help-49

[carmine sigil]

You have 3 forces acting on this point, yes?

If this is the case, why do I only see two forces written out?

[lime oracle]

ooooo

[lime oracle]

waittt may i ask what u meaan?

[lime oracle]

this was my work for vector resolution parttt

[carmine sigil]

Your F_R should be F_1 + F_2 + F_3

[lime oracle]

OHHHH REALLY

okayokayy i will change rn rn

[carmine sigil]

Well, hold up

[lime oracle]

F3 will be added fr?

[carmine sigil]

Well, I'm still orienting myself in the problem

So my understanding is that F_3 was chosen by experiment to balance F_1 and F_2 right?

[lime oracle]

okay i rlly appreciate it

[lime oracle]

yess this is what we got from our force table experiment

its our own values. then my other groupmate is assigned to make a percent error in which she already did

[carmine sigil]

So you have a theoretical value for F_3 which should have 0 = F_1 + F_2 + F_3

And then you have the experimentally obtained value which might be slightly different

[lime oracle]

basically what im answering rn is Part I (i did part II in google docs)

Part I is in paper and part I is about putting in data

[lime oracle]

OOO

our teacher put this in Part II, so F1 + F2 will be added in Part 2 because im looking for the calculated RF right

while in part 1 i add F3 too?

[lime oracle]

this one

[carmine sigil]

Ok so, let's go back to the assignment and check

[lime oracle]

okaaayy ^^

[carmine sigil]

We're on this step right?

[lime oracle]

ahhh the pic wont load hold onn

[carmine sigil]

This part in particular

I believe from context that they want F_1 + F_2 here actually

Which in this case, you're fine

Let me check the rest of your work

[lime oracle]

supposedly but im confused bcs part 1 is data table (ive been confused about this for daysss bro)

[lime oracle]

okaayyy gotchuu

[carmine sigil]

I think you understand what's going on better than you give yourself credit for.

[lime oracle]

OHHH does it mean that the final answers will be written in the dataaa

omgomg

[lime oracle]

im quite slow but if this is a compliment im so happy lolol

[carmine sigil]

So can you share how you got F1 = 0.9i + 0.5j ?

[lime oracle]

bsaed on my solutionss here i used the answers there

(however i added a comment in google docs bcs one should be 1.3)

[carmine sigil]

Those look good to me then

1 sig fig is probably not the best though

You should use full precision for intermediate steps and round only at the end

Typically

[lime oracle]

yesyesss our teacher actually prefers us to do 1 decimal place round off, but i talked to my other friend and i agreed that ill talk to my teacher about the rounding part if full precision should be applied

[lime oracle]

thank youuu, but the angle relative to x axis is rlly 46.6 rightt

133.4 was only from the videoooo

[carmine sigil]

Well, let's do this:

,w exp(i pi *30/180) + 1.5 exp(i pi 60/180) in polar form

[grand pond]

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[carmine sigil]

,w 0.8389 * 180/pi

[grand pond]

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[carmine sigil]

The angle is closer to 48 degrees

(the error is due to early rounding)

[lime oracle]

THANK YOUU

yesyesssz

aghhh something just added to this bro 😭

[lime oracle]

my friend said if we use this (46/48)

it will give 80 percent error

but we need 50 lowerrrr

[carmine sigil]

,w (48-46)/48

[grand pond]

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Requested by omnipotententity

[carmine sigil]

That's 4% error

[lime oracle]

waitt nono my bad i wasnt clearrr, i meant if we use 46.6 angle or 48.8 angle (theres a slight diff in error of rounding)

it ends up with 80 percent error but

[carmine sigil]

48.1° is closer than 48.8°

[lime oracle]

my groupmate made this one, its less, but the values are different

[lime oracle]

Oooohh

[carmine sigil]

Ok so 48.1° is the way the force is pointing from the first two pulleys

[lime oracle]

yesyess i will write that downn

[carmine sigil]

In order to find the direction of the force from the counter pulley you have to add 180° to it

[lime oracle]

OHH

[lime oracle]

so this step is only for the percent error?

[carmine sigil]

This is because the first two weights are pointing up and left

So the other weight is pointing down and right to balance them

[lime oracle]

OMGGGG

WAITTTTTTT

I UNDERSTAND

genuinely our teacher hasnt even taught us this part yet but this is due soon so we're left to learn by ourselves eueue

[carmine sigil]

Well, hopefully you can figure it out 😃

[lime oracle]

thank yyouuu 😭 idk which to use 46.6 or the one in percent error agh

[carmine sigil]

Also don't be so harsh on your teacher. It's tough being a teacher these days.

[carmine sigil]

So 48.1° is the direction of FR

F3 was determined from experiment, so it's whatever you actually found

But the theoretical value of F3 should have a direction that is FR + 180°

(because it has to balance FR, so it should be pointing in the opposite direction)

[lime oracle]

yessss dont worry 🙏 i swill still ask her nicelyyy

[lime oracle]

im ngl im already starting to get lost 😢

[lime oracle]

wait im slwoly getting ittt

it will be subtracted to 180 which is 133? what i got earlier from the paper? eueue

[carmine sigil]

Not subtracted from

Added to. The subtracted from thing is a different operation

[lime oracle]

thank youuuu so much 😭

[carmine sigil]

You ever heard the expression, "I'm gonna turn my life around 180 degrees"?

