it can't be
#help-17
wanton shuttle
its not even a quadratic equation
i think there is a problem with this question
i asked 3 ai to do this
digital lake
what
wanton shuttle
and no one got it right
vocal sleetBOT
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wanton shuttle
@digital lake ?
digital lake
let me just
wanton shuttle
ok
digital lake
digital lake
how come
wanton shuttle
idk thats the answer
thats why
i think its not possible
i did this question like 4 times
@digital lake i think they wrote x squared by a mistake
wanton shuttle
?
oh
lets try without the x squared because i think its a mistake
nvm still not possible
digital lake
im gonna do the verification
wanton shuttle
ok
digital lake
wanton shuttle
mhm
torn timber
what is the problem?
digital lake
the answer
wanton shuttle
only 2?
digital lake
1+1= 2
wanton shuttle
mhm
digital lake
wdym
wanton shuttle
its should be a quadratic equation
wanton shuttle
what is the problem?
torn timber
ok i see
digital lake
yea I just followed that to do verify
wanton shuttle
i think its just not possible
torn timber
lets see
wanton shuttle
oh
digital lake
to verify if the answer is correct or not, then you use verifications
wanton shuttle
and its got correct?
digital lake
im doing x2
digital lake
wtf
wanton shuttle
yeah
digital lake
3?
torn timber
$$\frac{1}{(x-3)^2}+\frac{4}{x(x-3)}=\frac{2}{x-3}$$
digital lake
thats just the problem
wanton shuttle
yeah
digital lake
try to replace 3 into x
torn timber
now we do $$\frac{1}{(x-3)}+\frac{4}{x}=2$$
twin meteorBOT
convergence
torn timber
$now we get:5x-12=2x^2-6x$
twin meteorBOT
convergence
digital lake
yea
torn timber
then its simple from here
twin meteorBOT
convergence
wanton shuttle
mmh
digital lake
uhhh
digital lake
**convergence**
we already solved this?
torn timber
alrihght then its done right
torn timber
yeah
digital lake
but his answer shows that it is x1 = 3, x2 = -4
torn timber
its an error
digital lake
💀
torn timber
x cant be 3 at all
wanton shuttle
how?
wanton shuttle
wdym
digital lake
btw i tried doing it by replacing x into 3
torn timber
**convergence**
this step is done assuming $x\noteql 3$
digital lake
and it doesnt work
wanton shuttle
so its not possible?
torn timber
yes
digital lake
answer 3 shouldnt work
torn timber
yes x cannot be 3
wanton shuttle
i see
wanton shuttle
get what?
digital lake
like do u understand it
torn timber
why it not x=3
wanton shuttle
no
torn timber
if x=3 is a solution then we would have 0/0 case
digital lake
epic
wanton shuttle
ok thanks guys for trying
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grand raven
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grand raven
Draw the two lines in the same coordinate system
In the same coordinate system find an equation for a new line that is parralel with the x axis so that the area of the limited space = 12
2 solutions ofc
I got the answers im just stumped on how to set up an equation for it
sturdy seal
hmm
correct me if im wrong
u wanna take this?
alr
so they gave u 2 lines
and asking u to draw a third line parallel to x axis such that area enclosed is 12?
@grand raven
grand raven
Yup, 2 isoceles triangles both negative and positive
I got the answers graphically but i cannot for the life of me find an equation
sturdy seal
wdym u got the answers graphically?
grand raven
It wants an equation that can be reused lmao
sturdy seal
thats it?
sturdy seal
It wants an equation that can be reused lmao
y = -5/3
simple?
grand raven
y = -5/3
simple?
I mean an equation that can be used for the same problem with different areas
sturdy seal
thats fine
i think i have an idea
lemme try it on paper first
@grand raven is this the figure u got?
sturdy seal
with this the area of triangle is the matrix determinant of (3+c/2,c) (3-c,c) and (2,1)
sturdy seal
ah thats also fine
did u take variable line y = c or u just took for area 12?
with the points u can take half of det of the points and equate it to any area u want
and u can find the c accordingly
i think thats what u want
?
grand raven
Yeah probably, having a hard time interpreting this, classmate got some insane set of equations
grand raven
Yup hahaha
grand raven
Yeah somewhat but idk if i have an equation for it
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tough prairie
Can anyone help me get started on this problem/give me a big hint? Been stuck on this for a long time
vocal sleetBOT
Can anyone help me get started on this problem/give me a big hint? Been stuck on...
tough prairie
Need to prove QM-AM>=GM-HM for two positive integers
sturdy seal
hmm
ionic merlin
Maybe try making it so anything that has square root is on one side, and anything that doesn't is on another side. Then simplify it you can, let's see if we can get to anything from there
ionic merlin
u need to square twice tho
Yes, but the divisions become nice on numerator
tough prairie
Oh so square twice?
ionic merlin
Or do they
sturdy seal
Oh so square twice?
after getting roots on one side
ionic merlin
Oh so square twice?
