||It is, but you should ask yourself why||
#help-39
dusky widget
stoic imp
surely it is sir
dusky widget
Alright, then if you proved it, you can be certain!
tropic saddle
you just wanted to prove that?
why didnt you just switch A and B in the first line then
stoic imp
you just wanted to prove that?
no
stoic imp
you just wanted to prove that?
it was part of a exercise wanted to check if it's commutative
quickly did this and got lost in the last step
but that's just another definition of the sym diff
stoic imp
why didnt you just switch A and B in the first line then
alright thank you for the help lads
.close
pearl pondBOT
shell brook
you're welcome
sterile python
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static isle
Helo
static isle
How did this happen
heavy onyx
is there more context
mortal seal
Please don’t divide by 0
static isle
How did this happen
I don't get how they wrote it in normal form
heavy onyx
is the comma , meant to be an equal to =?
static isle
It would've made sense if it was = instead of comma
Is that just a printing error
Oh okaay, thank youu
pastel umbra
No, the line is both equations (separated by a comma) being true at the same time
heavy onyx
i see
pastel umbra
It's the intersection of two planes
mortal seal
No, the line is both equations (separated by a comma) being true at the same tim...
Wait nvm you’re right
I’m tired sorry
pastel umbra
This is what Google says
Incidentally, it's why Google just threw up here and divided by 0 like that were a sane thing to do 
mortal seal
Another proof of why not to trust ai in maths
static isle
Another proof of why not to trust ai in maths
Ikrrr
heavy onyx
in that case, you already know it (the lines) intersects at y=1 so try using (x-1)/3 = -y and find the x coordinate too. the z coordinate is fixed by z+1=0
to find a line perpendicular to both lines, try to get the direction vectors of line 1 and line 2 by converting it into vector form
static isle
Okaay
halcyon fjord
I found ef to equal 2 and vertical height to be 12. how do I find the area of adm. I think I need vertical height but idk how ot get that.
pearl pondBOT
I found ef to equal 2 and vertical height to be 12. how do I find the area of ad...
@halcyon fjord Has your question been resolved?
autumn fossil
I found ef to equal 2 and vertical height to be 12. how do I find the area of ad...
Do you know this red length?
halcyon fjord
i do not know that yet
halcyon fjord
not really, i'm guessing to get line ac, you'll need to cord bash?
autumn fossil
I dont think that'll be necessary, ill try to figure sth out
oh, ADM and BMC are equal
so we only need to find the height at which M is
and that seems doable by similar triangles
halcyon fjord
oh, ADM and BMC are equal
how do we know that?
autumn fossil
its a property of trapezoids, the "left and right" triangles formed by the diagonals are equal in area
halcyon fjord
ah ok
autumn fossil
you can look at the triangle BCD and ACD
they are clearly the same area, and if you subtract the overlap, you get BMC and AMD. So they must be equal in area too
halcyon fjord
so we need to find the height from m to the 2 bases?
autumn fossil
for that, try looking for some similar triangles
halcyon fjord
and the area of triangle bad is 48?
autumn fossil
,calc 8 * 6
jolly parrotBOT
Result:
48
yeah
halcyon fjord
for that, try looking for some similar triangles
I think you can prove it by just aaa thanks to the diogonals? and the scale factor is 14/8
autumn fossil
indeed
halcyon fjord
is the height from m to ab 4.36? x + (14/8)x = 12
autumn fossil
,calc 12 * 8/22
jolly parrotBOT
Result:
4.3636363636364
ok ya, the answer is 336/11 ft
jolly parrotBOT
Result:
30.545454545455
damn it
jolly parrotBOT
Result:
30.545454545455
alr thanks
autumn fossil
np
halcyon fjord
how do u close a channel?
autumn fossil
.close
halcyon fjord
.close
pearl pondBOT
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shadow elm
pearl pondBOT
shell brook
what have you tried
tardy flicker
!status
pearl pondBOT
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
jolly parrotBOT
random ermine
try u = 1/x
random ermine
anywhere
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wooden vessel
How do I find the Area of the trapezoid?
pearl pondBOT
How do I find the Area of the trapezoid?
hollow jolt
@wooden vessel Do you remember the condition for a line to be tangent to a circle?
(/what that statement that I made even means)
lethal crane
jolly parrotBOT
M0UNT_Z10N
jolly parrotBOT
M0UNT_Z10N
pearl pondBOT
@wooden vessel Has your question been resolved?
wooden vessel
<@807187212494962708> Do you remember the condition for a line to be tangent to ...
Hey, sorry for the late response
pearl pondBOT
Hey, sorry for the late response
wooden vessel
I don't exactly get what you're asking
The intersection has to be perpendicular is guess
Making a right angle
Nevermind, figured it out
The sum of the parallel sides of a rectangle in which a circle is drawn have to be the same
.close
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green aurora
pearl pondBOT
green aurora
How do you switch from the photo on the left to the one on the right?
rough forge
Do you know what this means?
green aurora
No
rough forge
You can read it from right to left; take the first row and multiply it by 0.6, subtract it from the second row. Then replace this result with the current second row
green aurora
You can read it from right to left; take the first row and multiply it by 0.6, s...
Yes
tropic saddle
what is your message "yes" supposed to indicate. do you now know whats going on?
tropic saddle
0.10 times what = 0.06 ?
rough forge
When you do stuff with matrix, your goal is often to row reduce into row echelon form and derive from that properties off the matrix
green aurora
0.10 times what = 0.06 ?