[lime oracle]

nono but after this i will learn that because im so stressed HAHAHHAHA

[carmine sigil]

"He turned 180 degrees and walked away"?

[lime oracle]

OO YES

[carmine sigil]

Essentially, if you turn 180° you are facing exactly the opposite direction that you started

If someone is pulling a rope

And the rope is attached to you

Then in order to fight back, you'll want to turn away from the rope and walk away

That's turning 180° degrees

And that's what is happening in this problem

F1 and F2 are combining to make FR

And F3 is pulling against FR trying to find the balance

[lime oracle]

bro you are blessed from the heavens bro

im not even kidding ure saving lives right now

[carmine sigil]

So if FR has some angle θ, then we want to turn 180° away from θ which is 180° + θ

And that's where we should find F3

In reality, because experiments are messy and imprecise, we won't actually get the experimental angle we want for F3, there will be some error

And the error between those two angles, where F3 is, and where it should have been, is the error you want to find

Does this help?

[lime oracle]

THANK YOU

IT DOES SO MUCH.

IM eternally grateful bro

[carmine sigil]

Anyway, it's getting a bit late here, so I'm going to go to bed. Best of luck!

[lime oracle]

thank youuu!!! 😭 goodnight omggg

im so so grateful i hope u have sweet dreamss

[carmine sigil]

Thanks 😊

[midnight plank]

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[desert siren]

Whatchu tried

[lyric charm]

jee ahh question 💀

[desert siren]

Is just long working i think

Two lhls two rhls

Equate

[lyric charm]

yeah and you're supposed to do it in like 50 nanoseconds right

[woeful turret]

yes i was thinking along these linse

[desert siren]

Depends

[lyric charm]

rather than 10 which is the standard time limit

[desert siren]

This is prolly adv

Not mains

So u can take like 6 mins

[woeful turret]

this is just a worksheet

[desert siren]

Kek

[desert siren]

Yh try it then

[woeful turret]

im not getting how to find the rhl and lhl

wont they be the same

[desert siren]

How

[lyric charm]

F(x) = something when -1 < x < 1, something else when x = ±1 and something else when |x| > 1

find this representation

[woeful turret]

ohk let me try

[woeful turret]

oh yes this makes sense

x>1 i got F(x) = g(x)

x=1 i got F(x) = f(x) + g(x)/2

now i equate these 2 at x=1 right

ohk and then do same condition for -1

got it

thanks @lyric charm im surprised i wasnt able to do this

.close

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[woeful turret]

@desert sirenthx g

[lyric charm]

parentheses

[desert siren]

How did u get an F(x) variable wrt x at a constant x

[woeful turret]

i didnt put x=1 in the functions

[desert siren]

O

[woeful turret]

that x^2n becomes 1 thats all

[desert siren]

Alright

[woeful turret]

(f(x)+g(x))/2

[midnight plank]

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[vale zephyr]

is this mathematically correct? like does it not mess with the notation?

[slow thorn]

yes

[vale zephyr]

aren't we supposed to first use the dot product and then proceed?

[slow thorn]

distributive property

[vale zephyr]

wym

[lyric charm]

you know you can pull constant factors out of an integral

[vale zephyr]

this is how it should be right?

[vale zephyr]

yea

[vale zephyr]

i'm just not sure if this is the right way

like

can we do this?

or do we first expand it using the dot product formula and then pull out all the constant stuff?

[slow thorn]

exactly

it's just that a lot of steps are hidden

[vale zephyr]

ah ok

i see

thanks

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[pallid skiff]

how do i find eccentricity of an ellipse with semi major axis a and semi minor axis b

[pallid skiff]

how do i find the directrix

of a conic

basically

because eccentricity is defined as ratio of any point from a fixed point to a fixed line

that fixed line is directrix if i am correct

[midnight plank]

@pallid skiff Has your question been resolved?

[fallen sparrow]

The definition of an ellipse states that it is a locus of points whose ratio of distance from a given point to distance from a fixed line is less than 1

And constant

Called eccentricity

[pallid skiff]

why is it defined like that

[fallen sparrow]

e = sqroot(1-b^2/a^2)

[pallid skiff]

and how do i find the focus and directrix in terms of eccentricity

[pallid skiff]

i know this

they teach this in highschool

[fallen sparrow]

More like we found it shaped like that when we applied the condition that I told

And it popped up very frequently in astronomical physics so we decided to name it

[midnight plank]

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[fallen sparrow]

Not that it wasn't there before astronomy

[fallen sparrow]

If its the standard ellipse, then foci are (ae,0) and (-ae,0)
Directix has eqn x = a/e and x = -a/e

[midnight plank]

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[pallid skiff]

why

[pallid skiff]

ah

i get it

we probably used to

minor and major axes

to come up with that

[fallen sparrow]

Yeah

Coordinates of focii were found to be in terms of a and b which simplified to ae when we defined e

[midnight plank]

@pallid skiff Has your question been resolved?

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[severe grail]

have i done this correctly

[jaunty ivy]

sqrt(t^2) = |t|

[silent zodiac]

Ye

But other than that

Its good

[gaunt nimbus]

its square of sqrt(t)?

[gaunt nimbus]

is there a need for absolute here?

[severe grail]

yeah

im confused why he used absolute

[civic gazelle]

no

[gaunt nimbus]

sqrt(t) is always positive, no?

so op is correct

[severe grail]

so i got this answer correct?

