@vapid shuttle 0 isn't an eigenvector
#help-0
echo socket
I think by k they mean the eigenvalue
vapid shuttle
excluding 0
vale wigeon
this has already become a trainwreck
lime sluice
(4k,k) and (-4k,-k) both are the same eigen vector?
echo socket
Eigenvalue can be equal to 0 btw
vapid shuttle
yes, because they are constant multiples of the vector <4,1>
which is inside of the nullspace
of A-lambda I
(assuming you found that correctly)
lime sluice
So here k will not be 0 right?
vapid shuttle
right
echo socket
Depends on the matrix
vapid shuttle
if the nullspace is spanned by <4,1> then k will not be 0
echo socket
Eigenvalues can be 0, yes, but idk why you are using k in (4k, k)
If by k you mean the eigenvalue
lime sluice
Suppose (-1,1) is eigen vector
echo socket
Right
lime sluice
Then (1,-1) also eigen vector?
echo socket
Yes
lime sluice
Hmm i understand now
Thank you all
Now my focus on austin explanation about null space
vapid shuttle
A matrix only has a 0 eigenvalue when it does not have full rank.
An eigenvalue of 0 indicates that when you input a vector, it comes out scaled by 0, meaning it gets sent to 0. This implies that the vector is in the nullspace of the original matrix, which equally implies that this nullspace is nonempty, which equally implies that the matrix does not have full rank. Of course the 0 vector always lives in the nullspace so we do not actually consider this.
lime sluice
Because i have no idea what is actually null space is
vapid shuttle
The null space, is the space of vectors that when you input them into your matrix, get sent to 0
lime sluice
Determinants is zero?
vapid shuttle
determinant being zero means the nullspace is non empty yes
lime sluice
Like we find zeroes and we put back into it
echo socket
determinant being zero means the nullspace is non empty yes
Nontrivial you mean? Because nullspace is nonempty always by default
vapid shuttle
yes excluding 0 of course
lime sluice
It will be null space surely
vapid shuttle
It will be null space surely
Consider the matrix,
$$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}$$
ocean sealBOT
Austin
vapid shuttle
think about which vectors you can put into this matrix, that come out only being scaled
can you put a vector that has a nonzero 3rd component into this matrix, and have it come out being only scaled?
vapid shuttle
consider it being scaled by zero also Arjunn
we can put any vector of the form <0,0,b> with b nonzero, and it will come out being scaled by 0
right?
lime sluice
Because determinant is zero by 3rd row
vapid shuttle
Can you try to just follow along with what I said, do you agree with that last part?
lime sluice
we can put any vector of the form <0,0,b> with b nonzero, and it will come out b...
How??
vapid shuttle
perform the matrix multiplication $$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 \ 0 \ b \end{bmatrix}$$
ocean sealBOT
Austin
vapid shuttle
what do you get?
vapid shuttle
so when we input <0,0,b>, we get out, <0,0,0>, and this is our input vector, <0,0,b> scaled by 0. Now do you see?
lime sluice
If last 3rd row is 0,0,0 then we cannot get any scaled matrix
lime sluice
Nope it's not
lime sluice
After multiplying by that 3×3 matrix no?
vapid shuttle
no forget about everything
lime sluice
Okay done
vapid shuttle
forget about your question and everything you've ever known
and consider just this one thing
lime sluice
Sure sure
vapid shuttle
Scale the vector <0,0,b> by 0. What is the resultant vector?
vapid shuttle
Incorrect
Do you know what I mean by scale?
I mean multiply
If I asked you to scale by 2
vapid shuttle
Yes okay
surely <0,0,0>
so anyways let us go back now to the example at hand
I asked you
is there any vectors you can put into this
that come out, scaled by 0
Now, what do you think?
lime sluice
I didn't understand the question
vapid shuttle
Do you understand now?
lime sluice
that come out, scaled by 0
Only 3rd row
vapid shuttle
This is the entire premise of an eigenvector, it is a vector that goes into your matrix and comes out in the same direction it went in (with the possibility of being scaled- scaled meaning multiplied by a constant-)
vapid shuttle
Only 3rd row
so any vectors of the form <0,0,b> correct?
lime sluice
Why b
vapid shuttle
because we can put in
<0,0,1>
or
<0,0,2>
or <0,0,10>
and they all come out
lime sluice
It can be like (b,0,0) (0,b,0) too
vapid shuttle
<0,0,0>
no
it can't
because when we input <b,0,0>
we get out, <b,0,0>
this is not a version of the input vector that was Scaled by zero
it is scaled by one instead
lime sluice
Wait a minute i am confused here a little bit... because we are first talking about 3×3 matrix or 3×1 matric or their multiplication result?
vapid shuttle
$$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}$$ This is our matrix
ocean sealBOT
Austin
vapid shuttle
We are considering vector inputs to it, $$\begin{bmatrix} a \ b \ c \end{bmatrix}$$ like this
ocean sealBOT
Austin
lime sluice
Okay fine.
vapid shuttle
Tell me if you understand the following
lime sluice
I understand these two explanation
vapid shuttle
A vector is an eigenvector of a matrix, if when you input it into the matrix it comes out only being scaled by a constant value.
