So its 4 because-2*-2 there’s two - so it’s a +
#help-23
naive wyvern
plucky elk
plucky elk
So its 4 because-2*-2 there’s two - so it’s a +
so what does x^2 - (-2)^2 = ?
plucky elk
no idea what you're saying
plucky elk
so you should be able to find 6 roots now
mild sluice
just use bidmas ✌️
plucky elk
you should be able to find the domain of this.
naive wyvern
Same as what I said b4 no?
plucky elk
right
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muted solar
Is 2i equal to i^2?
safe radishBOT
Is 2i equal to i^2?
muted solar
What do you mean sometimes 😭
Cause I have a question that asks
Imaginary or sqrt-1
sudden grotto
Root of minus 1
unique ocean
Is 2i equal to i^2?
No?
muted solar
Oh okay
So how would I go about finding the absolute value of 4+4i
So 4^2+4^2= absolute value
unique ocean
So 4^2+4^2= absolute value
No
muted solar
So then what is it
sudden grotto
So 4^2+4^2= absolute value
Root it
muted solar
Okay so sqrt of that is the absolute value
unique ocean
Then its root 32 which could also be 4root2
safe radishBOT
sudden grotto
🥀
bold prawn
I look at it and I am immediately confused
I looked at these questions like 20mins ago before coming here
Did the ones I could
vast sequoia
i see
safe radishBOT
bold prawn
<@1082091701511917748> did you figure it out?
I did for that one, Idk how to do my last 2 tho
mighty mango
I did for that one, Idk how to do my last 2 tho
then feel free to keep them
here
until someone is willing to help
sudden grotto
Did you solve this one too? @bold prawn
a, b and x are given, you just need to find y
bold prawn
a, b and x are given, you just need to find y
Yeah, I just had to plug those in, I accidentally did positive instead of negative
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stone ore
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stone ore
How would I complete d?
winged flare
How would I complete d?
$$\frac{dy}{dx} = 2x - y$$
$$\frac{d}{dx}(mx+b) = 2x - y$$
$$m = 2x - y$$
$$m = 2x - (mx + b)$$
substitute y = mx + b
stone ore
then substitute m=2x-y into y=mx+b?
flat frigateBOT
winged flare
then substitute m=2x-y into y=mx+b?
oh no thats not what i meant
stone ore
ah alright
stone ore
dy/dx = m correct?
winged flare
yes
wild cape
u need to find values of m and b so that the equation is true for all x
winged flare
get it all on one side and set equal to 0?
yeah
winged flare
u need to find values of m and b so that the equation is true for all x
the x coefficient would be 0 since it should be true for all x
same for the constant term
stone ore
i got m=2 and my b=-2
winged flare
right
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desert pasture
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desert pasture
I'm confused here, using FTC gives g'(x)=0
do I like pull out the x
oh right
in that case how would I do this
using sin(a-b)?
stuck lotus
nimble wyvern
hmm
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stoic mirage
can someone help me with the Cartesian Plane
safe radishBOT
can someone help me with the Cartesian Plane
stoic mirage
pls
stoic mirage
i dont understand the concept itself 
im stuck at coordinates and origin 0,0
my first time learning it btw
someone help pls
timid ridge
2d number line
stoic mirage
yes ik its a graph with plotting coordinates but idk
anyone
pls
fine.....
.close
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idle anvil
yes ik its a graph with plotting coordinates but idk
like itd help if you explain what u dont understand
do you know how to read a graph? like if I show u this, can u read it?
stoic mirage
.reopen
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✅ Original question: #help-23 message
stoic mirage
my internet connection was nogt good
yes pls i wold love to understand it jordanhost
so iuts like a graph with plotting coordinates
idle anvil
Aight then lets start with that
U see theres some numbers from left to right and theres some numbers from below to up
idle anvil
Those are treated as two different number the one that goes from left to right and from below to above.
stoic mirage
or sort of like basic?
idle anvil
is it advanced algerbraic geometry
no, its pretty basic
idle anvil
now the graph has a lot of points, this green line, one point represents those 2 different numbers. e.g. u can see the point (1,1) because the green line hits both the number 1 from left to right and 1 from below to above
stoic mirage
yes is it origin point?
idle anvil
yes is it origin point?
origin is (0,0) usually, teh point where the line that goes from left to right and the line that goes from below to above intersect
stoic mirage
So we could intersect below and above points
idle anvil
yes is it origin point?
Now lets say I give you (2,x) 2 is the number from the line to left to right and x is the number rfom the line of below to above
idle anvil
Aight then lets start with that
based on this grpah can u figure out now what x is?
line that goes from left to right is called x-axis btw and from below to above is called y axis
usually
idle anvil
based on this grpah can u figure out now what x is?
but here Im asking for the number on the y axis where the green line lands
idle anvil
yeah like the point im asking for is at 2 at the x axis
and now just go up and figure out what the number at the y axis is
idle anvil
yeah
so the point is (2,4)
now u know how to read a graph on cartesian coordinate system
stoic mirage
i see
idle anvil
need anything else?
safe radishBOT
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pine shoal
I've learned that sine is the y axis of the rotating point that goes counter clockwise around a circle (of radius 1 unit)
And cosine is the x axis.
and that these are helpful in the forming of waves (sine waves and cosine waves).
I need some help in clearly understanding:
How these form the waves I mean, do u take the crest and trough as 1 unit or something and as sine reaches 1 u go to crest and - 1 u go to trough or something? not really sure how that works. If there's a significance of taking the radius of the circle of more than one unit as well.
And, what exactly is cot, tan, cosec, sec (I might know them in terms of fractions 1/sine, sine/cos etc.. But what are they)
What significance are they of
safe radishBOT
I've learned that sine is the y axis of the rotating point that goes counter clo...
rich gazelle
I've learned that sine is the y axis of the rotating point that goes counter clo...
Imagine that you have a wheel of radius 1 turning. When the point is at the top you reach the crest. When the point is completely down you reach the valley. Right?