[civic gazelle]

ye

[potent veldt]

I assume so, like the latex symbol \sqrt

[severe grail]

where is the mistake?

[jaunty ivy]

nevermind, it is correct

[severe grail]

i keep confusing my self with f(g(t)) and g(f(t))

[silent zodiac]

It is fine

[pine wave]

some reservations

[pine wave]

to show that the final expressions are not the same, you can subtitute a specific t into the expression, say t=1

and checking the outputs are obviously different

[severe grail]

can u texit an example

if u dont mind

[pine wave]

substitute $t=1$

$\frac{6}{1+1}=3$ and $\left(\frac{6}{1^2+1}\right)^{1/2}=\sqrt{3}$

[grand pond]

Element118

[severe grail]

so t could be any number

[pine wave]

nonnegative number

since you get some issue with negative numbers

[severe grail]

oh yeah the function on the left cannot be undefined

[pine wave]

or you can talk about when the functions aren't defined

because it asked you "and hence show that", you might want to be a bit more verbrose with your working

so at least do something instead of simply asserting the two expressions aren't the same

[severe grail]

so show the outcomes are different?

[pine wave]

to show the two expressions are different

because sometimes the same expression just looks different

[severe grail]

can u clarify this real quick

why non negative number? i mean ik u cant put -1 as it wil be undefined

[pine wave]

you can put 0 right

nonnegative means $\geq0$

[grand pond]

Element118

[pine wave]

read it as non-negative

[pine wave]

if the expressions were, say $\sin(3\theta)$ and $3\sin\theta-4\sin^3\theta$ it's not so obvious they are same or different

[grand pond]

Element118

[severe grail]

I havnt learned trig yet.. 😭

[pine wave]

oh hmm maybe more relevant example

if the expressions were, say $\frac{x^2-2x+1}{x-1}$ and $x-1$, it's not immediately obvious they are the same

[grand pond]

Element118

[severe grail]

thats true, simplify to see if they are the same

thanks guys

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[queen verge]

someone help me w my homework 😓😓

[slender walrus]

what have you tried

[queen verge]

uhh

nothing yet, imt rying to find my notes rq

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[neat silo]

Prove that $f(x) = \log_{2} {\left(t + \sqrt{t^{2} + 1}\right)}$ is an odd function

[grand pond]

acgn

[neat silo]

Wait im gonna type mu progress

[rose trout]

I guess the ts are xs?

[neat silo]

Iyes

I tried $f\left(-t\right)$ but couldnt prove that its the same at $-f\left(t)$

[grand pond]

acgn
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[rose trout]

Hint : ||conjugate||

[neat silo]

Stuck with $\log_{2} {\left(-t + \sqrt{t^{2} + 1}\right)}

[civic gazelle]

Try f(t) + f(-t)

[neat silo]

If its 0 then its odd?

[gaunt nimbus]

said suspiciously hyperbolic-sine-shaped function

[civic gazelle]

yep

[neat silo]

LMAO

f(t) + f(-t) deletes it

[neat silo]

Alr nice

Thank you lads

The original question is to evaluate this integral

$\int_{0}^{2} {\log_{2} {\left(x + \sqrt{x^{2} - 2x + 2} - 1\right)} dx$

[civic gazelle]

lmao

[grand pond]

acgn
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[neat silo]

I modified it by substituting t = x - 1

[civic gazelle]

using odd equation to make it 0 is creative as hell

[neat silo]

The substitution turns it from 0, 2 to a -1, 1

I just guessed its an odd function LoL

[civic gazelle]

ye I get it

funny

[neat silo]

Alr ty

Closing

.close

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[twilit jetty]

theres a few ways to go about this

[fierce canyon]

can you explain one

[twilit jetty]

the most barebones one is to let angle L be x

look for isosceles triangles in the problem, then use that to find all of the angles for each triangle

that way you can find angle LMN directly

this is not at all the intended solution, but you wont need to think to work through it

[fierce canyon]

wouldnt that be

110+2x+something=360

[fierce canyon]

what would we need to do after that

[twilit jetty]

presumably you should be looking for isosceles triangles

[fierce canyon]

but we dont know if 110 is split in half so how would we find it?

[twilit jetty]

wdym by that

does KM have to split the angle in the middle to see an isosceles triangle?

[fierce canyon]

i meant like there are 2 given but we only know the combined angle of the 2

2 isosceles triangles

[twilit jetty]

can you figure out the ? angle

[fierce canyon]

oh

wait is it just x

[twilit jetty]

yes

[fierce canyon]

wait i have another question im not sure if its possible to do or not tho

if u could help

[twilit jetty]

what happened to the original question

[fierce canyon]

i got 125

[twilit jetty]

thats not correct

[fierce canyon]

isnt it 110+4x=360

[twilit jetty]

oh that does work nvm

I made a mistake

[twilit jetty]

go post it

[fierce canyon]

i got the star as 4 or 9 but when i put them in it goes as a decimal

[twilit jetty]

why is the x on the right

[fierce canyon]

yeah idk

[twilit jetty]

also,

youre saying the * always stands in for the same digit

but this is plural

hmmmm, what do you think that means

[fierce canyon]

so it can be different digits right

[twilit jetty]

yep

[fierce canyon]

wait could you help me

im a bit stuc

[twilit jetty]

what are you stuck on specifically

[fierce canyon]

cause the first star can be 4 or 9 but i cant seem to get the 2 to be the second digit for the answer

