For the second one, ||A||, I do 2*A[1][1]\*B[1][1] + A[1][2]*B[1][2] + A[2][1]*B[2][1] + A[2][2]*B[2][2]
#help-49
fringe onyx
You missed the 2 on the last term
bitter wharf
A[2][2]*B[2][2]?
bitter wharf
Yeah
bitter wharf
For the second one, \|\|A\|\|, I do 2*A[1][1]\\*B[1][1] + A[1][2]\*B[1][2] + A[2...
I accidentally but B for the second one
But on the scrap paper I didn't make that mistake
fringe onyx
It's ok no worries
bitter wharf
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fringe onyx
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lusty cave
Wouldn't this be true only sometimes?
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Wouldn't this be true only sometimes?
lusty cave
my logic is results could be -1, 1
So it cant always be -1 cause it could be 1
it cant never be -1 cause -1 is viable
but it could sometimes be -1, and other times be 1
Or would 1 maybe not be applicable?
hearty rune
you are correct
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lusty cave
Can someone check this answer for me?
dire thorn
its right
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twilit field
could I have this verified please
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could I have this verified please
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trey
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carmine sigil
@split gust yes, imagine if included any a^n with n>=d. Then you could use the polynomial to reduce your a^n terms into smaller terms.
Like if you had x^8 - x^2 + 1 as your polynomial, then you know, because a^8 - a^2 + 1 = 0, that any time you have a^8 you can replace it with a^2 - 1
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ionic thicket
ideally i would do A^nB times (x,2x) = (x,2x)?
But that means 2^nx = x? or did i do smth wrong
ancient ridge
Where are you getting 2^nx = x from
ionic thicket
Where are you getting 2^nx = x from
ancient ridge
Then yes, you did something wrong.
ionic thicket
Then yes, you did something wrong.
what did i do wrong?
ancient ridge
I don't know, for one, that is not the product of A^nB[x,2x]^T and even if it were, you didn't show any work to indicate how you would go from that to 2^nx = x.
ionic thicket
I don't know, for one, that is not the product of A^nB[x,2x]^T and even if it we...
why is it not the product of A^nB[x,2x]^T? if the 2nd column is 0 0
<@&286206848099549185>
hybrid crane
ideally i would do A^nB times (x,2x) = (x,2x)?
When the question says y = 2x is invariant under the transformation, I believe it means all points on the line are mapped to other points still in the line
The equation you have would be the case of a line of invariant points rather than an invariant line
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tough spindle
You want to ensure that the new vector $\binom{\left(2^n \right)x}{\left(2^n + 32 \right)x}$ lies on the line $y = 2x$, or in other words $\left(2^n + 32 \right)x = 2\left(2^n \right)x$ for every $x \in \mathbb{R}$. Just solve for $n$ now.
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PowerUp
ionic thicket
You want to ensure that the new vector $\binom{\left(2^n \right)x}{\left(2^n + 3...
ohh
ionic thicket
You want to ensure that the new vector $\binom{\left(2^n \right)x}{\left(2^n + 3...
so basically i compare the top (x) and bottom (y) components of the matrix?
is that the essence of "invariant line"
tough spindle
"Invariance" here means that, given any point on the given line, the image must also lie on the line. So in this case, $A^n B \binom{1}{2}$ must equal $\binom{x}{y}$ satisfying $y = 2x$ because $\binom{1}{2}$ lies on the line as well.
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tough spindle
So your goal must be to find the value of $n$ that makes this possible.
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tough spindle
And it should not only be true for $\binom{1}{2}$, but also for ANY point on the line.
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ionic thicket
"Invariance" here means that, given any point on the given line, the image must ...
ohh i see
thank you
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bleak pier
Numbers between 200-800 which are not divisible by 5 and7
midnight plankBOT
Numbers between 200-800 which are not divisible by 5 and7
bleak pier
200/5 =40
200/7 = 28
200/35=5
62 numbers 1-200 which are divisible ny 5 and 7
Now 800/5 =160
800/7=114
800/35=22
274-22=252
251
Now 251
ruby sapphire
What’s with the subtracting 1
bleak pier
Because we want not to unclude 800
5 divides 800 so
@ruby sapphire
251-62=189 numbers
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untold gust
Find the limit of In as n approaches infinity.
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Find the limit of In as n approaches infinity.
untold gust
On the interval 3 4 the function decreases from 1 to 0 . Is the min value 0 enough to show that the limit is 0?
ill post my work in a sec
the functions becomes sqrt(-x^2+6x-8)
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stiff viper
can someone check this for me? thanls
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stiff viper
<@&286206848099549185> pls thanks
stiff viper
is saying ACXB is cyclic enough to say that X lies on (ABC)?
i read online they said no idk if that's right
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bleak hawk
If, for example, lim(x→7) a^x = 10, is a⁷ = 10?
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If, for example, lim(x→7) a^x = 10, is a⁷ = 10?
bleak hawk
Probably a bit of a dumb pointless question, but still, it does confuse me a bit
bleak hawk
(in this case)
When is the value found with the limit different from the real answer?
fresh sparrow
if you have a discontinuity
fresh sparrow
no
for example $f(x)=\begin{cases} x + 2 \text{ if } x\neq 3 \ 9 \text{ if } x=3\end{cases}$
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artemetra
fresh sparrow
$f(3) = 9$, but $\lim_{a\to 3} f(a) = 5$
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artemetra
bleak hawk
Ohhh makes sense yeah
Both lateral limits point to the same value (so the limit exists), but in this function the real value is different
Right?
bleak hawk
this is called a removable discontinuity if you are wondering
And in the cases like f(x) =
1/x, if x ≠ 0
0, if x = 0
Does it have a name?
