show the EXACT sequence of buttons pressed
#help-27
pseudo basin
sour spire
oops guys
the calculator didnt show it anymore
i think i pressed smth wrongly again several times
pseudo basin
again, reproduce it
sour spire
sorry sorry sorry
sour spire
again, **reproduce** it
wait whats reproduce
pseudo basin
do the same calculation again but be careful to do it with the exact same steps as the first time
pseudo basin
2.3 is correct
opal mantle
How do you get BD? Is there some sort of theory I'm missing out on here? I'm in 12th grade & this is a bit embarrasing that I'm stuck on something like this lol. Shouldn't have slacked off in 9th grade...
I went & solved for BC using a^2 + b^2 = c^2, but I can't seem to figure out how to get BD. The explanation says some angle similarity theory but I have no idea what that is & would appreciate some more detailed explananation!
devout snowBOT
How do you get BD? Is there some sort of theory I'm missing out on here? I'm in ...
@opal mantle Has your question been resolved?
winter patrol
have you drawn a pic/sketch
fervent helm
devout snowBOT
fervent helm
how to solve this ?
do I need to substitute values of n and just try random hit and trials?
thick schooner
do I need to substitute values of n and just try random hit and trials?
not really
light basin
do I need to substitute values of n and just try random hit and trials?
Idts it seems like a recursion based problem
fervent helm
i found few values like $a_2=1 , a_3=4 , a_4=11 , a_5=26$
woven radishBOT
Prathmesh
thick schooner
u can rewrite the recurrence to note something about the nature of the sequence
fervent helm
wdym by recurrence?
ebon coyote
This sort of thing, where new terms are defined by old terms, is called a recurrence
fervent helm
ohh
gloomy aurora
It can be solved. Have you been taught how to solve it?
fervent helm
It can be solved. Have you been taught how to solve it?
no
gloomy aurora
Have you searched up any online resources? Khan academy can be quite useful.
faint hearth
if possible, you dont want to do that here, since the solutions might be complicated exponentials
fervent helm
there maybe some pattern
fervent helm
u can rewrite the recurrence to note something about the nature of the sequence
can you help if you noticed it?
thick schooner
can you help if you noticed it?
,, a_{25} (a_{23} - 2a_{22}) - 2a_{24} (a_{23} - 2a_{22}) = (a_{25} - 2a_{24}) (a_{23} - 2a_{22})
woven radishBOT
mmmm7
thick schooner
so you want to learn something about a_n - 2a_{n - 1}
rewrite that to accomodate for that
fervent helm
$a_{n+2} -2a_{n+1} - 1 = a_{n+1} - 2a_n$
woven radishBOT
Prathmesh
thick schooner
$a_{n+2} -2a_{n+1} - 1 = a_{n+1} - 2a_n$
yep
so you can imagine defining a sequence like t_n = a_{n + 1} - 2a_{n}
to then have t_{n + 1} - 1 = t_n
fervent helm
i don't understand
thick schooner
which part?
$\underbrace{a_{n+2} -2a_{n+1}}{t{n +1}} - 1 = \underbrace{a_{n+1} - 2a_n}_{t_n}$
by definition^
for convenience sake we define a new sequence t_n and we note that the difference between consecutive terms in the sequence is 1
fervent helm
won't it be t_n+1 in the lhs
thick schooner
yep typo
thick schooner
to then have `t_{n + 1} - 1 = t_n`
^
fervent helm
ohh
woven radishBOT
mmmm7
thick schooner
that's it
i mean you're tasked with finding t_{24} and t_{23}
can you find any t_i for some i from given info
if you can do that then you can find t_{24} or t_{23}
fervent helm
can you find any t_i for some i from given info
not for any i but I can find few particular values with the given info
fervent helm
**Prathmesh**
like ik this so i can plug in and calculate few values
thick schooner
you have a_0, a_1, and possibly some more values
surely you can find t_0 or something
fervent helm
yea
thick schooner
fervent helm
btw how did you think all this up ?
thick schooner
well since this looks like a jee problem or smth
you're certainly not supposed to "solve" the recurrence
so factorizing and hoping that there's some pattern dictating a_n - 2a_[n - 1} was the next natural step
fervent helm
damnnnnn genius
thick schooner
far from it 😭
fervent helm
still that was so slay
thick schooner
💅
vivid summit
devout snowBOT
hollow ice
well, you already know that its an arc of a circle
and its centered at B, coz thats given
so, all that remains is to find the angle ABC to determine how much of a circle that arc forms
vivid summit
then bcs is 90 its a quarter?
hollow ice
i dont understand how the other unshaded region is a quarter of a circle
And importantly, they are talking about the semicircle + shaded region,
hollow ice
then bcs is 90 its a quarter?
yes, coz whole circle is 360 deg, 90 is quarter of that
hollow ice
type .close
dusty fox
Hey, do i write 3sinx³ or 3x², the last line where i just wrote 3
fair storm
fair storm
I think this will help you resolve your question.
devout snowBOT
@dusty fox Has your question been resolved?
dusty fox
I think this will help you resolve your question.
No, it doesn't. Just explain pls 🙏
devout snowBOT
No, it doesn't. Just explain pls 🙏
dusty fox
I thought u put the constant at the front and then write it as is
But the answer had -3x² -not 3x³
fair storm
But the answer had -3x² -not 3x³
That is correct.
Which part are you confused about?
You wrote: $y' = cos x^3 \cdot 2 sin x^5 \cdot cos x^5 \cdot 5x^4 - sin x^3 \cdot 3x^2 \cdot sin^2 x^5$
woven radishBOT
Erebus
dusty fox
Hold on
fair storm
$y' = 10 : cos x^3 , sin x^5 ,cos x^5 , x^4 - sin x^3 \cdot 3x^2 \cdot sin^2 x^5$
woven radishBOT
Erebus
dusty fox
Which simplifies down to...?
-3sinx³•x²•sin²x⁵ ?
fair storm
YES
dusty fox
Hih???
fair storm
That's correct.
dusty fox
Really
fair storm
The last part.
dusty fox
I put the u then v
The answer I'm looking at had v then u
fair storm
$- sin x^3 \cdot 3x^2 \cdot sin^2 x^5 = - 3 :sin x^3 , x^2 , sin^2 x^5$
dusty fox
So i did u•v' + u'•v
woven radishBOT
Erebus
dusty fox
So i did u•v' + u'•v
They did v'•u+u'•v
Could it be because of that?
dusty fox
**Erebus**
Oh
fair storm
They did v'•u+u'•v
Could it be because of that?