The Cosine is exactly the same, but it measures the horizontal position. For example, if you graph the position x with respect to time, you get the same wave, but a quarter of a turn ahead.
rich gazelle
I've learned that sine is the y axis of the rotating point that goes counter clo...
For the tangent Imagine a vertical wall touching the circle right at the point on the right. If you project the radius from the center until it hits that wall, the height on that wall is Tangent.
The Secant is the length of that line you drew from the center to the wall of the tangent.
The Cosecant and Cotangent are the brothers of the previous ones, but instead of using a vertical wall on the right, they use a horizontal roof above the circle.
Do you understand it better, now?
lean otter
I've learned that sine is the y axis of the rotating point that goes counter clo...
unit circle is helpful in understanding basic trigonometry, the reason radius of the circle being 1 is so that you can apply the sum of squares of distance of coordinate from x axis (sin theta) and from y axis (cos theta) = 1
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patent temple
Hello, the topic is normal distribution in statistics, I have no idea how to do 1 and 3 (Used google translate for the text, hope it's understandable enough)
safe radishBOT
Hello, the topic is normal distribution in statistics, I have no idea how to do ...
patent temple
.close
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knotty blade
Did you figure it out
patent temple
Solved it, had to do P(59,5 ≤ X ≤ 60,5)
Kept doing it with 59 and 61 🙂
Same goes for 3
rich gazelle
Solved it, had to do P(59,5 ≤ X ≤ 60,5)
Good! Have a nice day.
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broken forum
Does there, perchance, exists a way to do this which isnt iterative hell
wet wigeon
!nopings
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Please do not ping individual helpers unprompted.
broken forum
fathom jewel
safe radishBOT
Does there, perchance, exists a way to do this which isnt iterative hell
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.
broken forum
wet wigeon
(he was asked to ask here)
broken forum
primal gazelle
Does there, perchance, exists a way to do this which isnt iterative hell
multiply by denominators first maybe
or multiply everything by (n+3)!
and then set b_n = n! c_n
broken forum
was -8 not -22 mb
primal gazelle
and then set b_n = n! c_n
nvm this isnt very nice
broken forum
and then set b_n = n! c_n
i mean this made it look prettier at least
-8 not 22
omg
pine wind
broken forum
,tex $$y''' - 9y'' + 15x^2y' - 7xy = 0$$
$$y(x) = \sum_{n=0}^{\infty} a_n x^n$$
$$y' = \sum_{n=1}^{\infty} n a_n x^{n-1}, \quad y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}, \quad y''' = \sum_{n=3}^{\infty} n(n-1)(n-2) a_n x^{n-3}$$
$$\sum_{n=3}^{\infty} n(n-1)(n-2) a_n x^{n-3} - 9 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + 15x^2 \sum_{n=1}^{\infty} n a_n x^{n-1} - 7x \sum_{n=0}^{\infty} a_n x^n = 0$$
$$\sum_{n=0}^{\infty} (n+3)(n+2)(n+1) a_{n+3} x^n - 9 \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n + \sum_{n=2}^{\infty} 15(n-1) a_{n-1} x^n - \sum_{n=1}^{\infty} 7 a_{n-1} x^n = 0$$
$$n=0: \quad 6a_3 - 18a_2 = 0 \implies a_3 = 3a_2$$
$$n=1: \quad 24a_4 - 54a_3 - 7a_0 = 0 \implies a_4 = \frac{54a_3 + 7a_0}{24}=\frac{54(3)a_2+7a_0}{24}$$
$$n \ge 2: \quad (n+3)(n+2)(n+1) a_{n+3} - 9(n+2)(n+1) a_{n+2} + (15n - 22) a_{n-1} = 0$$
$$a_n = \frac{9(n-1)(n-2) a_{n-1} - (15n - 67) a_{n-4}}{n(n-1)(n-2)}, \quad n \ge 5$$
$$b_n=a_n n!$$
yeah i dont think this can be done non iteratively
big sad
.close
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torpid glacier
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edgy breach
by constant speed for the first part does it mean initial speed?
vocal dove
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vocal dove
The fourth integral is for me atleast literally impossible
I need someone to help me try to evaluate it
I tried everything
Yes
flat frigateBOT
BeautifulSoup
vocal dove
Yes this basically
I don't know where to start
I tried relationships between H.F and lograthims
U mean using ln?
I tried everything
Tried getting help else where but no result
Tried it,but still stuck
Well its midnight can it wait for tomorrow (since my work is far away)
I hope so
Pretty much
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flat frigateBOT
BeautifulSoup
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normal moss
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swift geyser
Need help with question 1
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Need help with question 1
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swift geyser
This problem seems fun
I figured it, thanks
safe radishBOT
I figured it, thanks
swift geyser
spark sentinel
Do .close
vague phoenix
<:catthumbsup:614540188747563008>
Do .close
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swift geyser
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solemn yoke
help i need to trace this sydney opera house picture for a math assignment and im having trouble getting the curve, i asked claude how to get it and it gacve me these. I dont have to use a curve line it can be any equation because i need 5 different equation kinds and i've already used Line segment equations which gives me the base. if anyone also has a good video or something that would be appreciated
solemn yoke
;-;
plss
im tired
i just need how to do this explained
or a better way
to do it
stoic torrent
well what about marking some points in the picture?
solemn yoke
like where it should start and stop?
stoic torrent
And then use desmos inbuilt curve equation generatorand choose as you prefer
stoic torrent
like where it should start and stop?
no in general some points lying on eqch curve
solemn yoke
And then use desmos inbuilt curve equation generatorand choose as you prefer
how would i do that?
stoic torrent
make a table for each such curve
solemn yoke
okay
stoic torrent
And first inout atleast 4 points coords which lie on that curve
But again mark 4-5 such points on the curve such that connecting them can describe the nature of your curve
so maybe 1 at each end and 3 in the middle at equal distance
But again this may not be the best idea out there!
let me just see if there is anything useful online
solemn yoke
I got here somehow now how would I cut it off?
stoic torrent
I got here somehow now how would I cut it off?