I meant why did you subtract R1 and put a coefficient of 0.6 multiplied in front?
tropic saddle
well to get the zeros
green aurora
Yes
I suppose the same thing was done here
So the matrix is independent because for example rows 2 and 3 are not multiples of the 1st
Right?
tropic saddle
the second row is 0 times the first row
pearl pondBOT
@green aurora Has your question been resolved?
pearl pondBOT
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desert jackal
I have a question. Why can't I distribute and mutiple exponent with it's +,-.
desert jackal
prime bramble
I have a question. Why can't I distribute and mutiple exponent with it's +,-.
well, there's not much of a reason in as much as it simply isn't true 
(x + y)^2 is not equal to x^2 + y^2; you can try a few examples with small numbers to see this 
the right thing to do here instead is to use the distributive law 
(x + y)^2 = (x + y)(x + y) and then distribute (or FOIL, if you've heard that before :p)
desert jackal
I'm kinda stupid with it, I always fail with something easy such as forgot to conjugate with the portion is radical and many more, Lol. Whatever, Thank for answer.
left steppe
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
left steppe
,tex .freshman
jolly parrotBOT
! sniper3d107
shell brook
No Junping Xi you are not stupid
left steppe
**! sniper3d107**
why can’t math be like this 😞
desert jackal
Whatever, I will try my best. Thanks
tame topaz
why can’t math be like this 😞
$(x + y)^2 = x^2 + y^2$ is true over $\bZ / p \bZ$ :D
jolly parrotBOT
left steppe
$(x + y)^2 = x^2 + y^2$ is true over $\bZ / p \bZ$ :D
what if p = 3?
worldly jewel
did you mean (x+y)^p
jolly parrotBOT
higher!
prime bramble
,, 1^2 + 1^2 = 2 \equiv 2 \mod 3
jolly parrotBOT
higher!
prime bramble
did you mean (x+y)^p
that's what I suspect they did 
amber pagoda
ye
left steppe
I'm kinda stupid with it, I always fail with something easy such as forgot to co...
!done
pearl pondBOT
I'm kinda stupid with it, I always fail with something easy such as forgot to co...
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@desert jackal Has your question been resolved?
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sonic sleet
pearl pondBOT
sonic sleet
How do I start it?
pearl pondBOT
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lyric glen
<@&268886789983436800>
quick venture
pearl pondBOT
quick venture
can someone explain how we get fxy
worldly jewel
f_xy is y then x
but since the function is twice-differentiable, f_xy = f_yx anyway so it does not matter which order you take them
quick venture
but how do i do it though?
becus wouldnt it be this:
like wouldnt it be 2 not 3?
becuase we differentiate 2x
summer imp
You do once differentiated with x and then with y
fx = 2x + 3y
Differentiate that in y
3
quick venture
oh y then x
summer imp
The point is you do once with one variable and once with the other
As the other helper pointed out for twice differentiable functions those will actually always match
So $(f_x)_y=(2x+3y)_y = 3 = (2y + 3x)_x = (f_y)_x$
jolly parrotBOT
Azyrashacorki
quick venture
ah okay
pearl pondBOT
@quick venture Has your question been resolved?
south ice
Hello I have a question
In problems like projectile motions with vector functions, why is it that we work with components separately when solving for things? For example if I’m tryna find the maximum height why do I take the derivative of the j component and not the whole thing? Isn’t the entire function what’s responsible for tracing the projectile motion’s path?
clear pivot
projectile motion is a combination of motions in x and y plane (z plane too in 3d)
here, the height is defined on y plane (j vector), so you would take derivative of the j component only if you need maxm height
@south ice did you understood what i said
south ice
<@1402083806034268161> did you understood what i said
Mb I didn’t see your message
south ice
here, the height is defined on y plane (j vector), so you would take derivative ...
The height isn’t defined by the motion of the object ?
Which that in and of itself is defined by the vector function as a whole?
Like the entire path is dependent on x and y no?
clear pivot
when an object moves in a plane it can have various accelerations and velocities. To make the computation easier we divided the plane in various axes and do the computation on those axis.
imagine a ball, if you give it a push in direction of i vector, it will move in that direction
clear pivot
what is f(x)
clear pivot
yeah, could be
south ice
<x,f(x)>
Converting it to vector form
Then you’d have to derive the j component here as well
If im not wrong
south ice
you are confusing with them
Wdym
clear pivot
I mean that the i component and j component do not depend on each other in this case.
They depend on the time t
clear pivot
(x,f(x))
In this statement you were claiming that the j component depends on i component but it not be true
summer imp
<x,f(x)>
The position has components s(t) = (x(t), y(t)). Those two aren't, in general, dependent on each other, except by virtue of both being functions of t.
If you frame this as a maximization problem, you're just trying to find the time, which maximizes the height (which is by definition y(t), a function from R to R). This has no inherent link with the horizontal component.
So really you're just solving for t where the derivative of the height is 0, i.e. where y'(t) = 0, just like you would maximize any function from R to R.
south ice
The position has components s(t) = (x(t), y(t)). Those two aren't, in general, d...
But j on its own can’t graph that path right
summer imp
But j on its own can’t graph that path right
The path is just two coordinates which are both functions of t
You’re just maximizing the second one
south ice
You’re just maximizing the second one
Right
Okay why do we differentiate the second one specifically
summer imp
Because that’s the height.
The height is the second component
You maximise y(t) just like any function. Set derivative to zero
south ice
I see
south ice
Because that’s the height.
But if I graph j on its own I won’t get the same curve right
Or would I ?
summer imp
If you graph y(t) it’ll just give you a function giving the height at every moment
south ice
If you graph y(t) it’ll just give you a function giving the height at every mome...
Will it look the same though ? It wouldn’t right
summer imp
It wouldn’t be the curve because it’s just the second component of the curve
Graphed against t
south ice
It wouldn’t be the curve because it’s just the second component of the curve
Right exactly
So our original curve needs the i component as well
summer imp
The curve does because that gives you a point at every time.
