so x3 needs to be able to vary
#help-10
wanton kraken
it has to be k?
wanton kraken
right
wary vigil
if you can vary x3 as you wish
without problems
and you can only vary x1, x2, and x4 in that way
(-1, 6, 0, 1)
what would the basis be?
wanton kraken
just three more rows with 0s?
wanton kraken
😂
warm shaleBOT
Katharine
wanton kraken
(0, 0, 1, 0) ?
wary vigil
exactly
wanton kraken
so the basis is (-1, 6, 0, 1) and (0, 0, 1, 0)
wary vigil
any linear combination of those two
will give a vector that is in the kernel
and all vectors in the kernel
wanton kraken
oh, alright
wary vigil
are able to be represented by those vectors
to be super rigour
you'd have to prove that
but i'm a physics students
student
so don't ask me
:D
wanton kraken
eh, I got the basis for the kernel, that satisfies me
but then I have to find the basis of the Im(ψ)
eh
tomorrow ig
tysm for the help @wary vigil
wary vigil
np
wanton kraken
.close
abstract charm
Good day, how do I choose an appropriate continuous distribution for the kernel density in approximating the distribution of a given discrete data set? I have tried the gamma kernel density (with a bandwidth of 50) on my data set however, I'm not sure if it's a great estimate.
My data set is household sizes. Attached are the empirical PMF and its gamma kernel density that I've computed. PMF showed that 4 has the highest frequency, however, the graph of the kernel density didn't show that. Is this normal or should I go find a different continuous distribution?
obtuse pebbleBOT
Good day, how do I choose an appropriate continuous distribution for the kernel ...
remote skiff
You usually just try 30 something distributions and then select the most likely distribution from that selection using the Kolmogorov Smirnov test. This isn't really something you would ever do by hand.
abstract charm
I see
Since my data is discrete, is finding the kernel density still necessary?
I've already chosen the Poisson distribution in modeling my data, however I am still tasked to find the kernel density, yet Poission is a discrete distribution. That's why I'm asking if it is still necessary?
remote skiff
I don't really understand what you are trying to do, if you already think your data is poison distributed, I'm not sure what the purpose of that step is without more context.
abstract charm
Oh okay okay. We are just applying all the lessons discussed in our class to a data set we were assigned. These lessons include (1) choosing an appropriate distrib. to find the model of a given data, (2) approximate the empirical distribution using kernel density, and etc.
In (1), I have already determined that my data is poission distributed. And right now I'm having a hard time applying (2) since I'm not even sure if it is correct to approximate the distribution of the household sizes using a continuous distribution.
I hope this is clearer T_T
but yeah, my concern now is I'm not sure if it is necessary to apply kernel density to a data set that is known to be discrete in the first place
remote skiff
That's not really a concern if the approximation is good. It's often easier to have a continuous distribution and have access to tools from Calculus.
abstract charm
Right
Thank you for this!
May I ask if you know any continuous distributions that I can try for my data set? I'll be starting to produce a lot of graphs but any leads to good cont. dist.'s is much appreciated! Preferably a continuous counterpart of Poisson if there's such a thing sksks
obtuse pebbleBOT
@abstract charm Has your question been resolved?
abstract current
After t seconds, a ball tossed in the air from the ground level reaches a height of
= h(t) = 144t-16t^2
whats the maximum height the ball reaches?
I know I have to use -b/2a but I dont know what b and a represents
obtuse pebbleBOT
After t seconds, a ball tossed in the air from the ground level reaches a height...
meager gale
After t seconds, a ball tossed in the air from the ground level reaches a height...
standard form
abstract current
standard form
elaborate please
meager gale
yes i will
abstract current
thanks
abstract current
ok
meager gale
h(t) = bt + at^2
abstract current
i see
meager gale
so t = -b/2a gives you the time value
so at time (-b/2a) the ball reaches the highest value, what is its height
well just plug it in
alr bye
meager gale
np
abstract current
.close
rich urchin
I have no idea how to do this let alone start it
obtuse pebbleBOT
I have no idea how to do this let alone start it
@rich urchin Has your question been resolved?
burnt thunder
obtuse pebbleBOT
abstract charm
i think the problem has something to do with the decibel scale. can you provide the decibel scale from your book?
burnt thunder
hm
this?
it doesnt have anything
abt it
this is the answer
i dont get how
they got it
.close
timid silo
I dont think im understanding the intuition behind: Cantor-Schroeder Bernstein theorem
obtuse pebbleBOT
I dont think im understanding the intuition behind: Cantor-Schroeder Bernstein t...
bronze mica
Can you clarify what you don't understand? what it's saying, or why (one would hope) it's true
timid silo
The idea of two injections implying a bijection confuses me.
bronze mica
can you understand why there being an injection would mean (or at least we would want it to mean) that the domain is 'smaller' (or same size)?
timid silo
Its sort of like only some of the elements in the domain exist and map to unique values in the codomain for both f: X -> Y and g: Y-> X hence a bijection?