Do you understand that?
The constant value that the vector comes out scaled by, is called the eigenvalue for that eigenvector.
Example:
$$\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$$ Any vector we put into this, comes out exactly the same. This is to say, all vectors come out scaled by 1. Which tells us that, by our definitions, all vectors are eigenvectors of this matrix, and their eigenvalues are 1.
ocean sealBOT
Austin
lime sluice
Yes this is the definition i understand it but i didn't understand <0,0,b> explanation what was that?
vapid shuttle
when we input <0,0,b> into our matrix we get out <0,0,0>. This is <0,0,b> * 0, meaning that the output we get is the vector scaled by 0. This is true for all values of b
I can't explain it in any simpler terms
vapid shuttle
We are considering for the sake of example, if any vectors come out scaled by 0. The only vectors for our matrix that do this, are of the form <0,0,b>. That is why we are considering them.
lime sluice
If we choose 0,b,0 or b,0,0 the. It will also scaled by 0 because 3rd row is 0,0,0 which makes all the Vectors zero and determinant 0
vapid shuttle
If we choose <0,b,0> we get out <0,b,0>. This IS NOT SCALED BY ZERO IT IS SCALED BY 1 Same for <b,0,0>
lime sluice
Let me check by multiplying
lime sluice
Ohh yes left side we have 1
That will not be 0,0,0
I understand now
Why you were taking 0,0,b so next step now
lime sluice
For null space 🚀
vapid shuttle
to demonstrate when a matrix has an eigenvalue of 0
you understand now that, that matrix has an eigenvalue of 0 right?
because there are vectors, of the form, <0,0,b> that come out scaled by 0.
and this is equivalent to the matrix's nullspace being non trivial
lime sluice
and this is equivalent to the matrix's nullspace being non trivial
I will need to learn these terms
Non trivial means not 0 right?
vapid shuttle
yes
lime sluice
Because 0 always satisfy
lime sluice
Yes true
So null space of a matrix A is set of all the possible vectors x which lead it to 0,0,0
Or any given form of matrix
vapid shuttle
precisely
lime sluice
Is this thing connected to nullity,rank?
vapid shuttle
very connected yes
lime sluice
If you have sometime time please explain
vapid shuttle
all I have time for is this
when the null space is trivial, the matrix is full rank, the matrix's nullity is 0
when the null space is non trivial, the matrix is not full rank, and the matrix's nullity is not 0
lime sluice
I'll need to learn more for these statements. Btw thanks for all the simplest explanations
When null space is trivial then we can take matrix as x= (0,0,0) and it will lead our any matrix A to 0
.close
languid gust
i don't understand this
lone heartBOT
i don't understand this
languid gust
i've made the triangle and found cos x = √7/4
subtle birch
!status
lone heartBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
2
subtle birch
i've made the triangle and found cos x = √7/4
what can you say about its sign?
x lies in the interval pi/2 to pi
heady void
i've made the triangle and found cos x = √7/4
use the fromula for sin(2x)
languid gust
what can you say about its sign?
x lies in the interval pi/2 to pi
its positive right
subtle birch
NO
modern kelp
guys, I finally get into a top-class university.
languid gust
OH WAIT
subtle birch
guys, I finally get into a top-class university.
languid gust
positive or negative?
subtle birch
What do you think
modern kelp
I am going to study courses combined with calculus and that would be a bit hard as what I have heard from my senior schoolmates
subtle birch
Yes
subtle birch
no its negative
so now apply sin(2x) formula
winter light
I am going to study courses combined with calculus and that would be a bit hard ...
Write in #discussion !!
modern kelp
okok
languid gust
hold on hold on
isn't sin positive in the second quadrant
or is it because of the cos
subtle birch
Yes it is
sin(x) is positive since x lies in (pi/2,pi)
2x lies in (pi,2pi)
which is third and fourth quadrant
halcyon monolith
What are the right options and why?
lone heartBOT
What are the right options and why?
vale wigeon
a and c are horribly worded
halcyon monolith
my main doubt was regarding a)
is it true. I know the trace of a matrix is a fixed natural number (right?)
But it's somewhat vague.
b) d) is true too
halcyon monolith
a and c are horribly worded
thank you @vale wigeon exactly my point
vale wigeon
if they meant tr(I_n) = n then it's true, if they meant tr(I_n) = 69420 no matter what n is then it's false
halcyon monolith
they gave some natural number, you went with 69420
sus lol
thank you again
btw, c is true too right?
because the trace will always be k*n where n is the order of the scalar matrix
vale wigeon
yeah ig
halcyon monolith
you have a nice day! this is the second question you have solved for me! I am closing this as soon as @winter light finishes typing 🥲
winter light
But also (b) is correct, right?
winter light
Alright thanks
halcyon monolith
.close
tough lintel
whys the principle argument of z, a complex number, defined between -pi < z < pi?
winter light
Complex numbers cannot appear in inequalities
They are not an ordered field
Hence -π < z < π is meaningless
Unless z is purely real, then it would be ok to write that
tough lintel
Complex numbers cannot appear in inequalities
im trying to question 1 right
and i go okay tan^-1(1/-1) = tan^1(-1)
winter light
Yes
tough lintel
and i get -pi/4
winter light
What are the angles that has that tangent?