It's the same.
dusty fox
Ok but they had -3sinx² is that correct too?
fair storm
They did v'•u+u'•v
Could it be because of that?
A and B, B and A, the same
fair storm
Ok but they had -3sinx² is that correct too?
The last part should include that part.
But it should not be the entire last part.
dusty fox
The last part should include that part.
I didn't include that part...is it still correct?
dusty fox
The last part should include that part.
I didn't include that part...is it still correct?
dusty fox
That's the WHOLE second part.
But i wrote what u wrote
fair storm
But i wrote what u wrote
Then, you're correct.
I mean, from this conversation, your idea is correct and your answer is also correct.
Let me write it out fully
dusty fox
Then, you're correct.
Ok but -3x² is a diff value then none of us wrote
fair storm
$y' = 10 : cos x^3 , sin x^5 ,cos x^5 , x^4 - 3 :sin x^3 , x^2 , sin^2 x^5$
woven radishBOT
Erebus
fair storm
Ok but -3x² is a diff value then none of us wrote
What do you mean by that?
The postioning of the terms?
You could write $- 3 :sin x^3 , x^2 , sin^2 x^5$ as $- 3 : x^2 , sin x^3 , sin^2 x^5$ or $- 3 : x^2 , sin^2 x^5 , sin x^3$
woven radishBOT
Erebus
pure flower
what the hell is going on here
fair storm
It just depends on your preference.
fair storm
what the hell is going on here
I don't know.
I'm just hitting all the dialogue options for now.
pure flower
we have $\dv{}{x} \left(\cos(x^3)\sin^2(x^5)\right)$, yeah?
lemme rewrite that
ok.
woven radishBOT
Περσυ
fair storm
Hey, do i write 3sinx³ or 3x², the last line where i just wrote 3
@pure flower
pure flower
yeah idk what the fuck they did
that derivative is so wrong
ok first term is right sure
whats the second term
3
fair storm
Which part was wrong?
pure flower
..whats the 3.
fair storm
This part?
pure flower
weird
pure flower
Hey, do i write 3sinx³ or 3x², the last line where i just wrote 3
...do you mean prefered order
like it super doesnt matter
pure flower
but clean order is $-3\sin(x^3)\sin^2(x^5)x^3$
woven radishBOT
Περσυ
pure flower
or maybe $-3 x^3 \sin(x^3)\sin^2(x^5)$
woven radishBOT
Περσυ
devout snowBOT
@dusty fox Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
charred turret
Hey, this is the proof of finding coordinates of a excenter , Im confused at the highlighted part i dont understand how that is true can anyone please help prove the highlighted point
devout snowBOT
Hey, this is the proof of finding coordinates of a excenter , Im confused at the...
@charred turret Has your question been resolved?
devout snowBOT
@charred turret Has your question been resolved?
lunar harbor
Hey, this is the proof of finding coordinates of a excenter , Im confused at the...
Note that $BI_1$ is the external angle bisector at $B$. So, by the exterior angle bisector theorem,
$$\frac{AI_1}{I_1 L}=\frac{AB}{BL}=\frac{c}{\frac{ac}{b+c}}=\frac{b+c}{a}.$$
woven radishBOT
Civil Service Pigeon
lunar harbor
You can also do this with barycentric coordinates if you want to
pastel rose
devout snowBOT
pastel rose
How would I go about proving this ?
So say I assume N is a normal subgroup of G then show that gng-1 is in N
But surely that’s just obvious
Cause of the definition of the normal subgroup
which is gNg^-1 = {gng^-1 such that g in G and n in N}
Then the other way, if gng-1 is in N then this just implies that it’s a property of a normal subgroup
Idk what I’m missing 😭
spark sleet
What is the definition of normal subgroup that was given in your class? Was it a subgroup N such that gN = Ng, for all g in G?
If so, you are right, there is almost nothing to do for your exercise, but your writing above is a bit inaccurate
pastel rose
What is the definition of normal subgroup that was given in your class? Was it a...
I was given gng-1 one
Not the gN = Ng
pastel rose
If so, you are right, there is *almost* nothing to do for your exercise, but you...
Where would u say it’s flawed?
spark sleet
which is gNg^-1 = {gng^-1 such that g in G and n in N}
Here for example, you claim this is the definition of N just before
And you use N in the definition itself
pastel rose
Here for example, you claim this is the definition of N just before
Well I mean is that not it, like a normal subgroup contains its conjugates
Idk how else to prove this for th exercise
spark sleet
Can you show the definition of a normal subgroup exactly as it is stated? Assuming you are doing this for a course
pastel rose
Can you show the definition of a normal subgroup exactly as it is stated? Assumi...
Yea sure
spark sleet
Ok, can you see the difference with the definition and the exercise?
spark sleet
No, the definition tells you $N$ is a normal subgroup of $G$ if (and only if) $gNg^{-1} = N$ for all $g \in G$. \
The exercise, however, asks you to show that $N$ is a normal subgroup of $G$ if and only if $gNg^{-1} \subseteq N$, for all $g \in G$.
woven radishBOT
all matrices are invertible
spark sleet
In other words, for all g in G, it is sufficient to only check for the inclusion of gNg^{-1} into N, to conclude that N is a normal subgroup.
Well, this is what the exercise is asking you to prove.
pastel rose
Oh I see
OHHH bruh
Nah that works 😭
I think I was overthinking it
lol tysm
.close
crude anvil
I’m not sure if this is so suitable for this channel, but I am to write up a proof of any geometry theorem using vectors. any suggestions?
devout snowBOT
I’m not sure if this is so suitable for this channel, but I am to write up a pro...
@crude anvil Has your question been resolved?
inland carbon
cosine law is a good place to start
prime vigil
how do i determine all the continous function f:D->R where f(2x)=f(3x) for every real x
prime vigil
i rewrote that as f(x) = f( (3/2)^n * x) for every n natural
harsh sierra
hint: take a limit
prime vigil
for n or for x
harsh sierra
what do you think
dire iris
limit as x aproaches a where a is real?
limit of n, not x
this is kind of spoiling but you are trying to show that f(x) = f(0) for all x, thus the solutions are the constant functions
how would you do that with a limit?
prime vigil
oh
i understand
i rewrite
it as
f(x) = f((2/3)^n * x
.closed
done it thanks
.close
devout snowBOT
devout snowBOT
Available help channel!
weak lichen
Hi, can someone help me with this?
devout snowBOT
Hi, can someone help me with this?
