like g(x)?
#help-36
sonic crystal
sonic crystal
im using it on F(x,y)?
bold turtle
I was literally getting ready to leave for work tho lol 
drowsy epoch
Me too 
little python
it will
little python
vvv
soft zealotBOT
drowsy epoch
You should expand this and compare
to this
ok weird notation actually
but i mean like compare theirs to your expansion and the left overs should vanish by lagrange
soft zealotBOT
sonic crystal
but i dont know how they converted everything to matrix form
drowsy epoch
the idea is instead of trying to get their form, you expand their thing and show if you subtract this from your result it's 0
edgy stump
I need help please
final saddleBOT
I need help please
sturdy wasp
with what?
worldly spruce
Don't we all
final saddleBOT
I need help please
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
edgy stump
with what?
Polynomial division
I don't know long division BTW but I am willing to learn it by force a
tropic crest
Polynomial division
what is the problem?
sturdy wasp
have u got a specific question?
worldly spruce
I don't know long division BTW but I am willing to learn it by force a
Well you should learn long division first
Polynomial division is long division generalized to algebraic expressions
tropic crest
Polynomial division is long division generalized to algebraic expressions
you divide variables instead of numbers
edgy stump
edgy stump
Polynomial division is long division generalized to algebraic expressions
Can you teach me tho
leaden moon
Can you teach me tho
Khan academy basically covers all of this
tropic crest
is this even divisible lol
edgy stump
Khan academy basically covers all of this
Seriously
edgy stump
Yea i took that lesson b4
OK I'll go watch it
worldly spruce
Can you teach me tho
This environment is suitable more for direct questions, not teaching whole concepts, as that takes a large amount of time. Maybe somebody has the time right now. I don't, sorry.
tropic crest
edgy stump
https://en.wikipedia.org/wiki/Polynomial_long_division
Thanks
tropic crest
np
final saddleBOT
@edgy stump Has your question been resolved?
onyx peak
and here too <@&268886789983436800>
final saddleBOT
old quarry
spare summit
final saddleBOT
Available help channel!
left trail
Can someone help me find the coefficients for 5. I have that the integral is equal to
final saddleBOT
Can someone help me find the coefficients for 5. I have that the integral is equ...
final saddleBOT
strange pelican
have a good day
final saddleBOT
Available help channel!
jagged flare
final saddleBOT
jagged flare
yeah confused again on the meaning of the problem
like, are there any orders for the bulldozers? does A and B have tk be adjacent to sweep? and for the orders, is it like for some ordering of the movements of the bulldozer, there would always be one left over? or whatever ordering it is, a specific town will be left over?
my reading comprehension is maybe just ass idk
final saddleBOT
@jagged flare Has your question been resolved?
signal vector
like, are there any orders for the bulldozers? does A and B have tk be adjacent ...
Regarding your first question:
In general you shouldn't assume something if it isn't explicitly said. So the towns don't need to be adjacent for sweeping (you can also infer this from what they have asked you to prove).
And regarding your other question, it's definitely the case that you have to show it for any ordering, because there obviously exists some ordering in which exactly one town cannot be sweeped away by the rest. (Put the bulldozers in descending order from left to right..)
jagged flare
Regarding your first question:
In general you shouldn't assume something if it i...
mm
for diffrent orderings (like left to right and right to left), the unsweepable town can be diffrent right?
signal vector
Yeah
The way they have written the question is:
You are given some ordering of towns, there is exactly one which is unsweepable
This is the same as:
Prove that for any ordering of towns, there exists exactly one town which is unsweepable
jagged flare
ok ty
final saddleBOT
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orchid scroll
is p and r labled correctly???? because i dont get how this makes sense
orchid scroll
because the bearign of p (plane) should be 78
however the bearign of the resultant is 78 instead
worldly spruce
<@&268886789983436800>
final saddleBOT
@orchid scroll Has your question been resolved?
orchid scroll
mb ir ead the qeution wrogn
sonic cairn
is 0 a natural number?
final saddleBOT
is 0 a natural number?
deep condor
depends
some ppl say yes, some say no, it's up to definition
but it's a natural number i dont care
sonic cairn
bruh i thought math can always be definied
sturdy flax
but haseeb never said math can't be defined. it's just that some people work with different definitions.
deep condor
bruh i thought math can always be definied
i guess the question is, what's convenient/makes sense?
for most things, there is only one definition that makes sense, but sometimes its more convenient to include zero and sometimes to exclude it
sonic cairn
.close
final saddleBOT
sonic cairn
logicians and set theorists tend to treat it as natural
0 isnt positive or negative though
ripe jewel
.reopen
final saddleBOT
✅ Original question: #help-36 message
ripe jewel
0 isnt positive or negative though
or it's both positive and negative, depending on your definition
but that's not really relevant to your question
sonic cairn
adding a zero to a number wont move move the number to the right or left on a scale
ripe jewel
adding a zero to a number wont move move the number to the right or left on a sc...
sure
so what
sonic cairn
what would a catholic say
ripe jewel
they'd defer to the Pope
sonic cairn
is zero a natural or a whole number
ripe jewel
"whole number" isn't really a term used much in highery level math because it's unclear what it shouls mean
sonic cairn
its positive and negative numbers actually
without pi, fractions or square roots
or powers
PURE number
sturdy flax
you're probably looking for the term integer.
ripe jewel
its positive and negative numbers actually
but whether 0 is positive is based on convention
as i said
gentle zephyr
final saddleBOT
soft zealotBOT
Renato
barren pebble
|{segments connecting 2 vertices}|-|{edges}|=|{diagonals}|
You count the segments connecting 2 vertices, then minus n (which is the number of sides)
mortal girder
or just count it directly like a diagonal need 2 points, so u have n points to choose for point A, then you have (n-3) choices for point B because point B must differ from point A and the two next to point A, then divide by 2 because the order doesnt matter ( AB same as BA )
barren pebble
Yeah that too
I put it the other way because I assumed the OP just learned about combinations
So the number of "segments connecting 2 vertices" is easier to calculate
gaunt raft
n choose 2 - n
final saddleBOT
@gentle zephyr Has your question been resolved?