bronze mica
I'm not entirely sure what you're saying
the intuition is that if f: X -> Y is an injection, then X is in some way 'smaller' than Y
so CSB is saying if X is 'smaller' than Y, and Y is 'smaller' than X, then they are the same size (which is an intuition with bijection, essentially saying sets are the same up to whether we've called the elements x or f(x))
timid silo
Yeah that is weird
bronze mica
(I've put smaller in quotes depending on how much you've seen cardinality discussed because with infinite sets it's kind of unintuitive e.g. natural numbers and integers are the same size with this interpretation, so that might not be the obvious word to use)
the injection meaning 'smaller' sort of intuitively comes from the bijection thing, because a set is in bijection with it's image under an injection (shrink the codomain / range to be only the elements you hit), so having an injection means it's the same size as a subset
timid silo
The set of integers is countable right?
bronze mica
yes
timid silo
because you can just "make the natural numbers negative" to map to the negative integers right
by defining some function f(x) = x if x >= 0, and f(x) = -x if x < 0
bronze mica
this f (if I've understood correctly) is a surjective, but not injective, function from the integers to the natural numbers
timid silo
Oh yeah, okay but there is some unique inverse
okay im ngl im brain dead
thank you for helping, i should take a break
I appreciate you.
.close
sage zephyr
Need help in discrete Maths/binary relations.
I think this symmetry is true for all numbers in integer set (Z). But the value of k will have fractions for some numbers and that can't be part of Z but R set.
For Real number its correct.
obtuse pebbleBOT
Need help in discrete Maths/binary relations.
I think this symmetry is true for...
sage zephyr
obtuse pebbleBOT
@sage zephyr Has your question been resolved?
unreal musk
Not entirely sure your question? They show you symmetry holds when n and m are integers, are you asking about whether the symmetry holds when n and m are general real numbers, or are you asking about whether all pairs of real numbers are such that n - m = 3k (with k being either integer or real), or?
sage zephyr
🤔
I agree that symmetry holds for n and m integers but the conditions k belongs to Z. I don't agree.
Because for number 4 and 6, k will be 2/3
twilit bay
Given the eight letter word "PROBLEMS", how many ways can you arrange them such that each letter is not in its original spot? For example, the letter P is not in the first position, R is not in the second position, and so on.
Can someone explain how to do this?
obtuse pebbleBOT
Given the eight letter word "PROBLEMS", how many ways can you arrange them such ...
twilit bay
sorry
obtuse pebbleBOT
@twilit bay Has your question been resolved?
blazing sapphire
Given the eight letter word "PROBLEMS", how many ways can you arrange them such ...
I can answer this.
First note that all the letters are distinct
So there are 8! ways to arrange the letters if we have no restrictions
so let's count the complement and subtract from the total
Assume 1 of the 8 letters is in its original spot. (8C1)7! ways for this to occur...(notice that this doesn't mean that only 1 of the letters will be in its original spot).
now assume 2 are in their original spots...(8C2)6! ways for this to occur...., but notice that there is some overlap between this case and the previous case!
So we must use the principle of inclusion and exclusion
Similar can be said for the next cases as well soo what do u think the answer should be @twilit bay ?
ping me when u reply pls
hazy elk
i don't think it's that complicated
it's an 8 letter word
so there are 8 possibilities for the first letter, 7 for the second.... 1 for the last
so that's 8!
i actually didn't read the question fully
meager gale
yeah that is the total before
now subtract the possibilities of "PROBLEMS"
so
7! ?
hazy elk
yeah
meager gale
ping me when u reply pls
@blazing sapphire
hazy elk
pretty sure the way logician said is fine then
meager gale
ok
blazing sapphire
yes what I said is completely fine lol
hazy elk
ye
meager gale
didnt know had a name
blazing sapphire
yea it's fun
blazing sapphire
ok and then
hazy elk
and this just doesn't make any sense either guys....
yeah
blazing sapphire
nooooo
brave bramble
blazing sapphire
Why not
I'll explain before u type anymore lol. let me explain why this doesn't work
meager gale
Ok
twilit bay
Similar can be said for the next cases as well soo what do u think the answer sh...
what you said makes sense yes but now what do i exclude and include
in the asnwer page what u said is written exactly but its like 8C1x7!-8C2x6!+8C3x5! and so on but why do we minus once and then again add
i dont understand
blazing sapphire
Suppose one of ur 7 choices for P is the second place. And suppose once of ur choices for R is the fourth place. Then does the O really have 5 choices?...no because it can't go where the O originally was and it could have gone where the R was sent, but now u didn't let it happen with ur counting
meager gale
7! Still makes sense to me
meager gale
Suppose one of ur 7 choices for P is the second place. And suppose once of ur ch...
On
Oh
blazing sapphire
in the asnwer page what u said is written exactly but its like 8C1x7!-8C2x6!+8C3...
the alternating (-1)^n sorta thing going on is from PIE.
meager gale
Suppose one of ur 7 choices for P is the second place. And suppose once of ur ch...