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
Show your work, and if possible, explain where you are stuck.
rain summit
Hi, can someone help me with this?
it's not normal trust me
earnest lagoon
1
weak lichen
it's not normal trust me
wdym
rain summit
if i remember correctly, 1 appears more than other digits as you go along
devout snowBOT
@weak lichen Has your question been resolved?
devout snowBOT
@weak lichen Has your question been resolved?
shut trail
Hi, can someone help me with this?
Isn't base 10 just the base we use? 0-9???
So it's not normal
And for the second part.
As you go farther and faster.
Consider 11111 and so on.
You will see that the numbers 1 and 2 will appear more frequently than any other number.
short hare
Hi, can someone help me with this?
what level of math are you studying? I've never been taught to do this yet this seems elementary but tricky
shut trail
So no. They don't have a limiting frequency of 1/10
haughty monolith
Prove it bro
shut trail
Prove it bro
Best I could come up with is that if you take an interval from 1000-1999. Then all numbers between that interval can be represented as a sum of 1000 + k. if 1 appears only 1/10 of the time. It means that only 1/10 of the numbers starts with 1000. Which implies that 9/10 of the numbers have a number greater than 1999. Which means that between the interval 1000-1999. There's a number, such that 2000 + k = 1000 + k. Which would imply that 2000 = 1000, Which is a contradiction.
(This is the best I could come up with, I know it work for all number 1-9)
Can't come up with anything better
😭🙏
But wait, would it work if you take this approach on every place?
Oh my fuckin god.
Not the leading digit.
AAAAAAAAAAAAAAAAAAAAAAAAA
haughty monolith
I mean, i don't feel like trying it
But i'd start by considering the frequency of the digits 0-9 for n-digit numbers
shut trail
But i'd start by considering the frequency of the digits 0-9 for n-digit numbers
What about uhh.
What if you start by trying to like. Make a pattern or somethin.
Like. 9 appear 100 times through 1-1000
haughty monolith
Not sure what kind of pattern you're thinking of making.
shut trail
Not sure what kind of pattern you're thinking of making.
Idek myself. 😭
haughty monolith
Like. 9 appear 100 times through 1-1000
It appears more than that
I mean it's just an exercise in counting
weak lichen
what level of math are you studying? I've never been taught to do this yet this ...
1st year
shut trail
It appears more than that
Yeah fuu
haughty monolith
You have 100 occurrences for 900-999, sure, but there's at least 101-th appearance in, say, 9 itself
That's why restricting to digits is probably the right move
haughty monolith
I don't understand your question
shut trail
Aaaaaaaaaaa
shut trail
I don't understand your question
I think the number of nines should be that.
haughty monolith
9.11?
haughty monolith
I haven't done any calculation but I'm almost 100% sure that's not it
shut trail
Yesh
haughty monolith
How many 9 digits appear in two digit numbers?
It's easy: we can start with the 90's
shut trail
How many 9 digits appear in two digit numbers?
19
haughty monolith
It appears eleven times.
Then you have seven other starting digits, in which 9 appears once, and the ones, where 9 appears once
So yeah, 19.
there's a recursion
There ya go.
devout snowBOT
devout snowBOT
Available help channel!
severe anvil
Theoretically, what would the result be if this did work? Does this even make sense?
devout snowBOT
Theoretically, what would the result be if this did work? Does this even make se...
severe anvil
(such that i is the square root of -1)
severe anvil
I am only more confused
but like theoretically wouldn't it just be 16+30i, because the integral from a to b on 2x is just b²-a² i think
but also how does this extend to other functions
dire iris
but like theoretically wouldn't it just be 16+30i, because the integral from a t...
no, because complex numbers lie on a plane, unlike real numbers
think about what it means to go from 0 to 5+3i on the complex plane
versus going from 0 to 5 on the real number line, for instance
severe anvil
still travelling in a line tho right
dire iris
well yeah the answer is right in this case, but you cant always just integrate like that
it doesnt generalize
dire iris
if its path independent, you can essentially integrate normally
so for f(x) = x^2 it works as well
so technically it works for almost all functions you would want to integrate, but it doesnt apply to any function
and its very sloppy in general, because it doesnt give you the proper intuition of what complex integration actually means
devout snowBOT
@severe anvil Has your question been resolved?
dire iris
if you're curious, i would reccomend you looking at the link riemann posted above
severe anvil
i think where we're confused is that im thinking of integrate this function from 0 to x which works normally when x is a real number, so why wouldn't that work if x is complex
dire iris
it does work (so long as the function is analytic everywhere)
supple knot
You should also read about line integrals first
dire iris
yes
supple knot
dire iris
@severe anvil integration for functions like this works because the integral is path independent, so you can choose the straight line path from 0 to z, and integrate normally
devout snowBOT
devout snowBOT
Available help channel!
echo dragon
How do I do E
devout snowBOT
How do I do E
rapid merlin
What have you tried to solve it
devout snowBOT
@echo dragon Has your question been resolved?
echo dragon
What have you tried to solve it
Nothing
coarse flume
You know what sin(pi/2) is?
scenic adder
What property/axiom should I cite to justify multiplying by (a - 1) so that the left hand side is divided by (a - 1)?
devout snowBOT
What property/axiom should I cite to justify multiplying by (a - 1) so that the ...
olive snow
sly oyster
What should be the limits?
devout snowBOT
What should be the limits?
hazy leaf
the limits of what?
sly oyster
the triple integral, using cylindrical cordinates
hazy leaf
oh
surely you don't have to triple integral, im not super well-versed in calculus but it looks as if thats excessive
my intuition tells me to shuffle the two non-trivial equations to look nicer
sly oyster
i dont understand?
sly oyster
the triple integral, using cylindrical cordinates
should it be: z: 0 to r^2/4; r:0 to 8sin(theta) and theta: -pi/2 to pi/2?
dire iris
no
cloud nymph
Mb
sly oyster
well the answwer i get from the limits is 96pi but the answer given is 64/3 pi
restive river
well the answwer i get from the limits is 96pi but the answer given is 64/3 pi
96 pi is cuz you converted it into the sin integral I presume?
dire iris
should it be: z: 0 to r^2/4; r:0 to 8sin(theta) and theta: -pi/2 to pi/2?
