Yes
#help-23
leaden harness
twilit moat
so you do that for x and y and equate your remaining constants
twilit moat
Oh? But don’t we need to make it = 1
you would convert 3y^2 to 3(y^2+2y)
twilit moat
Oh? But don’t we need to make it = 1
and also your subtraction here should be on the LHS not RHS
leaden harness
Oh but don’t we need to make it 1?
leaden harness
Oh? But don’t we need to make it = 1
Accidentally changed it
twilit moat
Oh but don’t we need to make it 1?
yes but you can divide both sides by the remaining constant
at the end
to make it that anyway
leaden harness
I see
twilit moat
I was just pointing out you had your signs wrong
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@leaden harness Has your question been resolved?
west hedge
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west hedge
I would like someone to help me solve this one
- Differentiate it
The derivative of h(x) has a factor of (x+1)
west hedge
The derivative of h(x) has a factor of (x+1)
Am I right?
west hedge
The derivative of h(x) has a factor of (x+1)
I actually stuck here
I found trouble to factorize h’(x)
twilit moat
I found trouble to factorize h’(x)
simplest way is to factorise out -12x
hard crest
twilit moat
and then it should be a way simpler binomial
but also you can observe that x = -1 is a factor ffrom looking at it
west hedge
are you guys familiar with the method that I used to factorize it.
twilit moat
removing -12x should give you x^2 -2x-3
twilit moat
sub x = 1 right
west hedge
are you guys familiar with the method that I used to factorize it.
the method that I used to divide h'(x) by x+1
twilit moat
and it doesn't equal 0 so not factor
west hedge
and it doesn't equal 0 so not factor
yes.
west hedge
and it doesn't equal 0 so not factor
that's what baffling me
twilit moat
that's what baffling me
well yeah cos you subbed in x = 1 which is (x-1) not (x+1)
and x = -1 or (x+1) is the factor
west hedge
well yeah cos you subbed in x = 1 which is (x-1) not (x+1)
I see.
thank you for pointing out the error.
let me try it again
twilit moat
all g
west hedge
After complete the factorization of h'(x)
we should be able to draw the graph of h'(x)
twilit moat
we should be able to draw the graph of h'(x)
yes
personally I avoided graphing and just solved using the concavity given by second derivative though.
also note that because the original function is -x^4 it will go off to negative infinity instead of infinity so we will have a non-infinite absolute max
which you might have already surmised already
west hedge
personally I avoided graphing and just solved using the concavity given by secon...
I found that the possible absolute maximums occur at x=0 and x=-3
Am I right
twilit moat
I found that the possible absolute maximums occur at x=0 and x=-3
I got points of inflection at 0,1 and 3 with only 3 being a local max
and subbing in x = 3 gives height of 135
west hedge
how should I interpret the graph?
How do i know where the extremes occur?
from the graph
twilit moat
how should I interpret the graph?
the first derivative graph tells you wehre gradient is increasing or decreasing
with your intercepts being where gradient = 0 or your stationary points
you can see that at x = -3, as you pass the point your gradient is negative. This means that you will be at a local max as gradient will go from increasing --> stat point --> decreasing
same for x = 0
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ocean tapir
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ocean tapir
,rccw
flat frigateBOT
ocean tapir
Can anyone verify b
tired kiln
Does the speed that the boat is moving downstream affect the time it takes for it to cross the river? (rhetorical question)
ocean tapir
Does the speed that the boat is moving downstream affect the time it takes for i...
Uhh idk how to respond to that but I’m still confused lol
tired kiln
Okay how'd you calculate the answer in b
Sorry I think it looks pretty close to what I got
ocean tapir
Okay how'd you calculate the answer in b
Uhh yk how the boat travels 13ms^-1 and at an angle of 69 degrees?
tired kiln
I don't like where this is going lol
tired kiln
Seems like it
So if the river was flowing faster the boat would take the same time to cross the river right? It would just finish further downstream
Since the crossing time only depends on the speed in the direction of the shore which is perpendicular to the direction of the river's flow
So if the river wasn't moving at all the boat would cross in the same time
ocean tapir
Oh
tired kiln
In the case of zero river velocity then the answer is pretty simple right?
ocean tapir
Yea I get it
tired kiln
Noice
ocean tapir
So it would just be 169/12?
tired kiln
Yes!
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deft kraken
Any idea on this ? I already figured out by some symmetry that if k is not a multiple of 4 then s_{n,k} = 0. Otherwise I've been trying to describe the underlying set to no avail...
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Any idea on this ? I already figured out by some symmetry that if k is not a mul...
@deft kraken Has your question been resolved?