Wait what
blazing sapphire
the principle of inclusion and exclusion
meager gale
I don’t see why O doesn’t have 5 options
twilit bay
Suppose one of ur 7 choices for P is the second place. And suppose once of ur ch...
I didnt understand :(\
meager gale
Not to the third place, not 4th, not second
I c
Ok I found a different flaw
You have 7 options for the first place. Say you pick “R”
Cuz it can’t be “P”
You still have 7 choices for the second place
R is already taken it isn’t to be worried about
blazing sapphire
The problem with ur counting is that not every one of those 7x6x5x... choices all lead to the same number of possibilities for each letter
twilit bay
also do u guys have a good series in yt i can watch to completely finish and understand permutation and combination?
meager gale
Idk
blazing sapphire
The problem with ur counting is that not every one of those 7x6x5x... choices al...
ur counting completely is dependant on where previous letters are sent!
blazing sapphire
Maybe it fixes itself up at the end
it doesn't lol
blazing sapphire
Think about each choice for the letters that ur saying can go
meager gale
Yea I get it
blazing sapphire
it's not always true that the third letter has 5 places to go lolll
twilit bay
also do u guys have a good series in yt i can watch to completely finish and und...
@blazing sapphire :)
blazing sapphire
I didnt understand :(\
nvm this I was explaining why @meager gale counting isn't the answer for this problem
meager gale
So what’s the answer
blazing sapphire
<@856623557685674005> :)
hmm I didn't learn from youtube. And this is going to sound cliche but I honestly learned how to do this just by assigning myself counting problems that were beyond my courses
blazing sapphire
So what’s the answer
I'll type it out in LaTeX
$8!-\left(\sum_{i=1}^8(-1)^{i-1}\binom{8}{i}(8-i)!\right)$
warm shaleBOT
logician
twilit bay
brother
meager gale
What in the
twilit bay
what.
blazing sapphire
oh I'm not relying on the formula LOL
I explained the processes for the first couple cases
$8!-\left(\binom{8}{1}7!-\binom{8}{2}6!+\dots-\binom{8}{8}0!\right)$
warm shaleBOT
logician
blazing sapphire
choose i of the 8 letters to be in their original positions
and then arrange the 8-i letters in (8-i)! ways
the alternating +/- is from PIE
@twilit bay u following?
blazing sapphire
and yet i still dont understand
the best way to learn this is
think about venn diagrams
twilit bay
yes i watched exactly that video
twilit bay
nA + nB - n(ANB)
blazing sapphire
just like how NB and BN overcounted NBN in your other question
yes
those were the 2 sets
in that example
twilit bay
but how does that make sense here
blazing sapphire
so for this one
notice that when I say 8C1 x 7!
I meant that one of the 8 was in it's original spot
and the others moved around in 7! ways
however
it could've been
that one of those 7! ways
made another letter
go into it's original spot
in fact
twilit bay
yes thats upto how much i understand
what i dont understand is
after that why do we continue on -8C2 x 6! then add +8C3x5! - 8C4 x 4! until - 8C8 x 0!
how does doing that exclude everything
cause again arent we adding +8C3x5!
shouldnt we just be excluding from 8C1 x 7!
blazing sapphire
how does doing that exclude everything
no no. it is not just the principle of exlcusion...it's the principle of inclusion and exlcusion
twilit bay
as in that case all of the letters could be in their original position
8C1 x 7! is just 8! which means all possibilities
blazing sapphire
yes but look at the 8C2 x 6! case. It could be that P and B are the two letters in their original spot. but then the 6! means that L could also be in it's original spot. So, for counting the complement, we've done (8C1)7! - (8C2)6! however this case where P, B, and L are in their original spots shouldn't be deleted so we must add (8C3)5!
if u can look at the venn diagram for the case of 3 sets, I think this will make sense if that venn diagram makes sense. This problem is definitely a jump from the other question u asked with BN and NB and NBN
that is honestly my best advice for grasping the PIE.
twilit bay
this problem is question 1 the BN was the last question 8 🤡
twilit bay
yes
blazing sapphire
lovely
twilit bay
let me try and understand it myself using the word FUN
blazing sapphire
let me find u a link for this PIE. Because the problem isn't that ur not getting PIE in this problem, it's that ur not getting the PIE in general. If u understand it in its generality, u'll understand it's application here
twilit bay
yes exactly
thank you
I looked at this
but it doesnt explain PIEs derivation fully
blazing sapphire
yes exactly
this is why ppl often refer to the visual aid of overlapping sets in a venn diagram to grasp the PIE in it's abstractness
twilit bay
so i dont get it
blazing sapphire
I'm not gonna click that, I'm gonna find another one rn
read the portion here talking about three sets A,B,C
specifically, I'm asking u to look at how they calculated |A U B U C|
they gave 6 steps. I encourage u to actually write in each portion of the diagram the number of times it's been counted. The goal is to count every region exactly once.
blazing sapphire
mhm
twilit bay
3C1x2! represents that one of the letters of FUN (perhaps two or even three) are in their original place right?