just check your end points
And you can put inequality on y or on x
stoic torrent
Is that good engh
solemn yoke
yeah
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woeful mauve
hi, im studying for the ACT and saw this in a video. can someone explain the binomial theorum to me? first time seeing it.
safe radishBOT
hi, im studying for the ACT and saw this in a video. can someone explain the bin...
woeful mauve
btw, you will need to ping me or i wont see your message!
quick crater
hi, im studying for the ACT and saw this in a video. can someone explain the bin...
i mean, it is the theorem for how you would expand (a+b)^n
do you want the explanation for how to use it? or where it comes from? or what exactly?
quick crater
hmm
woeful mauve
i'll probably have more time over the summer to learn more in depth about it, but i just wanted a uhhh short and kinda blanket basic explanation
quick crater
ig for starters like, its basically a "skip" so you dont have to expand (x+1)^9
<@&268886789983436800>
woeful mauve
ig for starters like, its basically a "skip" so you dont have to expand (x+1)^9
ooooooo
yeah i like it already
quick crater
i'll probably have more time over the summer to learn more in depth about it, bu...
so basically, from (a+b)^n=(a+b)(a+b)...(a+b), if we say, want the coefficient for like a^2, then from all of that, we choose 2 of the (a+b) out of the n to be a, and the rest b, which would be represented as n choose 2. but in general, if you want the coefficient for a^i, then it would be represented as n choose i
woeful mauve
i notice k in there, what does it stand for? and the greek symbol i am not sure about (reading what you just sent sorry one sec)
quick crater
if you do that for every single power of i, you can add them all up, and it would equal the original (a+b)^n
woeful mauve
ohhh another question i had, what is the meaning of stacked numbers in the parenthesis again? i used to know this lolll
woeful mauve
if you do that for every single power of i, you can add them all up, and it woul...
thats so cool
formal grail
i notice k in there, what does it stand for? and the greek symbol i am not sure ...
it means sum from k=0 to k=9
so it would be 9C0 * x^(9-0) * 1^0 + 9C1 x^(9-1) * 1^1 etc
changing k from 0 to 9 as to add 10 terms in total
woeful mauve
o wow
quick crater
i notice k in there, what does it stand for? and the greek symbol i am not sure ...
that is called a summation. so , $\sum_{k=a}^{b}f(k)=f(a)+f(a+1)+\dots+f(b-1)+f(b)$
flat frigateBOT
ihave<skissue>
quick crater
you basically plug k=a, k=a+1, and so on until k=b, then add them all together
formal grail
but here since 1^k is always 1, the question is asking what is attached to x^5 when you do that expansion
woeful mauve
okay
gonna have to play with it more in depth at some point
thats cool
thank you guys for introducing me to the little bits of the binomial theorum
(not closing channel yet in case theres more explanation)
quick crater
$\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k=\binom{n}{0}a^nb^0+\binom{n}{1}a^{n-1}b^1+\dots+\binom{n}{n-1}a^1b^{n-1}+\binom{n}{n}a^0b^n$
flat frigateBOT
ihave<skissue>
quick crater
essentially, the expansion looks something like this
formal grail
but do you know what the ( 9 k) looking thing is? (im on mobile not gonna bother w latex sorry)
formal grail
it's what you need to find the answer
woeful mauve
essentially, the expansion looks something like this
oooo
looks complicated to just work it out like that (like the mechanics of the math is cool and complicated)
woah
love how math is like a machine
formal grail
it has to do with probability but if you know what factorial is it's calculated as n! / (k! * (n-k)!)
and m! when m is a natural number is the product of all integers from 1 to m (so 4! = 1*2*3*4)
I imagine you can calculate that with a calculator though
in some calculators it looks like nCk with a big C
woeful mauve
i vaguely know what a factorial is
formal grail
the point is they're asking what number is multiplying x^5, so to solve that kind of question you would need to find the correct k that makes the term be x^5 then calculate the number
( 9
k )
woeful mauve
ohhhhhhh
i may need to look at this another time, because its a little much for my brain atm LOL. didnt realize how tired i was
thank you guys!!
.close
safe radishBOT
safe radishBOT
✅ Original question: #help-23 message
quick crater
to add, n choose k is just how many ways you can choose k from n
woeful mauve
ohhh okay
quick crater
it has to do with probability but if you know what factorial is it's calculated ...
which is calculated by this
woeful mauve
gonna do more research on this another time, but ill for sure reference what you guys said when i do
thank you!
.close
safe radishBOT
blazing swallow
i may need to look at this another time, because its a little much for my brain ...
. personally, i think an easier introduction to n choose k is pascal's triange
. the k-th entry of the n-th row is nCk (so the second entry of the third row is 3C2)
formal grail
. personally, i think an easier introduction to n choose k is [pascal's triange]...
I mean yes that's an easy way to know it without calculating and I did almost say it but it doesn't really give any understanding as to what or why it is
but it is cool too
woeful mauve
if i had more time and wasnt as tired id for sure dive in with you guys deep into this lol
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fathom jewel
Finding the normal vector of S1 is equivalent to finding the gradient of the level surface F(x,y,z)=2x²+y²-z=0 which is grad(F)=(4,2,-1)
echo gazelle
is this related to dot product
fathom jewel
no, F is not a vector field
echo gazelle
isnt the gradient a vector tho
fathom jewel
Yes but F is a scalar
In a sense though, the dot product helps to argue why the gradient is perpendicular to the surface. Take a random curve r(t), then F(r(t))=0 implies grad(F)•r'(t)=0 implies that F is normal to the tangent vector r'
molten acorn
is this related to dot product
ds/dx and ds/dy are tangent to the function, so you can find the normal line by finding a vector perpendicular to them
and that uses dot product
fathom jewel
rather the cross product
molten acorn
rather the cross product
yep mb
echo gazelle
Finding the normal vector of S1 is equivalent to finding the gradient of the lev...
can u explain dis part pls so then the grad is a formula that tells u the normal vector for any point on the plane?
molten acorn
can u explain dis part pls so then the grad is a formula that tells u the normal...
