it cud also be -300
#help-49
shell fractal
visual tiger
like you're solving for when the angle is between -180 and 180
shell fractal
60 -60
shell fractal
ye
visual tiger
if we choose the (a) correct branch to solve cos(...) < 1/2 on
like
knowing where theta is
if we solved the inequality for some random branch like 360 < theta <= 720
it might not be helpful considering we're solving for -pi/6 < theta < 7pi/6
shell fractal
but
marble creek
midnight plankBOT
marble creek
ew hold on
A hemispherical tank of radius 2 feet is positioned so that its base is circular. How much work (in ft-lb) is required to fill the tank with water through a hole in the base when the water source is at the base? (The weight-density of water is 62.4 pounds per cubic foot. Round your answer to two decimal places.)
better, anyways here my way of think through the problem...it didnt work. the formula for hemisphere according to google is 2/3pi r^3. i know radius is 2.. so i though the bounds would be 0 to 2 62.4 (density of water) * (2 - y) * (2/3 pi (2)^3
that didnt work
midnight plankBOT
@marble creek Has your question been resolved?
marble creek
<@&286206848099549185>
midnight plankBOT
@marble creek Has your question been resolved?
robust forge
Why doesn't cantor's diagnolization argument work for rationals, ie using his argument can't we conclude that rationals are also uncountable ?
midnight plankBOT
Why doesn't cantor's diagnolization argument work for rationals, ie using his ar...
runic hamlet
we cant freely choose the digits of rationals that way
rationals have to repeat eventually
robust forge
rationals have to repeat eventually
ahh ok
Also one more thing some no can have 2 diff decimal expansions like .5 and .4999...
doesn't this like disturb the bijection from n to r in his argument
runic hamlet
a little bit you can argue around that. depends on the specific way that you write the argument
for example in your list you can assume that only one of those is allowed
robust forge
for example in your list you can assume that only one of those is allowed
Hmm ic
I have another doubt from set theory itself could you also clear that
It's regarding how can I prove that there exists a bijection from (a,b) to R
runic hamlet
biject (a,b) to for example (0,1). and from (0,1) its easier to find a bijection to R
e.g a scaled version of tan
robust forge
e.g a scaled version of tan
could you pls elaborate on that
runic hamlet
you know the graph of tan?
robust forge
you know the graph of tan?
Yea I do
runic hamlet
its pretty clear from the graph that its bijective for each period
midnight plankBOT
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Available help channel!
alpine gyro
Is there any way to factorize this further?
$$2 \frac{\cos^{2}\theta(1-r^{2})-r\sin^{2}\theta(r^{2}+1)}{(r^{2}+1)^{2}}$$
midnight plankBOT
Is there any way to factorize this further?
$$2 \frac{\cos^{2}\theta(1-r^{2})-r...
grand pondBOT
Good
alpine gyro
Originally from
$$\frac{2\cos^{2} \theta}{(r^{2}+1)^{2}}(1-r^{2}) - \frac{2r\sin^{2} \theta}{r^{2}+1}$$
grand pondBOT
Good
alpine gyro
.close
midnight plankBOT
alpine gyro
.reopen
midnight plankBOT
@alpine gyro Has your question been resolved?
plush fossil
So far I've worked backwards and got 0 =< (a-b)^2. Im stuck trying to tie it all back to the statement because I can't find anything that tells me that a and b are positive numbers based off of what I have. This is only the second hw assignment and we're not allowed to prove by contradiction. Is there something im missing here, or is there still more that needs to be done on what I have so far?
midnight plankBOT
So far I've worked backwards and got 0 =< (a-b)^2. Im stuck trying to tie it all...
carmine sigil
@plush fossil you got to the point you needed to get to
What can you say about (a-b)^2?
I can't find anything that tells me that a and b are positive numbers based off of what I have
This question I don't understand, it's a premise of this problem that a and b are positive and real.
plush fossil
> I can't find anything that tells me that a and b are positive numbers based o...
my prof gave us like a bunch of definitions, and I was wondering if I needed to use one of them
carmine sigil
Well, you want to prove the proposition right? You can do it with the work you already have. No need for anything other than recognizing that (a-b)^2 is non-negative, because the square of any real number is non-negative.
You can do the following to prove this:
Consider the value (a-b)^2. Because it is a given that a and b are real numbers, the quantity a-b is also a real number, and the square of any real number is non-negative. Thus we have (a-b)^2 >= 0
Then we can multiply both sides by ab. We do not need to swap the direction of the inequality because both a and b are positive.
plush fossil
oh ok ok
carmine sigil
Then we can expand to get a^3b - 2a^2 b^2 + ab^3 >= 0
then we can move the 2a^2 b^2 to the other side and add a^4 + b^4 to both sides
and so on until you get back to the prompt
plush fossil
ok yeah i get it
my prof worked backwards a lot when we started doing this, but he never really showed us how to actually write the proof from there so I just felt like that was all I had to do
carmine sigil
well, you need to be a little careful that all of your steps are bidirectional implications.
for instance, if you have a > b and b is positive, then you have a > 0
but if you have a > 0 then you can't declare a > b just because you know that b is positive.
As long as you can work backwards, and you make sure of it, then providing the forwards proof is also sufficient.
plush fossil
ah well ok, thank you so much
carmine sigil
or swap forwards and backwards.
