Channel closed
Closed by @north fulcrum
Use .reopen if this was a mistake.
#help-23
safe radishBOT
safe radishBOT
Available help channel!
opaque hamlet
can someone give me a hint on how to do number 2.1?
safe radishBOT
can someone give me a hint on how to do number 2.1?
opaque hamlet
mmm lemme think
cobalt thunder
I would recommend checking whether the terms converge to 0
normal summit
I would recommend checking whether the terms converge to 0
1/n
cobalt thunder
for 2.1
woven hound
Oh you asked 2.1 I thought you asked 2.2 oops
opaque hamlet
I would recommend checking whether the terms converge to 0
I dont think so, i think the sequence will just swing between -1 and 1
opaque hamlet
Oh you asked 2.1 I thought you asked 2.2 oops
ohh
cobalt thunder
Right, and have you seen that for a convergent series, the terms must converge to 0?
opaque hamlet
Right, and have you seen that for a convergent series, the terms must converge t...
ye
normal summit
Right, and have you seen that for a convergent series, the terms must converge t...
?
opaque hamlet
as far as i remembered the terms need to converges to 0 to be a convergent series
ohhhh
i see
woven hound
?
If you're trying to show a series diverges it suffices to show the terms don't converge to 0
opaque hamlet
so i need to use the convergence test by taking the limit of sin(2n) as n approuch inf right?
normal summit
If you're trying to show a series diverges it suffices to show the terms don't c...
1/n^2
cobalt thunder
I dont think so, i think the sequence will just swing between -1 and 1
though be careful, this is sin(2n), not sin(2pi n), so the terms never assume the values Β±1
opaque hamlet
and then show that since the limit doesnt approach 0 we can conclude that the series diverge
normal summit
1/n they converge to 0
opaque hamlet
smth like that?
normal summit
But diverges
cobalt thunder
smth like that?
Exactly!
opaque hamlet
1/n they converge to 0
If im not wrong, for a series to converge the limit as n approach inf needs to be 0 but it doesnt neccesary mean that if the limit is 0 then the series will converge
opaque hamlet
though be careful, this is sin(2n), not sin(2pi n), so the terms never assume th...
thx
woven hound
If im not wrong, for a series to converge the limit as n approach inf needs to b...
Yes
normal summit
If im not wrong, for a series to converge the limit as n approach inf needs to b...
Yea, it's an and
safe radishBOT
opaque hamlet
.reopen
safe radishBOT
β Original question: #help-23 message
heavy glade
have you decided on any test?
opaque hamlet
have you decided on any test?
yes, i used the convergence test
glacial cairn
That's not a justification
normal summit
No. 0 is in that bound
heavy glade
i think its called the divergence test. if a_n doesnt converge to 0, the series diverges.
i didnt understand ur justification either
all u need to show is the limit isnt 0
opaque hamlet
i think its called the divergence test. if a_n doesnt converge to 0, the series ...
oh mb i think i remembered the name wrong
opaque hamlet
That's not a justification
but if the series converges wont it be 0 <=lim sin(2n) <= 0?
heavy glade
yes
opaque hamlet
since -1 <= lim sin(2n) <= 1 doesnt that mean that the series doesnt converge to 0?
heavy glade
but is it true?
glacial cairn
since -1 <= lim sin(2n) <= 1 doesnt that mean that the series doesnt converge to...
No
woven hound
I can say for f(x) = 0, -1 <= f <= 1 but it does converge to 0
glacial cairn
sin(2pi n) converges to 0
opaque hamlet
sin(2pi n) converges to 0
ohhh i see
do i do smth like lim sin( inf) != 0 to justified that?
or should i use smth else
normal summit
ok? My claim was if the terms don't converge to 0, the series diverges
I can have alternating terms and result into the series converging while terms don't converge to 0
cobalt thunder
no you can't
glacial cairn
do i do smth like lim sin( inf) != 0 to justified that?
No that's nonsense
For what values of x does sin(x) = 0?
opaque hamlet
pi or 2pi
normal summit
no you can't
?
glacial cairn
That's it?
opaque hamlet
n*pi if n is 0,1,2,3,4,...
opaque hamlet
damn
glacial cairn
Use your words, what is n
opaque hamlet
integers?
glacial cairn
Yes
So sin(x) = 0 if x = pi n for any integer n
And sin(x) =/= 0 otherwise
Do you agree?
opaque hamlet
yes
glacial cairn
Now, can pi*n be an integer?
opaque hamlet
mmm at some point, but for most parts, no
glacial cairn
Can you elaborate on that?
opaque hamlet
like if n is 7 or -7 for example. then we get (7)(22/7) = 22
the same goes for 14 21 ...
glacial cairn
pi doesn't not equal 22/7 though
glacial cairn
Just so we're clear, pi is an irrational number
opaque hamlet
then no
glacial cairn
Tbh I don't think you're expected to justify than sin(2n) does not converge to 0
But anyway, pi*n is never an integer except when n=0
Simply because if it were, then pi could be written like a/n with a and n integers
pi being irrational makes that impossible
So now that you know pi*n cannot be an integer when n is large, you can argue that sin(2n) cannot equal 0
It could approach 0, but then sin(2n+1) would not
Hence sin(2n) does not converge to 0
safe radishBOT
brave wolf
It could approach 0, but then sin(2n+1) would not
wait wdym by this btw?
why cant it converge to 0 just because it never = 0?
glacial cairn
Well if sin(x) is very close to 0 then sin(x+1) is not
brave wolf
oh okay, fair enough. how would that show that sin2n doesnt converge tho, if it does then sin(2n+1) does not, wheres the issue?
glacial cairn
Oh yeah I meant sin(2(n+1))
Same argument, if sin(x) is close to 0 then sin(x+2) is not
brave wolf
btw just an interesting fun fact, we dont even need irrationality of pi, we're essentially walking 2 units along the unit circle in each step, so we will always step in this region at least once per cycle (its length is 2, which is < pi). Hence the limit cant be 0. By drawing the same thing in the negative, we could prove that the limit doesnt exist at all, since therell be this oscillatory behaviour
glacial cairn
Yeah true, I just think the process of showing that sin(2n) doesn't converge to 0 is more understandable if you first show sin(2n) can never be 0, and then show that even if a term approaches 0 the next one does not
But yours is a shorter proof 
safe radishBOT
Available help channel!