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steep magnet
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steep magnet
Hmmm
Seems like we have 1 too many unknowns here
I cant just sub in one of the points because im missing the x coordinate of the centre and the radius
So we need to use the fact that I have 2 points on the line to get one of those unknowns
And im not really sure how to do that...
lean otter
A circle is defined by three real numbers, x,y, coordinates of the centre and the radius
steep magnet
I guess the distance between the 2 points and the radius is the same? So its an isoceles or however you spell it
main mural
there are infinitely many circles that pass through those two points and their centers lie on one line
lean otter
but since the centre is on the x-axis it now only has 2 variables
main mural
there are infinitely many circles that pass through those two points and their c...
the intersection of that line and the x-axis is your center
steep magnet
Ok I understand you both but I still dont see the next step sry
main mural
find the line i am talking about
the line is a normal to the line AB that goes through the midpoint of AB
timber karma
let the centre be on point O(x,0) and use formula for distances
steep magnet
let the centre be on point O(x,0) and use formula for distances
Oh I see
timber karma
use pythagoreom and set them equal to eachother
steep magnet
let the centre be on point O(x,0) and use formula for distances
Ok I think I know how to solve using that
Thank you so much everyone!
❤️
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steep magnet
Hmm I thought I knew how to do it but I got the wrong answer and I dont know what I did wrong
First I used the distance formula like you said to find the radius
Using points (1, 2) and (x, 0) I got a radius sqrt(x^2 - 4x + 5)
I then put the point on the circle (1, 2), centre of the circle (x, 0) and the radius of the circle (squared) x^2 - 4x + 5 into the circle formula (x - x1)^2 + (y - y1)^2 = r^2
That gave me this
With the x coordinate of the centre of the circle being the only unknown
But it gave me the solution x = 0, I was supposed to get x = 17 according to the answer sheet...
Ohhh I see my error I just made a mistake with my algebra
😅
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steep magnet
Its funny that I didnt use the other point at all to get my answer 😝
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steep magnet
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steep magnet
Oops forgot the actual question 💀
Centre of the circle is (17, 0) from part A but I dont think thats important
Is there any way to do this other than getting the derivative of the circle? Cause that only gets 1 half of the circle or something right? Im not too familiar with that method...
steep magnet
You can also implicitly differentiate
Whats that sry
steep magnet
You can also get the other half
How?
fathom jewel
When you take the square root you get two function solutions
and inplicitly differentiation means not solving for y in order to diff.
for example: y² = x
can be differentiated implicitly
2y * dy/dx = 1
dy/dx = 1/2y
quiet plume
(If you really want to avoid differentiating, another way would be to consider a general line going through (0,0), so y=kx, and plug that into the circle equation.
You want to find the values of k that ensure the (quadratic) equation you're left with has a single root.)
steep magnet
You want to find the values of k that ensure the (quadratic) equation you're left with has a single root
Why sry?
fathom jewel
Why are you apologizing btw
steep magnet
Idk sometimes people get annoyed with me for asking questions or act like im bothering them
I dont wanna bother anyone
quiet plume
> You want to find the values of k that ensure the (quadratic) equation you're l...
Nw! Hum, tangents to a circle only touch it once, so if it had more than one root, the line wouldn't be tangent to the circle, since it would intersect it twice.
steep magnet
Nw! Hum, tangents to a circle only touch it once, so if it had more than one roo...
Ahhh ok I see
So what from y=kx do i actually sub into the circle equation and where?
kx for the y coordinate?
quiet plume
kx for the y coordinate?
Yes exactly
quiet plume
Keep it like that. If you expand everything and gather like terms (k is constant), you should have a quadratic equation in x
steep magnet
But k is the other unknown? How can i solve?
quiet plume
x is unknown, but you don't want to solve for it.
You want to solve for the value of k that yields a single solution to the quadratic equation. This you can check using the discriminant.
steep magnet
Hmm I dont really understand yet but I will just get started and see where I get stuck
quiet plume
Yeah it might be clearer once you get the equation set in the form
Ax^2 + Bx +C = 0
steep magnet
(x - 1)^2 + (y - 2)^2 = 5
(x - 1)^2 + (kx - 2)^2 = 5
x^2 - 2x + 1 + k^2x^2 - 4kx + 4 = 5
x^2 - 2x + k^2x^2 - 4kx = 0
x(x - 2 + k^2x - 4k) = 0
So already we have 2 solutions no?
Or do we just need to make sure (x - 2 + k^2x - 4k) also equals 0
quiet plume
Or do we just need to make sure (x - 2 + k^2x - 4k) also equals 0
Yeah you can do it that way and ensure that the root of this is x=0.
quiet plume
x^2 - 2x + k^2x^2 - 4kx = 0
The other way is to group terms together and use the discriminant, but I think your method is easier, since now you just need to plug in x=0 and solve for k.
steep magnet
Yeah you can do it that way and ensure that the root of this is x=0.
Which square root tho sry? And am I solving for x or for k?
steep magnet
The other way is to group terms together and use the discriminant, but I think y...
Ohhhh
steep magnet
x(x - 2 + k^2x - 4k) = 0
I can just plug x=0 straight into this?
((0) - 2 + k^2(0) - 4k) = 0 ?
quiet plume
Yeah well just in the right factor, since we already know the x factor out front yields a root at 0
quiet plume
((0) - 2 + k^2(0) - 4k) = 0 ?
Yes this exactly
steep magnet
Ok I see
So the slope is -1/2
Then I can get the equation of the line easily
Alright I get it now
quiet plume
Good work
steep magnet
Just so I know, did this only work because the y intercept is 0?