Which ever convention we're using lol
you're very welcome
frank aspen
Hello, can anyone help out with this equation? So the basically I need to solve (x,y,z)
midnight plankBOT
Hello, can anyone help out with this equation? So the basically I need to solve ...
frank aspen
This equation tends to be tricky and messy 🤣
uneven sandal
is there any specific method that you need to use to solve this?
frank aspen
Any methode elimination, substitution.
fallow scarab
matrices would be fastest
cobalt swan
Eliminate x from 1 and 2 to get new equation 4
Eliminate x from 2 and 3 to get new equation 5
Solve 4 and 5 to get y and z
Put y and z in one the equations to get x
cobalt swan
matrices would be fastest
Or yea this
uneven sandal
inverse 3x3 matrix is fastest yeah
midnight plankBOT
prime garden
This equation tends to be tricky and messy 🤣
This specific equation?
midnight plankBOT
Available help channel!
main glade
Hello, can someone say if its correct? I had to find the limit and derivative. I did it with the h-method.
midnight plankBOT
Hello, can someone say if its correct? I had to find the limit and derivative. I...
elder spoke
Hello, can someone say if its correct? I had to find the limit and derivative. I...
at the top, it seems you have written 4^2 as 76?
elder spoke
Its 16
ah mb, when i saw 48 on next line i thought maybe that was why
1st one is right
main glade
ah mb, when i saw 48 on next line i thought maybe that was why
Thank you
elder spoke
Thank you
it shud be 17 for second
im checking ur expansion
i think u forgot the x^3 subtraction on first line
main glade
i think u forgot the x^3 subtraction on first line
So I did the binomial formula wrong?
elder spoke
So I did the binomial formula wrong?
are u asking about the (x+h)^3 expansion? it's not wrong but, since the formula is f(a+h)-f(a) ... it seems you forgot to subtract x^3 i.e ** (x+h)^3 - x^3** for d 1st paer
part
main glade
are u asking about the (x+h)^3 expansion? it's not wrong but, since the formula ...
Can you correct the first line?
elder spoke
Can you correct the first line?
main glade
Ahh I See I forgot the other x… thank you!
elder spoke
Ahh I See I forgot the other x… thank you!
no problem!
elder spoke
Ahh I See I forgot the other x… thank you!
if u no longer need help, u can . close the channel with no spaces
.close
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Available help channel!
final palm
can someone help me w this question? i know i have to find the derivatives of each but then im supposed to use chain rule and substitute and idk how to apply it in this situation
midnight plankBOT
can someone help me w this question? i know i have to find the derivatives of ea...
rose trout
You are given $R(f)$ and $f(s)$.
The chain rule will look something like $\frac{dR}{ds} = \frac{dR}{df}\cdot \frac{df}{ds}$
grand pondBOT
Azyrashacorki
final palm
how do i know what to substitute as my “u”
tribal temple
What's "u" supposed to refer to - are you given the chain rule as something like "dy/dx = dy/du * du/dx" and you're trying to work with that?
tribal temple
**Azyrashacorki**
If so, notice that R takes the place of y, f the place of u, and s the place of x
midnight plankBOT
@final palm Has your question been resolved?
final palm
i meant like after you get the derivative, i have to use a letter to substitute because im using leibniz notation
or do i not need to use leibniz notation
buoyant yoke
,rccw
grand pondBOT
final palm
nevermind i think i got it, thanks !
radiant sparrow
Hello! here to find the general solution can I leave it like that or do I have to eliminate the -1/8 ?
midnight plankBOT
Hello! here to find the general solution can I leave it like that or do I have t...
hearty rune
its fine like that
though you could define a new constant A=-c/8 to make it neater if you wanted
radiant sparrow
may I ask, what is the purpose of a new constant? or when would I need it?
hearty rune
no purpose its just that a constant * constant is another constant
so you can just roll it into one
then when finding a specific case you can just find A rather than finding c with -c/8
the only functional point is its neater, beyond that theres not any significant effect really
last slate
When do you use the disk/washer v.s. shell method?
midnight plankBOT
When do you use the disk/washer v.s. shell method?
@last slate Has your question been resolved?
midnight plankBOT
@last slate Has your question been resolved?
main current
disc integrates in the same direction as the axis of rotation.
shell integrates in the opposite direction
It's often impossible to use one of disc or shell, and you are forced into one of the choices
Or, sometimes it's easier to use one of the two.
kindred dagger
It's often impossible to use one of disc or shell, and you are forced into one o...
When is shell impossible?
main current
One case would be where a function doesn't have a clear inverse
So for example, revolving y = xe^x around the x-axis
Trying to shell that is trying to integrate the W-function, which would be unfun, probably impossible but I ain't checking right now lol
Actually simpler is revolving y = x^3 - x around the x-axis, which can't shell because an inverse doesn't exist
Well, you can split it up and maybe construct some weird shell
But that's a lot of work when you can just quickly get it with a disc
@last slate
last slate
<@456226577798135808>
Opposite direction. So, you can have a function f(x) where it is in terms of x, and the question prompts you to integrate with respect to the y axis? I would use shell?
midnight plankBOT
last slate
.reopen
midnight plankBOT
✅
last slate
Wait no. What's the rotation axis?
Well, This is professor Leonard's question
I can't just imagine myself using shells over disks.
main current
You could use either here
disc would be an integral in terms of dy, given by the revolution direction
last slate
disc would be an integral in terms of dy, given by the revolution direction
.Can you give me a question where I can't possibly use any other method other than shell? Just so I can thinker with it
main current
What I was saying earlier, but with the revolution direction changed
y = x^3 - x, where we are revolving around the y-axis. We need shell for this one
Uh, I guess let the bounds be between x = 0 and x = 1
Solid would look like a sliced mini-donut
last slate
y = x^3 - x, where we are revolving around the y-axis. We need shell for this on...
Think I get it.
This one doesn't have an inverse so we can't use washer or disc as you are revolving on the y-axis which implicates you because you have to have a function where the terms are with respect to y and not x.
Is there any other cases?
main current
Similar idea for "An inverse exists but its integral doesn't"
Similar idea for "An inverse exists but it's a pain to work with"
Similar idea for "Integral needs to be split up into multiple parts when working in one of the directions so choose the easier direction"
etc
last slate
Alright, that clears everything up. All I have to do now is practice problems and see this in action. Thank you Kaynex 🙂
.close
last slate
I am pretty confused.
midnight plankBOT
I am pretty confused.
last slate
Recieving two different answers. I am assuming my setup is wrong (ChatGPT said I should've done it wrt the y axis.) Could the solution manual be wrong?
main current
They should at the very least agree
last slate
Here is the solution manual answer
It differs because the solution manual has (1- sqrt(y)) where, I can say, I have no idea where this came from.
main current
There's my single addition of a red line
That red line represents the height of some shell
The length of the red line is 1 - sqrt(y)
The area of that shell is (base)*(height)*2pi = 2pi*y(1 - sqrt(y))
There's a blue line now!
