it's asking to rewrite the integral over y
#help-10
versed pecan
i don't know why we need the second part of the equation
my answer was the first part only.
I have drawn the entire thing and suppose there's a division because of some triangle, but I thought the first integral covered the entire volume
obtuse pebbleBOT
@versed pecan Has your question been resolved?
left jetty
how did they get the circled 60 degree angle?
obtuse pebbleBOT
how did they get the circled 60 degree angle?
brisk matrix
do you have the original question?
left jetty
yeah one sec
left jetty
do you have the original question?
knotty gulch
brisk matrix
the only thing you’re given is that the angle between the two vectors is 60
left jetty
they just chose 60
wait why can they choose any angle?
brisk matrix
you weren’t given any absolute directions
just the relative angle between the two vectors
if you were told emily kicked it 10 degrees above the horizontal, then you could make something out
but from what you were told you don’t know the absolute direction for either of them
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
fresh copper
just want to check answers ( i dont have an answer sheet)
obtuse pebbleBOT
just want to check answers ( i dont have an answer sheet)
fresh copper
i got -5, -5/5, -5/15
also i want to know if my method is correct
(if theres a better method)
heres how i worked it out:
ax+5, since a is a positive integer the x roots go like -5/1, -5/2, etc
3x+b, same thing, -1/3, -2/3, etc
Thus, we can narrow down to > -5, and since 3x+b does hit every negative root integer
we have -5, and -5/5 (-1)
Now all possibilities are -1/3 and -2/3, which only -1/3 can be converted into -5/15 (with the -5 on the top)
Therefore, the possible x coordinates are -5, -1, and -1/3
obtuse pebbleBOT
@fresh copper Has your question been resolved?
oak aurora
the better way to solve it would be to notice that ab=15
in addition to your answers, i got -5/3 too
your method should have gotten you -5/3 too, you just have to list it out abit more
-5/1,-5/2,-5/3
-1/3,-2/3,-3/3,-4/3,-5/3
oak aurora
y=ax+5
y=3x+b, we do some algebra here, and we get ab=15
fresh copper
yes
oak aurora
if ab=15, and a b are positive integers, there are only 4 possible combinations
a=3,b=5
a=5,b=3
a=15,b=1
fresh copper
oh i see
oak aurora
a=1,b=15
fresh copper
yep yep
oak aurora
then solve from there, its better imo because theres less guessing involve, so you just know theres 4 answers
noble pilot
is this just saying to rewrite this but using i instead of n
obtuse pebbleBOT
is this just saying to rewrite this but using i instead of n
brisk grove
I guess so
primal falcon
I need help with this... are there any mathematical geniuses?
Consider a hypothetical mathematical construct called a "dimensional transcendental curve." This curve exists in an n-dimensional space, where n is an integer greater than or equal to 3. The curve is defined by a set of complex equations involving advanced mathematical concepts such as hypercomplex numbers, fractal geometry, and non-Euclidean geometry.
The question is as follows:
Given the specific equation that defines a dimensional transcendental curve in a 7-dimensional space, along with a set of initial conditions, determine the exact number and positions of all points where the curve intersects a 5-dimensional hyperplane defined by another complex equation involving differential forms, Lie groups, and tensor calculus.
This question requires an advanced understanding of multiple branches of mathematics, including higher-dimensional geometry, complex analysis, differential geometry, and abstract algebra. Only individuals with extensive knowledge and expertise in these areas would have a chance to solve such a complex problem.
obtuse pebbleBOT
I need help with this... are there any mathematical geniuses?
Consider a hypoth...
obtuse pebbleBOT
@primal falcon Has your question been resolved?
obtuse pebbleBOT
@primal falcon Has your question been resolved?
obtuse pebbleBOT
@primal falcon Has your question been resolved?
trail musk
you're using a lot of words but not really saying anything
haha
you've not provided any clear definitions
give me an example of this so called "DTC"
craggy raptor
this some chat gpt bs
forest sinew
timid silo
they are chat gpt
obtuse pebbleBOT
@primal falcon Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
quaint finch
obtuse pebbleBOT
high lily
!status
obtuse pebbleBOT
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
rich silo
high lily
yeh, diagram is wrong
rich silo
Same angle, why did you elevate one
quaint finch
cuz I didn't understand what it meant by "angle of elevation from the receiver to each disc"
rich silo
i see!