blazing sapphire
yes and that "perhaps" part is induced by the 2! yes
blazing sapphire
mhm
twilit bay
and in UNF and NFU none of the letters are in their original position
blazing sapphire
right
twilit bay
3C1x2! represents that one of the letters of FUN (perhaps two or even three) are...
so if this was the case 3c1x2! should be 4 right not 6 cause UNF and NFU shouldnt be included as none are in their original position
so like
what the fuck
blazing sapphire
so if this was the case 3c1x2! should be 4 right not 6 cause UNF and NFU shouldn...
no wrong. and here's why
twilit bay
yes
blazing sapphire
3c1 represents us choosing which letter will be in its original position. Suppose that letter is F. (same can be said for any letter so talking about F is fine and we're not losing generality.)
noe the 2! represents the remaining letters flopping around
twilit bay
yes
blazing sapphire
but
twilit bay
so still
it means one of the letters has to be in its original position
by 3C1
then what really is 3C1 if not that
blazing sapphire
FUNis double counted
when
we do the 3c2 part
and is counted yet again in the 3c3 part
twilit bay
then what really is 3C1 if not that
yes but then what does 3C1 represent
blazing sapphire
so if this was the case 3c1x2! should be 4 right not 6 cause UNF and NFU shouldn...
so the reasoning here for 3c1 x 2! being 4 makes no sense
blazing sapphire
yes but then what does 3C1 represent
3c1 represents us choosing which of the 3 letters will be in it's oringinal spot. Remember, this doesn't mean that ONLY one letter will be in it's oringinal spot...because of the 2!
we could have more than one
this is why PIE
is necessary
because the choosing just isn't enough
it doesn't give us
"exactly one"
or exactly 2
etc.
twilit bay
3c1 represents us choosing which of the 3 letters will be in it's oringinal spot...
but bro that still MEANS ONE of the letters has to be in its original place and the maximum value of that is 4 not 6
blazing sapphire
but bro that still MEANS ONE of the letters has to be in its original place and ...
I'm not sure where ur getting 4 and what do u mean max value
twilit bay
or exactly 2
im not saying how many it gives us im saying shouldnt it exclude the words in which none of the letters are in place as you literally said 3c1 represents us choosing which of the 3 letters will be in it's oringinal spot meaning at least 1 of the letters HAS to be in its original position
blazing sapphire
4 of the 6 arrangements have at least one of the letters being it their original spot, yes. However this is all the more reason to believe 3c1x 2! isn't 4.
twilit bay
whyyyyyy
blazing sapphire
im not saying how many it gives us im saying shouldnt it exclude the words in wh...
we are counting the complement. We are only interested in when at least one of the letters is in the oringinal place. THEN we subtract from the total (in this case, 3!)
twilit bay
we are counting the complement. We are only interested in when *at least* one of...
brother
i understood that much
still doesnt
explain
what i asked
sorry if im being annoying im just very confused about this now
blazing sapphire
im not saying how many it gives us im saying shouldnt it exclude the words in wh...
it's because it overcounts that is why
look at this
suppose F was in it's spot, and suppose the 2! gave FUN.
then
now
look at
if N was in it's original spot
and suppose teh 2! then gave FUN as well
etc.
blazing sapphire
hang on
twilit bay
OH
blazing sapphire
one of the choosings fixes F.
twilit bay
THAT MAKES SO MUCH MORE SENSE
blazing sapphire
yes
fixing F is counted different as fixing U and fixing N
in 3c1 x 2!
all of those choices could lead to FUN being the sequence for instance
once the 2! is done
twilit bay
this counts FUN more than once.
FUN is counted 3 times right?
blazing sapphire
yes.
fix F, only one of the 2! ways would give _UN
fix U, only one of the 2! ways would give F_N
fix N, only on of the 2! ways would give FU_
each lead to FUN
twilit bay
OKAY
I GOT IT
AT LAST
YE THE PROBLEMS ONE ALSO MAKES SENSE
THANK YOU SO FUCKING MUCH
I LOVE U
blazing sapphire
I see why u were confused about "total- at least one in original place=none are in original place"
that equality is true
so long as
we really are couning at least one
correctly
blazing sapphire
which 3c1 x 2! doesn't as we discovered
twilit bay
makes so much sense
blazing sapphire
and here's this
read this...it talks about an example using 4 letters.
might be interesting
twilit bay
can i add you
blazing sapphire
sure
twilit bay
i did number 1 and 8 now ima do the rest 💀
blazing sapphire
I typically answer the counting questions here
twilit bay
aight thanks again bro
blazing sapphire
if I got work, and if u need a reply fast, someone else might be able to answer
but I wouldn't ask counting questions in a help channel
I'd ask them in the discrete math channel if ur not dm'ing me the question
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
fringe trellis
Determine the x-intercepts of the curve that passes through the point (2,-5) and has a slope defined by y = 4x - 5
I found the antiderivative( y = 2x^2 - 5x), now what do I do?