theta would be from 0 to pi, no?
sly oyster
96 pi is cuz you converted it into the sin integral I presume?
what is sin integral?
restive river
what is sin integral?
what limits did you get?
sly oyster
theta would be from 0 to pi, no?
both still give the same answer
restive river
both still give the same answer
no they won't
for this integral sin must be positive cuz r≥0
sly oyster
for this integral sin must be positive cuz r≥0
i dont understand?
restive river
i dont understand?
lemme explain
since r = 8sintheta
you can say that this is a circle of radius 4 centered at 0,4
and since r is a distance
rare kernel
What should be the limits?
hey 👋
restive river
and cannot be negative
r≥0
because by definition r is equal to the root of x2+y2
restive river
yes
that's why -pi/2 gives an incorrect answer
try 0 to pi and you'll get the right answer
sly oyster
wait a min, i will try
restive river
and substitute u = r2
mkay I calculated the integrals
in cartesian and cylindrical both give the answer to be 96 pi, I believe the book is wrong
restive river
yeah i did too, its still 96pi
The answer given is certainly wrong or I'm missing sumn
dire iris
yeah i think so too
restive river
mkay
sly oyster
.close
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Available help channel!
blazing hollow
Solve the following equations by using the given substitution. Leave your answers in exact form.
Q1. x^2 + 2x - 2root(x^2+2x) = 3. Let u = rootx^2+2x
Q2. x^2 + x + 12/x^2+x = 8. Let u = x^2+x
devout snowBOT
Solve the following equations by using the given substitution. Leave your answer...
thorny bloom
show ur work
blazing hollow
main focus currently is q1 right now
working is up to (for q1
u^2 - 2u - 3 = 0
skipping a step
u = 3, -1
woven radishBOT
LordFelix
wheat pawn
okay, so you have u
blazing hollow
u got the equation
solve for x
blazing hollow
no i'm kinda confused let me type my workkng out
wheat pawn
Case 1: $3 = \sqrt{x^2+2x}$
woven radishBOT
LordFelix
blazing hollow
yep
calm blade
you need to define u >= 0
wheat pawn
no you dont
thorny bloom
you need to define u >= 0
no u just need to square so it doesn't matter
calm blade
k
wheat pawn
what he DOES need to do is check all the solutions of x when he gets them
blazing hollow
can you solve that?
nope i'm really confused right now, skipping a few steps i'm up to x^2 + 2x - 9 = 0
wheat pawn
nope i'm really confused right now, skipping a few steps i'm up to x^2 + 2x - 9 ...
that's correct (for case 1)
can you solve that?
blazing hollow
do i use the formula x=-b rootb^2-4ac/2a ?
wheat pawn
well, it is a quadratic, right?
mystic scarab
Use whatever method you like/have been taught for quadratic equations
blazing hollow
okay so would the answer be -1 plus minus root10?
i'm trying to figure out if i went wrong or not bc my friend is asleep and her answer is not simular
wheat pawn
that sounds correct for case 1
blazing hollow
okay thank you
wheat pawn
now you need to solve case 2
blazing hollow
and for case two i got -2 plusminus root8 over 2
wheat pawn
Case 2: $-1 = \sqrt{x^2+2x}$
woven radishBOT
LordFelix
wheat pawn
and now you need to check all those 4 solutions
you have to check that, first off, the original equality holds for those values, and second, that x^2+2x is positive (assuming you're in R)
mystic scarab
**LordFelix**
Pay attention to the signs here. Is this even solvable 😉 ? @blazing hollow
blazing hollow
u = -1
wheat pawn
he's not asking that
wheat pawn
can the sqrt of a number be negative in R?
blazing hollow
i'm so confused rn
wheat pawn
in the real numbers
wheat pawn
i dont remember the latex code for it
mystic scarab
\mathbb{R}
blazing hollow
huh
woven radishBOT
LordFelix
wheat pawn
there we go
blazing hollow
i'm so confused can you elaborate on how -1 = rootx^2 + 2x is unsolvable
mystic scarab
i'm so confused can you elaborate on how -1 = rootx^2 + 2x is unsolvable
In other words, can you solve $\sqrt{x} = -1$ ?
blazing hollow
**LordFelix**
this isn't
woven radishBOT
Alberto Z.
wheat pawn
this isn't
precisely
mystic scarab
this isn't
There you go
wheat pawn
it doesnt matter what is inside the root
blazing hollow
but how is it the same
blazing hollow
it doesnt matter what is inside the root
so -1 equaling to
OHHHH
thank you very much
wheat pawn
so both values obtained from u= -1 are not gonna hold on the original equation
blazing hollow
that makes more sense
wheat pawn
but you still need to check the values that you obtained from u=3
thorny bloom
are we not taking complex numbers
blazing hollow
so u can only be -1 plus or minut root 10?
mystic scarab
are we not taking complex numbers
I don't think so at all
wheat pawn
those CAN be solutions, but you do not KNOW they are until you check them in the original equation
blazing hollow
those CAN be solutions, but you do not KNOW they are until you check them in the...
okay thanks
blazing hollow
are we not taking complex numbers
?
wheat pawn
are we not taking complex numbers
the variable is x, not z. I'd assume we arent
blazing hollow
currently in y10 so sorry
mystic scarab
Don't worry
blazing hollow
y10 tutor math at least
wheat pawn
that tells us very little, we dont know your school system
blazing hollow
that tells us very little, we dont know your school system
australian
thorny bloom
the variable is x, not z. I'd assume we arent
oh no like in quadratic equations the answer can be in form of i without z
blazing hollow
oh no like in quadratic equations the answer can be in form of i without z
yeah we aren't using i, i tried googling a few of my questions that came up with the answer 2i or something similar but i figured it was unsolvable -- idt im up to that level
thank you guys
wheat pawn
you should get to that level in 1 or 2 years probably, but just not yet
what i tell my students is that they should keep in mind that those exist, but that we're not gonna use them yet
blazing hollow
kill me now
wheat pawn
because knowing they exist is helpful for other things, like factoring polynomials
so when they get a result where they cant continue, it's because one of those appeared, and they dont need to continue
and not because they got it wrong
blazing hollow
ohhh okay
wheat pawn
that way some theorems are easier for them to understand as well
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mellow nest
hi, would someone be able to explain how to sketch sinusoidal waves??
hazy leaf
you need two things:
- the range, or minimum and maximum
- the starting state
mellow nest
how do you find those??
calm blade
how do you find those??