When you force it to be continuous
fresh sparrow
it's not continuous
bleak hawk
Oops
fresh sparrow
it's just defined for all reals now
bleak hawk
Yeah, wrong example
fresh sparrow
And in the cases like f(x) =
1/x, if x ≠ 0
0, if x = 0
Does it have a name?
maybe it does but to be honest i don't know. note that the limit as x goes to zero doesn't exist
bleak hawk
Yeah
fresh sparrow
non-removable discontinuity i guess lol
bleak hawk
Lol
fresh sparrow
bro's cooking something
bleak hawk
So I was kinda struggling to get why 0,999... = 1
I get that lim(n → +inf) 0,[n digits of 9] = 1
So that's why I asked you if the limit is equal to the real value. So that is a valid way of thinking of that 0,999... = 1, or nah? Or is there some discontinuity that makes the limit different from the actual value??
bleak hawk
bro's cooking something
I was just confused on wording and also talking about random things with the ppl around me 😂
fresh sparrow
So I was kinda struggling to get why 0,999... = 1
I get that lim(n → +inf) 0,[n...
no, there is no discontinuity, although dealing with the "0.[n digits of 9]" things may be unrigorous
bleak hawk
Idk if my train of thought is making any sense to begin with lol
bleak hawk
no, there is no discontinuity, although dealing with the "0.[n digits of 9]" thi...
There are probably more mathematical correct ways to express that
But I don't think I have learned them yet?
fresh sparrow
quick and easy proof of this\
consider the sum\ $0.9 + 0.09 + 0.009 + \cdots = 0.9 + 0.9 * 0.1 + 0.9 * 0.1^2 + \cdots$\
this is an infinite geometric series, with first element 0.9 and ratio 0.1\
so using the infinite geometric series formula we get\
$\frac{0.9}{1-0.1} = \frac{0.9}{0.9} = 1$
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artemetra
fresh sparrow
There are probably more mathematical correct ways to express that
what i wrote is an example of a more rigorous proof
fresh sparrow
go for it
bleak hawk
So I get that some infinites are larger than others. Is it possible to quantify that difference in some cases?
Ex1:
x = +inf
y = x + 5
(by consequence) y = +inf
Is it correct to say:
- Although both are positive infinites, infinity y is exactly 5 units larger than infinity x.
Ex2:
(informally) We know that for every natural number (except 0) there is exactly one negative counterpart.
Be n the number of elements in the set of the natural numbers (N)
And z the number of elements in the set of the integers (Z)
Is it correct to say:
- Although both are positive infinites, z is exactly twice as much as n, minus one. That is: z = 2n - 1
fresh sparrow
ex1: sorta!
ex2: nope
bleak hawk
Why? Especially ex2
fresh sparrow
okay i'm going to try to be brief but it's a topic on which there are hundreds of videos on youtube
bleak hawk
No problem
fringe onyx
Even 1 is kind of nope
fresh sparrow
with ordinal arithmetic it's sorta doable
fringe onyx
Oh yeah, I was thinking we are using the extended reals
fresh sparrow
No problem
there are two types of numbers commonly considered, cardinals and ordinals
cardinals are numbers that measure sizes of sets: 1 apple, 2 banas, 5000 fighting jets, etc.
ordinals are numbers that represent the position of something: 1st, 2nd, 5000th
bleak hawk
Hm yeah I'm aware of that
fresh sparrow
with finite numbers there's a direct and one-to-one correspondence between cardinals and ordinals, they are basically the same thing
the moment we start dealing with infinities they become different things
(i will not go into the reason why)
so your statement in ex1 is actually doable within ordinal arithmetic, but not cardican arithmetic
the smallest infinite ordinal is called omega ($\omega$)
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artemetra
fresh sparrow
the smallest infinite cardinal is called aleph-zero $\aleph_0$
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artemetra
fresh sparrow
within ordinals $\omega$ and $\omega + 5$ are distinct things: indeed, if we have something (somehow) at the $\omega th$ place, the next thing after would be on the $(\omega+1) th$ place, they are clearly distinct
grand pondBOT
artemetra
fresh sparrow
that is not the case though for cardinals: $\aleph_0 + 1 = \aleph_0$
grand pondBOT
artemetra
fresh sparrow
if we consider the set of all positive integers $({1,2,3\cdots})$ and add $0$ as another element, we get ${0,1,2,3,\cdots}$
grand pondBOT
artemetra
fresh sparrow
do these two sets have the same size (cardinality)? yes!
why?
the very definition of two sets having the same cardinality is if there is a bijection between them: basically a way to map each element from one set to a unique element in a different set
i.e. pair them in a one-to-one correspondence
we can do that with the aforementioned sets:
we can map 1 to 0, 2 to 1, 3 to 2 and so on to infinity
each number from set will be associated with another number in the other set, and vice versa - so we used up all of them and managed to use each element just once - this is only possible if these two sets have the same cardinality
fresh sparrow
So I get that some infinites are larger than others. Is it possible to quantify ...