obtuse pebbleBOT
Determine the x-intercepts of the curve that passes through the point (2,-5) an...
autumn adder
narrow needle
u forgor the c
fringe trellis
u forgor the c
That damn c....
fringe trellis
subsitute values of x and y given
And then I simply and find the x intercepts?
cloud belfry
the curve isnt necessarily linear
oh shit
fringe trellis
Yeah
cloud belfry
yea so follow procedures accordingly
cloud belfry
So what would I use then
anti der
narrow needle
so
you want to find c such that the point exists on the curve
after that, set y = 0
and factor to find roots
fringe trellis
you want to find c such that the point exists on the curve
Example:
y = 2x^2 - 5 + c
Plug in (2,-5)
-5 = 2(2)^2 - 5 + c
-5 = 3 + C
C = -8
For c
fringe trellis
Yeah
fringe trellis
Oh true
fringe trellis
so -5=2(2)^2-10+c
And the I plug it back into the C in the antiderivative equation and find the x-intercepts?
Then*
narrow needle
yes
karmic hedge
obtuse pebbleBOT
karmic hedge
,tex ((Divide[Power[(40)5(41),3],3]+Power[(40)5(41),2]-3(5))-(Divide[Power[(40)-1(41),3],3]+Power[(40)-1(41),2]-3(-1)))-((Divide[Power[(40)-1(41),3],3]+Power[(40)-1(41),2]-3())-(Divide[Power[(40)-3(41),3],3]+Power[(40)-3(41),2]-3(-3)))
HI CHARTBIT!! ❤️
I am still struggling with latex 😭
warm shaleBOT
Odie
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
karmic hedge
Wait
I got the answer
My method was correct I just made a typo
Sorry haha
Thank you anyways!
❤️
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
weak apex
bam
true granite
okay so the key point here is the chain rule
narrow needle
i would like some help w this pls
dy/dx = (dy/du)(du/dx)
chain rule
derivative of f(g(x)) is f'(g(x)) * g'(x)
true granite
if we need h’(9), it would first make sense to deduce what h’(x) should be equal to, based on the equation we were given for h(x)
where would you start?
true granite
3ln(g(x))^2 - ln(g'(x))
okay so we’re on the right track
our first step should be to write out 3(ln(g(x)))^2, but then we need to multiply this by the derivative of ln(g(x)), according to the chain rule
however, we are going to need the chain rule AGAIN to differentiate ln(g(x))
weak apex
nooo
true granite
g(x) would be the “inner” function and ln(x) the “outer” function
true granite
right! multiplying we get g’(x)/g(x) for the derivative of ln(g(x))
true granite
our first step should be to write out 3(ln(g(x)))^2, but then we need to multipl...
finally, we just need to multiply this to what we got earlier
true granite
wait wait
true granite
we still need to plug in the values, but I think you can do that
weak apex
yes
true granite
yeah
weak apex
i got it from here thank u sososos much
true granite
np!
weak apex
.close
ember frost
$\int \frac{x+1}{x^2+x\ln{x}}dx$
obtuse pebbleBOT
$\int \frac{x+1}{x^2+x\ln{x}}dx$
ember frost
how to integrate this by subtitution?
warm shaleBOT
FungusDesu
ember frost
i tried taking x out in the den and let t = lnx+x or t = lnx but both get me nowhere
elfin burrow
t = x + ln(x) ?
sweet gulch
i tried taking x out in the den and let t = lnx+x or t = lnx but both get me now...
taking x out of the den in the bottom will not give you x(ln(x) + 1) if this is what you are saying
ember frost
oh yeah i mistyped that, i meant lnx + x
sweet gulch
t = x + ln(x) ?
it's this
ember frost
oh wait i think i got it
sweet gulch
$\int \frac{x + 1}{xu} * \frac{1}{\frac{1}{x} + 1} du$
warm shaleBOT
dragonbreath
worthy bramble
obtuse pebbleBOT
brazen viper
I don't see a mistake in your work
worthy bramble
I don't see a mistake in your work
i think the question should be, solve for x instead of proving
brazen viper
But even so, that's not a solution in the range given
worthy bramble
But even so, that's not a solution in the range given
yeh
brazen viper
,w arccot( (sqrt(1 + sin x) + sqrt(1 - sin x) ) / (sqrt(1 + sin x) - sqrt(1 - sin x) ) ) = pi/2 - x/2
warm shaleBOT
worthy bramble
branch cut ?
worthy bramble
so the questions wrong ?
hearty terrace
could it be the restricted range of cotangent
brazen viper
Yes
hearty terrace
inverse* cotangent
brazen viper
Hmm... Maybe there's a sign error
hearty terrace
actually nevermind i take that back
brazen viper
I think It's both things
worthy bramble
this could have been the actual question, the teacher just modified it blindly
hearty terrace
but the proof is true
worthy bramble
but the proof is true
where did i go wrong then
worthy bramble
can the proof be done on the level of grade 12 mathematics ?
warm shaleBOT
hearty terrace
oh right
brazen viper
On (pi/2, pi) cos(x) is negative
So you should have (1-cos(x))/sin x
So you get tan(x/2) instead.