To fully get rid of square roots, but it'll be ugly
sturdy seal
To fully get rid of square roots, but it'll be ugly
yes the correct word
im also wondering abt the signs of inequality
ionic merlin
Here me out, I think the square root one side division one side, simplifying might be helpful
-2/(1/a + 1/b) = -2ab/(a + b)
You take the left one to right
sturdy seal
im also wondering abt the signs of inequality
is there a guarantee the values wont be in fractions?
ionic merlin
You do the subtraction
sturdy seal
then squaring will change inequality
ionic merlin
That might be the case
So wouldn't ie be better if we simplify as much as we can before squaring? 
sturdy seal
So wouldn't ie be better if we simplify as much as we can before squaring? <:bre...
yes
tough prairie
Here me out, I think the square root one side division one side, simplifying mig...
I tried putting the square roots on one side and (a+b)/2 -2ab/(a+b) on the other and squared but things didn't simplify out
The inequality signs would also stay the same way
ionic merlin
I tried putting the square roots on one side and (a+b)/2 -2ab/(a+b) on the other...
what was result of subtraction?
ionic merlin
Still annoying, but looks better
Let's see what happens if we square
And take right side to left
So you'd have something bigger equal to 0
For now we don't need to expand (so like just keep in (a-b)^4 for example)
Let's see if it can be of any help
tough prairie
Well a^2+b^2/2+ab = (a+b)^2/2, idk if that helps
ionic merlin
Oh I forgot about ab
tough prairie
So we'd have (a+b)^2/2 - (a-b)^4/[4(a+b)^2] - QM*GM*2 >= 0
ionic merlin
What is QM, GM?
ionic merlin
Well a^2+b^2/2+ab = (a+b)^2/2, idk if that helps
Also what is this, I didn't understand
Like left side fine, what is right side
tough prairie
sqrt[(a^2+b^2)/2]*sqrt(ab)
ionic merlin
Oh ok
tough prairie
Also what is this, I didn't understand
Since sqrt[(a^2+b^2)/2]^2+sqrt(ab)^2
ionic merlin
You just added them
tough prairie
Yea
ionic merlin
What we have now certainly looks like something
But like what thing is what we should figure out
tough prairie
Yeah
I was originally thinking of breaking things apart and simplifying them into squares/fourth powers, then move them all to one side to prove it must be >= 0 by the trivial inequality
ionic merlin
I thought of that too, couldn't find an answer
tough prairie
I might just check the solution 😭
ionic merlin
So we'd have (a+b)^2/2 - (a-b)^4/[4(a+b)^2] - QM*GM\*2 >= 0
Let's try mixing the first two parts before QM,GM
ionic merlin
I might just check the solution 😭
Also valid
tough prairie
It'd be 2(a+b)^4/4(a+b)^2 - (a-b)^4/4(a+b)^2
tough prairie
Let's try mixing the first two parts before QM,GM
Trying to put the first two together
ionic merlin
So like now let's see
If we can simplify the numerator
So we'd have
((a+b)^4 + 8a^3b + 8ab^3)/4(a+b)^2
tough prairie
I got a^4 + b^4 + 12a^3b + 12ab^3 + 6(a^2)(b^2)
tough prairie
sqrt[(a^2+b^2)/2]*sqrt(ab)
ionic merlin
The QM,GM?
tough prairie
Yea
ionic merlin
Oof
tough prairie
Yea
ionic merlin
Let's look through that
tough prairie
Damn it's long af lmao
tough prairie
Ooooh
Ok ic
The first parts were good but instead of continuing to bash it we should've noticed/applied inequalities in our toolbox
ionic merlin
I forgot this question is a proof...
We went well for like half of the way
Still good enough tbh
tough prairie
Yeah
ionic merlin
İn these types of questions I feel like unless you've solved a similar question, it's not really achieveable to solve it fully
But if you can even do some part of it yourself and then learn from solution how it should be done, you'd progress
tough prairie
Or you do lots of experimentation
ionic merlin
Or you do lots of experimentation
Yes that also
But like sometimes it needs a specific way of thinking
tough prairie
Mhm, especially in this case
Didn't notice the connection between the GM/radical and the left side
Anyways tysm for helping me 🙏
.close
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rough pilot
when it says " the topology generated by the borel sets B " what it means ? The topology that make the colection of sets B, to be borel or the topology that the sets in B are open?
vocal sleetBOT
when it says " the topology generated by the borel sets B " what it means ? The ...
cobalt crypt
the second one is usually what you mean when you say generate
take the smallest topology such that every set in B is open
vocal sleetBOT
@rough pilot Has your question been resolved?
rough pilot
oh ok thank you
wide olive
Hi, Im doing a grade 12 mathematics investigation about waterslides.
sharp shell
Lol funny question
wide olive
I need to optimise it so i need to use alot of decimals the maximise the accuracy
glacial osprey
that is a lot of decimals
sharp shell
So just a linear function or what?
glacial osprey
do you have the original problem with you
wide olive
it must be in this form
I initially was going to use a interpolation calculator
but it wouldnt be in this form so
idle sparrow
it must be in this form
put this in Desmos and adjust the 3 sliders until it looks good I guess
That’ll be the fastest way without any calculation
idle sparrow
I have to optimise it though
this is solvable, just put the step to be 0.0001 or something :3
idle sparrow
and I need to write a report on how I calculated it
Just
wide olive
guessed it xD
idle sparrow
I mean the other thing u can do is use wolfram alpha to solve
So like
Forget about the t
U have two points that u can use (start and end point)
Put that in to solve for u and v and just take t as 1
idle sparrow
Lol
If u want t just estimate another point on there
I’m pretty sure it’s doable with wolfram alpha
Unless ur teacher specified no tech
wide olive
We can use tech
idle sparrow
Yeah then just do that
wide olive
would that find the values of u and v?
idle sparrow
would that find the values of u and v?