the gradient is normal to the level surface of a function
r(t) is a kind of path along the level surface
F(r(t)) = 0 taking the derivative on both sides you get
F_x. x' + F_y. y' + F_z. z' = 0
grad F • r'(t) = 0
(in which r(t) can be decomposed into x(t), y(t), and z(t))
also, since r(t) is a path on the level surface, r'(t) is tangent to the level surface
echo gazelle
o
ok i think i see
thx so much
u can represent grad f . r(t) as like grad f . dr right?
when we set F(r(t)) = 0 what does it represent?
or can it be anything because the derivative will still be 0
.close
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open wedge
<@&268886789983436800>
eternal breach
channel was closed but I would like help solving this
safe radishBOT
channel was closed but I would like help solving this
eternal breach
I used the approximation π(x) = x/ln(x)
peak estuary
if you already used that approximation then what exactly are you hoping for
you can of course continue approximating the function and get ln(x)/x^2
which is easy to integrate
eternal breach
you can of course continue approximating the function and get ln(x)/x^2
I'm not getting ln(x)/x² though I'm getting 1/ln(x)(x²-1) 😭😭
eternal breach
can't find any way to integrate it
I keep getting integrals with hyperbolic functions
eternal breach
I keep getting integrals with hyperbolic functions
to be more precise it's int from ln2 to +inf of 1/tsinh(t)
I tried using Feynman but I still get int of 1/e^(t(1+s)) - e^(t(1-s))
eternal breach
I tried using Feynman but I still get int of 1/e^(t(1+s)) - e^(t(1-s))
maybe use half sums to get this back into a sinh ?
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@eternal breach Has your question been resolved?
tropic summit
channel was closed but I would like help solving this
Have you ever done an integral with a floor function in the integrand
near sky
Have you ever done an integral with a floor function in the integrand
hello
do you have a question
tropic summit
There is a common trick used to solve those that can be employed here
near sky
Sorry this is refering to the above question
bro is green could have used .reopen
tropic summit
Yeah I forgor..
junior smelt
bro is green could have used .reopen
(not if you're too late to the channel and it becomes available again, you can't 
)
eternal breach
Have you ever done an integral with a floor function in the integrand
yes
thanks for reopening it too !
obsidian oracle
thanks for reopening it too !
Ok, so more specifically, you can write pi(x) as a sum of indicator/characteristic functions
The characteristic function of some set $A$, is given by
$$\mathbf 1_A(x) = \begin{cases}1&\text{if } x\in A,\ 0&\text{if }x\notin A\end{cases}.$$
flat frigateBOT
Rafilouyear2026
obsidian oracle
So remember now that the prime function counts how many primes are below x, so it counts how many primes x is above
For each prime p, if p <= x (so if x is in ...), we add 1 to the count. Otherwise, we add 0 to the count
So $\pi(x) = \sum_{p\text{ prime}}\mathbf 1_{[p,\infty)}(x)$
flat frigateBOT
Rafilouyear2026
obsidian oracle
One last thing to know about characteristic functions is that they behave well with integrals
multiplying the integrand by a characteristic function is the same as restricting the domain of integration
$\int_B \mathbf 1_A(x) f(x)dx = \int_{A\cap B} f(x)dx$
flat frigateBOT
Rafilouyear2026
obsidian oracle
If you have any questions about all of those, feel free to ask for some more clarification
but you should have enough tools to solve your integral now
safe radishBOT
@tropic summit Has your question been resolved?
obsidian oracle
@eternal breach if your question is still unanswered I suggest you open a channel of your own so Acman isn't forced to keep this one up for you
We'll link what was said here to the new one
eternal breach
<@611977038126907392> if your question is still unanswered I suggest you open a ...
yeah sure wait
done
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tropic summit
.close
sullen cargo
hi
sullen cargo
Wrong channel mb
rich gazelle
Wrong channel mb
Np, you can see the information in #info.
rich gazelle
Wrong channel mb
!done
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Wrong channel mb
If you are done with this channel, please mark your problem as solved by typing .close
normal moss
.close
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austere goblet
<@&268886789983436800>
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austere goblet
<@&268886789983436800> round 2
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verbal cloud
Let $x_1, \dots x_n \in \mathbb{R}^m$, and $A$ be the set of rank 1 matrices. || || is the euclidean norm. Show that $$\Psi: M \mapsto \sum_{i=1}^n ||x_i - M x_i|| ^2$$ has a minimum on $A$ and express it in terms of $\sum x_i x_i^T$
safe radishBOT
Let $x_1, \dots x_n \in \mathbb{R}^m$, and $A$ be the set of rank 1 matrices. ||...
flat frigateBOT
bloubbloub
verbal cloud
the hint is that for any unitary vector $x$ and any vector $y$, $\Psi(xy^T) \geq \Psi(xx^T)$ which I was able to show
flat frigateBOT
bloubbloub
verbal cloud
I tried taking the differential but did not find success
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@verbal cloud Has your question been resolved?
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@verbal cloud Has your question been resolved?
@verbal cloud Has your question been resolved?
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@verbal cloud Has your question been resolved?
left gyro
Let $x_1, \dots x_n \in \mathbb{R}^m$, and $A$ be the set of rank 1 matrices. ||...