But they’re not asking about the curve, they’re asking specifically about the second component.
south ice
The curve does because that gives you a point at every time.
But the curve modules height
No?
I don’t see how it’s only the second component and not both
Since the graph is traced via the two components
summer imp
The graph is traced by two components. They are asking you to maximize the second component
The second component is a function of t
You maximize it like any function
summer imp
But the curve modules height
The height is just y(t)
The module would give you the distance from the origin to the point (x(t),y(t))
south ice
The height is just y(t)
It’s just a visualization thing I’m stuck on
But I guess I’ll figure it out
Do you know here why in part b they didn’t just set the vertical component = 0?
Why did they set it = -100
And not just 0
summer imp
The way they’ve set up the problem would be with the x axis at the level of the cliff rather than the ground.
south ice
The way they’ve set up the problem would be with the x axis at the level of the ...
I thought so too
Btw if I’m asked how far a distance traveled before reaching the ground
You find the time when the vertical component = 0?
south ice
Btw if I’m asked how far a distance traveled before reaching the ground
Horizontal distance btw
summer imp
Well in b you found the time time it took to reach the ground.
south ice
Yea
summer imp
Then if x(0) = 0 was the initial horizontal position,
x(the time you found in b) is the final horizontal position, which is also your horizontal distance.
south ice
Then if x(0) = 0 was the initial horizontal position,
x(the time you found in b...
Do you plug for x only or the whole thing?
Cuz they plugged for the whole thing
Which is why I’m asking
summer imp
Well plugging that in s(t) just gives you the point at the end, so the x component of it gives the horizontal distance
south ice
Well plugging that in s(t) just gives you the point at the end, so the x compone...
Ohhh
Yea
pearl pondBOT
@south ice Has your question been resolved?
south ice
For this problem, after finding t such that v_0cos(θ)=0, why do we plug t back into the vertical component?
I think I figured it out
pearl pondBOT
@south ice Has your question been resolved?
pearl pondBOT
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keen plaza
I'm trying to figure out the number and type of roots for a problem " x^5 + 4x^3 = 0". I keep getting 3 roots but i thought the number of roots was supposed to be the highest exponent? so far i factored to x^3 (x^2 + 4) =0 and solved from there to get roots of 0, 2i and -2i. I'm unsure whether this is right and im being stupid, or if theres supposed to be 5 roots
summer imp
I'm trying to figure out the number and type of roots for a problem " x^5 + 4x^3...
The degree of the polynomial gives the number of complex solutions up to multiplicity.
glacial bluff
there are 5 roots, but not all of them are distinct.
summer imp
Since x^3 is a factor of you polynomial, x=0 is a triple root (it counts as 3 roots).
keen plaza
OHHH that makes alot of sense
thank you
i was sitting here wondering what i did wrong lol
.close
pearl pondBOT
keen plaza
.reopen
pearl pondBOT
✅ Original question: #help-39 message
keen plaza
wait would the category for those be imaginary roots or do they just not get mentioned?
like its not three real roots because its the same number
split void
You can't really differentiate between the three equal real roots.
If you do factorize the polynomial, they'll show up as factors
split void
theres at least 2 imaginary from the +-2i
yes its degree 5
keen plaza
ill just leave it blank and if my teacher gets on me about it i'm gonna ask what she wants me to do
glacial bluff
it's 3 real roots and 2 complex, but of course those 3 real roots are the same (multiplicity 3).
pearl pondBOT
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Available help channel!
keen plaza
pearl pondBOT
keen plaza
What do I do from here? I’m so confused as this doesn’t seem factorable
glacial bluff
,rccw
jolly parrotBOT
summer imp
Do synthetic / long division since you know a root
keen plaza
Would I divide by three? Teacher barely covered this and we’re quizzing on it tommorow
tardy flicker
Divide the function by x-3
keen plaza
Oh no it’s factors again
I can do that
Just annoyed she didn’t tell us that’s what we were supposed to do
lowk just doomed at this point
i'm trying to synthetic divide, is the remainder/final number a root?
or do i now have to write it out again and then factor more
pearl pondBOT
@keen plaza Has your question been resolved?
pearl pondBOT
shell brook
i unironically ended up helping 😞
keen plaza
was lowk very helpful
pearl pondBOT
Available help channel!
iron basin
how is this what
pearl pondBOT
how is this what
iron basin
would (sum of squares of roots) x (sum of roots) work?
shell brook
it should
iron basin
and how would i evaluate the weird residue $\sum \alpha \beta^2$
jolly parrotBOT
pearl pondBOT
pearl pondBOT
✅ Original question: #help-39 message
iron basin
i think i got the answer but it seems to mismatch
so $(\sum \alpha^2)(\sum \alpha) = (\sum \alpha^3) + (\sum \alpha \beta ^2)$
jolly parrotBOT
iron basin
and then $(\sum \alpha)(\sum \alpha \beta) = 2(\sum \alpha \beta^2)$
jolly parrotBOT
iron basin
then $\sum \alpha^3 = 0$ but the given answer is 9
jolly parrotBOT
pearl pondBOT
@iron basin Has your question been resolved?
plush bramble
Did you try using sum root^3 = sum (3 - 1/root) and then vieta's to find sum(1/root)
iron basin
ooh
i forgot that works
i think thats the intended method since it matches
ill troubleshoot from here
thank you
.close
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void flume
I really need someone to explain how to antiderivative
pearl pondBOT
I really need someone to explain how to antiderivative
void flume
I don’t get the whole theirs over ours thing
I understand regular anti derivative but I missed a couple days and now I’m posts
Lost*
versed mica
I don’t get the whole theirs over ours thing
theirs over ours?
