wdym?
#help-17
echo plaza
lapis marten
There are 100 vertices, each vertex can correspond to a number between 1 and 100
and a diagonal is defined by 2 vertices
Actually there is simpler answer then I had before: 4 vertices define 2 diagonals, right?
echo plaza
Actually there is simpler answer then I had before: 4 vertices define 2 diagonal...
yes
lapis marten
How many ways are there to connect those 4 vertices to get intersecting diagonals?
echo plaza
How many ways are there to connect those 4 vertices to get intersecting diagonal...
this does depend if any of them are adjacent right?
lapis marten
You can assume they are all different, don't think adjacency matter
lapis marten
They have to intersect inside the 100-gon though
You can connect them in 3 ways
but not all of them will produce intersections
echo plaza
You can connect them in 3 ways
wait how did you get this
lapis marten
lapis marten
Yup, so that reduces the question to finding how many tuples of 4 vertices there are in the 100-gon
echo plaza
Yup, so that reduces the question to finding how many tuples of 4 vertices there...
100 choose 4?
lapis marten
Yes
echo plaza
thanks for the help!
lapis marten
👍
echo plaza
.close
vocal sleetBOT
vocal sleetBOT
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rocky lodge
i fear that this question risks to be fuzzy and unprecise but i'm wondering if the following reasoning is sound:
so let T be the quotient space R / Z such that real numbers whose difference is an integer are equivalent.
i oberse that each equivalence class is represented by canonically by a number in [0, 1) and observe that for any sequence x_n the sequence of canonical representatives x'_n is a limited real sequence and therefore has a convergent subsequence and i wish to conclude that therefore x_n has a converging subsequence
vocal sleetBOT
i fear that this question risks to be fuzzy and unprecise but i'm wondering if t...
wraith venture
no
not for the usual metrix
xn = n is constant in R/Z, but clearly xn doesn't have any convergent subsequence
wraith venture
i fear that this question risks to be fuzzy and unprecise but i'm wondering if t...
btw the term is not limited, but bounded. Also technically you'd need compactness to use Bolzano-Weierstrass, and yet [0, 1) is not closed as a subset of R. So this would require a bit more detailing, but the conclusion holds.
However the conclusion can not be brought back to R
wraith venture
i fear that this question risks to be fuzzy and unprecise but i'm wondering if t...
so all but the final step is a sound reasoning
vocal sleetBOT
@rocky lodge Has your question been resolved?
rocky lodge
btw the term is not limited, but bounded. Also technically you'd need compactnes...
ah okay i see
rocky lodge
xn = n is constant in R/Z, but clearly xn doesn't have any convergent subsequenc...
hmm i don't see the issue with this example
because we just the constant 0 sequence right
and that is convergent
like, if we observe the sequence of canonical representatives of all sequence elements
rocky lodge
okay but then isn't the real sequence of canonical representatives also just the 0 sequence
rocky lodge
sorry, i'm a bit confused, do you mean xn in R or xn in T
because it seems to me that for xn in T this argument holds
wraith venture
i fear that this question risks to be fuzzy and unprecise but i'm wondering if t...
I'm using your notations
xn in R
x'n in T
rocky lodge
oh my bad i missphrased my question, sorry
what i mean to show is that any sequence x_n in T must have a convergent subsequence
wraith venture
btw the term is not limited, but bounded. Also technically you'd need compactnes...
then this is the idea
rocky lodge
by constructing the real sequence of canonical representatives
ah, i see now i misread that answer aswell at first
wraith venture
because you can use the homeomorphism between T and [0, 1) to your advantage
rocky lodge
ah, so then it is sound to observe the real sequence of canonical representatives to conclude properties any sequence in T?
rocky lodge
d(x, y) = inf { | x - y - k | : k is integer }
wraith venture
so it's not the same metric as the usual one on [0, 1)
That's a problem you can solve quite easily though, I let you figure out how
rocky lodge
hmm, i'll ponder a bit and try to properly write my idea down, thanks for the assistance
.close
vocal sleetBOT
wraith venture
so it's not the same metric as the usual one on [0, 1)
That's a problem you can ...
the idea is to argue ||there's a convergent subsequence in [0, 1) with the usual metric, and it's also convergent for the metric on T||
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vast shale
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vast shale
Have I done this correctly?
peak shard
yes
vast shale
vocal sleetBOT
flat whale
,rotate
vast shale
How do I go about solving 1st question
twin meteorBOT
vast shale
I thought of assuming a parametric point on the ellipse but I didn't get anything pretty
vast shale
wow
but that wouldn't be general would it
how would we go about solving it for any point
copper rapids
parametric is prbly the way to go
vast shale
yeah i thought so too
copper rapids
I am assuming you were having bad equations for the normal?
copper rapids
then what was the problem with parametric form?
vast shale
it dosent simplify
its just a trigonometric expression
in the end
after we calculate it
i dont get a cosntant
constant*
copper rapids
what expression do you get?
vast shale
Something ugly like this
copper rapids
damn thats ugly
vast shale
yes very
stiff crown
if you simplify the value of CF using perpendicular distance formula u get 2root6/](2 + sin^2(theta))^(1/2)]
vast shale
if you simplify the value of CF using perpendicular distance formula u get 2root...
got it
lol
now
does PG value simplify to something good
?
copper rapids
I think I have an idea to simply the process a bit
I got it down to just one thing
I think
its correct
you only need to find point G slope of tangent
and you should be done
stiff crown
vast shale
what
how did you get
G as 2/3costheta
i got it as root3costheta/6
equation of normal at parametric point is ax/costheta + by/sintheta = 1
if you substitute y = 0 in this you dont get 2/3 costheta
i think
@stiff crown
stiff crown
Is it right?