unrelated: you should do \| instead of || to properly render norms
also \|\cdot\| for the usual || . || notation
left gyro
also Ive got a dumb idea
left gyro
if somehow you can reduce a member of A to be xy^T, where x and y are vectors,
(which should be the case since a rank 1 matrix has only one dimension in its column space which really suggests the columns are just scalar copies of each other)
you can redefine (x, y) := (x/||x||, y||x||) so that x is unitary then the above shows to minimize with y -> x
so that reduces this problem to minimizing across all matrices M = xx^T where x^T x = 1
then ||xi - M xi||^2 = (xi - M xi, xi - M xi) = (xi, xi) - 2(M xi, xi) + (M xi, M xi)
= (xi, xi) - 2(xx^T xi, xi) + (xx^T xi, xx^T xi)
= (xi, xi) - 2 xi^T xx^T xi + xi^T xx^T xx^T xi
= (xi, xi) - 2 xi^T xx^T xi + xi^T xx^T xi
= (xi, xi) - xi^T x x^T xi
= (xi, xi) - (xi, M xi)
so the sum is (x1, x1) + ... + (xn, xn) - (x1, M x1) - ... - (xn, M xn)
idk what youre going to do with that sum though
raven heart
yeah that checks out, and (xi, M xi) = (xi, x)^2
the (xi, xi) don't depend on M so we don't give a shit about them
left gyro
that seemed more useless so I left it out
raven heart
so you're left to maximize sum (xi, x)^2
left gyro
I still dont see what you can do with that, like how are you going to prove that x = xi would maximize it it doesnt nvm
nvm made the same mistake again
raven heart
well q(x) = sum (xi, x)^2 is a quadratic form
if we call A = [x1 ; x2 ; ... ; xn] (each row is an xi),
then q(x) = (Ax)^T(Ax) = xA^TAx
and optimizing a quadratic form on the unit sphere is a pretty standard thing, just check the max/min eigenvalue of the matrix in the middle (A^TA here)
left gyro
thats some black magic with that x^T voodoo x sphere stuff but yea
raven heart
also I think A^TA is that sum xi xi^T the question mentions so it checks out
well hopefully I voodoo'd correctly
gonna go sleep
left gyro
gn voodooman
I believe the first half , defining A like that you do get Ax = (a column vector with elements (xi, x))
then ||Ax||^2 or x^T A^T Ax would be the sum (xi, x)^2
wait
nvm I dont know how this works
safe radishBOT
@verbal cloud Has your question been resolved?
verbal cloud
well q(x) = sum (xi, x)^2 is a quadratic form
if we call A = [x1 ; x2 ; ... ; ...
I got to that point but didn't see the matrix thing, nice one
safe radishBOT
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glacial plume
Someone help
safe radishBOT
Someone help
rich gazelle
Someone help
What have you tried, can you show better the original question?
glacial plume
What have you tried, can you show better the original question?
It's just finding the angle measures of this quadrilateral and I am lwky lost on what to start with
Ignore the numbers
That a diff thing
glacial plume
Someone help
Question 11
rich gazelle
It's just finding the angle measures of this quadrilateral and I am lwky lost on...
Opposite angles add up to 180°, right?
glacial plume
Ye
glacial plume
I recommend that you add the expressions of angles D and F and set them to 180.
They don't have expression
S
Ignore the writing
It just the angles d c e f with no writing or anything
glacial plume
I recommend that you add the expressions of angles D and F and set them to 180.
Here is a better picture
rich gazelle
Here is a better picture
You're right, in this photo you can see that the original drawing is clean. What happened is that in the first photo you sent there were some handwritten expressions.
rich gazelle
Here is a better picture
They give you any value?
verbal wigeon
Um..so we don't even know wot sort of quadrilateral is that..?
Rectangle..or.. trapezium
glacial plume
Here is a better picture
Well I looked at the answer key the answer is 90° for all the angles why is that tho
warm warren
Someone help
for the first one: hint: DF is a diameter.
what angle does a diameter subtend?
similarly work out the second one too.
glacial plume
what angle does a diameter subtend?
Right angle?
warm warren
which diameter is it, for (II)?
rich gazelle
similarly work out the second one too.
Yeah, they need to use that.
rich gazelle
Right angle?
Right!
verbal wigeon
Right angle?
Exactly
warm warren
similarly do (III) and (IV)
warm warren
!done
safe radishBOT
If you are done with this channel, please mark your problem as solved by typing .close
glacial plume
.close
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glacial plume
Have a nice day!
You too
safe radishBOT
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white pebble
why do we have to put an - after the 1
safe radishBOT
why do we have to put an - after the 1
brave wolf
why do we have to put an - after the 1
1^- means that you approach it from the left side (from the negative x axis, the blue arrow)
1^+ means that you approach it from the right side (from the positive x axis, the green arrow)
acoustic frigate
why do we have to put an - after the 1
Because we are approaching towards 1 from left side
white pebble
oh
brave wolf
why do we have to put an - after the 1
if you look at it from the left side, the curve goes all the way down to -infinity (look at the blue arrow, pointing from the left side towards right)
white pebble
so that means if x goes towards 1 from the left side f(x) goes towards - infinity?
fathom jewel
so that means if x goes towards 1 from the left side f(x) goes towards - infini...
ja
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white pebble
<@&268886789983436800>
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blazing harness
hello im new here and i rlly need help w this
safe radishBOT
hello im new here and i rlly need help w this
plucky elk
should translate if you want help unless you want to wait for a french speaker
severe slate
hello im new here and i rlly need help w this
Umm so i translated some part
So you got any idea what to do @blazing harness
blazing harness
no not at all
severe slate
Have u find g(x) and f(x)?
blazing harness
yes i think g(x)=2x but not sure
severe slate
g(x) is area in red right ?
blazing harness
yes
blazing harness
f(x)= (5-x)^2 / 2
blazing harness
idk how to write it prbly
frozen marlin
i'm not entirely sure tho, bit rusty
severe slate
f(x)= (5-x)^2 / 2
Are i sure that correct bec i having trouble reading data i am just guiding you as per traslater
frozen marlin
In the diagram given below, we trace the lines D1 and D2. M is movable along the x axis (you can vary the x-coordinate of M), presumably in the interval [0, 5].
- Express the area of the red rectangle, denoted g(x), as a function of x.
- [same thing but for blue triangle, f(x)]
- Using a calculator, conjecture the values of x for which the areas are equal.
- Show that the equation f(x) = g(x) is equivalent to the equation (x-7)^2 - 24 = 0.
- Deduce a solution to the conjecture from Q4 graphically.