I mean, if u put in the value instead of x, yes
Like u know the start and end point from ur previous two functions
wide olive
yes
idle sparrow
(Also, I’ve never seen so many decimals in a function 
wide olive
(Also, I’ve never seen so many decimals in a function <:opencry:5860786148654448...
neither
Im gonna lose my brain
idle sparrow
I tried solving by hand
Would be easier if Ramanujan is around 😭
wide olive
something like this?😭
idle sparrow
Yeah
wide olive
becz when x=4.79070543365, y should basically equal to = 10,
and vise versa
idle sparrow
(Does it…solve? Or did it broke wolfram alpha
wide olive
idle sparrow
becz when x=4.79070543365, y should basically equal to = 10,
and vise versa
^ yep
wide olive
I got two real answers
I guess ill try putting those values in and check if they actually work
idle sparrow
I got two real answers
Make sense cuz they r within a term that’s being squared
idle sparrow
I guess ill try putting those values in and check if they actually work
Okie
idle sparrow
Ooooops
wide olive
I shouldve assumed that would happen without the t
idle sparrow
Pick another point
wide olive
bahaha
idle sparrow
how do i show working out to this?
I guess ur thinking process
Like
U picked points
wide olive
"I kinda guessed through trial and error and guessed points"
idle sparrow
So that u can solve the system of equations
wide olive
im gonna fail lol
idle sparrow
"I kinda guessed through trial and error and guessed points"
Noooooo
Like just say
Since there’s three unknowns
wide olive
apparently its possible
idle sparrow
We can solve the system of equations if we have 3 distinct known points
wide olive
you have to cancel out the t though
idle sparrow
We can solve the system of equations if we have 3 distinct known points
And then go on to show process
wide olive
actually im not sure let me check
vocal sleetBOT
@wide olive Has your question been resolved?
wide olive
idk
rough pilot
the second one is usually what you mean when you say generate
sorry i am coming back to this. if we dont have a topology how we define the borel sets ? So then we will have the topology generated from the borel sets
vocal sleetBOT
sorry i am coming back to this. if we dont have a topology how we define the bor...
half imp
borel sets are for sigma algebras, right
Just think of any collection of sets which is closed under countable intersection and union, those can be borel sets
cobalt crypt
borel sets are for sigma algebras, right
no theyre for topologies 
half imp
hmm
rough pilot
Borel sets are the sets who belongs to the sigma algebra generated by the open sets of a space, If I have understood correctly. But if I don't have a topology how I define them ?
half imp
then they're just sets which belong to a sigma algebra
ig
but you can have a topology
have it generate borel sets
and then take the topology generated by those borel sets and it's not the same topology you started with
cobalt crypt
Borel sets are the sets who belongs to the sigma algebra generated by the open s...
you need a topology in the first place to talk about the borel sigma algebra
yeah the topology generated by the borel sets will be in general a larger topology
dull pine
no theyre for topologies <:kekehands:1130985995102322719>
both, actually
borel sets always form a sigma algebra in any topological space
dull pine
neat thing about topology is that you don't need to know you're using topology for most of it :D
rough pilot
I see. But I read about lemmas which start like let's take the borel sets of X and then generate a topology that do something etc. But it doesn't define a first topology to specify which are the borel sets and I am confused
cobalt crypt
theyre putting a different topology on X
rough pilot
Yeah a topological space. But I guess I need to know the topology? Or it means this lemma for example works for any topology and its borel sets ?
proven whale
damn
dull pine
I see. But I read about lemmas which start like let's take the borel sets of X a...
they're saying "Take any topological space X.
Take its Borel sets.
generate a (presumably different, maybe larger) topology from those using some algorithm.
etc."
ie. the algorithm they describe works for any such X
cobalt crypt
you dont need to know the exact sets of the topology to work with it so to speak
its just given to you when we say X is a topological space
dull pine
Yeah a topological space. But I guess I need to know the topology? Or it means t...
works for any (unless they specify otherwise, like "Take an X with this or that property")
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vast shale
having problems answering this problem
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having problems answering this problem
vast shale
this is what i have, i do not know how to get F1
if im able to get k i should be able to get the rest of the questions, any hints?
i also know that F1 would equal ma but where do i get the acceleration here?
vocal sleetBOT
@vast shale Has your question been resolved?
vast shale
<@&286206848099549185>
vast shale
<@&286206848099549185> anyone know anything about springs?
gloomy nacelle
Springs?
surreal ledge
What kind of springs?
surreal ledge
Hmm…
vast shale
this is what i have, i do not know how to get F1
i do not know how to get F1 here, can anyone help me out?
surreal ledge
Where did you get that you need F1 from? I cannot find a single mention of F1 in the question.