Allright, so now you have the right one
What do you suggest to find the height difference?
keen lodge
question says: The apothem of a regular hexagon of side length 8 cm a√3, where a is a constant to be determined. Give the value of a.
obtuse pebbleBOT
question says: The apothem of a regular hexagon of side length 8 cm a√3, where ...
latent fjord
in order to solve this you need , first the angle from the property of hexagon which will help you to find the hypotenuse and then by pythagoras theorem you will find value of apothem
keen lodge
is the angle from the property 120?
latent fjord
yeah but you need a angle bisector to form the triangle
which gives the angle to be 60
keen lodge
ohhh okyokay
latent fjord
if the two angle are 60 then the third will be too thus giving you a equilateral triangle
silk epoch
obtuse pebbleBOT
wintry swift
are there twox in the sketch (or one x and a mark for the center)?
silk epoch
the angle x
not the centre
I've tried some of the circle theorems I still can't find the answer
rich plume
is this the centre?
slow topaz
silk epoch
is this the centre?
Yes
slow topaz
use the bottom right one and you can probably solve it easily ya
slow topaz
<3
rich plume
Find this angle
rich plume
no
silk epoch
not that one
rich plume
no
rich plume
noo
silk epoch
because angle at the circumference is half the angle at the center
rich plume
This is 2x
rich plume
you're assuming the wrong angle to be 2x
rich plume
Yes
obtuse pebbleBOT
@silk epoch Has your question been resolved?
wind sierra
obtuse pebbleBOT
wind sierra
How do I do the first part?
I know I can multiply out the matrix and get rid of an x
But that doesn't seems to help
fathom flicker
So, when you multiply out row1/row2/row3 * <x, y, z>^(T) what are the equations that you get?
Like if you were to perform that matrix multiplication, and then set it equal to that column vector, what are the 3 equations that form?
wind sierra
2x-9y-8z=48 (1)
5x+10y+6z=-10 (2)
2x+y+kz=13 (3)
the i do (3)-(1)
which gets me 10y+(k+8)z=-35
that's where i got up to
fathom flicker
I would focus only on equation 3
it is the only equation that involves k
actually, nvm
wind sierra
ummm
timid silo
Row reduce the augmented matrix
wind sierra
Uhhhh, lemme just search those terms up
fathom flicker
the augmented matrix is just when you attach the column vector that the matrix is set equal to, using a line through the matrix
this would represent 3x+5y-z=10..... etc in the vocab of your problem
can you see?
wind sierra
i get it
but, is there another more algebraic way. As we definitely have not been taught this method
well first, ill acc wait and see where this goes
fathom flicker
this is completely algebraic
fathom flicker
by row reducing, he means to perform row operations (scaling and addition/subtraction) to the current rows
to take it into RREF
yes
fathom flicker
timid silo
To get into rref
fathom flicker
the words sound complicated, but the idea behind the methods is to actually simplify the amount of work needed to be done
wind sierra
So its essentially what I did at the beginning, but just get it to only be z?
fathom flicker
you'll see that this is just a much more efficient and neat way of solving the system
fathom flicker
So its essentially what I did at the beginning, but just get it to only be z?
yes. Find one of the variables (it could be z) in terms of k, and then anywhere you see this one you can substitute back in its value in terms of k and start solving for one of the other two variables left, and then repeat
wind sierra
right. Thank you both very much!
fathom flicker
np!
timid silo
You're welcome
wind sierra
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
sleek fulcrum
i got -9x^2 + 176x -560 = 0
obtuse pebbleBOT
i got -9x^2 + 176x -560 = 0
obtuse pebbleBOT
@sleek fulcrum Has your question been resolved?
sleek fulcrum
.close
thick phoenix
obtuse pebbleBOT
thick phoenix
I do not understand a).
cerulean root
E
I think it's easier if you were to rearranged the equation of a) to the original equation
But let me try to solve it first
crimson breach
<:kongouDerp:483988392099446794> I do not understand a).
you have to just show the given equation is true
whats the matter ?
crimson breach
just write the first equation in standard form and then multiple it by a phi
thick phoenix
is phi the symbol?
thick phoenix
oh
limber crane
limber crane
ur welcome
sage geyser
obtuse pebbleBOT
sage geyser
for part b, i dont understand why it isnt a horizontal translation to the left by 3 units
the function that has been graphed is 2sqrt(4-x)-2
4-x means it is shfted to the right by 4 units
in order to obtain the graph y = 2sqrt(1-x)-2 shouldnt the original graph be translated by 3 units to the left?