vast shale
let me do it again
celest forge
vocal sleetBOT
celest forge
if I have to rotate that shape 90 degrees anticlockwise
I know I can do that method of like
runic otter
Rotate points A,B and C 90 degrees anticlockwise
celest forge
subtract centre, swap both places + then change x sign, add centre
runic otter
subtract centre, swap both places + then change x sign, add centre
This would be too complicated
runic otter
Btw is it rotate with respect to point I ?
celest forge
yes
runic otter
Alright so first rotate point A
celest forge
now when the point is (0,0). I use the compass and protractor steps to find rotation
here
i first tried subtracting centre, marked those points and then on those centre-subtracted points, I applied those steps with the compass and protractor
got the new points and added centre in them
but it is wrong apparently
runic otter
Wdym by subtracting the centre ?
magic wasp
subtract centre, swap both places + then change x sign, add centre
Do that for point B, step by step, and write down each step here
celest forge
Do that for point B, step by step, and write down each step here
yeah but why can't I do this without that rule
like just with the compass and protractor rule
magic wasp
What's the compass and protractor rule
magic wasp
I'm not watching that
celest forge
that thing works when point of rotation is at the origin (0,0) but dosen't work when the point of rotation is not origin (0,0)
celest forge
I'm not watching that
you can click it and it opens the right timestamp so dosen't waste your time
i have added the timestamp thingy in the link already
magic wasp
that thing works when point of rotation is at the origin (0,0) but dosen't work ...
If a method works when the center of rotation is the origin then it works when it's not, you just have to pretend that the center of rotation is the origin
For example by "subtracting" it and then "adding" it back
vocal sleetBOT
@celest forge Has your question been resolved?
meager root
How would you rewrite this without trig functions? And yes I used tan(2x) identity already but then there's square roots in the end and I'm sure there's a simpler way bcs my teacher wouldn't make it like that, I'm sure there's a trick.. or not.. so I'm curious, is there?
vocal sleetBOT
How would you rewrite this without trig functions? And yes I used tan(2x) identi...
neon kernel
You should use their exponential expression
meager root
Could you show me?
neon kernel
The thing is tho that you might be working with imaginary numbers
Im looking up their formulas bcs i forgot 🙂
meager root
ohh that's okay.. could you sent them here as well? just so I make sure we're looking at the same stuff 😅
barren falcon
you might need this: $
\tan(\arcsin(x)) = \frac{\sin(\arcsin(x))}{\sqrt{1-\sin^{2}(\arcsin(x))}}$ and $\tan(\arccos(x)) = \frac{\sqrt{1-\cos^{2}(\arccos(x))}}{\cos(\arccos(x))}$
twin meteorBOT
Alisia
neon kernel
ah yes
compact wren
cedar kernel
occupied @compact wren
#❓how-to-get-help
if you have a channel stick to it please
barren falcon
**Alisia**
first of all make these formulas in terms of x
then use them in your task
meager root
so I can still use those formulas even though its tan(2arcsin(x)) and not tan(arcsin(x)).. What happens with the 2?
neon kernel
You should look up how to prove these formulas and see what changes
barren falcon
you have to keep in mind that $tg(2x)=\frac{2tg(x)}{1-tg^{2}(x)}$
twin meteorBOT
Alisia
meager root
oh and then I just use those formulas on this one?
barren falcon
yes
neon kernel
Yea
neon kernel
just for reference
meager root
I just did the top and bottom of the fraction separately, I hope its not confusing
tell me if you need a clearer picture?
it looks kinda scuffy
neon kernel
I took it trough photomath and it doesn't seem easier tho
meager root
ik
I feel like there is a different way to approach this, I just don't know what i tis
neon kernel
what is the actual problem?
meager root
well
usually when we get these kinda problems on the exam, it expands nicely without any square roots in the end (without trig functions) but this one has them in the end and idk I feel like theres a diff way to simplify this
bcs then we also have to draw the graph
neon kernel
Indeed when you look at the solution provided by photomath it doesn't look easier tho
lemme try with my uni programms
meager root
can I show you something?
neon kernel
sure
meager root
in one of my recent exams, we had this problem
and yeah I used the identities and got it correct but
the teacher did it in such a simpler way look
sorry it's in my language but yeah
that second row..
I haven't seen that before
neon kernel
is that the ln(cas(a) + i*sin(a))³ on line 2?
neon kernel
you undergrad?
meager root
yes
neon kernel
this is what I get if i put it in the program
meager root
damn really
neon kernel
its Im as in imaginary I think?
hmmh i have no idea haven't seen it
neon kernel
neon kernel
yea indeed
meager root
like in comparison to the previous one
um okay thanks anyways I'll try figuring it out or asking someone from my school maybe.
neon kernel
Yea i have no idea how you would approach ir
neon kernel
yea gl
sterile bobcat
Ok guys, My book wants me to find the area (IN SQ IN) of a room that is 12ft 9in by 8ft 8in. So I converted it to 153in x 104in, right? That comes out to 15,912. My book says its wrong, says its. 12,168 sq in. Am I crazy?
vocal sleetBOT
Ok guys, My book wants me to find the area (IN SQ IN) of a room that is 12ft 9in...
shadow raft
!original
vocal sleetBOT
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.
shadow raft
do you know how to go from feet to inches?
sterile bobcat
Should be that number times 12
On this problem... 12 x 12 + 9
8 x 12 + 8
right?
That came out to 153in & 104in
shadow raft
did you get 2a correct?
sterile bobcat
yes
shadow raft
which is?
dark kiln
book's wrong yeah
sterile bobcat
When I do the example problem the exact same way I get it right just like the example, but when I do this one its off.
shadow raft
it just seems like the book is wrong
the book isnt always correct
most of the time it is
but even authors of a textbook make mistakes
sterile bobcat
Ok, thanks. I just thought I was missing something.
GREATLY appreciate your help guys!
