help-39

[sudden badger]

i might need more help with vectors later

.close

[pearl pond]

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[pearl pond]

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[pale frost]

<@&268886789983436800>

[sterile python]

Fourth line 2nd letter on the left

[pearl pond]

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[void flume]

Hey I’ve got a question on the washer method

[void flume]

Basically my teacher was talking to us and said you do upper minus lower

Which sounds right to me

But then kind of contradicts himself and says if ur not given a graph then u just do fx-gx even if its not upper minus lower

I’ma go find the problem

I forget the rotate command sorry

[summer imp]

,rccw

[void flume]

But problem one would be like 57.59 if I did it the other way

[errant badger]

and this is strictly speaking for a rotating about the x axis?

or do you mean like in general you’re confused with the wording

[void flume]

This problem yes

[void flume]

No like do I do upper minus lower or fx -gx

[errant badger]

it really really depends

[errant badger]

for your interval, put in any x value to both f and g of x. whichever gives the higher value is your upper function

hes probably implicitly defining f(x) as being the greater function

your teacher

[void flume]

Ok

I’d guess he labeled it wrong here then

[limber flume]

errrm i have a kinematics question that i have no clue how to do?

[errant badger]

!occupied

[pearl pond]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[void flume]

Thanks

.close

[pearl pond]

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[errant badger]

your first giveaway is that volume is negative

it should never be negative

@void flume

[void flume]

Yes

That makes sense

It’s just weird because he told us that’s right

So I was confused

[errant badger]

did he tell you the negative answer was correct?

[void flume]

Yep

And I even asked him that’s same. Thing

Like volume can be negative?

He just said that means it’s under the axis

[pearl pond]

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[marsh sapphire]

I'm having trouble understanding what exactly I did wrong.

[pearl vigil]

reading a sec

the original piece of cardboard is say, this

they cut away 8 inches

this blue bit represents that they took away 16

yours says x+16, but the 8 is from the original cardboard, they didn't add more cardboard to make it equal

[marsh sapphire]

So what would make v=l*w•h be?

[pearl vigil]

v=8(x-16)(x-16)

i believe

so the height of the box is 8

the length and width should both be x-16

i think?

[marsh sapphire]

I'm not really sure if I'm doing this right.

[pearl vigil]

sorry, chatting in another help channel rn and wrapped up another earlier

uhhh

my first step after finding the expression was dividing by 8

and i think u expanded (x-16)(x-16) wrong

[marsh sapphire]

Uh.

[pearl vigil]

Okay so uh

You get like

the dividing by 8 part on both sides? or no

oh yeah also remember that we're given its supposed to be like

24200

[marsh sapphire]

8(x-16)(x-16)=24200

I'm not exactly sure what to do with the (x-16)(x-16)

[plush bramble]

treat t = x - 16

Then solve for t first

[pearl vigil]

i would move it all to one side

idk if u sub / t sub whatever is neccesary...

[plush bramble]

From here

You don't need to expand it out with t

[pearl vigil]

er... i factored the expanded version

and remember the original piece of cardboard was x by x right?

so.. i got 71 by 71

[marsh sapphire]

Does the first part say x^2-32x-2769=0?

[pearl vigil]

yup

sorry messy handwriting

multitasking a lot rn

[marsh sapphire]

I restarted the equation to do the problem on my own.

So different problem same equation.

I'm having a hard time factoring with this one.

[pearl vigil]

okay, ill look

288/8 is equal to er... 36

[marsh sapphire]

Oh that's how you got that.

[torn flicker]

it can be factored

[pearl vigil]

yea ik

i dont wanna js like

hand them the answer yk

[torn flicker]

oh mb

[cyan nymph]

Isn’t 288 divisible by 8?

[pearl vigil]

ye

i divided out the 8

which made it 36

then i moved the 36 to the other side

[cyan nymph]

Yup

[pearl vigil]

^^

[cyan nymph]

X - 16 = 6

X = 22

[pearl vigil]

yeah i have that as my answer too

[marsh sapphire]

I restarted the problem again.

[pearl vigil]

factoring got me 22 and 10

but you can't use 10 cus u need to take 16 away

[pearl vigil]

i see

[pearl vigil]

oh yeah i think btw. like. you shouldn't factor like when its not equal to 0

[marsh sapphire]

How do you know if it's equal to 0?

[pearl vigil]

um

where did 1156 come from

similar to the last problem

you have

8(x-16)(x-16)

= (some number divisible by 8 convienetly)

and you dont have to know if its equal to 0, it should be if you manipulate it properly, balancing it as an equation...?

[marsh sapphire]

Oh it's a new problem.