The blue line represents where the shell would need to be, in order to get a height sqrt(y).
If we were taking the volume to the left of the curve like that, then sqrt(y) would work
You showed that the volume to the left is 4pi/5.
But the inner volume is pi/5
midnight plankBOT
@last slate Has your question been resolved?
last slate
There's a blue line now!
That makes sense haha, thanks again 🙂
midnight plankBOT
midnight plankBOT
Available help channel!
lone hound
a and b are real positive numbers
midnight plankBOT
**a** and **b** are real positive numbers
midnight plankBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
autumn canopy
@lone hound
lone hound
k
lone hound
they aren't equal to anything.
they are all equal to each other
let me try to make some more equations
winged maple
oh, so you have 3a + 18 = the second one
in which case no, because in general if you have N unknowns you need N equations
grand pondBOT
bagelguy3
lone hound
how bout this for a third equation?
winged maple
yeah you can solve these as simultaneous equations
grand pondBOT
bagelguy3
winged maple
and you can do that = 3a+18
lone hound
can u solve it
winged maple
and the second set would be 3a + ab/2 + 3b = 3a + 18
midnight plankBOT
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Available help channel!
tidal turret
midnight plankBOT
tidal turret
Find, if possible, a polynomial ( P \in \mathbb{R}[X] ) of minimal degree that simultaneously satisfies:
-
All ( z \in \mathbb{C} ) that simultaneously satisfy ( z^4 + z = 0 ) and ( \text{Re}(z) > 0 ) are roots of ( P ).
-
It has at least two roots in common with ( Q = X^3 - X^2 + 4X - 4 ).
-
( P(0) = -1 ).
Factor ( P ) in ( \mathbb{Q}[X], \mathbb{R}[X] ), and ( \mathbb{C}[X] )
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
buoyant yoke
tidal turret
Re(z) > 0 ==> -π/2 < arg(z) < π/2
buoyant yoke
did you find the roots of 1. and 2.?
tidal turret
did you find the roots of 1. and 2.?
z is not given
buoyant yoke
you can find the roots of z^4+z=0
tidal turret
did you find the roots of 1. and 2.?
for 2 u can use rational root theorem because the polynomial is in R3[X]
tidal turret
you can find the roots of z^4+z=0
how?
buoyant yoke
youve had complex numbers, no?
tidal turret
you can find the roots of z^4+z=0
one root is z = 0 but that has Re(z)=0
tidal turret
z = 0 is not interesting
for z^3 we can use demoivre (trigonometric form)
z^3 = |z|^3 (cos(3θ) + isin(3θ))
-1 = |-1| . (cos(0) + isin(0))
|-1| = 1
tidal turret
huh?
Just expressing -1 using trigonometric form then equal the modulus and equal arguments
buoyant yoke
\begin{align*}z^3=-1\\z^3=e^{i(\pi+2k\pi)}\\z=e^{\frac{i(\pi+2k\pi)}3}\end{align*}
grand pondBOT
Bonk
buoyant yoke
from here you get all the solutions
tidal turret
bro
buoyant yoke
$z_1=e^{\frac{i\pi}3},z_2=e^{i\pi},z_3=e^{\frac{5i\pi}3}$
grand pondBOT
Bonk
buoyant yoke
much simpler (imo)
tidal turret
which ones have Re(z) > 0
you need to translate to cartesian or to trigonometric form at least
tidal turret
much simpler (imo)
idk why you prefer operating with exponential form
buoyant yoke
which ones have Re(z) > 0
you just need to look at the argument
buoyant yoke
idk why you prefer operating with exponential form
then its much easier to take a root
tidal turret
pi/3, pi, 5pi/3
instead of taking root, I use demoivre, but is okay, both path works
buoyant yoke
pi/3, pi, 5pi/3
Re(z)>0 => -pi/2 < arg(z) < pi/2
tidal turret
pi = 180°
pi/3 = 60°
-pi/2 = -90°
pi/2 = 90°
true or false:
-90° < 60° < 90° TRUE
buoyant yoke
is this srsly your internal thought process?
tidal turret
true or false:
-90° < 180° < 90° FALSE
grand pondBOT
Result:
300
you need to draw the angles in the argand plane
buoyant yoke
,w graph z^3+1=0
tidal turret
!no wolfy
grand pondBOT
buoyant yoke
!no wolfy
bro i do this in my head
its just for yu to visualise
this is what i see in my head
tidal turret
bro i do this in my head
we need the arguments to be in first and fourth quadrant, a simpler problem is finding in which quadrants our arguments are in
buoyant yoke
when asked to solve z^3=-1
buoyant yoke
this is what i see in my head
and from this you can instantly see z1 and z3 have Re>0
tidal turret
arg(z1) = 60°
tidal turret
if i do this angle 3 times, im at -1
why 3 times tho
tidal turret
i,-i, 1, -1
tidal turret
bruh what
tidal turret
this is z^5=1
how to do you know the quadrants for this one
buoyant yoke
no?
buoyant yoke
how to do you know the quadrants for this one
its 5 equally spaced points
tidal turret
ohh
buoyant yoke
i assume that the magnitude for all of them is 1
then the angle of z1 is 0, z2 is 2pi/5, z3 is 2 * 2pi/5, z4 is 3 * 2 pi/5 and z5 is 4 * pi/5
kinda make sense?
tidal turret
it does but u do it so quickly bro
tidal turret
then the angle of z1 is 0, z2 is 2pi/5, z3 is 2 * 2pi/5, z4 is 3 * 2 pi/5 and z5...
I got it bro
tidal turret
this doesnt happen out of nowhere
wdym?
buoyant yoke
like, being able to see this so quickly
buoyant yoke
pi/3 and 5pi/3 are valid
buoyant yoke
😄
tidal turret
bruh, why is complex numbers so hard haha
buoyant yoke
practice 🙂
grand pondBOT
Bonk
tidal turret
exponential form -> trigonometric form -> cartesian form
?