a similar proof can be constructed to disprove the claim in ex2: there is a way to uniquely associate every number in N with a number in Z, and every number in Z with a number in N
so they are both of the same cardinality, namely aleph-null
@bleak hawk i know it was a lot so if anything ask lol
although i don't know if i will reply, it's 2 am here
bleak hawk
@fresh sparrow It is more complex than I imagined, but I think I got a part of it at least
fresh sparrow
there's a great vsauce video on this
i think it was called "how to count past infinity", probably more digestible than what i wrote here
bleak hawk
Out of curiosity, what area of mathematics studies this type of things? The only math classes I have taken in college (for now) is Calculus I and Discrete Maths
fresh sparrow
logic and set theory
bleak hawk
there's a great vsauce video on this
Gonna watch it later then
fresh sparrow
Out of curiosity, what area of mathematics studies this type of things? The only...
i don't know if people study this in college
bleak hawk
Hm probably not in computer science lol
Maybe in math
Anyway, thanks a lot for the attention and explanations 😄
fresh sparrow
Hm probably not in computer science lol
oof most likely not
bleak hawk
Have a good night
fresh sparrow
you too
bleak hawk
oof most likely not
Yeah lol 😂
fresh sparrow
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grave locust
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grave locust
hello, im doing #8
not sure what im doing wrong
i chose 2 points in Plane 1
then found the vector connecting them
distant vigil
can you show any working out?
distant vigil
From what I remember
grave locust
im using the formula $d(A,P) = \frac{\vec{QA}\vec{n_2}}{||n||^2}\vec{n}$
distant vigil
to show two planes are parallel you find their normal vectors
grave locust
yes i did that
but i have trouble with the second part
my teacher told us, to find the distance between 2 planes, u take one point on any of the planes and find the distance using the formula i gave
distant vigil
if I remember correctly there are two ways to find it
you can either find two corresponding points on each plane and find the distance between those two points
or you can find the distance of each plane from the origin and then posit that since each plane is parallel, the distance from the origin will give you the distance between each plane
distant vigil
you can either find two corresponding points on each plane and find the distance...
apologies, let me revise this: You find one point on one plane then find the distance from point to the other plane
my fault
grave locust
yes thats the method im using
the problem is in the formula
theres 2 points
Q and A
and im confused
do i take 2 random points?
grave locust
so n should be of plane 2
distant vigil
Well think about how you would find the distance
the point of the process of picking one point from the plane is because you know the planes are parallel
so the distance should be equal from any point
grave locust
yes
distant vigil
as such you take the point and find the distance from the point on one plane to the other plane
grave locust
yes
grave locust
for parallel planes?
distant vigil
but both should be the same
distant vigil
for parallel planes?
the formula should function regardless of whether the planes are parallel or not
because it is the shortest distance from a point in space (which we assign one point on our first plane to be this point) to a plane
think about it like this diagram
grave locust
ive never seen that formula
distant vigil
**marco.py**
I believe it should relate to this formula (admittedly I have not seen this)
distant vigil
I'm not sure what your formula directly denotes unfortunately
grave locust
my teacher gave us this formula to find the distance between a point A and a plane
and so Q is a random point
distant vigil
what does A, P and Q refer to here?
grave locust
P is the plane
distant vigil
oh
grave locust
Q is a point on the plane,
distant vigil
**marco.py**
then this should be equivalent to
grave locust
and P is the plane
distant vigil
which is this formula according to my textbook
this
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uncut plume
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pearl idol
Please only ping Helpers after 15 min
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this image is wrong
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@uncut plume Has your question been resolved?
uncut plume
this image is wrong
circle is a bit wrongly made
you can have a look at the original question
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uncut plume
@pearl idol can you help plz
uncut plume
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burnt flame
uncut plume
oh thank you but I think we need similarity and construction here
carmine sigil
@uncut plume please do not DM me unless I request it.
supple pilot
if you look at the line AB and knowing the centers are in distance of 1 and 2 from A and B respectively it follows that the distance between centers of two small circles is 2
which means that center of the circle with radius 1 lies on the second circle
then you get sth like this and if i didn't make a mistake then you can get y and z from this
by using pythagorean theorem
rest is quite easy
uncut plume
then you get sth like this and if i didn't make a mistake then you can get y and...
is it x next to 1 ( next to the other x)
supple pilot
the little one between 1 and x is z
uncut plume
oh what is that
supple pilot
it's the distance from center of circle 1 to the PQ line
y is distance from center of circle 2 to PQ line
uncut plume
yea got it thanks a lot I will try solving it and inform you
uncut plume
by using pythagorean theorem
i got these but the prob is that there are 3 variables now
supple pilot
yup, but you know that y+z = 2 right
it's a bit tricky but if you subtract the two lines it should be more evident
uncut plume
it's a bit tricky but if you subtract the two lines it should be more evident
oh i got z=1/4 I think now it has become much easier
supple pilot
that's what i arrived at too
uncut plume
that's what i arrived at too
if we assume ab bisects pq then as per my calculations half of pq is √5.6
supple pilot
if we assume ab bisects pq then as per my calculations half of pq is √5.6
i can dm you later if it's correct as someone called me just now
uncut plume
yes sure thanks a lot
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uncut plume
how?
civic moth
how?