Or could this method work with a point that isnt at the origin
quiet plume
It would work with any point, you would just have to write the equation of the line accordingly for a line that goes through that point.
quiet plume
It's quite interesting indeed
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halcyon light
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halcyon light
To use the formula l = r theta
I don't know how to decide which length I should use
I have angle BAC
but how do I know if the length I use is 12 m or 10 m
timber karma
are you stuck on (b)?
halcyon light
are you stuck on (b)?
yeah
halcyon light
but how do I know if the length I use is 12 m or 10 m
well Its just this I dont get
I gotta work out r to use l + 2r
timber karma
well i dont think you have to use it right?
halcyon light
i need r
plain furnace
i think u have to subtract the whole area to the area of the part of the circle
safe radishBOT
halcyon light
12 c
I don't know why the bounds of 1 and 16 are wrong
Plus idk where this triangle comes from
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agile lava
Find all the suits of integer such as $u_n \mid 2^{u_{n+1}}-1$
safe radishBOT
Find all the suits of integer such as $u_n \mid 2^{u_{n+1}}-1$
flat frigateBOT
Gamma
agile lava
I see that u_n is odd
then 2 and u_n are coprime
I think I should use multiplicative order or fermat/euler theoreme but I don't see how
tropic summit
Find all the suits of integer such as $u_n \mid 2^{u_{n+1}}-1$
what do you mean by suits?
agile lava
what do you mean by suits?
a sequence
lean otter
I imagine this problem has something to do with the property that 2^x - 1 is a factor of 2^{xy} - 1 for x,y naturals.
agile lava
is this the original question?
that's the original question, I just translated it
agile lava
I imagine this problem has something to do with the property that 2^x - 1 is a f...
I'll search with that hint
can u help? I'm stuck
lean otter
do you manage to see the property I mentioned?
agile lava
do you manage to see the property I mentioned?
I don't see how I can use it tbh
Ah I see !
ku_n= ...
ku_n+1=2^(un+2)-1
ku_n divides 2^(ku_n+1)-1
And k u_n+1=2^u_(n+1)-1
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@agile lava Has your question been resolved?
wheat temple
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wheat temple
Hey ive tried graphing this and i dont know how
If someone could just walk through how they would do it that would be great
I dont need you to ask me questions just explain your step by step process
mossy lotus
start from basics
plot sinx
then sin(x+pi/4)
then sin3(x+pi/4)
then 2*sin3(x+pi/4)
versed wave
do you know what a sinusoidal function is and what does each component convey?
mossy lotus
The curve just shifts, then it becomes higher frequency, then higher amplitude in those steps
versed wave
what i could do is to bring the 3 in the parentheses, and identify ||the amplitude, the angle frequency and the intial phase||
wheat temple
ok i have one question how do i plot sin(x + pi/4)
lean otter
plotting sin(x+3pi/4) is more useful
because you just need to relabel the axes at the end
safe radishBOT
versed wave
how do i find the points
...?
wheat temple
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versed wave
which axis?
wheat temple
both
versed wave
a sinusoidal function either intersects the horizontal axis at infinite points, or never intersects at all
why would you want to do so?
versed wave
what i could do is to bring the 3 in the parentheses, and identify ||the amplitu...
were my explanation not clear enough?
wheat temple
the answer for the question has it plotted at a specific place
versed wave
a sinusoidal function either intersects the horizontal axis at infinite points, ...
as for the vertical axis, well what do you think?
wheat temple
i’m just confused why they chose that one area of the graph to plot
wheat temple
as for the vertical axis, well what do you think?
one intersection
versed wave
i’m just confused why they chose that one area of the graph to plot
which area?
usually the function is plotted in the domain of 0 to some arbitrary numbers, like 10 or 15
versed wave
either way, have you completed the original question yet? can you show your work?
versed wave
seems like an error on their part
wheat temple
hmm
it’s alright i will ask my teacher tomorrow
it’s 1am i am not thinking right
thanks for helping
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long sorrel
help please 😭 idk where to get started
safe radishBOT
help please 😭 idk where to get started
fathom adder
So its defined on the integers, lets say between 1 and 3
What do you think, can you graph it as a one line, without jumping ?
Or would it be multiple points with no line joining them ?
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@long sorrel Has your question been resolved?
long sorrel
So its defined on the integers, lets say between 1 and 3
woudnt it be -1 and 3?
fathom adder
woudnt it be -1 and 3?
Its an exemple
Take the integers you want
The thinking is the same
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dawn crest
Hey all, first time poster. I'm self studying mathematics and currently on going a course about algebra. I have a exercise that requires me to solve for x. I have the equation and my solution posted below(as a picture). Any help appreciated!
safe radishBOT
Hey all, first time poster. I'm self studying mathematics and currently on going...
proud dust
what seems to be the issue?
icy lance
dividing by x+1 is risky business
dawn crest
What concerns me is that should i treat a - 1 as one, i.e wrapping it inside parenthesis. With my solution i just moved -1 to right side and left the a on the other side
icy lance
youd be better off using the quadratic formula if youre familiar
proud dust
i'd do x^2 +(a)x+(a-1) = 0
gusty trench
just wrap a - 1 up as a single constant, maybe call it p if you want, and then just quadratic formula
obsidian oracle
and remember that you're not allowed to divide by something if that something is...