since 1-x means that it is shifted, from the origin, to the right by 1 unit
timid silo
the function that has been graphed is 2sqrt(4-x)-2
for b
if f(0) = 0
and f(x-3) = ?
what value of x would you need for f(x-3) to equal zero
sage geyser
x = 3
yes
sage geyser
but it is asking for the transformations required from the graph of y = f(x)
doesnt that mean that f(x) must be messed around with to get y = 2sqrt(1-x)-2
plain stag
I think I agree that it should be 3 units left.
f(x-3) = 2sqrt(4-(x-3))-2 = 2sqrt(4 - x + 3) - 2 = 2sqrt(7 - x) - 2
timid silo
doesnt that mean that f(x) must be messed around with to get y = 2sqrt(1-x)-2
$$ f(x) = 2\sqrt{4-x} -2 $$
$$ g(x)= 2\sqrt{1-x} -2 $$
$$ f(x-3)= 2\sqrt{4-(x-3)} -2$$
$$ f(x-3) = 2\sqrt{4-x +3} -2$$
$$ f(x-3) = 2\sqrt{1-x} -2$$
plain stag
4 + 3 =/= 1
timid silo
wtf?
timid silo
I guess i'm confused on what their question is even asking 😭
fluid snow
Can anyone give me a one-variable definite integral that WA/Mathematica/... has difficulties with determining the closed form?
obtuse pebbleBOT
Can anyone give me a one-variable definite integral that WA/Mathematica/... has ...
fluid snow
Thanks
fluid snow
Does that have a closed form?
$\sin(\pi \cdot \frac{4}{35})} - \sin(\pi \cdot \frac{11}{49})$ is close but not it
drifting wraith
yeah
fluid snow
elfin burrow
try the integral of e^(x^2) or sin(x)/x
drifting wraith
wolfram knows them obviously
elfin burrow
,w integral 0 to 1 sin(x)/x
warm shaleBOT
drifting wraith
Does that have a closed form?
$\sin(\pi \cdot \frac{4}{35})} - \sin(\pi \cdot \f...
brave oxide
Why is this converging? Because we get a number as a value and not infinity?
brave oxide
I’m not sure actually if it’s diverging or converging
The only thing I know is that when it’s infinity it’s diverging
Can come one help?
lean ore
Why is this converging? Because we get a number as a value and not infinity?
yes, if you get a number as a value, it's converging if you get infinity or negative infinity, it's diverging
brave oxide
Just that
lean ore
yes, that is the definition of converging/diverging
brave oxide
Wouldn’t you only get infinity if one of the bounds is infinity?
How can you get infinity with finite bounds
lean ore
for example 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
you're adding infinite terms
but it converges to 2
because it gets small very fast
1 + 1 + 1 + 1 + 1 + ... diverges
brave oxide
That’s not what I’m saying
I know that a function with infinity bounds can be converging
My question is how can a function without infinity bounds be diverging
lean ore
oh, wait, my bad
that's also possible
like 1/(x^2)
from -1 to 1
it has infinite area
for example, you know that the integral of 1/x from 1 to infinity diverges right?
this shows that an integral with finite bounds exist, that has infinite area, and that it's almost the same like integrals with infinite bounds that have infinite area
obtuse pebbleBOT
@brave oxide Has your question been resolved?
obtuse pebbleBOT
brave oxide
Thanks @lean ore
obtuse pebbleBOT
Available help channel!
weak meadow
obtuse pebbleBOT
@weak meadow Has your question been resolved?
candid yarrow
Do you know what the graph looks like at that point?
fluid snow
A similar question has been asked on Math StackExchange, here:
https://math.stackexchange.com/questions/2034169/point-of-discontinutinuity-fx-x-lfloor-frac-1x-rfloor?noredirect=1
The first answer states that the left limit to 0 is the same as the right limit to 0 and gives a proof, thus, it should be a removable discontinuity at 0.
@weak meadow
weak meadow
A similar question has been asked on Math StackExchange, here:
<https://math.sta...
Did you mean 1 ?
fluid snow
Did you mean 1 ?
I mean at x = 0
weak meadow
I should also find the type of discontinuity here
I can make an educated guess and say that it's jump but not sure how do I prove it?
fluid snow
I can make an educated guess and say that it's jump but not sure how do I prove ...
Why do you guess it's a jump discontinuity?
Have you looked at the graph?
weak meadow
no
I just guessed because it seems to me that for x>1 the limit is e and for x<1 the limit is e^-1 = 1/e
I am not sure since I can't prove it
( the limits exist and they're different )
fluid snow
I just guessed because it seems to me that for x>1 the limit is e and for x<1 th...
