or maybe dont idk
#help-23
brave wolf
youll probably want to express r^5 + 5q + s^5 in some simpler way
or just solve for q
thats fast enough
safe radishBOT
safe radishBOT
Available help channel!
flat frigateBOT
TheArrasGunner
harsh condor
just rearrange the terms
glacial cairn
That needs simplification
harsh condor
continously square both sides
i mean square root
yes but then you can further simplify
give it a try and try to notice a pattern or trend
a lot of times in contest math the best way to tackle a problem is just trying basic algebraic manipulation techniques
flat frigateBOT
TheArrasGunner
harsh condor
and make sure you do it fast because if it doesn't work you can try another method
yeah
simplify the terms inside
i can't do everything for you
well look inside the radicals
idk u tell me
while that is true, sometimes expanding things out help us see the bigger picture
glacial cairn
I don't see where this is going
flat frigateBOT
TheArrasGunner
harsh condor
ok good
now what do you notice
well
i suppose yes
when you come across solutions midway and notice dead ends, it's okay to backtrack a bit
i say lets try squaring both sides
and then compute the constants
just to make it a little cleaner as you like
flat frigateBOT
TheArrasGunner
harsh condor
ok good
now we could have done this step earlier, but can you evalue 60^2 and 165^2?
what do you think we should do
there is no wrong answer
yes good
i agree
however i would take it a step further
and compute every constant
that is the 20, 60, 165 on both sides,
this would help us simplify down further to then isolate the variable
flat frigateBOT
TheArrasGunner
harsh condor
uh
ok i think i misinterpreted what you said
i was thinking just evaluate the constants first
not expanding the 180n + 90
i really think you're overthinking this question
after you compute the constants, simply rearrange the terms to isolate for n
glacial cairn
And I think you're leading him into dead ends...
flat frigateBOT
TheArrasGunner
harsh condor
ok now you can evaluate the left side
and then subtract 27225
yes correct
now what do you think we should do after that
ok we can try that
but if it were me, i would factor the expression
because notice 180n + 90 = 180 (n + 1/2)
flat frigateBOT
TheArrasGunner
harsh condor
so (180n + 90)^2 = 180^2 (n + 1/2)^2 = 32400 ( n + 1/2)^2
flat frigateBOT
TheArrasGunner
harsh condor
ok i see what you did here
if ur doing it this way
you probably will be using the quadratic formula for this
because we have the form ax^2 + bx + c = 0 yes?
not directly
because this is not just n^2 but rather n(n+1)
flat frigateBOT
TheArrasGunner
harsh condor
but since n(n+1) is approximately n^2 for large n then we can estimate
i suppose that may work
but if i am to be honest
had you factored it as i recommended earlier, you could have ended up with an expression of 379.715 < (n + 1/2)^2
in which you can easily solve for n > 18.985 or around n > 19
yes do double check your work
good job
no problemo 
.close
safe radishBOT
safe radishBOT
Available help channel!
winged fulcrum
my sci. cal says its 400 but the answer is different from the internet
light shoal
using e as a variable
light shoal
my sci. cal says its 400 but the answer is different from the internet
show what the internet said
400 is obviously correct
winged fulcrum
i searched up Slovin's Formula on the internet, and its different compared to what my teacher showed us on our study filew
I don't know which one to trust
i input 1000/1000(0.05)^2 on my sci. cal and the answer is 400 yeah
light shoal
the screenshot seems to be solving a different problem
that said, are you sure this formula is right?
what's the point of not cancelling the N's?
and just saying n = 1/e^2
winged fulcrum
that said, are you sure this formula is right?
i dont know thats what my taecher sent on the pdf and i searched it up online
and this one popped up
light shoal
seems like your teacher's is wrong then
winged fulcrum
that said, are you sure this formula is right?
yea, seems like shes wrong, ill stick with this one on our quiz. right after were done, ill talk to her aboyt it cuz i honesstly dont kniw
appreciate the help bro
frigid bramble
<@&286206848099549185> Explain its solution in simplest way possible.
earnest nacelle
!15m please.
safe radishBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
round egret
<@&286206848099549185> Explain its solution in simplest way possible.
I think you can rewrite the sum in terms of cot as well because there is some synmetry and complementary angles stuff going on if you use the identity: $$\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$$
flat frigateBOT
MxRgD
frigid bramble
I think you can rewrite the sum in terms of cot as well because there is some sy...
Is it a standard way to approach this type of problem
round egret
Is it a standard way to approach this type of problem
I'm not sure, but it helps simplify things.
I think this problem requires you to use De Moivre's Theorem, specfically expressing tan(nθ) as tan(θ)
frigid bramble
I have done it but I cant see any way to move forward
Like its a finite summation and there is no apparent telescopic summation type thing
Did you mean this ,but now what?
round egret
Did you mean this ,but now what?
,rccw
flat frigateBOT
round egret
yeah that looks right
frigid bramble
Is this any known format of summation with some known value because I don't remember anything in trig except the basics
round egret
Is this any known format of summation with some known value because I don't reme...
no not really
have you been taught de movire's theorem?
because I think it might just be that
frigid bramble
have you been taught de movire's theorem?