@worthy bramble
worthy bramble
It's abs(cos x)
what is abs(cosx)
warm shaleBOT
worthy bramble
wait a min
brazen viper
Tricky problem
brazen viper
Yes, because -cos x is positive
brazen viper
yw!
worthy bramble
.close
warm shaleBOT
Mycobacterium
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
crisp glen
obtuse pebbleBOT
crisp glen
can someone please help me with this
hearty terrace
speeding up means increasing velocity (i guess the absolute value )
crisp glen
yes
hearty terrace
what about (0,1)
crisp glen
I tried that and its wrong also
crisp glen
let me try
hearty terrace
or you can do a comma or however your teacher told you to
hearty terrace
(1,2) is right
there are more places
speeding up is whenever the value of velocity is going farther away from 0
so from 0mph to 10mph
but ALSO from 0mph to -10mph (i.e., 10mph traveling backwards)
crisp glen
alright
hearty terrace
so where is it speeding up
crisp glen
speeding up is from (1,2),(3,4)
hearty terrace
exactly
crisp glen
so (2,3) is slowing down
hearty terrace
(3,4) may look like its going down but imagine it as going backwards, and going more backwards
crisp glen
i meant 0,1
hearty terrace
both
crisp glen
oh
hearty terrace
so (2,3) is slowing down
at (2,3) you are still moving forward because v>0, but your velocity forward is decreasing
its kind of weird to think about at first that (2,3) is slowing down but (3,4) is speeding up
but you'll get the hang of it
crisp glen
alright
hazy root
Hello,
I don't understand how the integral of cos(x)sin(x) gives 1/2sin(x)^2?
Could anyone be so kind and smart to explain? 🙂
obtuse pebbleBOT
Hello,
I don't understand how the integral of cos(x)sin(x) gives 1/2sin(x)^2?
Co...
royal basin
do you mean like "idk what's going on" or "i think that's wrong"?
hazy root
Yeah more like "what the f is going on?" 
royal basin
there's at least two ways to do it:
\begin{enumerate}[a)]
\item write $\cos(x) \sin(x)$ as $\frac12\sin(2x)$
\item substitute $u := \sin(x)$ and integrate $\int u \dd{u}$
\end{enumerate}
warm shaleBOT
Ann
golden night
if this is a MCQ, you probably substituted the wrong function
royal basin
which one first and what exactly is your doubt
hazy root
Yeah first one has gotten something to do with the geometric functions, but I don't see which one.
You obv got cos(x) = 1 - 2sin(x)^2 . It's close but not it?
Oh yeah I see the second method now indeed. ✅ 🫶🏻
hazy root
Oh yeah I see the second method now indeed. ✅ 🫶🏻
Oh wait, maybe I do not. How do you end up with the 2x between the brackets?
golden night
Why dont you use substitution
also integrals will eat you alive if ur trigo is weak
hazy root
Why dont you use substitution
Could you elaborate how you would get the 2x in this case please?
golden night
Could you elaborate how you would get the 2x in this case please?
you dont
you say t = sin x
then you differentiate
which will be ?
hazy root
Right?
huh
Because you have gotten t * cos(x) right?
If you take t = sin(x)
oh noooo, wait nvm
Yeah so you end up with the integral of u (or t) which gives 1/2u^2.
And that gives 1/2sin(x)^2
Right?
not 1/2sin(2x)^2?
hazy root
**Ann**
Ah yes cheers. @royal basin could you also confirm that this 2x should be just x?
Thanks for the help
I don't get it anymore 
royal basin
Ah yes cheers. <@183668144404037632> could you also confirm that this 2x should ...
no, the 2x is 2x
double angle identity
sin(2x) = 2 sin(x) cos(x)
hazy root
Yeah so you end up with the integral of u (or t) which gives 1/2u^2.
And that gi...
But then how does this work?
royal basin
method b)?
hazy root
Yeah
royal basin
didn't you yourself say exactly how it works
hazy root
Yeah but that only gives x? Not 2x???
As you get 1/2u^2 with u = sin(x) , and NOT u = sin(2x)
royal basin
are you trying to mix the two methods together...
hazy root
are you trying to mix the two methods together...
Oh wait I see. Obviously 1/2sin(2x) is not the final answer of method a. But how would that give then give 1/2sin(x)^2 after taking the integral?