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Available help channel!
late gazelle
yo how do I do this i've got no clue
final saddleBOT
yo how do I do this i've got no clue
@late gazelle Has your question been resolved?
lofty sinew
yo how do I do this i've got no clue
The question says you need to find the sum of these two lines
lofty sinew
Why 3d
rugged merlin
there is elevation
lofty sinew
But there's no depth
rugged merlin
map coordinates describe x and y, and z describes the change in height
lofty sinew
Oh damn you're right I skimmed that part
rugged merlin
@late gazelle
here is your hint to get going: your first hilltop can be described with $\mat{500 \ 700 \ 400}$ in metres. Likewise, the second hilltop is at $\mat{800 \ 350 \ 320}$
soft zealotBOT
rugged merlin
calculate the displacement vector, the difference of the two, and the straight-line distance between the two
final saddleBOT
late gazelle
.reopen
final saddleBOT
✅ Original question: #help-36 message
late gazelle
sorry, i had to go for a few minutes
late gazelle
here is your hint to get going: your first hilltop can be described with $\mat{5...
alright lemme see what i can do
so thats distance between the two mountains right
oh wait i made a slight mistake
i forgot the height
d=468m roughly
where do i go from here
i cant really say the "slack" is halfway along can I ?
rugged merlin
d=468m roughly
yeah right
rugged merlin
i cant really say the "slack" is halfway along can I ?
its an isosceles
each equal side is cable length / 2
late gazelle
so i can, nice
so like 234 meters ish then
and the "slack" is $= \frac{5}{100} 468$ ?
soft zealotBOT
KB
rugged merlin
wait but like
you are kinda done lol
cable length is 468 and slack is what u wrote
late gazelle
oh slack is what we were finding
i thought it was the cable length including slack
so the two similar lines
oh if that's all i needed that wasn't as bad as what i thought
rugged merlin
yeah
late gazelle
or is it just 8+2-3
desert mantle
yes
late gazelle
oh so we omit the i, j, k then
abstract bloom
$\mathbf{a}\cdot\mathbf{b} = \sum_{i}a_ib_i\$
$\mathbf{a} = \begin{bmatrix} a_1\a_2\a_3\\dots\end{bmatrix}$
$\mathbf{b} = \begin{bmatrix} b_1\b_2\b_3\\dots\end{bmatrix}$
soft zealotBOT
Flash09A14m
tranquil pine
dot product is <x,y,z> • <l,m,n> = xl+ym+zn, which is a scalar
multiply each is, js, ks together and sum them up
dot product returns the projection of one vector onto another
late gazelle
yea ik what it is and how to do it but we haven't used i, j, k notation yet so i got confused
abstract bloom
dot product returns the projection of one vector onto another
Scaled* projection
late gazelle
and its also 3d now
abstract bloom
yea ik what it is and how to do it but we haven't used i, j, k notation yet so i...
i, j, k represent the unit vectors for each direction
tranquil pine
i, j, k are unit vectors of the x, y, z coordinates respectively
late gazelle
i understand it but not in calculations
tranquil pine
if you understand it now then ggs
abstract bloom
dot product returns the projection of one vector onto another
The projection is scaled
I like to call it the "effect" of the projection
Pure projection of a onto b would be a dot b divided by |b|
abstract bloom
Yes
late gazelle
so $a \cdot b = |a||b|\cos(\theta)$
soft zealotBOT
KB
tranquil pine
yeah
south dirge
,w 40 cos(50
soft zealotBOT
tranquil pine
does wolfram auto does trigonometric functions in degrees
late gazelle
for this would $\theta=\arccos(\frac{\overrightarrow u \cdot \overrightarrow v}{|u||v|})$
soft zealotBOT
KB
south dirge
ye
tranquil pine
look at the cosine graph and see when cos theta = 0 and most of the time bind yourself to 0 to 360 or 0 to 2pi (in radians)
that should tell you that arccos(0) = 90 or pi/2
late gazelle
what is this ew
south dirge
Length Qsn
late gazelle
projection was my #1 opp last year when doing 2d vectors
south dirge
Pure projection of a onto b would be a dot b divided by |b|
use this
tranquil pine
what is this ew
anytime you see 'projection' in a problem, it wants you to dot product
if you want to add a+c, just add the ijk together
late gazelle
hmm i remember it having unit vectors tho no? or was that something else
south dirge
yeah
south dirge
This would give you the mag
for the vector form,
You need to apply
(a . b / (|b|^2) ) * b
this is a method to find the projection of vector a onto vector b
late gazelle
can proj be negative?
south dirge
as in?
late gazelle
bc i got $\frac{-18}{\sqrt{30}}$
soft zealotBOT
KB
south dirge
if you take the positive value for this you would get the magnitude of the projection of vector a+b on c
But the question isnt asking you the magnitude
tranquil pine
it can be negative
late gazelle
doesn't projection give me another vector
south dirge
doesn't projection give me another vector
late gazelle
ahhh
south dirge
this is how you calc the projection vector
what you found was the magintude of the projection
late gazelle
is $\proj_(a+b){c} = \frac{(a+b) \cdot c}{|c|}$
soft zealotBOT
KB
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
late gazelle
for magnitude ?
south dirge
**KB**
Compile Error! Click the <:errors:504266556054962196> reaction for more i...
as in take absolute value of this
magnitude cant be negative!
late gazelle
sooooooo. Projection of a on b is $= |\frac{a\cdot b}{|b|}| b hat$
soft zealotBOT
KB
late gazelle
b hat b ^ thingy
late gazelle
unit vector of b
late gazelle
ah interesting
south dirge
wait
wheres the minus sign
you remove the minus sign if you are calcing the magnitude
but you are finding the vector so minus sign is imp
late gazelle
so we do want the - now
i vaguely understand this but not entirely
ok nvm idk how to even start
tranquil pine
ok nvm idk how to even start
okay, make a displacement vector v = xi + yj + zk
so |v|² = x²+y²+z²
then take v dotted with x unit vector (here, it's denoted as i), you get
v dot i = |v| * |i| * cos(alpha)
|I| is just 1
and then you repeat for the rest of the x y z for the displacement vector
and them simplify final answer to get the equation they got
abstract bloom
hmm i remember it having unit vectors tho no? or was that something else
Yes
The magnitude of the projection is a dot b divided by |b|
The actual vector would be that multiplied by the unit vector of b
late gazelle
and then you repeat for the rest of the x y z for the displacement vector
and t...
uh huh
where x = cos^2(alpha) and so on?