Yes but its algebraically heavy for me
round egret
Yes but its algebraically heavy for me
yeah that's a problem
frigid bramble
because I think it might just be that
Show me how?
round egret
it's pretty messy
toxic stratus
use the identity [ \sec^2(x) + \csc^2(x) = 4\csc^2(2x) ]
flat frigateBOT
frigid bramble
Is this something useful?
frigid bramble
But where?
toxic stratus
add 7 to the sum to change it to sec^2(kpi/16)
then change the second half of it to csc^2
frigid bramble
then change the second half of it to csc^2
Oh let me try
toxic stratus
repeated application of this should then simplify it down to only 3 terms
frigid bramble
repeated application of this should then simplify it down to only 3 terms
Let me try one thing at a time ,I forgot many things since I learnt it 2 years earlier
How to proceed further?
toxic stratus
no you need to only turn half of them into csc
you should get sec^2(pi/16) + csc^2(pi/16) + sec^2(2pi/16) + csc^2(2pi/16) + sec^2(3pi/16) + csc^2(3pi/16) + sec^2(4pi/16)
frigid bramble
you should get sec^2(pi/16) + csc^2(pi/16) + sec^2(2pi/16) + csc^2(2pi/16) + sec...
Yes
toxic stratus
the first 6 terms give you 4csc^2(pi/8) + 4csc^2(2pi/8) + 4csc^2(3pi/8)
and then you do it again on 4csc^2(pi/8) + 4sec^2(pi/8)
frigid bramble
frigid bramble
Like this?
toxic stratus
no i dont see how you have that
frigid bramble
This euqals 2S + 14
frigid bramble
How to proceed further?
I did it from here
toxic stratus
right
you doubled it
okay i see now
not sure why there's an 8 in front
you combined the last half i guess
okay
well you have to do the thing again
frigid bramble
you combined the last half i guess
Yes
But now there is not symmetry about middle term,how?
toxic stratus
well when you combined in the second half it should've been sec^2
something's wrong here
flat frigateBOT
toxic stratus
csc^2(pi/8) + csc^2(3pi/8) + csc^2(pi/4)
csc^2(pi/4) is just 2
the first two are csc^2(pi/8) + sec^2(pi/8)
frigid bramble
But do we have to evaluate each term?
toxic stratus
and you use the identity again
frigid bramble
and you use the identity again
Which ?
toxic stratus
this
frigid bramble
frigid bramble
Let me check
frigid bramble
csc^2(pi/8) + csc^2(3pi/8) = csc^2(pi/8) + sec^2(pi/8)
But how they are not complementary angles
toxic stratus
,w pi/2 - pi/8
flat frigateBOT
toxic stratus
understandable
safe radishBOT
Available help channel!
versed wave
safe radishBOT
versed wave
for the first question i can calculate that f(2) = p^2, f(3) =f(4) = (p^2)(1- p), but i'm unable to express f(x) as a recurrence relation
solar hazel
i do not like calling the pmf p here 🥀
versed wave
ok let's rename it to pmf(x)
solar hazel
f would have been fine
versed wave
for the first question i can calculate that f(2) = p^2, f(3) =f(4) = (p^2)(1- p)...
for reference to calculate f(5), i would first assume the last three start-up results are .....XOO (O indicates success), then i just need to multiply it with g(2), where g is the probability for x startup results to not have 2 consecutive successes
but i'm stumped with determining g
my current g is [
(1+p)(1-p)\left(\frac{1-p+\sqrt{(1-p)(1+3p)}}{2}\right)^{x-2}+(1-p)(1+p-p^2)\left(\frac{1-p-\sqrt{(1-p)(1+3p)}}{2}\right)^{x-2}
]
flat frigateBOT
versed wave
but i'm not confident and god damn that's long
peak estuary
I would maybe try counting strings of length n-2 made from the building blocks X and OX
but I'm not sure whether thats nice either
versed wave
can't O be the last character?
safe radishBOT
@versed wave Has your question been resolved?
peak estuary
(any valid string) = (any of my string)+OO
versed wave
for reference to calculate f(5), i would first assume the last three start-up re...
here i could instead calculate f(5) as (1-p)p^2*(1 - p(2)), since for p(x = 3) and higher x, that means the trial process requires higher number of trials
that's synonymous with calculating g(2), since it doesn't have consecutive Os either (otherwise it would have ended no later than 2 trials)
safe radishBOT
@versed wave Has your question been resolved?
viscid quiver
anyone know how to solve this
safe radishBOT
anyone know how to solve this
crisp mortar
anyone know how to solve this
what is struggling you
hard crest
anyone know how to solve this
well so like
can you write out x1, x2, x3, etc?
viscid quiver
idk what x:xm even is tbh
hard crest
its kinda weird notation lowkey
crisp mortar
i think it should be written like
hard crest
but the idea is that you take positive integer m and calculate 2m + 3
crisp mortar
{x|x_m=2m+3}
hard crest
and that's xm
hard crest
{x|x_m=2m+3}
i'd probably write it as A = {2m + 3 | m \in N}
crisp mortar
{x|x_m=2m+3}
it is ranging the x
viscid quiver
im so sorry to say this but im not really a notation guy
i have no idea how to use it
or what it means
most of the time
hard crest
but anyway so finding $x_1$ is just plugging 1 in for $m$ in the form $x_m = 2m + 3$
flat frigateBOT
schrödinger's kitten
crisp mortar
i'd probably write it as A = {2m + 3 | m \in N}
it has many variations, colons, bars, etc.
viscid quiver
**schrödinger's kitten**
oh so its just the same as like tn = something n-1 bla bla bla
but like writen differently
viscid quiver
how do i find a then?
hard crest
well so like
can you write out x1, x2, x3, etc?