Unless you added it in.
vast shale
its the force i represented when the box is pushed
and displaced by 0.05m
in theory it should equal the force of the spring, unless theres another way to get it?
surreal ledge
I was never good at physical math and more mental math. I mean like I cannot do math with physical objects or word problems and have always been better at math without objects.
I cannot help, sorry.
vast shale
surreal ledge
I hope you find help!
placid spear
if im able to get k i should be able to get the rest of the questions, any hints...
I think you could use the first sentence of your problem to find k?
If I interpret it right. Then the spring is stretched 0.200m when 450-g mass is attached to it means the force of gravity affects it
So you could use that information to find k?
vast shale
oooh, but question tho why does gravity affect this if its on a horizontal plane?
placid spear
oooh, but question tho why does gravity affect this if its on a horizontal plane...
I think it has two parts
vast shale
so fs would equal to mg, but im still confused why we would use g
placid spear
F = ma but when force is because of gravity then a = g so it is F = mg
vast shale
F = ma but when force is because of gravity then a = g so it is F = mg
i see, so i would just use g then...
fs = -kx
-mg = -kx
k = mg/x right?
placid spear
fs = -kx
-mg = -kx
k = mg/x right?
wait, we need to be careful here
vast shale
oohh
placid spear
because spring force will be opposite to force of gravity
so you need to choose whether your positive side is into the Earth or away from the Earth
placid spear
If into earth then mg = -kx
placid spear
if out i assume its -mg = -(-kx)?
Yeah, then it would be -mg = kx
vast shale
Yeah, then it would be -mg = kx
i see i see, so k is always negative?
placid spear
i see i see, so k is always negative?
No
Actually, the sign of k is not important now
we just care about its magnitude
vast shale
i see i see
with k here im pretty confident i can get the rest
thank you!! @placid spear
placid spear
<:sugoi:556557078924099612> <:sugoi:556557078924099612>
Just remember to plug it in the equation to test, though
In physics, most of the time sign is just for direction
so you can always choose if you should set it to negative or positive
depends on your problems
vast shale
since fs is pulling the box to the left, it would be negative right?
if we set left as negative and right as positive?
placid spear
since fs is pulling the box to the left, it would be negative right?
Do you mean spring force or force that push the mass to the left?
placid spear
if we set left as negative and right as positive?
Let's say we choose this as our frame of reference
The force that pushes 450-g mass to the left is negative
But the spring force is the opposite to it so it is to the right
Means spring force is positive
vast shale
Do you mean spring force or force that push the mass to the left?
spring force here
pushing the box to the right
vast shale
But the spring force is the opposite to it so it is to the right
OHHHHHH
since the spring is compressed, it wants to decompress (to the right) so its positive
placid spear
since the spring is compressed, it wants to decompress (to the right) so its pos...
Btw, the original F = -kx still works here
We don't need to think that much 😂
because it depends on x
If pushing it to the left means x will be negative
so F will be positive
vast shale
ah i see it depends on x
ok ok i got it
i just got all the values for the problem, thanks man
.close
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wary mantle
\begin{problem} Determine a function for $f(x) = \left|(x-2)^2 - 4\right|$ without absolute values and graph $f$ . \end{problem} So, I graphed it, but how do you determine a function for it without absolute values? Do you just make it casewise defined?
vocal sleetBOT
\begin{problem} Determine a function for $f(x) = \left|(x-2)^2 - 4\right|$ witho...
hard atlas
what do you even mean with "determine"
wary mantle
wary mantle
what do you even mean with "determine"
hard atlas
well then I suppose piecewise
vocal sleetBOT
shadow raft
,w graph |(x-2)^2-4|
twin meteorBOT
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vocal sleetBOT
@sand ember Has your question been resolved?
solar barn
you want all conditions $x>0$, $x\neq 1$, $x>1$ or $x<-2$, and $x<-1-\sqrt{5}$ or $x>-1+\sqrt{5}$ to be true at once, since else $x$ would violate at least one of these. This means that you most likely want the intersection of all of these.
twin meteorBOT
Crystopher
solar barn
If it is somewhat challenging to figure out their intersection you can draw the real line and each interval on this, then find the interval where all are valid.
the intersection should be $x>-1+\sqrt{5}$, as shown in Desmos
twin meteorBOT
Crystopher
solar barn
Here are 3 real parallel lines drawn, the blue segments are different intervals, the red is the intersection
upper is for $x>0$, middle is for $x<-2$ or $x>1$ and the last line is for $x<-1-\sqrt{5}$ or $x>-1+\sqrt{5}$. If you 'merged' all of them you would see that the red one is the intersection.
twin meteorBOT
Crystopher
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red finch
hey, so i got this new topic and i was just wondering how do you solve this, like where do i start, whats not possible and and and (its about volume of a rotational body and i think this is the way to get the volume)
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hey, so i got this new topic and i was just wondering how do you solve this, lik...
red finch
this is the root function
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<@&286206848099549185>
red finch
.close
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.reopen
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@red finch Has your question been resolved?
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@red finch Has your question been resolved?
granite drum
hey, so i got this new topic and i was just wondering how do you solve this, lik...
what do you know about integration?