[pearl vigil]

OH

wait i didn't look at it oops

lmao

alr im looking now srry was answering questions in another help channel

[pearl vigil]

srry someone is asking me to help with checking in another help channel, multitasking. like this, do u see how i manipulated it ot be 0

[marsh sapphire]

The 8 this time is 4.

[pearl vigil]

OH

sorry im distracted

[pearl pond]

@marsh sapphire Has your question been resolved?

[marsh sapphire]

Would someone be able to help me?

[plush bramble]

Did you fix your picture with the new measurements

[pearl pond]

@marsh sapphire Has your question been resolved?

[pearl pond]

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[pearl pond]

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[iron basin]

unable to see how they got c/b and a/c in that form

[iron basin]

question indirectly asks for values of m and n

[pearl pond]

@iron basin Has your question been resolved?

[pearl pond]

@iron basin Has your question been resolved?

[eager jewel]

C=1-alpha^4 and B=1-alpha^2

For c/b

[iron basin]

ohhh

right thanks

.close

[pearl pond]

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[eager jewel]

And a=1-alpha^6

C=1-alpha^3

For a/c

[pearl pond]

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[terse patio]

Can somebody guide me through 3 questions?

[green aurora]

the letter a)?

or also other?

[terse patio]

A and others right after

[dire tapir]

do you know your exponent laws?

[green aurora]

use power rules

[green aurora]

its not 0

[terse patio]

How? Wouldn’t it be multiplied by 0?

[green aurora]

10/10 = 1 right?

[terse patio]

Yeah

[exotic seal]

a power to 0 is always ?

[green aurora]

so 10^(1-1)= 10^ 0 = 1

ok so far?

[terse patio]

Why does 0 equal 1?

[exotic seal]

because

10^1 = 10 right

[terse patio]

Yeah

[exotic seal]

10^0 = 1 because if you go down a power you divide the number

in this case

10/10 = 1

[terse patio]

Ah

I think I get it now

[exotic seal]

and 10^2 = 100

10 times 10 = 100

get it?

[terse patio]

Yeah

[exotic seal]

alright

[terse patio]

B?

Hold on

I’ll write

[exotic seal]

@terse patio i gotta go

someone else will help you probably

[terse patio]

Aight take it easy

[exotic seal]

aight cya

[terse patio]

This?

[smoky gull]

thats wrong!

[green aurora]

10^0 = 7???

[smoky gull]

6^-2 isnt same as 6^2

[visual ruin]

Be careful, 6^-2 is not 36

[smoky gull]

also this

[terse patio]

It’s supposed to be 1 bru

[visual ruin]

When you have a negative exponent it equates to 1/the term

[terse patio]

Huh explain

[smoky gull]

2^-1 can be rewritten as (1/2)^1

[visual ruin]

So 6^-2 is equivalent to 1/6^2

[terse patio]

Ahh

I remember that

[smoky gull]

The role of the negtative sign is just to invert it

as in reciprocal

[visual ruin]

Remember that rule, it’s very useful

[terse patio]

Ight im gonna change it

Uh

[green aurora]

why?

[timber quarry]

Good morning/good afternoon everyone, could someone help me with statistics? I'm in the 4th year of secondary school.

[terse patio]

Because I’m pretty sure we can ignore the 1/ because of the exponent?

[green aurora]

so 6^2 = 1/(6^2) = 36 ?

[terse patio]

Yeah

That’s the idea

[green aurora]

can you show how

[terse patio]

Idek tbh

[ashen ivy]

this channel is occupied, please see #❓how-to-get-help :)

[terse patio]

What am I doing wrong though

[visual ruin]

It would be 1/36 not 36.

[green aurora]

1/6^2 = 1/(6 * 6) = 1/36

[terse patio]

Okay

1/36

That’s where I’m supposed to be at?

Uh

Actually ill be back in 10-20 minute

Not sure if there’s a special rule against this

Wait

I got it

.close

[pearl pond]

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[pearl pond]

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[humble steppe]

I want to find the vertex of this quadratic, and since its negative it will mean thats the highest point where it works but I just dont really understand how its connected to figuring out the width that will give most area

[humble steppe]

<@&286206848099549185>

[plush bramble]

do you have a picture of what's going on

[valid nova]

yeah a pic would be helpful

[plush bramble]

you should draw a picture of the fence and house

[humble steppe]

after

did I not explain my question well? I want to figure out the value of y that will lead do the most area, I am thinking of doing it by finding the vertex of the quadratic function -2y squared-20y

.closed

.close

[pearl pond]

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[jolly parrot]

Renato

[summer imp]

Is this in relation to the problem you just finished?