.
write out the values for x1, x2, x3, x4
then figure out what goes in this box:
x3 = x2 + _______
viscid quiver
2
safe radishBOT
Available help channel!
dreamy marlin
I need help with Quadratic Equations I have a huge test tommorow and I need some help on the topic
safe radishBOT
I need help with Quadratic Equations I have a huge test tommorow and I need some...
frozen marlin
I need help with Quadratic Equations I have a huge test tommorow and I need some...
whatcha need
plucky elk
I need help with Quadratic Equations I have a huge test tommorow and I need some...
do one problem at a time and show your work here
plucky elk
there are two things that you'll need to know. for any numbers a, and b, if a * b = 0, then one of a is 0 or b is 0
the other is substituting 10 for x so x - 27 = 10 - 27 = -17 and x - 10 = ?
the first has a fancy name called "zero product property"
try using those two facts to answer the question. or ask if you don't understand what i've explained so far
ivory fable
You could also take the 60 to the other side, develop the product, and obtain the root(s) using the quadratic formula, but that's the long way.
safe radishBOT
safe radishBOT
Available help channel!
floral raven
why does the sequence a^n (mod m) always become periodic?
safe radishBOT
why does the sequence a^n (mod m) always become periodic?
floral raven
it's not a general homework problem, but i've been solving the "find the last digit of a^n" by looking for cyclical patterns, but I don't think this solution is rigorous enough
signal crane
there are only a finite number of remainders modulo m
mortal sandal
as soon as you hit the same value twice you know you have to start repeating
because you're multiplying by a each time
signal crane
yes
safe radishBOT
floral raven
hmm in that case where does it become periodic? sometimes i've had at n=0 and so on and sometimes it is definitely not at n=0
and in that case how does one know that you've entered a cyclical pattern, provided that the cycle is very large?
ugh
safe radishBOT
Available help channel!
north orbit
why do we do pythag to find the resultant force at A?
why not just use R?
i thought that was the resultant force
winter whale
A has 2 reactions
R doesn't inherently mean resultant force
You can call the vertical reaction Ay and horizontal Ax
worldly lantern
why not just use R?
R here is the normal reaction by ground on point A and F is due to friction
since force is vector, it is vectorially added
safe radishBOT
safe radishBOT
Available help channel!
vast socket
https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/dbc17cbfdac61daaff75056992d0af9a_section15.pdf
I am having trouble following through and still stuck on 2nd page. Anybody that coud assist walking through this lecture?
safe radishBOT
https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/dbc17cb...
@vast socket Has your question been resolved?
austere goblet
!nopdfs as a reminder, OP. it might be more helpful for other helpers to step in if you screenshot the specific page and explain what you do and don't understand!
safe radishBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
vast socket
i understood everyghing till the derivative part. but the the folowing is wehre i do nto understand how the beta estimator is deduced. The formulas have shortcuts in between so I am trying to fill in the gap
safe radishBOT
@vast socket Has your question been resolved?
flat frigateBOT
pola_touche
vast socket
yes and another thing
warm lodge
k, i can try to explain it
vast socket
And also i neither understand the derivative respect to B_i why it would not apply to B_j, technically B_i and B_j are from the same vector B.
i.e. if i=j, then why derivative with respect to B_i is not applied on B_j as well?
warm lodge
we want maximize the likelihood L as a function of the parameters of the model B_0 to B_p meaning L(beta) is a function from R^p+1 to R if you have p variables. To optimize such a function you set the gradient of L equal to 0 i.e. (B_0, … B_p) = (0, … ,0)
warm lodge
And also i neither understand the derivative respect to B_i why it would not app...
Does that make sense?
Maybe having only 2 B_i B_0 and B_1 will make it simpler
And then from those equations coming from this minimization criterion gradL=0 you can get the matrix equation you want to understand
warm lodge
Maybe having only 2 B_i B_0 and B_1 will make it simpler
here to optimize i took the natural log of L to make the computation easier. Taking the log won’t change the optimal (B_0,B_1)
warm lodge
**pola\_touche**
typo there X^tY=X^tX\beta
safe radishBOT
@vast socket Has your question been resolved?
vast socket
Thanks! I didn't quite get the 1st picture, but the 2nd one made sense.
safe radishBOT
Thanks! I didn't quite get the 1st picture, but the 2nd one made sense.
@vast socket Has your question been resolved?
west hedge
Foooo
vague phoenix
Ask maths related question only
solar hazel
good question
lone void
what is this bro
winged flare
ive seen this before but apparently these are called nodal/antinodal lines
https://www.youtube.com/watch?v=CZ4Z8H7CgBQ&t=5s
due to interference of the ripples
im not sure how the math works out for them to be lines though
vague phoenix
good question
Ohh i thought blud was asking why its all chinese my bad😭😭😭
safe radishBOT
@west hedge Has your question been resolved?
solar hazel
pcmg only asks good questions
vague phoenix
safe radishBOT
safe radishBOT
Available help channel!
stiff sundial
Can I get some help with bool algebra pls
safe radishBOT
Can I get some help with bool algebra pls
flat frigateBOT
@shut hound
shut hound
@stiff sundial this is how
iron tapir
<@1379097960314441788> this is how
wut r u showin me
safe radishBOT
@stiff sundial Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
loud ember
For triangle with length of a,b,c
What kind of triangle is it when
pow(a,2) * b + a * pow(b,2) + pow(b,2) * c - b * pow(c,2) - pow(c,2) * a - c * pow(a,2) == 0 is true?
safe radishBOT
For triangle with length of a,b,c
What kind of triangle is it when
pow(a,2) * b ...
primal gazelle
For triangle with length of a,b,c
What kind of triangle is it when
pow(a,2) * b ...
what have you tried?
lean otter
For triangle with length of a,b,c
What kind of triangle is it when
pow(a,2) * b ...
is pow just power, pow(a,2) = a^2?
round egret
i'm pretty sure it is
loud ember
what have you tried?