$z_1=e^{\frac{i\pi}3},z_2=e^{i\pi},z_3=e^{\frac{5i\pi}3}$
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
e^(i(pi/3)) = 1.(cos(pi/3) + isin(pi/3))
e^(i(5pi/3)) = 1 . (cos(5pi/3) + isin(5pi/3))
tidal turret
$re^{i\theta}=r\cos(\theta)+ri\sin(\theta)$
then to cartesian?
we need a calculator bro
otherwise?
grand pondBOT
Bonk
tidal turret
bro
buoyant yoke
i go:
0 1/6 1/4 1/3 1/2
1 sqrt(3)/2 sqrt(2)/2 1/2 0
numerator goes sqrt(4)-sqrt(3)-sqrt(2)-sqrt(1)-sqrt(0)
if that kinda makes sense
anyway, brb
10 mins
gotta buy some fries
tidal turret
gotta buy some fries
eet smakelijk
tidal turret
i go:
0 1/6 1/4 1/3 1/2
1 sqrt(3)/2 sqrt(2)/2 ...
bro, trig is hard
I'm just bad
XD bro. . .
I struggle with trig
AI can't think for themselves
human brain is superior
do you know the AI singularity
still will take like 100 years bro
until AI surpasses human brain
if they do
AI as of RN is very stupid
it will probably hit a plateau
deepseek is decent but still worse than GPT 4 let alone gpt o1 or o3 (comparing R1 to o1)
buoyant yoke
Im back
buoyant yoke
numerator goes sqrt(4)-sqrt(3)-sqrt(2)-sqrt(1)-sqrt(0)
Did this make any sense?
tidal turret
numerator goes sqrt(4)-sqrt(3)-sqrt(2)-sqrt(1)-sqrt(0)
yes, i know that one from a song in Portuguese
tidal turret
Learn it in pi
ok
tidal turret
sorry for interrupting but doing it in pi is better
buoyant yoke
🤣
buoyant yoke
sorry for interrupting but doing it in pi is better
Do not use degrees
You want to get out of degrees asap if you can
tidal turret
ok will do
brisk iris
tidal turret
yes, 2 roots for P we found
buoyant yoke
There is an easy integer solution
tidal turret
use rrt
buoyant yoke
But you need a root first
tidal turret
gauss lemma is called here in Argentina
tidal turret
But you need a root first
of Q?
tidal turret
Just find the roots
use rrt to find a root
then perform polynomial division
we get a quadratic
then apply quadratic formula
tidal turret
You can find a root by inspection
1
tidal turret
No need to be so complicated
nice
tidal turret
$\polylongdiv{x^3-x^2+4x-4}{x-1}$
buoyant yoke
Bruh
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
buoyant yoke
You know how to do this by hand right?
buoyant yoke
Ok 
tidal turret
Q(x)=(x-1)(x^2+4)
tidal turret
Ping when youre back
ok, I'll ping you in a sec
if and only if i manage to graduate
same bro
wdym?
do what u can otherwise prepare for it better next time
I can retake indefinitely many time my courses tho
can you just take the ones you already failed
and don't take new courses
you are in a sticky situation then
otherwise you will miss like a full year
that sucks man, do what you can, otherwise better luck next year
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
buoyant yoke
Sure ig
tidal turret
x^2 + 4 = 0
buoyant yoke
Or diff of squares
tidal turret
Or diff of squares
(x-2)(x+2)
tidal turret
ohh
tidal turret
Sqrt(-4)=?
2i
buoyant yoke
Yes
tidal turret
+2i and -2i? or only 2i?
buoyant yoke
Both
tidal turret
buoyant yoke
x^2-(-4)=(x+2i)(x-2i)
tidal turret
ok
buoyant yoke
Understand why?
tidal turret
(-2i)^2 = (-2)^2 . i^2 = -4
(2i)^2 = 4 . i^2 = -4
tidal turret
Understand why?
Q(x) = (x-1)(x-2i)(x+2i) then
tidal turret
Understand why?
I think soo
buoyant yoke
Q(x) = (x-1)(x-2i)(x+2i) then
Yup
tidal turret
we have 3 roots of P + how many other ones from the complex number?
3 solutions from Q for P
2 solutions from the complex number for P
P is a quintic
tidal turret
So for 1. we found z=e^(+- i pi/3)
2 roots from here for P
3 roots from Q for P
but we need to factor it in Q, in R and in C
oh my
buoyant yoke
Which one should we start with
tidal turret
Which one should we start with
I think doing it for C is the simplest, only factorization in linear factors
then we can think of Q and R
tidal turret
Which one should we start with
it sounds complicated but I think is not, just depends if you keep a quadratic factor or linear I think
tidal turret
Which one should we start with
if we are familiarized with factoring over R it should be similar for example the complex conjugate we keep them as quadratic factors
but factoring over C should be keeping the complex roots as linear factors, you see what I mean
over Q, is the hardest, but it will depend on the roots we get for the complex number in 1)
tidal turret
Which one should we start with
just, lets focus on bringing from exponential form to cartesian, and then continue from there
buoyant yoke
just, lets focus on bringing from exponential form to cartesian, and then contin...
okay, do that
also remember the condition P(0)=-1
graceful ferry
Hey guys sorry to interrupt i just got my algebra grade and i passed im so happy 😆
buoyant yoke
Hey guys sorry to interrupt i just got my algebra grade and i passed im so happy...
congrats
graceful ferry
Ty !