versed mica
pretty straightforward
void flume
I thought it was 3x^4/2
versed mica
just power rule after rewriting this
void flume
+c
versed mica
yes
void flume
Wtf
versed mica
🤔
void flume
So 2x-1)^3 /3
void flume
Why though
versed mica
since the derivative of 2x-1 is 2
torn flicker
Yeah
void flume
Like I got a problem 2x(3x^2)
Why not here?
torn flicker
You differentiate a constant its 0
void flume
Yes?
versed mica
Why not here?
because the derivative of x is 1
the thing being raised to the power is what mattered
in that case it was just x
here it’s 2x - 1 so it’s different
void flume
But the last one 2 x’s were raised to powers
versed mica
🤔
void flume
2x(3x^2)
versed mica
$2x(3x^2) = 6(x^3)$
jolly parrotBOT
knief
versed mica
constant multiple rule allows us to bring out this 6
versed mica
and as you see it’s just x being cubed
torn flicker
Then dy/dx = 18x^2
versed mica
so 6 times x^4/4 or 3/2 x^4
void flume
Why’s it divide on one and multiple on the other
versed mica
don’t know what you mean
versed mica
so 6 times x^4/4 or 3/2 x^4
here?
void flume
Actually no
I get that
But why don’t you divide by the derivative then
Because it’s not a chain?
versed mica
where?
void flume
2x(3x^2)
versed mica
what should i be dividing?
we divide by 1
but again like not really
we divide when we perform a substitution
versed mica
here it might be easier to just expand
versed mica
why
void flume
X^4/4
versed mica
go on
void flume
Ok
Idk if my teacher is putting the answers down wrong or smt
But this sucks
Left is answers he gave us
So I’m slowly going through them trying to understand how
And his whole there’s over ours thing too
void flume
Like he makes us take the first part of the function
And then second part
Just the x’x
X’s*
Then divide them by each others derivatives
Then solve and multiply by that at the end
versed mica
a bit vague
versed mica
what’s this x/2x
void flume
Can u see prob 7
versed mica
yes
void flume
Ya that’s where he loses me
versed mica
where is 2x coming from
void flume
The derivative of the second x
versed mica
so the x^2 + 2
void flume
Ya
versed mica
ok so he’s just complicating u sub
void flume
Please god less complicate it😭
versed mica
in all of these integrals you have some function being multiplied by its derivative or a constant times its derivative which looks exactly like what happens in the chain rule.
so you just have to identify these pairs to find your right u sub
void flume
How do u know it’s one of these pairs
Well ig if the x^s r the same power when doing it
If they aren’t can u not do them?
versed mica
- x is a constant times the derivative of x^2 + 2
- 6x is the derivative of 3x^2 + 1
- 5x is a constant times the derivative of 4x^2 + 7
- x + 1 is a constant times the derivative of x^2 + 2x
- 3 is the derivative of 3x - 5
- here just expand, no u sub
- 1 is a constant times the derivative of 2x - 1, but you can also just expand if you want
in all of these cases we look at the composition of functions and identify the inner function and see if we multiply by its derivative or a constant times its derivative
remember what the chain rule is about, it’s about differentiating a composition of functions
multiplying by the derivative of the inner function
if we see that happen in our integrand then we can easily identify what must have been differentiated
versed mica
we have other techniques
and sometimes the u sub isn’t as straightforward
and it actuakly takes multiple substitutions along with algebraic manipulation to get it into a form we like
this is why i don’t like your teachers explanation
i think he should just teach u sub the traditional way
the way he teaches it assumes it’s always in this precise form
pearl pondBOT
@void flume Has your question been resolved?
vagrant kraken
How do u do this q
pearl pondBOT
How do u do this q
vagrant kraken
and also i cant rlly visualise why for a circle to intersect, r1-r2 occurs but i can understand why r1+r2 is required
sterile python
How do u do this q
It is form (x-a)^2 + (y-b)^2 +(z-c)^2 =r^2 where centre (a,b,c) and radius r
If d> r1+r2 then the circles dont contact = is contact at 1 point
then to intersect the distance must less then sum of 2 radius
vagrant kraken
If d> r1+r2 then the circles dont contact = is contact at 1 point
yea i get that
sterile python
For example you have 2 circles and they touch at 1 point, d =r1+r2
vagrant kraken
is r1-r2 like this
sterile python
vagrant kraken
sterile python
Correct
sterile python
Could be other way around like u drew
sterile python
If d <|r1-r2| then the circles inside eafh other
Basically the same i guess but have absolute value since d cant be negative
vagrant kraken
ahh ok
sterile python
Nw
vagrant kraken
i usually get qs w ith two spheres that have the same x,y or y,z or x,z tho so it's a simultaneous eqn wioth only 1 variable coordinate
would u just expand everything?
and then equate?
sterile python
Its like formula (x-a)^2 + (y-b)^2 = r^2
This formula u given have z, seems more like a sphere to me
Like this
vagrant kraken
This formula u given have z, seems more like a sphere to me
how do u find centre and radius?
sterile python
how do u find centre and radius?
Chat. I sent u the photo
The radius is square root of rhs
And stick to the formula😭
vagrant kraken
like r=4 r=5 for s1 s2 respectively?
pearl mauve
like r=4 r=5 for s1 s2 respectively?
u must subtract the equation then simplifiy to get the plane of intersection first
vagrant kraken
u must subtract the equation then simplifiy to get the plane of intersection fir...
o-o
what eqn do i subtract
like i got soemthing in terms of x y z and constnat =0
what do i do after?
vagrant kraken
u must subtract the equation then simplifiy to get the plane of intersection fir...