Is there another angle that has that tangent?
Remember that tangent has a periodicity of π
winter light
yeah but why do i need to go pi - pi/4
Did you draw the number -1 + i in the complex plane?
tough lintel
yeh its the 2nd quad
winter light
Exactly, and π - π/4 is in the 2nd quad and has tangent = 1
Therefore that's the correct angle
tough lintel
its at 3pi/4
winter light
Yes
tough lintel
the orgional drawinhg
winter light
Wdym with original?
tough lintel
Did you draw the number -1 + i in the complex plane?
this
tough lintel
AHHH okay so arctan is defined on that range
winter light
And so adding π or -π allows you to reach all angles between -π and π
tough lintel
Okay so how am i supposed to realize this without graphing it
Do i need to always think about this when finding this
winter light
That's why it's very important to first draw the complex number on the plane and then you can compute the argument
winter light
Do i need to always think about this when finding this
Yes
There's actually a math function that returns the arg taking this thing into account but I never remember it
tough lintel
You can use cos and sin to solve these angles as well and the same problem would apply like cos is defined same as tan and sin is quad 1 and 2 right
anyways thanks for the help alberto. Gracias!!!
winter light
Let me search it
Here it is, if you'd like @tough lintel
But it's pretty uglier than drawing the number on the plane for me
frosty gull
lone heartBOT
winter light
Better with 1/2
Because with taking 1/3 power you wouldn't cancel any exponents
At least with ^½ you cancel the ² in the x
frosty gull
winter light
oh okay thanks
You're welcome
mellow grail
And now you can cube both sides
*or be mad and try use cubic formula by letting x = a^9
winter light
Lol
alpine sable
instead of trying to remove the power on the right side , try to do it on the left side
so after for second step , just raise both to the power of 3/2
$x^{\frac{2}{3}}= 3^2\(x^{\frac{2}{3}})^{\frac{3}{2}}=(3^2)^{\frac{3}{2}}$
ocean sealBOT
male-blu
alpine sable
from this you get x = 3^3= 27
lost mica
.reopen
How does this work? I understand 91 and 84 is divisible by 7. But where or how did 48 become 4? All I can think of is 48/12 = 4, and the 12 came from 84/7 = 12... i'm confused.
vale wigeon
shoddy phrasing on their part
here is how to write the same idea but better
$\frac{48 \cdot 91}{84} = \frac{(4 \cdot 12) \cdot (13 \cdot 7)}{12 \cdot 7} = \frac{4 \cdot \cancel{12} \cdot 13 \cdot \cancel{7}}{\cancel{12} \cdot \cancel{7}}$
ocean sealBOT
Ann
lost mica
my God, did you just write that in free hand and the bot generated it?
vale wigeon
my God, did you just write that in free hand and the bot generated it?
yes i did
it's a markup language called LaTeX
lost mica
It's been a while since my mind has been blown. Incredible, Ann. Thank you.
So they just* chose factors of the original numbers and cancelled out?
vale wigeon
yeah could put it that way
lost mica
What stops me picking 6 and 8 for (48)?
vale wigeon
$\frac{6 \cdot 8 \cdot 13 \cdot 7}{4 \cdot 21} = \frac{(2 \cdot 3) \cdot (2 \cdot 4) \cdot 13 \cdot 7}{4 \cdot 21}$
ocean sealBOT
Ann
lost mica
ahh i see
so get the denominators cancelled out ASAP
so you can simply multiply what's left on top
Wonderful!
Thank you again. I think that's it.
.closed
.close
frosty gull
lone heartBOT
frosty gull
can i use this method?
frosty gull
what should i write then
gray isle
what were you applying to get from
$$\frac1x = \frac13$$
to
$$\red{-}x = \red{-}3$$
ocean sealBOT
ℝam()n()v
weary quartz
lone heartBOT
weary quartz
Could someone check question 5
ocean sealBOT
Result:
270
no degrees sign, because it's given in the function
Ah okay
Thank you
Whilst your here
How would I do question 6
I’ve not done exact values when it’s sin^2
nimble fern
let's see
since
-1≤sin(x)≤1
what are the ranges for sin²(x)
now
ohhhhh
you just need to calculate
my bad
let's start again
sin²(x) is just sin(x) times sin(x)
@weary quartz
ocean sealBOT
Dyssrupt
nimble fern
so
sin²(750°)=sin(750°)*sin(750°)
weary quartz
so
sin²(750°)=sin(750°)*sin(750°)
ahh that’s kinda what i was thinking just couldn’t put it into words
nimble fern
no problem
weary quartz
.close
lone heartBOT
lone heartBOT
Available help channel!
bright remnant
how come the graph here doesnt go down to infinity like the other ln graphs?
gray isle
note that you have **-**ln(x)
which is a reflection transformation of ln(x) about the x-axis
so instead the graph shoots up to (+)inf there
bright remnant
so instead the graph shoots up to (+)inf there
i understand but why is the range limited for ln(x+1)
when it is not limited for other ln graphs such as ln(x-1)
gray isle
if you're referring to that, its just a shitty graph
lone heartBOT
@bright remnant Has your question been resolved?
grand berry
Problem: Each successive row of a triangular stack of bricks has one less brick until there is only one brick on top. How many rows does the stack have if there are 210 total bricks?
