wolf
I see
crisp mural
but instead here i'm only considering 11
crisp mural
This seems equivalent to A and B getting to add numbers 1, 2 or 3 to an accumula...
yeahhh and my approach was similar
crisp mural
Here's my approach;
Minimum moves a game may have is 4 (3,3,3,2)
maximum is 11(1,1,1,...(11 times))
so, to generalise, the equation i solve for is:
a1+a2+...+an=11 for n E {4,11} and i only consider whole number solutions
This helps me find number of possible games per 'n'
and I add up the number of solutions i get for each 'n' for all n from 4,11 including both
for example
brave wolf
Here's my approach;
Minimum moves a game may have is 4 (3,3,3,2)
maximum is 11...
Not really
each of the a's must be between 1 and 3
crisp mural
well yes
brave wolf
and that's the condition that kinda sucks
crisp mural
what's wrong with my method tho
crisp mural
say, the game were to have 5 turns, then two of the possible solutions will be
crisp mural
you need to add the condition 1 <= ai <= 3
but the condition is being encompassed in my solution?
that's why i chose n=4,11
brave wolf
that's why i chose n=4,11
n = 4:
(8, 1, 1, 1)
brave wolf
anyway
you get at least n = 11, n = 10 and n = 9 I think
which are 10 choose 10, 10 choose 9 and 10 choose 8
there could be some recursive formula dependent on n maybe
crisp mural
hmmm
brave wolf
how?
crisp mural
and i have to repeat this process for all possible number of turns
all the way from 4,11; right?
this is basically an example for a game which had 4 turns
brave wolf
what exactly is this process?
I see
so this solution is essentially systematically listing out all games?
crisp mural
you know the multinomial theorem; The sum of powers for every single general term is equal to the original power (in this case 11) and i'm only limiting it to the cases where ti is between and including 1,3
crisp mural
so this solution is essentially systematically listing out all games?
yeahhh
brave wolf
you know the multinomial theorem; The sum of powers for every single general ter...
that sounds really exhausting
but yes, that will work
brave wolf
it's essentially just listing out everything (unordered) and applying combinatorics to count number of permutations
crisp mural
yessss
brave wolf
is there some other, possibly shorter way
i didnt find one
crisp mural
actually im trying to code a game where I play 21 dares with a computer, but the computer tries to win against me by looking at the best possible way to make me reach 11
and well that can only be done if it knows all possible ways so yeah~
brave wolf
Can't you use a computer to solve this problem then?
this is 3-minutes python program
to do the exhausting work for you
crisp mural
:o
brave wolf
for each n, generate all n-length sequences (with elements from 1 to 3) and check if they work
desert swallow
actually im trying to code a game where I play 21 dares with a computer, but the...
I don't think you need all possible ways actually
brave wolf
very inefficient, while efficient enough
crisp mural
I see
desert swallow
I don't think you need all possible ways actually
I solved a problem once (though it was a subtraction game rather than an addition one)
crisp mural
but then, how else will the computer find the most efficient way to beat me without knowing all the outcomes-
desert swallow
IIRC, there exists a game state the computer can lock the player into
brave wolf
this looks more like something that should be attacked through minimax
or something of that sort
desert swallow
IIRC, there exists a game state the computer can lock the player into
Gimme a sec. It's a been a while since I've looked at this code
crisp mural
alright alright
brave wolf
actually, if someone is forced to say 21, they must have come from 20. So whoevever gets to 20 wins
and you can get to 20 iff you start at 17, 18 or 19
so those are the losing positions
crisp mural
if the computer gets to 11, it can just keep saying only 1 number every turn
desert swallow
Yeah, that's the idea
crisp mural
and then it's trivial to figure out how it can make me reach 21 and win
brave wolf
if the computer gets to 11, it can just keep saying only 1 number every turn
you can exactly determine what numbers are winning and losing
the winning numbers should be 4n I think
so 4, 8, 12, 16, 20
so your goal is to say the winning number everytime
if you do that you win
let's play a game, I wanna see if it works
you start
desert swallow
1,2
brave wolf
3, 4
desert swallow
5,6,7
brave wolf
8
desert swallow
9,10
brave wolf
11, 12
desert swallow
13,14,15
brave wolf
16
desert swallow
17,18
crisp mural
damn
desert swallow
21
brave wolf
damn
similar games can be solved by looking at what positions are winning and what are losing
if you say 21, you lose
so 21 is losing position
if your opponent can get you to losing position, they win
brave wolf
meaning that winning position is 20 and losing positions are 17, 18 and 19
crisp mural
wish i knew that in middle school
brave wolf
meaning that winning position is 20 and losing positions are 17, 18 and 19
wait this is stupid
desert swallow
One thing I remember from when I implemented this. If the player starts first and the compuuter plays this strategy, the computer wins
crisp mural
yeah same thing happened w me💀
brave wolf
Oh, I see.