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outer flax
outer flax
shadow raft
Same thing again?
velvet zephyr
find factors of -20 such that f1 + f2 = -1
outer flax
Same thing again?
Q: How long do you think I've been stuck on this exactly?```but...
that doesnt matter to me anymore...
it just is...
ive given up all pleasures
for me all thats left in this world is me and what i am trying to do
thats all that matters in regards to attention
A:
outer flax
Same thing again?
Q: I need to know what the function represents.```
red spruce
if you're really unable to factor quadratics with integer solutions, you could just memorize the quadratic formula and then put in the values that way
or even better, factor it
(x - a)(x - b) = x^2 - x - 20
which means
ab = -20
a + b = 1
outer flax
Q: If you factor the polynomial, it, apparently, gives you a lot of information relating to it. This implies that A: when graphed the only real reason it extends infinitely is because that would be the representation of the function repeated endlessly, and that the function itself is not infinite.```how do you type the messages in this format?
red spruce
3 `, both at the start and at the end
velvet zephyr
oh
ikrn
outer flax
or even better, factor it
(x - a)(x - b) = x^2 - x - 20
which means
ab = -20
a ...
I need to know what it all represents, otherwise all knowledge presented will be repetitive, and or useless.
red spruce
it represents what meaning you give it
outer flax
Incorrect.
velvet zephyr
you mean what ax^2+bx+c means?
Keep your questions short and concise.
outer flax
I need to know the structure of the function itself.
And why it appears the way it does in response to change or graph.
If I do not have the structure, then the numbers are meaningless, just like if I didn't know it was a function, this would be nearly impossible.
red spruce
there is a coefficient of x^2, coefficient of x and a constant
velvet zephyr
,w graph x^2-x-20
twin meteorBOT
outer flax
damn sugar withdrawal symptoms... I swear that stuff is like drugs.
worthy citrus
its very unclear exactly what it is youre asking here
outer flax
velvet zephyr
he wants to know everything
outer flax
```you mean what ax^2+bx+c means?```
Keep your questions short and concise.
Sorry, I did not see this.
Finally, for once in my life I can focus. I am finally truly perfect... I can resist temptation... I can cancel it before it becomes a problem... My mind can finally be clear of everything... And so now I can simply process without fault.
hexed orchid
bro whatt
outer flax
Just some personal information, ignore that. I worked for quite some time to achieve that.
hexed orchid
what is the qs here
velvet zephyr
What shall be known by you?
outer flax
So, relating to this, why is it that it does not start at -20 or lower and subtract? How are the numbers being interpreted?
outer flax
what is the qs here
Q: (question)
A: (answer)
G: (guess)```you need to sit down and actually plot one of these for yourself if you dont understand why they look like that. take x^2-x-20, pick your favourite x values, plug them in and plot the corresponding (x,y) points on some paper
outer flax
done
outer flax
elaborate?
worthy citrus
thats desmos, not you manually plotting points so see why they are where they are
velvet zephyr
for x =1, y = -20
x= 2, y= -18
x= 3, y = -14
x= 4, y =-8
x = 5, y= 0
so x=5 is a root.
after x=5, y=+ve
x= 6 , y = 10
outer flax
thats desmos, not you manually plotting points so see why they are where they ar...
....
outer flax
thats desmos, not you manually plotting points so see why they are where they ar...
X^2
x
OOPS
velvet zephyr
x and x^2
outer flax
velvet zephyr
wdywtk?
outer flax
Why was X a combination of the x and y axis?
X was an unknown value, an input value, now what?
red spruce
desmos is plotting the line y=x with blue and y=x^2 with red and y=x^2 - x - 20 with green
outer flax
yeah i showed that earlier
outer flax
red spruce
Why was X a combination of the x and y axis?
so why are you asking this?
velvet zephyr
,w graph -x
twin meteorBOT
velvet zephyr
,w graph -20
twin meteorBOT
outer flax
so why are you asking this?
You do understand that the color of the line doesnt really have anything to do with the core concept right?
Which ive been asking for this whole time.
velvet zephyr
yes!
outer flax
But nobody seems to know how to explain this probably because they forgot what it was like not to understand it so they dont have reference.
which is why teachers look insane most of the time.
because nobody bothers to remember its hard to remember e v e r y single problems points of confusion.
but who knows
velvet zephyr
You do understand that the color of the line doesnt really have anything to do w...
that's just desmos.
red spruce
Which ive been asking for this whole time.
please form your questions better, it's causing quite a bit of confusion
outer flax
You are asking me to ask that which I don't understand myself.
outer flax
I am asking for the core concepts behind how these numbers interact, are represented, and are formatted.
red spruce
"quadratic equations" are equations with a variable, most commonly x, with 2 as their highest power
they cannot go into negative powers of the variable, they can have coefficients attached to them
outer flax
That isn't what I asked for.
outer flax
I am asking for the core concepts behind how these numbers interact, are represe...
I asked for
I'm sure you know what I'm talking about, you probably just don't understand the perspective of the question.
velvet zephyr
go to this to see how changing the coefficients effect the graph
try to understand it yourself
put some time into observing.
i also don't know much as i am in 9th. but i can help a little.
outer flax
erm
Ok, one second
?