tranquil pine
|v|cos(alpha) = x
(|v|cos(alpha))² + y² + z² = |v|²
you are looking at the v projection on x axis
you'll end up with something like this
|v|cos(alpha) = x = |v| • i
|v|cos(beta) = y = |v| • j
|v|cos(gamma) = z = |v| • k
(|v|cos(alpha))² + (|v|cos(beta))² + (|v|sin(gamma))² = |v|²
and then you divide all size by |v|² to get
cos²(alpha) + cos²(beta) + cos²(gamma) = 1
tranquil pine
we made a displacement vector v = xi + yj + zk
what's the magnitude of the displacement vector? |v| = ? @late gazelle
late gazelle
$\sqrt{(v_1)^2+(v_2)^2+(v_3)^2}$
soft zealotBOT
KB
tranquil pine
no
late gazelle
magnitude of a vector is not that?
tranquil pine
$\sqrt{(x)^2+(y)^2+(z)^2}$
soft zealotBOT
polar
late gazelle
oh same thing, v_1, v_2, v_3 are just the x, y, z components of v
tranquil pine
yeah, and you know where i got those from right?
just by dotting the displacement vector V by i j k unit vectors
late gazelle
if x^2 = cos^2(alpha), x = cos(alpha)
tranquil pine
yeah, and we got that because we took the displacement vector and we saw how much we projected through the x axis
and we did that with a dot product
late gazelle
if x^2 = cos^2(alpha), x = cos(alpha)
so where/how does the |v| part come into this then?
tranquil pine
|v| • i = x
x = |v||i|cos alpha
|i| = 1
x = |v| cos alpha
x² = |v|² cos² alpha
late gazelle
and that only holds equality to our original if |v|^2 is = 1 and if v = cos^2(alpha) + cos^2(beta) + cos^2(gamma) its equal to 1?
tranquil pine
Okay, let me explain properly now that I'm on a computer.
The question wants us to show that cos^2 (alpha) + cos^2(beta) + cos^2(gamma) = 1 is true for any directional cosines of any line in 3D space.
Before, I begin, I'm going to introduce the notion of a unit vector.
A unit vector, often times denoted as **i j k** are vectors with a magnitude of one that point to the **x y z** axis on a 3D cartesian coordinate systems plane and are often used to describe direction of a vector.
Let's start off with the question. The question first wants to 'use' a displacement vector.
We can make a sample displacement vector by only considering the points P1 (0,0,0) and P2 (x,y,z)
We can use the idea of unit vectors to define direction to the displacement vector, and we can call this vector 'V'
V = (x-0)i + (y-0)j + (z-0)k = xi + yj + zk
So that we can perform a scalar product, we will find |V| which happens to be
|V| = sqrt(x^2 + y^2 + z^2)
Hopefully that makes sense so far, all we did was defined a displacement vector and took it's magnitude.
Now, since we know what the magnitude of the displacement vector is, we can now take the scalar product of it with respect to the i j k unit vectors. This will describe the 'projection' of the displacement vector on x y z, so basically it's contribution to the x, y, z coordinates.
So,
Using the formula u dotted v = |u||v|cos(theta)
|v| dotted i = |v||i|cos(alpha) = x
|v| dotted j = |v||j|cos(beta) = y
|v| dotted k = |v||k|cos(gamma) = z
Now, recall how I mentioned that unit vectors *are of magnitude 1?* (It's often good to know that whenever I say 'unit', I mean anything of value 1.) We can then say |i| = 1, |j| = 1 and |k| = 1
|v| dotted i = |v|cos(alpha) = x
|v| dotted j = |v|cos(beta) = y
|v| dotted k = |v|cos(gamma) = z
Now, from that, we can recall our displacement vector and square it to get
|v|^2 = x^2 + y^2 + z^2
Plug everything in
|v|^2 cos^2(alpha) + |v|^2 cos^2(beta) + |v|^2 cos^2(gamma) = |v|^2```woah, this is gonna take a minute
tranquil pine
Notice how |v|^2 is a common term in |v|^2 cos^2(alpha) + |v|^2 cos^2(beta) + |v|^2 cos^2(gamma) = |v|^2
We can divide the entire equation by |v|^2 which is mathematically sound and we get
cos^2 (alpha) + cos^2(beta) + cos^2(gamma) = 1
Which is what the queston wants.
late gazelle
that makes a lot more sense
tranquil pine
Yeah, try the question out and see if you truly understood it - without any help
tranquil pine
@surreal grove blessing us with a virus 🙏
final saddleBOT
final saddleBOT
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gentle zephyr
final saddleBOT
zinc bluff
mmm, this sounds like the euler theorem for the diagonals of a polygon
gentle zephyr
Care to elaborate?
zinc bluff
I'm sorry I cannot help that much here, I just wanted to tell you that it looks like that theorem😅
wary juniper
Does that say prove the number of diagonals of an n sides polygon is n(n-3)/2
gentle zephyr
ye
wary juniper
First, think about how many diagonals you can draw from 1 point
gentle zephyr
First, think about how many diagonals you can draw from 1 point
n
blissful meadow
n
So if you're sitting at the vertex of a polygon, you can draw n lines to other vertices that aren't already sides of the polygon?
In particular, if you fix a vertex, how many vertices can you draw a line to that aren't already sides.
gentle zephyr
So if you're sitting at the vertex of a polygon, you can draw n lines to other v...
n-1
blissful meadow
So is your guess of n-1 diagonals from a given vertex correct?
gentle zephyr
So is your guess of n-1 diagonals from a given vertex correct?
n-3
blissful meadow
Yes. If you fix a vertex then you can't draw a line to any of the adjacent vertices, since that would just be a side of the polygon. You also can't draw a diagonal from the vertex to itself. That leaves n-3 vertices to draw a diagonal.
gentle zephyr
Yes. If you fix a vertex then you can't draw a line to any of the adjacent verti...
then wat
wary juniper
Now do the same for all points on the polygon
gentle zephyr
why / 2
wary juniper
why / 2
Cause youre counting a diagonal twice, for example for the diagonal AC you counting from A to C and then again from C to A
gentle zephyr
i see because for it to be a diagonal two edges need to connect so we are double counting
gentle zephyr
Cause youre counting a diagonal twice, for example for the diagonal AC you count...
ye
can u help me write the proof
blissful meadow
You've pretty much got it served on a platter now.
gentle zephyr
I am not really sure how we got to the n - 3 formula
blissful meadow
Go back and read what we talked about.
blissful meadow
Yes. If you fix a vertex then you can't draw a line to any of the adjacent verti...