dawn crest
I am familiar with it, i actually considered it but couldn't manage to apply it since there is this a variable
gusty trench
just because the a is also an unknown doesn't mean you can't solve for x using the formula
you'd just get x in terms of a
obsidian oracle
I am familiar with it, i actually considered it but couldn't manage to apply it ...
it doesn't really matter, just get x in terms of a
dawn crest
So the correct approach would be to plug it into quadratic formula as -> b=a, c=a-1, a=1?
icy lance
yup
dawn crest
so is my approach completely wrong here? I did plug x = -a+1 back into original equation(its not shown in the picture tho)
icy lance
its just that when you divided by x+1, you assumed that x+1 is never 0, this isnt necessarily true
obsidian oracle
and remember that you're not allowed to divide by something if that something is...
yep
dawn crest
in case i had stated the domain to be x != -1 would it had made a difference?
icy lance
yeah, then nothing you did was wrong
there are two solutions
the one you got
and x=-1
dawn crest
ok, thanks! I think it was meant to be done this way then. I'm trying to plug it into quadratic formula but its getting awfully complicated(atleast for me)
icy lance
just take it slow, youll get used to these things in time
dawn crest
its looking like this currently with the quadratic formula
dawn crest
woops, thanks
icy lance
consider how you could factorise the expression inside the root
dawn crest
i see!
many thanks, i think i will be going with quadratic formulate approach after all!
could you guide me with one other exercise as well?
so the assigment goes like: "Find a value for constant c which can be used to reduce equation f(x)". So the idea is to factorize the equation. I got the right answer but the method was some what sketchy.
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@dawn crest Has your question been resolved?
icy lance
the only issue is your actual value of c is -30
when you wrote (x+3)(3x+c) you created a new c, different from the original
original c=3*new c
dawn crest
right, should have picked a different variable for that. However is there a more mathematical way to solve this? Like utilizing x-intercept points of the 3x^2-x, or the vertex x-coordinate of 3x^2-x
icy lance
if we assume (x+3) is a factor of 3x^2-x+c=0
then if i sub in -3 i should get 0
so 3(9)-(-3)+c=0
27+3+c=0 c=-30
is perhaps more straight to the point
youd still have to factorise after though
before you could cancel
the way you went about it was actually quite nice though
dawn crest
god damnit, im an idiot. Took me like 2 hours to solve that. I will definitely use the 3(9)-(-3)+c=0 method you mentioned, that's way more cleanier. Thanks a lot!
icy lance
no worries
youre not an idiot though
you still got the answer in a correct method besides the little blip
dawn crest
thanks sensei. Really appreciate your help, i gotta be going now tho. See you later ✌️
icy lance
cya
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elfin mortar
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crude blade
@elfin mortar
crude blade
Do you have a questiom
tired kiln
Treat the 4x and 9x as the sidelengths of the big square, and I'm guessing the -45x and -35 are supposed to be the areas of the other rectangles? But I'm not 100% sure what this diagram means out of context
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clikz
kindred fjord
rain galleon
yes, it's correct
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rough pelican
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magic junco
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half canyon
can someone help me find the gradient
safe radishBOT
can someone help me find the gradient
half canyon
i got the answer 0 but i dont think thats right
river oriole
whats your 1st derivative looking like
half canyon
i got 2(x+2) x 1
half canyon
yea so would it be 0?
river oriole
2(-1+2)=0??
river oriole
lol
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glass field
how can one solve this
safe radishBOT
how can one solve this
prisma mauve
Id say introduce the lengths around as unknown and that should just be a system to solve 🤔 after all you just care about the length of the 72 rectangle and the height of the 128 rectangle 🤔 wait a sec
winged flare
how can one solve this
didnt you ask this that day?
glass field
didnt you ask this that day?
yea
glass field
i sort of ignored this problem cuz i was stuck
now im back at it and still am confused
winged flare
make the triangles into rectangles and then find areas in terms of common lengths and breadths
glass field
then itd be the same
it just gives me x=x
winged flare
what method did you use?
glass field
what method did you use?
i took the height of the rectangle and triangle as x
l as length of rectangle
b as base of triangle
hence b*x=96
hence l*x=36
hence 2l=3b
winged flare
l as length of rectangle
length of which rectangle?
glass field
the top left
winged flare
b as base of triangle
what about height of bottom rectangle and triangle?
glass field
what about height of bottom rectangle and triangle?
didnt give value to them
winged flare
didnt give value to them
youd need that to find the area too
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past star
Hello, I would like some help and understand how to do this.
I have an economics question and it is asking me to differentiate (First Order Condition FOC) the profit formula.
This isn’t an exam question or homework, it is just tutorial questions. I have the answer but I don’t know how to get to the answer.
safe radishBOT
Hello, I would like some help and understand how to do this.
I have an economics...
@past star Has your question been resolved?
crystal palm
you are pretty much done i think
we want to maximise pi with respect to q1
so if we solve the last line for q1, we get the value of q1 when q2 is given such that pi will be maximised
this is it visually. the equation 1-3q1-q2=0 is the straight plain. so given q2, the plain gives us q1. and the value of pi at that (q1,q2) will be the max
past star
you are pretty much done i think
we want to maximise pi with respect to q1
so if...