[stoic imp]

no

we have a parametrization

and they give us a isomorphism

[summer imp]

This is d) in a series of exercises.

[stoic imp]

your point?

[summer imp]

My point is that they're talking about a parametrization sigma and you didn't provide the context with respect to this parametrization.

[stoic imp]

my bad

[stoic imp]

can u help

[summer imp]

Have you tried something?

Or thought about something

[stoic imp]

yes

(x - 1)^2 = (-(x-1))^2

@summer imp

(-(x-1))^2 = (1 - x)^2

do you understand?

[summer imp]

So you've noticed squaring is an even function. That could help. Anything else?

[stoic imp]

thats all the tricks I have under my sleeve

jaja

[summer imp]

Suppose you did have this function g(t).

What does the requirement that $\bar{\sigma}(g(t)) = \sigma(t)$ mean explicitly?

[jolly parrot]

Azyrashacorki

[summer imp]

In particular, what is $\bar{\sigma}(g(t))$?

[jolly parrot]

Azyrashacorki

[summer imp]

Explicitly.

[stoic imp]

function composition

[summer imp]

Okay. What would that look like if you plug it into sigmabar?

[stoic imp]

we dont know what g is

[stoic imp]

care to elaborate?

[summer imp]

I just want you plug g(t) inside sigmabar.

Directly

[stoic imp]

we dont know what g is

g(es - 1)
g((es - 1))^2

@summer imp

[summer imp]

So if you replace every s in sigmabar with g(t) you get that?

[stoic imp]

what are you talking about

wdym replace s with gt

where is that coming from

[summer imp]

You applied g to both coordinates for some reason

[stoic imp]

ohhh sigmabar uses s and g uses t

[summer imp]

That is not what plugging g in sigmabar means

[stoic imp]

s = g(t)

?

[summer imp]

Yes

[stoic imp]

how?

[summer imp]

LIke you would do it if I asked you to plug in 1 for s.

[stoic imp]

e^{g(t)} - 1

(e^{g(t)} - 1)^2
@summer imp

are you here?

[summer imp]

Ok, so now you have an explicit expression for $\bar{\sigma}(g(t))$. You want this to be equal to $\sigma(t)$.

[jolly parrot]

Azyrashacorki

[summer imp]

What does this say if you look just at the first coordinates?

[stoic imp]

dude this shit is tough, correct?

e^{g(t)} - 1 = (t^3, t^3)
(e^{g(t)} - 1)^2 = (t^3, t^3)

@summer imp

[summer imp]

Where did cubes come from?

And why are you equating a function to a tuple?

[stoic imp]

e^{g(t)} - 1 = (t, t^2)
(e^{g(t)} - 1)^2 = (t, t^2)

@summer imp now it is correct?

[summer imp]

No, you're still setting functions and tuples.
The equality should say that the first coordinates of both sides are the same and similarly with the second coordinates.

[stoic imp]

sigmabar grabs values in R and spits stuff in R2

[summer imp]

But the x-coordinate function of sigmabar doesnt.

[stoic imp]

ok

[stoic imp]

e^{g(t)} - 1 = t^2
(e^{g(t)} - 1)^2 = t^2

@summer imp now yes?

[summer imp]

t in the first coordinate

But yes.

[stoic imp]

care to elaborate?

[summer imp]

Just look at it for a second.

[stoic imp]

(t, e^{g(t)} - 1) = (t, t^2)
(t, (e^{g(t)} - 1)^2) = (t, t^2)

@summer imp like this?

[stoic imp]

care to elaborate? I dont think I follow

[summer imp]

The x-coordinate of sigma is not t^2.

[stoic imp]

e^{g(t)} - 1 = t
(e^{g(t)} - 1)^2 = t

@summer imp now yes?

[summer imp]

Well now the y-coordinate doesn't match..

In any case, from the first coordinate, what could you do to get g(t)?

[stoic imp]

what do you mean?

I dont think I am following what is going on

.close\

.close

[pearl pond]

Channel closed

Closed by @stoic imp

Use .reopen if this was a mistake.

[summer imp]

I've taken 3 minutes to respond and you close

Why do I even bother lol

[stoic imp]

do I reopen?

[summer imp]

Solve for g(t) in terms of t.

[stoic imp]

I was about to take a shower and take a break, this exercise seems like a pain in the ass

[summer imp]

Then you'll be done.

[stoic imp]

but I dont understand

[summer imp]

If it was y instead of g(t)

What would you do?