.
gentle zephyr
Yes. If you fix a vertex then you can't draw a line to any of the adjacent verti...
seems out of the blue
blissful meadow
In what sense? If you draw the situation, as you should in pretty much any case, it's pretty clear both what I'm saying and why it's true.
gentle zephyr
In what sense? If you draw the situation, as you should in pretty much any case,...
yes but I was looking for an formal argument rather than a visual proof
blissful meadow
It's a formal argument which you can convince yourself works using a visual aid.
It doesn't mean it's a visual proof.
gentle zephyr
It's a formal argument which you can convince yourself works using a visual aid.
what would be the formal argument be?
gentle zephyr
You've pretty much got it served on a platter now.
I still need help with the induction
blissful meadow
what would be the formal argument be?
It's formal enough in that a polygon is already a rather geometric object in nature.
Induction would work but this would require that a polygon in n vertices can be used to construct a polygon in n+1 vertices. It can, but this construction in itself would be geometric...
It should be a rather intuitive fact that from a given vertex you can draw n-3 diagonals. This is where the geometric intuition ends.
If you want to avoid this at all cost you can also compute the number of pairs of vertices, hence giving you the total number of lines you can draw between any two of the n vertices in the polygon.
Then you could subtract the number of such lines we know are not diagonal.
gentle zephyr
If you want to avoid this at all cost you can also compute the number of pairs o...
how do I do this?
blissful meadow
Well you need to compute those two quantities.
How many pairs of vertices can you pick from the n vertices in the polygon?
How many lines in the polygon don't count as diagonals?
gentle zephyr
Well you need to compute those two quantities.
How many pairs of vertices can yo...
How many pairs of vertices can you pick from the n vertices in the polygon?
I dont know
blissful meadow
Think about it. You spent less than a minute thinking about what I wrote before saying you don't know.
gentle zephyr
n - 1
blissful meadow
Did you check n-1 fits with that number or did you write it down because it sounds nice?
Does it make sense that n-1 is the number of pairs of n vertices?
gentle zephyr
Does it make sense that n-1 is the number of pairs of n vertices?
wdym by number of pairs of n vertices?
you mean the combinations of pairs of vertices?
blissful meadow
The number of ways you can pick 2 vertices from n vertices.
gentle zephyr
for example if the edges of a triangle are 1 2 3
then I have (1,2) , (1,3), (2,3)
gentle zephyr
The number of ways you can pick 2 vertices from n vertices.
I dont think I understand, care to explain with other words?
blissful meadow
How many ways can you select 2 people from a group of 5?
gentle zephyr
5 c 2
blissful meadow
Ok. How many ways can you select 2 people from a group of n?
gentle zephyr
n c 2
blissful meadow
So how many ways can you select 2 vertices from a set of n vertices?
gentle zephyr
n c 2
blissful meadow
So there are nC2 possible lines you can draw between vertices in the polygon.
How many of them do not qualify as diagonals?
blissful meadow
Out of all the lines between vertices.
There's a number of lines between vertices we don't count as diagonals.
What are those lines?
How many of them are there?
gentle zephyr
There's a number of lines between vertices we don't count as diagonals.
the adjacent edges
no?
blissful meadow
Adjacent to what? We don't have a fixed vertex here.
If you think of a polygon with n sides with its diagonals drawn, what is the number of lines in the polygon that aren't diagonals?
If you try with a square (and its diagonals) first, it should be apparent what this number is and what those lines are.
gentle zephyr
Adjacent to what? We don't have a fixed vertex here.
If you think of a polygon ...
If you think of a polygon with n sides with its diagonals drawn, what is the number of lines in the polygon that aren't diagonals?
the sides, n
@blissful meadow
blissful meadow
Yes. So now we have nC2 pairs of vertices (and therefore lines between those), and we want to get rid of the n lines in there which aren't diagonals.
What's left?
gentle zephyr
Yes. So now we have nC2 pairs of vertices (and therefore lines between those), a...
what does n C 2 represent?
blissful meadow
The number of pairs of vertices you can pick from amongst the n vertices of the polygon.
You can imagine that any such pair gives a line between two vertices.
gentle zephyr
The number of pairs of vertices you can pick from amongst the n vertices of the ...
You can imagine that any such pair gives a line between two vertices.
okay so nC2 lines.
gentle zephyr
Yes. So now we have nC2 pairs of vertices (and therefore lines between those), a...
we want to get rid of the n lines in there which aren't diagonals.
nC2 - n, got it
@blissful meadow
blissful meadow
Yes, so now you can write down what nC2 means computed and combine those.
gentle zephyr
Yes, so now you can write down what nC2 means computed and combine those.
how?
care to elaborate?
blissful meadow
nC2 is some expression.
gentle zephyr
,w simplify binomial(n, 2) - n
soft zealotBOT
gentle zephyr
@blissful meadow
blissful meadow
What?
gentle zephyr
n(n-3)/2
gentle zephyr
Well, the first one was more like a visual representation, I wouldnt call it a proof since it was kind of non formal argument
gentle zephyr
I didnt realized until it was too late
gentle zephyr
https://tenor.com/view/skeleton-falling-falling-over-gif-9881376952697054490
however, the argument should be fairly similar, right?
first we use the base case n = 3, then assume it holds for n = k, and show that it holds for n = k + 1
timber leaf
Is this weak induction
gentle zephyr
ye of course
timber leaf
Why not use strong induction
gentle zephyr
strong iirc is used when the inductive hypothesis depends on other previous hypothesis
for example when you ahve a sequence an = a_{n-1} + a_{n-2} , if you assume P(n) then you are assuming P(n-1) and P(n-2), I think?
@blissful meadow
timber leaf
I see there is no need to find polys that have 4 or 5 sides but you only need a k side
Make senses
gentle zephyr
care to elaborate?
timber leaf
Base case is n=3 which is correct
Inductive step for k also good
I think for k+1 is basically add 1 more vertex, which is keep same number of diagonals for k vertices and add (k-2) vertices
Because this new vertex doesnt connect itself and the 2 vertices next to it
gentle zephyr
I think for k+1 is basically add 1 more vertex, which is keep same number of dia...
what
drowsy epoch
. @proud raft
bold turtle
. Note your first message will be pinned by the bot, so ensure that your first message is the question you want help on.
proud raft
I have a doubt .