The limit has been asked here https://artofproblemsolving.com/community/c7h1109059p5043304 and here https://math.stackexchange.com/questions/1568430/limit-of-lim-x-to-1-x-frac1x-1?noredirect=1, it's e from both sides, thus it should be a removable discontinuity
drifting wraith
he doesn't it's a free service from alphabet or something
fluid snow
how do you find these im impressed
This post on Math StackExchange describes how to search efficiently: https://math.meta.stackexchange.com/questions/29265/how-to-search-on-this-site
Mostly Approach0 and google
Alphabet?
There is also SearchOnMath
I use the former two usually, though
drifting wraith
third largest company in us apparently
fluid snow
he doesn't it's a free service from alphabet or something
Is that a search engine?
weak meadow
I see
drifting wraith
it's a company that provides a free search engine
fluid snow
it's a company that provides a free search engine
Haven't heard of that yet, not sure if that offers LaTeX support for searching.
Usually googling with site:math.stackexchange.com is enough, if not, then Approach0 is usually more accurate and also searches through ArtOfProblemSolving
drifting wraith
lol ok
fluid snow
Math StackExchange also has a chatroom dedicated to searching
fluid snow
Haven't heard of that yet, not sure if that offers LaTeX support for searching. ...
Approach0 also has some search operations like \exact{}, ..., I think it's described in the post I linked
obtuse pebbleBOT
@weak meadow Has your question been resolved?
frail depot
Probably a really simple question, but how was this calculated lmao?
obtuse pebbleBOT
Probably a really simple question, but how was this calculated lmao?
frail depot
I tried to put everything on the right side
just plussing the 1/9 matrix and subtracting the other one but thats not the same, so how did they do it like on the picture
median dome
write the first matrix as 1/9 (9 0 -9 0)
median dome
yes
median dome
Why?
so that you can take 1/9 common from both terms and then subtract
the two matrices
vectors*
frail depot
Might be stupid
But why couldn't i just ignore the 1/9
subtract the two
and then after put the 1/9 in
median dome
well
frail depot
so that you can take 1/9 common from both terms and then subtract
Because that sounds like fraction stuff hahaha
worn lily
What am i doing wrong
median dome
the 1/9 changes the inside
median dome
What am i doing wrong
make another channel
worn lily
Sorry bad connection
median dome
Because that sounds like fraction stuff hahaha
you can multiply the 1/9 inside the vector if that seems easier
frail depot
the 1/9 changes the inside
Nope xD
oh and then you mutiply the other vector with 9 to get it to have the same denomitor
and then you can subtract them?
Is that what's going on?
Or wait
frail depot
no but i confused myself again
lmao
So like
You don't just want me to multiply with 9 here
Wouldn't that be illegal
Sionce it's no longer the same vector
I could multiply with 9/9
so something like this?
frail depot
But the -2/9
median dome
now you can subtract
frail depot
is wrong
$\left[\begin{array}{c}
\frac{9}{9}
\
0
\
-\frac{9}{9}
\
0
\end{array}\right]-\left[\begin{array}{c}
\frac{5}{9}
\
-\frac{2}{9}
\
-\frac{4}{9}
\
\frac{6}{9}
\end{array}\right]=\left[\begin{array}{c}
\frac{4}{9}
\
0
\
-\frac{5}{9}
\
0
\end{array}\right]$
warm shaleBOT
SimonWin
frail depot
Nono
I completely messed that up
but watch
Actually.. no almost
haha
0 - (-2/9)
that's + then right?
frail depot
what you wrote
$$
\left[\begin{array}{c}
\frac{9}{9}
\
0
\
-\frac{9}{9}
\
0
\end{array}\right]-\left[\begin{array}{c}
\frac{5}{9}
\
-\frac{2}{9}
\
-\frac{4}{9}
\
\frac{6}{9}
\end{array}\right]=\left[\begin{array}{c}
\frac{4}{9}
\
\frac{2}{9}
\
-\frac{5}{9}
\
-\frac{6}{9}
\end{array}\right]
$$
warm shaleBOT
SimonWin
frail depot
AAh! And then that's why it isn't -1/9 anymore
$$
\left[\begin{array}{c}
\frac{9}{9}
\
0
\
-\frac{9}{9}
\
0
\end{array}\right]-\left[\begin{array}{c}
\frac{5}{9}
\
-\frac{2}{9}
\
-\frac{4}{9}
\
\frac{6}{9}
\end{array}\right]=\left[\begin{array}{c}
\frac{4}{9}
\
\frac{2}{9}
\
-\frac{5}{9}
\
-\frac{6}{9}
\end{array}\right]=\frac{1}{9}\left[\begin{array}{c}
4
\
2
\
-5
\
-6
\end{array}\right]
$$
warm shaleBOT
SimonWin
frail depot
obtuse pebbleBOT
@frail depot Has your question been resolved?