$e^y - 1 = t$

[jolly parrot]

Azyrashacorki

[summer imp]

"Solve for y in terms of t"

[stoic imp]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[stoic imp]

e^(g(t)) - 1 = t

[stoic imp]

just ln

ey = t + 1

y. ln(e) = ln(t+ 1)

y = ln(t+1)

[summer imp]

Right, so verify that g(t) = ln(t+1) has the desired properties.

[stoic imp]

which properties

[summer imp]

Well I guess it just says observe

But you can see it's a bijection and C1

[stoic imp]

i dont know

[summer imp]

If you don't know then check.

[stoic imp]

I mean

how do you know that ln(t+1) is a bijection

are you certain that it is injective and surjective

seeing the graph perhaps but without it?

[summer imp]

As I said, if you don't know then check that it is.

[stoic imp]

I dont remember

.close

[pearl pond]

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[pearl pond]

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[finite narwhal]

I can’t seem to figure out the height,

[solid ivy]

wishful thinking makes you think Pythagorean triples

Thats the ez way to guess the height without actually algebraically deriving the height or w/e

[finite narwhal]

ah that makes a lot more sense 😭 tyty

[solid ivy]

npnp

[finite narwhal]

.close

[pearl pond]

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[celest yoke]

Oh wait is d -2

[pearl pond]

@celest yoke Has your question been resolved?

[mint scarab]

you can check by graphing if you need

i wish i had the time to walk u thru it

[celest yoke]

So jsut plug in a point

It’s not equal so where did I go wrong?

[crimson rain]

hi

[celest yoke]

Hi

[pearl pond]

@celest yoke Has your question been resolved?

[dense jasper]

dropped negative

[pearl pond]

@celest yoke Has your question been resolved?

[celest yoke]

Oh thanks

[pearl pond]

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[dense jasper]

[pearl pond]

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[jolly parrot]

Renato

[pearl pond]

@stoic imp Has your question been resolved?

[errant cedar]

What have you tried, i believe this should be three computations with formulas from your notes

[stoic imp]

im not sure

how the arc length is measured

[errant cedar]

it should be the integral of speed

[stoic imp]

im not sure how speed is measured

[hard relic]

Its literally written there?

[errant cedar]

speed is the norm of velocity so $\sqrt{x’(t)^2+y’(t)^2}$

[stoic imp]

i sent the message b4 he sent his picture

[jolly parrot]

pola_touche

[hard relic]

Have you done any intro physics? When you are stuck with some of these things it's helpful to recall some of the intuition from there. The terminology is mostly borrowed from the physics context.

[errant cedar]

intuition can be gained over these 3 formulas, indeed from physics

[stoic imp]

i am new to physics

but idk why they have this physic esque exercises in my calculus class

[stoic imp]

am I supposed to memorize?

[hard relic]

Well you would never guess how Calculus originated lol

[errant cedar]

you can for now and imo that would be enough for the exercise, but we can try to give some intuition

[hard relic]

Here's the bare bones intuition. If you know the position of an object as a function of time, then it's derivative at any point tells you the velocity (rate of change) of the object at that point. The speed at any point of an object is considered the magnitude of the velocity, aka the norm.

[stoic imp]

im not isaac newton

[hard relic]

Nobody but Isaac Newton is Isaac Newton.

[errant cedar]

You two are antagonistic, that’s no good for the purpose of explaining the thing imo…

just saying

[stoic imp]

wdym?

[stoic imp]

ok

what about the arclen

[hard relic]

If you know the speed at any given point, then by definition, that is also the rate of change of distance with respect to time. The total distance covered in an interval of time is equivalent to the length of the curve described by the position in that interval. I'm sure you can now translate this to the required integral?

[hard relic]

Agreed but idt there was any antagonising here. You're reading too much into it.

[stoic imp]

sorry for being your nemesis

[errant cedar]

I will add smt more geometric: think of the the integral as a sort of smooth summation making precise and infinitesimal this picture summing the lengths of the line segment (an approx of arc length). As the segment becomes smaller, they become more and more tangent to the curve i.e they get close to v(t) and also their length

hence making $\int_a^b||v(t)||dt$ more intuitive maybe?

[jolly parrot]

pola_touche

[patent vessel]

Speed = distance/time so distance = speed x time is layman language

Just do that for infinitesimals - speed = (very small displacement)/(very small time), so very small distance = speed x (very small time). Now if you want the total distance, add up all the “small distances”, that is add up (speed)x(very small time)

Which is popularly called an integral

[daring nebula]

Line integral of vector field

[errant cedar]

maybe i should have removed the arrows heads , imo the picture still conveys the idea

[stoic imp]

i dont really picture it

im not sure of what I am looking at

[errant cedar]

an approximation of arc length by a bunch of line segments

[stoic imp]

why norm

[errant cedar]

because norm is the length of the segment, the length of the small displacement in space made in “dt” seconds

the segments becomes closer and closer to the curve and aligned with v(t) as the partition of point used to make the secant line is refined

[stoic imp]

this is a lot to sink in

[stoic imp]

integral is area between a and b or summation

[errant cedar]

it’s area under the graph of | |v(t)| | as a function of t but also the length of the curve in R^2 as r(t)=(x(t),y(t))

[hard relic]

I wonder, did they not teach you this stuff in lectures or in text? Or do you just dive into problems without reviewing the theory?