21 is losing. 20 is winning, because your opponent can only get to a losing position - 21
crisp mural
wait what is the fallacy u found in ur strat?
brave wolf
Oh, I see.
21 is losing. 20 is winning, because your opponent can only get to a ...
17, 18, 19 are losing, because they allow the opponent to get to a winning position 20
16 is winning, because your opponent can only get to 17, 18 or 19 which are losing
and by induction (or pattern recognition), 4n is winning
crisp mural
so all we gotta do is somehow get to 20 which can be accomplished by staying at 4n, everytime, right?
brave wolf
so all we gotta do is somehow get to 20 which can be accomplished by staying at ...
Yes.
that's the easiest possible proof ig
crisp mural
so in a game of two players, whoever starts second, wins
desert swallow
I recall playing this at school and it tripped up so many people
brave wolf
btw this winning-losing position approach is very, very common
you can solve many games through this
even tic-tac-toe, if implemented smartly
crisp mural
yeah i was thinking of the same thing, if you were to generalise this
brave wolf
This is a lite version of algorithm called "minimax" if you want to learn more about it
crisp mural
even tic-tac-toe, if implemented smartly
i remember some guy posted a video about it, he basically solved tic tac toe by that winning losing approach
crisp mural
This is a lite version of algorithm called "minimax" if you want to learn more a...
oooh i see, thanks again
brave wolf
in theory, with sufficient computational resources, this could solve any 2-player game that's independent on randomness
crisp mural
oh damn yeah I just googled it
desert swallow
This is a lite version of algorithm called "minimax" if you want to learn more a...
Been a while since I looked at minimax but I guess the special case for this game is that we only need to look one move(perhaps level) into the future to find the next best one, right?
compared to say tic-tac-toe
brave wolf
I think the main difference is that classical minimax looks both forward and backward, because there are great many ways to end a tic tac toe game
yeah, minimax looks in the future
here, there is just one losing position, so we can instead go only "backward" in time
desert swallow
Ah I see
brave wolf
minimax starts by computing all possible game-plays until all games in the tree end
then it assumes the opponent is perfect player
meaning that if he can get you to a losing position, he will do it
and it moves backward from all the ending positions through the tree
in similar way as we did here
the main difference is that there are far more ending positions
desert swallow
I think I've gotta take another look at minimax then. I remember implementing it(for tic-tac-toe) by incrementally building the tree on every move. In retrospect, that makes sense because of what you mentioned (more ending positions) -> can't determine which one at the start of the game
brave wolf
I think I've gotta take another look at minimax then. I remember implementing it...
i think that it's first incrementally built, by going through all possible moves, and then once everything reaches the end, it starts evaluating from bottom up.
desert swallow
Mhmm yeah that seems familiar. Another thing I recall was that I never built the tree to the end, different game states were scored and it only went down to a certain depth before bubbling back up
brave wolf
Mhmm yeah that seems familiar. Another thing I recall was that I never built the...
that's when you do more difficult game
like chess
tic-tac-toe can actually be fully scanned within ~50ms on my laptop
Here is an example tree.
on red states, it's "Red" player's turn. On blue states, "blue" player plays.