4x−3y+4z−10=0?
pearl mauve
4x−3y+4z−10=0?
Find the vector eq of the line between the centres next right since the plane is perp to line between centres
And so this would imply something right do yk what?
Or what do u think it implies
pearl pondBOT
@vagrant kraken Has your question been resolved?
vagrant kraken
Find the vector eq of the line between the centres next right since the plane is...
vagrant kraken
Find the vector eq of the line between the centres next right since the plane is...
wait so does this equation 4x−3y+4z−10=0 repreesent a plane that the circle lies on or perpendicular to the circle plane
next step would be to sub in i j k components of line eqn into the plane equation?
so i get lambda
and then i sub it back into r
?
vagrant kraken
wait so does this equation 4x−3y+4z−10=0 repreesent a plane that the circle lies...
wait why does this work tho
to obtain the centre
pearl mauve
wait so does this equation 4x−3y+4z−10=0 repreesent a plane that the circle lies...
Yea basically the intersection circle is inside the plane
pearl mauve
and then i sub it back into r
Yea bc the intersection of the vector line of centres of both spheres with the plane is exactly the centre of the intersection circle
vagrant kraken
Yea bc the intersection of the vector line of centres of both spheres with the p...
ohh
is the line eqn perpendicular to the plane then?
pearl mauve
is the line eqn perpendicular to the plane then?
Yes
vagrant kraken
or does it run along the plane
pearl mauve
or does it run along the plane
No perp
Its perpendicular
U can check urself look at the direction vector and the normal of the plane
vagrant kraken
like this?
vagrant kraken
But remember the intersection circle lies on the plane
like this?
vagrant kraken
Hope i helped! 🥹
YESSS thank u
pearl mauve
like this?
Yupp
vagrant kraken
also h o wwould u find x?
pearl mauve
also h o wwould u find x?
Do u mean radius?
vagrant kraken
pearl mauve
You could just take the difference between the sphere radius and distance from sphere centre to the intersection centre
vagrant kraken
You could just take the difference between the sphere radius and distance from s...
THANK U 🙏 🙏
pearl mauve
np! Lol
sterile python
Nice
sonic sleet
pearl pondBOT
honest oyster
Whats your question?
sonic sleet
How do I prove line p and m are parallel?
honest oyster
Prove BOP+OPD=180°
sonic sleet
why?
Because parallel lines interior angles sum?
Ahha nice
I can see easily it would be 100
So sum is 180°
Thanks
.close
wild fable
i am confused if we want to compute alpha * beta why do we start from beta first, like right to left why dont we do alpha first then beta
pastel umbra
i am confused if we want to compute alpha * beta why do we start from beta first...
cf. matrices, you're applying a transformation from the right
ABv is the matrix B applied to the vector v, then A applied to the result Bv
unborn abyss
if we want to compute f º g we start with g
pastel umbra
Similarly, ... ^
wild fable
if we want to compute f º g we start with g
oh yeah
pastel umbra
It's a composition of permutations
wild fable
ABv is the matrix B applied to the vector v, then A applied to the result Bv
but these arent matrices right
theyre just notation for function
pastel umbra
No, but they're the same sorta thing in that they get applied onto something
wild fable
.solved
pearl pondBOT
pastel umbra
so beta would apply on an arrangement, then alpha would
unborn abyss
they can be expressed as matrices if you want, like swapping row elementary operations
pearl pondBOT
Available help channel!
silent fractal
uh hi everyone
pearl pondBOT
uh hi everyone
silent fractal
i need help bout math
narrow matrix
just post the question, it is easier for everyone
pearl pondBOT
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.
sterile python
What was the question about?
sterile python
I know but what question asking, identifying elements or..?
honest oyster
The union doesnt remove the elements that exists in both P and Q tho
placid geyser
maybe they confused it with n(A U B) = n(A) + n(B) - n(A intersection B) lol
honest oyster
Probably
pearl pondBOT
@silent fractal Has your question been resolved?
flint basalt
pearl pondBOT
flint basalt
i got to $\int \int_D \sqrt{(u^2+v^2)(u^2+1)} dA$ but dont know where to go from there
jolly parrotBOT
Krish
flint basalt
well, $\int_0^1 \int_0^{\frac{\pi}{2}} \sqrt{(u^2+v^2)(u^2+1)} dv du$
jolly parrotBOT
Krish
leaden cape
sqrt(x^2 + y^2) transforms to u not sqrt(u^2 + v^2)
flint basalt
is it not $(u \cos v)^2 + (u \sin v)^2$
jolly parrotBOT
Krish
flint basalt
which factors u^2(cos^2 + sin^2)
leaden cape
So now it should be a nice integrand in the end with that fix
flint basalt
for my final answer i got $\frac{\pi (2\sqrt2 -1)}{6}$
jolly parrotBOT
Krish
flint basalt
or $\frac{\pi \sqrt2}{3} - \frac{\pi}{6}$
jolly parrotBOT
Krish
flint basalt
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pearl pondBOT
pearl pondBOT
Available help channel!
royal harness
Can someone please walk me through this exercise. I've been sick and absent during class.
https://chatgpt.com/share/69b0953f-6d70-8012-ab8c-cc8d27fbe55e
latent quail
rare scaffold
here are the non translated exercises
is there any terminology that you do not recognize?
royal harness
is there any terminology that you do not recognize?
i recognize every terminology except vector spaces since we didn't get to that part yet. the problem is i don't know how to write and justify things correctly
rare scaffold
take (1) as an example; we know what a subgroup is, so what we ought to show is that all the conditions for A being a subgroup of M_2(R) are met
component by component
royal harness
take (1) as an example; we know what a subgroup is, so what we ought to show is ...