.close
vast shale
vocal sleetBOT
vast shale
I assume the answers are wrong, or maybe there is some flaw with my understanding
but apparently, period is $\frac{\pi}{3}$
twin meteorBOT
mj
vast shale
doesn't make sense to me
because the k-value here is 3
and period=2pi/k, in this case 3. So wouldn't the period be 2pi/3?
median crane
I’ve no idea what the 2pi/k is supposed to be but normal tan has a period of pi so if you stretch horizontally it by a factor 3 (or 1/3 rather), you’d have totimes the period by 1/3.
vast shale
it comes from the formula k=2pi/period
sage dew
it comes from the formula k=2pi/period
this is for sin and cos
period of tangent is pi as pure said
vast shale
this is for sin and cos
how come?
shadow raft
,w graph sin(x),cos(x),tan(x)
twin meteorBOT
vast shale
period is shorter
okahy
so base period of
sin and cos
is 2pi
for tan it's pi
so would you use period=pi/k(3)
alright thanks
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
marsh coyote
Let $P(n)$ be the statement:
For $n\ge 1$ , if $|A|=n$, then the size of the set of all bijective functions from $A$ to itself is $n!$.
$P(1)$ is true since the only function from a singleton set to itself is the identity which is bijective.
Suppose inductively, we have shown that $P(n)$ is true.
If $|A|=n+1$, $|A|$ is non-empty and thus there exists $a \in A$. Then we have that $|A/{a}|=n$ and so by our inductive hypothesis, the size of $K$(set of all bijective functions from $A/{a}$ to itself) is $n!$. For $b \in A$, let $A_b$ be the set of functions from $A$ to itself where $f \in A_b$ iff $f(b)=a$ and $f(a)=\rho(b)$ and $f(c)=\rho(c)$ if $c\not=a,b$ where $\rho \in K$.
Next, we show that:
- The elements of $A_b$ are bijective.(Skip)
- $|A_b|=n!$ (skip)
- $A_b\cap A_{b'}$ is empty if $b\not=b'$. (Skip)
Then since $|A|=n+1$, we have $|\bigcap_{b \in A}A_b|=(n+1)!$.
marsh coyote
Im having some trouble with the final step
of showing that $\bigcap_{b \in A}A_b$ contains every bijective function from $A$ to $A$
hybrid flicker
hint : for f a bijective function on A, let b = f^-1(a)
marsh coyote
oh okay i will thikn about it and oops, in my working it should be unions instead of intersections*
hybrid flicker
no problem, union is \bigcup
marsh coyote
hint : for f a bijective function on A, let b = f^-1(a)
Ok i think i got 1 way. If $f$ is bijective on $A$. we can have $g$ from $A{a}$ to itself defined by $g(c)=f(c)$ if $c\not=a,b$ and $g(b)=f(a)$. We have to show that this $g$ is bijective, then $g\in K$.
Finally, we have that $f(b)=a$. $f(a)=g(b)$, $f(c)=g(c)$ for $c\not=a,b$ and thus $f \in A_b$
twin meteorBOT
Iusgnol
marsh coyote
**Iusgnol**
oh and also I need to show that f(a) is indeed in A/{a} but this should be easy enough since f is bijective and f(b)=a
vocal sleetBOT
Available help channel!
halcyon turtle
vocal sleetBOT
halcyon turtle
<@&286206848099549185>
vocal sleetBOT
@halcyon turtle Has your question been resolved?
halcyon turtle
<@&286206848099549185>
unreal oriole
What is your question?
A 3x3 matrix times a 3x1 matrix gives you a 3x1 matrix. The multiplication was performed and the values in each row were added together. I'm not sure why it was written like that though
vocal sleetBOT
@halcyon turtle Has your question been resolved?
halcyon turtle
A 3x3 matrix times a 3x1 matrix gives you a 3x1 matrix. The multiplication was p...
but here they added the row elements..is that even an operation?
or is it a formula to find x y z?
unreal oriole
but here they added the row elements..is that even an operation?
I'm not sure why they wrote it like that, but for example, the first value in the 3x1 matrix should be (9x18)+(-38x13)+(37x20). It should just be standard matrix multiplication
the variables are found using the equation Ax = b
Ax = b
A^-1(Ax) = A^-1b
Ix = A^-1b
x = A^-1 * b
unreal oriole
The same thing that I said above still applies. It's just that the inverse of A is found using cramer's rule, or X = (Adj(A) / |A|) * D, rather than standard matrix inversion
halcyon turtle
i do understand that..my doubt is in the last 2 step..can row elements just be added like that?
unreal oriole
I'm not sure why they wrote it like that, but for example, the first value in th...
^
there should be addition symbols between the values but I'm guessing they were left out for some reason. let me right out the multiplication rq
strange crater
unreal oriole
^^ Just a mistake in the book. Here’s my work though
vocal sleetBOT
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vast shale
vocal sleetBOT
vast shale
I'm trying to determine the period of this function
f = 1/T, where T = period, and f = frequency
use this
oh sorry didn't even explain what I tried to do lol
okay, so I basically realized it was 3 rotations / 5 minutes
so I plugged it in to find time for 1 rotation
giving me 0.6 minutes / rotation
okay
1 rotation being 1 period
not sure if that's correct though
its more common to express in seconds
but otherwise you agree period in this instance is 0.6 minutes?
because that's where I'm getting confused
T will be 5/3
not 3/5
vast shale
T will be 5/3
how come?