Hello. I don't know how to start solving this problem. I really want guidance on what I can do so I can solve this problem, here are the information I have gathered (please correct me if there's something wrong):
A1 = ?
Nth term (what the question is looking for) = ?
Sn = 210
An = 1
d = -1
granite badger
we can start backwards
grand berry
Backwards?
granite badger
if row 1 has 1 brick, row 2 has 2 bricks and row n has n bricks
grand berry
Yes
grand berry
Oh wait, row with 210 bricks? But it should be total of bricks
granite badger
like
an = an-1 + 1
where n is the number of bricks on row n
if a1 = 1
a2 = a1 + 1 = 1 + 1 = 2
grand berry
Yes
granite badger
do you understand the general pattern
grand berry
Yes I do, but you said the row with 210 bricks. But 210 should be the total of bricks, no?
grand berry
Yes
grand berry
Oh alright
granite badger
ok so
to find the sum of the first x natural numbers, we can use $\frac{x(x+1)}/2$
$\frac{x(x+1)}2$
ocean sealBOT
aysob_ay212
granite badger
(note we will scrap the backwards approach i was talking about)
we know that we have x total number of bricks
in a row
and then to find all number of bricks in each row before x
we can do x + (x-1) + (x-2) + ... + 1
because 1 is the base case
granite badger
we can do x + (x-1) + (x-2) + ... + 1
we know that this must equal 210
granite badger
**aysob\_ay212**
and using this formula
we can do x(x+1)/2 = 210
x(x+1) = 420
x^2 + x - 420 = 0
then solve for x
x = 20, or x = -21, since x > 0, therefore x = 20
does that make sense
grand berry
Yes it does
But how did you know that we can use the formula x(x + 1) / 2 so we can find the sum of the first x natural numbers
or is that just commonly known?..
granite badger
i feel like its commonly known
grand berry
Oh
grand berry
Actually before you were going to present the formula I was doing a trial of error of for example let's say n is 10, n is 20 and so on
But yeah it wouldn't be efficient I guess
grand berry
you should've learnt this in your sequences/series
I see
granite badger
yeah it wouldnt, because if i gave you a huge number, it would take forever
grand berry
10th grade hasn't started for me, I'm just learning in advance
I was learning in math libretext
grand berry
https://www.youtube.com/watch?app=desktop&v=aaFrAFZATKU
Thank you for the help
.close
lone heartBOT
lone heartBOT
Available help channel!
zinc pier
Hello everyone, I urgently need help with this example. There are no errors in the example and the answer should work. (Text on the photo: Solve the equation. If there are several solutions, then indicate the largest root.)
vale wigeon
Hello everyone, I urgently need help with this example. There are no errors in t...
а срочно-то почему?
vale wigeon
просто тут часто так бывает, что люди пытаются здесь на экзаменах/контрольных списать
ладно, проехали. прогресс какой-нибудь с решением уравнения есть?
zinc pier
короче решить мог бы и сам
но походу решения нет
посмотрел график, даже пересечений нет
но на счет условия ничего сказать не могу, что есть пока, то есть
задание из подготовки для экзамена, списать ничего не пытаюсь
vale wigeon
и что он тебе выдал?
vale wigeon
скрин есть?
zinc pier
ща
vale wigeon
решается вполне себе, только надо некоторые нюансы с корнями
zinc pier
zinc pier
я же все правильно ввел
zinc pier
все верно
vale wigeon
да-да, я неправильно прочла
zinc pier
так-то мы здесь таким не занимаемся
а чем тут занимаются
vale wigeon
а чем тут занимаются
ну модельно помощь проводится так: у тебя спрашивают, в чем проблема (не можешь начать / застрял / ответ не тот / не понимаешь какое-то понятие и т.д.), а затем уже именно эта проблема прицельно лечится, чтобы ты мог сам задачу решить
короче да, ты прав, мн-во решений тут пусто
zinc pier
какое множество
vale wigeon
множество решений уравнения
или если просторечно, но нет их
таки все равно photomath'ом некошерно пользоваться
zinc pier
ну так а как я должен был понять то
vale wigeon
please open your own channel. read #❓how-to-get-help for instructions.
this channel is occupied.
vale wigeon
ну так а как я должен был понять то
ну например можно начать с того, что одну корнюгу отнять с обеих частей: $$\sqrt{x + 2\sqrt{x+1}} = 2 - \sqrt{x - 2 \sqrt{x-1}}$$ потом возвести в квадрат, еще похимичить, и в конце концов свести к полиномиальному уравнению. здесь вроде бы всего 2-й степени будет, так что решится все
ocean sealBOT
Ann
alpine sable
please open your own channel. read <#488120190538743810> for instructions.