1 - Red wins
0 - tie
-1 - blue wins
and black is ending position (it has defaultly assigned number)
once the last black row is filled, the row before it is fileld
the red one
since red player tries to win, he will choose the maximum of the nodes he is connected to
so they end up being as drawn
then the 2nd layer is evaluated
blue player tries to minimize red's chance to win, so he takes the minimum
and then the 1st red layer just takes the maximum of 0 and -1, which is 0
desert swallow
tic-tac-toe can actually be fully scanned within ~50ms on my laptop
😅 Yeah, you're right! I just tried with my implementation. Can't even notice a difference in response times
desert swallow
blue player tries to minimize red's chance to win, so he takes the minimum
Ohhhh right I guess my code sorta confused me. This is what I believe happens as my minimax unwinds(bubbles up), i.e. it is indeed walking backwards through the tree
safe radishBOT
@crisp mural Has your question been resolved?
crisp mural
you can continue ur discussion here till whnever the channel closes by itself
native robin
how to do
safe radishBOT
how to do
@native robin Has your question been resolved?
Available help channel!
haughty yarrow
B intersection C means common to both B AND C
B (compliment) intersection C would mean just C? so everything in C excluding B? like C - B
safe radishBOT
B intersection C means common to both B AND C
B (compliment) intersection C woul...
fathom jewel
B intersection C means common to both B AND C
B (compliment) intersection C woul...
yes
actually it should C \ B
just C would be wrong, cause there is a part of B in C
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@haughty yarrow Has your question been resolved?
haughty yarrow
@fathom jewel and if A, B, C are 3 sets, where A = {1, 2, 4, 5}, B = {2, 3, 4, 6} and C = {4, 5, 6, 7}
then (A̅ - C) would mean what?
fathom jewel
<@587715738484211754> and if A, B, C are 3 sets, where A = {1, 2, 4, 5}, B = {2...
Complement of A without C
haughty yarrow
so nothing?
fathom jewel
draw it
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fathom jewel
B \ (A or C)
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haughty yarrow
B \ (A or C)
yes why do we include B like the ans is 3. why cant it be just Φ
haughty yarrow
just draw it
no i get that but my question why r we considering B at all when the question only says (A̅ - C)
doesn't include B
fathom jewel
no i get that but my question why r we considering B at all when the question on...
it does
if you would get it you wouldn't ask that lol
A̅ is everything outside of A
which includes B (without A)
Then you take away C you have Universe \ (A or C) and that one includes B \ (A or C)
haughty yarrow
so B has to be in the same universe?
fathom jewel
yes
haughty yarrow
my question doesn't say that B is in the same universe
thats why im confused
safe radishBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
haughty yarrow
hmm
i'll have to ask my teacher for a photo cause i jjst wrote it in my way
but if the question doesn't say anything about the universe, do i just consider them in the same universe?
haughty yarrow
otherwise it doesn't make sense
yes i understand now
the universe part was confusing me lol
thank you man
fathom jewel
no problem sam
haughty yarrow
.close
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Available help channel!
cunning roost
[OFF-TOPIC QUESTION]:
Is there a map or website that indicates me every field of math?
I mean, if i'd like to build again my math knowledge from the start, where should i start and what guide-map should i follow?
safe radishBOT
[**__OFF-TOPIC QUESTION__**]:
Is there a map or website that indicates me every ...
soft matrix
[**__OFF-TOPIC QUESTION__**]:
Is there a map or website that indicates me every ...
From the start ? Like first learn 1234..
Then basic operators
silent token
Not sure what you're looking for exactly, but this kinda fits?
Some context for where you are and what exactly you want would be nice
cunning roost
From the start ? Like first learn 1234..
Yeah, that would be too deep but i'd need a map that let me deeply understand Analysis 1 from Engineering field
soft matrix
Not sure what you're looking for exactly, but this kinda fits?
That just highlights important fields of math and also misses alot other
But still I feel would be pretty efficient for the op
cunning roost
Not sure what you're looking for exactly, but this kinda fits?
Yeah something similar
silent token
That just highlights important fields of math and also misses alot other
But st...