9 different objects should be distributed to 7 different person .
Atleast one should be distributed. .no of ways is 8⁹-1 .
Why the most common explaination is take an empty box to avoid counting exact cases.
But I don't understand it .
Can anyone explain logically
Derive the formula mechanism wise why is this that .
final saddleBOT
I have a doubt .
9 different objects should be distributed to 7 different person...
proud raft
Star
I really appreciate your help
People like u make the world a civil society
Thanks
rugged merlin
i guess it is just to avoid casework, if i understood your question correctly?
like it would be nine different cases would it not
exactly 1 object was distributed, ...
proud raft
I don't understand the reason
The formula has 8⁹
.the formula should have had 7⁹ because objects are only seven
@rugged merlin ???
gritty chasm
Please don't ping individual helpers unprompted.
rugged merlin
I don't understand the reason
The formula has 8⁹
.the formula should have had ...
sorry i like
dont understand what that is saying
proud raft
No problem
Anyone else that can help ?
Doubt closed
.closed
. Close
. close
.close
final saddleBOT
drowsy epoch
.closed
. Close
. close
.close
this is also combinatorics
strange pelican
I have a doubt .
9 different objects should be distributed to 7 different person...
8^9-1 does not feel correct
bold turtle
.reopen
final saddleBOT
✅ Original question: #help-36 message
bold turtle
I don't understand the reason
The formula has 8⁹
.the formula should have had ...
"The formula has..."
Can you cite your source on this? What formula?
Because conceptually I agree with Paradox, that this doesn't feel right
final saddleBOT
@proud raft Has your question been resolved?
final saddleBOT
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proper lion
Hijack
runic needle
modping?
proper lion
The length of the shortest path that begins at the point (-1, 1), touches the x-axis and then ends at a point on the parabola (x - y)^2 = 2(x + y - 4), is:
Mb wrong ping
heady portal
The length of the shortest path that begins at the point (-1, 1), touches the x-...
Wait I've seen this ques...
It's from the black book-
proper lion
please dont delete modpings <@1483446811619496059>
Alr i was going to edit it but it pinged
proper lion
It's from the black book-
True
heady portal
That's a really good ques... I'd have loved to help u but I need to go right now...sorry
proper lion
Alr no worries
final saddleBOT
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
@proper lion Has your question been resolved?
final saddleBOT
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brave moon
This is college algebra rational equations. I already know to move all the numbers to one side, and to make a sign chart, but im not so confident on the details.
final saddleBOT
This is college algebra rational equations. I already know to move all the numbe...
sturdy flax
have you attempted this and gotten stuck? if not, since you have some idea, maybe try it first and we'll see what else needs brushing up on.
brave moon
got it.
ok, i tried to solve it. Is the solution set (-∞, -5) U (4, ∞) ?
i put it to one side, then combined the equations, then tried to find what makes the numerator and denominator 0, then put those numbers into a number line with negative and positive infinity and tried to find what numbers make the solution negative or positive.
final saddleBOT
@brave moon Has your question been resolved?
brave moon
.close
final saddleBOT
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fresh pendant
So with this question... does the sample size increasing affect the confidence interval?
final saddleBOT
So with this question... does the sample size increasing affect the confidence i...
fresh pendant
Sorry for the ping
I feel bad about that now
I didn't realize I was pinging 1500 people
I would like some help tho
crystal pawn
Sorry for the ping
!15m
final saddleBOT
Sorry for the ping
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
zinc verge
i dont know what to do in here sorry
crystal pawn
I didn't realize I was pinging 1500 people
so you’re fine.
fresh pendant
so you’re fine.
alright
fresh pendant
So with this question... does the sample size increasing affect the confidence i...
Does anyone know about statistics?
I'll just have to come to this question later
I'm wasting a lot of time on it
junior token
So with this question... does the sample size increasing affect the confidence i...
could you send the entire question?
there’s context missing, like the meaning of the variable s
fresh pendant
there’s context missing, like the meaning of the variable s
fresh pendant
there’s context missing, like the meaning of the variable s
sorry about that
fresh pendant
could you send the entire question?
I just realized I made a mistake
I failed to consider the changed mean and SD in part 2(a)
I'll have to redo that
junior token
I just realized I made a mistake
Nice
junior token
I’m not familiar with statistics so I can’t really help, but have you tried looking up the relevant sections in the textbook/course materials where they mention confidence intervals?
final saddleBOT
@fresh pendant Has your question been resolved?
fresh pendant
I’m not familiar with statistics so I can’t really help, but have you tried look...
I'd have to dig through a fair amount of content
but it's achievable I suppose
fading bramble
How do I do part b
final saddleBOT
How do I do part b
brazen breach
How do I do part b
notice that part of f'(x) is similar to g(x)
namely the xarcsinh(x^2) part
fading bramble
Yes yes
brazen breach
fill in the missing parts
brazen breach
If that's allowed
integral F+G dx = integral F dx + integral G dx
it's like
forgot if whatever it was called FTC 1 or 2
fading bramble
integral F+G dx = integral F dx + integral G dx
So I can split the integrals apart? Do them separately and add them?
I'm so confused lol
brazen breach
So I can split the integrals apart? Do them separately and add them?
you can split integrals appart that's part of the fundamental theorem of calc
fading bramble
Haha I can read it if I focus enough 😅
fading bramble
Tytyty
brazen breach
So was I on thr right path?
keep going w integrating
fading bramble
keep going w integrating
Kk I kinda gor stuck as well
I don't know I coukd integrate x³/(x⁴+1)½
full jungle
I don't know I coukd integrate x³/(x⁴+1)½
substitution
brazen breach
I don't know I coukd integrate x³/(x⁴+1)½
u-sub
x^4+1=u
fading bramble
substitution
I see
brazen breach
Let me try
wait if you didn't know about the adding thing how were you integrating things like x^2+x
fading bramble
wait if you didn't know about the adding thing how were you integrating things l...