Thank you for the answer. I think my question was worded poorly…
What I wanted to ask was how do I, step by step, differentiate the Pi1 equation (after expanding it multiplying it by q1)
crystal palm
though the notation and implication here is definitely not always clear when talking about partial derivatives
i think this video shows that quite well: https://www.youtube.com/watch?v=mICbKwwHziI&t=517s
BUT: If you don't wanna get confused, maybe not watch that right now haha
stable flint
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mossy lotus
check signs for all terms. I just used division rule for the derivative, and it seems to me bc should have a -ve sign
stable flint
check signs for all terms. I just used division rule for the derivative, and it ...
i think it's correct
lime stone
wouldnt this work better
stable flint
uh
stable flint
wouldnt this work better
can you explain how you got it
lime stone
lime stone
i feel like this would be a bit simpler
stable flint
fr
lime stone
what should I do after last step
i found it
the error is in the last 3 lines
try those again
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tacit atlas
For each of the following requirements, write a verse in a language that has a two-local relation sign R and a relation sign =, so that the verse satisfies the structure M=(W^M,R^M) if and only if it fulfills the requirement -
A. R^M equivalence ratio on W^M
B. R^M Partial order relation on W^M
C. R^M linear order relation on W^M
any help?
safe radishBOT
For each of the following requirements, write a verse in a language that has a t...
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flat frigateBOT
l'agit
buoyant shadow
it is
flat frigateBOT
l'agit
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fervent hatch
What am I doing wrong from the $$M_x/M$$ part
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What am I doing wrong from the $$M_x/M$$ part
flat frigateBOT
Nathan
fervent hatch
It should be 10/33
It’s a center of mass problem but density is constant so you’ll see I just ignored it
safe radishBOT
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fervent hatch
<@&286206848099549185>
crude blade
Hello
tough rapids
What am I doing wrong from the $$M_x/M$$ part
may I ask how you arrived at your Mx equation?
because the calculations look correct
fervent hatch
may I ask how you arrived at your Mx equation?
Plug my integral into a calculator and it agrees
I genuinely don’t know how I’m wrong
@tough rapids
tough rapids
mb
uh lemme see
I'm taking physics this coming year, so my knowledge of center of mass is very limited
but I was just wondering how you got the integral
cause, as you said, if you plug it into a calculator it checks out
so it probably has to be a problem surrounding the original integral
fervent hatch
Well okay
So he dx and the x^4-x^5 gets the area
Density times the area would get the mass but since the density is constant it will cancel later on
The other bit is writing the y position of the center of mass of the given strip in terms of x
So just dividing it by two since the center of a rectangle is the middle
fervent hatch
Yes the density which is irrelevant times the area will be mass here
It’s a 2d region
tough rapids
ah ok
fervent hatch
yeah so I don’t see how my integral is wrong
tough rapids
then idrk, you might have to consult someone more versed in physics
if it were four months later I would be able to help 😅 cause I would have learned it by then
fervent hatch
Calculus 1
fervent hatch
this is the first unit of calculus 2 so that’s probably why you’ll learn it when school starts again
tough rapids
yeah probably
fervent hatch
But I mean the integral is cal 1 it’s just the application you gotta learn
safe radishBOT
@fervent hatch Has your question been resolved?
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@fervent hatch Has your question been resolved?
white yew
can someone help me this please
safe radishBOT
can someone help me this please
white yew
<@&286206848099549185>
white yew
im not familar with what that is?
fathom jewel
Yes or no
drifting knot
No i mean
It's a part of linear programming?
Or it's a simple case of maxima minima
Ya ok I saw it now it's maxima minima
Oh
I get it
It's the application of calculus part
Revenue function =Price x No right ?
white yew
is the answer $4.64?
fathom jewel
,w -2*1500x+12500=0
flat frigateBOT
white yew
?
fathom jewel
,calc 25/6
flat frigateBOT
Result:
4.1666666666667
yes
fathom jewel
idk
white yew
is there any way you can check?
fathom jewel
Give me context
Is this some profit formula
Also it says maximize
So i assume you will need differentiation at some point
white yew
P(x) = R(x) - C(x), where P(x) is profit, R(x) is revenue, and C(x) is cost
flat frigateBOT
fathom jewel
,w -3000(p-4.64167) = 0
flat frigateBOT
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molten flume
P(A)=0.4
P(AUB) 0.7
A and B are independent
P(B)=?
safe radishBOT
P(A)=0.4
P(AUB) 0.7
A and B are independent
P(B)=?
lean otter
what does it mean for two events to be independent?
lean otter
AUB=A+B-AÛB
AÛB=0 because independent
so shouldn't P(B) be 0.3?
that’s not it
that’s disjoint/mutually exclusive events
molten flume
but answer is 0.5
lean otter
not independent events
molten flume
what does it mean for two events to be independent?
intersection null?
lean otter
no
again that’s the definition of mutually exclusive or disjoint events
they said it’s independent here
molten flume
so what should I do
lean otter
can you check your notes for what “condition” you have for independent events?