Does each variable take on a different representation depending on how many are added?
boreal wren
can someone explain a point function using mathematical proof, that this particular certain quantity is a point function
vocal sleetBOT
can someone explain a point function using mathematical proof, that this particu...
boreal wren
I understood path function using area under curve. because initial and final positions are same but paths can be different.
vocal sleetBOT
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gritty monolith
i tried to simplify first and then find the derivative instead of using the product rule. where did i go wrong?
vocal sleetBOT
i tried to simplify first and then find the derivative instead of using the prod...
wary mantle
i tried to simplify first and then find the derivative instead of using the prod...
You didn't send a picture
red spruce
put a dot at the start
gritty monolith
.close
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gritty monolith
thank u lol
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hexed furnace
how to do this
vocal sleetBOT
how to do this
hexed furnace
someone help pewasse
hushed pewter
someone help pewasse
Okay, let's do the first one
It wants you to find f(g(3))
Any thoughts?
hexed furnace
Okay, let's do the first one
??
hushed pewter
Oh sorry. Was in another channel
hushed pewter
first find g(3)
Yes. You have it exactly right
You need to find g(3) first. Do you know how?
hexed furnace
You need to find g(3) first. Do you know how?
im not sure but i think like
3 is the input
and you find the output
hushed pewter
you just look at the graph
yes
hushed pewter
so g(3) = 3
yes
hushed pewter
yes. Precisely right
hexed furnace
which is 6?
hushed pewter
yup. good work
hushed pewter
np
vocal sleetBOT
hushed pewter
it was all you
hushed pewter
doesn't seem like it. You did that totally fine
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wary mantle
How do you determine which system belongs to each figure? (2 figures will be left)
vocal sleetBOT
How do you determine which system belongs to each figure? (2 figures will be lef...
chrome fern
by using the normal vectors i guess
lethal heart
How do you determine which system belongs to each figure? (2 figures will be lef...
Do you know what those figure means? in terms of solutions of a system of equations
wary mantle
by using the normal vectors i guess
Exactly, but that still doesn't help that much?
wary mantle
Do you know what those figure means? in terms of solutions of a system of equati...
Yeah
magic wasp
You have equations of planes
chrome fern
well if the normal vectors are pairwise linearly dependent then it's figure 7 for example
wary mantle
well if the normal vectors are pairwise linearly dependent then it's figure 7 f...
They never are
Fig 7 will be left, probably
magic wasp
Pairwise they are
wary mantle
Pairwise they are
Where?
magic wasp
Well I don't know if we mean the same thing
I mean that in some cases two equations have linear dependencies while the third doesn't
wary mantle
Ah
Well I thought you meant all three are pairwise linearly dependant (thus all three linearly dependant)
magic wasp
Do you see how in (1), the second and third equations are basically the same, up to a displacement factor?
lethal heart
Ok, so figure 10 means that there is a unique solution, so is there any of the 3 systems that has unique solution?
wary mantle
Ok, now (2)
lethal heart
a solution to a simultaneous system of linear equations, by definition, has to satisfy every one of the equations. graphically, a point will be in the solution set if it lies on every one of the planes represented by the individual equations. the red sets in figures 8 and 9 do not depict points that lie on every one of the planes so they are not going to be the answers
as you seem to be starting to point out, the other figures depict situations where there are no, exactly one, or infinitely solutions, respectively, so you can calculate with those systems and see which of those possibilities you get
i guess i misspoke, the 'no solutions' case could "look like" either figure 7 or figure 8 or figure 9 graphically, depending on whether pairs of the equations have a solution or not
but the use of the red coloring in figures 8 and 9 just really puts me off, in suggesting that the red points are the solution set
wary mantle
i guess i misspoke, the 'no solutions' case could "look like" either figure 7 or...
Oh, true
magic wasp
i guess i misspoke, the 'no solutions' case could "look like" either figure 7 or...
They are just highlighting intersections, although figure 10 is inconsistent
magic wasp
Ok, now (2)
Do you see anything?
lethal heart
you might have to be more careful depending on whether the red coloring in the figure is just there, or is intended to represent points in the solution set
wary mantle
Do you see anything?
The first and second lines look really similar
lethal heart
I edit some messages
magic wasp
You can also see that by taking the normal vectors
If the normal vectors are colinear, the planes are parallel
lethal heart
the more interesting interpretation of the problem would be to assume that the red coloring has no meaning
chrome fern
2 is either figure 9 or 11
wary mantle
2 is either figure 9 or 11
why
magic wasp
Oh sorry I added a - sign in my head somehow
wary mantle
Oh sorry I added a - sign in my head somehow
Might be a misprint tbh
chrome fern
in the case of 9 or 11 the normals all lie on some plane, therefore they are linearly dependent
magic wasp
Might be a misprint tbh
No I don't think it is, it gives a different figure for each system
chrome fern
but since none of the normals in 2 are pairwise linearly dependent it can't be fig 8
wary mantle
in the case of 9 or 11 the normals all lie on some plane, therefore they are lin...