It provides a pretty nice general overview tho
cunning roost
I think that it would be good to start from expressions or equations again, is it fine?
I've found this btw:
https://encyclopediaofmath.org/wiki/Special:AllPages
https://mathworld.wolfram.com/
.close
turbid ember
I just have a quick question
safe radishBOT
I just have a quick question
hearty geyser
Yes
turbid ember
when I cross over 2 to 2(x-3) how do I multiply it
do I multiply 2 to all of it or just 2 then distribute 2 to x-3
sinful iris
factorized the denominator
hearty geyser
Don't you multiply 2 to both sides
sinful iris
why?
hearty geyser
To move the 2
sinful iris
when I cross over 2 to 2(x-3) how do I multiply it
u can just skip the factorizing
cross multiply
2(2x-6) = x(x-4)
hearty geyser
I forgot about cross multiply
sinful iris
2(2x-6) = x(x-4)
this might be easier
turbid ember
4x-12=x^2-4x
X^2+8x-12=0
sinful iris
yea
turbid ember
what adds to 8 that multiples to -12?
turbid ember
i don’t get it
sinful iris
do you know the cuadratic formula
turbid ember
no
sinful iris
alright
safe radishBOT
safe radishBOT
Available help channel!
molten hawk
safe radishBOT
molten hawk
how do I start this
safe radishBOT
@molten hawk Has your question been resolved?
crimson trellis
Hi, I think im supposed to answer these questions algebraically but idk how without a graph. I know the values of A h and k for each function, but im not sure if it will help me with this
safe radishBOT
Hi, I think im supposed to answer these questions algebraically but idk how with...
empty pebble
sup
alr lets see
the domain is essentially what numbers you are allowed to plug into the function
broken cloud
fill in the blank and how
crimson trellis
oh yeah ik what the question are asking
broken cloud
for b.
empty pebble
fill in the blank and how
yo please claim a new help channel
under the "math help (available" category
helps 35, 39, and 48 are available
alr thanks
empty pebble
oh yeah ik what the question are asking
ok so back to this
what numbers can you plug into -3|x - 2| + 5
crimson trellis
yup
crimson trellis
lol
so is this a extra paper kind of question or is the work small enough to do in the page?
empty pebble
small enough for me, but thats personal preference
empty pebble
what numbers can you plug into -3|x - 2| + 5
ok so can you just plug in any real number into this and always get a real output?
crimson trellis
what numbers can you plug into -3|x - 2| + 5
idk how to tell
it's probably not a line
empty pebble
are you familiar with absolute value (the '|'s around the x - 2)
crimson trellis
yes
crimson trellis
ok
crimson trellis
whys that
empty pebble
youll see
crimson trellis
also this is my graph
empty pebble
oh ok thats roughly it
empty pebble
oh yes they are i just realized i meant that the shape is accurate
alright so you notice that the graph extends infinitely in both directions right
you drew the little arrows
too
which is good
crimson trellis
ya
empty pebble
so this graph goes on forever in both directions, right
crimson trellis
well i can tell everything from the graph, but r u saying these problems can't be solved algebraically?
empty pebble
oh ok good cause i wasnt exactly sure how to explain it properly lol
um algebraically makes it easier
so we have f(x) = -3|x - 2| + 5
think about the equation y = -3x + 5
we can plug anything into x and get something out, right
crimson trellis
yes
empty pebble
so the domain of -3x + 5 is all real numbers
crimson trellis
ok
empty pebble
now, this is not exactly our function
we have
-3|x - 2| + 5
so we have an |x - 2| instead of an x
empty pebble
so the domain of -3x + 5 is all real numbers
but from this, we already know that the x used here can be anything
so |x - 2| can be anything and we will get an output
since |x - 2| is just the distance from 2, it is defined for all real inputs, and the domain of |x - 2| is all real numbers
clear lark
crimson trellis
hi
empty pebble
ok
go to
the math help (available) category
and choose one of help-10, help-12, and help-21
and post your problem there
empty pebble
those are the available help channels
np
ok my explanation probably didnt make sense
crimson trellis
so what exaclty do i do to aglebraically find the domains and stuff
empty pebble
well you have to consider the types of functions involved
in our case, we only have absolute value
and the domain of absolute value is all real numbers
crimson trellis
what abotu inf
empty pebble
sorry what?