sleek wagon
!help
safe radishBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #βhow-to-get-help for instructions.
sleek wagon
I have a question
normal moss
You can post it here
normal moss
It's number three btw
!status
safe radishBOT
It's number three btw
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
!status
safe radishBOT
It's number three btw
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
normal moss
Why did you just repeat that
Also, sorry, but I will let someone else help because I gotta go
sleek wagon
Why did you just repeat that
π€·ββοΈ
normal moss
Best of luck
sleek wagon
Best of luck
Ight he better explain good
signal crane
reduced by 18.6% means it has now become 81.4% of the original
sleek wagon
reduced by 18.6% means it has now become 81.4% of the original
Bro why did it go lower
signal crane
it has been decreased by 18.6%
sleek wagon
reduced by 18.6% means it has now become 81.4% of the original
And do u know what's the problem
signal crane
find 81.4% of rhe x
if your quantity was x, ut says it has been reduced by 18.6%
so we subtract 18.6% of that x
to get the answer as x - (18.6%ofx)
sleek wagon
This is grade 9 work bro
signal crane
π₯
plucky elk
My question is so bad my instructor left
stick to this channel
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1 did this already
frozen marlin
so ydk where to begin on which q
safe radishBOT
@sleek wagon Has your question been resolved?
flat marlin
This one
So they've given by how much the price has decreased in percentage, first thing you can do is convert that to how much actual money is being reduced
do you know how to do that?
(ie converting from percentage to actual cost as they've given total cost)
safe radishBOT
@sleek wagon Has your question been resolved?
mellow grove
How to solve this question ?I'm stuck,
I think I read that Baire's theorem is needed to solve this but I'm not sure
cobalt thunder
Do you have a guess?
mellow grove
I think it is false
cobalt thunder
Yeah! So I think a hint can be: obviously one disk won't work, and notice that R^2 minus an open disk is closed
So you can try to argue by contradiction
mellow grove
Ok i'm going to try, thank you
safe radishBOT
@mellow grove Has your question been resolved?
flat frigateBOT
woody6978
mellow grove
juste one last question,for the first question I can apply Baire's theorem's only if the number of circles is countabble, how can I show this ?
cobalt thunder
I think you can use the fact that the intersection of a decreasing chain of non-empty closed sets is non-empty
cobalt thunder
**woody6978**
This is what I had in mind!
mellow grove
I think you can use the fact that the intersection of a decreasing chain of non-...
to show the coutability ??
cobalt thunder
No, to directly show the statement
I'm not sure whether there is another way which involves Baire's category theorem
cobalt thunder
I think you can use the fact that the intersection of a decreasing chain of non-...
mellow grove
yes i do agree with you but I don't see how it can solve the problem
cobalt thunder
This is kinda the idea
If you try to cover R^2 with circles, you will always miss a point, which is the intersection of the disks delimited by the circles
At least if you assume the disks are in a decreasing sequence like shown
mellow grove
ooooooh I think I got it, i'm going to come back with a proof
cobalt thunder
At least if you assume the disks are in a decreasing sequence like shown
I have to go, so I'll leave some extra hints
If you assume by contradiction that the statement is true, then you will always have such a configuration; to see this, you can start with a random circle, ||and notice that for the center of the circle to be covered, you need a circle inside it whose radius is less than half than that of the original circle. Repeat this and you get a decreasing chain of open disks whose radii tend to 0. For each of those open disks, you can take a closed disk with the same center and half the radius, to get a chain of decreasing closed disks whose radii tend to 0. The radii tends to 0 part allows you to say that any circle (more precisely, the open disk delimited by the circle) in your (falsely assumed) partition will either be disjoint with, or contain one of the closed disks from before, and so the intersection of the closed disks is not contained in your partition.||
Good luck!
mellow grove
Thank you π
flat frigateBOT
woody6978
mellow grove
Last things remaining are to prove rigorously some assumptions I have made without proof , I will do them on my own.Baire was so useless. Thanks again for the help
.close
safe radishBOT
safe radishBOT
Available help channel!
waxen prawn
safe radishBOT
waxen prawn
help
median vigil
,rccw
flat frigateBOT
waxen prawn
thanks
anyway
can i have help with problem #3
im not understanding it
the stupid problem doesnt tell me if the decagon is regular or irregular
and i js set up a system
and is it not just 45,45,135
for all the measures of the angle?
i feel like it is but it probably isnt because why would it tell me that seven of the angles of a decagon have measures that have a sum of 1220 degrees
median vigil
there are three unknown angles, so how do you only have two variables?
manic radish
the stupid problem doesnt tell me if the decagon is regular or irregular
it doesn't matter, the angles add up to the same
waxen prawn
so is it x+y=90,then x+z=180
shouldnt there be another equation then
bro this problem is so stupid
manic radish
shouldnt there be another equation then
yes, the sum of internal angles
manic radish
correct, you can't. but you don't need to
waxen prawn
wth am i supposed to do then
manic radish
you're given the sum of the seven other angles.
waxen prawn
but its not 7x=1220
manic radish
do you know the sum of internal angles in a 10-sided convex shape?
manic radish
(tbh, you're not told it's convex which is a big assumption, but we'll roll with it)
manic radish
so, you know the sum of all angles, and you know the sum of the seven other angles.
waxen prawn
could it be 1440-x+y+z=1220 or is that dumb
manic radish
pretty much. just be careful with which way around.