I didn't think too much about it
I should've 😭
I'm still stuck😭
Inded did something wrong
Sorry for the messy working out
twin pivot
the derivative is not correct
full jungle
recall power rule
lilac bison
no worries
fading bramble
Is this rightt
fading bramble
no worries
Ty😭
twin pivot
why -2?
lilac bison
no negative sign
fading bramble
Oh true kk
Kk im going to try working out the integral in the question with this
So when I solve it, am I supposed to get a constant value -1/4
Gtg so
Ty for all the helppp
<333
.close
final saddleBOT
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left trail
Can someone explain why for $\int\frac{dx}{2+a\cos x}$ I can't say $a=\cos x$ and say $\frac{\arctan(\cos x)}{-\sin x}$ I see that it doesn't hold if I take the derivative. But I want to understand what is wrong with the sub
final saddleBOT
Can someone explain why for $\int\frac{dx}{2+a\cos x}$ I can't say $a=\cos x$ an...
soft zealotBOT
BigBen
night raft
The main issue is the x variable you are integrating with respect to
when you say let a = cos(x), you are trying to treat a part of the function as a constant while it is actually changing as x changes
instead you can do something like u = cos(x)
desert mantle
imagine instead of a it said 7
left trail
Ok so the issue is a is a constant that is between 0 and 1 and I set it equal to a function that is between 0 and 1
desert mantle
putting 7=cosx is obvious nonsense
final saddleBOT
final saddleBOT
Available help channel!
cedar lichen
I am supposed to use continuous convolution on this function with itself (the second image is the given formula continuous convolution; I basically have to calculate (f * f)(x) )
I used it on f(x) which gives me an Integral over R with f(x-t) f(t) dt inside (notationally I use t here instead of x')
Now I have already established that t would need to be within [-1 , 1] for the integral not to be 0 (as the f(t) part of the integral would otherwise be 0 so that part is not of interest) and that since f(x-t) also needs to not be 0 only the area -1<=x-t<=1 is of interest as well
(Which I then summarized to t needing to be within [-1 , 1] and x needing to be in [-2 , 2] (which I got from the maximum and minimum of x-t) for the integral to not be 0)
Now I am stuck though since if I don't really know how to define over which parts to integrate to get the resulting convolution function (for which I think I at least already have (f*f)(x) = 0 if x∉[-2,2] ). Are the borders just -1 to 1?
Sorry if this is like a weirdly specific topic or something
final saddleBOT
I am supposed to use continuous convolution on this function with itself (the se...
@cedar lichen Has your question been resolved?
final saddleBOT
@cedar lichen Has your question been resolved?
muted prairie
The integral should be over all of R
But you could replace the bounds with -2 and 2 here
Since it's always zero outside of that
cedar lichen
that is basically what I did but why -2 to 2 (I tried it with -1 to 1 )? Since I integrate over t wouldn't it be entirely 0 for any x bigger than 1 or smaller than -1?
muted prairie
Yeah -1 to 1 should work too
Just note that the function won't always not be 0 inside that interval, it depends on what x is
You have the integral of f(x-x')f(x') dx'
The f(x') precisely had support [-1,1] so you can factor it out if you set the bounds
final saddleBOT
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simple flower
I am not really sure what to do next, or if this is complete or not. The question is: Suppose a,b,c are integers. If a|b and a|c then a|(b+c)
I am new to proofs, so sorry if the structure is off.
I have done this so far:
Proof. Suppose a|b and a|c, if b=ak and c=az, integers k,z
Then b+c = ak+az, so a|(b+c) if ak+az=aq, integer q...
final saddleBOT
I am not really sure what to do next, or if this is complete or not. The questio...
solar crest
I am not really sure what to do next, or if this is complete or not. The questio...
...
Then b+c = ak+az = a(k+z), which is divisible by a
simple flower
...
Then b+c = ak+az = a(k+z), which is divisible by a
Ohh, that makes sense. Thank you!!
.close
final saddleBOT
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full compass
final saddleBOT
full compass
why is there a dx term in the denominator?
in the second equation
when did that get introduced
nvm, i see
forced factor
.close
final saddleBOT
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knotty pier
this might go a bit beyond the scope of the question, but how would I go about the actual transformation to the function g?
final saddleBOT
this might go a bit beyond the scope of the question, but how would I go about t...
knotty pier
my original idea is that I could take that limit equation i have, and convert it back to cartesian coordinates
blissful meadow
You mean how you would define g?
knotty pier
but I couldnt figure out how to get rid of cos^2 theta
knotty pier
You mean how you would define g?
i believe so
blissful meadow
Because then it's just about "adding" in the point at (0,0) which isn't initially defined on f.
You can write it down piecewise.
knotty pier
is that with the polar equation?
also arent there issues with r being only positive despite the only restriction being xy != 0
blissful meadow
I mean either would work, but there's nothing stopping you from defining g(x,y) = f(x,y) when (x,y) != 0 and g(0,0) = 0. This is continuous as you wrote.
blissful meadow
Like $$g(x,y) =\begin{cases} \frac{7x^{2.5}}{x^2+y^2} & (x,y) \ne (0,0)\
0 & (x,y) = (0,0)\end{cases}$$ is just $f(x,y)$ with the hole "plugged" where it isn't defined.
soft zealotBOT
Azyrashacorki
knotty pier
im realizing now that was the point of finding the limit in polar coordinates
so if my limit were say, DNE, then the mapping from f to g wouldnt be possible?
blissful meadow
Yep
It's like how $f(x) = \frac{x(x+1)}{x+1}$ is just the line $y=x$ with a hole at $x=-1$. The limit exists so you could define $g(x) = x = \begin{cases} \frac{x(x+1)}{x+1} & x\ne -1 \ -1 & x=-1\end{cases}$
soft zealotBOT
Azyrashacorki
final saddleBOT
final saddleBOT
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high agate
I need help, guys.
final saddleBOT
I need help, guys.
tranquil totem
i.e that minus * minus = +
Rotate one minus by 90° then add to first - then u get +
tranquil totem
but what
gritty chasm
Well it depends on what structure you're proving this for. If it's for R, assume field axioms for R, and hint: you need to use that if $x \in \mathbb{R}$ then $x \cdot 0 = 0$ and $1+(-1)=0$
soft zealotBOT
Annie Maqionde
tranquil totem
we gotta prove it using algebra
Are geometrical proofs invalid?
high agate
Are geometrical proofs invalid?
yeah
high agate
**Annie Maqionde**
thank you annie
do you think that you need to be very intelligent to get a math major
that's what I am trying to do now...
gritty chasm
im the wrong person to ask
We'll wait until another helper comes by or you can post this in #study-discussion or #discussion or #serious-discussion
high agate
why the wrong person to ask buddy
also how much do you ask for chess lessons
I wanna reach 1800 elo by the end of 2026
final saddleBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
gritty chasm
If you're done close the channel 
high agate
I am not done with this channel...
severe canyon
I am not done with this channel...