I understand method 2 now. Cheers.
royal basin
method a) would give you -1/4 cos(2x)
which looks different at first but actually only differs by a constant
because -1/4 cos(2x) + 1/4 = 1/2 * (1-cos(2x))/2 = 1/2 sin^2(x)
stone totem
Let's call the group $PSL(2, \mathbb{C})$ the automorphisms of $( \mathbb{C} $ expressed by fractional-linear transformations:
$[ z \mapsto \frac{az + b}{cz + d} ]$
where $( a, b, c, d \in \mathbb{C} )$. Prove that for any chosen three points $( (x_1, x_2, x_3) )$ there exists a unique element $( g \in PSL(2, \mathbb{C}) )$ transforming them into $( (0, 1, \infty) )$. Is it correct to speak about the isomorphism:
$[ PSL(2, \mathbb{C}) \cong \mathbb{C}^3 ]?$
obtuse pebbleBOT
Let's call the group $PSL(2, \mathbb{C})$ the automorphisms of $( \mathbb{C} $ e...
warm shaleBOT
nana
stone totem
how can i prove that there's unique such element in PSL?
obtuse pebbleBOT
@stone totem Has your question been resolved?
stone totem
<@&286206848099549185>
obtuse pebbleBOT
stone totem
.reopen
obtuse pebbleBOT
✅
stone totem
<@&286206848099549185> 😢
obtuse pebbleBOT
@stone totem Has your question been resolved?
stone totem
<@&286206848099549185>
stone totem
.close
floral dagger
could someone help?
obtuse pebbleBOT
could someone help?
floral dagger
no idea......
golden night
buddy...
floral dagger
just started doing the homework after not paying attention in class for like a week straight 😭
golden night
What are Zeros of a Function? The zeros of a function f(x) are values of the variable x such that the values satisfy the equation f(x) = 0.
we good
golden night
could someone help?
!status
obtuse pebbleBOT
could someone help?
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
1
robust bloom
just use the wavy curve
robust bloom
method
golden night
man
sage geode
Recall that a fraction is equal to zero if and only if the numerator is 0 and the denominator is nonzero
floral dagger
mf what wavy curve
golden night
i haev said this so many times
dont just drop a random method unless asked
explain why
robust bloom
dont just drop a random method unless asked
ok
floral dagger
Recall that a fraction is equal to zero if and only if the numerator is 0 and th...
aa yeah i know that
golden night
aa yeah i know that
thats 60% of ur question
floral dagger
but I have no idea on what to do or how to start
robust bloom
but I have no idea on what to do or how to start
factor numerator
floral dagger
yeah I don't know how to apply that knowledge
sage geode
E.g., to solve (x^2 + 2x - 3)/(x - 1) = 0, you need to find x such that x^2 + 2x - 3 = 0 and x should be different from 1 so that x-1 is not zero
floral dagger
how the hell do you factor 😭
robust bloom
wth
sage geode
Can you solve x^2 + 2x - 3 = 0?
floral dagger
I know how to do derivatives but still dont know how to factor
its overrrrrrrrrr for me
robust bloom
I know how to do derivatives but still dont know how to factor
whaaaaaaaaat
floral dagger
Can you solve x^2 + 2x - 3 = 0?
no
golden night
also
floral dagger
MANNN 😭
golden night
just use some logic
floral dagger
im literally retaking this course guys
from last year
its so basic but i dont understand
sage geode
In case you don't want to factor or complete the square, just use the quadratic formula
Are you familiar with it?
golden night
if the denominator has an expression then obviously the numerator would also include the expression, whats the point of the question if the denominator has no use
golden night
yeah but I don't think thats allowed?
what.
robust bloom
yeah but I don't think thats allowed?
whaat
sage geode
yeah but I don't think thats allowed?
You were told that you can't use the quadratic formula?
robust bloom
??
floral dagger
no i meant
robust bloom
u were slepin in clas?
robust bloom
hmm
golden night
they advocate using the shortest and easiest method
then divide the numerator with the denominator
sage geode
Using quadratic formula is the shortest and easiest way
robust bloom
Using quadratic formula *is* the shortest and easiest way
no
floral dagger
my classmates literally factorise in their heads
robust bloom
Using quadratic formula *is* the shortest and easiest way
factoring by splitting middle term is
floral dagger
which one is truly shorter
golden night
which one is truly shorter
realising that the denominator is 0 at 1 so that MUST be the root of the numerator
floral dagger
im so confused
golden night
x-1 is a factor without work
for the second one, +-3 one of which satisfies the numerator, check and then break
robust bloom
realising that the denominator is 0 at 1 so that MUST be the root of the numerat...
thats remainder theorem
golden night
cancel factors
sage geode
realising that the denominator is 0 at 1 so that MUST be the root of the numerat...
Where does that implication come from
floral dagger
do i just kms
golden night
<:thonk:406575732563902485> Where does that implication come from
for this question only
golden night
do i just kms
no
you learn.
it will take you 40 mins
max
dont crib when you can learn
youtube it and watch a lecture
solve some questions
apply
done
wtf.
floral dagger
it will take you 40 mins
it took me 1 hour and a half doing 1 question
from business math yesterday
golden night
it took me 1 hour and a half doing 1 question
so?
floral dagger
another course
golden night
you think people in this server were born smart?
floral dagger
im saying that 40 minutes seems so unrealistic
golden night
Most of us here suck at maths
floral dagger
im soo slow
golden night
we just like it
floral dagger
me too
golden night
im soo slow
no youre not
golden night
theres no thing as slow and fast, theres curiosity and theres boredom
if youre curious, you will be fast
floral dagger
true
golden night
but i genuinely suck
do you suck at your hobbie too?
floral dagger
my hobby is video games 😭
golden night
If you were slow you would be diagonsed and would be slow in life
floral dagger
you cant possibly suck at playing games
golden night
my hobby is video games 😭
nothing bad
golden night
you cant possibly suck at playing games
you can..