[runic hamlet]

it would be helpful to give the stars better names

p7ab * 6 = c2d84

[twilit jetty]

you could also consider erasing stars you dont need to consider

[fierce canyon]

wait i think i got it

i did a wrong calculation

[twilit jetty]

@fierce canyon its been 5 minutes

do you want to know a shortcut or to keep going

[fierce canyon]

mb i was going through the other questions i got 8 as my answer tho is it correct

[twilit jetty]

yep

[fierce canyon]

wait i needa eat dinner rn but i have a question im stuck on, ill send it rn and if the channel closes before i finish do u think u could dm me abt it

[twilit jetty]

if it closes, you could just open up another channel

[fierce canyon]

o alr

heres the questio tho

@twilit jetty

[twilit jetty]

Im figuring this out

[fierce canyon]

O alr

[twilit jetty]

@fierce canyon did you figure it out

I can tell you how to solve this if you are stuck

[fierce canyon]

Sry I was busy doing it idk how tho

Could u tell me

[twilit jetty]

first, do you know how the square PQRS rotates around LMNO

[fierce canyon]

It’s like one on each side right

[twilit jetty]

yep

it rotates all the way around LMNO to get back to where it started

thats four rotations

now in each rotation, you need to measure the length that P travels with

[twilit jetty]

since we know how the square rotates, we can simplify what we are looking for

so far we only really need to know this much

taking a look at the picture, which corner would P be afterwards

[fierce canyon]

Top right one?

[twilit jetty]

like this?

[fierce canyon]

Ye

[twilit jetty]

think about this

if it was in the upper-right corner, that would instead look like this

you only did half of the rotation

you need to do a 180-degree rotation to fully place the square back on the other square

so instead, where would P be after rotating this square

[fierce canyon]

Wait wdym by that?

[twilit jetty]

you know how the square rotates, right

[fierce canyon]

Arojund that one point

[twilit jetty]

sure

so imagine this setup in your mind ok?

then rotate the top square around "that one point" until its to the right of the center square

where would point P on the square end up?

its not the upper-right corner - you cant just guess its a 90 degree rotation, thats wrong

[fierce canyon]

I’m kinda lost idrk 😭

[twilit jetty]

you said how the square rotates

do you know what that means

yes or no

[fierce canyon]

ig not rly

[twilit jetty]

then why did you say it

[fierce canyon]

i thought i did

but idrk now

[fierce canyon]

i thought it was just like that

[twilit jetty]

that doesnt even make any sense

look at the square before and after it got rotated around the point

first, can you tell me where the point is in the picture

[fierce canyon]

top left?

[twilit jetty]

lets read the original problem closely

in particular, one square stays in place and the other square is moving, right?

[fierce canyon]

yeah

[twilit jetty]

which square is moving?

[fierce canyon]

pqrs

[twilit jetty]

now when its rotating, which point stays in the same place

[fierce canyon]

for the first one its R

[twilit jetty]

now

go take a look at this picture again

draw on the picture where point R is

[fierce canyon]

i cant rly draw on it but that bit where the 3 squares are close

[twilit jetty]

thats good

now think about this

the square started up there

then ended down there

youre saying this is a 90-degree rotation?

yes or no

[fierce canyon]

yeah

[twilit jetty]

let me draw the situation again

still a 90 degree rotation?

[fierce canyon]

o wait its 180 degrees

[twilit jetty]

yea

it ended up on the opposite side

thats not 90 degrees bro

just because its 1/4th of the way along the center square doesnt mean its a 90-degree rotation

thats why you actually gotta visualize the square moving instead of guessing 90 degrees

[fierce canyon]

so p is bottom right

[twilit jetty]

yea

alr now for the next rotation

this time Ive colored the center square white so you can tell it apart easier

shouldve done that earlier whoops

now where would P end up for the bottom square

[fierce canyon]

should be top left right

[twilit jetty]

yep

two 180-degree rotations = a 360-degree rotation

so P ends up back where it started

it still should feel a bit weird that you only make it halfway around the square and P's back where it began

for now though, you can see that its a 180-degree rotation each

3rd rotation, where would the P be

[fierce canyon]

bottom right again?

[twilit jetty]

yep

alr now we have a sort of a good idea on how P moves

now keep in mind this isnt an exact picture

for one, the square is floating off of the center square which isnt too accurate

so youll have to stick to what makes sense when we then consider what path P is making

lets begin with this

we know this should be a 180-degree rotation around the corner of the white square, right?

[fierce canyon]

yeah

[twilit jetty]

now how far away is P from this corner?

or how far away is P from the center of the rotation?

[fierce canyon]

1m?

[twilit jetty]

thats correct

now knowing that:

• P is 1m away from the center of the rotation
• the rotation is 180 degrees
  is enough to figure out how far P traveled from this particular rotation

how do you think we calculate that?

[fierce canyon]

isnt it just 2m

oh wait

it rotated right

[twilit jetty]

what do you think

you can let me know if you are unsure on how to read the original problem

[fierce canyon]

is it half pi

[twilit jetty]

ok now youre forgetting the formula for circumference

[fierce canyon]

wait i got mixed up with area

oops

2 pi

[twilit jetty]

2pi?