molten flume
I'll see wait
pallid crystal
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
flat frigateBOT
Heartuary
lean otter
pallid crystal
lean otter
it’s P(A|B) = P(A)
But also tbh not what i’m looking for, but you can derive it from here
anyway what i wanted you to say was
molten flume
oh yes
lean otter
P(A n B) = P(A) * P(B)
lean otter
it’s P(A|B) = P(A)
which can also follow from here using the definition of “conditional probability”
molten flume
yes if they independent then multiplication give intersection
lean otter
yes now u can use that
lean otter
**Heartuary**
here
molten flume
which can also follow from here using the definition of “conditional probability...
yes this topic is under conditional probability
so if A and B and independent then AÛB will exist
lean otter
$P(A \cup B) = P(A) + P(B) - \underbrace{P(A) P(B)}_{P(A \cap B)}$
flat frigateBOT
!Kiz__
lean otter
so if A and B and independent then AÛB will exist
not sure what that’s supposed to mean
molten flume
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molten flume
you helped i got answer
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jaunty trench
What's "trend"?
safe radishBOT
What's "trend"?
mossy lotus
trend is behaviour of the y values dependent on x values in relation to commonly known functions
so trends can be exponentially increasing, logarithmically increasing, quadratically increasing and so on
since e^x, log x, x^2 are well known functions and considered to be good enough approximations for many functions
safe radishBOT
@jaunty trench Has your question been resolved?
jaunty trench
i dont understand <@&286206848099549185>
safe radishBOT
@jaunty trench Has your question been resolved?
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@jaunty trench Has your question been resolved?
safe radishBOT
@jaunty trench Has your question been resolved?
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@jaunty trench Has your question been resolved?
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@jaunty trench Has your question been resolved?
dusty merlin
A bit confused on what happened here between the encircled steps
compact rampart
A bit confused on what happened here between the encircled steps
oh , they just decomposed the fraction
(1+x+x^2)/x^2 = 1/x^2 + x/x^2 +x^2/x^2
which is equal to
x^-2+x^-1+1
hopw it helps
compact rampart
u need anymore ?
safe radishBOT
pearl pollen
How do i simplify this one?
safe radishBOT
How do i simplify this one?
thin bridge
your last step is wrong
pearl pollen
Can you explain
thin bridge
only the -2 should be multiplied to the fraction
pearl pollen
only the -2 should be multiplied to the fraction
I have a question
When do we only put 1 as a denominator?
When it's not multiplication?
thin bridge
whenever you feel like
thin bridge
putting /1 makes absolutely no change to the value of an expression
it may help people multiply fractions, you can explicitly write it if you want but it is completely unnecessary
the issue with the way you did it is that
(1-2) isn't actually being multiplied to 1/x
thin bridge
by putting /1 over that entire component, you are essentially putting() around the 1-2
which changes the value of the expression
you are however allowed to do
$$1 - \frac21 \br{\frac 1x}$$
if you want
flat frigateBOT
ℝαμΩℕωⅤ
thin bridge
Putting 1 still arrives to the same answer but a longer solution?
if you do it properly yes
whether you do it usually depends on how you are taught to multiply fractions
lean otter
find the greatest common divisor of 53667 and 25527 using euclids algorithm
lean otter
53667 = 25527 * 2 + 2613
25527 = 2613 * 9 + 2010
2613 = 2010 * 1 + 603
2010 = 603 * 3 + 201
603 = 201 * 3 + 0
Why do we divide by 2 on the first number and why do we divided by 9 on the second number
lean otter
oh why
obsidian oracle
that's literally euclid algorithm
lean otter
oh
obsidian oracle
if you wanna find gcd of a and b
lean otter
you divide the first by the second number
lean otter
wow
obsidian oracle
take remainder r'
r/r'
etc...
until remainder is 0
and gcd is last non-zero number
lean otter
you are very helpful thank you
how would you determine the gcd of 3 numbers
a,b,c
gcd (a,b) = d
obsidian oracle
and then you do gcd(d,c)
lean otter
gcd (a,b,c) = gcd (d,c)
okay
i can also do then gcd(b,c)=d to do gcd(a,d)
?
and get the same number
thank you
.close
safe radishBOT
obsidian oracle
and get the same number
yes
safe radishBOT
✅
obsidian oracle
there are two parts to this
gcd(a,b) = 1 and lcm(a,b) = ab when a and b are coprime
and gcd(kc,kd) = k*gcd(c,d), lcm(kc,kd) = k*lcm(c,d)
obsidian oracle
you can just google lcm times gcd
cerulean haven
How do I simplify this thing?
safe radishBOT
How do I simplify this thing?
cerulean haven
The answer is cot 2x
lapis shadow
Have u tried anything
lapis shadow
U can factor
cerulean haven
How?
cerulean haven
a(b-1)
lapis shadow
🙂
cerulean haven
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lean otter
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lean otter
ion get it still
safe radishBOT
@lean otter Has your question been resolved?
lean otter
A first step would be to label things
ye I did label em
glacial cairn
ye I did label em
Show
lean otter
I can't click a pic rn but thisis what my teacher had sent me earlier
glacial cairn
I can't read that
Anyway it seems pretty useless to label the points
Does the question specify that the main figure is a parallelogram?
lean otter
Does the question specify that the main figure is a parallelogram?