Note for future helpers, to avoid confusion: all areas are positive
i'm fairly certain this is the translation, but i'm not a native speaker, and i'm fairly rusty, so yeah
blazing harness
In the diagram given below, we trace the lines D1 and D2. M is movable along the...
yeah this is correct
frozen marlin
alr
frozen marlin
In the diagram given below, we trace the lines D1 and D2. M is movable along the...
there
severe slate
@blazing harness you there buddy ?
blazing harness
yeah
severe slate
So f(x) you found is correct ?
blazing harness
idk im asking you guys...
frozen marlin
!status
safe radishBOT
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
!show
safe radishBOT
Show your work, and if possible, explain where you are stuck.
frozen marlin
there
severe slate
idk im asking you guys...
Alr let me calculate
blazing harness
f(x)= (5-x)^2 / 2
this is f(x)
frozen marlin
anyway, im off
blazing harness
anyway, im off
noo
blazing harness
```
What step are you on?
1. I don't know where to begin.
2. I have begun but go...
2 and 4
blazing harness
ok now the 3 and the ones after it are the ones i have most trouble in
blazing harness
It is isosceles triangle right ?
correct
severe slate
So 3 one is asking value of x for which both area equal you know what that means right ?
F(x)=g(x) bec both are area
blazing harness
yeah but i dont know how
blazing harness
yeah but i dont have the scientific calculator that lets me do that
severe slate
Write f(x) equations and equal to g(x) equation
blazing harness
and i dont know for which value of x they are both equal
severe slate
Oh can u tell what u got after simplyfing this equation?
You have to solve 2x=(5-x)^2/2
blazing harness
yeah how do you do that
severe slate
Take denominator 2 left side then Expand right side then merge both ?
severe slate
you know what ill do this in the morning
?
blazing harness
i rlly cant focus
severe slate
i rlly cant focus
Oh k it's on u do whenever u want
blazing harness
thanks tho you all were rlly helpfull
severe slate
Wc
blazing harness
.close
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blissful burrow
hi; I am (still) struggling with proving the Ramsey's theorem true.
"0.14 Ramsey’s theorem. Let G be a graph. A clique in G is a subgraph in which every two nodes are connected by an edge. An anti-clique, also called an independent set, is a subgraph in which every two nodes are not connected by an edge. Show that every graph with n nodes contains either a clique or an anti-clique with at least 0.5*log2(n) nodes."
My line of reasoning is as follows:
(1) The maximum amount of cliques (or anti-cliques) is 2^n when all (or no) vertices are connected, where n describes the amount of vertices
(2) From (1) it follows that the amount of vertices is log2(x), where x denotes the amount of cliques (or anti-cliques). This holds only when all vertices are adjacent to each other (or the graph is edgeless). This is where the theorem's log2(n) comes from.
(3) Adding (or removing) edges to a graph reduces (or increases) the amount of anti-cliques (or cliques) by 2^e where e denotes the amount of edges. I am unsure of this statement.
(4) Due to (3) the amount of cliques is 2^(n-e), and the amount of anti-cliques is 2^(e-n). Maximum value of e is n(n-1).
(5) This sum is minimised when e=n. I'm not exactly sure if I'm right to be honest.
This can't be right though...
Dunno, I think the 0.5 term in the theorem must have something to do with getting a square root of n – the amount of vertices in graph G. But I don't know how to get to that point?
-# also apologies if I'm overusing the help channels ;-; I'm trying to self-learn & I can't quite just ask someone
safe radishBOT
hi; I am (still) struggling with proving the Ramsey's theorem true.
"0.14 Ramse...
@blissful burrow Has your question been resolved?
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@blissful burrow Has your question been resolved?
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mighty mango
yes.
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obsidian palm
hello hello hello
safe radishBOT
hello hello hello
obsidian palm
this is trigonometric integrals. im trying to integrate the following:
and i keep getting a wrong answer. i'm using u-sub and the pythagorean identity to do it. i remembered to change the bounds for u-sub. i don't think i messed up my arithmetic any
but my answer keeps coming out as .157 and i dunno why
lethal shoal
ur expansion of $$(1-u^2)(u^3)^{1/2}$$ is wrong
obsidian palm
let me try again
wait. don't i add the 2 and 3/2 when i expand the second term?
2 + 3/2 is 7/2. i am missing something
lethal shoal
wait wait holdup
oh wait ur subbing back u as sin(x) at the end
if u alr changed the bounds no need to do that
obsidian palm
ohhhhhhhhhhhhhh
let me try one more time
wait. for clarity. if i did not change the bounds, then i would have to sub u back in right?
obsidian palm
okay okay. thanks
lethal shoal
unless it’s an indefinite integral
safe radishBOT
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hoary seal
Im getting last box wrong
safe radishBOT
Im getting last box wrong
clever tinsel
where did this come from?
vague phoenix
what is a
hoary seal
vague phoenix
dude a is not 3 4
vague phoenix
the question gave a (3,4) as an example
clever tinsel
alright I see that I'm just repeating what the other helper is saying, so I'll step out. all the best.
hoary seal
ohh
safe radishBOT
@hoary seal Has your question been resolved?
lilac relic
I need help with this question. I don't know how to work out the bottom row of that table.
safe radishBOT
I need help with this question. I don't know how to work out the bottom row of t...
lilac relic
pwease
near sleet
covert yoke
That being said, if you've waited for 15 minutes, you can try pinging helpers, but before you do that it would be useful to know what you've tried and what thoughts you have about how to approach this problem.
lilac relic
That being said, if you've waited for 15 minutes, you can try pinging helpers, b...
I have thought about it
two unknown values won't work in the same scenario that would work for one value
covert yoke
I'm not saying you haven't, just asking you to share those thoughts with us so we know where you are at and what exactly you are stuck on
lilac relic
I'm not saying you haven't, just asking you to share those thoughts with us so w...
I'm at the beginning
OH I SEE
well
lilac relic
I need help with this question. I don't know how to work out the bottom row of t...
I don't know how to work out the bottom row of that table
covert yoke
So here's something that might help, there are 52 cards in the deck, how many cards will get you the $5 reward? What about the $10 one? Etc.