crimson trellis
infinitiy
empty pebble
well infinity isnt a number
more like a concept
you cant "plug" infinity into a function
crimson trellis
oh ok
i was thinking of a domain for exmaple of (-inf, 5) or something
for a ray
empty pebble
oh yes good example
ok so you know the difference in interval notation between '(' and '['
crimson trellis
oh yeah
empty pebble
parentheses vs square brackets
crimson trellis
[ and { and (
empty pebble
parentheses mean it is not inclusive, and square brackets mean inclusive
the '{' is used for when you have like a set of numbers, but that is also good
crimson trellis
i was thinking of a domain for exmaple of (-inf, 5) or something
(-inf, 5]
crimson trellis
its an idea
empty pebble
you cant actually plug -infinity into the function
its not in the domain
however, any number greater than -infinity and less than or equal to 5 works
however, since -infinity is a concept that represents the smallest possible value (im iffy on this wording), all numbers are greater than -infinity
so (-inf, 5] just means all numbers less than or equal to 5
crimson trellis
yup
so where do i find the domain in the abs function
is it anything to do with A h k?
empty pebble
im not exactly sure what you mean by A h k
h and k are the vertex coordinates
and i presume A is the coefficient of the absolute value
crimson trellis
f(x) = A|x-h|+k
empty pebble
yes good
actually the domain of anything of that form is all real numbers
as long as A, h, and k are real
crimson trellis
wait i was just thinking is the domain for -3|x-2|+5 (-inf , inf)
empty pebble
yes
crimson trellis
amazing
empty pebble
range depends on A, h, k though
ill walk you through finding the range if you want me to
crimson trellis
wait i jut had a revelation
crimson trellis
the graph is open down so the low in the domain is -inf and the highest is the vertex at (2,5) so the range is (-inf, 5
]
?
crimson trellis
bruh i literally wrote the ansewr in my example
crimson trellis
we can also do it algebraically
ok sweet
empty pebble
ok so we have f(x) = -3|x - 2| + 5
what is the range of the absolute value of something
crimson trellis
no idea
empty pebble
absolute value makes it nonnegative, right? if it is negative, make it positive, and |0| = 0.
so |x| >= 0
crimson trellis
yes
empty pebble
the range of |x| is [0, inf)
crimson trellis
🧠
crimson trellis
oh yeah
empty pebble
you sure?
crimson trellis
i dont htink the +5 changes the range
empty pebble
it actually does, and we'll see why
so the range of |x - 2| is [0, inf). what do you think the range of 3|x - 2| is?
crimson trellis
the same thing? its always going to be positive
empty pebble
*nonnegative, but yes
crimson trellis
ok
empty pebble
remember it can equal 0
crimson trellis
yes
crimson trellis
umm all real numbers?
because it can go negatigve
oh wait
it cant go positive
empty pebble
but can -3|x - 2| be positive? if -3|x - 2| were positive, then 3|x - 2| would be negative
yes
so what do you think the range of -3|x - 2| is
empty pebble
*0]
crimson trellis
oh ok
empty pebble
-3|x - 2| can equal 0 if x = 2
finally, we just add 5
-3|x - 2| + 5
what is the range of that
the range of -3|x - 2| is (-inf, 0], so if we add 5, how does the range change
crimson trellis
(-inf , 5] if the greatest value of -3|x - 2| is 0?
empty pebble
yes correct
crimson trellis
AMAZING
empty pebble
the range of f(x) is (-inf, 5]
and you found the range without even having the graph it!
graphing tends to be a bit of a hassle when youre running low on time
at least for me
crimson trellis
agreed
empty pebble
Hi, I think im supposed to answer these questions algebraically but idk how with...