I already found the solution for this online
But
:(
Ion get it
glacial cairn
Then your question should be about that
lean otter
This is the question
glacial cairn
What don't you get?
glacial cairn
If that's the whole question then there is no exact solution and you can only try measuring the area
If you can assume that it's a parallelogram, then there is a pretty easy solution, but really the question should state that
safe radishBOT
@lean otter Has your question been resolved?
dense moon
worked on this problem a bit earlier I got 12
safe radishBOT
worked on this problem a bit earlier I got 12
dense moon
as it was the lowest possibility because any other source of numbers is just divisible by it
glacial cairn
What's your question?
mossy lotus
8 seems like it can be a factor too
dense moon
right but we want the highest divisible factor
and so far it seems 12
I just need a scientific way of proving this
mossy lotus
yea so with 8 and 12 you get 24
dense moon
or matematical way
mossy lotus
thats what i meant to say
dense moon
ok but doesnt it have to be a multiple of n
and if n is 24 how does d=4,c=3,b=2,a=1 work
glacial cairn
You have four integers. Two of them are necessarily separated by a multiple of 3. Additionally, two pairs of them are necessarily separated by a multiple of 2. That gives you 12 as a lower bound
mossy lotus
wait no mb
mossy lotus
yeah, it gotta be 12
dense moon
But I still need a mathematical way of always proving its 12
But I like the idea of seperation as it will always be a multiple of 3 and two multiples of 2
mossy lotus
express a, b, c, d as mod p^n numbers and prove that for mod p^n+1 it doesnt work
dense moon
Like it needs to be based off a formula and something that works everytime and it needs to be achieved by doing that not just guessing and checking
glacial cairn
You have four integers. Two of them are necessarily separated by a multiple of 3...
This is not "guessing and checking"
dense moon
Ok now I see that
cause I tried with a couple more cases and it worked
so I do think its 12 because of the seperation
dense moon
no like i meant chose a random set of numbers
yeah thats what i did so I came to the conclusion that your answer is right
with the logic you have making sense
because c-a is a multiple of 2 d-a is a multiple of 3 and d-b is a multiple of 12
that multipled together always equals 12 minimum
glacial cairn
You have four integers a,b,c,d
All integers are of one of three forms: 3k, 3k+1, 3k+2
dense moon
ok
glacial cairn
So at least two among a,b,c,d have the same form
Hence their difference is a multiple of 3
dense moon
ok
glacial cairn
Additionally, all integers are of one of two forms: 2k and 2k+1
Among a,b,c, at least two have the same form
dense moon
ok
glacial cairn
If you consider d as well, either it has a different form but then a,b,c all have the same form (and so you get three differences that are multiples of 2), or it has the same form as at least one of a,b,c, and so you get yet another difference that is a multiple of 2
dense moon
ok
glacial cairn
In total, you have a difference that is a multiple of 3, and two differences that are multiples of 2
That means the product of all differences is a multiple of 12
dense moon
ooh so thats how we establised this conclusion
that makes more sense now that product is a multiple of 12 for the difference
Thank you for the explanation
safe radishBOT
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dense moon
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remote zephyr
hey guys im thinking there's an error in this solution. inside the |s_n - s_n*|, i think s_n should have been just s. is that correct?
safe radishBOT
hey guys im thinking there's an error in this solution. inside the |s_n - s_n*|,...
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.close
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keen furnace
does this multiply the determinant's size by 3?
safe radishBOT
does this multiply the determinant's size by 3?
keen furnace
i don't know what rotating by pi/3 is
is that just the angle of rotation or does it stretch or squash it?
if the former, then it would be same determinant
timber moss
is that just the angle of rotation or does it stretch or squash it?
that is the angle of rotation; you see these rotation matrices very often in dynamics courses that give you your 'x' vector expressed in a different coordinate system
keen furnace
ok since rotation doesn't increase or decrease the area of T, it's the same determinant right?
timber moss
yes, because the determinant of A is just 1
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lean otter
is this 0?
safe radishBOT
is this 0?
lean otter
not sure on h o w I could transform the bottom one
i know that the top one could be turned into 2sin^2x-2sinx
eternal carbon
is this 0?
yes
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fossil path
Hi
fossil path
Can you help me with two questions?
I answered one just need it to be reviewed to see if I did it wrong
And the second question I'm not sure what equation to use to solve
First Question ^
2
junior smelt
Shade the region that's represented by these?
fossil path
"Graph the solution set to the following system of inequalities." I think so
junior smelt
For that one, your second (in)equation is 3 - 6x^2, but it looks like you read it as 3 - 6x? (at which point, even then, that would need to be steeper)
So there should be two boundary parabolas you'd want right?
fossil path
Yes
junior smelt
2
(anyways, as for this, you want your total mass of coffee, made of those two blends, to be 20lb, but you want the price to work out to $6.30 per lb...)
fossil path
Is there an equation to follow I've looked online and couldn't find one
junior smelt
Is there an equation to follow I've looked online and couldn't find one
For the second one? we can work through it together if you want?
fossil path
Yes for the coffee one
Please it's a little confusing ^^'
I just don't know what steps to take
junior smelt
Alright, we'll go step by step, most of these are similar, but you want to create and then solve a set of simultaneous equations, from effectively two pieces of information that's in the question
So, a hopefully easy question, if I charge $6.30/lb, and you want 20lbs, how much will you have to pay?
fossil path
126?
junior smelt
Yep 
fossil path
Phew LOL
junior smelt
(anyways, as for this, you want your total mass of coffee, made of those two ble...