FPS shooters
but anyways
focus here
learn the concept
floral dagger
okk so
golden night
dont stop because you are stuck at a level man
golden night
thing is that ive watched 2 videos
i see
robust bloom
they explain so well
try khanacademy
golden night
well, good luck with that
floral dagger
but out of curiosity, is the quadratic formula and factorising the only way to solve rational equations?
golden night
stop being an ass man
golden night
but out of curiosity, is the quadratic formula and factorising the only way to s...
to solve a polynomial ?
you can also guess work
robust bloom
but out of curiosity, is the quadratic formula and factorising the only way to s...
no
golden night
ik it dosent sound mathematical
floral dagger
guess work?
golden night
guess work?
try +-1,+-2 and 3
floral dagger
noo i hate that so much
golden night
A LOT of times, they work
golden night
noo i hate that so much
then learn the method
floral dagger
plus you can't really do that in the final exam
robust bloom
A LOT of times, they work
for cubic i do that
floral dagger
you require like reasoning and proof when solving equations or else u get 0 or minus points
floral dagger
then learn the method
so you suggest I learn factorising
golden night
so you suggest I learn factorising
yes?
robust bloom
so you suggest I learn factorising
Yes
golden night
You should know it
floral dagger
I know idk how ive gotten this far
golden night
like you arent passing calc without it
floral dagger
😭
golden night
I know idk how ive gotten this far
stop being so dependent man.
floral dagger
on?
robust bloom
so you suggest I learn factorising
Have some affirmations + positive vibes while watching videos
floral dagger
cringe
golden night
Do you seriously like asking for help to factor online on reddit and discord
floral dagger
I do not
floral dagger
i dont ask for help that often
golden night
learn it
golden night
i dont ask for help that often
thats not the point
floral dagger
its not enjoyable yeah I get where ur coming from
golden night
you CAN ask for help and you should, but it should be something which is genuinenly good
floral dagger
I lack resources I think
robust bloom
You got access to internet
floral dagger
I already mentioned watching yt, access is clearly not the issue
robust bloom
Then learn how to learn
floral dagger
I just don't have digestable resources at hand
robust bloom
That really works ^
floral dagger
we only have one textbook for this course
factorising was taught so long ago, there isn't a specific chapter or area where it explains
but yeah so factorising and then?
is there a formula I have to apply after that? @golden night
robust bloom
is there a formula I have to apply after that? <@645275087313960970>
no logic
floral dagger
im asking for steps
robust bloom
Just select the zeroes for which numerator is zero but denominator isnt
golden night
is there a formula I have to apply after that? <@645275087313960970>
well then you need to know why denominator = 0 is a problem
robust bloom
wait dont i have to find the domain first
why?!?!?!
floral dagger
thats the method we use here
robust bloom
Where you from?
robust bloom
you get 1-2 points from just pointing out the domain
ok you get the domain after factoring
not before it
floral dagger
aa okeei
summer robin
obtuse pebbleBOT
summer robin
I'd like a walkthrough on how to solve Sturm-Liouville problems please, im very lost
dont understand much of this
obtuse pebbleBOT
@summer robin Has your question been resolved?
obtuse pebbleBOT
@summer robin Has your question been resolved?
digital sedge
does anyone have tips or suggest on how to do this? ive applied every single bit of my knowledge it just doenst work
obtuse pebbleBOT
does anyone have tips or suggest on how to do this? ive applied every single bit...
@digital sedge Has your question been resolved?
median dome
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(y+zf(x))=f(y)+xf(z)\forall (x,y,z)\in\mathbb{R}^3$$
obtuse pebbleBOT
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(y+zf(x))=f(y)+xf(z)...
median dome
are there any errors in this solution?
median dome
yes
spice citrus
P(x, 0, 1): f(f(x)) = x * f(1), right?
median dome
Ohhh yeah
Wait how would I solve that then
Oh P(1,0,x) gives me f(x)=xf(1)
Not P(x,0,1)
spice citrus
f(zf(1)) = zf(1), there should be a f(1) in the inside of the function as well
spice citrus
You can prove f(1) = 1 though
median dome
f(zf(1)) = zf(1), there should be a f(1) in the inside of the function as well
f(zf(1))=f(z) no?
median dome
You can prove f(1) = 1 though
Yes exactly
spice citrus
f(zf(1))=f(z) no?
oh, yes
spice citrus
Does it?
spice citrus
P(x, 0, 1): f(f(x)) = x * f(1), right?
I got it by plugging in x = 1 here, that might actually be the same, not sure
median dome
Yes same thing