[errant cedar]

yeah i agree

[hard relic]

Translation?

[hard relic]

I see. Then you should consider doing as they say and refer to either the provided material or an alternative. Because this is covered in nearly all textbooks that talk about these topics.

If you do that and look at solved examples, perhaps you'd be less confused.

[stoic imp]

fuck my life dude

we still having exams even if no classes

[daring nebula]

Arc length of differentiable curves is easy

You know Pythagoras?

,w Pythagoras theorem

[jolly parrot]

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[stoic imp]

a^2 + b^2 = c^2

a,b,c in Z+

[hard relic]

That Greek guy who stole results from the East?

[daring nebula]

ℝ+

(Well, if we consider negative to be just, the other direction then ℝ)

[hard relic]

Only if a, b, c, are the sides of a right angled triangle.

[daring nebula]

@stoic imp

[stoic imp]

what about it

[daring nebula]

Well

Think about a curve

In 2D space

Defined from [a,b]

We can represent this curve as ( x(t), y(t)) for some t on [a,b] right?

[stoic imp]

then what

[daring nebula]

Then consider a small change in the input

t+Δt

This is another point.

We can draw a straight line between this point and (x(t),y(t)) right?

[stoic imp]

why

[daring nebula]

Well, we can consider the change in both variables individually

and it makes a right triangle

So we can use Pythagoras

So:\
(tiny change in arc length)$^2=(\frac{dx}{dt})^2+(\frac{dy}{dt})^2$

[jolly parrot]

YeetusDeletus5

[daring nebula]

(The tiny change in arc length is usually represented as ds)

[stoic imp]

i dont get it

[daring nebula]

Where’d I lose ya?

[stoic imp]

here

.close

[pearl pond]

Channel closed

Closed by @stoic imp

Use .reopen if this was a mistake.

[pearl pond]

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[north talon]

Let $p$ be an odd prime and $S = {1,2,3,\dots, p}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that$$B\cdot U(U(a)) - a : \text{ is a multiple of} : p : \text{for all} : a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.

[jolly parrot]

Copter

[north talon]

im kinda stuck, probably not even halfway

we have $B * U(U(a)) \equiv a \pmod p$ and for $a=p$ since we cant have $p \mid B$ then $U(U(p))=p$, if we consider $(B,p)=1$ we get $\$ $U(U(a)) \equiv \frac{a}{B} \pmod p$. Since $U$ is a bijection there exists some $k$ such that $U^k (n) \equiv n$, If $k$ is even you get $B^{\frac{k}{2}} \equiv 1 \pmod p$ If $k/2$ odd we get the result so imstuck on $k/2 even$

[jolly parrot]

Copter

[north talon]

but idk how to continue from there

[tropic saddle]

I'm not seeing where you are getting U(p)=p from or U(U(a))=a/p

[north talon]

ooops wait

[tropic saddle]

fwiw, B^(p-1)/2 is 1 mod p iff B is a square

[north talon]

should be a/B

[tropic saddle]

that might help

[jolly parrot]

Copter

[north talon]

fixed

[north talon]

wdym?

[tropic saddle]

also I'm not seeing what k has to do with (p-1)/2

[tropic saddle]

quadratic residue

[north talon]

does k/2 not divide p-1? or am i doing smth completely wrong

[north talon]

aaaa

[tropic saddle]

not in general

[north talon]

how would we show something is a quadratic residue?

[tropic saddle]

sometimes we could get lucky and find an element x with x^2=B

but I'm not sure here

I'm just mentioning it cause (p-1)/2 is certainly not a random exponent

[north talon]

probably stuff to do with order

bwaa

[thorny radish]

since ℤ/pℤ is a field we can reformulate the question like so: we are given a permutation U and a number A such that for all x, Uº²(x) ≡ Ax mod p. then we are tasked to show that A is actually a square.

recall that multiplication of ℤ/pℤ by a nonzero number is a bijection. so conceptually we are *almost* trying to show that in fact U corresponds to multiplication by a number. however square roots of permutations (when they exist) are not unique, so we have to do a bit of work.