im just replying to this so i can quickly navigate up here
crimson trellis
ok thank you a lot, I know how to read the function now
crimson trellis
ok
empty pebble
oh yeah and we must do the y-intercept for f(x) first
the y-intercept is when _______
crimson trellis
x = 0
empty pebble
so plug 0 into f(x) and see what you get
empty pebble
yes
correct
but remember, dont just write what f(0) is. write (0, f(0)). the problem wants coordinates
the y-intercept isnt just a value, its a point
so you would put (0, -1) as your answer
cool?
empty pebble
ok so now try g(x)
crimson trellis
i was thinking all real # for both because its a line
empty pebble
yes, awesome
empty pebble
i was thinking all real # for both because its a line
is this for both domain and range or what
crimson trellis
yeah
empty pebble
good
that is correct
and now h(x)
similar to f(x) (same form)
im sure you can do it
crimson trellis
(-inf , inf) for domain
crimson trellis
[-8/3 , inf) for range?
crimson trellis
is it (0,1)
empty pebble
yes
good
im glad you put it in coordinate form (i lost so many points from putting just a number a couple years ago)
crimson trellis
rip
empty pebble
ok so you got another question?
crimson trellis
nope, thats all thx
crimson trellis
anything i have to do to close the room
empty pebble
ok just use .close if youre done
safe radishBOT
ember raven
Troy's truck has a 30 gallon gas tank and gets an average of 21 miles per gallon. Write an equation to represent the amount of gas in Troy' s truck after driving a certain number of miles (assuming he starts with a full tank). Define your variables.
I am confused on this question and have other questions like this. I just want to know how to do these type of questions to solve other ones.
safe radishBOT
Available help channel!
ember raven
Troy's truck has a 30 gallon gas tank and gets an average of 21 miles per gallon. Write an equation to represent the amount of gas in Troy' s truck after driving a certain number of miles (assuming he starts with a full tank). Define your variables.
safe radishBOT
Troy's truck has a 30 gallon gas tank and gets an average of 21 miles per gallon...
sly path
Troy's truck has a 30 gallon gas tank and gets an average of 21 miles per gallon...
y=mx+b
ember raven
What would the equation be
sly path
so you wanna solve for the amount of gas, thats y
and you want to know that amount after driving a certain distance, x
b is the amount of gas that you start with
and m is the miles per gallon
so you sub in the values, you get y = -21x + 30
m is negative in this case because youre losing gas
ember raven
ok so if his tank is half full, approximately how many miles has he driven since last filling up his tank?
sly plank
,tex $ 15 = -21x + 30 $
flat frigateBOT
luxzi
sly plank
i.e. this 👆
ember raven
ok so subtract 30 on both sides and divide by -21 on both sides and you find x which is how many miles he drove since last filling up his tank?
sly path
exactly 👍
safe radishBOT
safe radishBOT
Available help channel!
woven axle
What is multiplicity?
safe radishBOT
What is multiplicity?
icy lance
how many times that root appears if you factorise the polynomial
eg if you had a (x-5)^3 then the root x=5 has multiplicity 3
icy lance
but it has two distinct roots
5 and -5, both of which have multiplicity 1
if you sum the multiplicities of all the roots you should get the degree of the polynomial, which is what i think you have on your mind
woven axle
Can you provide one more example
icy lance
f(x)=(x-2)^2 (x+3)(x+7)^3
roots:
2 multip 2
-3 multip 1
-7 multip 3
woven axle
Okay Is it becuase they all have diffrent squares
woven axle
I'm lost again
woven axle
it has x^2
Going back to this, it has an exponent of 2
icy lance
it does
woven axle
how come its 1 and not 2 when in your example the (x-2) has a multiplicity of 2
icy lance
if a quadratic has two distinct roots they both can only have multiplicity of 1
the multiplicity is about the roots, not the polynomials themselves as a whole
woven axle
for example, (x-5) and (x+5)
icy lance
if it was (x-5)^2 (x+5)
then it would be a cubic with roots 5 (multiplicity 2) and -5 (multiplicity 1)
woven axle
Oh I get it
it has to be an (x) to the exponent of smt to be the exact number
but'
if it was lke (x^2+9) you need to break it down
icy lance
that has no roots
not real ones anyway
the key point is when you factorise a polynomial, the number of times a root appears in that factorisation is its multiplicity
woven axle
I dont unddrstand that terminology is there like an example where you can just point an arrow to it so I can get a better understanding
icy lance
if it was (x-5)^2 (x+5)
then it would be a cubic with roots 5 (multiplicity 2) a...