fresh copper
obtuse pebbleBOT
obtuse pebbleBOT
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
1
im on step 1
- I don't know where to begin
what is happening
<@&286206848099549185>
loud socket
What’s wrong
fresh copper
i legit have 0 clue how to start
i dont know how to solve it
can somebodyhelp me
i have the translation on top of the problem
i have no clue where to start
bronze osprey
lets say x for the longer side of the rectangle
fresh copper
yes
bronze osprey
and y for the other side
fresh copper
mhm
bronze osprey
can you express an equation that involves both x and y
so we can write y in terms of x
it is given that we have 96m of fence
fresh copper
6x + 2y = 96
bronze osprey
wait
are fully doing all the rectangles or if we already did one side for another rectangle before, we are not making it again?
like EF
it is a long side for both of the rectangles there
are we taking it as x or 2x
what do you think
fresh copper
wait
fresh copper
are fully doing all the rectangles or if we already did one side for another rec...
dym
fresh copper
are we taking it as x or 2x
as just x
fresh copper
but its not that easy because
bronze osprey
we can say x=2y too
fresh copper
we need 3 equal rectangles with a permieter for 96, while all 3 rectangles must have the maximum area possbile
bronze osprey
by looking at the visual
fresh copper
i think the visual is misguiding
bronze osprey
hmm
fresh copper
x doesnt nessisarily have to be 2y
or its too simple
also it should be solved with quadratics
because its a set on quadratic problems
fresh copper
yes
fresh copper
yep
bronze osprey
y=16-x
fresh copper
i still dont see how quadratics would be involded though
fresh copper
yes
fresh copper
triangle???
fresh copper
0, 16
bronze osprey
so x*(16-x)
fresh copper
-x^2 + 16x
bronze osprey
this is the function of our area
fresh copper
still same thing
bronze osprey
now we should find the maximum
fresh copper
0, 16
fresh copper
now we should find the maximum
oh
bronze osprey
these are the zeros
fresh copper
8, 64
bronze osprey
we should take derivative and set it equal to zero
or just arithmetic mean of zeros
fresh copper
y -= 8
bronze osprey
then y=8
fresh copper
y = 8*
couldnt we have done this faster
since we know that the total is 16
and 16/2 = 8
and the least difference creates the greatest area
thus 16/2 = 8 yields a difference of 0
bronze osprey
yes but the proof is by this
fresh copper
meaning an area of 64 is the greatest
bronze osprey
it is how its done
fresh copper
oh
fresh copper
okay
fresh copper
A to D is 16?
bronze osprey
there is something really wrong with the visual
bronze osprey
if we go by the visual then the total area is (y+x)(2y)=(16-x+x)(32-2x)
fresh copper
okay
bronze osprey
yeah something really wrong with the visual
or i am missing something
it says that 2y=x
if we go by the visual
fresh copper
yeah
sorry my firend translated it, i have no clue if he missed infromation
lets say 2y = x
bronze osprey
and are we fully making the rectangles
thats a point too
do we go by the same lines again or just make them once
fresh copper
once
bronze osprey
if we are just making them once and not going again then it becomes even more complicated
we can say x+2y+2y+x+x+x+2y=96
6y+4x=96
fresh copper
uh
fresh copper
but the visual is misguiding
but then why is our answer wrong
it says its 18
@bronze osprey apparently
three chicken kernels…
wait
4x + 6y = 96
2/3x + y = 16
y = 16 - 2/3x
what the shit
oh
times
-16x^2+384x
12, 2304
x = 12
y = 8
huh????