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
Hey guys sorry to interrupt i just got my algebra grade and i passed im so happy...
well done, gg brother
Q(x) = (x-1)(x-2i)(x+2i)
tidal turret
also remember the condition P(0)=-1
P(x) = α(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
heavy falcon
**938c2cc0dcc05f2b68c4287040cfcf71**
question : why are you taking P to be quintic
tidal turret
ohh is a quartic
tidal turret
yeah my baad
heavy falcon
np i just wanted to point this out so that you dont have to repeat alot of steps after you go further
tidal turret
P must have at least 2 roots from Q
can we repeat the root = 1
no, just pick both complex roots of Q
so we dont care about complex conjugates
heavy falcon
can we repeat the root = 1
probably no since 1 is not a root of multiplicity 2 of Q(x)
heavy falcon
no, just pick both complex roots of Q
tidal turret
yeah exactly, it says use 2 roots of Q, but it doesnt specifically say they have to be distinct
is best just to pick both complex roots of Q to get out of the hassle
heavy falcon
yeah exactly, it says use 2 roots of Q, but it doesnt specifically say they have...
i dont think that is necessary but since 1 is only repeated once as a root of Q(x) , ie (x-1) is a factor of Q(x) but (x-1)^2 is not then you cant take 1 2 times for P
tidal turret
P should then, be a quartic, nice observation, I missed that
heavy falcon
if (x-1)^2 was a factor of Q(x) then you probably could take 1 twice for P(x)
because the condition doesnt require 2 distinct roots of Q(x)
but any 2 roots of Q(x)
tidal turret
if (x-1)^2 was a factor of Q(x) then you probably could take 1 twice for P(x)
even if thats the case I will still pick the two complex roots to avoid ambiguity
then. P is a quartic I missed that, thank u
P(x) = α(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
,w solve -1 = a × (-2i)(2i)(- (1/2 - isqrt(3)/2))(-(1/2 + isqrt(3)/2))
grand pondBOT
tidal turret
P(x) = (-1/4)(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
this is in linear factors
now we need to factorize in Q, in C and in R
heavy falcon
P(x) = (-1/4)(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
so this is the factorization in C
tidal turret
where is C
tidal turret
so this is the factorization in C
C is bigger than R then
heavy falcon
surrounding R
tidal turret
tidal turret
yes since a=a+0i for all a in R
quite precisely
heavy falcon
P(x) = (-1/4)(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
so this ends the factorization in C
what about R ?
grand pondBOT
tidal turret
P(x) = (-1/4)(x^2+4)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
heavy falcon
well you had the factors before factorizing in C
heavy falcon
P(x) = (-1/4)(x^2+4)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
you still need to deal with the other 2 paranthesese
grand pondBOT
tidal turret
well you had the factors before factorizing in C
where
oh right after doing polynomial division
when we got a quadratic
heavy falcon
earlier you moved from x^2+4 to (x-2i)(x+2i)
and the other one comes from x^3+1=(x+1)(x^2-x+1)
tidal turret
earlier you moved from x^2+4 to (x-2i)(x+2i)
yes but the one from the the complex number
heavy falcon
a^3+b^3=(a+b)(a^2-ab+b^2)
heavy falcon
sum of cubes?
yes
tidal turret
I got this one in exam and I would die
I dont know difference of squares let alone sum of cubes or difference of cubes
well you can always perform polynomial division
is okay I think
$\polylongdiv{x^3+1}{x+1}$
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
you still need to deal with the other 2 paranthesese
what about Q?
over Q is the same as over R
we are Done
so in summary:
over C is:
P(x) = (-1/4)(x-2i)(x+2i)(x - (1/2 - isqrt(3)/2))(x-(1/2 + isqrt(3)/2))
over Q is:
P(x) = (-1/4)(x^2+4)(x^2 - x + 1)
over R is:
P(x) = (-1/4)(x^2+4)(x^2 - x + 1) in R
tidal turret
over Q is the same as over R
we literally got rid of the irrationals when we got P factored over R
thats why
midnight plankBOT
heavy falcon
we literally got rid of the irrationals when we got P factored over R
thats because there werent irrational numbers to begin with
heavy falcon
over R is:
P(x) = (-1/4)(x^2+4)(x^2 - x + 1) in R
great job all of these are correct
tidal turret
thats because there werent irrational numbers to begin with
disguised irrational numbers as complex, maybe can it be said(?
or maybe I am hard tripping, idk
I think factoring over Q step was to make u scared
idk
heavy falcon
disguised irrational numbers as complex, maybe can it be said(?
well irrational numbers are real numbers which are not rational
tidal turret
right
I think I got it, though it was a little bit confusing some parts, specially the trigonometry and factoring over Q[X]
heavy falcon
what confused you in each of these 2
tidal turret
why is factorization of P(x) over Q[X] the same as P(x) over R[X]?
heavy falcon
because there were no irrational coefficients in the factorizaion of P(x) over R[x]
so if the factorization of $P(x)$ over $\mathbb{R}[x]$ was something like \$(x-1)(x-2)(x-\sqrt 2)(x+\sqrt 2)$\ for example then the factorization of $P(x)$ over $\mathbb{Q}[x]$ would be \$(x-1)(x-2)(x^2-2)$
tidal turret
right
grand pondBOT
tidal turret
and factorization over R[X] would be just linear factors
tidal turret
so if the factorization of $P(x)$ over $\mathbb{R}[x]$ was something like \\$(x-...
yeah I think I understand
just confusing differentiating Q and R and C
heavy falcon
np if you see some examples which illustrate the difference between the 3 you will get used to it
heavy falcon
I think I got it, though it was a little bit confusing some parts, specially the...
what about the trigonometry
tidal turret
what about the trigonometry
I think trigonometry was fine but got a little bit stuck when finding |z| when I knew that |z|^3 = -1
I mean Trigonometry Is NOT fine
got a little bit stuck when finding cos(-π/3)
same for cos(π/3)
heavy falcon
I think trigonometry was fine but got a little bit stuck when finding |z| when I...
|z|^3=-1 cant happen
tidal turret
|z|^3=-1 cant happen
|z|^3 = 1, typo
tidal turret
is hard to explain but
heavy falcon
you need z^3+1=0
tidal turret
we had z^3 = -1
heavy falcon
use a^3+b^3=(a+b)(a^2-ab+b^2)
heavy falcon
we had z^3 = -1
yes i get this part since i saw some of the work above
tidal turret
use a^3+b^3=(a+b)(a^2-ab+b^2)
yeah. what if it was z^100 tho, u need to know demoivre theorem
heavy falcon
use a^3+b^3=(a+b)(a^2-ab+b^2)
and then find the roots using quadratic formula
heavy falcon
yeah. what if it was z^100 tho, u need to know demoivre theorem
well in that case exponential form is better than trig form
or maybe trig form too
both work
tidal turret
both work
yeah
bonk uses exponential
but then u still need to translate to trigonometric form
so, i domt see why not just stick with trig form all the way
tidal turret
but why would you want to do that ?