[fierce canyon]

omg wait its kinda late

pi

[twilit jetty]

yep

so P traveled a distance of pi for this one rotation

so that explains how far for two of the moves

now this next one is sneaky

how far did P travel for this rotation?

[fierce canyon]

should just be 0 right

[twilit jetty]

yea

ok but really the next one's sneaky

we know where P is rotating around, and that its 180 degrees

what we now need is the radius, so how far P is from the X

can you figure that out

[fierce canyon]

√2?

[twilit jetty]

yep

so radius sqrt(2) and rotation 180 degrees,

how far of a path did P take here?

[fierce canyon]

√2 times pi

[twilit jetty]

yep

so adding it all together

whats the total length of path that P traveled across the 4 rotations

[fierce canyon]

2pi plus √2 pi

[twilit jetty]

or (2 + sqrt(2)) pi

[fierce canyon]

ic tysm :D i have like 6 more questions left hopefully i can do them

[twilit jetty]

np

[midnight plank]

@fierce canyon Has your question been resolved?

[midnight plank]

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[inland patio]

Recall Tychonoff's theorem (any arbitrary product of compact spaces is compact). One proof goes by showing that any net $\langle x_i\rangle_{i\in I}$ in $X=\prod_{\alpha\in A}X_\alpha$ has a cluster point. The way this is done is by examining cluster points of the nets $\langle\pi_B(x_i)\rangle$ in the subproducts of $X$ (here $\pi_B(x_i)$ is the restriction of $x_i$ to $B\subset A$). To this end, define $$\mathcal{P}=\bigcup_{B\subset A}\left{p\in\prod_{\alpha\in B}X_\alpha: p\text{ is a cluster point of }\langle\pi_B(x_i)\rangle\right}.$$ Let ${p_l:l\in L}$ be a linearly ordered subset of $\mathcal{P}$, where $p_l\in\prod_{\alpha\in B_l}X_\alpha$. Let $B^\ast=\bigcup_{l\in L}B_l$ and let $p^\ast$ be the unique element of $\prod_{\alpha\in B^\ast}X_\alpha$ that extends every $p_l$. Why is $p^\ast$ unique?

[grand pond]

psie

[midnight plank]

@inland patio Has your question been resolved?

[undone heath]

Is this a part of the proof tychonoff’s? I actually haven’t reached that part and product topology but I can look for you

[midnight plank]

@inland patio Has your question been resolved?

[inland patio]

.close

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[normal furnace]

Hello can you please help me on how to do these questions?

[uncut cradle]

what part of the question are you struggling with?

[normal furnace]

basically graphing them

so ik for a. the amplitude is 2

[uncut cradle]

do you know how to calculate the period?

[normal furnace]

yes

it is 360 divided by 1/3 for a.

but i dont know how to graph it

[uncut cradle]

something ive seen people use when graphing trig functions, just to wrap their head around it

is they will graph the default function

so for a, just graph cos(x) for 0 to 1080

then do it, but for 2cos(x) for 0 to 1080

and then again for 2cos(x/3) for 0 to 1080

so we can see the steps we take easier

[normal furnace]

do you know how to do it on a CAS graphing calculator

im not sure how to graph it on that either and we need to know that for the test on this chaptr

[uncut cradle]

ive never used a graphing calculator personally

[normal furnace]

oh ok thats fine

the main thing im stuck on is how to implement the period

like for this its 360 divided by 1/3

how would i graph that

[uncut cradle]

well we need to find when it cuts the x axis right?

[normal furnace]

oh so thats the period

when it intersects the x axis

or how frequently it insersects it

[uncut cradle]

the period is when the wave repeats itself

[normal furnace]

so how often there is a peak?

or trough

so for this graph the period was 1080 degrees

[uncut cradle]

well not exactly, just the distance between two maximum points, or minimum points

[normal furnace]

oh ok

so if the perid was 1080 degrees

[uncut cradle]

this is also the period u see

[normal furnace]

ok

i think i get it now

[uncut cradle]

okay lmk if u need any more help, i understand how confusing trig functions can be to understand

once you understand it, it will make a lot more sense

[uncut cradle]

also just thought id point out, for a) the period isnt 1080 degrees, the domain is [0, 1080] though

period should be 360/k for sin and cos, and 180/k for tan

where y=asin(kx)

for example

[normal furnace]

oh ok

wait for b.

it has a negative apmplitude

how do i work out this

[gaunt nimbus]

wdym "negative amplitude"

[normal furnace]

like the amplitude is -3

[gaunt nimbus]

ahh ok

consider the 3sin(2x) but reflected about x axis

(x,y) -> (x,-y)

anyhow

whats the progess w/ b

[normal furnace]

so im not sure if the graph starts at 0 or not

but if its a sin graph it should start at zero

[gaunt nimbus]

(0,0) -> (0,-0)

which is still (0,0), no?

[normal furnace]

yes

but instead of going to a positive number

it will go to a negative

[gaunt nimbus]

yes

for example

at x=45 deg

normally its (45 deg, 3)

but becuz of the negative sign its (45 deg, -3)

[normal furnace]

oh ok

yep

yes now i 100% get it

i think that will be all

thanks

.close

[midnight plank]

Channel closed

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[normal furnace]

.reopen

[midnight plank]

✅

[normal furnace]

nvm

i need help with c.

it has a period of 720 degrees

i dont know how to implement it into the graph

because it has a range of -180 to 180 degrees

[hushed mauve]

if i understand correctly, you don't need one full repetition of the graph? just need to state the period separately

[normal furnace]

wait wdym?