1220 + x + y + z = 1440
waxen prawn
oh yeah
but
does it matter the order
as long as i solve it with the 3 variables
wait so could my3 equations be
x+y=90
x+z=180
manic radish
or in your case - 1440-(x+y+z)=1220 (don't forget the parentheses)
median vigil
*(tbh, you're not told it's convex which is a big assumption, but we'll roll wit...
the formula is true regardless
waxen prawn
1440=1220+x+y+z
manic radish
yep
manic radish
move on, and come back to it when you've answered the ones you do know
answer what you can
waxen prawn
what if i dont know the entire test
manic radish
yeah, not quite right there
manic radish
try substituting x+y=90 into 1220+x+y+z=1440
waxen prawn
πbro what how am i supposed to do that
i thought i take 2 of the same variables
eliminate it
then use that
and take another variable
manic radish
hint: 1220 + (x+y) + z = 1440
manic radish
π
waxen prawn
but fhat wouldnt work for all equations
whats the usual way to do a three variable system
manic radish
I mean, you could have also rearanged to get x = 90-y and then substituted it into 1220+x+y+z=1440 and the ys will cancel
manic radish
.close
safe radishBOT
manic radish
best of luck with the exam
safe radishBOT
β Original question: #help-23 message
waxen prawn
ignore my terrible lighting
but
for #4
shouldnt i have another equation
but i cant figure out what it could be
<@&286206848099549185>
ancient kernel
what is the measure of the exterior angle of a hexagon
humble thunder
?
ancient kernel
yes but we need that to solve problem 5
ancient kernel
whats a pentagon?
ancient kernel
no i know what a pentagon is
but like what do you mean by pentagon it never mentioned pentagon once
waxen prawn
what r u saying dude
u told me i had to solve #5
when im asking for help on #4
i solved problem #5
its a pentagon
ancient kernel
whats your question then
waxen prawn
no i need help on #4
waxen prawn
so i set up one equation
but im assuming i might need 2
to do a system
but i cant figure out wth the second one could be
ancient kernel
so i can see you got the equation 180(n-2)=180x-720
humble thunder
So what is the problem, you are doing correct
ancient kernel
isnt there multiple value of n
humble thunder
The equation then simplies to "n= x-2" which is answer
humble thunder
but i need to find the number of sides
I don't think you have to find numeric value
waxen prawn
oh
ancient kernel
yea
waxen prawn
so its literally just that?
humble thunder
but im not even given an x value
Yeah, so why I think nyou just had to write n=x-2
humble thunder
unless there is extra information
Yeah
waxen prawn
bro thisbis a stupid friggin packet yo
humble thunder
so its literally just that?
Btw which standard?
waxen prawn
wdym which standard
humble thunder
Which class/standard you stufying
humble thunder
Are you a high schooler??
humble thunder
Oh ok
waxen prawn
r u not from the us
ancient kernel
yea
humble thunder
No
waxen prawn
sorry im slow
humble thunder
me when i discover not everyone is from the us
Whatt??
waxen prawn
thanks for the help anyway
humble thunder
Really
waxen prawn
.close
safe radishBOT
humble thunder
thanks for the help anyway
Ok
safe radishBOT
Available help channel!
kind quartz
Can someone help with this? ill post my work in a minute i gotta take a photo
kind quartz
i keep getting this specific type of question incorrect so its definitely something im doing wrong
like the way im trying it is take both sides, ofc isolate y', etc etc. but im certainly making a mistake somewhere in the process bc yeah.
( i didnt write down the last step, but i did divide 5x^4 etc by (x-4y+4y (which i know cancels but whatever) and use that for my final answer)
(To isolate y'
spare sable
Wait I suggest multiply the denominator on LHS to both side so terms with x can be clubbed and terms Γ½ can be clubbed too
kind quartz
wdym
spare sable
Wait I show my work in some time
kind quartz
???
anyways im pretty sure the mistake im making is in differentiating y/ x-4y
but im not sure how
royal kiln
He meant multiplying (x-4y) on both sides
spare sable
Ya that's why take the term x-4y to other side
plucky elk
You need to show how you evaluated at the point (1,1/3)
kind quartz
You need to show how you evaluated at the point (1,1/3)
okay just a minute then
Hm i lowkey got a diff answer, but ill check if thats right and if it is then ill just take the l to my ego
damn ok doing it again gave me the right answer, i hate when that happens bc i then feel really awkward
kind quartz
You need to show how you evaluated at the point (1,1/3)
thanks lol, i probably just input smthn in my calculator wrong
(the correct answer was 8/9)
.close
safe radishBOT
safe radishBOT
Available help channel!
honest idol
Can someone help me understand the concept here
safe radishBOT
Can someone help me understand the concept here
honest idol
What does it really mean to write them in Cartesian coordinates
plucky elk
Set x to be the stuff multiplying by i then solve for y in terms of x
e.g. x = 2t then y = t^2 = (2t^2) * some number= x^2 * some number
errant rapids
So solve for $f(x)$ in ${\mathbf r}(t)=(2t,t^2)=(x, f(x))$
flat frigateBOT
flynger
errant rapids
??
honest idol
I just donβt understand the point of all this
All Iβm seeing rn is a vector function that graphs vectors
plucky elk
What are you asking exactly
plucky elk
Well you should read a book then
errant rapids
instead of x and y depending on t they want you to eliminate the parameter t and have x and y depend on each other
i guess
yea it isnt really that interesting
honest idol
Well you should read a book then
Why are you presupposing I havenβt read bro
Do you think I wanna waste peoples time for no reason
plucky elk
Why are you presupposing I havenβt read bro
Indeed
plucky elk
Do you think I wanna waste peoples time for no reason
Lots of people do yes
honest idol
Not me
errant rapids
I think its fair to ask what the point of something is
plucky elk
errant rapids
even if you understand it
honest idol
Also I have tried looking for it in the book but it simply doesnβt talk about it or the **intuition behind it **
plucky elk
Also I have tried looking for it in the book but it simply doesnβt talk about it...
That's not what you were asking
honest idol
instead of x and y depending on t they want you to eliminate the parameter t and...
Like for example I understand you fine here but Iβm having trouble understanding the why
plucky elk
Can someone help me understand the concept here
.
honest idol
Yes I wanna understand the concept not be told the concept
plucky elk
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/11%3A_Param...
Explained here in the first few paragraphs
honest idol
Anyway
errant rapids
Like for example I understand you fine here but Iβm having trouble understanding...