I have no idea because this is too complex
loud ember
is pow just power, pow(a,2) = a^2?
Yes
manic patrol
$ba^2 + ab^2 + cb^2 - bc^2 - ac^2 - ca^2 = 0$?
flat frigateBOT
trilunar arithmetic (Columbina)
round egret
one thing you can do is to group them
manic patrol
I don't suggest writing powers in this Python-like manner, can be rather confusing to non-programmers at first.
round egret
one thing you can do is to group them
into $(a^2b - ca^2) + (ab^2 -c^2a) + (b^2c -bc^2) = 0$
flat frigateBOT
MxRgD
round egret
then factor out the common terms from each group
loud ember
There is no greatest common divisior other than 1 on all terms
round egret
huh?
manic patrol
you don't need a, b, and c to have a non-trivial GCD to do what MxR told you to try.
round egret
What i meant, is looking at each individual bracket and factoring out there common terms. So for example $(a^2b -ca^2) = a^2(b-c)$
flat frigateBOT
MxRgD
loud ember
Does it group like this?```
a * a * ( b - c) + a * (b * b - c * c) + b * c * (b - c)
yeah that's right
now notice the $(b^2-c^2)$ part
flat frigateBOT
MxRgD
round egret
does it look familiar to you?
loud ember
It gets simplificated to (b - c) * (b + c) as I remember
flat frigateBOT
MxRgD
loud ember
(b - c) * (a² + a * (b + c) + b * c) == 0 still does not tell you what kind of triangle is it
round egret
(b - c) * (a² + a * (b + c) + b * c) == 0 still does not tell you what kind of t...
not right no, but we're close
so now we have $(a^2 + a(b+c) + bc) = 0$
flat frigateBOT
MxRgD
round egret
we can get $(a^2 + ab + ac + bc)$ = $(a^2 + a(b+c) + bc)$
flat frigateBOT
MxRgD
round egret
now try to factor it
round egret
into $(a^2b - ca^2) + (ab^2 -c^2a) + (b^2c -bc^2) = 0$
it's a similiar idea we did to to the grouping here
round egret
Does ((a + b) * a + (a + c) * b) work?
that wouldn't work because you'd get a^2 + ba + ba + bc
which isn't our orginal
but you're close
it's just the b(a+c) that is wrong
loud ember
(a + b + c) * a + bc looks better for some reason although it is not symmetrical
round egret
(a + b + c) * a + bc looks better for some reason although it is not symmetrical
Right yeah, that's wrong and not the one I had in mind
we want to get ac + bc
and what common factor do they share?
loud ember
Is it
a * (a + b) + c * (a + b)
``` Then?yep
and how can you factor that out even further?
loud ember
(a + c) * (a + b)
flat frigateBOT
MxRgD
round egret
which is very helpful to find out what kind of triangle it is
I'll let you try figure it out
loud ember
I think it isn't even a triangle because the length of a,b,c has to be zero
round egret
I think it isn't even a triangle because the length of a,b,c has to be zero
Why would a,b,c have to be 0?
loud ember
Since (a + b) and (a + c) is never zero when all lengths are 0
All length has to be zero
(b - c) would be zero when b == c
round egret
okay, but for the product to be zero, at least one of the factors must be zero. So this doesn't hold
round egret
(b - c) would be zero when b == c
since (b-c) would equal 0
the others (a+b) and (a+c) don't necessarily have to be 0
you are correct that the both of them can't be 0, however you're reasoning is incorrect
loud ember
Is the useful information just b == c and that is only one information?
round egret
Is the useful information just b == c and that is only one information?
there's two more
that you can deduce from (a+b) and (a + c)
loud ember
There is no negative length so it never equals to 0
round egret
right
so (a+b) = 0 and (a+c) = 0 can't be 0 as a, b, c > 0
and since we know b = c from (b-c) = 0
then a can not equal b or c
so the triangle must be?
loud ember
It must be the isosceles triangle then
loud ember
Hold on. a can be equal to b or c because
a = 1
b = 1
c = 1
And the expression is true
round egret
Hold on. a can be equal to b or c because
a = 1
b = 1
c = 1
And the expression ...
does either of (a+c) = 0 and (a+b) = 0, hold true however?
if they were all equal to 1
loud ember
(b - c) is 0 so it is still true
round egret
it satisfies that yes
oh wait i get what you mean
right so yeah this is valid
but an equilateral is a special type of isoceles
we don't know what a can be
round egret
~~then a can not equal b or c~~
so yeah this is wrong actually
a could be equal to b or c but we don't know
we don't have any info on what a should be
except for a > 0
if the equation required a = b =c to be true, then it would be strictly equilateral.
safe radishBOT
@loud ember Has your question been resolved?
pure acorn
here , cant we take 4 as 2^2 and multiple 2^2 wiht root 2 which will give us 2*2
pure acorn
<@&286206848099549185> pls its urgent
low echo
Hey
civic musk
what is ur question
round egret
here , cant we take 4 as 2^2 and multiple 2^2 wiht root 2 which will give us 2*2
you can't do that
pure acorn
why?
low echo
4 is √2•2•2•2 yeah
waxen mason
Parentheses, exponents, multiplication, division, addition, and subtraction
Well...