[hushed mauve]

your concern is that the range is less than the period, right?

[normal furnace]

yes

[hushed mauve]

yeah i think you don't have to have one full period in your sketched graph

it's separate

[normal furnace]

so like we are basically just graphing a section of the graph

[hushed mauve]

yes

because the period > the range

i don't think any sane person would expect one full period when the period > range

[normal furnace]

so what i got was -180 is 3

and 180 is -3

and since its a sin graph it intersects the x at 0

[normal furnace]

yea im just a lil confused on how to graph it

[hushed mauve]

everything you said is correct

[normal furnace]

oh ok

so im wondering on how to do d.

so basically the period is 120 degrees

[hushed mauve]

right

[normal furnace]

and its a negative amplitude and since its a cos graph

it will start at -1

so will i have to seperate the degrees in increments of 30 degrees

so like 30, 60, 90, 120

etc

or is there a simpler route

[hushed mauve]

i feel like there is

so we know the graph starts at -1

[normal furnace]

yes

[hushed mauve]

it repeats every 120deg

so we will have three periods

[normal furnace]

yes so there is a distance of 120 degrees between each maximum and minimum of the graph right>

[hushed mauve]

no

[normal furnace]

oh wait

[hushed mauve]

after 120 deg it will be -1 again

you know when it will be -1

now just find when it will be 0 and 1

then draw the wave

you should get 3 "waves"

[normal furnace]

so basically it starts at -1

when it reaches 120 degrees

will it be 0

and then when it reaches 240

it will be 1?

[hushed mauve]

no

no to both

ok so ignoring the negative sign

we have cos(3x)

we know cos starts at 1

[normal furnace]

yes

[hushed mauve]

and we know cos 360 = cos 0

so for what x would 3x be a multiple of 360?

[normal furnace]

yes

60?

[hushed mauve]

60 x 3 = 180

is 180 a multiple of 360?

[normal furnace]

yes

no wait

multiples are like 360

720

etc

[hushed mauve]

what integer multipled by 360 gives you 180 then?

[normal furnace]

yea wait nvm

0.5

[hushed mauve]

0.5 isn't an integer

[hushed mauve]

you are right here though

[normal furnace]

wait lets go back

yea so 720 is a multiple of 360

same with 1080

[hushed mauve]

but my question was this

for what x would 3x be a multiple of 360

including 1 x 360

actually including 0 x 360

[normal furnace]

wait im confused on the question

would it be 1080?

[hushed mauve]

okok

so you know multiples of 360 are 0, 360, 720, etc.

[normal furnace]

yes

[hushed mauve]

but now you have 3x

not x

[normal furnace]

yes

[hushed mauve]

so if we let y be the multiples of 360

so y = 0, 360, 720, etc.

[normal furnace]

yes

yes

[hushed mauve]

now we are saying

3x = y

what values of x satisfy this?

if not, let's take a concrete example

if 3x = 0, what's x?

[normal furnace]

0

[hushed mauve]

good

so cos 3x = 1 at x = 0

first one

if 3x = 360, what's x?

[normal furnace]

120

[hushed mauve]

so cos 3(120) = 0 also

what about 3x = 720?

[normal furnace]

240

[hushed mauve]

mhm

[hushed mauve]

edit, cos 3(120) = 1

cos 3(240) = 1 also

get the idea now?

[normal furnace]

yes

[hushed mauve]

your answers are the answers to this question

[normal furnace]

oh ok

[hushed mauve]

but rmb the negative sign

so now you know where your graph will be at -1

at x = 0, 120, 240, 360

[normal furnace]

os the points are (0,1) (120, 0) (240, -1) (360, 0)

[hushed mauve]

no

[normal furnace]

no wait

but it will be inverse

coz it is

[hushed mauve]

at x = 120, cos 3x = 1

at x = 240, cos 3x = 1 also

so the points are (0, 1), (120, 1), (240, 1), (360, 1)

[normal furnace]

oh ok

[hushed mauve]

and make the 1s negative ofc to account for the -cos

[normal furnace]

so when will the graph have an amplitude of 0

[hushed mauve]

amplitude of 0?

[normal furnace]

so like when will the y become 0\

[hushed mauve]

the only graph that has an amplitude of 0 is a flat straight line on the x-axis

[hushed mauve]

aha, good question

for a normal cos graph

when is cos = 0?

[normal furnace]

at 90

and 270

[hushed mauve]

so at 90 degrees, then every 180 degrees afterwards, correct?

so just set 3x to those values and find the corresponding values of x

eg: 3x = 90, x = 30
cos 3x crosses y = 0 at x = 30

[normal furnace]

yes

[hushed mauve]

etc.

you can do the same for the other 1

[normal furnace]

yes

i get it

so basically (60, 1)

(180,1)

and you can add 120 to the x value to find out the other times the y will be 1

[hushed mauve]