[north talon]

yeah, i saw this

idk what id do though

[thorny radish]

strong hint: ||think about where U takes 1||

[north talon]

it would be at x = B mod P?

[thorny radish]

what does that mean

[north talon]

U(x) isnt 1 unless x=B, maybe

[tropic saddle]

when in doubt, give names to things. U(1) is some number, call it y

[thorny radish]

sorry, the solution i had in mind doesn't work. think about what Uº² being multiplication tells you about the cycle lengths of U

[feral sedge]

I think ||even vs odd permutations|| is a useful concept here

[north talon]

wdym?

[thorny radish]

actually it is still helpful to think about U(1) i guess

[feral sedge]

guys what if B is 0

[thorny radish]

or rather, what happens when you apply U to 1 a bunch of times

[feral sedge]

nvm

[thorny radish]

lol

[feral sedge]

||Show that multiplication by x is an even permutation iff x is a quadratic residue||

[north talon]

even permutation is the same thing as U(U(x)) right

[feral sedge]

hmm, you were talking about fields right, I feel like you would have seen them before

[north talon]

i dont think that was me

[feral sedge]

rip

an even permutation is one with an even number of even cycles

composing two evens or two odds gets you an even, while composing an odd with an even or vice versa gets you an odd

not every even permutation is a square, ex: (12)(3456)

but every square is even

since it's either odd*odd or even*even

[pearl pond]

@north talon Has your question been resolved?

[thorny radish]

actually it's easier to start from the cycles of A

so consider just (ℤ/pℤ)× here. then multiplication by A is a permutation, and furthermore all the cycles induced by this permutation are of the same size

now think about how you could build U on (ℤ/pℤ)× such that Uº² = (A as a permutation)

[north talon]

you lost me 😭

[thorny radish]

from where

[north talon]

here

i dont really get what youre saying

;-;

[thorny radish]

recall this is my definition of A

if you apply a permutation A to an element repeatedly, since your set is finite eventually you will return to yourself. you can do this thing for the other elements not in this list until your entire set looks like circles of arrows

(this is called the cycle decomposition of a permutation; each circle of arrows is called a cycle)

i mentioned the cycle decomposition of a permutation that corresponds to multiplying by a number has all the cycles of the same size. do you see why?

well i guess i should ask first, do you know why multiplying by a nonzero number gives a bijection on (ℤ/pℤ)×?

[north talon]

if you multiply by a you can just pick the element with the inverse of a so its a bijection?

[thorny radish]

that's sort of circular reasoning since if you know multiplying by a is a bijection then there must exist an element that gets sent to 1 which is then the inverse

like why does an inverse exist

[north talon]

if you have ax = ay then p | a(x-y) but a nonzero so x = y

[thorny radish]

yes

so it's injective and since finite it is also bijective

[thorny radish]

next is this

this will also tell us that A^(p-1)/2 ≡ 1 is equivalent to there being an even number of cycles in the cycle decomposition of A

[north talon]

multiplicative order is the same

[thorny radish]

can you elaborate

[north talon]

the size of the cycles is just order of a modulo p, i think?

[thorny radish]

exactly

[north talon]

why is this😭

[thorny radish]

there are p - 1 total elements, so (size of a cycle) × (number of cycles) = p - 1

@sharp vigil

[north talon]

number of cycles is p-1/ ord(A)

[north talon]

ohh cause cycles cant overlap

[thorny radish]

yeah the structure is very simple

[north talon]

if this is even, then ord A divides (p-1)/2 and we get the result

[thorny radish]

so somehow the constructing of U has to tell you this

i have to go soon but here is an illustration that is hopefully enlightening rather than confusing:

there is a minor argument where you need to show that if U takes 0 somewhere then A is just 1 but otherwise you don't need to worry about it

[vestal tapir]

did you solve this?

[north talon]

forgot about that

[vestal tapir]

you can't have 3 if you strat with 1,2,3,4, maybethey meant {m,m} is fine

[north talon]

oof, by the construction the problem means 1,2, 2^2,2^3,...

[north talon]

hmmmm

[vestal tapir]

by what?

i didn't think about that

[north talon]

the numbers arent 1,2,3,...n

[vestal tapir]

it's trvial then

the sum is 111111111111, you subtract e.g. 16 to decrease the sum by 32

as long as you can subtract in both directions at least

no that's not a problem

[pearl pond]

@north talon Has your question been resolved?

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[burnt torrent]

Did I mess up somewhere here trying to find an x intercept?