Well just as an example, there are things like functions we can describe using x and y = f(x)
while things like circles are better off parameterized since there can be multiple values of y for each x
plucky elk
Yes I wanna **understand** the concept not be **told** the concept
This is too vague to help you
shut hound
Can someone help me understand the concept here
Which concept? Parametric and Cartesian co-ordinate systems and how they relate to each other?
honest idol
Which concept? Parametric and Cartesian co-ordinate systems and how they relate ...
Pretty much yeah like how do we go vector coordinates to Cartesian coordinates with the substitution and everything like why is that the same thing
Why does it work out
errant rapids
Why does it work out
because when you graph ${\mathbf r}(t)$, you graph the points as $(x(t),y(t))$ on the $xy$-plane
flat frigateBOT
flynger
shut hound
Pretty much yeah like how do we go vector coordinates to Cartesian coordinates w...
I mean all it is, is a way of assigning identifiers to points in an n-dim space, which we're atp applying to 2 dims right
errant rapids
So ${\mathbf r}(t)=(x(t),y(t))$ on our $xy$-plane
flat frigateBOT
flynger
covert yoke
@honest idol we have y = f(x) where x is our independent variable, and y is the dependent variable. If we envision this as a pen drawing a graph, this means that we always continue moving our pen from the left to the right. There's no opportunity to double back on itself.
A perimetric equation instead makes both x and y dependent on another variable t, so this "unlocks" our pen movement, which had been confined to only moving from left to right. We can draw more complicated curves this way.
Hopefully, this gives you a bit of insight as to why we bother with this
errant rapids
<@1402083806034268161> we have y = f(x) where x is our independent variable, and...
good explanation! you can think of the third variable t here as time for example
and you can move your pen around anywhere in the 2d plane as time progresses
honest idol
<@1402083806034268161> we have y = f(x) where x is our independent variable, and...
Can you please give an example of this
Also they called it βCartesian coordinates β
covert yoke
Cartesian coordinates are just the standard square coordinate system. It's named after some dude
shut hound
Rene Descartes I believe
honest idol
Cartesian coordinates are just the standard square coordinate system. It's named...
In your explanation, where is the Cartesian coordinates part?
covert yoke
This is in contrast to polar coordinates which is a different way of labeling the plane
honest idol
Oh I see
covert yoke
But both systems I described can use either method of labeling the plane. Generally we refer to Cartesian coordinates with (x,y) and polar with (r, ΞΈ). But if we let r = f(ΞΈ) vs perimetric plots with (r(t), ΞΈ(t)) we can see it works both ways again
Just in this case your pen goes "around in a circle" vs "left to right" in the non-perimetric case
honest idol
Just in this case your pen goes "around in a circle" vs "left to right" in the n...
Algebraicly how does it look like
covert yoke
Algebraicly how does it look like
I'm not sure what you mean.
honest idol
I'm not sure what you mean.
Like here for example
Can we just do this problem and see how it goes ?
covert yoke
Well, the hint tells you straightforwardly what to do, yes?
We have (x(t), y(t)) = (2t, t^2)
honest idol
Well, the hint tells you straightforwardly what to do, yes?
Yes but the problem is that itβs still not clicking for me what the point is π
covert yoke
Like here for example
Start with (x(t), y(t)) = (2t, t^2)
covert yoke
Now we know that x = 2t, so t = x/2
Then substitute into y = t^2
You should get (x, x^2 / 4)
honest idol
Now we know that x = 2t, so t = x/2
Okay from here I get lost, why solve for t and substitute?
covert yoke
Which is just all points (x, f(x)) such that f(x) = x^2 / 4
covert yoke
Okay from here I get lost, why solve for t and substitute?
Because the idea is we want to transform from r(t) = x(t) i + y(t) j into y = f(x)
In other words, eliminate t and solve for y
This step eliminates t
honest idol
Because the idea is we want to transform from r(t) = x(t) i + y(t) j into y = f(...
Okay so
Arenβt 2t and t^2 separate equations ?
Are they?
covert yoke
They are
honest idol
So why is their t the same
covert yoke
Well, more specifically, they are expressions
The equations are x = 2t and y = t^2
Through t we can relate x and y
covert yoke
So why is their t the same
Does the above make sense?
honest idol
Because the idea is we want to transform from r(t) = x(t) i + y(t) j into y = f(...
So basically we want to express the curve made by the vector equation as a function on its own ?
covert yoke
The t is the same because these two equations were pulled from the single equation, r(t) = 2t i + t^2 j
So when we select any particular value of t we select it for both x and y
covert yoke
So basically we want to express the curve made by the vector equation as a funct...
Yes, exactly
honest idol
Yes, exactly
And is that function called a Cartesian coordinate or Cartesian equation?
covert yoke
The coordinates are called Cartesian coordinates. I've never personally heard of a Cartesian equation. Generally, I hear this referred to as parametric or standard.
Parametric being the vector valued equation as a function of time, vs standard as the y = f(x)
Perhaps they are using Cartesian equation to refer to what I know as standard, which is fair.
It would make sense
honest idol
Perhaps they are using Cartesian equation to refer to what I know as standard, w...
I get this, but thereβs something still not clicking about how this just magically works out this way
errant rapids
Theres a problem with your $t=\sqrt{y}$ technically
flat frigateBOT
flynger
covert yoke
I get this, but thereβs something still not clicking about how this just magical...
Is your doubt that you don't understand why the steps were taken or is it that you don't understand why the steps necessarily work?
honest idol
Is your doubt that you don't understand why the steps were taken or is it that y...
I think both tbh π
covert yoke
Well, for the second, they don't necessarily work. There are some parametric equations that cannot be faithfully reproduced in the standard form, (recall the locked vs unlocked pen from earlier), you can often use a few different equations in the standard form to describe the curve, but in general you might lose some information.
But typically you will be given examples that do work to solve
For instance, the equation you posted earlier does work.