Hm
Can you elaborate?
chrome fern
look at the picture and imagine the normals for 9 and 11 (and also 8)
lethal heart
in the more interesting case, you would look at the system with no solutions and see if every choice of pairs of equations has solutions (figure 9), only some pairs do (figure 8), or none of them do (figure 7)
chrome fern
you will see that they all lie on some plane, and are therefore linearly dependent
wary mantle
chrome fern
the red intersection lines are orthogonal to the plane spanned by the normals
magic wasp
Assuming these intersection lines are parallel (which they are, otherwise these figures would be equivalent to figure 10)
wary mantle
the red intersection lines are orthogonal to the plane spanned by the normals
True
Yes
chrome fern
fig 10 is the case where all normals are linearly independent
wary mantle
you will see that they all lie on some plane, and are therefore linearly depende...
Why "therefore linearly dependant"
I see how the red line is perpendicular to the plane spanned by the normals
chrome fern
because you have three vectors in R^3 spanning a 2 dimensional subspace
wary mantle
But I don't really know how you conclude the linearly dependant thing
magic wasp
They span a plane instead of the whole space
They aren't completely linearly dependent, as in all colinear, but they are linearly dependent in the sense that you cannot form all vectors with a linear combination of them
wary mantle
How do you know they span a plane
chrome fern
by the usual definition of linear dependence, a nontrivial linear combination that's equal to zero, they are linearly dependent
chrome fern
How do you know they span a plane
by looking at the picture and seeing that the red lines are always orthogonal to the normals
magic wasp
How do you know they span a plane
That's just how the figure looks
chrome fern
i don't have a rigorous proof in case you want one
lethal heart
in figuring out the picture for the 'no solutions' case (which is one of 7-9) looking at the normals is one way to do it. 7 is all the normals are multiples of one another, 8 is a pair of normals are multiples of one another, 9 is the normals are not multiples of one another, but all lie in a plane. 11 is like 9 in this last respect, but those are distinguishable based on the solution set (nonempty in 11 vs. empty in 9)
wary mantle
Oh
magic wasp
Some intuition: if you take a random set of three planes in space, you'll get figure 10
That's the general case
wary mantle
Wait let me just pull down the picture:
wary mantle
in figuring out the picture for the 'no solutions' case (which is one of 7-9) lo...
Ok, yes
magic wasp
Figure 9 has fewer degrees of freedom, figures 8 and 11 even fewer, then figure 7
(I think 8 has fewer than 11 but whatever)
lethal heart
Are there any other doubts?
magic wasp
When you have linear dependency, you basically lose a degree of freedom
wary mantle
So for 9, all normals lie in a plane
How do we deduce which system that is
Or atleast the possibilities
chrome fern
you take some point of intersection of two planes and check if it's contained in the third plane
wary mantle
So in the end, we can't say which belongs to which by just observation?
We have to do some calculation?
wary mantle
Might as well just solve all the systems then..?
magic wasp
So in the end, we can't say which belongs to which by just observation?
Well, for (2) and (3), if you ignore x1, the normal vectors are all colinear
That means figure 9 or 11 depending on whether there are solutions or not (or 8 or 7 but we ruled those out)
wary mantle
Well, for (2) and (3), if you ignore x1, the normal vectors are all colinear
Well but we can't just ignore x1 can we?
chrome fern
Might as well just solve all the systems then..?
you don't need to solve all the systems tho
wary mantle
you don't need to solve all the systems tho
Well ok for (1) we know it's fig 8
We can just solve (2) and (3) quickly
Isn't that the "best" thing to do here
We will have to do some calculation anyways
magic wasp
Well but we can't just ignore x1 can we?
I mean you can if you just want to build intuition
But yes you would need to actually do calculations to verify
wary mantle
But yes you would need to actually do calculations to verify
Well then I find it's better to right away after concluding (1) belongs to 8, just solve (2) and (3)
And we will probably right away know to which they belong
magic wasp
Sure
wary mantle
Without having to argue about anything with normals or anything like that
Maybe we can "headsolve" (2) more or less
2x1 = 3 will be the second line
x1 = 3/2
magic wasp
The whole exercise can be answered by just solving systems of equations (of 3 or 2 equations)
wary mantle
Ok now -3x2 + 2x3 = -3.5 and this means (2) will have no solution
It can't be 7 nor 8
chrome fern
well in the case of 10 checking for linear indepence of the normals would be easier then just solving, no?
magic wasp
well in the case of 10 checking for linear indepence of the normals would be eas...
I think it's basically the same amount of effort
chrome fern
I think it's basically the same amount of effort
but the right side would just be zero
magic wasp
but the right side would just be zero
I guess but you can stop as soon as you know there is a unique solution, you don't need to compute it
It's more or less the same process
magic wasp
No
magic wasp
It's not linearly independent
If you solve it you get an infinite number of solutions
wary mantle
If you solve it you get an infinite number of solutions
Well I guess its enough to get it to REF
Then conclude it's 11
Since free variable
It can be 10 or 11 anyways
Nothing else
magic wasp
Sounds right
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tropic plume
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shadow raft
its the determinant of the matrix
tropic plume
yes
shadow raft
this is the determinant of a 3x3 matrix
if you want to prove it then you can use the leibniz formula for determinants
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@tropic plume Has your question been resolved?