Ok, thanks
vale wigeon
хотя может вылезти и четвертая степень. вот тогда будет грустно.
alpine sable
@vale wigeon how do i create one for myself
vale wigeon
!help
lone heartBOT
Please read #❓how-to-get-help
alpine sable
Channel
vale wigeon
@zinc pier что характерно, если бы вместо x+1 там было x-1 (как сначала подумала я), то уравнение было бы в разы приятнее. а сейчас мне мерещится какая-то жуть полнейшая.
zinc pier
<@431819465278226433> что характерно, если бы вместо x+1 там было x-1 (как снача...
возможно, не отрицаю
vale wigeon
ты же написала нет решений
нет решений ≠ уравнение невозможно решить
решить уравнение -- значит найти все его решения или убедиться, что таковых не существует
zinc pier
ну так мы убедились же что их нет
vale wigeon
пока еще нет
oak perch
Inside your second square root is a square , so your second square root =|sqrt(x-1)-1|. Since you are looking for the maximal solution we can now let it be (sqrt(x-1)-1). Then your question in the end becomes sqrt(x+1)+3sqrt(x-1)=4. Let t=x-1 this can become quadratic in terms of t
vale wigeon
ну так мы убедились же что их нет
в смысле да, но способ некошерен
там написано черным по белому, что решение (в смысле письменные вычисления) надо прикрепить
думаю, что за скриншот из photomath тебе влетит
zinc pier
да это просто пробник, его никуда не отправляю
я ж просто готовлюсь
ну и хз на счет прикрепление решений, в других заданиях протсо ответ например: х = 2
vale wigeon
не ну ладно, тогда просто надо написать "решений нет". я хз, как в твоей системе это сделать.
oak perch
Can’t understand what you are talking about. Are you saying it’s 2? Because I got 4+1/sqrt(2)
vale wigeon
Can’t understand what you are talking about. Are you saying it’s 2? Because I go...
i am saying that there are no solutions at all
and both photomath and desmos confirm it
oak perch
No solution? Okay I need to check my steps…
Oh indeed … sqrt(x-1)>=1 doesn’t work since x<=2, I had to take sqrt(x-1)<=1, then x+2sqrt(x+1)=(1+sqrt(x-1))^2
sqrt(x+1)=sqrt(x-1), no solution indeed…
ocean sealBOT
Result:
2.3375417889607
I checked 2 is not even a solution
vale wigeon
кажись, в условии опечатка.
oak perch
Sum of two roots=2, so the first root<=2 right, so x<=2, meaning |sqrt(x-1)-1|=1-sqrt(x-1).
vale wigeon
ну это и так понятно было)
-____-
zinc pier
Sum of two roots=2, so the first root<=2 right, so x<=2, meaning |sqrt(x-1)-1|=1...
Condition: enter the largest root in the answer
vale wigeon
largest positive integer root.
oak perch
$$\sqrt{x+2\sqrt{x+1}}+|\sqrt{x-1}-1|=2$$
$$\sqrt{x+2\sqrt{x+1}}+1-\sqrt{x-1}=2$$
$$\sqrt{x+2\sqrt{x+1}}=1+\sqrt{x-1}$$
$$x+2\sqrt{x+1}=(1+\sqrt{x-1})^{2}=x+2\sqrt{x-1}$$
$$\sqrt{x+1}=\sqrt{x-1}$$
No solution
ocean sealBOT
Cogwheels of the mind
zinc pier
Ладно, рас уж мы закончили, всем спасибо) Я просто для друга решение искал, он не мог решить, а потом там оказывается кнопка проверить ответ была)
ну и ну нас третий друг математик сидел и сразу сказал что там ошибка в условии и если второй знак заменить, то выйдет 2
lone heartBOT
@zinc pier Has your question been resolved?
near lion
lone heartBOT
alpine sable
Answer will be D) I can explain in DMS if you want
near lion
Sure
somber eagle
!nosols
lone heartBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
pale kestrel
@alpine sable @near lion this server is not for giving solutions, and please do not offer to dm users.
For future reference
alpine sable
<@456226577798135808> <@827046935331471390> this server is not for giving soluti...
Oh
Then what should we do ?
I'm new so idk what to do I thought we are supposed to help them
pale kestrel
I believe
has some guidelines
uhhh
@real gazelle could u link up guidelines for helping people, not loading for me
real gazelle
!nosols
Bot seems to be slow today
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
pale kestrel
The goal of any helper should be to enable the helpee understand how to do a similar problem, so they don't come back again with it.
alpine sable
As a helper, please do not give out answers that could be copied as a homework s...