6x + 2y = 96
y = 48 - 3x
-3x^2 + 48x
x = 8
y = 24
WHAT THE FUCK BRO
HOW TF
<@&286206848099549185> please help me i cant solve it at all, i have the answer
the answer is 18, and ive gotten 16 and 20 closest
do y know
how u solt it
maximum area
i dont know why google translate
gives width
i think so
unless my friends korean sucks
i think u have to follow the diagram of layingn them out
..?
no, 3 saame rectangles
yes but the 3 rectanles are placed
fresh copper
<@471009973208350720> apparently
like this
so some sides dont count
thanks
did u get 18 becuas
the andwer is 18
length of AD
i also got 16
i dont know how on earth its 18
yes
i think its the way the problem is translated
jm not korean
my friend translated and hes not really a specific or precise nor accurate person
im gonna find skmeine else
just for fun
as a prActice cuz ive never seen a quadratic priblme like this
yeah
i dont see how its 18
yea
did u use quadratics?
its meant ti be a quadratics roblem
yea i see
lets try that
what
he jusy told me all 3 same size
oh well
well that makes it even harder
i have no ckue
clue
how on earth
also the diagram is very confusing
yea
also whetehrr to omit the extra lines????
i have absolutely 0 clue
i dknt think its too overly complex, ive tried somilar level problmes
they are just manipulating quadratics and their equations
ues
yes*
fresh copper
ohhh
when yoy say derivative cant you find vertez form?
i havent learned derivaties yet
so just get the vertez?
the maximum
which is…?
can u give me the equation of the parabola pls
i xan get it
x = 18
18, 288
ur right
kind
kinda
ur A to D is
3/3x or x
and x is 18
can you run me through again
please
thank u so mich
that too please
alright
yes
yes
alright
okay
warm shaleBOT
marc
fresh copper
mhm
warm shaleBOT
marc
fresh copper
3x = a?
warm shaleBOT
marc
fresh copper
other wya around
yes
yeah
sorry its modnight her
yes
hes
yes*
genius
yes
ans tben get a parabola
warm shaleBOT
marc
fresh copper
when you multiply
so you sepereate y
or x
and u mutioly
which gets you a parabola
warm shaleBOT
marc
fresh copper
yes
3y = 96 - 8x/3
yes
and now
you can dk
yes
so this is the side lengbrs
lengths* of the entire rectangle
mhm
mhm
warm shaleBOT
marc
fresh copper
3y = 96 - 8x/3
this becomes
y = 32 - 8x/9
and then you did
x * (32-8x-9)
which got you the paarabaola
now convert to vertez from whcih is the maxmimum (what we want)
thus we get x = 18
great
thanks!!! so much
i leanred a ton
thanks again, bye’ have a good day or night
.close
supple epoch
Why is the characteristic polynomial $\chi_A (t)$ of a nilpotent matrix $A \in \mathbb{C}^{n\times n}$ equal to $t^m$ for a $m \in \mathbb{N}^+$? Like I get that the eigenvalues are equal to zero but why is the characteristic polynomial $t^m$?
obtuse pebbleBOT
Why is the characteristic polynomial $\chi_A (t)$ of a nilpotent matrix $A \in \...
warm shaleBOT
Levens
kind hawk
well what else should it be
if it had any other factor then it would have a corresponding eigenvalue
supple epoch
idk like how do you get to that point
supple epoch
the minimal polynomial is clearly t^k
wait why could u explain
kind hawk
if m_A is the min poly then clearly m_A divides t^k cause A^k=0 by definition of nilpotent
where k is the relevant number so that this is satisfied
k minimal
and then m_A cant have a smaller degree cause otherwise k wouldnt be minimal
so m_A=t^k
supple epoch
could u explain minimal polynomials to me? Or if u have good links on it cause I still dont get them; or can't really imagine what they are
kind hawk
well they are the smallest (degree-wise) polynomial that give 0 if you plug the matrix into it
I'm not sure what you mean with "imagine"
they are just a useful tool
supple epoch
well they are the smallest (degree-wise) polynomial that give 0 if you plug the ...
hm i see.. so yeah i get why it would be t^k with k \leq m.. but for the characteristic polynomial-- could u elaborate on how its a "mulitple" of the min polynomial
kind hawk
by cayley hamilton you have that p_A(A)=0 where p_A is the char polynomial
and then by a theorem you have that m_A divides all polynomials f with f(A)=0
by euclidean division
supple epoch
But then isnt e.g. (t-2)t^k divisible by t^k?
kind hawk
if it had any other factor then it would have a corresponding eigenvalue
yes but then A would have an eigenvalue of 2
obtuse pebbleBOT
@supple epoch Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
thick torrent
could someone explain to me how my book integrated this equation to get this answer.
obtuse pebbleBOT
could someone explain to me how my book integrated this equation to get this ans...
thick torrent
p_s, z, and epsilon are constants - I know how they moved those out. a_z is a vector so it's not included in the integration.