Ok PQ =✓21
For solving this u have to use a little geometry,u will get that the circle omega2 passes through the centre of small circle hence u can find the distance between A and intersection of AB by PQ
By using this u can calculate the distance between intersection of AB by PQ and the centre of big circle ohm
Then apply Pythagoras theorem and solve @uncut plume
uncut plume
is the final answer 2.1
echo eagle
what is 9-10
uncut plume
what is 9-10
11
civic moth
is the final answer 2.1
It is √21 not 2.1 ,u can check this answer because calculation mistake happens
uncut plume
It is √21 not 2.1 ,u can check this answer because calculation mistake happens
it's √75/2
very close to your answer
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craggy yew
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craggy yew
in the solution of the question, why are a^2 and b^2 replaced with 4^x and 4^y?
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bleak pier
What is wrong @craggy phoenix in big letters?
craggy phoenix
What is wrong <@1185744654474412155> in big letters?
nothing wrong, but i really cant understand what you wrote. thats all
bleak pier
You can not read it or you are not understanding what method i used?
craggy phoenix
You can not read it or you are not understanding what method i used?
cant read
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drifting root
For part a:
I did it as remainder theorem a=3,b=-1 and then I did
(-1/2)a + b, which is -2.5, but the answers are 2.5
did I do smth wrong?
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drifting root
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drifting root
oh sorry for a
remote glade
For part a:
I did it as remainder theorem a=3,b=-1 and then I did
(-1/2)a + b, w...
the answer is -2.5. you are right
remote glade
you're welcome
drifting root
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twilit field
using the definition of a definite integral prove that $\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{n+n}\le\log\left(2\right)\le\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n-1}$
midnight plankBOT
using the definition of a definite integral prove that $\frac{1}{n+1}+\frac{1}{n...
grand pondBOT
ƒ(Why am. I here)=I don't Know
twilit field
$\frac{1}{n}\sum_{r=1}^n\frac{n}{n+r}\le\log\left(2\right)\le\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n-1}$
grand pondBOT
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twilit field
$\int_0^1\frac{1}{x+1}\le\log\left(2\right)\le\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n-1}$
grand pondBOT
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twilit field
right so far?
wind oxide
the finite sums is an inegral only in the limit as n goes to infinity right?
if i am not wrong
twilit field
yeah
wind oxide
ah okay
twilit field
I'm assuming n tends to 0
wind oxide
not infinity???
twilit field
the partition must tend to zero, no
wind oxide
i think the goal is to find integral(s) whose riemann sum should be equivalent to the lower bound sum and upper bound sum
wind oxide
i think the goal is to find integral(s) whose riemann sum should be equivalent ...
if its not this then probably euler-maclaurin formula
twilit field
what now
wind oxide
what now
uhhh try finding something similar for the upper bound and then work it back from those integrals for the actual proof. Also pretty sure if the limit holds to infinity, in this case, it will hold for some finite constant as well, so then ur proof would be done
altho i cant really remember why it'll hold for n constant in this case
wind oxide
**ƒ(Why am. I here)=I don't Know**
its not that tough - 2n-1 - (n+1) = n-2, so just take the sum till there
twilit field
yeah, I know
wind oxide
oh wait I am so god damn dumb, u gotta prove log 2 >= or <= than those integrals 🤦♂️
twilit field
trying to figure out the bounds of the integral
twilit field
oh wait I am so god damn dumb, u gotta prove log 2 >= or <= than those integrals...
basically the squeeze theorm
sort of
twilit field
yeah
wind oxide
the integral of 1/(x+1) = ln|x+1| + C. so just to definite integration on that
twilit field
we're just using the defn of the definite integral
wind oxide
yes
twilit field
I'm trying to learn the defn by solving problems as this isn't a proof based book, is that a bad idea?
wind oxide
**ƒ(Why am. I here)=I don't Know**
pret sure this holds for finite n as well, which follows from the fact that a finite riemann sum will always be <= than an infinite riemann sum ie def integral
hard umbra
that's false
wind oxide
I'm trying to learn the defn by solving problems as this isn't a proof based boo...
not really, tbvh i still cant recognise riemann sums. My brain cant ever see it, and its probably due to less practice
wind oxide
that's false
why?
hard umbra
a riemann sum for something like x^3 - x could go either way
wind oxide
ur adding on a positive quantity (atleast in this case)
wind oxide
a riemann sum for something like x^3 - x could go either way
oh right
hard umbra
the position you choose to evaluate the point matters
wind oxide
the position you choose to evaluate the point matters
its n >= 0 it seems so it holds
hard umbra
no it's just false in general
twilit field
$\int_0^{1^+}\frac{1}{x+1}\le\log\left(2\right)\le\int_0^{1^-}\frac{1}{x+1}$
grand pondBOT
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twilit field
pretty sure i've messed this up somehwere
wind oxide
what do those integrals have to do with log(2)
hes trying to prove something using definite integrals
twilit field
**ƒ(Why am. I here)=I don't Know**
I feel like the signs should be reversed though
hard umbra
have you seen the darboux sums?