60 isn't 1

cos 3(60) = cos 180

i think you should use a table

[normal furnace]

wait but its a negative amplitude

when u do -cos(180)

its 1

[hushed mauve]

oh wait yea i miscalculated

continue

[normal furnace]

yep so i sketched the graph

so the y is at 0 at 30, 90, 150, 210, 2 70, 330

[hushed mauve]

yep

[normal furnace]

wiat im on question e rn

so for this basically does it go from (0, 5) to (180, 0)

no wait

i made a mistake sorry

so it goes from 0,5 to 180,5

and for f

this goes from (0,0) to 180,0

[hushed mauve]

mhm

same strategy applies

[normal furnace]

yep thank you for your help

i understand it now

[hushed mauve]

don't mention it

[normal furnace]

im gonna close it now thanks again

.close

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[inland patio]

Let ${X_\alpha}{\alpha\in A}$ be all compact and consider $X=\prod{\alpha\in A}X_\alpha$ as well as a net $\langle x_i\rangle_{i\in I}$ in $X$. Let $\pi_B(x_i)$ be the restriction of $x_i\in X$ to $B\subset A$ (recall, an element of $X$ is a function from $A\to\bigcup_{\alpha\in A}X_\alpha$).\

Now suppose the net $\langle \pi_B(x_i)\rangle$ has a cluster point $p$, where $B\subset A$ is a proper subset. This means it has a subnet $\langle \pi_B(x_{i(j)})\rangle_{j\in J}$ that converges to $p\in\prod_{\alpha\in B}X_\alpha$. Let $\gamma\in A\setminus B$ and consider the net $\langle\pi_{{\gamma}}(x_{i(j)})\rangle$. This net lives in $X_\gamma$, which is compact, so $\langle\pi_{{\gamma}}(x_{i(j)})\rangle$ has a convergent subnet $\langle\pi_{{\gamma}}(x_{i(j(k))})\rangle_{k\in K}$ converging to $p_\gamma\in X_\gamma$. Let $q\in\prod_{\alpha\in B\cup{\gamma}}X_\alpha$ be the unique extension of $p$ and $p_\gamma$.\

Question: Is it true $\langle\pi_{B\cup{\gamma}}(x_{i(j(k))})\rangle_{k\in K}$ converges to $q$? How does one show this?

[grand pond]

psie

[midnight plank]

@inland patio Has your question been resolved?

[midnight plank]

@inland patio Has your question been resolved?

[zealous heart]

The product ( \prod_{\alpha \in B \cup {\gamma}} X_\alpha ) is equipped with the product topology, so convergence in the product space is equivalent to coordinatewise convergence.

That is, a net ( \langle y_k \rangle_{k \in K} ) converges to ( q \in \prod_{\alpha \in B \cup {\gamma}} X_\alpha ) if and only if, for every ( \alpha \in B \cup {\gamma} ), we have:
[
\pi_\alpha(y_k) \to \pi_\alpha(q).
]

Now:

• For every ( \alpha \in B ), we have by construction
  [
  \pi_\alpha(x_{i(j(k))}) = \pi_\alpha(\pi_B(x_{i(j(k))})) \to p_\alpha = \pi_\alpha(q).
  ]
• For ( \alpha = \gamma ), we have
  [
  \pi_\gamma(x_{i(j(k))}) \to p_\gamma = \pi_\gamma(q).
  ]

Therefore, for all ( \alpha \in B \cup {\gamma} ), the projections ( \pi_\alpha(x_{i(j(k))}) \to q_\alpha ), hence
[
\pi_{B \cup {\gamma}}(x_{i(j(k))}) \to q.
]

[grand pond]

Clint

[inland patio]

awesome 🙂 that actually makes total sense. Thank you very much!

.close

[midnight plank]

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[steel crest]

<@&268886789983436800>

[midnight plank]

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[twilit field]

Why isn't the cardinality of the null set $1$.
\
We can have a bijection from say ${1}$ to ${}$. Where 1 maps to nothing

[grand pond]

wai

[twilit field]

Like why isn't this a bijection

[rugged saffron]

‘nothing’ is not an element of the empty set, you cannot have a function with codomain empty set

Because by definition I have no options to send 1 to

[steel crest]

unless the domain is empty too

[twilit field]

so that's the reason there's no bijection

[steel crest]

in fact that's one of the reasons why it's a good idea to define $0^0=1$

[grand pond]

gfauxpas

[steel crest]

because

[lyric charm]

and what, pray tell, is this bijection

[steel crest]

$\vert \varnothing^\varnothing \vert = \vert \varnothing \vert^{\vert \varnothing \vert} = 1$

[lyric charm]

"1 maps to nothing" isn't... valid

[twilit field]

I see

okay

[grand pond]

gfauxpas

[steel crest]

fixed

[twilit field]

got it

[rugged saffron]

Is there actually a morphism there in the category Set?

[steel crest]

i dont know category theory, but this is an ordinary function

set-theoretical

[twilit field]

This is RA 😭 ( build up to cantor's)

Thanks

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[grave kernel]

need to prove for any real number x>0, there exists irrational number alpha such that 0<alpha<x

i said that lets take n as a rational number and lets take sqrt(2)
so sqrt(2)/n is such a number where i can adjust the n to make it in the interval
is this a valid proof

[steel crest]

its the start of one!

do you know the Archimedean Property?

let's make rigorous the part where you find the correct n

that will complete your proof

[grave kernel]

i dont

whats that

[steel crest]

it says

for every real number r>0

there is an integer n > r

[grave kernel]

right

makes sense

[steel crest]

so is that good enough as your n? explain why it is

[grave kernel]

i mean this basically explains the point that
no matter what x is
ill always have a n to divide it with so that i can make the number smaller right
?