so i should first show that the elements are stable in A and that every element has a symetric
is that right ?
rare scaffold
yeah, the english terms are different but it sounds close
royal harness
yeah, the english terms are different but it sounds close
finding that it is stable is easy but what about the symetrics, what should i do ?
it should be M + M' = the matrice where there's only zeros ?
rare scaffold
take an element M of A, and determine if it's inverse, -M, is in A as well
royal harness
take an element M of A, and determine if it's inverse, -M, is in A as well
ohhhh alrightttt
what about question 3)a
rare scaffold
what about question 3)a
we need to know what an isomorphism is, and what the underlying spaces are
in particular, C^* and A^* are C and A, but removing the additive identity from both
and the operation is typical complex number / matrix multiplication respectively
royal harness
ugh i don't really understand especially since english isn't my first language, can you please do the question so that I can get a proper demonstration please
pearl pondBOT
The purpose of this server is to help you learn; please don't ask for direct answers. Ask for guidance, explanations, or feedback instead.
fluid axle
c'est quoi le souci ? @royal harness on parle de l'exo en haut right ?
royal harness
c'est quoi le souci ? <@611977038126907392> on parle de l'exo en haut right ?
oui question 3
a
fluid axle
ok qu'est-ce que tu piges pas sur cette question ?
t'as vu ce que c'était un groupe déjà ou c'est exactement ça que t'as loupé en étant absent ?
royal harness
t'as vu ce que c'était un groupe déjà ou c'est exactement ça que t'as loupé en é...
je galere avec la redaction, mais sinon ca va pour le reste
fluid axle
ok t'as un brouillon de réponse ou qqch dans ce cas ?
si qqun me dit qu'il/elle galère juste à la rédaction, c'est qu'au moins la personne pige ce que la question lui demande, c'est ok de côté-là pour toi?
@royal harness
royal harness
si qqun me dit qu'il/elle galère juste à la rédaction, c'est qu'au moins la pers...
il faut que f(z x z') = f(z) + f(z') et que la fonction soit bijective. c'est ca ?
fluid axle
il faut que f(z x z') = f(z) + f(z') et que la fonction soit bijective. c'est ca...
l'opération c'est le produit pour les 2 groupes, donc f(z x z') = f(z) x f(z')
a gauche le produit de complexes, à droite le produit de matrices
royal harness
injection et surjection, le truc classique
c'est la meme chose qu'avec les fonctions normales ?
meme si on a des matrices
fluid axle
il y a un truc en plus qu'il vaut mieux que tu montres aussi, on te donne une fonction C -> A, c'est pas dit que la définition marche forcément bien de C* -> A*
i.e. si z != 0, que t'aies bien f(z) != 0
fluid axle
c'est la meme chose qu'avec les fonctions normales ?
ça reste une fonction oui, injection bijection surjection ça change pas
royal harness
est ce possible de me montrer comment faire pour montrer qu'elle bijective stp, j'ai besoin d'un exemple pour savoir comment rediger
fluid axle
c'est toujours pareil,
pour l'injection
soit z1 = x1 + i y1, z2 = x2 + i y2 dans C* tel que f(z1) = f(z2)
.....
tu bosses jusqu'à en déduire z1 = z2
et puis pour la surjection
soit M dans A*, et tu veux trouver un z dans C* avec f(z) = M
et puisque M dans A* ça veut exactement dire que il existe x, y avec M = [[x -y] [y x]], ton z il est vite trouvé
royal harness
qlq chose comme ça ?
royal harness
la rédaction est correcte ? j'ai l'impression que j'ai peu écrit
j'espère que c'est bon 😭
fluid axle
je disais que c'était ok pour les ... que je te laissais faire
c'est vrai que là on voit pas directement quel est le rapport entre ton travail et la fonction f
royal harness
c'est toujours pareil,
pour l'injection
soit z1 = x1 + i y1, z2 = x2 + i y2 dan...
je peux pas faire l'injection et la surjection en meme temps ? je suis oblige de les faire separemment ?
royal harness
c'est vrai que là on voit pas directement quel est le rapport entre ton travail ...
😭
royal harness
mais bon tu me fais ça ça me va très bien
oh oui c vrai
mais puisque que c'est vrai pour quelque soit x y ca veut dire que la fonction est surjective aussi
fluid axle
je peux pas faire l'injection et la surjection en meme temps ? je suis oblige de...
ça dépend si ton prof est assez sympa pour juste te laisser écrire l'inverse de f et montrer que c'est bien l'inverse, ça marche aussi
royal harness
ça dépend si ton prof est assez sympa pour juste te laisser écrire l'inverse de ...
il est tout sauf sympa pour le coup
fluid axle
c'est pour ça que je préfère faire un truc très standard là
royal harness
mais bon tu me fais ça ça me va très bien
et pour montrer qu'elle est surjective ?
ca fait un an qu'on a fait le cours sur les applications desole 😭
fluid axle
surjective je t'ai quasiment donné la preuve en entier là
royal harness
et puis pour la surjection
soit M dans A\*, et tu veux trouver un z dans C\* av...
j'ai pas vraiment compris comment je suis sense trouver le z
fluid axle
et puis pour la surjection
soit M dans A\*, et tu veux trouver un z dans C\* av...
il y a toutes les infos là dedans, j'ai fait exprès d'écrire la dernière ligne pour être 100% explicite
t'as pas besoin d'introduire de nouvelles notations ou quoi, les x et y sont pas là pour faire joli
c'est encore plus simple que pour l'injection honnêtement
faut pas te casser la tête, il y a un choix bien évident pour ton z, prends-le
@royal harness
royal harness
comme ça ?