alright, let me write it like this
you have (x-5)(x-5)(x+5)
the (x-5) appears twice, so root x=5 has multiplicity 2
(x+5) appears once so root x=-5 has multiplicity 1
that make any more sense?
im not sure how i can really break the concept down more
woven axle
Okay I think I understand that
so the amount of times a root appears is the multiplicity liek yo usaid
like you said*
but
I'm supposed to factor it first get multiplicity right?
icy lance
yeah
woven axle
okay can you give me a practice problem that I can work on?
woven axle
1: 3, -2 (multiplicity 1)
2: 13, -13 (multiplicity 1)
3: 0, -5 multiplicity 1
icy lance
x^2-32x+256
woven axle
okay
alright thank you
I think I understand now
actually
have a nice day
tthanks for the help
.close
safe radishBOT
safe radishBOT
Available help channel!
hollow elbow
what exactly is the role of linear combination of random variable
hollow elbow
like
what we are calculating using it
safe radishBOT
@hollow elbow Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
neon sandal
rugged marsh
yo
rugged marsh
can someone help
viscid garden
?
rugged marsh
I need help in geomatry
soft prism
!da2a
safe radishBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
soft crown
I need help in geomatry
Share question here
rugged marsh
Share question here
do I paste problem
soft crown
Yes
viscid garden
do I paste problem
yeah
rugged marsh
soft crown
So u have to refer this diagram and answer the following. Like can u identify points and lines?
rugged marsh
So u have to refer this diagram and answer the following. Like can u identify po...
can I say points on the lines
E,D,C,K
?
@soft crown
soft crown
Yes...they all are points
soft crown
Do u know the difference between lines and rays?
rugged marsh
Do u know the difference between lines and rays?
a ray has 1 endpoint and keeps on going on 1 side
soft crown
a ray has 1 endpoint and keeps on going on 1 side
Yes
soft crown
do lines keep on going and never end
Yes
rugged marsh
and line keeps going on 2 sides
soft crown
And line segments?
soft crown
Yes
rugged marsh
rugged marsh
What about rays?
excuse me sir can you please help me
ig u want to ignore me
alr
<@&286206848099549185>
I call the crew
rugged marsh
How about CD?/
it wont be a line
viscid garden
I am talking about rays
viscid garden
Yep
rugged marsh
not CD
rugged marsh
It will work...
1 side is not going unlimited
viscid garden
1 side is not going unlimited
Are you talking about lines?
I was talking about rays
viscid garden
No...
rugged marsh
oh
rugged marsh
oh
viscid garden
When we generally talk about ray CD, it's not the small segment, but the whole ray
viscid garden
LA is not really a ray...
viscid garden
LA is a line
Both of it's ends are infinite
Both of the ends have arrows at the end
rugged marsh
I am not saying B a
viscid garden
And if BA is a line
rugged marsh
BA"
viscid garden
The third ray is DE
rugged marsh
IT STOPS
viscid garden
Because they are a line...
They lie on a line..
Not a ray...
You agree that BA is a line?
viscid garden
No it doesn't stop on one end...
Ignore B
And then see
Does it stop?
B is just a point...
Don't consider it...
Now does LA stop on one point?
rugged marsh
Just take help from someone else...
just anwser my question
are these rays
ZX and ZY
rugged marsh
WRONG
viscid garden
What?
viscid garden
Yeah..
my brain is dead
I dunno the definitions...
In geo
We never assume this as rays
They are lines...
But I dunno about the critical definitions..
Just google them up
rugged marsh
<@&286206848099549185>