But at least one of the others you posted in your image will require a few equations in the standard form to describe completely
errant rapids
I think both tbh π
im gonna do my best here but sorry if this isnt the best way for you to understand
so you understand what functions are correct?
honest idol
so you understand what functions are correct?
They are a relation where a set maps to another set
errant rapids
yes!
most important
every element of the input set maps to exactly one in the output
honest idol
Yes
errant rapids
The special type of relation that does that
ok cool
then uh
You have that 2t = x
if you were to describe that using functions, do you see how it goes both ways?
like x is a function of t and vice versa
its bijective if you know what that is
honest idol
like x is a function of t and vice versa
Whatβs the vise versa
honest idol
If you solve for t then yes
flat frigateBOT
flynger
errant rapids
But if you were to try to relate t in terms of y
$t=\pm \sqrt{y}$
since the square rooting a square could lead to either pos or neg, we dont know
flat frigateBOT
flynger
errant rapids
sorry typo
honest idol
Yeah I agree you can solve for either variable and the output can map to the input right
errant rapids
thats not a function since two values of t per y
errant rapids
**flynger**
we have $t=\sqrt{y}$ and $t=-\sqrt{y}$ would have to be mapped to from y
flat frigateBOT
flynger
errant rapids
In this case its much much easier to just leave it as $y=t^2=f(t)$
flat frigateBOT
flynger
errant rapids
if $t=g(x)$
flat frigateBOT
flynger
errant rapids
$y=f(t)=f(g(x))$
flat frigateBOT
flynger
errant rapids
And now you have $x$ and $y=f(g(x))$ and no more t
flat frigateBOT
flynger
errant rapids
so we tried to make one a function of t ($y=f(t)$ in our example)
and make the other $t = g(x)$
does that kinda make sense?
errant rapids
then $y=h(x)$
flat frigateBOT
flynger
errant rapids
because we plugged t in
honest idol
we have $t=\sqrt{y}$ and $t=-\sqrt{y}$ would have to be mapped to from y
So youβre saying t canβt be a function of y here ?
errant rapids
exactly!
because there would be two values of t per y
thats why we avoid that and keep it as $y(t)$
flat frigateBOT
flynger
honest idol
because there would be two values of t per y
Yes
errant rapids
because there would be two values of t per y
remember that a function of y here would only have one value of t per y
flat frigateBOT
flynger
errant rapids
$y=f(t)=t^2$
flat frigateBOT
flynger
flat frigateBOT
flynger
errant rapids
$y=f(t)=f(g(x))$ and the $t$ is completely gone
since $f(g(x))$ is only in terms of $x$
errant rapids
we're swapping out the input t for some expression of x
leaving no t once it evaluates the function
and x is just... well... x
honest idol
Iβm having trouble because I imagine x = 2t and y = t^2 as two different functions
Like 2t is a line
errant rapids
thats completely fine!
errant rapids
but what happened if you paired the two graphs together
one as a the x coordinate and one as the y
errant rapids
Iβm having trouble because I imagine x = 2t and y = t^2 as two different functio...
you are not wrong at all!
its just that we're combining the two graphs by t
think of your pen
drawing on the paper
honest idol
Ok
errant rapids
what if i only told you how much you're moving to the right and left?
maybe the graph looks like x=2t
in a diff scenario, i onlyt show you how much you're moving up and down
you can visualize that as y=t^2
but it turns out you were moving both vertically and horizontally at the same time
so if you combine those two graphs by the same scale (time) we can figure out the motion of your hand overall and not just up and down and left and right
I think maybe you're confused because you need to try visualizing graphs on a 1d scale
honest idol
Wait so we have a line and a parabola right
errant rapids
what does the graph of a line look like in 1d? on a number line
honest idol
I think maybe you're confused because you need to try visualizing graphs on a 1d...
Iβm confused on the following tbh:
We have a vector function with coordinates being literally two different curves..
But then each output of those curves generate a new curve for the vector function?
So then from there.. crashes
errant rapids
Instead of graphing a function in 2d, consider visualizing it like the above
as a point thats moving left and right
as t changes
does that make sense?
its the output
at a time t
Sort of like the frames of a video
the value of x at a given time t
honest idol
Okay
honest idol
But on two separate curves right
errant rapids
sure individually you could
errant rapids
So you could draw y like this
another number line
and a dot going across it in some pattern over time
going up and down in some pattern
honest idol
another number line
Those number lines are supposed to resemble the functions right
honest idol
going up and down in some pattern
Yes
errant rapids
Those number lines are supposed to resemble the functions right
yea
so what they're doing then
since these are two indeed DIFFERENT functions
theyre saying but what if they both were moving according to the same input (like time)
honest idol
Yes
errant rapids
and the dot was the same dot
now since theyre different, we need TWO axes
one for the movement of the function x and one for y
honest idol
Okay
errant rapids
but if its the SAME dot
moving according to the SAME time variable
we can track x and y at the same time
errant rapids
See how if you ignore y and go left to right its the same as the number line for x?
and if you ignore the left to right axis its the same as the number line for y
honest idol
Yea?.
errant rapids
So we're creating a new trajectory using the trajectory in two independent directions (x and y)
by combining them
Thats it. thats your vector function of t
honest idol
So we're creating a new trajectory using the trajectory in two independent direc...
So are you saying that the line 2t and the parabola t^2 have outputs for t
The line output is a coordinate point and the parabola curve is the second coordinate of the vector function ?
errant rapids
yes thats one way to put it
honest idol
yes thats one way to put it
And then what from here? My bad I donβt mean to make you explain too much itβs just Iβm trying to connect things
errant rapids
welp it looks like i failed...
If that isnt sufficient i have no idea
what is missing
honest idol
welp it looks like i failed...
I really appreciate your help man but for some reason itβs not clicking..
Iβm not not tracking because thereβs a lot of things Iβm not understanding yet
Can we start from the very beginning I guess
errant rapids
welp
i dont know how to explain it then...
i have a question though, has everything made sense until you arrived at this topic
in the math you've learned so far
honest idol
https://www.desmos.com/calculator/it2coxxjiq
Does this help a little bit?