fluid axle
yes indeed
royal harness
yes indeed
enfin
dernier truc, pour la question 3)d est ce que je dois montrer qu'elle reversible ?
et si oui comment je fais
MxM-1 = e (avec e l'element neutre) ?
fluid axle
dernier truc, pour la question 3)d est ce que je dois montrer qu'elle reversible...
vaut mieux oui
fluid axle
et si oui comment je fais
le déterminant c'est cool
royal harness
le déterminant c'est cool
mais comment je vais trouver M-1 avec le determinant
fluid axle
tu me demandais comment montrer que M est inversible, le déterminant ça te donne une réponse direct
et sinon il y a une formule bien classique pour l'inverse d'une 2x2
qui utilise le déterminant
royal harness
tu me demandais comment montrer que M est inversible, le déterminant ça te donne...
ah deso la question c'etait de trouver l'inverse de chaque element
royal harness
et sinon il y a une formule bien classique pour l'inverse d'une 2x2
c'est quoi
royal harness
ça m'étonnerait que t'aies jamais vu ça
j'etais absent lors du cours j'ai rate qlq truc
royal harness
ah c plutot simple alors
il y a une formule pour les matrice de rang 3 ?
je suis bete je peux rechercher sur google
sinon merci beacoup @fluid axle pour ton aide tu me sauve !
bonnenuit a toi et merci encore une fois !
.close
pearl pondBOT
fluid axle
il y a une formule pour les matrice de rang 3 ?
oui avec les matrices adjointes etc...
https://en.wikipedia.org/wiki/Adjugate_matrix#3_×_3_generic_matrix ça devient vite moche
déjà juste le déterminant 3x3 va commecer à être un peu emmerdant
- tout le reste
pearl pondBOT
Available help channel!
lean niche
I need someone to prove my calculations
A drop-off pocket for public transport and taxis, located in front of a supermarket entrance, is shaped like an isosceles trapezoid ABCD. The length of the larger base AD is 38 m, and the width (height) of the pocket is 5 m. Safety bollards are planned to be installed along the smaller base BC and the side legs AB and CD at a distance of 1 m from each other. Some of them have already been installed (see figure). How many bollards in total should there be according to the plan along sides AB, BC, and CD of this pocket, if 15 bollards have already been installed along BC?
lean niche
pearl pondBOT
@lean niche Has your question been resolved?
lean niche
<@&286206848099549185>
vernal jasper
Length of CD = Length of AD
So, no. of bollards along the slopes is 2x the legnth of either CD/AD.
Also you know BC already has 15 bollards, which i suppose is 14m (n-1) in length.
Then we can find the sloped length by taking (38-14)/2 = 24/2 = 12m.
So by the Pythagorean Theorem,
CD^2 = 5^2 + 12^2
CD = sqrt(12^2+5^2)
CD = 13m
No. of Bollards along slopes = 2(13+1)
= 26 bollards.
Therefore, total no of bollards would be 26 + 15 = 41
@lean niche
lean niche
Length of CD = Length of AD
So, no. of bollards along the slopes is 2x the legn...
Ty
.close
pearl pondBOT
vernal jasper
🙂
pearl pondBOT
Available help channel!
green aurora
$f(x)+f(\frac{1}{1-x})=x$
pearl pondBOT
$f(x)+f(\frac{1}{1-x})=x$
jolly parrotBOT
Lups
green aurora
Determine f(10)
eager jewel
What have u tried
green aurora
I actually would like a hint
eager jewel
Sub x-->1/(1-x)
green aurora
okay
neat salmon
hello, im new at math, can you guys share the best course to learn calculus from basic to advanced details?
calm wing
hello, im new at math, can you guys share the best course to learn calculus from...
check out #❓how-to-get-help and #book-recommendations too :)
neat salmon
alright tq guys
green aurora
f(10)+f(-1/9)=10
f(-1/9)+f(9/10)=-1/9
f(9/10)+f(10)=9/10
2f(10)=991/90
f(10)=991/180
eager jewel
yes thats correct
green aurora
thanks!
eager jewel
no problem
green aurora
.close
pearl pondBOT
pearl pondBOT
Available help channel!
primal viper
So I have an object and a plane here and their equation, if I get z in term of x then integrate z wrt x would it give me the cross section area in this case?
pearl pondBOT
So I have an object and a plane here and their equation, if I get z in term of x...
@primal viper Has your question been resolved?
rough forge
So I have an object and a plane here and their equation, if I get z in term of x...
That doesn't seem quiet right...
primal viper
I do have a feeling that it's not right either but I'm not sure
midnight haven
by cross sectional area you mean the one of the shape formed by intersection of plane and object ?
primal viper
Uhh.. yeah
jolly parrotBOT
primal viper
Wao... uh
I'm still in highschool so I don't understand a bit of what you just wrote
rough forge
The idea is you have a surface, and on that surface you create normal vectors across the surface. If you sum up their magnitudes, you get roughly an approximation of that surface.
primal viper
Holy cow 
primal viper
So I have an object and a plane here and their equation, if I get z in term of x...
Is there another way that I can find the area in this case
It's my hw, I just want to finish my hw
Also they probably won't accept that sol even if I understand it
rough forge
Well I spoke for general cases, your case is simple enough to consider the area formula for circles
rocky socket
An ellipsoid and a plane intersect in an ellipse, generally. Try getting the equation of this ellipse in the plane? The area would be pi a b, where a, b are the semi-axes lengths
Or is it a circle in this case?
rough forge
Probably an ellipse
primal viper
They never said anything about ellipsoid and a plane intersect is an ellipse
rough forge
since z is being scaled
rocky socket
If you forget the 2 in front of z it’s simpler
Then it’s a sphere and a plane intersecting in a circle as said above
primal viper
An ellipsoid and a plane intersect in an ellipse, generally. Try getting the eq...