No.
waxen mason
4 is √2•2•2•2 yeah
Because multiplication is done first
2^3 = 8
sqrt(2) = 1.4142...
So therefore, 8*sqrt(2)
low echo
Mhm
waxen mason
Therefore, not equal to four
low echo
What?
gritty tartan
or like 2^(5/2)
low echo
4 is √2•2•2•2 yeah
Those are 4 2s
waxen mason
or like 2^(5/2)
That's equivalent to sqrt(2^5)
woven fjord
What's the question lmao
low echo
√2•2•2•2 and another √2 seperately
woven fjord
What are u tryna solve
waxen mason
2(2^2*sqrt(2)) = 2(4*sqrt(2))
pure acorn
just add all the exponents
??
waxen mason
But this also means this is equivalent to 2*4*sqrt(2) = 8*sqrt(2)
waxen mason
There's no arguing
woven fjord
after this it becomes 2*2
No it doesnt 2[2^{2+1/2}] = 2[2^{5/2}] = 2^{1+5/2} do it urself do u mean this
?
pure acorn
No it doesnt 2[2^{2+1/2}] = 2[2^{5/2}] = 2^{1+5/2} do it urself do u mean this
no i mean 2[2^2(2^1/2)] this
how did 3/2 come ther lol
woven fjord
no i mean 2[2^2(2^1/2)] this
I simplified using exponent laws
woven fjord
how did 3/2 come ther lol
I meant 5/2
pure acorn
uhh im not getting it
woven fjord
2^a * 2^b = 2^{a+b}
pure acorn
yes thats true
woven fjord
Thats what i did
ebon hatch
$4=2^2$ and $sqrt(2)=2^{1/2}$. Multiplying together gives $2^2*2^{1/2}$ but you add the exponents when you multiply, so you don't get a nice whole number from that.
woven fjord
2^2 * 2^(1/2) = 2^{2+1/2}
ebon hatch
sqrt for square root. I don't remember how to make the symbol appear..
lean otter
$a^b \cdot a^c = a^{b+c}$ and not $a^{bc}$
flat frigateBOT
hail
woven fjord
2^2 is equal to (2)^2 right
Yes ofc
safe radishBOT
safe radishBOT
Available help channel!
timid mulch
If the square root of any number is always positive, then will the nth root of any number, where n is an even number, also always be positive?
safe radishBOT
If the square root of any number is always positive, then will the nth root of a...
summer coral
can you find any case where this would turn out to be false?
timid mulch
I don't think so
covert yoke
By convention we take the positive root for even powers.
timid mulch
Okay
covert yoke
Are you familiar with complex numbers?
timid mulch
Yes I am
covert yoke
So if you have z^n = 1, the solutions to this form what we call the roots of unity
timid mulch
Hmm
covert yoke
So if we have z = r e^{it}
timid mulch
My knowledge about complex numbers is superficial sorry 😐
covert yoke
oh, it's ok
timid mulch
Thanks
covert yoke
anyway, let me see if I can't make a visualizer one moment
I just found one, turns out it's probably a common want.
timid mulch
Oh
covert yoke
So this will show you the locations of the roots of unity
timid mulch
I'll check it out
covert yoke
they just are evenly spaced points on a circle.
So z = r e^{it} is just another way of writing a complex number, z = a + ib or z = r e^{it}
The r e^{it} way is where the complex number has a magnitude and a rotation away from the positive real axis
timid mulch
I notice that for n it gives an n sided regular polygon
covert yoke
this is accurate!
timid mulch
The r e^{it} way is where the complex number has a magnitude and a rotation away...
Ohh I get it now
covert yoke
so if we try to take the nth root of r e^{it} what we get is:
$\sqrt[n]{r e^{it}} = \sqrt[n]{r} \sqrt[n]{e^{it}} = \sqrt[n]{r} e^{it/n}$
flat frigateBOT
OmnipotentEntity
covert yoke
which because r is real, we have the nth root of r which we take to be the positive root,
timid mulch
It makes sense now
covert yoke
and then e^{it/n}, which is going to be some complex number with magnitude 1
here's the confusing part though
because e^{it} repeats every 2pi, we also need to consider the other roots
$\sqrt[n]{r e^{it}} = \sqrt[n]{r} e^{it/n + 2\pi k/n}$
where $k$ is some integer. We can prove that if $n$ is an integer then we only need to consider the values of k from 0 to n-1.
timid mulch
I'm a bit overwhelmed actually 😅
covert yoke
and these will give us different roots based on the roots of unity
just imagine that we rotate our polygon by t/n radians counterclockwise
in the roots of unity
timid mulch
Hm
covert yoke
by convention we take the root of unity closest to the real axis but above it.
so in other words, once you get out of high school (assuming that you're in high school) the cube root of $-8$ is not going to be $-2$, that's not the principal root, it's going to be $2 e^{\pi i / 3}$
flat frigateBOT
OmnipotentEntity
timid mulch
I really appreciate your efforts 😊
timid mulch
Yeah I'm all ears
covert yoke
dang, I can't find it anymore.
anyway, it was an animation where they tried to show in a 3d plot how the "negative tail" of a nth root worked.
tender orchid
The r e^{it} way is where the complex number has a magnitude and a rotation away...