I understand there's different ways to say it
in fractions
5 minutes / 3 rotations
or 3 rotations / 5 minutes
actually it's making sense
because one is dependant on the other
the more time goes by, the more rotations go by
rotations doesn't subjectively make time go by
f = 1/t
f = 1 /3/5
f = 1 * 5/3
f = 5/3
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worn bobcat
How do you guys think about why (AB)^T = (B^T)(A^T) for matrices A and B of appropriate dimensions?
vocal sleetBOT
How do you guys think about why `(AB)^T = (B^T)(A^T)` for matrices `A` and `B` o...
scenic ravine
this may help
this even more so https://math.stackexchange.com/questions/1279861/why-intuitively-is-the-order-reversed-when-taking-the-transpose-of-the-product/1282499#1282499
vast shale
How do you guys think about why `(AB)^T = (B^T)(A^T)` for matrices `A` and `B` o...
basic idea is to think of [
\trans{(AB){ij}} = (AB){ji}
]
twin meteorBOT
vast shale
and think what that really means
vocal sleetBOT
@worn bobcat Has your question been resolved?
worn bobcat
The way I see matrix multiplication at the moment is that you have the definition which one can visualize however one chooses, and then from that definition you can interpret AB as either
- a matrix where the columns of B dictate what kind of linear combinations of the columns of A to take as the columns of the resulting matrix
- a matrix where the rows of A dictate what kind of linear combinations of the rows of B to take as the rows of the resulting matrix
And the fact that those two interpretations arise from the same general definition is essentially what (AB)^T=(B^T)(A^T) is trying to capture.
So if you think about just AB as taking linear combinations of the columns of A as dictated by B, then if you flip and transpose the matrices to get (B^T A^T) then you can change your viewpoint to think about taking linear combinations of the rows of A^T, as dictated by B^T. But these rows were previously the columns of A, and likewise the columns of B are now the rows of B^T. Therefore you're taking the same linear combinations of the same vectors, except this time they're row vectors, and thus (B^T A^T) = (AB)^T
I don't know if that makes any sense or if you think that's a good way to think about this.
I used to think of AB as only having the interpretation of taking linear combinations of the columns of A (disregarding the row interpretation), but the LU decomposition made me realize that the row interpretation can be just as valuable as the column interpretation, because (at least the way they do it in Gilbert Strang's lectures) you use lower triangular matrices to denote row operations, which basically can be interpreted as saying "take this amount of the first row, this amount of the second row, etc."
And combining those two interpretations gives rise to the famous (AB)^T = (B^T)(A^T) result as far as I can tell. Or at least it seems to make sense in my head.
Thoughts?
vocal sleetBOT
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urban laurel
When f=o(g) or f=O(g) are we comparing the growth rate or the order of magnitude of the functions ?
vocal sleetBOT
When f=o(g) or f=O(g) are we comparing the growth rate or the order of magnitude...
paper depot
growth rate if you're looking at x->infty
order of magnitude if you're looking at x->c for some finite point c
well i guess not order of magnitude in the power of 10 sense
urban laurel
growth rate if you're looking at x->infty
Wouldn't comparing the order of magnitude give the same result ?
paper depot
ok wait
what is the difference between order of magnitude and growth rate according to you
urban laurel
Growth rate means how fast the value of f(x) increasing or decreasing as the value of x increases.
Order of magnitude is how big the value of f(x) is ? (i'm not sure if this is the right definiton)
vocal sleetBOT
@urban laurel Has your question been resolved?
urban laurel
.close
urban laurel
What's the difference between growth rate and order of magnitude of a function ??
vocal sleetBOT
What's the difference between growth rate and order of magnitude of a function ?...
vast shale
growth shows how fast it is going
order of magnitude is its size relative to a power of 10
growth rate emphasizes the relative speed of function growth, while the order of magnitude focuses on a coarse measure of size relative to powers of 10
urban laurel
order of magnitude is its size relative to a power of 10
What does that mean ?
Size meaning the value of f(x) ?
vast shale
yes
urban laurel
When f=o(g) or f=O(g) are we comparing the growth rate or the order of magnitude of the functions ?
vast shale
if a function's order of magnitude is 10³ it means the function's value is on the order of thousands
vast shale
When f=o(g) or f=O(g) are we comparing the growth rate or the order of magnitude...
growth
urban laurel
if a function's order of magnitude is 10³ it means the function's value is on th...
It means the value of f(x) is between 1000 and 9999 ?
vast shale
correct.
vast shale
not exactly
similar
but not exactly
if f and g have the same order of magnitude, it means that the most significant terms in their expressions are of the same power of 10. This doesn't necessarily mean that they grow at the same rate, but they share a similar scale
if any other questions ping me
related to this
urban laurel
if f and g have the same order of magnitude, it means that the most significant ...
But if f is dominated by g
We can say that g has a bigger order of magnitude or that g grows faster than f
They mean the same thing to me...
vast shale
But if f is dominated by g
We can say that g has a bigger order of magnitude or...
it implies one of them
either magnitude either growth
urban laurel
Why not both?
vast shale
its possible for both to happen
but it might happen that it grows faster but it stays in the same magnitude as the other function
and vice versa
to grow faster and not be in the same magnitude
urban laurel
but it might happen that it grows faster but it stays in the same magnitude as t...
How is that possible ?
vast shale
do we consider strict monotone functions
Two functions can have the same order of magnitude if their leading terms are of the same power of 10, even if one grows faster than the other
do you want an example?
vocal sleetBOT
@urban laurel Has your question been resolved?
urban laurel
do you want an example?