I used u-sub to sub u=r^2+z^2
brave bramble
You do the inner integral first
thick torrent
Did not get same result as them though
Yea, but I don't know how they integrated that to get -1/sqart(r^2+z^2)
brave bramble
Might need to see your work
thick torrent
Sure
brave bramble
Should be u^(-3/2)
thick torrent
Oh, good catch!
I will fix that up quickly and see if I get the same result as the book
I tried again and I got the same result 
Question, I think switching the order of integration would still give the same answer here, but is this true for double/triple/etc integrals in general?
And also, what would be a relevant theorem answering this question?
obtuse pebbleBOT
@thick torrent Has your question been resolved?
analog coyote
obtuse pebbleBOT
analog coyote
not really sure where to go from here
pretty sure i have the right approach though
obtuse pebbleBOT
@analog coyote Has your question been resolved?
analog coyote
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
flint kelp
obtuse pebbleBOT
flint kelp
I can't seem to get the right combination here
since the graph shown is f'(x) I'd assume that means
f'(x) is greatest at x=4 and is lowest at x=2... because the graph visually shows that
f''(x) should be greatest where the slope is steepest positive... so x=3
and least where slope is steepest negative, so x=1
not sure about f(x) ones though
actually I see now that the whole function is positive so f(x) is greatest at x=6 and least at x=1 I think
but this is still wrong
omg they want you to write x=x1 etc 🙄
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
winter gorge
someone explain for me pelase
obtuse pebbleBOT
someone explain for me pelase
undone wing
yeh i got it
winter gorge
however i dunno how it reached the very last d/dx function with 3-8(f(x))
undone wing
but without completing the square u can also get the answer
winter gorge
its too long tho
well id be usimg quiotent
btw this is ther answer booklet not my own answer
undone wing
use quotient rule
undone wing
dude did not do quotient rule bruh
winter gorge
but wait its not the derivative thats hard to find its the
3-8f(x)
im not seeing how it got to that
from 3f(x)^2-8/x+1)^2
makes it neater
more impressive for examiner
warm shaleBOT
marc
undone wing
quotient rule would be fast in this case
winter gorge
ahhhhh!!!
undone wing
idk
winter gorge
the base
undone wing
so i can do a/b and b/b
u did that already
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
thick torrent
I've been asked to prove this proposition using induction.
My base case was letting n=2 (as you can take intersection with minimum 2 sets), and then proved that the intersection of A_1 and A_2 would contain a.
However I am confused by how to do the induction step. Am I supposed to replace the n with k, and say assuming that P(k) is true, try to prove P(k+1)?
obtuse pebbleBOT
I've been asked to prove this proposition using induction.
My base case was let...
thick torrent
But if n (or k here I guess) is the set of all natural numbers, what's k+1 supposed to be then?
gleaming ridge
use induction hypothesis
zenith raft
treat the intersection as one set
thick torrent
Well I know to plug in k+1. I just don't know what this is supposed to represent
A set with 1 element more than the natural numbers?
gleaming ridge
notice that $\cap_{i=1}^{n+1} A_i = \cap_{i=1}^{n} A_i \cap A_{n+1}$
warm shaleBOT
numbpy (anti glomed)
gleaming ridge
you want to show that a in in the lhs
which is same as showing a in in rhs
use induction hypothesis now
thick torrent
Ok thanks, let me give it a try
rocky goblet
But if n (or k here I guess) is the set of all natural numbers, what's k+1 suppo...
n isn't the set of all real numbers, it's a natural number
$n \in \mathbb{N}$ means $n$ is a natural number
warm shaleBOT
bee [it/its]
thick torrent
But it said for all n in N, doesn't that mean all natural numbers?
fathom flicker
yes, and all natural numbers is not the same as the real numbers
zenith raft
tbh
fathom flicker
for example, -1 is a real number and not a natural number
zenith raft
this is just true by definition of intersection
thick torrent
Wait, but I didn't mention real numbers-
rocky goblet
But if n (or k here I guess) is the set of all natural numbers, what's k+1 suppo...