Is this the multivalued func?
covert yoke
r e^{it} only has a single value.
you might be thinking of taking the logarithm of this
or indeed nth roots
tender orchid
Im talking abt the euler's identity of complex numbers
covert yoke
there are multiple representations of the same value, for instance if z = r e^{it} then also z = r e^{i(t + 2pi)}
tender orchid
The cosx+isinx
covert yoke
yes, that is another way to write this, z = r (cos t + i sin t)
tender orchid
Ah ok thx
covert yoke
coolio
timid mulch
.close
safe radishBOT
safe radishBOT
Available help channel!
safe radishBOT
north orbit
.close
safe radishBOT
safe radishBOT
Available help channel!
clear torrent
Hello guys. I've been stuck with this problem right here
safe radishBOT
Hello guys. I've been stuck with this problem right here
clear torrent
I am supposed to calculate a in b)
But i just can't seem to understand what to do
Could anyone help me?
simple gazelle
what have you tried so far
clear torrent
well i figured out like
It's hard to say in english
Wait
I already figured out that the side whose part is a = 15 centimeters.
I've tried many other things that have led me nowhere.
silk tendon
Its not 15
simple gazelle
yea
silk tendon
The 90 angle isnt there
simple gazelle
the side lengths should be smaller than the hypotenuse
simple gazelle
use square roots man
clear torrent
Oh yeah i made a mistake
silk tendon
Because $ \sqrt{12^2-9^2} $
simple gazelle
this entire side is $\sqrt{63}$
flat frigateBOT
MarcoMa210
clear torrent
c²-a²
silk tendon
Xd
simple gazelle
find the blue side and you're almost there
clear torrent
But
Like
The part where it's 12cm
How do i split correctly if you understand what i mean
silk tendon
You let one be x other be 12-x
simple gazelle
you don't need to split it to find a
simple gazelle
You let one be x other be 12-x
yea it's this
clear torrent
clear torrent
yea it's this
Oh wait
silk tendon
Yea let that be x
clear torrent
Let me try
clear torrent
160-x²?
silk tendon
What??????
elfin comet
And what do i make of that?
u squaring the brackets wrong
(12-x)² = (12-x)(12-x)
u have to expand that
and also why did u guys introduce x
nimble helm
if you know what the red line is here, what else do you need to know to calculcate a?
elfin comet
use pythagoras to find blue
u mean in terms of a?
just for context the orange + yellow = 12
and the angle the blue line makes with the red line at their intersection is not 90 degrees
lean otter
u mean in terms of a?
oh mb i took orange one to be 12
elfin comet
i have an answer i think i over complicated it way too much lmao
lean otter
indeed there seems no need for pythagoras
lean otter
if you look, side with 9 and 4 are altitutdes of the triangle
equate the areas and we will find a
elfin comet
I dont see this with the 9
im sure theres a trick for this question but i went the long way
and then i added the pink and dark green which should be 12
and then solved for a
but im very sure there is an easy way to do this but im just too stoned
lean otter
I dont see this with the 9
9 is the altitude and side perpendicular to it is the base, a
lean otter
😭
lean otter
solve until a comes to be 16/3
elfin comet
yea
clear torrent
Should i just quit atp 😭
elfin comet
Should i just quit atp 😭
no no i think i see it
the trick
theres similar triangles in that picture
elfin comet
triangles are similar
clear torrent
I was supposed to get ready for a test and ended up trying to figure this out the whole time
So like 2 hours
clear torrent
Yeah i moved on after an hour but it couldn't leave my head lol
elfin comet
but yea u see the small triangle
with a and 4
sides a and 4
that triangle is similar to the big triangle with side 9 and 12
they both have right angles and both share the same angle
these 2 triangles are similar
elfin comet
yea 16/3
but u wanna know how to get it tho
do u see how they are similar?
they have 2 of the same angles
they both have right angles and they share an angle on the top right of the triangle
clear torrent
I just 4:9
elfin comet
do u see it
clear torrent
So the big triangle is 2.25 times bigger
And then 12:2.25
But is it actually it
Dang
Thank you bro
elfin comet
ok nvm
i see it now
u can work out the area of the big triangle 2 different ways
so first u can just do 0.5 x 9 x root(63)
u could also do 0.5 x 12 x h
where height is that dotted green line
and when u do that u can equate them to get the vertical height of the big triangle and u get (9root7 /4) for the height
elfin comet
and now from before the red triangle is similar to the green triangle
so what u can do now is using the vertical height from earlier 9root7/4 and the vertical height of the green triangle given by 4cm
find the ratio as they are both similar and this ratio is 4 / ( (9root7)/4)
this gives u a ratio of 16/(9root7)
and now because u know the side from earlier is root63
root 63 x (16/9root7) = a
and doing that gives u 16/3
@clear torrent
safe radishBOT
@clear torrent Has your question been resolved?
cosmic fjord
safe radishBOT
cosmic fjord
i dont understand how to form the original functions
like i know to let x represent number of passengers
and that i should set another variable to represent the change in smthn but like im not sure
im fine w finding the derivatives tho
vast sequoia
so you know that for each member you have to pay 30 dollars
and you start with a constant 225 dollars
vast sequoia
so you know that for each member you have to pay 30 dollars
and there is x members
cosmic fjord
right
vast sequoia
so how can you write that
cosmic fjord
225+30x
vast sequoia
nice
now you also know that for every empty seat you lose 5 dollars
so maybe you could introduce another variable
how would you account for the loss of the 5 dollars for each empty seat
cosmic fjord
22-5x?
cosmic fjord
oh
vast sequoia
yup
cosmic fjord
22-5y then?
vast sequoia
so y would be passengers missing right
for each number of missing you lose 5
vast sequoia
225+30x
add it to this equation
there is no 22 bte
what do you know about the sum of x and y
cosmic fjord
225x+30x-5y
vast sequoia
exactly
vast sequoia
wdym, i dont think you can isolate cuz you dont have it set equal to anything
cosmic fjord
oh right
cosmic fjord
oh right
vast sequoia
how do you calculate marginal profit and profit?