twilit field
no
wind oxide
no it's just false in general
its false in this case too btw? for n >= 0
twilit field
it isn't atleast explicitly mentioned in my book thusfar
hard umbra
well have you even defined integrals so far
hard umbra
what's the definition that you're working with
twilit field
The definite integral of f betwnee a and b is the unique number which is greater than or equal to every lower sum and less than or equal to every upper sum
hard umbra
those are the darboux sums
hard umbra
you need to pick a partition of [0, 1], and apply the definition
twilit field
hmm, so I could take a partition of 1/n in both cases
hard umbra
no you dont need "n tends to infinity"
The definite integral of f betwnee a and b is the unique number which is greater than or equal to every lower sum and less than or equal to every upper sum
it doesnt say anything about n tending to infinity here
twilit field
ok, yeah
hard umbra
so what are the points of the partition
last slate
no
twilit field
there are n points
hard umbra
tell me the points
grand pondBOT
ƒ(Why am. I here)=I don't Know
hard umbra
if you want to evaluate the integral
[ \int_0^1 \f 1 {x + 1} \dd x, ]
you are partitioning $[0, 1]$
grand pondBOT
hard umbra
the partition is a sequence of points
[ P = (0 = x_0 < x_1 < x_2 < \dots < x_n = 1) ]
grand pondBOT
twilit field
so 1/n,2/n
grand pondBOT
hard umbra
write out the definition
twilit field
$\sum_{i=0}^{n-1}f(a)((x_{i+1} - x_i)$
grand pondBOT
ƒ(Why am. I here)=I don't Know
twilit field
where a is the point at which the function attains its minimum in a given interval
hard umbra
okay so
apply this to our current partition
,, L(f, P) = \sum_{i = 0}^{n - 1} \min_{x \in [x_i, x_{i + 1}]} f(x) \cdot (x_{i + 1} - x_i)
grand pondBOT
twilit field
okay, I don't see how the helps here though
hard umbra
,, \map L {\f 1 {x + 1}, P = (x_j = \tf jn)} = \sum_{j = 0}^{n - 1} \min_{x \in [\f jn, \f {j+1}n]} \f 1 {x + 1} \cdot \parens [\bigg] {\f {j + 1} n - \f jn}
grand pondBOT
hard umbra
this is a literally rewriting of the definition, with everything plugged in
twilit field
hmm,ok
and the upper sum would be ,, \map L {\f 1 {x + 1}, P = (xj = \tf jn)} = \sum{j = 0}^{n - 1} \max{x \in [\f jn, \f {j+1}n]} \f 1 {x + 1} \cdot \parens [\bigg] {\f {j + 1} n - \f jn}
right?
$\map L {\f 1 {x + 1}, P = (xj = \tf jn)} = \sum{j = 0}^{n - 1} \max{x \in [\f jn, \f {j+1}n]} \f 1 {x + 1} \cdot \parens [\bigg] {\f {j + 1} n - \f jn}$
grand pondBOT
ƒ(Why am. I here)=I don't Know
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
twilit field
TeX fail oops
hard umbra
yeah ive got quite a couple of macros in here
,, \map U {\f 1 {x + 1}, P = (x_j = \tf jn)} = \sum_{j = 0}^{n - 1} \max_{x \in [\f jn, \f {j+1}n]} \f 1 {x + 1} \cdot \parens [\bigg] {\f {j + 1} n - \f jn}
grand pondBOT
hard umbra
here you go
twilit field
so this would be the right hand side of the inequality, right?
hard umbra
what we know for sure is that
[ \map L {\f 1 {x + 1}, P} \le \int_0^1 \f 1 {x + 1} \dd x \le \map U {\f 1 {x + 1}, P} ]
grand pondBOT
hard umbra
so this would be the right hand side of the inequality, right?
yeah so hopefully what ive just written here ^^^ will be exactly the inequality you're looking for
after you expand out the definitions and the dust settles
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slate ferry
I dont understand why pi is 180 when its in an angle of a trig function
midnight plankBOT
I dont understand why pi is 180 when its in an angle of a trig function
dim lotus
I dont understand why pi is 180 when its in an angle of a trig function
By definition, 2pi radians is 360 degrees. Therefore pi radians must be 180 degrees
We define radians to act like this
slate ferry
By definition, 2pi radians is 360 degrees. Therefore pi radians must be 180 degr...
I thought angles are supposed to be in degrees
Like sin(70) that 70 is in degrees
tacit rose
No degrees is a unit, same with radians
zealous schooner
Radians is a much better unit
Which is why we ditch degrees altogether after a point
slate ferry
So why sin(pi 70) is sin( 18070) like why did we turn pi in degrees to pi in radians
Hmmmmm
tacit rose
Because pi radians and 70 degrees are two different units, so it's not compatible to do the calculation like that so you have to convert one of the values
So pi radians = 180 degrees
It's similar to ft and inches, if you wanted to find area, you don't do 1 ft times 12 inches because the units are different
You'd convert the units first
slate ferry
Because pi radians and 70 degrees are two different units, so it's not compatibl...
Ooooh so 180 is pi in degrees not radians…
Ok that makes sense
But still- why do we assume that pi in an angle is in radians not degrees
Like sin(pi) why did we assume that its sin(180) not sin(3.14…..)
zealous schooner
$180^\circ=\pi^c$
grand pondBOT
kheerii
zealous schooner
Like sin(pi) why did we assume that its sin(180) not sin(3.14…..)
Because as I said, after a point we completely ditch degrees
slate ferry
I see
zealous schooner
It’s much more convenient to think of angles in radians
tacit rose
Like sin(pi) why did we assume that its sin(180) not sin(3.14…..)
Because the input for trig functions is either units of radians or units in degrees. Most of the time, when you see pi involved, it's radians
zealous schooner
They connect very nicely to how we define trig functions using the unit circle
slate ferry
It’s much more convenient to think of angles in radians
Radians are rlly small and i hate their formula
Like
X/180 = theta/pi
I remember this
zealous schooner
Radians are rlly small and i hate their formula
You’ll get used to it eventually
It’s a bit hard in the beginning
zealous schooner
X/180 = theta/pi
Uhh, yeah
I think that’s right
X is the angle in degrees and theta is in radians
slate ferry
Yes!
slate ferry
Because the input for trig functions is either units of radians or units in degr...