so we can set the modulus and the arguments equal
is hard to explain
heavy falcon
well i see why you want to find |z| but it can be found from the equation itself
for example if you have z^51-3=0 then you know that |z|^51=3 directly
and then |z|=3^{1/51} no ?
tidal turret
for example if you have z^51-3=0 then you know that |z|^51=3 directly
yeah
heavy falcon
so where is the problem that you are facing with |z|
tidal turret
well
is hard to see that |z|^3 = 1 implies |z|=1
for example
x^3 = 1
has three roots
but only one is in R
tidal turret
since |z| is in R
exactly
we ignore the complex solutions
idk, I just forgot that |z| is in R i guess XD
heavy falcon
then np we all forget something like this in times
so now do you still have problems with anything related to the original question like the ones you had rn ?
midnight plankBOT
Available help channel!
tidal turret
so now do you still have problems with anything related to the original question...
how do you find cos(π/3) by hand
midnight plankBOT
how do you find cos(π/3) by hand
grim vector
how do you find cos(π/3) by hand
Draw
lyric charm
pi/3 is the angle of an equilateral triangle
tidal turret
soh cah toa?
lyric charm
draw one of those, split it in half with an altitude from one vertex, and you have yourself a right triangle with angles pi/2, pi/3 and pi/6 in which you can calculate everything
with pythagoras and yes eventually SOH-CAH-TOA to pin the trig values down
tidal turret
pi/3 is the angle of an equilateral triangle
how do you know is equilateral
grim vector
how do you know is equilateral
Definition
heavy falcon
well other than just memorizing it you can draw a semi equilateral triangle and use the properties of the sides of the triangle to find cos(π/3)
tidal turret
Definition
only one of them is 60 deg
lyric charm
how do you know is equilateral
wdym?
heavy falcon
huh , when did you all have the time to send all of these messages
lyric charm
are you trying to ask "how do i know that in an equilateral triangle all angles are 60°"? @tidal turret
heavy falcon
it was so fast 
tidal turret
guys I am trying to find cos(pi/3) by hand
grim vector
it was so fast <:opencry:586078614865444865>
They call me math luck in the west
tidal turret
the basics
lyric charm
im telling you:
heavy falcon
They call me math luck in the west
and what do they call you in the east
lyric charm
start with an equilateral triangle
all of its internal angles will be sixty degrees
do you follow thus far? yes or no @tidal turret
grim vector
and what do they call you in the east
They don't call me, my phone credit can't lmao
lyric charm
@grim vector @heavy falcon if yall wanna banter go to #「helpers-lounge」 please
tidal turret
yes but why equilateral i thought one of the angles was 90 for soh cah toa
lyric charm
patience
my next step is explaining how you get the right angle
it'll also be convenient to take the side length of your triangle as 2
you'll have to trust me on this one
lyric charm
tidal turret
hip^2 = cat1^2 + cat2^2
tidal turret
yeah
tidal turret
yes
grim vector
cos(pi/3)=1/2
Wunderbar
lyric charm
and there you have it
tidal turret
what about -pi/3 for cosine and sinee
grim vector
what about -pi/3 for cosine and sinee
Use cos proprety
lyric charm
then you have to move up a step in complexity and look to the unit circle
cos(-x) = cos(x)
in other words cos is an even function
cos(-pi/3) = cos(+pi/3) = you already know this
tidal turret
Use cos proprety
cos(-x)=cos(x)
tidal turret
cos(-pi/3) = cos(+pi/3) = you already know this
oookay
grim vector
Sin is odd
tidal turret
but sine
lyric charm
in this picture you can use pythagoras to get AK = sqrt(3).
tidal turret
2^2 = 1^2 + h^2
tidal turret
2^2 +1^2 = h^2
tidal turret
in this picture you can use pythagoras to get AK = sqrt(3).
how are you getting sqrt3
heavy falcon
in the triangle BKA, what is the length of the hypotenuse
tidal turret
in the triangle BKA, what is the length of the hypotenuse
sqrt5
heavy falcon
no
tidal turret
why
heavy falcon
look at the triangle carefully
heavy falcon
no you are not. everything is fine
heavy falcon
and cos(-x)=cos(x)
tidal turret
and cos(-x)=cos(x)
I always forget both of this
is so annoying
do you know how they are called?
heavy falcon
you mean that sin(x) is odd and cos(x) is even ?
tidal turret
you mean that sin(x) is odd and cos(x) is even ?
yeah I never understood that
heavy falcon
well if a function is even then its graph has a symmetry wrt to the y-axis
and if it is odd then its graph is symmetric wrt to the point (0,0)
tidal turret
wdym?
grim vector
If you fold the paper on the y axis it will be one over the other
The easiest example is constant function
,w graph f(x) = 4
grand pondBOT
heavy falcon
wdym?
here is the graph of cos(x)
look at the y-axis
notice that the part of the graph before the y-axis and the part after it are the same
tidal turret
notice that the part of the graph before the y-axis and the part after it are th...
so?
how is odd even related
tidal turret
If you fold the paper on the y axis it will be one over the other
I see
heavy falcon
it is an even function
tidal turret
it is an even function
symmetric = even?
heavy falcon
If you fold the paper on the y axis it will be one over the other
if this is true for the graph of the function you have then this function is even
tidal turret
if this is true for the graph of the function you have then this function is eve...
what about odd
you said something about the origin
tidal turret
look at the 2 parts that i highlighted
what now?
heavy falcon
lyric charm
how are you getting sqrt3
4 = 1 + h^2
=> h^2 = 3
heavy falcon
notice that these 2 points have the same distance from the x-axis
and that one of them is below the x-axis and the other is above it
tidal turret
here is the graph of cos(x)
cos(x)=cos(-x)
heavy falcon
cos(x)=cos(-x)
yes , the graph is symmetric wrt to the y-axis so its graph on the right of the y-axis and of the left of it are exactly the same
and so the y-coordinate of x and -x is the same
and the y-coordinate of x is given by cos(x) here so cos(x)=cos(-x)
i am not sure if this is confusing you more than being useful
tidal turret
i am not sure if this is confusing you more than being useful
so we dont need to memorize the unit circle
we only need to use soh cah toa + symmetry of sine and cosine
why is tangent used in soh cah toa though?