cosmic fjord
derivative of profit function
nd for revenue its derivative of profit minus cost i think
vast sequoia
derivative of profit function
hmm
if you find the derivative of a minear function you obviously get a constant, is there something jm missing?
cosmic fjord
and i see whrre yiu were going with this now, but ut would be -5(22-x)
yhis would be what y equals?
vast sequoia
mate
cosmic fjord
ur good lol jus go to #❓how-to-get-help
vast sequoia
derivative of profit function
is it normally a constant in these kinds of question?
vast sequoia
yeah
cosmic fjord
vast sequoia
also do you know how to the account for the 60?
cosmic fjord
yeah i cant figure tht out icl
like would it just be a restriction
or a seperate equation with a restriction
vast sequoia
i think a seperate equation, but then you have straight constants and no variables
cosmic fjord
mabye it would be a constant in the price equation?
but idk how thatd work
cuz itd make sense that the derivative of that would be 0 since its jus constants
nd the question dont rlly ask anything abt tht so mabye its jus like a red herring
vast sequoia
aight ive searched some stuff up about this topic. yiu know about the price function and cost function right?
vast sequoia
what year you in
cosmic fjord
im in g12
i mean it was on someone who had it last years test
i got that one right but im lowk confused abt the marginal revenue
or jus how the revenue is profit times x
lethal plank
lmao whoever wrote this was on something
such a badly made problem
also they typed sally instead of shelly 😂
cosmic fjord
also they typed sally instead of shelly 😂
my school bro
the teachers be makin mistakes on the lessons too so much
vast sequoia
<@&286206848099549185>
i mean i see how you coukd formulate it, but accounting for that full seat case is weird
cosmic fjord
😭 idk man
everything except this is easy tbh
in this unit at least
and finding points on a curve with the og function and a point off the graph is also pretty hard for me
but its way easier than these application questions
vast sequoia
in our case, p(x)=225 + 30x -5(22-x)
vast sequoia
then we multiply that by x and find the derivative
cosmic fjord
wait why multiply by x?
vast sequoia
revenue equals the quantity multiplied by p(quantity)
fiery fractal
uhh i assume p(x) is ur profit function that u came up with
cosmic fjord
yeah
cosmic fjord
accounting for the case when x=22
vast sequoia
this is discrete tbf, if it was continuous then we would been fugged
cosmic fjord
isnt it impossible for it to be continuous since there is a jump discontinuity?
fiery fractal
accounting for the case when x=22
ok so.. it costs company f(x) = 225 + 30x ...
they charge g(x) = x(60+5(22-x)) yea???
profit gives p(x) = g(x) - f(x)
which i am too lazy to simplify
i might be wrong english is kinda hard 🥀
vast sequoia
ok so.. it costs company f(x) = 225 + 30x ...
they charge g(x) = x(60+5(22-x)) ...
where did the x come from on the g(x)
cosmic fjord
yeah
fiery fractal
where did the x come from on the g(x)
they said they charge 60+extra charge if seat r unfilled to EACH PERSON so... the x came from EACH part indiviual cost * number of folks
cosmic fjord
i think it saus 60 charge if the seats are filled
fiery fractal
i think it saus 60 charge if the seats are filled
welp x is ur number of folks in bus buut x =22 ... it is 60*x
so it works
cosmic fjord
ohh yeah
fiery fractal
there is no edge case i can think of .. even if bus is empty .. that gives 0* (60+5*22) = 0 (as x = 0 )
fiery fractal
welcome ig ;p
safe radishBOT
safe radishBOT
Available help channel!
warped crystal
I’m bored of math. Please help
safe radishBOT
I’m bored of math. Please help
fallen spruce
give a specific question
vague phoenix
just show the question man
fallen spruce
otherwise go to the help-forum
warped crystal
Bruh fuck you
normal moss
Bruh fuck you
<@&268886789983436800>
normal moss
No
warped crystal
Hahaha
polar lynx
Okie dokie
normal moss
Somebody to talk sense into you though, perhaps
wild cape
.close
safe radishBOT
covert yoke
You need to ban me
Done
fallen spruce
that was more than I expected yeah thanks mod
polar lynx
My phone lagged and I did not get to ban 🙁
normal moss
normal moss
My phone lagged and I did not get to ban 🙁
excuses
safe radishBOT
Available help channel!
indigo siren
https://www.desmos.com/calculator/xb0w0qh5nx how do i make the visuals follow the point e^iz?
safe radishBOT
https://www.desmos.com/calculator/xb0w0qh5nx how do i make the visuals follow th...
quiet plume
As in you want the green and blue lines to follow the purple dot?
indigo siren
As in you want the green and blue lines to follow the purple dot?
yes
please explain the math behind it if possible
quiet plume
For you segment on the horizontal axis, you want to draw the line y=0 from x=0 to x=cos(z).
Your vertical (blue) segment should be x=cos(z) because that's the fixed x-coordinate, and then you can extend it up to draw up to the y coordinate sin(z)
So it's x=cos(z) from y=0 to y=sin(z)
This will make stuff work for the first quadrant. An issue that arises is that those inequalities are never fulfilled when the coordinates are negative.