Find derivative
Or just plug max and min value of sin and cosx
hazy leaf
dont find derivative 💀
do you know the min and max of sin x?
mellow nest
is it 2?
hazy leaf
is what 2?
hazy leaf
i just want sin(x) lol
hazy leaf
soft umbra
Find derivative
not necesary at all
calm blade
Or just plug max and min value of sin and cosx
^
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Available help channel!
surreal orchid
Hello
surreal orchid
No
devout snowBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
surreal orchid
Ok 👌
calm blade
type .close
midnight echo
.close
pastel rose
Hi I’m so confused here
devout snowBOT
Hi I’m so confused here
pastel rose
These definitions are too much 😭
It’s just saying all the valid outcomes which have a probability of being greater then 0
So for the question
How can I even think abt it
😭
iron sun
What is an outcome that has zero probability?
pastel rose
What is an outcome that has zero probability?
Well a fish that’s not tagged?
According to X
iron sun
It could very well happen that some of the fish are not tagged, so the probability of throes outcomes are nonzero and in fact are apart of the support
pastel rose
And the probability of a fish being tagged is also non zero
Oh wait
Unless u get all 10 fish which aren’t tagged
Is that 0?
Nah no that wouldn’t be
That’s still in the support
Surely what ever type of fish you get, the chance of that being in the support is 100%
iron sun
It could very well happen that some of the fish are not tagged, so the probabili...
Wait uh
I think I read ur RV wrong
pastel rose
RV?
iron sun
X
pastel rose
What’s rvx?
iron sun
Random variable = RV
pastel rose
Ohh
iron sun
Okay outcomes in this case are numbers
late pivot
Support of X just means the different scenarios of X with non-zero prob.
pastel rose
Support of X just means the different scenarios of X with non-zero prob.
Yea so surely that’s just all possible outcomes no?
iron sun
So can you come up with an outcome (a number of fishes) that have probability zero?
pastel rose
Cause you either get a fish with a tag or a fish without
iron sun
There’s only 10 cases here
pastel rose
So can you come up with an outcome (a number of fishes) that have probability ze...
So a probability of 0 means no tagged fish?
pastel rose
And that’s the support?
iron sun
You’ve intuitively already answered the question in some sense
Is it possible for X = 11?
iron sun
So what’s the support?
iron sun
Yes (integers and >= 0)
pastel rose
Ah I have to state integers cause of them being while fish
Would I I write that formally?
pastel rose
Just the support so X = {0,1,2,3,4,5,6,7,8,9,10} ?
iron sun
It’s all the x such that P(X = x) > 0
pastel rose
Well what’s the definition of support?
Well its an outcome which have probability of more than 0 to occur
iron sun
Yes
pastel rose
Just the support so X = {0,1,2,3,4,5,6,7,8,9,10} ?
Oh so this is fine
iron sun
This was already in some sense clear to you before even thinking about the problem I bet
iron sun
Oh so this is fine
Yes
pastel rose
This was already in some sense clear to you before even thinking about the probl...
Yea its just this way of formalising which I’m trying to understand in some sense
iron sun
Mhm
iron sun
I’d say the key here was to recognise what the outcomes in the first place looked like
pastel rose
So give each outcome a probability then
iron sun
Not fishes but numbers
pastel rose
Oh alright
When we say probability mass function what do we actually mean
Is it like, in this case for each possible number of tagged fish x, what is the probability that exactly x tagged fish appear in the sample of 10?
And the probabilities of each of them summed is 1
Hm idk if that even makes sense
How would I use ur idea of thinking of them in terms of number rather than tagged fish as
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@pastel rose Has your question been resolved?
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lament schooner
Given an almost exact equation here, is it legal to like put the dy and dx back to dy/dx? aka y’
devout snowBOT
Given an almost exact equation here, is it legal to like put the dy and dx back ...
lament schooner
Ik we can change y’ to dy/dx and separate but someone told me if we started with dy dx we cant glue it back to y’ idk if it’s true
astral cargo
depends
lament schooner
On?
wicked rover
Given an almost exact equation here, is it legal to like put the dy and dx back ...
its legal. the issue is it doesnt seem separable
stuck star
well it depends if the function doesnt change like people in physics do du/dx*dx/dt= du/dt is a lil invalid but moreover in variable separable method (which is the method used to solve the given question) it is allowed
which is one of the method that CAN be used*
lament schooner
its legal. the issue is it doesnt seem separable
Yeah I’m giving an example that’s it
I would just like to know if legal
wicked rover
its legal
lament schooner
well it depends if the function doesnt change like people in physics do du/dx*dx...
Ah ok ok
Ty
Okay so in every case where you know dy/dx are linked it’s legal right?
stuck star
if u can see that y and x terms can be separated do for it
to solve a differential equation it is best to form dy/dx first
so yes it is legal pretty much
wicked rover
Given an almost exact equation here, is it legal to like put the dy and dx back ...
in fact the form with dx and dy on separate sides is a sort of abuse of notation (until you rigorously learn differential forms later) and is just meant to be a mnemonic for the form with dy/dx on one side
stuck star
in fact the form with dx and dy on separate sides is a sort of abuse of notation...
true
devout snowBOT
@lament schooner Has your question been resolved?
lament schooner
Ok thanks guys
surreal orchid
Hello I need help
devout snowBOT
Hello I need help
surreal orchid
Can someone pls look over this work and tell me if this is correct
If not tell me what I did wrong and correct it for me
Thxxx
Any body
knotty sage
could have been done easier by checking that
2 - sqrt(5) = - (sqrt(5) - 2).
mighty thicket
2 -root5 = (root5 -2).-1
surreal orchid
Where in the picture
mighty thicket
at the start
surreal orchid
Where should I change it
mighty thicket
u would have to change the whole proccess shlawg
surreal orchid
Im good
mighty thicket
yeah
calm blade
Where should I change it
1st step
surreal orchid
Do I just have to change that
mighty thicket
no
surreal orchid
Will I still end up with the same answer
mighty thicket
no
u went with the most complicated way lmao
u prob got something wrong in the middle of all that stuff
livid geyser
The answer he got is incorrect right?
devout snowBOT
@surreal orchid Has your question been resolved?
short hare
sqrt(5) - 2sqrt(5) is not 3sqrt(5)
1-2 = -1 in my book!
thats backtracking from the result btw, im not sure if you've made any other mistakes
@surreal orchid .
calm blade
Op went to sleep
short hare
Op went to sleep
as in you know that they went to sleep or is that a metaphor
calm blade
as in you know that they went to sleep or is that a metaphor
Assumption
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fair fjord
I'm working on Linear Algebra Done Right 4th edition question 6.A.19:
Given that this is meant to be true for any basis, my intuition is to phrase this question in terms of operations on T and basis vectors. However, I am stuck. Is this approach wrong, or could I have a hint at how to phrase it?
devout snowBOT
I'm working on Linear Algebra Done Right 4th edition question 6.A.19:
Given that...
stiff karma
Help. FAST, why in 3. Am i getting the line in wrong direction?
f(x) = 2-sqrt(1-x)
I did the table…
fickle moat
I'm working on Linear Algebra Done Right 4th edition question 6.A.19:
Given that...