Ok sorry
real gazelle
No worries, now you know!
lone heartBOT
@near lion Has your question been resolved?
near lion
No
potent quiver
Find h
potent quiver
angles are 15 and 18
i got 1.24 but im not sure if im correct and theres no answers
lethal belfry
don't you need some side elngths to find h?
lethal belfry
Ah, AB=6?
potent quiver
oak perch
Correct
quick laurel
I am looking for a quick sanity check. I did work breaking down the vectors into their components, but I cannot figure out why this is wrong.
quick laurel
I drew out the vectors as well. My assumption is that I misinterpreted the direction of the wind vector.
Alas, nothing else make sense to me at the moment
Reading the given information, the vector of wind is "S 30(deg) E at a speed of 37mph"
lone heartBOT
@quick laurel Has your question been resolved?
tardy stag
S30°E would be mostly south
like S1°E would be almost entirely south but just a smidge east
quick laurel
that makes sense, I did actaully get to that point and attempt it using 60(deg) from the x axis
and it still didnt work
tardy stag
would be -60°
quick laurel
This is what I have mathwise
So for the wind vector I have, 37cos(-60)i - 37sin(-60)j
and combining the two vectors gives me 131.637i + 145.18j
which apparently isn't the correct answer
tardy stag
why combine them yet? it looks like you just need the wind vector itseld
quick laurel
Because, I had only 5 attempts at that portion and I can no longer change it
and It did not give me the correct answer (annoyingly)
tardy stag
So for the wind vector I have, 37cos(-60)i - 37sin(-60)j
why the - between i and j
quick laurel
Isn't the direction of the vector in the y direction negative
since it points S
therefore, the j component is negative?
tardy stag
yeah but the sin(-30) already does that for you
lone heartBOT
@quick laurel Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
steep drift
am stuck
lone heartBOT
@steep drift Has your question been resolved?
remote heron
@lapis stream
lone heartBOT
<@564524057375277068>
lapis stream
Is c=16-4t-0.25t^2 then plug two into t?
lapis stream
Is c=16-4t-0.25t^2 then plug two into t?
This is wrong because it would entail that c=5
It probably doesn't work because you can't solve a quad equation when it equals anything other than zero
remote heron
this is your teachers work? they dont write very well
whats the left hand limit of B?
remote heron
why is t still there
the limit is along t
think of it like if i gave you f(x) and asked you what the value of f is at x=4
and you told me x+6
that doesnt make sense
f(4) cant be something involving x
4 replaces x
its kinda the same with a limit
if it exists
lapis stream
He just equaled the two sides then plugged two
remote heron
he evaluated the limit
remote heron
4-0.5t?
neither of these are right
whats the left hand limit?
$\lim _{t \to 2 ^-} \qty( 4-\frac 12 t ) = ?$
ocean sealBOT
jan Niku
remote heron
dont worry about me lol
as long as you understand it and can get the answer thats all that matters
lapis stream
I think that I can close this now
.close
Oh u are the one who can open and close channels
remote heron
.close
lone heartBOT
remote heron
lone heartBOT
Available help channel!
mellow skiff
Trying to understand graf (1/2)^x
lone heartBOT
Trying to understand graf (1/2)^x
vale wigeon
well if you raise 1/2 to higher and higher powers you get smaller and smaller numbers don't you
tardy stag
what's (½)³?
young finch
..
young finch
do you know what an exponent is
mellow skiff
Im tryna understand why it goes opposite way
tardy stag
do you know what an exponent is?
young finch
you need to know what an exponent is first
mellow skiff
Yeah ofc
young finch
whats 2^3
mellow skiff
Its a polynom with a positive realla number as bas with a exponent
young finch
Its a polynom with a positive realla number as bas with a exponent
doesn't have to be a polynomial
mellow skiff
8
young finch
do you know property of exponents then
mellow skiff
Can u give example
mellow skiff
Oh
mellow skiff
No i know already
tardy stag
,tex .exp rules
ocean sealBOT
hayley!
mellow skiff
I was thinking diffrently
A bit lost there
I was thinking with a diffrent question
young finch
so does it make sense 1/2^x decreases
mellow skiff
I understand but i dont understand why y is negative
young finch
y isn't negative
mellow skiff
In a graf
young finch
,w graph 1/2^x
ocean sealBOT
young finch
it asymptotes at 0
mellow skiff
Yeah you see that it goes left
young finch
,w compute 1/2^999
ocean sealBOT
young finch
large values of x give a very small but positive result
mellow skiff
,w graph 1/2^9999
ocean sealBOT
young finch
essentially y=0
mellow skiff
What
young finch
it's a line
mellow skiff
I feel dumb why does it go that way
young finch
you just graphed a straight line
young finch
what's 1
mellow skiff
1 should be it’s y
young finch
x=1?