heavy falcon
we only need to use soh cah toa + symmetry of sine and cosine
well yes you can do this but for the first part you would need more than one type of triangle for your approaches
so by drawing an equilateral triangle and dividing it into 2 triangles as ann showed you
you can get sin and cos of π/3 and π/6
for angle π/4 you need to draw a right isosceles triangle
tidal turret
for angle π/4 you need to draw a right isosceles triangle
90 45 45
heavy falcon
and for π/2, 0( or 2π),π,3π/2 they are easier than the others
tidal turret
trig is hard
tidal turret
this is the graph of sin(x)
sin is cos shifted half period
heavy falcon
sin is cos shifted half period
is it a half period ?
heavy falcon
and by how much is sin shifted from cos
heavy falcon
xD
tidal turret
I appreciate the help
heavy falcon
np dont mention it
tidal turret
.solved
midnight plankBOT
midnight plankBOT
Available help channel!
midnight plankBOT
How do i solve this integral
regal rampart
This looks like an e&m problem hahaha
midnight plankBOT
@gilded crow Has your question been resolved?
gilded crow
This looks like an e&m problem hahaha
Yeah i got this while calculating flux 😂
regal rampart
you might be able to just look it up because these something over ( , )^(3/2) integrals are pretty standard i think
midnight plankBOT
@gilded crow Has your question been resolved?
mortal sleet
midnight plankBOT
sleek cloud
!status
midnight plankBOT
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
Is the bottom triangle right?
mortal sleet
No
Theres gotta be some sort of proof that the bottom i a right triangle
Cuz the way im imagining it if we change the lenth of that diagonal it would mess up the fixed propotions
<@&286206848099549185> 🥲
sleek cloud
Is this all you got or any information missing?
Always good to show the original question
mortal sleet
The techer gave this to me from a book she has she was also stunned to see that it wasnt acc a right triangle
Or atleast it wasnt noted that it was a right triangle
Maybe i have to prove it myself but im struggling to find a way to do it
<@&286206848099549185> if anyone finds a solution pls dm me cuz im going to sleep
heady fog
gn @mortal sleet
mortal sleet
gn <@1125582542641315921>
Gn
mortal sleet
this is the question
Yeah
The bottor triangle is not noted to be a right triangle although if it is we gotta have a proof
dusk pier
What do you need to find?
midnight plankBOT
@mortal sleet Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
static lark
I’m stuck on what blunds to use
I got $\int_0^1t^xdt=\frac{1}{1+x},;x>-1$ as a given, and was asked to use this result to get an expression for $\int_0^1t^x(\ln(t))^ndt,;n\in\N$
My problem isn’t the actual problem, which is fairly straightforward.. it’s rather the tips in the task, to show that the condition for leibniz rule holds, and in particular choosing the bounds of [c,d]\times[0,1] to show $f$ and $f_x$ are continuous
midnight plankBOT
I’m stuck on what blunds to use
I got $\int_0^1t^xdt=\frac{1}{1+x},\;x>-1$ as a...
grand pondBOT
pax_ignis
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
static lark
That was long
I don’t have any work because i’m stuck on that single fuckin thing lol
fallow scarab
what does "tips in the task" mean
can you just show the original question. screenshot or picture is best
torpid garnet
static lark
the question is in norwegian
my understanding of this is that i have to show that the conditions hold
midnight plankBOT
@static lark Has your question been resolved?
midnight plankBOT
@static lark Has your question been resolved?
midnight plankBOT
@static lark Has your question been resolved?
static lark
<@&286206848099549185> 🙏
cosmic junco
Could you please give a translation?
static lark
use that $\int_0^1t^xdt=\frac{1}{1+x}$ for all $x>-1$ to find an expression for $\int_0^1t^x(\ln(t))dt$ where $n\in\mathbb{N}$
grand pondBOT
pax_ignis
static lark
Hint: Show that the conditions for moving the derivative under the integral holds
The task in and of itself is not too bad
but
i have to show that it holds
which i currently don't
These are the conditions my course has defined for it..
ancient ridge
I don't really want to think about all the a,b,c,d's but I would use the substitution u = -ln(x) and argue that any order derivative is a linear combination of gamma pdfs and hence that g(t) exists and constant.
autumn python
These are the conditions my course has defined for it..
wtf is this
Anyways let’s see
@static lark u here
midnight plankBOT
@static lark Has your question been resolved?
static lark
i was, had nap
limber nimbus
That would be hard
fleet moss
midnight plankBOT
midnight plankBOT
Available help channel!
shut canyon
pls check my proof
proof by calculation:
a + b = a + c
(-a) + a + b = (-a) + a + c
0 + b = 0 + c
b = c.
the original conditional statement can be written as
if b = c, b = c.
glossary:
p: b = c.
p ⇒ p is a tautology:
| p | p | p ⇒ p |
|---|---|---|
| T | T | T |
| F | F | T |
midnight plankBOT
pls check my proof
proof by calculation:
a + b = a + c
(-a) + a + b = (-a) + a ...
shut canyon
this is the original exercise, is this a good approach?
lyric charm
everything except the calculation part is basically unnecessary
shut canyon
everything except the calculation part is basically unnecessary
Thank you! Truth table is enough?
solid iris
seems like a field axiom problem in which case there should be no reference to stuff other than field axioms
lyric charm
Thank you! Truth table is enough?
truth table not needed.
keep only the calculation, dump everything else.
shut canyon
seems like a field axiom problem in which case there should be no reference to s...
Alright, thanks. Where do you see the fields?
solid iris
do you know what i mean by a field
shut canyon
do you know what i mean by a field
[Linear Algebra - Meckes], and the exercise I am working on is part of a section in another book on proving things for the integers
shut canyon
do you know what i mean by a field
Sort of, do you think this proof is still relevant to fields?
solid iris
Sort of, do you think this proof is still relevant to fields?
the addition cancellation law holds in any field or additive group
its proof is exactly your "calculation"
each step is really justified by a field axiom
midnight plankBOT
@shut canyon Has your question been resolved?
shut canyon
the addition cancellation law holds in any field or additive group
Okay, thank you, I will try to construct it by referring to the axioms at each step.