Yeah that makes sense
Well ty sir nova and kheerii
I have another question
There was a formula for the angle between two straight lines
Tan(theta)= absolute value of
M1-m2/m1+m1m2 smth like that
M is the slope of each line
zealous schooner
All your formulas remain the exact same
It’s just that you need to calculate theta in radians instead of degrees
slate ferry
M1-m2/m1+m1m2 smth like that
So- why cant an angle be negative? Why do we take the absolute value
slate ferry
It’s just that you need to calculate theta in radians instead of degrees
No its a completly different wuestion
zealous schooner
So- why cant an angle be negative? Why do we take the absolute value
Oh
The angle between two lines is defined to be the ACUTE angle between the two lines
Whenever two lines intersect, they describe two angles
One will be acute and one will be obtuse (except for the case when both lines are perpendicular, in which case both angles are right angles)
If you take the negative value then you would get the obtuse angle
slate ferry
Whenever two lines intersect, they describe two angles
I thought its 4
Oh you mean two of them will be acute or-
slate ferry
One will be acute and one will be obtuse (except for the case when both lines ar...
What foes obtuse even mean
zealous schooner
Oh you mean two of them will be acute or-
Yeah
zealous schooner
What foes obtuse even mean
$90^\circ<\theta< 180^\circ$
slate ferry
Where
grand pondBOT
kheerii
zealous schooner
Where
Here you made the two lines almost perpendicular, so it isn’t obvious
Make them more slanted
slate ferry
Ah- idk what slanted even means i just changed how they look
zealous schooner
They still look very perpendicular
clever coral
zealous schooner
slate ferry
Ok… what abt this i forgot 💀
Oj yes
Where id yhe acute angle
I love how this turned into an english terminology class
slate ferry
I dont see any acute angles i just see two obtuse ones
Theta1 and theta2 are def not acute… right
clever coral
This picture should be helpful
clever coral
Theta1 and theta2 are def not acute… right
they are acute
An acute angle is just a small angle
slate ferry
This picture should be helpful
Oh?! I thought acute angles are right angles
Nice
I thought its another word for right angles
clever coral
tbh all the words for angles are weird
summer ravine
acute angles are less than 90 deg.
obtuse more than 90 and less than 180.
slate ferry
So- getting positive value gives you the acute ones and getting negative value guves you the obtuse ones
This rule
My book says “if tan of theta is positive” how will it ever be anything but positive if we take the absolute value
slate ferry
The angle between two lines is defined to be the ACUTE angle between the two lin...
Why not the obtuse? And doesnt that mean that tan(of acute angle)= -tan(of obtuse angle)
.close
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iron wharf
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bleak pier
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bleak pier
59
It is continuous at x=1 as i got 2 from both function
For derivative
x-3/|(x-3)|=-2/2=-1
1-3/2=-1/2 so not same so not differtiable
fresh sparrow
For derivative
x-3/|(x-3)|=-2/2=-1
are you sure that's the derivative?
fresh sparrow
why are you dividing by |x-3|
near 1+, |x-3| is the same thing as -(x-3)
wait nvm
fresh sparrow
your answer is correct
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thorny pagoda
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thorny pagoda
how to do b ii
unreal fjord
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hollow harness
midnight plankBOT
hollow harness
these are all the questions
some more are at the top dk if i shou;d pin or what
some help would be greatly appriciated and i look forward to seeing what u guys cook up 👍 😄
fathom knoll
3 ideas are required here: first that the angles inside a triangle need to sum up to 180, then the angle of a full circle is 360 (use for the red one) and the angle spanning a line/half-circle is 180. So basically the complement of the red angle which corresponds to that angle inside the triangle is 360 - 287, the angle complementing the yellow one which is the angle inside the triangle is 180-119 and hence you can figure out what the blue angle should be. Does that make sense?
fathom knoll
For this the two lines around the yellow angle have the same length hence the two angles complementing the yello angle inside the triangle are equal.
cunning vault
The sum of an interior angle of a polygon is 180(n-2) where n is the side, you already had 3 given angles
cunning vault
For this the two lines around the yellow angle have the same length hence the tw...
There are also parallel lines so one of the angles can easily be found
cunning vault
Same reasoning. Parallelism
fathom knoll
There are also parallel lines so one of the angles can easily be found
yup
cunning vault
Just substitute 26 on the given formula
fathom knoll
Here draw a line from M to PN which is orthogonal let's say it hits PN at Q. So QMLP is a rectangle and angle MQN is 90 and angle QMN is 138-90 = 48. You can get angle MNP
midnight plankBOT
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radiant ore
Question 1 d)
midnight plankBOT
Question 1 d)
radiant ore
I found the equation of the plane but how do I limit it to just the triangle
Like what would the bounds of the double integral be
<@&286206848099549185>
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radiant ore
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craggy fjord
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last slate
What do you mean? It is just a slope formula.