This is what it's asking of you with the table
lilac relic
$10 is for any club that isn't the ace
12/52
covert yoke
Can you simplify this?
lilac relic
and
covert yoke
12/52 = 3/13
lilac relic
1/52 = 0.0192
lilac relic
12/52 = 3/13
I don't know why I didn't realize that before
I'm so slow at maths
Its because I didn't actually think about the question
covert yoke
Math is a skill you develop.
It's ok to not get it right away
Accurate to 6 DP is pretty gnarly, tbh
Why not just have you express as ratios?
Anyway, best of luck.
lilac relic
Anyway, best of luck.
I'll do the next half of the question now
that's that
I spend an hour on each question
when its unbelievably simple
So silly
covert yoke
If you want to know despair, this week I spent 2 days on a relatively simple question.
Math is tough no matter what level you're at, just it's also doable if you keep at it.
lilac relic
Math is tough no matter what level you're at, just it's also doable if you keep ...
Very wise
anywya
.close
safe radishBOT
lilac relic
Math is tough no matter what level you're at, just it's also doable if you keep ...
also thanks for the help
safe radishBOT
Available help channel!
sonic hound
why is the shaded area = 2(area of quarter circle) - area of square? isn't that the overlapping area in the middle?
safe radishBOT
why is the shaded area = 2(area of quarter circle) - area of square? isn't that ...
@sonic hound Has your question been resolved?
cunning pasture
The expression = 2(A+B) - s^2 = 2A + 2B - s^2
= (s^2 - the shaded region) + 2B - s^2
= 2B - the shaded region = 2B - (B + C) = B - C
But the shaded region is B + C
So it's incorrect.
stoic torrent
!nosols
safe radishBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
stoic torrent
Just guide him and make him solve certain parts on his own!
You may give him a step or two for progression!!
sonic hound
The expression = 2(A+B) - s^2 = 2A + 2B - s^2
= (s^2 - the shaded region) + 2B -...
is it possible to find the area of the shaded region if you know s?
cunning pasture
Of course.
cunning pasture
First find B in terms of s. Other quantities can be found thereafter in terms of s.
It is a famous problem to find B.
In the given image, the blue shaded region is just twice of B.
sonic hound
First find B in terms of s. Other quantities can be found thereafter in terms of...
how do I find this with r alone?
oh wait
sonic hound
First find B in terms of s. Other quantities can be found thereafter in terms of...
would it be 2(1/4 pi s^2 - (s^2)/2)
so the shaded region would be
s^2 - 2A
s^2 - 2(qrtr crcl - B)
B = 1/4 pi s^2 - (s^2)/2
s^2 -2(1/4 pi s^2 - (1/4 pi s^2 - s^2/2))
it cancels out?
safe radishBOT
@sonic hound Has your question been resolved?
sonic hound
@cunning pasture is there like some hidden angle or smt
cunning pasture
Whatchu doing
safe radishBOT
@sonic hound Has your question been resolved?
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@sonic hound Has your question been resolved?
sonic hound
Whatchu doing
hey bro can you elaborate further
near sky
Hm.. here you can draw a straight line passing through the intersection, it will become a rectangle, if you subtract area of quarter leave from the rectangle, you will get a shape.
Now subtract area of half of the leave from the quarter circle
You can now figure
sonic hound
Hm.. here you can draw a straight line passing through the intersection, it wil...
wdym by rectangle and leave?
sonic hound
How do I get area of leaf?
near sky
How do I get area of leaf?
Think
sonic hound
so the shaded region would be
s^2 - 2A
s^2 - 2(qrtr crcl - B)
B = 1/4 pi s^2 - (...
this was my attempt
near sky
Ig there are multiple ways, one way is take one half then connect radii of the circle so you have a sector then Subtract the area of triangle from sector you will find the area of leaf segment
warm sphinx
I hate maths
near sky
near sky
I hate maths
Good
sonic hound
Ig there are multiple ways, one way is take one half then connect radii of the c...
but we don't even know the area of the sector?
near sky
Idk why everyone love ts
non one likes
warm sphinx
stoic torrent
I hate maths
Says a pending postgraduatee
sonic hound
plz help me out lwkey
near sky
Hm.. here you can draw a straight line passing through the intersection, it wil...
.
if you find this gray area
sonic hound
if you find this gray area
I got it
near sky
subtract from green
sonic hound
subtract from green
I got a negative
my work :
lets say s = 14
I cut them in half just like you said and get rectangles
Then subtract the quarter leaf from the rectangle?
(14x7 - (1/4 pi 49)) approximately 59.5
so we got the grey area like you asked
Then I find the green area by subtracting the quarter circle from the square
14^2 - (1/4 pi 196) approx 196 - 154 = 42
but this is smaller than earlier
near sky
I got a negative
my work :
lets say s = 14
I cut them in half just like you said...
the quarter leaf is not a quarter circle hence you cannot do 1/4pi^2
sonic hound
yh I think I misunderstood the area of sector you meant
near sky
yh, you can get the area of quarter leaf if you subtract the area of triangle
from sector
sonic hound
so how do I know the area of the sector then
sonic hound
you first need to find the angle
shouldn't it be 90 degree? because we split it in half
near sky
shouldn't it be 90 degree? because we split it in half
it is not if it were theit wouldbe a quarter circle
and the perpendicular from D to AC would be the radius and foot will lie on the center of circle
use trigonemtry to find the angle
cos theta = adj/hyp = 7/14
so theta = ?
flat frigateBOT
Result:
1.0471975511966
there must be something wrong then what is the answer key
sonic hound
there must be something wrong then what is the answer key
112
sonic hound
thats way off from what we would get do they have a solution?
yeah but the solution is so weird
like it referenced set theory like this
A(P∪Q)=A(P)+A(Q)−A(P∩Q)
P is the quartr circle with its center on A
Q is the quartr cirlce with its center on C
A(PUQ) is the sum of area of both quartr cirlce that is equal to the area of the square
A(P∩Q) is the shaded region
then it basically told me that area of square = area sum of two quartr circle - the shaded region
vocal pollen
they must be trolling
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gentle edge
small doubt
what is an R to R funtion
safe radishBOT
small doubt
what is an R to R funtion
fathom jewel
a function thats takes real numbers and outputs real numbers
gentle edge
all real or..?