Is it always like that, or it's just that the plane are perpendicular to Oxy plane and only z component is scaled down?
rocky socket
You could have a circle intersection in a special case between an ellipsoid and a plane, sure. Let me see
primal viper
So only circle and ellipse ?
That's good enough
Thank you
I can do it from here
Ty both
.close
pearl pondBOT
rocky socket
The axes lengths are .63 and .89 sorry to say
pearl pondBOT
Available help channel!
fading nexus
I’m doing trig identities and some how got the answer but flipped and don’t know how to fix it
midnight haven
I’m doing trig identities and some how got the answer but flipped and don’t know...
Multiply numerator and denominator by 1-sinx
feral olive
no you multiply both top and bottom of the fraction by 1-sinx
if you expand it out you can simplify it
fading nexus
How do I know when I can do this? Sorry if that doesn’t make sense s it just feels random I don’t know how I would know this on a test
And what would be the reason for this
midnight haven
I can’t change stuff on both sides
If you multiply and divide a fraction by the same term the value of fraction remains same
It is like multiplying by a/a or 1
fading nexus
I don’t know how to simplify this though
midnight haven
I don’t know how to simplify this though
1-sinx
warm patio
I don’t know how to simplify this though
sinx-1 not sinx + 1
midnight haven
Do you know (a+b)(a-b) = a^2-b^2
midnight haven
Hm.. do you know how to simplify (a+b)(a-b)
fading nexus
No
midnight haven
or multiplication of any two expressions
midnight haven
No
Hm.. basically when you multiply a(b+c) you multiply a with each term b and c then add them
a(b+c) = a*b + a*c
fading nexus
14?
fading nexus
If I multiply the fraction both the denominator and numerator what is that called?
Would I just list the reason as algebra
midnight haven
Oh ic
midnight haven
Would I just list the reason as algebra
You don't need to list any reason tbf its just algebra
Well back to the original question you can remember this (a+b)(a-b) = a^2-b^2 as a standard result
It is easy to understand why it is after knowing multiplication of expressions
fading nexus
I’m looking at the ANSWR key and I’m still confused because how did they cancel if it is not matching
And then on like half of my problems I can’t finish them because I don’t know how to multiply cosx and sinx together
midnight haven
I’m looking at the ANSWR key and I’m still confused because how did they cancel ...
Looks like they multiplied by cosx in numerator and denominator
Then used cos^2x =1-sin^2x
Then they split 1-sin^2x as (1+sinx)(1-sinx)
midnight haven
Well back to the original question you can remember this (a+b)(a-b) = a^2-b^2 as...
Using this
midnight haven
They skipped few steps
midnight haven
Where did it go though
Where did what go
fading nexus
Like it look liked they cancelled it
midnight haven
I gtg rn
flint basalt
Like it look liked they cancelled it
they use the identity $\cos^2x + \sin^2x = 1$
jolly parrotBOT
Krish
flint basalt
therefore, $\cos^2x = 1 - \sin^2x$
jolly parrotBOT
Krish
flint basalt
and using difference of squares, $a^2 - b^2 = (a+b)(a-b)$, you can say that $1 - \sin^2x = (1+\sin x)(1-\sin x)$
jolly parrotBOT
Krish
flint basalt
then you can cancel from there
pearl pondBOT
@fading nexus Has your question been resolved?
jade skiff
And then on like half of my problems I can’t finish them because I don’t know ho...
theres no need to multiplu sinx and cos x js simplify it out cox(sinx+1)/ 1-sin2x and 1-sin2 x can be written as (1-sinx)(1+sinx) its an identity (a2-b2)=(a+b)(a-b)
sharp smelt
Really silly question, is the "In particular" part just refering to when HK=G
worldly jewel
yes
pearl pondBOT
worldly jewel
What are you sorry for? Don't be!
pearl pondBOT
Available help channel!
final smelt
1+1
flint basalt
<@&268886789983436800> troll
flint basalt
oop lol
sharp vigil
.close
warm elbow
\subsection{Problem 4.}
\textbf{For any set $x$, the \textit{transitive closure of x}, written $TC(x)$, is defined by: $TC(x) = \bigcup{h_n(x) : n \in \omega}$ where $h_0(x) = x$ and $h_{n+1}(x) = \bigcup h_n(x)$ for all $n \in \omega$.}
\\
\textbf{(a) Prove that for every set $x$, $TC(x)$ is transitive.}
\\
Let $x$ be any arbitrary set. Select $a \in TC(x)$. So $a \in \bigcup {h_n(x) : n \in \omega}$. There exists some $b \in {h_n(x) : n \in \omega}$ such that $a \in b$. Call $b = h_{n}(x)$, where $n \in \omega$.
warm elbow
lowkey stuck on this
jolly parrotBOT
toast
pearl pondBOT
@warm elbow Has your question been resolved?
autumn fossil
lowkey stuck on this
So your strategy is proving that a is a subset of TC(x)?
warm elbow
So your strategy is proving that a is a subset of TC(x)?
ye
autumn fossil
hmm so generally, you'd now pick some a' in a and try to prove that a' is in TC(x) ig
autumn fossil
that's just from the definition of subset
it's still just proving that a is a subset of TC(x)
jolly parrotBOT
toast
autumn fossil
Sometimes, yeah
warm elbow
i cant tell if this is the case o rnot
autumn fossil
That'd probably depend on the theorems you have proven before, but i cant see any direct way here
autumn fossil
hmm so generally, you'd now pick some a' in a and try to prove that a' is in TC(...
i believe this is pretty simple
warm elbow
kk