and obviously if x > sqrt(2)
i can just put n as 1

[steel crest]

oh i thought you were doing something harder

between any two positive real numbers

[grave kernel]

i mean, ill eventually get to it ig

im starting with number theory

[steel crest]

sure

[grave kernel]

that

seems hard

[steel crest]

anyway

[grave kernel]

i just wanted to know is this a valid proof tho

cuz in the solution
it said
if x is irrational, do x/2
if x is rational, do x/sqrt(2)

[steel crest]

if n> r, 1/n < 1/r, but that doesnt mean sqrt 2 /n < r, or does it

i wanted one answer for both lol

[grave kernel]

i did think about the x/2 part, which is where i got this idea, didnt do the rational part tho

[grave kernel]

no but im not saying that essentially

[steel crest]

anyway choose n > r + sqrt 2

[grave kernel]

considering the case where r < sqrt 2

i can have an irrational number sqrt 2/n which is smaller than r

where n will vary according to r

i mean like, will this be considered a valid proof, idts right

[grave kernel]

yeah

right

yeah

makes sense

[steel crest]

sure

but i prefer to have one answer for both

so im showing you how

[grave kernel]

right like a general answer

?

[grave kernel]

okay oaky

okay

makes sense

right

hmmmmm

[steel crest]

so if n > r+sqrt2

[grave kernel]

itll always fit in the interval then

[steel crest]

then 1/n < 1/(r+sqrt 2) < 1/r

< r

[grave kernel]

rigghhhhtttt

damn

[steel crest]

and also

1/n < 1/sqrt(2) < sqrt 2

anyway this is one of those things that you should see a few times and then get a feel for "oh, so this way we can do it in one step

[grave kernel]

right this was

really good thinking

lmao

never really thought it this way

[steel crest]

you're not gonna get it the first tim eyou see a problem like it

just like I did not the first time :)

[grave kernel]

yeah im like

REALLY bad at proofs lol

[steel crest]

jumping from casual math to rigorous math is a big jump

gambatte

you can do it

[grave kernel]

hahahaha

fr

[steel crest]

let's try the harder one

between every two positive numbers p , q

there is an irrational number

[grave kernel]

wait lemme think about it

p and q are real numbers right

hmmmm

damn

[steel crest]

yeah

[grave kernel]

wait

so

i could

possibly

take an irrational number

and multiply or divide to scale it between p and q

[steel crest]

good thinking

[grave kernel]

now i need to find the factor

[steel crest]

actually i might have made this too hard lol

[grave kernel]

nah come on

lets do it

[steel crest]

start easier

[grave kernel]

im bored anyway rn

bet bet

[steel crest]

between every two positive real numbers

is a RATIONAL number

you'll need the archimedean principle here

[grave kernel]

that does not sound easier

[steel crest]

it is because

[grave kernel]

r>0
n>r right

[steel crest]

once we find that

you can just multiply it by sqrt 2 for the other one

or some multiple of sqrt 2

to make sure it fits

yeah

[grave kernel]

OOOOOHHHH

RIGHT

[steel crest]

let's call the numbers a and b

b > a > 0

[grave kernel]

hm okay

now

rational

number

[steel crest]

well

let's first of all take a number in between the two

actually no

scratch that

pick a number sch that 1/n is between them

how does one do that

[grave kernel]

a<1/n<b
1/a>n>1/b ?

[steel crest]

oh boy this needs the well-ordering principle, did you learn that

[grave kernel]

no

😭

[steel crest]

well you have to know it for number theory

for any collection of natural numbers

0,1,2,3,4,5, .... (some authors omit 0)

the well-ordering principle says

any set of natural numbers has a smallest number

[grave kernel]

right

these are, pretty straight forward

proving them tho, thats hard

[steel crest]

so let's find a number bigger than 1/(b-a)

[steel crest]

this you can prove by induction

[grave kernel]

why 1/(b-a)

[steel crest]

because bigger than 1/b and bigger than 1/a individually might overshoot or undershoot

[grave kernel]

why so tho

oh

doing them case wise

okay

but how does 1/(b-a) fix that then

[steel crest]

because if a=1, b = 51.2

and I choose n = 500

then 1/500 is too small

[grave kernel]

right

mhm

[steel crest]

if n > 1/(b-a)

let' see, fdo we solve the problem? i copuld be wrong lol

bn-an > 1

bn > 1+an

b > 1/n + a

wait what am i doing

just invert them

n > 1/(b-a), then b-a < n

[grave kernel]

right

yeah

[steel crest]

we need the numerator and denominator seperately

i realize

im looking at a proof online

[grave kernel]

wait im just, raelly confused where we are rn

we are looking for a rational number between 2 positive real numbers

a and b

[steel crest]

yeah i picked one that was way harder than I intended sorry

[grave kernel]

and we are choosing a number

[steel crest]

finish it anyway?

[grave kernel]

1/n

which is

between a and b

so now

we are finding

n

and

if n > 1/(b-a)

[steel crest]

phone call sorry 😭

[grave kernel]

okay okay

[steel crest]

https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number but here

sorry this will take a while! have to give up on this for now

gl

[grave kernel]

okay okay

thank you tho

learned a lot

.close

[midnight plank]

Channel closed

Closed by @grave kernel

Use .reopen if this was a mistake.

[midnight plank]

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[real marlin]

can anyone give fast help pls