An easy workaround in Desmos is to plot another segment of the same color but with the flipped inequality
indigo siren
For you segment on the horizontal axis, you want to draw the line y=0 from x=0 t...
how would i put y=0 from x=o to x=cos z?
quiet plume
Kind of like you did for your black line already. Only you had the inequality end at sin(z) rather than cos(z)
indigo siren
oohhh
safe radishBOT
@indigo siren Has your question been resolved?
indigo siren
Kind of like you did for your black line already. Only you had the inequality en...
now how would i get the radius to do the same?
nvm its adding tan z in fron of x
.close
safe radishBOT
safe radishBOT
Available help channel!
long shore
safe radishBOT
long shore
for this question i know its trying to get the expected value
i added to the graph the values fo g(X) for each x value which were
25, 169, and 361
but when i went to calculate the expected value I used x * g(X) and got a different answer to the solutions
the solutions did f(x) * g(X) to calculate the expected value of the distribution
but i don't understand why and the notes didn't explain it either
safe radishBOT
@long shore Has your question been resolved?
flat frigateBOT
Civil Service Pigeon
tardy mango
But $P(x=X)=f(x)$, so you're just summing $f(x)g(X)$
flat frigateBOT
Civil Service Pigeon
tardy mango
@long shore
long shore
ok lemme think for a sec
long shore
$$\mu_{g(x)}=\sum P(x=X)g(X)$$
in my lecture notes it says that its $\sum x \cdot P(x=X)$
flat frigateBOT
suds
long shore
But $P(x=X)=f(x)$, so you're just summing $f(x)g(X)$
so does g(X) take the x spot?
polar pulsar
in my lecture notes it says that its $\sum x \cdot P(x=X)$
yes, g(X) takes the spot of X, f(x) is the probability
long shore
oh okay
polar pulsar
im not familiar with this notation but from what you're saying you want the expected value of the function g(x)? Which would be what i said before
long shore
yes
polar pulsar
then yes
long shore
E(g(X))
what would be the word for x then? like 0, 1, 2, 3 etc
the term
because g(X) turns into that basically?
OH
polar pulsar
because g(X) turns into that basically?
Expected value is basically
a way to know what is the average amount of something you get
to know that
long shore
values of the distribution, and g(X) is just modifying the function that gives you the x values
polar pulsar
you need to multiply the amplitude(value of x) with the probability(or frequency) that x gets that value
then sum them up
polar pulsar
values of the distribution, and g(X) is just modifying the function that gives y...
ye
long shore
so if X is say the probability of pulling a popsicle stick of a certain colour from a tub, g(X) is the transformation onto the amount of popsicle sticks kind of thing?
which is why the probability stays the same
polar pulsar
so if X is say the probability of pulling a popsicle stick of a certain colour f...
X is not a probab
it's a variable
polar pulsar
so g(X) adds more colours?
yea thats one example
g(x) would be more like asking, "how many different coloured sticks can i break this into?"
imo that'd be more accurate since g(x) isnt changing the colours themselves, it's just asking another question that is related to the colours
polar pulsar
because the probabilities stay the same
yup, but the number of sticks you can break them into are different depending on the colour(in my example)
polar pulsar
so the question of "How many coloured sticks do i get after breaking on average" gets changed(as in the probab is different to the mean value of X)
polar pulsar
is there like a number generator or dice example for this
idk what ya mean
safe radishBOT
long shore
.reopen
safe radishBOT
✅ Original question: #help-23 message
long shore
yup, but the number of sticks you can break them into are different depending on...
but if f(x) is the probability of picking the coloured stick randomly, then wouldn't the number of breakages be unrelated to the probability
polar pulsar
but if f(x) is the probability of picking the coloured stick randomly, then woul...
f(x) is more like the probability of getting a tub with coloured sticks
long shore
idk what ya mean
like if x is rolling a dice or something, and g(X) adds a certain amount of sides to the dice
are the coloured sticks mixed with all the colours or each tub has a different colour
polar pulsar
like if x is rolling a dice or something, and g(X) adds a certain amount of side...
g(x) in that case is like multiplying a side of the dice
with a larger number
long shore
g(x) in that case is like multiplying a side of the dice
ok so it like multiplies the number shown
polar pulsar
changing number of sides would affect f(x)
long shore
okay okay
dull sequoia
think about it like "if i rolled a 1, instead feed this 1 into my function g( • ), then say i rolled a g(1)"
long shore
so a real life example would be like population of a species and the discrete variables are like coat types or something, g(X) could be like the increase of the population in general
polar pulsar
so a real life example would be like population of a species and the discrete va...
didnt get that
long shore
think about it like "if i rolled a 1, instead feed this 1 into my function g( • ...
i get that part like transformations of equations, but im trying to relate it to how you get the expected value
polar pulsar
i get that part like transformations of equations, but im trying to relate it to...
as i said before
long shore
didnt get that
yeah that was not actually a great example
polar pulsar
and g(X) modifies the amount of "something"
yup
long shore
okay cool
polar pulsar
while keeping the probabilities the same
long shore
kinda like if everyone got a bonus at work
dull sequoia
there's $\mathbb E[g(X)] = \int_\bR g(x)f_X(x),dx$
flat frigateBOT
frosst
long shore
the lecture didn't talk about transformations in distributions at all so idk how we were supposed to know to do f(x) * g(x) instead of x * g(x)
polar pulsar
kinda like if everyone got a bonus at work
that could be an example ye, in that case, $g(x)$ could look like $M+Bx$ where x is the amount of hours they worked
flat frigateBOT
Supernova
dull sequoia
well $\mathbb E[X] = \int_\bR xf_X(x),dx$
flat frigateBOT
frosst