Yes please
urban laurel
x^2 obv
urban laurel
f(x) = 10x and g(x) = x²
But as x approaches inf we can't determine the order of magnitude
vast shale
both terms have the same order of magnitude which is x^1 or 10x
take another example
f(x) = 2x^2 + 5x+1
g(x) = 3x^3 +2x^2 + 4x
urban laurel
For x=100 : f(x)=10³ and g(x)=10⁴
urban laurel
both terms have the same order of magnitude which is x^1 or 10x
They don't have the same order of magnitude ?
vast shale
they have initially , as it approaches infinity it changes
that's what you seems confused about
f(x) = 100x
g(x)= x^2
x = 100
urban laurel
Why did u say that they have the same orderof magnitude then?
vast shale
Why did u say that they have the same orderof magnitude then?
they initially have
it depends on the function totally
urban laurel
f(100) = 10000
Try 100k
vast shale
there are counterexamples
yes
there are counterexamples
but for some point they still have the same magnitude
vast shale
but for some point they still have the same magnitude
INITIALLY
urban laurel
Yeah but in all the examples that u provided comparing the order of magnitude or growth rate give the same result which is that one function grows faster than the other
vast shale
i stated that "but it might happen that it grows faster but it stays in the same magnitude as the other function"
right?
urban laurel
Yes
twin meteorBOT
snow
vast shale
$g(x) = \sqrt(x) + ln(x)$
twin meteorBOT
snow
urban laurel
The order of magnitude differs for different values of x
vast shale
yes i know
but
what is it now
for both functions
there is a common magnitude without any values for x
urban laurel
there is a common magnitude without any values for x
Which is what ?
urban laurel
well idk it srry 😐
vast shale
the order of magnitude right now is $\sqrt(x)$
twin meteorBOT
snow
urban laurel
So order of magnitude is a function ??
vast shale
its a concept
not a function
when we note O(x^2)
it means the growth of f(x) is roughly propotional to the square of the input x for large values of x
same as sqrt(x)
urban laurel
I'm not following srry
vast shale
**snow**
here in this two functions for small values f(x) behaves like g(x) due to the dominance of square root, if x grows larger g(x) grows faster because of ln(x)
urban laurel
its a concept
Wdym it's a concept?
vast shale
what do you consider order of magnitude as being?
urban laurel
Can u give the definition of the order of magnitude of a function?
urban laurel
what do you consider order of magnitude as being?
idk
vast shale
Can u give the definition of the order of magnitude of a function?
The order of magnitude of a function is a way of characterizing its growth or size, particularly as the input approaches certain values (often infinity). It provides a simplified description of the function's behavior by focusing on the most significant term or dominant factor that influences its growth
urban laurel
How is it calculated?
vast shale
How is it calculated?
consider this function f(x) =3x^3 + 2x^2 +5x+1
what is the dominant term of the function
urban laurel
3x³
vast shale
correct
the focus will be on that term
now you express the magnitude on that term
on the dominant term
urban laurel
3x³ is the order of magnitude of f ?
vast shale
f(x) = O(x^3)
vast shale
3x³ is the order of magnitude of f ?
yes
the cubic term is the most significant for large values of x
so we will express magnitude as f(x) = O(x^3)
urban laurel
Ok
vast shale
do you understand?
urban laurel
$g(x) = \sqrt(x) + ln(x)$
So the order of magnitude of g(x) is ln(x) ?
vast shale
i have not said which one grows faster, i just asked you to tell me what magnitude each of them have, at the begging
both function have the same magnitude
your statement is correct, g(x) grows faster than f(x)
urban laurel
They don't
vast shale
why not?
urban laurel
What's the order of magnitude of g(x)
urban laurel
Well u said that they have the same order of magnitude not me
How can u say that if udk the order of magnitude of g(x)
urban laurel
the order of magnitude right now is $\sqrt(x)$
You said that the order of magnitude of both functions is sqrt(x)
vast shale
consider this function f(x) =3x^3 + 2x^2 +5x+1
based on this example
vast shale
You said that the order of magnitude of **both** functions is sqrt(x)
yes
is my statement true or false?
urban laurel
False cause based on the definition u provided if sqrt(x) is the order of magnitude of g(x) then g(x)=O(sqrt(x))
vast shale
$g(x) = \sqrt{x} + ln{x}$
twin meteorBOT
snow
vast shale
saying g(x) = O(g(x)) is not true
urban laurel
urban laurel
g(x)=O(sqrt(x)) is also wrong
vast shale
why?
urban laurel
vast shale
vast shale
wrong
gritty sage
sqrt(x) + ln(x) < 2sqrt(x) after x = 10, which is O(sqrt(x)), so it's O(sqrt(x)).
vast shale
hard atlas
big O notation does not care about small numbers. zoom out
gritty sage
urban laurel
sqrt(x) + ln(x) < 2sqrt(x) after x = 10, which is O(sqrt(x)), so it's O(sqrt(x))...
how'd u know that though
gritty sage
Well, ln(x) < sqrt(x) after x = 10.
So, sqrt(x) + ln(x) < sqrt(x) + sqrt(x) = 2 sqrt(x) after x = 10.
I just added sqrt(x) to both sides.
Then, constant terms are removed, so O(sqrt(x)).
gritty sage
Well, for big-O, it only needs to be always greater after any x value you choose.
So, it might be greater than the other after x = 0 too, so you can use that if you want.
I just picked x = 10 because that also works.
urban laurel
$f(x)=\sqrt{x} \ g(x)= \sqrt{x} + \ln (x) \ f=O(g)$ and $g=O(f)$
twin meteorBOT
Adam Chebil
gritty sage
Yes. O is really a set, and O(f) and O(g) are the same set.
You can do that by multiplying either f or g by 2.
urban laurel
so f and g are equivalent ?
gritty sage
That'll show that g is in O(2f) and f is in O(2g).
No, just their big O set.
You can even say that g is in big Theta(f) or vice versa.
vocal sleetBOT
gritty sage
You're welcome.
vocal sleetBOT
Available help channel!
vocal sleetBOT
vocal sleetBOT
@upbeat maple Has your question been resolved?
sudden compass
sub x=1 and x=-1 and then see if it converges
vocal sleetBOT
@upbeat maple Has your question been resolved?
limber wedge
What did you try?
I can see two subsequences of the k-th sum forming that converges at two different points. So you need to divide the series into two parts
For the evens we have $\sum_
{n=0}^\infty x^n$ and for the odds we have $\sum_{n=0}^\infty (x/3)^n$
sudden compass
For the evens we have $\sum_
{n=0}^\infty x^n$ and for the odds we have $\sum_{...