So, now, I can rephrase this a tiny bit, let's call the amount of the Kenyan blend be k, and the other one be s
junior smelt
(anyways, as for this, you want your total mass of coffee, made of those two ble...
We want here, to rephrase this slightly:
- that the total mass of the new blend you make to be 20lb
- to be able to charge $126 for it, given the Kenyan one is $7 per lb and that the other one is $6/lb
Do you think that gives you any idea as to how to create either equations for them? (the $126 comes from the fact you're charging $6.30 for it, but are making 20lbs of it)
fossil path
7k+6s=20?
7k+6s=126?
7k+6s timss (20 -6.30)=126?
7k+6s *(20-x)=126?
7k+6(20-x)=126?
Idk I'm trying haha ^^''
fossil path
Okay 😂
junior smelt
But for the first one, remember that k is the mass of Kenyan coffee in lbs, the s is mass the other coffee (I just know I'm gonna spell it wrong 
), and you want the mass of both coffees combines to be 20lbs 
fossil path
So wouldn't it be 7 times 2 and 6 times 1
126÷14=9 and 126÷6=21?
So 9 of Kenya and 21 of S?
junior smelt
Not quite like that, waaaiitttt, let's slow it down a tiny bit 
junior smelt
Another new question, note that the price of the coffee doesn't really matter that much for this one, but
If I use a mass of k pounds for the Kenyan coffee, and a mass of s pounds of the other one, how much in total, in terms of k and s, will my new coffee weigh?
fossil path
13?
junior smelt
13?
Not quite, that would be the case if I used 7lbs Kenyan and 6lbs respectively, but that isn't the case, rather it was k lbs of the Kenyan and s lbs of the other, soooooo?
fossil path
Divide? :0
junior smelt
Remember that the 7 and 6 are the cost in dollars for each lb, so if you used 2lbs, you'd charge $14 and $12 respectively, same that 5lbs is $35 and $30
junior smelt
Divide? :0
Nooo 
not quite-
fossil path
Ohh
junior smelt
13?
Anyways, what was your thought process to get this number you thought before? What was the idea?
fossil path
For 13 I just added for, 9 and 21 I divided ^^'
junior smelt
Awwww
Anyways, for the mass equation we'd find, we wanted as before to find the total mass of both the k pounds of Kenyan coffee, and the s pounds of the other coffee, to work out to 20 pounds...
Of course it probably is a massive spoiler as it is, that you want to take the k pounds of the Kenyan coffee, and then add the s pounds of the other coffee to it 
For which, I'll let you have that another equation is k + s = 20 to you for free
fossil path
Haha thank you!
junior smelt
So anyways, you then want to solve k + s = 20 and 7k + 6s = 126
Hopefully doing that will be fine? 
fossil path
Knowing me and my me it might be a mess thank you
junior smelt
Knowing me and my me it might be a mess thank you
your me 
but awwww, as long as you know the process for it at least?
fossil path
So divide by 7 get 18 and input it?
junior smelt
Not quite 
have you done simeultaneous equations before?
fossil path
Wait
Hm
Nope if I have I forgot ^^'
What would be the steps to take if not division
K=6 and s=14?
junior smelt
,w k + s = 20, 7k + 6s = 126
flat frigateBOT
fossil path
Yay 😁 thank you so much for the help forgive me for my dumbness
junior smelt
Awwwww, you're fine, don't worry about it 
junior smelt
Anyways, for the first plotting one, are you fine with that one? that you wanna plot the other parabola and work with that one?
fossil path
Could you help with it?
So I misplaced the points?
Sorry it confuses me as well haha
junior smelt
Sure 
and yep, you plotted x^2 - 4 correctly, but not 3 - 6x^2
You know how to plot the second one, right?
junior smelt
It isn't linear, or a line
If I rewrite it a tiny bit, do you know how to plot -6x^2 + 3?
junior smelt
That one is a bit more "interesting"
junior smelt
Would two points be at (-2,-6), and (2,6) or no
Ohhhh you can choose points to put in in which case, not quite these
fossil path
Okay ^^
junior smelt
If you choose x = ±2, then you put that into -6x^2 + 3, and work out what it becomes
Which isn't 6, but 21 for both 
fossil path
junior smelt
Looks much better think that should be fine
fossil path
Okay thank you so much for being patient and helping me 😄 you're awesome!
junior smelt
Awwww 
thank yooooouuu 
fossil path
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worn relic
Hi. What will be the sum of all the positive factors of 1?
How about 4? Thanks in advance