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split plank
this formula just It's just the definition of the derivative
of point a
It describes the rate of change of the function in a delocentric domain of point A
An infinitesimal neighborhood
I'm sorry my English is shit
QAQ
slate ferry
What is a perfect square
midnight plankBOT
What is a perfect square
slate ferry
And how do we turn equations into perfect squares
lapis citrus
waxen schooner
how to solve an linear equation when y is minus a number?
midnight plankBOT
how to solve an linear equation when y is minus a number?
waxen schooner
I know how to solve them normally but not when it's something like y-5=
humble torrent
More context?
waxen schooner
More context?
yeah sorry I didn't think to screenshot
humble torrent
So you know how to find the equation in the form y = kx + m, right?
waxen schooner
I know y=mx+b
humble torrent
Alright yeah same thing
waxen schooner
like I know how to find it from the two points
I just don't know how to handle the -5
like what do they want me to do with the other side
waxen schooner
ok lemme solve it real quick and try it
waxen schooner
so I got
y=-3x+-11
which is just y=-3x-11
so is it now y-5=-3x-16? @humble torrent
humble torrent
Yup!
waxen schooner
it's wrong according to kahn academy 💀
wait I did the slope wrong
it's normal 3
not negative
waxen schooner
yeah but I'll get it wrong if thats the format they want
humble torrent
They’re just doing it directly, rather then subtracting afterwards
humble torrent
yeah but I'll get it wrong if thats the format they want
No?
So the correct equation is y = 3x + 23
Now subtracting 5 we get
y - 5 = 3x + 18
Which is correct
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half ingot
Is this legit
midnight plankBOT
Is this legit
half ingot
<@&286206848099549185>
midnight plankBOT
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.
half ingot
Oh I thought it closes after 15 minutes
My bad 💀💀
Someone @ me if they answer my question, I’m turning my phone off
midnight plankBOT
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half ingot
<@&286206848099549185>
pearl idol
Is this legit
More or less, but it doesn't explain that the key step here is linearity of expectation. Are you familiar with that property?
pearl idol
We don't really give out answers on this server, we can only help guide you to the answer
half ingot
So you’re not allowed to tell me if they ai is correct?
pearl idol
I'd rather go through its answer with you so that you're able to do the problem yourself, and if the AI is wrong, you can catch it
half ingot
Alrighty
pearl idol
So, the thing we want to figure out is E f(n) (where E means the expected value)
You can expand this out as E (|a1 - a2| + |a3 - a4| + |a5 - a6| + |a7 - a8|)
The key step here is that you can break up the sum into four smaller calculations, this is called linearity of expectation
So it's equal to E|a1 - a2| + E|a3 - a4| + E|a5 - a6| + E|a7 - a8|
Does that make sense?
half ingot
Not really
pearl idol
Okay let's take it slower
half ingot
What’s linearity of expectation
pearl idol
Yeah, so linearity of expectation is the idea that if you want to find the expected value of a sum, you can find the expected value of each part of a sum instead
So, E(x + y) = Ex + Ey
This is useful because sometimes it's easier to find the expected value of each part of the sum rather than the whole thing at once
half ingot
Ohh
pearl idol
It's a simple idea, but it helps you solve a ton of expected value questions
pearl idol
You can expand this out as E (|a1 - a2| + |a3 - a4| + |a5 - a6| + |a7 - a8|)
Here we're taking this sum here and breaking it into four smaller parts
pearl idol
Yeah
half ingot
Sorry I’m tired 😭😭
pearl idol
All good haha
half ingot
Anyways 🥲
pearl idol
So now the next observation that makes this problem a ton easier
Is that permutations are symmetric, so for example the number of permutations with a1 = 3 and a2 = 2 is the same as the number of permutations with a3 = 3 and a4 = 2
Do you see why?
half ingot
So when you calculate the sum it’s 84?
pearl idol
We're not quite there yet, but we're getting there soon
half ingot
Ok
pearl idol
Is that permutations are symmetric, so for example the number of permutations wi...
Does this part make sense to you though
half ingot
Do you see why?
Yea
pearl idol
Alright
So basically that means that E|a3 - a4| will be the same thing as E|a1 - a2|
because it doesn't really matter if I switch a3 and a1, or a4 and a2
pearl idol
yup exactly
so then this simplifies the problem even more
E|a5 - a6| and E|a7 - a8| are also the same thing as E|a1 - a2|
which means the answer to the problem comes down to
E|a1 - a2| + E|a1 - a2| + E|a1 - a2| + E|a1 - a2|
half ingot
I love symmetry
pearl idol
the third position of the permutation isn't any different from the first position
like, 5 isn't more likely to be first than third or anything
pearl idol
which means the answer to the problem comes down to
E|a1 - a2| + E|a1 - a2| + E|...
so now adding this up, we get 4 * E |a1 - a2|
half ingot
Wait so what exactly is a1 and a2? Does it matter since it’s summery
Symetrry
equal
pearl idol
a1 and a2 are the first and second elements of the permutation
but yeah, since they're symmetric it could really be any two elements and we'd get the same answer at the end, that's right
half ingot
So like 8 and 3 or 4 and 2?
pearl idol
yeah it could be like 4 * E |a8 - a3| or 4 * E |a4 - a2|
and because of symmetry, we would still get the same answer at the end
pearl idol
why frowning, lol
half ingot
I thought that’s shocked?
pearl idol
oh haha
half ingot
🤯
pearl idol
so then now we just have to calculate E |a1 - a2| which is way simpler than our original problem
if you remember the definition of average value, it's just
(sum of |a1 - a2| for all different possibilities of a1 and a2) / (the number of different possibilities of a1 and a2 there are)
does that make sense?