This is the second coordinate
Right
Second coordinate of the vector function
covert yoke
I changed it slightly
honest idol
t^2
covert yoke
So the red line is the parametric function where t is allowed to vary from -3 to 3
The green dot is a specific value of t
The orange and black dots are just the x and y coordinates vs time
And finally, the dotted line is the standard form of the equation
@honest idol does the above make sense?
Due to technical limitations, I had to use "a" as the variable for the points instead of t.
covert yoke
It is!
honest idol
Is that the combination of 2t and t^2
covert yoke
yes
honest idol
Okay alright
honest idol
<@1402083806034268161> does the above make sense?
Yes
covert yoke
Does the green vs orange and black points also make sense?
As one of the friction points seemed to be that you didn't understand why/how the two equations were combined to make a single curve
honest idol
Does the green vs orange and black points also make sense?
Sure
covert yoke
Alright, so can you do the next problem then?
honest idol
Alright, so can you do the next problem then?
I can but bro idk why Iβm still stuck on intuition
The problem is some things that I donβt understand are being explained
Wait can I tell you my thought process
We have a vector function r(t) = 2ti + t^2j
With coordinates (2t, t^2)
But I see this as just the normal plotting for the vector function
Not necessarily two separate functions
x = 2t and y = t^2
But we label them as such because they are the x and y coordinates on a Cartesian space
We have the same input t
So if t is the same
And t is = x/2
this β that x/2 is also the t in the other equation
Because t is the same
In both equations
Up to here it makes sense
I see why we can plug in x/2 in the other equation
But I donβt understand how it all of a sudden represents the curve created by the vector function
Like you just expressed the y coordinate in terms of the x coordinate
honest idol
Like you just expressed the y coordinate in terms of the x coordinate
@covert yoke
covert yoke
I apologize, some mod shit needs my attention, I'll be gone for like 5-10 taking care of it
honest idol
I apologize, some mod shit needs my attention, I'll be gone for like 5-10 taking...
Itβs all good take your time
covert yoke
Like you just expressed the y coordinate in terms of the x coordinate
This is the key though
Any Cartesian equation is an equation in the form (x, y) and recall that y = f(x) so (x, f(x))
In this case y = x^2 / 4 is all points (x, y) such that y = x^2/4. In other words, it's all points (x, x^2/4)
You went from (2t, t^2) found a mapping t -> x/2 and wound up with (x, x^2/4)
honest idol
You went from (2t, t^2) found a mapping t -> x/2 and wound up with (x, x^2/4)
can I ask something
covert yoke
Of course
honest idol
Of course
So originally we had 2t = x
If we solve for t.. what does this mean in algebra? It might be a dumb question but there are some fundamentals I probably forgot
covert yoke
I'm not sure what you mean by "what does this mean in algebra?"
honest idol
I'm not sure what you mean by "what does this mean in algebra?"
Because I see that we solved for t, which is 2/x then we plugged it for the t in the other equation , pre asserting the fact that the two ts are the same . So x/2 is t in both equations
covert yoke
t = x/2 not 2/x
honest idol
Yes mb
covert yoke
Well, let's consider all pairs of numbers (x,t) such that 2t = x.
So we can start choosing values of t and then figuring out what x should be, so if we choose t = 1 we get 2, so (2, 1) is a point. If we choose t = 2 then we get (4, 2) in a point. If we choose t = 3/2, we get (3, 3/2). If we choose t = Ο we get (2Ο, Ο).
Does this make sense? Can you envision a bag where all of these points exist, and we can reach out and pick out a random point that fits the equation, (42, 21) for instance?
@honest idol
honest idol
Does this make sense? Can you envision a bag where all of these points exist, an...
Sure yes
covert yoke
Ok. Now if we solve instead for t, we get t = x/2
Does our bag of points change when we do this?
@honest idol
honest idol
Does our bag of points change when we do this?
Uh
Yeah thatβs what Iβm trying figure out
covert yoke
Would you like the answer to the riddle? Or would you like to reason it through?
I wouldn't want to deny you the joy of figuring it out, after all.
honest idol
Reason it through
covert yoke
Ok, so if I pick a point (2, 1) this is in our original bag with the rule x = 2t.
We have 2 = 2(1)
What if the rule is t = x/2 does this point (x, t) = (2, 1) still follow this rule?
honest idol
What if the rule is t = x/2 does this point (x, t) = (2, 1) still follow this ru...
Ohhh wait no itβs the same points
Because the equation is still true
covert yoke
This is what defines a valid manipulation of an equation.
The set of all of the points satisfying the equation before the function manipulation takes place is the same as the set of all points after satisfying the equation after.
covert yoke
Because the equation is still true
If the equation wasn't true, then we have made a mistake in our algebra
honest idol
The set of all of the points satisfying the equation *before* the function manip...
Okay let me ask this
Why is t = x/2 the same value as t in y = t^2
covert yoke
I explained it earlier, but here's another try:
Our original equation was a single equation: r(t) = 2t i + t^2 j. From this we extracted two separate equations, one for each vector component. x(t) = 2t, and y(t) = t^2.
Because they came from the same original equation, these two equations are linked together by the variable t. We don't just have two equations where t can be different between the two. We have two linked equations, and t is shared between them.
@honest idol ^ does that make sense?
honest idol
Yes I get that part
Lemme ask it this way
If you solve for t in either equation
Will you get the value of t you first plugged in?
Or am I asking the wrong question
covert yoke
Ok, so let's do it this way. We have y = x^2 / 4, yes? We know that the point (x, y) = (2, 1) satisfies this equation.
Your question is, can we find a value t0, such that there is some (t, x, y) = (t0, 2, 1) that satisfies the equation r(t0) = 2 i + j, yes?
We know x = 2t and y = t^2, so we have 2t = 2, so t = 1. And we have 1 = t^2 so t = Β±1. Both equations must be true, so we find that such a t0 is 1.
@honest idol sorry if it's confusing
It seems as if you are running out of steam, that is ok. It is also late here as well
honest idol
It seems as if you are running out of steam, that is ok. It is also late here as...