Your intuition is correct, but the question is much easier to solve maybe if you think about the coordinate vectors F instead of thinking about the T operator.
devout snowBOT
@fair fjord Has your question been resolved?
fair fjord
Your intuition is correct, but the question is much easier to solve maybe if you...
I'm not sure that helps - I've been playing around with 2x2 real matrices trying to get matrix multiplication w/ a vector to look anything like a sum of squared entries, and I think I'm missing something
fickle moat
Yes, your steps are absolutely correct. Have you thought about trying the Couchy Schwartz inequality next?
fair fjord
I'm thinking I need to end up there eventually, but in order for that to get something relevant I need vectors v, w w/ one of <v, w>, ||v||, ||w|| looking like something in the question
which is where Im stuck
fickle moat
Your intuition is correct, but the question is much easier to solve maybe if you...
thats why I said this actually, in this way we can get what we needed as vectors.
fair fjord
ok so I get that the original sum is the same as $$ \sum_{i=1}^{n} || T\vec{v}_i ||$$ - is this what you mean
woven radishBOT
Plaisk
fickle moat
That's right
fair fjord
OH - so with some eigenvector v of norm 1, we have $<Tv, Tv> = |\lambda|^2$ and these terms are all in the right sum somewhere if we break v up to basis vectors?
woven radishBOT
Plaisk
fair fjord
nah that doesnt work nvm
fickle moat
When you decompose $v$ into basis vectors ($v = x_1 v_1 + \dots + x_n v_n$). We need to prove that the sum of the squared matrix entries on the right is greater than or equal to your $|\lambda|^2$. Then Cauchy-Schwarz inequality to get rid of those $x_k$ coefficients and establish the $\le$ sign.
woven radishBOT
ShaddowDagger
fickle moat
Since you've already reached this point I think you can handle the rest from here
fair fjord
prob eventually! thanks!
fickle moat
Im glad I could help 
slate kite
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slate kite
uhh my brain is throbbing cause my friend asked me this question
and he has to describe the transformation in words
solar goblet
i assume that how each component of g(x) transforms f(x), such as how "-3" transforms f(x)
slate kite
the thing that im going insane about is the f(x+1)^2
fair mantle
What does the ^2 notation mean in this context? f(x+1)*f(x+1)?
solar goblet
i'm going to assume that the notation $f(x)^2$ means $f(x)\times f(x)$
woven radishBOT
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
solar goblet
in my opinion there's really no other way to word the transformation f(x)^2 without just saying it turns the parabolic graph into a quartic-shaped graph
slate kite
yeah uhh apparently his answer key says otherwise... its looking at it without the ^2
yeah uhh
is it just a typo
fair mantle
so it's 2f(x+1)-3?
solar goblet
without ^2 it looks more reasonable
slate kite
uhh apparently thats what the answer suggests...
but he is even more confused about this...
apparently its x cannot be equal to -1..
solar goblet
one question at a time please
slate kite
ah mb
solar goblet
so i assume g(x) here means 2f(x + 1) - 3
slate kite
yeah
solar goblet
think about what each constant does here, they must be transforming the graph in a way
what does multiplying by 2 mean? what does adding 1 to the variable x mean? what does subtracting 3 from the whole function mean?
if it helps, play around in desmos
slate kite
i know what it actually is without the ^2, was kinda just puzzled by the notation..
vertically stretched by a factor of 2 about the x axis, horizontally translating it 1 unit left and vertically translating it 3 units down
solar goblet
yup that's the way
slate kite
.close
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Available help channel!
formal wraith
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formal wraith
would it be 4050 instead of 4900 after optimization because it cannot be a square?
because the barn takes a side
fossil locust
that's correct, cause if the area was 4900, you would need 70 + 70 + 70 + 70
but then 100 metres would be longer than one of the sides
I think you might need to check 4500 actually
devout snowBOT
@formal wraith Has your question been resolved?
echo dragon
How do I solve
devout snowBOT
How do I solve
woven radishBOT
echo dragon
Yes
soft umbra
Can you briefly explain the concept of unit circle?
echo dragon
I got to csc 1/sqr5 over 3 but I don’t know where to go from there
soft umbra
And after that, try to find the value of k based on the explanation
soft umbra
I got to csc 1/sqr5 over 3 but I don’t know where to go from there
oh wow, lemme check
coral dragon
reminder: $cos(\theta) = \frac{\text{adjacent side}}{\text{hipotenuse}} = \frac{2}{3}$
woven radishBOT
Tales78945
coral dragon
call "adjacent side" as "a" and "hipotenuse" as "h" for better notation 🙂
soft umbra
I got to csc 1/sqr5 over 3 but I don’t know where to go from there
Assuming your calculation is all right, that’s the answer, no?
They’re asking csc theta
echo dragon
Assuming your calculation is all right, that’s the answer, no?
It’s reciprocal on answer key
3/ sqr5
soft umbra
Could you show your work?
echo dragon
If u can decipher
soft umbra
,rccw
woven radishBOT
soft umbra
This looks correct to me, what is the problem?
coral dragon
I got $\frac{1}{\sqrt{(\frac{3}{2})^2 -1}}$.
@echo dragon did you too?
woven radishBOT
Tales78945
echo dragon
Are you having problem with a or b
Okay for A I got 1/sqr5 over 3
Do I need to simplify that?