half ingot
Would the different number of possibilities be |8- 1| + |7 - 2/ + /6 - 3/ + /4-2/
pearl idol
why?
half ingot
Idk I have this random memory of collecting the largest and the smallest and like repeating that and finding something maybe it was relevant
I dunno man I was dealing with a bunch of 4 year olds for like 8 hours I’m dead 💀💀
So I just need to find out how to find the number of possibilities
Different possibilities*
This makes sense
Thank you 😭😭
pearl idol
I dunno man I was dealing with a bunch of 4 year olds for like 8 hours I’m dead ...
oh no 😭 are you a camp counselor or smth
half ingot
Yea I volunteer
pearl idol
that's cool
half ingot
They were so hyper and they kept wanting to go play during snack time
ONE KEPT RUNNING OFF I LITERALLY HAD TO CATCH HIM 💀💀
Anyways I can finally solve my math now, thank you 🫶
pearl idol
If you're curious how to find E |a1 - a2|, it's basically what the AI did
the number of different possibilities is 8 choose 2, which is 28
and then you can find the numerator of the fraction by doing the weird sum thing it does in step 2
so the AI answer ends up being right 👍
pearl idol
haha yeah, but don't trust it too much
half ingot
I like giving it math questions cause sometimes it’s right and soemtkwms it’s so far off it’s kinda funny
half ingot
Yea it seemed accurate like it knew what it was doing but I was iffy
pearl idol
like doing basic arithmetic wrong
half ingot
yeah I've seen it doing some crazy mistakes
Once it just stopped working on me 💀
half ingot
like doing basic arithmetic wrong
It’s an ego boosters knowing you’re better 😍
Better than goofy ai and ones and zeros
Okay I gotta finished some computer science hw and then I’ll finish this question up and go sleep and brace myself for these little rascals
pearl idol
haha good luck
half ingot
THANK YOU FOR HELPING ME 😍🫶
pearl idol
you're welcome
midnight plankBOT
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prime salmon
midnight plankBOT
iron whale
use similar triangles
prime salmon
use similar triangles
how
prime salmon
I knew I needed pythagorean theorem but I do not know what to do
prime salmon
how do I do that?
iron whale
use the radius to make a right triangle
acoustic frost
no
it's a property
of triangles into a circle
when the triangle cut the circle in the middle
the biggest angle is a right angle
prime salmon
oooooooooo
acoustic frost
and it's what the diameter will do
acoustic frost
sorry for the english, i'm brazilliam and don't know too much
acoustic frost
no
iron whale
thats what im saying
acoustic frost
i think other way
prime salmon
no cuz
iron whale
@acoustic frost are you thinking of this theorem?
prime salmon
after I find x, I subtract it by 6.5
acoustic frost
make a triangle
prime salmon
guys
acoustic frost
oh
acoustic frost
i understanded
prime salmon
its 0.7
acoustic frost
yes
prime salmon
thanks
acoustic frost
it's right
iron whale
dont forget the hundredths place
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sly fiber
The central span of the Golden Gate Bridge has 84 pairs of equally spaced vertical suspenders that connect the bridge to the main cables.
The function, f(x), roughly approximates the curve of the main cables. Hence determine the total length of the vertical suspenders that connect the bridge to a suspension cable in the central span of the Golden Gate Bridge.
midnight plankBOT
The central span of the Golden Gate Bridge has 84 pairs of equally spaced vertic...
sly fiber
im gonna keep it real, im not too sure what to do
could summation be used here?
but i dont know how to take the equally spaced intervals into account
fringe onyx
yeah I think it is just going to be the sum of the f(x) evaluated at the position of the 84 suspenders
you can simplify it a bit using the fact that f(x) is even
the exponentials are just a cosh
fringe onyx
yeah I think it is just going to be the sum of the f(x) evaluated at the positio...
times two I guess
since they come in pairs
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twilit field
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@twilit field Has your question been resolved?
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bleak pier
period of |sin3x|+|cos3x|+|tan4x|
midnight plankBOT
period of |sin3x|+|cos3x|+|tan4x|
bleak pier
2pi/6,2pi/6,pi/8?
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civic moth
it's √75/2
I think I did some calculation mistake,but I have told u the method to solve so u can try
uncut plume
I think I did some calculation mistake,but I have told u the method to solve so ...
yea the answer is √75/2
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glad merlin
Hello
glad merlin
when writing formal proofs where i want to assume a lot of things from the start, what is the best way to go about that
like here, i am kinda repeating the word 'lets'
would i have to keep on using these connecting words such as 'additionally', 'similarly' or is this all just nuances and nobody cares
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glad merlin
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jagged saffron
Primarily nuances and no one really cares. But also you can just drop superfluous language like "additionally" if you want. And if you don't want to use "let" again that second sentence could just read "We represent these elements...".
But again I don't think people overly care, these are words they gloss over and barely notice
Unless you have something wild like 20 sentences in a row that start with "Let us"
glad merlin
lmao
yeah
i just have like threeish sentences
im a first year so hopefully they will be nice
.close
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glad merlin
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glad merlin
oops
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brisk crag
How come that's the answer rather than M^4+8m^3+24m^2+32m+16
midnight plankBOT
How come that's the answer rather than M^4+8m^3+24m^2+32m+16
tribal tartan
thats the same thing you wrote there tho