fathom jewel
yes
gentle edge
both input and output?
fathom jewel
yes
safe radishBOT
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gentle edge
where did i mess up?
safe radishBOT
where did i mess up?
gentle edge
assuming xsqr as t?
gentle edge
where did i mess up?
oh full question aint visible
find range of x^2/x^4+1
severe slate
What u done in 4th step ?
gentle edge
Can u check again
wait a sec
severe slate
wait a sec
U messed up in 4th step it look to me ig
gentle edge
U messed up in 4th step it look to me ig
i just checked
it seems fine to me
quadratic in t
severe slate
It's b^2-4ac right ?
severe slate
Oh mb I look to y
fathom jewel
The issue is y>=0
fathom jewel
You assumed it can be negative but you have squares
fathom jewel
x²/(x⁴+1)>=0
gentle edge
yep mb
but how do i catch this mistake
if i do make in exam
or do i just have to be extra carfull when taking x^2 as t
fathom jewel
So the interesection of y>=0 and -1/2=<y<=1/2
gentle edge
So the interesection of y>=0 and -1/2=<y<=1/2
yeah
fathom jewel
or do i just have to be extra carfull when taking x^2 as t
Thats valid
gentle edge
however i worrie
i might miss an alltime +ve when finding range
and get a question wrong
fathom jewel
Just keep in mind t cant be negative so you need to exclude real solutions where t<0
gentle edge
Just keep in mind t cant be negative so you need to exclude real solutions where...
ight
severe slate
i might miss an alltime +ve when finding range
I also missed this thing I just missed 😭 need practice ig
gentle edge
I also missed this thing I just missed 😭 need practice ig
i had done question like this earlier😭
but forgot most of it so was doing revision
anyways ill leave you free
thanks a lot @severe slate @fathom jewel
.close
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gentle edge
what do i do here?
safe radishBOT
what do i do here?
gentle edge
(im not taking multiple channels its a new question)
i think we should see the min value of the sum of first 2 terms
which i think would tend to 0 as x tends to 0
then range would be (1,inf)
brave wolf
which i think would tend to 0 as x tends to 0
as x tends to 0?
brave wolf
its not as x tends to 0, its as x tends to -infty
gentle edge
but yeah, 2^x and 4^x tend to 0
ight thanks a lot x2
gentle edge
so your range is correct
i mean if it works it works
(i really shouldnt do that)
ill be back soon enough
cya
.close
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normal moss
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steep relic
hello people
safe radishBOT
hello people
steep relic
if a function has an oblique asymptote of slant m at infinity , is it correct to say
$\lim_{x \to \infty} f'(x) = m$
flat frigateBOT
voltive
steep relic
?
stoic dune
Yeah that sounds right to me
median vigil
i'm not sure that has to be true in general
it has to get infinitely close to the line mx + b, but that doesn't mean it approaches tangency
steep relic
doesn't it approach the slant, thus being parallel with the line?
median vigil
consider f(x) = sin(x^2)/x + x
it has an oblique asymptote of y = x, but its derivative doesn't have a limit
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small rampart
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small rampart
I came across this when i was looking up the shoelace formula
Source: Wikipedia
I thought determinants were only for square matrixes?
Is the notation on the last line correct and commonly understood?
astral glacier
The thing on the second line is simply for notational convenience
It isn't the determinant of that matrix
small rampart
The thing on the second line is simply for notational convenience
If i were to use that notation, would most people understand it?
Idk if that's standard, i've never seen it
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astral glacier
If i were to use that notation, would most people understand it?
. I doubt it personally. I wouldn't at least unless the notation was explained first
I'd suggest avoiding it personally. Or clarifying it when you use it
small rampart
Let $ABC$ be an acute scalene triangle with altitudes $AD, BE, CF$. From $A$, drop perpendiculars to the lines $EF, FD, DE$, and denote the feet by $X, Y, Z$, respectively. Let the line $BZ$ intersect the circumcircle $(BDY)$ again at $P$, and let the line $CY$ intersect the circumcircle $(CDZ)$ again at $Q$. Prove that the point $X$ has equal power with respect to the two circles $(YFP)$ and $(ZEQ)$.
safe radishBOT
Let $ABC$ be an acute scalene triangle with altitudes $AD, BE, CF$. From $A$, dr...
flat frigateBOT
Nerdyasianguy
small rampart
My approach so far:
Let XY ∩ (FEP) = M, XZ ∩ (ZEQ) = N
I need to prove that MNYZ is concyclic so that X becomes the radical center of (FEP), (ZEQ), (MNYZ) thus i get what i need
Here's the link to the diagram
small rampart
I need to prove that MNYZ is concyclic so that X becomes the radical center of (...
I'm struggling to use the angles of MNYZ though
Trig Ceva doesn't seem usable here
safe radishBOT
@small rampart Has your question been resolved?
safe radishBOT
@small rampart Has your question been resolved?
rich gazelle
sorry so much
Np.
stoic torrent
Quite a mess eh?
tiny forge
looks so clean wow
small rampart
Quite a mess eh?
I sent the GeoGebra version
The notation of M and N in the paper pic is different, it's anothe approach I tried that didn't work
small rampart
looks so clean wow
Thanks
stoic torrent
I sent the GeoGebra version
Yeah not that, your diagrams actually really really neat btw, just that there are too many intersecting points!
small rampart
Yeah not that, your diagrams actually really really neat btw, just that there ar...
I only drew points and lines needed in the statement of the problem 
I tried different configurations to make it easier to see but this was the best
The points are inevitably close to each other
stoic torrent
The points are inevitably close to each other
Yeah not your fault😭
Sorry btw I am not really good in these type of problmes, so cant help you out
small rampart
I need to prove that MNYZ is concyclic so that X becomes the radical center of (...
This seems very promising btw, my 2 other approaches were terrible
safe radishBOT
@small rampart Has your question been resolved?
safe radishBOT