mellow skiff
x=0
young finch
what's that have anything to do with this
young finch
and youre asking why it decreases as x increases
young finch
graph
young finch
between what
mellow skiff
Between 2^x
mellow skiff
Yeah
young finch
obviously
mellow skiff
Why’s that I just don’t get it
young finch
1/2^x is just 2^-x
mellow skiff
Oh shoot
young finch
you would know this if you knew exponent rules
ocean sealBOT
mellow skiff
Why is that so
mellow skiff
Is it too minuscule?
young finch
minuscule?
mellow skiff
,w graph 2^-55
ocean sealBOT
mellow skiff
Why is it a straight line
young finch
cause you graphed y=2^-55
ocean sealBOT
young finch
not a zoom in of y=1/2^x at a certain value of x
ocean sealBOT
young finch
,w graph y=2
ocean sealBOT
mellow skiff
Is it going to left because
The exponential is negative and the base get higher up to y when base is higher
,w graph -2^-x
ocean sealBOT
alpine sable
What does Instantaneous Velocity mean?
lone heartBOT
What does Instantaneous Velocity mean?
whole shell
What does Instantaneous Velocity mean?
if you were accelerating
instantaneous velocity is the velocity at one specific point in time
alpine sable
Is it calculated in real time?
tardy stag
speedometer
alpine sable
So technically it’s impossible to calculate instantaneous velocity
whole shell
So technically it’s impossible to calculate instantaneous velocity
well its the same as saying its impossible to know any objects true mass
you can still get very close
alpine sable
well its the same as saying its impossible to know any objects true mass
Oh ok
So how would you get instantaneous velocity?
Is there like a specific formula
tribal valve
winter light
Inst. velocity is the limit of that as b -> a
tribal valve
yeah i just realized
winter light
And it should be with vectors but yeah that's the definition of derivative
alpine sable
distance travelled / time taken is the basic way
Isn’t that Average Velocity
whole shell
yes
tribal valve
this is better
whole shell
but if you take average velocity
over a very small time frame
that becomes closer to instantaneous
alpine sable
I’m pretty confused what’s the difference between Instantaneous Velocity and Average Velocity?
whole shell
velocity
winter light
In other words, instantaneous velocity is the first derivative of position @alpine sable
tribal valve
thats a good way to put it
tribal valve
basically just take a super close point to the right and to the left and find velocity between them
whole shell
I’m pretty confused what’s the difference between Instantaneous Velocity and Ave...
instantaneous velocity is the average velocity over a infinitesimal time frame
its not possible to actually measure this over literally 0s time, but you can get close by measuring over 0.01s time or something
alpine sable
In other words, instantaneous velocity is the first derivative of position <@456...
You can calculate that with data right?
Previous data
alpine sable
Interesting
winter light
Wow interesting
whole shell
unfortunately quite literally everything you measure has a delay
because of the speed of causality
winter light
Wow interesting
Didn't know that
whole shell
so youd be measuring the instantaneous velocity of however long it took the light to go from your radar gun to the object and vice versa
alpine sable
unfortunately quite literally everything you measure has a delay
So like instantaneous
I could view it as a way of getting the exact velocity within a given set of distance precisely
Which is practically impossible
But you can still get very close
Am I right?
Also @whole shell @winter light thanks for helping me
If there’s anything I can do in return tell me
whole shell
I could view it as a way of getting the exact velocity within a given set of dis...
im lost as to what you mean by this
alpine sable
im lost as to what you mean by this
Like instantaneous is the velocity at a specific point in time and you can never get that velocity 100% accurately but still very close
Right?
whole shell
Like instantaneous is the velocity at a specific point in time and you can never...
yh
alpine sable
Ok thanks
whole shell
also not just accurately though
like even if yourr radar gun has 100% precision
youll only ever measure the velocity the object had a little while ago (well the time is very very very small, but it is a small lag)
so you can never measure it in "real time"
whole shell
@alpine sable if you were measuring the speed of a planet 5 light years away from you
youd be measuring the speed it had 5 years ago
alpine sable
<@456226577798135808> if you were measuring the speed of a planet 5 light years ...
Oh
That’s a pretty good way of putting it
That makes sense
Isn’t calculating average velocity the same then?
Because
Average velocity isng 100% accurate too
whole shell
we have 2 seperate problems
the fact that we cant get this accurate velocity to be instantaneous
because we cant measure in 0 time
and the fact that anything we do measure is from the past
whole shell
because we cant measure in 0 time
even radar guns cant detect the exact frequency change
thats just how life is
but other than those 2 issues that you will never really feel
yes
youll be very close
to the answer you want
as in your velocity in real time
alpine sable
as in your velocity in real time
So technically it’s also impossible to get a perfect average velocity too
whole shell
So technically it’s also impossible to get a perfect average velocity too
well its impossible to measure the time or distance youve taken 100% accurately
but the time delay issue doesnt work here, since average velocity already refers to something in the past
alpine sable
Okay thank you for your help it’s very much appreciated
@whole shell what song do you like?