[split void]

what's the original question

!xy

[pearl pond]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[burnt torrent]

Number 3

But in context that doesn’t make sense- so I guess for our purposes the original equation is

0= x^3 -4x/-4x^2+12x

[burnt torrent]

Were you gonna help me?

[pearl pond]

@burnt torrent Has your question been resolved?

[burnt torrent]

<@&286206848099549185>

I’m on a new problem but I still need some assistance

,rotate

[burnt torrent]

.pin

Shit

[burnt torrent]

.pin

Ugh

[burnt torrent]

,pin

Idk the pin command 🥲

[buoyant cypress]

there is a mistake with 8 in denominator

[burnt torrent]

In which part?

[buoyant cypress]

you dont have the permission

[burnt torrent]

Oh

I see

[burnt torrent]

Ohh

[burnt torrent]

I fixed that

[buoyant cypress]

ok '

[burnt torrent]

The x+2’s cancel and become a hole right

[buoyant cypress]

one more thing cancels out

[burnt torrent]

Uhmmmm

I’m not seeing it-

But if I redistribute that x on the outside and have x^2=4 could I square root and have the VA be x=2?

Wait shit nvm

Ignore that

The -4x’s cancel

[buoyant cypress]

are you sure

[burnt torrent]

No I see now if I redistribute the x there’s -4x /-4x

But where does that go- it’s not a hole is it?

[buoyant cypress]

you dont need to redistribute, just see the x is common

[burnt torrent]

Ah

So (x-4)/-4(x-1)

So….5-4x?

[buoyant cypress]

Yes

I gtg sorry

[burnt torrent]

No worries

I’ve made some progress and gotten to here (trying to find the y intercept)

What do I do about the 5 on the x^2?

<@&286206848099549185>

[hollow mason]

Idk

[burnt torrent]

Bruh then why did you respond 😭😭😭

[mossy lance]

Legit

Y intercept, as in y intercepting the x-axis?

[burnt torrent]

Yes

[hollow mason]

Bc i'm doing my long ahh homework so i decided to go on discord

[mossy lance]

Also for anyone who doesn't know you should type in help channels if you don't know how to help

[burnt torrent]

It’s a small part of a larger problem

The larger portion is kind of irrelevant- I just need to know what to do with the 5 there-

[mossy lance]

Over here if you put this equal to zero you could just have either x=0 or solve a quadratic equation to find 5x^2+2x-16=0 and that will give you 3 values of x

[hollow mason]

I joined the server bc i'm dumb mum said it's the only server i can join

[burnt torrent]

Ok I get that

But if you’re not going to be helpful you don’t need the helper role

You need to open up a help channel and get help

[mossy lance]

Please don't talk in the DAMN HELP CHANNELS

Go to #discussion

[burnt torrent]

Idk if I’m gonna get to factor this one I might have to go quadratic equation

[hollow mason]

Alr bay

[burnt torrent]

But there’s also an x on the outside that idk what to do with

[mossy lance]

Okay so the inside equation is technically a polynomial of sorts, correct?

We can just say

5x^2+2x-16=U or some other value

Then we just simply have

X(U)=0

What does this mean?

Either x=0 or U=0, what I'm trying to say is you don't have anything special to do with the x on the outside

Intact

Infact*

Let me tell you a more useful piece of information

An equation of degree 5 always has 5 solutions for x

An equation with degree 21 has 21 solutions

And so on

Your original equation is 5x^3+2x^2-16x, so what degree is this?

Uhm hello?

If you come back, ping me so I can continue explaining (hopefully better than before)

[mossy lance]

And answer this question

[pearl pond]

@burnt torrent Has your question been resolved?

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[naive hare]

is my solution correct?

[pearl pond]

@naive hare Has your question been resolved?

[buoyant cypress]

1/z is the work done by y men of team B per day so you need to divide 1/z by y to get the rate of one man/day ,
similar for team A

now you can compare individual rates z>x, y>9

[naive hare]

i used proportional reasoning to come up with 9x ><=? yz

is this step correct ?

from there , i can safely determine each 1,2 sufficiency or not

[buoyant cypress]

i dont know what is that

[naive hare]

9x ><=? yz ... i put ><=? because i dont know the association, it could be anyhting

[buoyant cypress]

you can determine it

based on the info

you know x,y,z are positive

[naive hare]

whats your answer.