What else do you need? Write it down then
final saddleBOT
@high agate Has your question been resolved?
tawny remnant
1+1=?
onyx peak
1+1=?
Please dont troll in the help channels
these channels are for genuine questions
if you dont have any real questions, you can close this with .close and move to #chill for joking around next time
final tangle
.close
candid pulsar
is there some clever matrix multiplication method to find sum of elements of a
lilac bison
oh damn
desert mantle
x^t A x for x=(1,1,1) but I dont think thats gonna help here
candid pulsar
well adj A is symmetric if that helps
old quarry
is there some clever matrix multiplication method to find sum of elements of a
adj(adj(A))=|A|^n-2 (A)
quaint rampart
use det(adj(A)) = {det(A)}^n-1 for det(A)
quaint rampart
adj(adj(A))=|A|^n-2 (A)
then plug det(A) into this to get A
quaint rampart
use det(adj(A)) = {det(A)}^n-1 for det(A)
choose the negative solutions, as det(A)<0
final saddleBOT
old quarry
thanks gng
i want that shoutout in june 😡
old quarry
yeah
"he was such a strong support throughout this journey especially with his funny jokes that kept me going"
candid pulsar
💔
strange pelican
https://tenor.com/view/ok-cat-okay-cat-cat-ok-cat-okay-thumbs-up-crying-cat-gif-14516071108819125230
final saddleBOT
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left trail
Regarding 1 how can I integrate it? I have tried parts and I tried making a sub. Nothing works.
final saddleBOT
Regarding 1 how can I integrate it? I have tried parts and I tried making a sub....
gritty chasm
Well I feel you dont have to integrate it for this one 
Maybe some kind of reciprocal trick
steep nest
Furthermore, it doesn't even have an antiderivative
left trail
Well I feel you dont have to integrate it for this one <:pandathink:524885344995...
Well I looked at that option too. I had $\int_1^x \frac{log t}{t+1}dt+\int_1^\frac{1}{x}\frac{log t}{t+1}dt$ I wasn't sure how to proceed
soft zealotBOT
BigBen
left trail
Furthermore, it doesn't even have an antiderivative
Really? no tricks of any sort for it?
left trail
Ok
kindred mortar
u = 1/t might give something dont you think ?
steep nest
Why is there a - in front of the 2nd integral if I may ask?
kindred mortar
yeah i'm pretty sure it works
left trail
Why is there a - in front of the 2nd integral if I may ask?
Sorry that should be a plus
left trail
u = 1/t might give something dont you think ?
Let me see
left trail
u = 1/t might give something dont you think ?
But we have neither u or du in our integrand?
steep nest
Do you know u substitution?
kindred mortar
So I see how I can get it for the t in the log
think bout log properties
left trail
But I then have $\int_1^\frac{1}{x} \frac{log\frac{1}{u}}{1+t} (-t^2du)$
soft zealotBOT
BigBen
kindred mortar
But I then have $\int_1^\frac{1}{x} \frac{log\frac{1}{u}}{1+t} (-t^2du)$
replace every t
kindred mortar
change the bounds
left trail
change the bounds
I did. For f(x)
kindred mortar
I did. For f(x)
oh alright
i would rather do it with f(1/x)
but it works
i suppose
final saddleBOT
@left trail Has your question been resolved?
rain sentinel
final saddleBOT
rain sentinel
can someone help me integrate this
vital crag
is that a 1 next to cos
rain sentinel
yes
vital crag
$\int \cos(x) dx = \sin(x) + C$ so you just need to do substitution
soft zealotBOT
pi_day
vital crag
do you know what u equals in terms of t?
vital crag
right. then solve for what dt equals in terms of du
vital crag
is that a separate question
rain sentinel
no
vital crag
or how is it related to this
rain sentinel
the question is to integrate that
vital crag
is the integral of 2npit a constant
where does this come from
rain sentinel
its the fraction numerator
vital crag
right. why do you need to integrate it?
vital crag
are you saying i should sub the fraction
write out exactly what u equals in your u sub
rain sentinel
ok
vital crag
i see. your du/dt is incorrect
$\frac{d}{dt} \lp \frac{n \pi t}{5} \rp = \frac{n \pi }{5} \frac{d}{dt} t = ?$
soft zealotBOT
pi_day
rain sentinel
oh yh because t would be 1
vital crag
the derivative of t with respect to t is 1 yes
vital crag
is this a seprate problem?
rain sentinel
yes
vital crag
looks fine. get a number and post it here or check with wolfram
,w int 0 to pi sin(x) dx
soft zealotBOT
vital crag
use that as a template
final saddleBOT
@rain sentinel Has your question been resolved?
vital crag
cos(n * pi) depends on the value of n. write out a few numbers and try to find a pattern. also you can't divide by 0 when n=0 so solve that case seprately.
severe canyon
Well, what happens if you substitute 0?
vital crag
dont i just sub in 5 and 0
for t yes. i was talking about the step after that
rain sentinel
if you sub zero the term becomes 0
final saddleBOT
@rain sentinel Has your question been resolved?
vital crag
for t yes. i was talking about the step after that
I already answered that here
final saddleBOT
@rain sentinel Has your question been resolved?
left trail
For 5b if we are generating a solid of revolution don't we need to know what the shape of the cross sectional area. Because can't the shape because a square or a circular disk, etc
final saddleBOT
For 5b if we are generating a solid of revolution don't we need to know what the...
formal trail
"rotating it" would imply a disk
any other shape would imply it stretches during that rotation
final saddleBOT
@left trail Has your question been resolved?
left trail
"rotating it" would imply a disk
Ok thank you for making it clear.
jagged flare
final saddleBOT
jagged flare
ABCD is a rectangle such that AB=68 and AD=27, E,F,G,H is shown like the diagram. AF=BE=CH=DG=54. find the shaded region
ok so i wanna know why this is inccorect (ik its convoluted but i just wanna know why)
... ok nvm i see why lmao
brah
.close
final saddleBOT
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high canyon
hey can anyone help with this question from div curl and all that
final saddleBOT
hey can anyone help with this question from div curl and all that
high canyon
i got till c part , can someone give me a hint for how to do d part
final saddleBOT
@high canyon Has your question been resolved?
final saddleBOT
pliant elk
gauss law is just $\phi = \frac{q_{en}}{\epsilon}$
final saddleBOT
Available help channel!
junior siren
final saddleBOT
junior siren
How do I check one-one property?
onyx peak
Uh, is it one to one nvm misread it
onyx peak
How do I check one-one property?