And for B I got 1 over -3/5
But the answer on the answer key is -5/3
soft umbra
Okay for A I got 1/sqr5 over 3
$\frac{\frac{1}{\sqrt5}}{3}$?
woven radishBOT
VerifiedCrank
safe knoll
echo dragon
Isn’t it 1
calm blade
Isn’t it 1
Which one you on?
devout snowBOT
@echo dragon Has your question been resolved?
fierce matrix
devout snowBOT
fierce matrix
.reopen
devout snowBOT
✅ Original question: #help-27 message
fierce matrix
∀B,C ∀S,T: Functor(S) ∧ Functor(T) ∧ B, C are groups →
(∃η: NatTrans(η,S,T) ↔ ∃h∈C ∀g∈B, T(g) = h·S(g)·h⁻¹)
-h->
<-h⁻¹-
g-S->
B--S-->C
| |
v v
B--T-->C
g--S-->h
g--T-->h
trying tto build the diagram and figuring out what this pattern is
g = h ∘ f ∘ h⁻¹
this is equal g ∘ h = h ∘ f ?
then this is imilar to the pattern of topological conjugacy
magic:
g ∘ h = (h ∘ f ∘ h⁻¹) ∘ h = h ∘ f ∘ (h⁻¹ ∘ h) = h ∘ f ∘ id = h ∘ f
* ──S(g)──> *
| |
η* η*
v v
* ──T(g)──> *
η* ∘ S(g) = T(g) ∘ η*
η* = h ∈ C
T(g) = h · S(g) · h⁻¹
* ──S(g)──> *
| |
h h
v v
* ──T(g)──> *
* ──S(g)──> *
| ^
h h⁻¹
v |
* ──T(g)──> *
∃ h ∀ g seems like local-global pattern!
h is maybe something global to S and T,
a natural transformation connecting them
∃ h ∈ C s.t. ∀ g ∈ B = global map
η: S → T = global structure map
this seems very similar to how functor is global to arrows
.close
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Available help channel!
heavy basalt
My friend had a question that baffled me.
Suppose we have f(x)=x^2 maped from R to R, can you change it's codomain to make the function injective?
From what we learned in the lessons, in order to make a function injective you need to limit its domain and not codomain.
But my friend suggests limiting the codomain to y<0 or even y<=0, and in both cases the function f(x) seemingly follows the definition of an injective function.
Is my friend correct?
devout snowBOT
My friend had a question that baffled me.
Suppose we have f(x)=x^2 maped from R ...
acoustic leaf
you can limit the codomain to make a function surjective, but it wouldn't change injectivity
if you had f(x) = x^2 as a function from R to (negative reals) that wouldn't be a well-defined function, since most inputs would produce outputs not in the codomain
devout snowBOT
@heavy basalt Has your question been resolved?
heavy basalt
if you had f(x) = x^2 as a function from R to (negative reals) that wouldn't be ...
Why? If we define x^2 this way, each line parallel to the x axis that we'll draw, will intersect the function at 1 point or not ay all
acoustic leaf
every input from a function's domain must be assigned to an output in the codomain
heavy basalt
Because the function essentially is just the origin point
acoustic leaf
your function must have a valid output for every input in the domain
if you want to restrict the domain to the origin as well then it works, but otherwise not
heavy basalt
every input from a function's domain must be assigned to an output in the codoma...
Why? There are functions with discontinuies, in which there is a value in the domain that doesn't have a valid output in the codomain
acoustic leaf
strictly speaking the domain does not include those values
e.g. the usual domain for the function f(x) = 1/x would be R \ {0}, the nonzero real numbers
heavy basalt
Or am I confusing domain with "preimage" I believe it's called in English?
fierce matrix
What about change codomain to 0?
∀x₁, x₂ ∈ ℝ, f(x₁) = f(x₂) → x₁ = x₂
f: ℝ → {0}
f(x₁) = f(x₂) ⟶ x₁ = x₂
¬(x²(x₁) = x²(x₂)=0 → x₁ = x₂) --- oh no it does imply that because the only thing is 0 that can be sent to 0 via x^2, maybe
x²(x₁) = x²(x₂)=0 → x₁ = x₂
∵ ∃!r∈ℝ r²=0, r=0
acoustic leaf
there is a notion of "partial functions" where not every point in the domain must have a defined output, but these are not used very often
fierce matrix
ℝ---x²--->0
acoustic leaf
What about change codomain to 0?
∀x₁, x₂ ∈ ℝ, f(x₁) = f(x₂) → x₁ = x₂
f: ℝ → {...
the problem remains that "x^2" is not a valid formula for a function R -> {0}
fierce matrix
Oh wait I don't think my logic holds there
heavy basalt
So the function 1/x can be mapped from R to R but its domain would be R \ {0}?
And not R?
acoustic leaf
yes, but we wouldn't say it's mapped "from R to R", we would say it's mapped "from R \ {0} to R"
fierce matrix
what do you think?
injectivity: ∀x₁, x₂ ∈ ℝ, f(x₁) = f(x₂) → x₁ = x₂
change co-domain to 0: x² : ℝ → 0,
so x²(x₁)=x²(x₂)=0 → x₁=x₂=0
∵ ∃!r∈ℝ r²=0, r=0
acoustic leaf
again, you are presupposing the function is well defined in the first place, which is precisely the problem (it is not well defined)
also your logic makes no sense, in the third line you just assert it's injective with no proof. i suggest writing out your proof in words
fierce matrix
also your logic makes no sense, in the third line you just assert it's injective...
I wrote I think, if for any input from ℝ you get the same output, then the inputs were the same:
x²(x₁)=x²(x₂)=0 → x₁=x₂=0
acoustic leaf
that's the definition of injectivity. you have not proven that the function satisfies that definition
fierce matrix
that's the definition of injectivity. you have not proven that the function sati...