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bleak prawn
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bleak prawn
The white is the question the green is the solution
How come in the first step they put everything over y? What is the thought / purpose behind doing that
astral pilot
multiplying numerator and denominator by y gives (x+xy)/(x-xy)
which gives x(1+y)/x(1-y)
which gives (1+y)/(1-y)
which can be written as
(1+y)/-(y-1)
or
-((1+y)/(y-1))
there is a easy way my friend
why complicate stuff
bleak prawn
I understand all of that, but im just trying to figure out why did they change x/y +x to (y-1)(x)/ y
And the denominator too
Oh wait
I see what youre saying
astral pilot
oh ok
bleak prawn
You did it a different way
astral pilot
i got it
multiply second term's numerator and denominator by y
so
x/y + (xy/y)
x+xy/y
x(1+y)/y
did you get it?
vocal sleetBOT
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opal yoke
Is my solution correct? N(x) is a function and you have to find derivative N(x)'
opal yoke
You can't use the product rule ?
stone gazelle
Product rule is the right call, only went wrong when differentiating (8x+9)^2 and (6x-7)^3 separately
opal yoke
I did differentiate them separately. I left the power outside the bracket alone and differentiated the inside terms. Should i do anything with the power term like multiply it outside + reduce -1?
stone gazelle
I did differentiate them separately. I left the power outside the bracket alone ...
But for the derivative of (8x+9)^2 you got (16x)^2 which is not correct
opal yoke
But for the derivative of (8x+9)^2 you got (16x)^2 which is not correct
I did differentiate (8x+9)^2 wrt.x and 8x+2 wrt.x separately. Does it seem correct now?
stone gazelle
I did differentiate (8x+9)^2 wrt.x and 8x+2 wrt.x separately. Does it seem corre...
Looks good now
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quartz beacon
Given three-dimensional space with a cartesian coordinate system xyz containing the plane v with equation x + y + z = 1 and the plane w with equation x - y = 0. The line l is the intersection of the planes v and w. The line m is the line through the point P(1, 1, 1), parallel to the line l.
Which of the following points lies on this line m?
(A) A(0, 0, 0) (B) B(0, 0, 1) (C) C(0, 0, 2) (D) D(0, 0, 3)
vocal sleetBOT
Given three-dimensional space with a cartesian coordinate system xyz containing ...
quartz beacon
so i did v and w together and then i got z = 1-2x and x=y
but for the 'direction numbers (not sure if its the right translation)' i thought you could just fill in 1 for example and then you'd get (1,1,-1) but this wont work for the final answer??
tawny hollow
Given three-dimensional space with a cartesian coordinate system xyz containing ...
parallel to m? Uh maybe you meant l here?
quartz beacon
Yeah ur right mb
tawny hollow
so i did v and w together and then i got z = 1-2x and x=y
ok so you probably want to parametrize this right?
as its maybe not obvious what the the direction vector here is
quartz beacon
yea
tawny hollow
So if im not wrong, i think we want a line of the form (x,y,z) = Q + Vt
where Q is the starting point, and V the direction vector youre after
and t some real number
so hm can we maybe just let t = x = y ?
vast shale
The line of intersection would be perpendicular to both the planes right ?
vast shale
the line of intersection of the planes
quartz beacon
why can’t we just let x=y=1?
tawny hollow
Yeah how does that imply that
tawny hollow
why can’t we just let x=y=1?
i dont see directly why not, but its best if you can argue why
tawny hollow
Oh yeah thats true, we can just find two such points on the line
vast shale
you just need the d.r's of l so why not just do the cross product of dr.'s of normal's of both the planes
tawny hollow
and then we're done
so let say x=y=0 and also maybe x=y=1
then you have two points on the line
then take their difference and voila you have a direction vector
quartz beacon
why do we always need two points
?
I keep forgetting to do it but sometimes I swear we only used one ?
tawny hollow
If you just had one point, how would you know in which direction the line goes?
tawny hollow
nice!
quartz beacon
you just need the d.r's of l so why not just do the cross product of dr.'s of no...
how would u do this?
vast shale
you can find the dr.'s of the normal to the plane from the plane equation
tawny hollow
cross product seems alot messier than just doing one substraction
vast shale
agreed
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manic lance
how do i prove rigorously that for every $a_n \geq 0$, that is converging to $a \geq 0$, i.e $a_n \rightarrow a$, that it's also true that: $\sqrt{a_n} \rightarrow \sqrt{a}$?
vocal sleetBOT
how do i prove rigorously that for every $a_n \geq 0$, that is converging to $a ...
twin meteorBOT
Ayanokoji
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.close
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silver tusk
hi
silver tusk
i need help
tame schooner
just send the question, and your attempt
vast shale
that is wrong
vast shale
Okay
vast shale
well the arrow mark says what to do ?
vast shale
vast shale
rearrange the terms such that it looks like
tan(x)tan(y)=1+tan(x)+tan(y)
vast shale
oh
silver tusk
im confused
vast shale
nvm
silver tusk
tan20 = tan 90-70
vast shale
on second thought