(use latex)
twin meteorBOT
Ericsson
limber wedge
Try to observe the question maybe, problem solving comes with practice
Okay tell me what will the numerator be when i plug in an even number
You forgot the x
You don't need to apply the geometric formula since we are interested in the interval on which the sum will converge
For the evens, what should be the interval for x so that the sum will converge?
Why not the evens?
twin meteorBOT
Ericsson
$\sum_{r=0}^\infty r^n=\dfrac{1}{1-r}$ given that $\abs(r)<=1$.
```Compilation error:```! Missing delimiter (. inserted).
<to be read again>
r
l.53 \end{document}
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.```
sudden compass
ratio test he mean ig
limber wedge
$\sum_{r=0}^\infty r^n=\dfrac{1}{1-r}$ given that $|r|<1$.
sudden compass
yeah thats a GP property
limber wedge
What's the command for an absolute?
sudden compass
|| i think
limber wedge
No abs?
twin meteorBOT
Ericsson
limber wedge
I always thought it was \abs
Yes
Take the intersection of both of these intervals and that's your answer
Yeah
Check the options again, option D have the point x=1 included
Bruh
What's the sum of infinite 1's?
If x=1, you will get infinity thanks to the even series
Close the channel if you are done with the question
tulip sleet
Calculating the volume of a solid revolution. What am I doing wrong? Im using the formula marked in green (disk method)
vocal sleetBOT
Calculating the volume of a solid revolution. What am I doing wrong? Im using th...
visual oracle
what’s wrong
tulip sleet
my final answer
vague arch
what is shared looks correct so far
visual oracle
^
tulip sleet
what is shared looks correct so far
The final ans should beis $ 144\pi - 2\pi \int_2^{0} (6-x4+x^2)^2 dx $
$ 144\pi - 2\pi \int_2^{0} (6-x4+x^2)^2 dx $
Hello latex bot?
vague arch
$ 144pi - 2\pi \int_2^{0} (6-x4+x^2)^2 dx $
tulip sleet
broke
but yeah
thats what it should be
Of coruse, i changed the limits of integration, which is allowed
but the final integrated answer is not the same
this should be the final answer
vague arch
what is the solid you are trying to compute?
tulip sleet
idk the name of it
vague arch
0<y<4-x^2 or 4-x^2<y<6 about the line y=6 ?
your 2:33 and 2:29 posts correspond to each case respectively
tulip sleet
0<y<4-x^2 or 4-x^2<y<6 about the line y=6 ?
🧐
I dont get it
Never really thought of that
Okay, i got it working with the shell method. Ill come back to the disk method later as its required to use both methods
vocal sleetBOT
@tulip sleet Has your question been resolved?
brittle ingot
vocal sleetBOT
brittle ingot
Does anyone know how to do this?
vocal sleetBOT
@brittle ingot Has your question been resolved?
winter bison
Hello! I have no idea how to validate these statements only with a graphic, any idea?
vocal sleetBOT
Hello! I have no idea how to validate these statements only with a graphic, any ...
winter bison
Thanks!
I think that the first objective is know how to get the first derivate and i don't know : (
<@&286206848099549185>
vocal sleetBOT
@winter bison Has your question been resolved?
winter bison
.close
vast shale
if i wrote it the other way around on a test, is it still correct?
vast shale
that is x goes from + to -
vast shale
so, not right?
vast shale
rip
paper depot
when writing an interval the left endpoint goes on the left
vocal sleetBOT
@vast shale Has your question been resolved?
formal bronze
omitting the abs(x) will decrease of sqrt(abs(xy)) and that will make it superior but omitting the sqrt with the abs (x) will not make it superior to (sqrt(abs(x)) + sqrt(abs(y)))^2 why they did it anyway in the correction this is confusing me
vocal sleetBOT
omitting the abs(x) will decrease of sqrt(abs(xy)) and that will make it superio...
median crane
Sorry, I don't really understand your confusion. Could you point out which step/line you're referring to?
formal bronze
Sorry, I don't really understand your confusion. Could you point out which step/...
okay first i have to prove this
median crane
Okay.
median crane
Yes, if abs(y)<=abs(y) then the inequality is trivially true.
formal bronze
Yes, if abs(y)<=abs(y) then the inequality is trivially true.
yeah the problem is with the second one
then
now we try to make (sqrt(x)-sqrt(y))^2 <= abs(x-y)
but the method they used in the correction semmed to me incorrect
median crane
\begin{align}
\p(\s{\abs{x}} - \s{\abs{y}})^2 &= \abs{x} + \abs{y} - 2\s{\abs{xy}}\\
&<=\abs{x}+\abs{y} -2\abs{y}\\
&<=\abs{x}-\abs{y}\\
&<=\abs{x-y}
\end{align}
which equation do you think is incorrect?
formal bronze
which equation do you think is incorrect?
but why eliminating the sqrt
twin meteorBOT
snow
median crane
so from (1) to (2)?
formal bronze
so from (1) to (2)?
yeah
median crane
You agree that in this case abs(x)>=abs(y)
formal bronze
You agree that in this case abs(x)>=abs(y)
yeah
median crane
then you should also agree that (\s {|x|} >= \s{|y|})
twin meteorBOT
snow
formal bronze
yes
median crane
so (2\s{\abs{xy}} >= 2\s{\abs{y^2}} = 2|y|)
twin meteorBOT
snow
median crane
because |x|>=|y|
formal bronze
because |x|>=|y|
yeah yeah
median crane
(2\s{\abs{xy}} = 2\77{\s{|x|}}\s{|y|} >=2\77{\s{|y|}}\s{|y|} = 2\s{\abs{y^2}} = 2|y|)
twin meteorBOT
snow
median crane
No problem.
vocal sleetBOT
@formal bronze Has your question been resolved?
oblique gazelle
The question is: How can 8 rooks be placed on a chess board so that they cover all the squares
vocal sleetBOT
The question is: How can 8 rooks be placed on a chess board so that they cover a...
oblique gazelle
My answer is: 8^8 - 8!