I’ll play it for you in return
whole shell
wdym
alpine sable
I play piano for anyone that helps me solve a problem
Of their specific song choice
whole shell
rush e 
alpine sable
I’ll try
whole shell
@alpine sable i was joking, the actual song would be twinkle twinkle little star, specifically variation 1
timestamped
once you see rush E youll know why i did the troll face
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@456226577798135808> i was joking, the actual song would be twinkle twinkle lit...
i cant really do it lol
but thanks for the help
.close
alpine sable
Ask the employee at the gun range
alpine sable
what country?
night geyser
lone heartBOT
Wtf mods
dull dove
Why are you entertaining the behavior
lone heartBOT
Available help channel!
tribal valve
(a) Prove that if the roots of
[x^3 + ax^2 + bx + c = 0]form an arithmetic sequence, then $2a^3 + 27c = 9ab.$
(b) Prove that if $2a^3 + 27c = 9ab,$ then the roots of
[x^3 + ax^2 + bx + c = 0]form an arithmetic sequence.
ocean sealBOT
dabbingpotato
tribal valve
Roots are z-r, z, z+r
$(x-(z-r))(x-z)(x-(z+r)) = \x^3-(z-r)x^2-zx^2+(z-r)zx-(z+r)x^2+(z-r)(z+r)x+(z+r)zx-(z-r)z(z+r)$
tardy stag
\\ for newline
ocean sealBOT
dabbingpotato
tribal valve
Now we use vietas?
a = -3z
b = 3z^2 - r^2
c = - z^3 + zr^2
am i on the right track?
tardy stag
b = 3z^2 - r^2
you might consider simplifying that a bit...
tribal valve
z^2 - rz + z^2 - r^2 + z^2 + zr
3z^2 - r^2
Now we substitute those values into $2a^3 + 27c = 9ab$?
ocean sealBOT
dabbingpotato
tardy stag
yep
tribal valve
$-54z^3 + 27zr^2 - 27z^3 = -81z^3 + 27zr^2$
ocean sealBOT
dabbingpotato
tribal valve
is that right?
Okay so now part b)
We know that this is true, so how can we use this to know that the roots form an arithmetic sequence?
oak perch
Prove that if they don’t form an arithmetic sequence then 2a^3+27c doesn’t equal 9ab
This time three roots are z-r, z, z+s where r-s is not zero
tribal valve
Prove that if they don’t form an arithmetic sequence then 2a^3+27c doesn’t equal...
I don't think thats what part b is asking though
oak perch
P->Q is equivalent to (not Q)->(not P)
Show 2(3z+s-r)^3+27(z^3+(s-r)z^2-rsz) doesn’t equal 9(3z+s-r)(3z^2+(2s-2r)z-rs)
When r-s is not zero
tribal valve
That doesn't prove B tho
Just because when 2a^3 + 27c doesn't equal 9ab doesn't form an arithmetic sequence
That doesn't mean that when 2a^3 + 27c is equal to 9ab always has to form an arithmetic sequence
oak perch
P: 2a^3+27c=9ab Q : form an arithmetic…
P->Q is equivalent to (not Q)->(not P)
Not Q: they don’t form an arithmetic sequence
Not P: 2a^3+27c doesn’t equal 9ab
tribal valve
P->Q is equivalent to (not Q)->(not P)
How do you know
tribal valve
what is logic
oak perch
tribal valve
can you give me another example of when this works
oak perch
x is an integer-> x is a rational number
x is not a rational number-> x is not an integer
oak perch
No
tribal valve
isn't that a contradiction
tribal valve
oh
okay i guess that works
$2(3z+s-r)^3+27(z^3+(s-r)z^2-rsz) = 9(3z+s-r)(3z^2+(2s-2r)z-rs)$
ocean sealBOT
dabbingpotato
tribal valve
pain
tribal valve
okay so we use vietas again?
tribal valve
**dabbingpotato**
how we get this tho
tribal valve
so veitas
oak perch
Yeah
So in the end it becomes proving 2(s-r)^3 doesn’t equal -9rs(s-r)
When s-r doesn’t equal 0. Obvious
tribal valve
okay
is this right for the vietas?
$a = - 3z + r - s$
$b = 3z^2 - 2rz + 2sz - rs$
$c = z^3 + z^2r + rzs - sz^2$
ocean sealBOT
dabbingpotato
tribal valve
**dabbingpotato**
also shouldn't it be a and c not -a and -c
tribal valve
So in the end it becomes proving 2(s-r)^3 doesn’t equal -9rs(s-r)
did you get this by just powering through the expandsion
oak perch
Show 2(3z+s-r)^3+27(z^3+(s-r)z^2-rsz) doesn’t equal 9(3z+s-r)(3z^2+(2s-2r)z-rs)
...
I expanded this
Coefficients of z^3,z^2,z are equal respectively
Only the constant term of LHS doesn’t equal the constant term of RHS left
Which is proving 2r^2+2s^2+5rs doesn’t equal 0 when s doesn’t equal r
tribal valve
is $2r^2+2s^2+5rs \neq 0$ just on one side $2(s-r)^3 \neq -9rs(s-r)$
ocean sealBOT
dabbingpotato