What else does it need?
maybe
proposition:
change co-domain of x² to 0: x² : ℝ → 0
→ ∀x₁, x₂ ∈ ℝ, x²(x₁) = x²(x₂) → x₁ = x₂
proof:
x² : ℝ → 0,
∀ x₁,x₂∈ℝ, x²(x₁)=x²(x₂)=0, because x² maps to 0
∃!r∈ℝ r²=0, r=0,
∴ x²(x₁)=x²(x₂)=0 → x₁=x₂=0
acoustic leaf
ok still we have the problem, your function remains not well-defined
fierce matrix
ok still we have the problem, your function remains not well-defined
∀x ∈ ℝ, ∃! y ∈ 0, x²(x) = y,
for x²: ℝ → 0,
∀ x, x²(x) = 0
every input from ℝ has a unique output, 0 in the codomain {0}
acoustic leaf
sure but that's just a different function at that point
because the formula is f(x) = 0, not f(x) = x^2
fierce matrix
because the formula is f(x) = 0, not f(x) = x^2
but then how do you change the codomain?
acoustic leaf
you can only change the codomain while keeping the same formula as long as the image is a subset of whatever codomain you choose
fierce matrix
can we change the codomain of x² : ℝ ⟶ ℝ so it becomes injective?
f : domain ⟶ codomain
codomain change ℝ to 0, f becomes injective
x²: domain ⟶ codomain
x²: ℝ ⟶ ℝ not injective
x²: ℝ ⟶ 0 injective
acoustic leaf
you cannot change the codomain of a function (while keeping the same formula!) to make it injective
only changing the domain can work for that
fierce matrix
you cannot change the codomain of a function (while keeping the same formula!) t...
what is the formula? x ⟼ f(x)?
acoustic leaf
something like that
fierce matrix
x ⟼ x² ?
acoustic leaf
strictly speaking the same graph, pairs (x,f(x))
devout snowBOT
@heavy basalt Has your question been resolved?
fierce matrix
strictly speaking the same graph, pairs (x,f(x))
Graph(x² : ℝ → ℝ) = { (x, y) ∈ ℝ × ℝ | y = x² }
Graph(x² : ℝ → 0) = { (x, y) ∈ ℝ × 0 | y = x² }
acoustic leaf
second one is not a valid graph of a function
fierce matrix
second one is not a valid graph of a function
Oh I think I see what you mean, every element in ℝ should have an output, but for example 1² doen't map to anything, because {0} has only one element
but is that not a slightly different concept?
f : ℝ ⟶ ℝ domain codomain are sort of types? so don't have to include all elements
preimage and image
acoustic leaf
all elements of the domain must be included for a relation to be a function
devout snowBOT
fierce matrix
all elements of the domain must be included for a relation to be a function
What if you change the codomain to something larger, where source is recorded?
like f⁺ : ℝ → ℝ × ℝ
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Available help channel!
fierce matrix
how to change codomain of x^2 : ℝ ⟶ ℝ to make it injective
devout snowBOT
how to change codomain of x^2 : ℝ ⟶ ℝ to make it injective
harsh sierra
Change the codomain to the image
topaz beacon
dont you want to change the domain
harsh sierra
Oh right injective
fierce matrix
What if you change the codomain to something larger, where source is recorded?
like f⁺ : ℝ → ℝ × ℝ
harsh sierra
Yeah so there is no one way to do that
fierce matrix
topaz beacon
at that point youre just straight up changing the function
fierce matrix
f⁺: ℝ → ℝ×ℝ
f⁺(x) = (x², x)
1 ↦ (1,1)
−1 ↦ (1,−1)
→ injective
stone stump
thats not wrong but not the intention
the idea is to replace the Rs with subsets
we still want the same formula f(x)=x^2
Oh I didnt see that you only want to change the codomain. then yeah you'll have to use something like that
harsh sierra
Oh I didnt see that you only want to change the codomain. then yeah you'll have ...
This seems kind of pointless
stone stump
yeah a bit
harsh sierra
Because you make the function injective while forgetting its original structure
fierce matrix
This seems kind of pointless
fierce matrix
My friend had a question that baffled me.
Suppose we have f(x)=x^2 maped from R ...
someone asked this question, and I am learning about injectivity so I am curious
you can now ask someone about the same question who can ask someone etc
heavy basalt
Damn my question gained some interest
fierce matrix
Damn my question gained some interest
your friend's question, maybe they were asked by someone also
heavy basalt
yes, but we wouldn't say it's mapped "from R to R", we would say it's mapped "fr...
I still dont get that
Since you still say about x^2 that it's mapped from R to R and not R to R+ and 0
heavy basalt
your friend's question, maybe they were asked by someone also
Nah it was his originally, we answered some questions together and then the question about x^2 came up and he asked the question I wrote
fierce matrix
I still dont get that
Since you still say about x^2 that it's mapped from R to R...
that's a good point
heavy basalt
I still dont get that
Since you still say about x^2 that it's mapped from R to R...
I know that this is matter of pure definitions, but I think I need to get better at that, especially in English because I don't study in English
Like there is domain, range, image, and something that relates to domain the same way image relates to range
fierce matrix
I know that this is matter of pure definitions, but I think I need to get better...
f : ℝ → ℝ
f(x) = x²
maybe this is to say, the domain is ℝ, the codomain is ℝ, where f(x)=x^2
harsh sierra
While the image is {x>=0}
fierce matrix
preimage ⊆ domain
image ⊆ codomain ?
stone stump
yes but you should be careful
there is an image of a function
there is no preimage of a function
only a preimage of a set under a function
fierce matrix
only a preimage of a set under a function
Thank you @stone stump for this! I don't yet understand this, but I find it interesting, and I found literature on this just now
fierce matrix
I know that this is matter of pure definitions, but I think I need to get better...
@heavy basalt check https://webhomes.maths.ed.ac.uk/~tl/ast/ast.pdf writes about just this , domain, codomain, image, etc, and look has the same example of changing the codomain of x^2
Thank you everyone 🙏
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pseudo basin
rip to that
hearty bison
they're a known spammer
ah
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rough sparrow
not a concrete question but why i the fuction f(x)= x^x only defined for postive numbers?
sand dove
not a concrete question but why i the fuction f(x)= x^x only defined for postiv...
well, for starters
(-1/2)^-1/2?
what is that supposed to be
sand dove
?
rough sparrow
its discontinious before 0?
unborn osprey
yeah
rough sparrow
why doesnt atleast show the points that do work?
sand dove
well for the points you can define
sand dove
why doesnt atleast show the points that do work?
because -1/3 isn't an exact value that a computer can store
unborn osprey
why doesnt atleast show the points that do work?
computers cant do that
rough sparrow
so its more of computational limitation and to make the fuction easier to understand?
sand dove
I guess you can technically define k^k when k is a negative integer
But when you plot functions on desmos
usually the argument is a float type
and so there's no good way to define x^x for any those values, even when there are rare exceptions
unborn osprey
my teacher told me functions like f(x)=1, when x is rational, 0 when x is irrational
cant be plotted by computers because they change values so rapidly in a discontinuous manner
rough sparrow
i see
rough sparrow
my teacher told me functions like f(x)=1, when x is rational, 0 when x is irrati...
oh thats the dirichlet fuction