Yeah itβs alright I really appreciate your help tho
covert yoke
No worries, perhaps a good night's rest will help you see this with fresh eyes. The mind has a way of subconsciously working out things during a sleep
safe radishBOT
@honest idol Has your question been resolved?
honest idol
No worries, perhaps a good night's rest will help you see this with fresh eyes. ...
Youβre right
Thanks again
half elk
safe radishBOT
half elk
GMT doesn't work
This doesn't work
Diameter of the small white circle
The height of the vertical line of segment 8 is β2*8 =4. R=2, d=4
Diameter of the whole semicircle
8+4+2 = 14
Radius = 7
Area of large semicircle
oi(7^2) = 49pi
49pi/2 = 24.5pi
Area of the white circle = 4pi
24.5pi-4pi = 20.5pi
red raft
The difference of radius of bigger and smaller circle will be 5
half elk
The difference of radius of bigger and smaller circle will be 5
Wait I'm sorry why
half elk
So
3^2 + r^2 = (R-r)^2
And R-r = 5
So r=4
red raft
Yup
safe radishBOT
safe radishBOT
Available help channel!
tropic hatch
How to construct a Cantor-like set that has Lebesgue
measure zero, yet Hausdorff dimension 1
safe radishBOT
How to construct a Cantor-like set that has Lebesgue
measure zero, yet Hausdorff...
@tropic hatch Has your question been resolved?
obsidian oracle
How to construct a Cantor-like set that has Lebesgue
measure zero, yet Hausdorff...
Do you know what the hausdorff dimension of the cantor sets are
tropic hatch
Do you know what the hausdorff dimension of the cantor sets are
Yes log2/log3
Ah you mean generally
No i don't
obsidian oracle
Ok, so call K_epsilon the cantor set where, in the first step, you remove from [0,1] the central interval of length (1-epsilon)/2
So that we get [0,epsilon] U [1-epsilon, 1]
And so on...
0 < epsilon < 1/2
And when epsilon = 1/3, you get back the regular triadic cantor set
Now, to find its hausdorff dimension, you can do the naΓ―ve reasoning "If I multiply all lengths by X, I end up with Y times as much quantity"
So X^d = Y
And you find the hausdorff dimension d
safe radishBOT
@tropic hatch Has your question been resolved?
tropic hatch
Now, to find its hausdorff dimension, you can do the naΓ―ve reasoning "If I multi...
Well those sets arent the cantor sets with constant dissection? I know for those sets that dim=log2/log(2/1-Ξ΅)
So cant be 1
obsidian oracle
So cant be 1
Yes, but you can have an idea of what you can do with it if you let epsilon -> 0 in the dim
tropic hatch
Yes, but you can have an idea of what you can do with it if you let epsilon -> 0...
Well this could make some problems
What if making a cantor-type set with a sequence?
The seq could be such that measure is 0 but i have always problem to show the Hausdorff meas is >0
obsidian oracle
The hausdorff dim can be 1 while the hausdorff measure is 0
As you know the 1-hausdorff measure in R is the lebesgue measure
And yes there's some idea in using a sequence of epsilons in each step of the creation of the cantor set
Such that epsilon_n -> 0
(Those epsilon_n are the relative sizes of the intervals you're removing)
Another idea is to just start with some pre-built cantor sets K_epsilon_n
And ||taking their union||
tropic hatch
And yes there's some idea in using a sequence of epsilons in each step of the cr...
And then how can i find its dimension using its union? Shouldn't be the intersection? I cant handle this
obsidian oracle
And then how can i find its dimension using its union? Shouldn't be the intersec...
It is union
If you take the union of two sets, do you have any idea of how the hausdorff dimension of their union is gonna be compared to their respective dims ?
And then same question for a countable union
tropic hatch
And then same question for a countable union
Well, till now ive that if dimEkβ€a then dim of union β€a. I think that dim of union is generally the sup of dims, right?
obsidian oracle
Well, till now ive that if dimEkβ€a then dim of union β€a. I think that dim of uni...
Well for countable union yes
obsidian oracle
Well, till now ive that if dimEkβ€a then dim of union β€a. I think that dim of uni...
you just said dim(countable union) <= sup(dims)
The other way is true because of sigma-subadditivity of every hausdorff measure (and so a countable union of sets of alpha-measure 0 is still 0)
so
$dim\left(\bigcup_{n}K_{\varepsilon_n}\right) = ...$
flat frigateBOT
Rafilouyear2026
obsidian oracle
while $\lambda\left(\bigcup_{n}K_{\varepsilon_n}\right) = ...$
flat frigateBOT
Rafilouyear2026
safe radishBOT
@tropic hatch Has your question been resolved?
tropic hatch
The other way is true because of sigma-subadditivity of every hausdorff measure ...
Ahhh now i got all of these. Yes, ur right. For a=sup etc with what u said we have that dimU=supdims and then for the union of cantor sets C_1/n we have what we need
safe radishBOT
safe radishBOT
Available help channel!
coral wing
I need help understanding how to do these. I understand what the congruent theorems are but Iβm not sure how to tell which one it is when the congruent angles/sides arenβt already marked. I know this is easy but itβs one of those things where Iβm missing something lol
safe radishBOT
I need help understanding how to do these. I understand what the congruent theor...
glacial cairn
They are marked though? That's the point of the exercise
coral wing
They are marked though? That's the point of the exercise
I mean not all 3 angles/sides are marked. for example the one marked SAS basically gives you the answer and I donβt understand this concept enough to get the answer unless itβs presented like that
okay so for example, #1
I canβt tell if itβs SAS or SSS or something else, I think it might be SSS but like Iβm not %100 sure
because idk if I can mark the angle as congruent between the two congruent sides
but they share the 3rd side so itβs probably SSS? Iβm just not confident
glacial cairn
Yes, it is
You just need to see what the triangles have in common
In 1 they share a side, so of course that side is the same length for both
coral wing
In 1 they share a side, so of course that side is the same length for both
waiiiiit this is looking a lot easier then
could u check my answers after i do them rq?
coral wing
Sure
- SSS 2) SAS 3) ASA 4) SAS 5) SSS 6) ASA
Iβm not sure if theyβre right cus I kinda only used 3 different ones lelll
glacial cairn
Looks good to me
coral wing
o okay I think the next questions use the other ones
let me answer the next slide and then Iβll be good
coral wing
Looks good to me
- AAS 8) ASA 9) HL 10) ASA 11) SAS 12) SSS
glacial cairn
Where are the two angles in (8)?
coral wing
Where are the two angles in (8)?