Well, usually you assume that T(f) = T(g) and then try to prove that f = g
worldly sandal
x+3=21
final saddleBOT
x+3=21
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
junior siren
actually i am not getting how do we start or deal such questions
final saddleBOT
x+3=21
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
drowsy epoch
actually i am not getting how do we start or deal such questions
Start with the definition
onyx peak
actually i am not getting how do we start or deal such questions
Do you know what a linear map is and what a kernel is? If yes, you can probably use bloubbloub's way, if not, look at the definition of one to one
shadow marlin
actually i am not getting how do we start or deal such questions
Assume T(f) is the zero function (that's what T(f) =0 means)
Can you do something to T(f) to get back f(t) ?
junior siren
Do you know what a linear map is and what a kernel is? If yes, you can probably ...
a kernel when all the maps goes to zero vector
junior siren
Can you do something to T(f) to get back f(t) ?
i am not sure what does it mean?
can we take any simple example?
mellow wolf
What is the derivative or how do you compute definite integral
junior siren
What is the derivative or how do you compute definite integral
by applying integration and then we put limits in the last integral
and derivative and integration are quite opposite things
y=x^2-->y'=2x
integration of y'= x^2+c
mellow wolf
So how do you differentiate the integral function
Or how would you compute the definite integral then two ways to solve it
junior siren
leibeneiz rule?
junior siren
Or how would you compute the definite integral then two ways to solve it
we put values of limits in functions
mellow wolf
we put values of limits in functions
Yeah I mean like can't you see the answer at that point?
mellow wolf
leibeneiz rule?
Indeed but there is something more fundamental
junior siren
i can't because it is my first time of solving such things
junior siren
Yeah I mean like can't you see the answer at that point?
why do we apply derivative but
i know you want to say that both the function would be same so we will get =0
it is constant function
so it is not one-one
am I right?
shadow marlin
$\forall x, \int_0^x f(t) dt =0$
soft zealotBOT
bloubbloub
shadow marlin
So what must f be
junior siren
0
mellow wolf
a kernel when all the maps goes to zero vector
What is the map that maps to the zero vector
shadow marlin
Explain further
junior siren
What is the map that maps to the zero vector
kernel
mellow wolf
Kernel is a set
junior siren
it should be kernel
mellow wolf
What function is that
junior siren
zero function
mellow wolf
zero function
You got it then
junior siren
i am not getting it
why do you guys not say things properly
you can see how i am stuck
mellow wolf
What are you not getting
junior siren
or maybe i need to understand all the things on my own via youtube only
mellow wolf
What is one-to-one?
junior siren
f(x)=f(y) implies x=y
mellow wolf
Well, usually you assume that T(f) = T(g) and then try to prove that f = g
^ Yes in this case you would do this
mellow wolf
**bloubbloub**
Then arrange it in this form
mellow wolf
Or how would you compute the definite integral then two ways to solve it
Then there are two ways to go about it
Essentially the end goal is to show that the function f and g are the same
mellow wolf
If the function is 0
You should write it down, there are two ways either by the definition of one-to-one or by showing the kernel only contains the zero map, I am not sure which way are you doing
junior siren
Second way
strange sparrow
Do you understand why to show that T is injective it suffices to show Tf = 0 ⇒ f = 0 (Tf identically zero implies is f identically zero)
junior siren
I will try
I will take two functions f and g
T(f)=T(g)==>f=g
T(f-g)=T(f)-T(g)
Sooo T(f-g)=0
Yeah Tf=0 then f=0
strange sparrow
I can see you understand why but you need to present your workings in a much more organised way
junior siren
I can see you understand why but you need to present your workings in a much mor...
I will I will it's my first time
Thanks
How do I check onto?
.close
final saddleBOT
strange sparrow
How do I check onto?
It is not true that T is onto
final saddleBOT
Available help channel!
autumn osprey
Need help with number 6
final saddleBOT
Need help with number 6
toxic flume
If u don't mind can ik which grade u are in?
autumn osprey
I am in uni
autumn osprey
If u don't mind can ik which grade u are in?
Am I still supposed to be in highschool to post here?
gritty chasm
Am I still supposed to be in highschool to post here?
no limitations of that sort.
autumn osprey
no limitations of that sort.
Cool
final saddleBOT
@autumn osprey Has your question been resolved?
royal gust
The fact that limits are brought up at all suggests to me that the limit definition of the derivative will be used
autumn osprey
The fact that limits are brought up at all suggests to me that the limit definit...
Yeah i was thinking of that like using the fact 1 is a particular limit of ln
royal gust
f has a nutty definition and the derivative existing is weird to me
autumn osprey
autumn osprey
f has a nutty definition and the derivative existing is weird to me
Bro!!!
royal gust
From another mother?
autumn osprey
From another mother?
Yes?
junior siren
Sry for asking same question again and pinging you back @onyx peak
If we take f(x) as constant function
Will it be one-one
royal gust
This channel is taken, grab a free one
junior siren
Opps sry sry
final saddleBOT
final saddleBOT
Available help channel!
rigid cedar
could someone with desmos experience help me graph this
final saddleBOT
could someone with desmos experience help me graph this
rigid cedar
never used the website
sinful fjord
The second one
blissful meadow
!nosols please
final saddleBOT
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.
blissful meadow
I understand that it's a bit harder in this case because the helpee is not familiar with Desmos, but chances are the point of their exercise is to get familiar with it.
sinful fjord
right, my bad
sinful fjord
-5 is the x if im not mistaken and y is 1
its the other way around
rigid cedar
its the other way around
how tho
sinful fjord
the first coordinate in the bracket is the x one
rigid cedar
ahh