The correct answer is: 2*(8^8) - 8!
Any ideas why? The rooks are all identical (I googled a bit and for some reason in some problems people consider colored rooks)
tidal dust
or at the A column or the h coloumn
tidal dust
I need to find the number of all the possible ways
ohhh , that would be 8^8 *4
oblique gazelle
The question is: How can 8 rooks be placed on a chess board so that they cover a...
.
all the possible ways to arrange them so they cover all the ways
8^8 is just all the possible ways
tidal dust
8^8 is just all the possible ways
hmmm , you're right its 8^8
oblique gazelle
I subtract 8! from that because thats the number of ways to arrange them so that they dont hit each other
so my answer is 8^8 - 8!
but it should be 2*(8^8) - 8!
oblique gazelle
8^8 is just all the possible ways
@wide sundial I explained there
wide sundial
yeah and that’s for all rows
oblique gazelle
cuz 8^8 is a very large number
well its in fact all the possible ways to arrange the rooks randomly
oblique gazelle
you can do one for all columns as well, no?
Doesnt 8^8 account for all that?
wide sundial
then you subtract instances where you have rooks in every column and every row
oblique gazelle
like 8^8 already includes the columns and the rows
tidal dust
i think its only 8!
wide sundial
8^8 only includes the rows individually or the columns individually imo
this is just inclusion exclusion
not both
oblique gazelle
8^8 only includes the rows individually or the columns individually imo
I understand what you mean here, but if I add another 8^8 am I not adding the same thing?
Because it doesnt matter, the placement is still the same
I calculate the same position twice if I add another 8^8
magic wasp
8^8 to cover all rows, 8^8 to cover all columns, - 8! to not count the intersection twice
oblique gazelle
8^8 to cover all rows, 8^8 to cover all columns, - 8! to not count the intersect...
yeah and I dont understand the rows and columns part
wide sundial
yeah there are some double counted instances which is why you subtract the 8! (so that every rook has a row and every rook has a column)
oblique gazelle
I understand what you mean here, but if I add another 8^8 am I not adding the sa...
this is my problem
magic wasp
Place a rook on each row, you have 8^8 ways to do that, and the board is always fully covered because all rows are
Now do the same with the columns, you won't get the same configurations
oblique gazelle
Place a rook on each row, you have 8^8 ways to do that, and the board is always ...
its not always covered, if you put the rooks in a diagonal the rooks dont attack each other
wide sundial
it still covers the entire board
magic wasp
Some of the configurations are the same, namely those where all the rooks are isolated in both columns and rows
That's the 8! part
oblique gazelle
it still covers the entire board
hmmm
magic wasp
its not always covered, if you put the rooks in a diagonal the rooks dont attack...
Why are you talking about rooks attacking each other?
wide sundial
rooks attack all vertical spaces above and below it
oblique gazelle
Why are you talking about rooks attacking each other?
because I need to cover all of the board
wide sundial
so it can all be diagonally configured and still cover the board
magic wasp
because I need to cover all of the board
That has nothing to do with rooks attacking each other
All you need is to cover either all rows or all columns, or both
oblique gazelle
That has nothing to do with rooks attacking each other
hmm I guess ur right. I just wanted them attack each other so I covered all the board xd
wide sundial
if the rooks are in the same row/column then they attack each other and they might not cover the board but otherwise
oblique gazelle
All you need is to cover either all rows or all columns, or both
I see
wide sundial
it does
oblique gazelle
okay so can we go over the row and column part again? I did not understand a single thing
8^8 is for all the rows
magic wasp
Place a rook on each row, you have 8^8 ways to do that, and the board is always ...
This?
oblique gazelle
Place a rook on each row, you have 8^8 ways to do that, and the board is always ...
I understand this
vast shale
guys give me a question
oblique gazelle
I dont get what comes after it tho
vast shale
I'll solve
wide sundial
I'll solve
!occupied
wide sundial
lol
vast shale
I need help actually
magic wasp
You can do this:
x _
x _
x _
_ x
_ x
x _
_ x
_ x
so all the possible ways to arrange rooks there is 2^2
magic wasp
4 ways to cover all the rows
oblique gazelle
yep
wide sundial
and this channel is occupied so #❓how-to-get-help
oblique gazelle
4 ways to cover all the rows
which also accounts for covering the coulmns so we dont need to add nother 2^2
vast shale
im really bad at math I only know 1st grade maths
magic wasp
You can also do this:
x x
_ _
x _
_ x
_ x
x _
_ _
x x
4 ways to cover all the columns
magic wasp
which also accounts for covering the coulmns so we dont need to add nother 2^2
It doesn't, as you can see above
oblique gazelle
slow down
vast shale
guys I'm really bad at math
oblique gazelle
wait
vast shale
help me please
oblique gazelle
You can also do this:
```
x x
_ _
x _
_ x
_ x
x _
_ _
x x
```
the middle two, how are they relevant to the columns
magic wasp
They still cover all the columns...
oblique gazelle
I see that they are repeated both for columns and rows
wide sundial
oblique gazelle
I dont get how you get that
magic wasp
Yeah, and that's the 8! part
oblique gazelle
how do you get this arrangement tho
magic wasp
What arrangement
oblique gazelle
You can also do this:
```
x x
_ _
x _
_ x
_ x
x _
_ _
x x
```
this
magic wasp
You can do this:
```
x _
x _
x _
_ x
_ x
x _
_ x
_ x
```
... same as this but sideways (rotated 90º)
vast shale
<@&286206848099549185> please help me
oblique gazelle
... same as this but sideways (rotated 90º)
oh its like changing the axis
Yeah I see
vast shale
I'm really bad at math
I just know 1st grade math
how can I become Einstein in 1 day?
oblique gazelle
so the middle parts are repeated twice
magic wasp
Yeah
magic wasp
Yeah
wide sundial
magic wasp
No, it's 2!
oblique gazelle
but we subtract 8! there
wide sundial
but its not 4!
it’s 2x2 so 2!
vast shale
I thought someone will help me
vast shale
it’s 2x2 so 2!