WAIT I think it would be sas
not asa
I was confused because ASS isnβt used so like I thought I could use the top angles instead
But SAS would make more sense maybe?
I mean
omg
SAS
I canβt type
glacial cairn
Is the angle in between the two equal sides?
coral wing
Is the angle in between the two equal sides?
This was my thought process
I feel like it might actually be SAS though
hmmm I dunno
glacial cairn
What do these triangles share?
coral wing
A side
glacial cairn
Yeah and what is marked?
coral wing
a side and an angle?
glacial cairn
Yeah, and so in what order do they come?
coral wing
well the only possible order is SAS
glacial cairn
Oh like in the options here?
glacial cairn
Why is that AAS weird
coral wing
cus there are multiple stacked lol
glacial cairn
Of the same?
coral wing
Itβs so u can use multiple if u feel like thatβs correct
yep
Itβs like that with all of them tho
Theyre just stacked better
coral wing
Itβs a right triangle with like
a congruent hypotenuse or something
it has a right angle and 2 congruent lines
like that
glacial cairn
Ok I found it, Hypotenuse-Leg
glacial cairn
No, 8 is definitely ASS or SSA (same thing)
Either whoever set this up messed up or there's an option hidden somewhere idk
Or maybe SAS is considered the same thing
These names are kind of stupid anyway
coral wing
ASS isnβt a thing Iβm pretty sure
like itβs not a valid thereom I think
neither is SSA
so itβs either not enough info or ASA
glacial cairn
Oh ok I think I know why that is
glacial cairn
Give me a minute I'll draw it out
coral wing
alright
glacial cairn
Right so these three angles I marked are equal
Additionally, BC = BD = AE
You can see that the triangles ABC and BAE are congruent, but not the triangle ABD
coral wing
hmm okok I see
glacial cairn
Yet AB = AB, BD = BC, and angle BDA = angle BCA
That's the reason SSA is not a valid rule
So in your exercise, 8 is "not enough info"
The issue I have with this is that the rule still works if the two triangles are either both acute or both obtuse
glacial cairn
And they are clearly both drawn acute there
coral wing
ASS or SSA definitely looks like it would work so that is weird
at least in this problem I mean
glacial cairn
The issue I have with this is that the rule still works if the two triangles are...
Because of that
glacial cairn
But anyway, the idea is that you can't determine that they actually are acute because you are not given the other side or angles
coral wing
the rest are correct though right?
coral wing
But anyway, the idea is that you can't determine that they actually are acute be...
u canβt do anything with ur eyes in geometry I swear ππ
glacial cairn
10 doesn't look right to me
glacial cairn
u canβt do anything with ur eyes in geometry I swear ππ
You can but I'm guessing they want to teach you not to rely too much on them
Like some angles can look right but aren't and so on
coral wing
10 doesn't look right to me
AAA isnβt a choice so I assumed since the angles were congruent the side between them would be as well
coral wing
Like some angles can look right but aren't and so on
I suppose
glacial cairn
AAA isnβt a choice so I assumed since the angles were congruent the side between...
No, that's the difference between congruent and similar
coral wing
No, that's the difference between congruent and similar
Iβm not familiar
we havenβt learned that yet
so Iβm not sure what the difference is
glacial cairn
I'll draw it hold on
AB and ED are parallel, so angle CAB = angle CDE, and angle ABC = angle DEC
And of course angle BCA = angle ECD
Yet the two triangles are clearly not the same size
coral wing
AB and ED are parallel, so angle CAB = angle CDE, and angle ABC = angle DEC
ooooohhhh I understand
that makes sense
ok well I have no idea what 10 would be then
glacial cairn
Well they are not necessarily congruent, just similar
glacial cairn
No, they are not marked as such
coral wing
soooo not enough info?
glacial cairn
Yes
coral wing
because ASA an AAS are the only options that would include both marked angles
alright
glacial cairn
AAA is sufficient for similarity, not for congruency
coral wing
Does this look good?
glacial cairn
(for congruency you need some kind of size, so a side length)
coral wing
I have 5 slides of proving with statements after this πππ so fun
its ok its not THAT many right
glacial cairn
Does this look good?
Yes that's correct
coral wing
Yes that's correct
okay thankyou!
okay well I have to go figure out the rest now
I really appreciate the help!! I understand it much much better now seriously
glacial cairn
You're welcome
coral wing
.close
safe radishBOT
safe radishBOT
Available help channel!
worn geyser
Show that $\cos(u)\cdot\cos(v) = \frac{1}{2}(\cos(u-v) + \cos(u+v))$ by using the formulas for $\cos (u+v)$ and $\cos(u-v)$.
safe radishBOT
Show that $\cos(u)\cdot\cos(v) = \frac{1}{2}(\cos(u-v) + \cos(u+v))$ by using th...
flat frigateBOT
dream
signal crane
you're given the formulas for cos(u-v) and cos(u+v) right
worn geyser
But zero wouldnt remove anything
signal crane
for nwo can you try just substituting the known formulae in the RHS
worn geyser
Oh okai
broken forum
<@&268886789983436800>
worn geyser
<@&268886789983436800>
vivid stream
Ty
gleaming moon
why would someone fall for this scam in a maths server lol
worn geyser
for nwo can you try just substituting the known formulae in the RHS
Ohh i got it
Thanks!
signal crane
np
worn geyser
.close
safe radishBOT
safe radishBOT
Available help channel!
glass talon
hi
safe radishBOT
@glass talon Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
lavish skiff
not understanding the last step
safe radishBOT
not understanding the last step
edgy breach
do you understand that cosy dy = dx
lavish skiff
no