yes you did
rocky goblet
anyway
n is still a natural number, not the set of all natural numbers
thick torrent
So basically, they mean n as in some random natural number
Huh ok lol I think my understanding of notation was fundamentally flawed there lol
gleaming ridge
n here means a fixed natural number
rocky goblet
"all n in N" just means the rest of the statement is true if n = 0 (if that's a natural number) and true if n = 1 and true if n = 2 and true if n = 3 and so on
thick torrent
Thanks everyone, I think I have an idea how how to proceed with the hypothesis now
gleaming ridge
zenith raft
no induction proof: $$a\in A_n \text{ for every } n\iff a\in \bigcap_{i=1}^\infty A_i,$$ and $$\bigcap_{i=1}^\infty A_i\subset \bigcap_{i=1}^n A_i$$ for any $n$
warm shaleBOT
💜𝓁𝒶𝓎𝓁𝒶💜
gleaming ridge
no induction proof: $$a\in A_n \text{ for every } n\iff a\in \bigcap_{i=1}^\inft...
proving a weaker statement using a stronger statement 
dusk pasture
numbpy help me pls
thick torrent
**💜𝓁𝒶𝓎𝓁𝒶💜**
Would proving the first line (for some n) be sufficient/equivalent to proving the same for all natural numbers?
The second line is kind of related to my question I suppose, since there are infinite natural numbers
thick torrent
Oh sorry, clarification, proving n for the original proposition I posted
zenith raft
wdym proving n?
zenith raft
proving a weaker statement using a stronger statement <:catThink:526983414226747...
well they give a strong statement lol
thick torrent
This, but instead of using n, we use N (all natural numbers)
zenith raft
wdym instead of n?
thick torrent
So union of (i belonging to N) A_i
zenith raft
intersection?
zenith raft
idk about "instead of n"
gleaming ridge
This, but instead of using n, we use N (all natural numbers)
Noooooo, n is a fixed natural number
thick torrent
No I get that I was just wondering if proving the original proposition could extend to another problem
warm shaleBOT
jdanvers
thick torrent
the latex thing it just generated
gleaming ridge
for example we know that $n + 0 = n$ for all $n \in \mathbb{N}$
warm shaleBOT
numbpy (anti glomed)
zenith raft
maybe this is easier... let $n\in \bN$. then $a\in A_1, A_2, \ldots A_n$ by hypothesis (the hypothesis just says something stronger, so we also have this)
warm shaleBOT
💜𝓁𝒶𝓎𝓁𝒶💜
zenith raft
the question is the same as this: if $a\in A_k$ for all $k\in\bN$ then for all $n\in\bN$, $a\in\cap_{i=1}^nA_i$
warm shaleBOT
💜𝓁𝒶𝓎𝓁𝒶💜
thick torrent
Wait nooo I do understand the original question
zenith raft
**💜𝓁𝒶𝓎𝓁𝒶💜**
do you understand this is the same thing?
thick torrent
**💜𝓁𝒶𝓎𝓁𝒶💜**
I was just wondering if proving that would also mean that this is true
thick torrent
do you understand this is the same thing?
Yes, I know that's true
warm shaleBOT
jdanvers
thick torrent
is true?
zenith raft
idk exactly what you're asking, but the hypothesis in the question is equivalent to what you wrote anyway
zenith raft
**jdanvers**
like these two things are exactly the same, by definition
zenith raft
kind of?
thick torrent
The latex phrase generated above, is it referring to the intersection of all natural numbers i?
zenith raft
sure?
zenith raft
like these two things are exactly the same, by definition
it's defined to mean exactly this though
thick torrent
Ah ok. The hard part for me would be actually proving that then
Well thanks! I have made this question more confusing through weird phrasing no doubt
zenith raft
lol np
thick torrent
.close
gleaming rose
can i get some help?
sudden garnet
can i get some help?
leave your math problem here and people will try to help if they can
royal basin
close your prev channel
royal basin
yes it is
gleaming rose
what do i need to change?
royal basin
can you post the original problem statement?
its a bit unclear from your work what youre even doing
ok so you got $V = \pi \int_0^3 \frac{4}{\sqrt{16+x^2}} \dd{x}$ and are trying to integrate that
warm shaleBOT
Ann (glomed)
gleaming rose
yes using trig sub
royal basin
,w 4pi * int[0,3] dx/sqrt(16+x^2)
warm shaleBOT
royal basin
well, it would appear you have the right answer.
gleaming rose
im lord sauron
royal basin
are we to understand this answer or work was rejected?
or were you simply unsure whether you'd made a mistake
gleaming rose
nah i was just unsure if i made a mistake
was trying to understand if that four PI was right
because when i put it into a AI after
it said 2 Pi
so it had me confused
viral blade
chatgpt can't do math
don't rely on it whatsoever for correct answers
its wrong more often than it's right
obtuse pebbleBOT
@gleaming rose Has your question been resolved?