2×2is4
oblique gazelle
8x8 so 8! i mean you could extrapolate the trend
oh makes sense
magic wasp
You pick one column for the first row, another column for the second, and so on
minor plaza
How do I get good?
practise ? i guess
oblique gazelle
yeah I think I got it
vast shale
like I legit know only 1st grade maths
vast shale
practise ? i guess
from where?
magic wasp
So 2*1 in 2x2, or 8*7*6*... in 8x8
oblique gazelle
So `2*1` in 2x2, or `8*7*6*...` in 8x8
yep
vast shale
2nd grade math
but I'll waste alot of time
oblique gazelle
would have been easier if some people didnt turn into a public chat tho @vast shale @minor plaza @vast shale
vast shale
would have been easier if some people didnt turn into a public chat tho <@456226...
this is public chat?
dark kiln
you're acting like it is
oblique gazelle
Thanks guys, all the best! @magic wasp @wide sundial
wide sundial
lol
vast shale
would have been easier if some people didnt turn into a public chat tho <@456226...
?
cause there are many helps
minor plaza
would have been easier if some people didnt turn into a public chat tho <@456226...
i just came
vast shale
I'm just asking for help
dark kiln
it's an exclusive channel, belongs to fen
vast shale
Why are you getting so angry
dark kiln
you can't
vocal sleetBOT
vast shale
even I'm asking questions
minor plaza
straight up trolling this is
vast shale
2-2 gives you zero
vocal sleetBOT
Available help channel!
vast shale
2-2 gives you zero
Yess
vast shale
I'm not trolling
minor plaza
how is asking 2-2 a genuine question
mild flower
i timed out syon, giving everyone else the benefit of the doubt because i am tired
.close
winter bison
Hello! I have no idea how to validate these statements only with a graphic, any idea?
vocal sleetBOT
Hello! I have no idea how to validate these statements only with a graphic, any ...
winter bison
Thanks!
I think that the first objective is know how to get the first derivate and i don't know : (
lone linden
Thanks!
I think that the first objective is know how to get the first derivate a...
For the first derivative, you just rlly need to know that
If f is decreasing -> f' < 0
If f is Increasing -> f' > 0
winter bison
For the first derivative, you just rlly need to know that
If f is decreasing -...
um....
winter bison
For the first derivative, you just rlly need to know that
If f is decreasing -...
following this, the first derivate of f(x) is f'(x)>0
lone linden
Yes, the function is always increasing, so f' > 0 for all x
so what can we say about this with regards to the options
winter bison
so what can we say about this with regards to the options
the second option is true
but it can be more than only one
lone linden
Do you know how to the second derivative relates to the original function?
winter bison
no
lone linden
f is concave up -> f'' > 0
f is conacve down -> f'' < 0
winter bison
I mean, i know how to calculate it, but i'm not sure abut the meaning in the function
lone linden
f is concave up -> f'' > 0
f is conacve down -> f'' < 0
If you mix up which shape is which, an easy way to figure it out is to consider x^2 and -x^2 (sketch the graphs and take the second derivatives to determine concavity)
winter bison
following this, the true statement the third one
So just to confirm, the TRUE statements are the first and third? @lone linden
winter bison
Hello! I have no idea how to validate these statements only with a graphic, any ...
about this
oh
lone linden
the second option is true
You said second here
idk if you just transcribed incorrectly
it should be (2) and (3)
winter bison
idk if you just transcribed incorrectly
ops sorry yes yes
thanks! @lone linden
.close
slender depot
need help with this functional equation g(x + y) = (g(x) + g(y))/(1 - g(x)g(y), g is defined on (-1, 1) and is continuous. I have found that g = 0 and g = tan(x) are solutions and I suspect they are the only solutions.
vocal sleetBOT
need help with this functional equation g(x + y) = (g(x) + g(y))/(1 - g(x)g(y), ...
lone linden
there should be a family of solutions with the tan
but I think this is what you're looking for
vocal sleetBOT
twin meteorBOT
paper depot
ok
and you think that for some reason these formulas could possibly NOT apply to some specific distribution, yes?
twin meteorBOT
paper depot
these formulas are universal
they apply to everything
discrete and continuous respectively
the other formulas you see further down the line are consequences of these
when you replace the general mass/density functions with the specific functions for particular classes of distributions
not sure if "make it easier" is a good angle for it
oblique raven
What is
6/2(1+2)
vocal sleetBOT
What is
6/2(1+2)
vast shale
what do you think it is?
depends on if you are taking that as [
\f62(1+2) \tqq{or} \f6{2(1+2)}
]
twin meteorBOT
oblique raven
9
vast shale
What is
6/2(1+2)
shit form
oblique raven
hey I didn’t write it 🤷♀️
paper depot
@outer warren drop that 6/2(1+2) fact sheet
oblique raven
Thx
vast shale
6/2(1+2) is 1
its like if ur told 6/2x ur not gonna do
6x/2
thats js dumb
For you
tidal dock
there's no correct or incorrect way
tidal dock
there's no correct or incorrect way
the syntax is ambiguous, period
vast shale
theres a reason why the divide symbol isnt used beyond grade 4 maths
you can definitely still use the division symbol appropriately given that you use sufficient parentheses
it can still lead to confusion so its best to steer clear
tidal dock
if you use sufficient parentheses you can completely skip PEMDAS lol
functional programming ftw
vast shale
ok
livid horizon
It's never ok
vast shale
do u need help
elfin surge
if no, type .close
vocal sleetBOT
@regal temple Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
vocal sleetBOT