I just noticed this pattern might be related to this graph, injectivity, and left cancellation and monic arrows, but I don't yet see clearly how this works.
still cairn
what pen are you using?
solid iris
I just noticed this pattern might be related to this graph, injectivity, and lef...
the cancellation law says that left/right composition with any group element is injective
shut canyon
what pen are you using?
Schneider Xtra 805 0.5 mm it's a cool pen
shut canyon
the cancellation law says that left/right composition with any group element is ...
Nice!
shut canyon
the cancellation law says that left/right composition with any group element is ...
a + b = a + c ⇒ b = c
This is the left cancellation law, the left side of the statement "any group satisfies left cancellation laws".
ℤ is an additive abelian group where addition:
(i): is associative
(ii): has a two sided unit u, with a + u = a = u + a for all a.
(iii): has to each element a two sided additive inverse a,a + a' = 0 = a' + a.
a' + (a + b) = a' + (a + c)(by hypothesis a + b = a + c)(a' + a) + b = (a' + a) + c(associative axiom)u + b = u + c(axiom (iii) for a')b = c(u is a left unit).
I used this reference to construct it [Algebra, Saunders Mac Lane, Garret Birkhoff]
solid iris
yes
midnight plankBOT
@shut canyon Has your question been resolved?
night jungle
Sorry just a very fundamental doubt
midnight plankBOT
Sorry just a very fundamental doubt
night jungle
Contours cant pass over singularities/poles right?
Like its not like "you should avoid them because they are hard to compute"
You just fundamentally cant pass Over them
Nvm it should be continous on the boundary to apply cauchy thereom
Damn it
.close
midnight plankBOT
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Available help channel!
twilit field
midnight plankBOT
stone leaf
well...
twilit field
On reversing the order of integration , I'm getting $\int_{0}^{ \pi} \int_{0}^{y} \frac{sin(y)}{y} dx dy$
Is this right
stone leaf
Is this right
i think is not
twilit field
You are integrating siny/y wrt to x ?
I mean, I'm changing the order of integration
twilit field
i think is not
what's wrong
grim vector
So why from 0 to x ?
grand pondBOT
What a wonderful world !
twilit field
my bad
stone leaf
that would describe a different region, where y is bounded by x
y should be the outer variable, as it naturally ranges from 0 to π, and then x is bounded by y
twilit field
y should be the outer variable, as it naturally ranges from 0 to π, and then x i...
Eh
That sounds wrong
dawn dagger
**What a wonderful world !**
It's correct
midnight plankBOT
grim vector
anyone got any ideas?
Open a new channel this one will close soon
midnight plankBOT
Available help channel!
twilit field
midnight plankBOT
twilit field
I'm trying to switch the order of intergration
I was thinking it's $\int_{0}^{2} \int_{0}^{y^3} \frac{dx dy}{y^4+1}$
grand pondBOT
What a wonderful world !
stone leaf
I'm trying to switch the order of intergration
how?
twilit field
how?
Sketching it and then seeing where horizontal lines enter the region, and where the exit the region
dawn dagger
**What a wonderful world !**
looks right
dawn dagger
try polar coordinates
twilit field
try polar coordinates
huh?
twilit field
any other questions?
i'll have some soon, I'm revising for my midsem
stone leaf
wow
grand pondBOT
What a wonderful world !
dawn dagger
,w Integrate[1/(y^4+1),{x,0,8},{y,x^(1/3),2}]
grand pondBOT
dawn dagger
,calc log(9)/4
grand pondBOT
Result:
0.54930614433405
,calc log(9)/4
grand pondBOT
Result:
0.54930614433405
,w ln(9)/4
carmine void
Why would it be 9?
grand pondBOT
carmine void
2^4+1 = 17
grand pondBOT
What a wonderful world !
dawn dagger
,calc log(17)/4
grand pondBOT
Result:
0.70830333601405
midnight plankBOT
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Available help channel!
sinful condor
If an asian passengers has a 60% chance of visiting F&B outlets, and passengers traveling for business have a 5% chance of visiting.
What is the chance of an asian qpassenger travelling for business, visiting F&B stores?
Is it 60%*5%?
midnight plankBOT
If an asian passengers has a 60% chance of visiting F&B outlets, and passengers ...
buoyant yoke
yes
dreamy lichen
is it?
dreamy lichen
A - asian
B - travelling for bussines
C - visiting F&B
P(C | A) = 60%
P(C | B) = 5%
P(C | (A & B)) = ?
i dont think we can compute that actually, but i might be wrong
ill try to construct a countermodel or whatever is it called
buoyant yoke
counterexample
dreamy lichen
yeah, we certainly cannot know it
it can be anywhere between 0 and 100%
we could have 100 passengers
buoyant yoke
huh? wdym
dreamy lichen
ill draw a pic, i dont wanna write 100 words just to describe what i mean
buoyant yoke
sure
dreamy lichen
nvm, drawing it is even harder because i cant find any suitable tool.
So, say we have total of 100 passengers.
50 are asian
20 are bussines
10 asians are bussines
now one non-asian bussines visits F&B
and 30 non-bussines asians visit F&B
in this case, the probability of asian bussines visiting F&B would be 0
sinful condor
Yikes,
Thanks for explaining this so clearly tho
dreamy lichen
we could also have
50 asian
20 bussines
1 bussines asian
now if 30 of the asians, including the bussines one, visit F & B, the probability would be 100% (because the only bussines asian had indeed visited F & B)
dreamy lichen
we could also have
50 asian
20 bussines
1 bussines asian
now if 30 of the asians...
i couldve explained this in a simpler way:
Say that there is a single asian in bussines (this is not against the constraints)
Now this guy could either visit F&B or not and that would result in either 100% or 0% probability
we could also achieve almost anything inbetween
next rover
the problem is that they sometimes want you to solve it anyway