[buoyant cypress]

you need to find it out

[naive hare]

well my answer is E.

you can try and convince me otherwise

<@&286206848099549185>

[buoyant cypress]

if i give you an example,

there are 4 positive integers a,b,c,d and now you have to compare product of a,b and c,d
ab<>=? cd
if you sub values such that a<c and b <d
eg : a = 2 , b = 3 and c = 4, d = 6
now compare the product

2 x 3 and 4 x6

which is bigger

[naive hare]

smaller × smaller vs bigger × bigger is not the setup in this problem though

smaller × bigger vs bigger × smaller is forced using the 1,2 conditions

[buoyant cypress]

the question says z>x and 1st statement says y>9 so it is small x small and big x big

[naive hare]

You are mixing up time with rate

[buoyant cypress]

are you sure

let me see once again

oh fuck i misread the question it asks is z>x

damn your analysis is right now i need to go hide somewhere deep in the dark

https://tenor.com/view/kratos-gif-26454802

[buoyant cypress]

sorry it is correct

[sterile python]

To know z>x we need to compare 9rA and yrB. 9<y, but rA>rB-> Contradiction. Therefore your work is correct @naive hare

Since we don't have the exact number to declare

[naive hare]

but i have not seen a solution that utilizes what i did:

so i dont know how mathematically correct i am

for instance:

[sterile python]

100%

[naive hare]

i did 9 = 1/x and y = 1/z

then i crossed multiplied

to get to 9/z ? y/x

[sterile python]

The symbols just a little bit different from the question but i see that is correct

[naive hare]

and then finally 9x ? zy

[sterile python]

Nah you on the right track

[naive hare]

alright thank you, ...

also

shouldnt there be an R aswell ?

let me show you:

[sterile python]

You mean "r"?

[naive hare]

the R's still cancel out , but i'm not sure they are "the same"

[sterile python]

You can simplify on both side tho thats fine

[naive hare]

im confused here because it says the rates are not the same

so if i am using r on both sides, its a contradiction

[sterile python]

Ah, yeah, thats wrong

They cant be cancelled out

You need to note as rA and rB separately

[naive hare]

so thats also why its E

[sterile python]

Then 9rAx = 1 and yrBz=1

Yep

[naive hare]

ok last question

if they ahve a different rate, instead can we not assume that its implying z <> x

since rate in this case is 1/x and 1/z respectively (we are talking about work problems so numerator is 1)

[sterile python]

Nah you cant tho

[naive hare]

ok

[sterile python]

You cant assume z neq x based on their different rates

Team A = 9.rA and B is yrB, we know products of 2 different numbers can also be the products of other 2 numbers

Like 2x4=1x8

Hope it helps

[naive hare]

thank you 🙂

[sterile python]

If you are done you can type .close

[pearl pond]

@naive hare Has your question been resolved?

[pearl pond]

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[limpid ginkgo]

1+1=؟

[autumn fossil]

Please don't troll in the help channels, use #chill

If you don't have any real questions, you can close this with .close

[hoary relic]

<@&268886789983436800> ?

[merry carbon]

I think it works out to 3

[hoary relic]

Can we close

Did that guy just leave

[autumn fossil]

he is still here it seems

[merry carbon]

They didn't, but they'll see why I said 3

.close

[pearl pond]

Channel closed

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[autumn fossil]

good for them that they use such low numbers

imagine asking what 67 + 420 is

[hoary relic]

,calc 67 + 420

[jolly parrot]

Result:

487

[hoary relic]

Trivial.

[merry carbon]

"I'm gonna go for years for this one"

[hoary relic]

Anyways imma leave cya

[pearl pond]

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[jolly parrot]

Renato

[rough stream]

Just the arclength formula used directly

[stoic imp]

this ?

[jolly parrot]

Renato

[rough stream]

That's typically taken as the arclength formula

<@&268886789983436800> scam bot

Surely banning these could be automated? It's always the same picture

[stoic imp]

oh so its not the supremum of this set then?

[rough stream]

It is!

Hmm. Maybe we could do it with some kind of Riemann sums approach?

Are you not expected to use the arclength formula?

[jolly parrot]

Renato

[stoic imp]

it was hidden as a proposition

[stoic imp]

we need to check if this parametrization is regular

hopefully it is otherwise we are doomed

.close

[pearl pond]

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[scarlet aurora]

Yo wanted to ask if someone could help me with maybe one of this topics in a call. If u are german it would be great.

[scarlet aurora]

tomorrow or thursday would be also great

[pearl pond]

@scarlet aurora Has your question been resolved?

[pearl pond]

@scarlet aurora Has your question been resolved?

[pearl pond]

@scarlet aurora Has your question been resolved?

[dense jasper]

We don't have voice channels for helping because of moderation reasons

I'm sure the mods can speak more on this

[pearl pond]

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[pearl pond]

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[sleek sandal]

hi

[sleek sandal]

is the picture he drew on the right side of the paper correct in term of the circulation. I thought it would have to go the other way because if my thumb points the direction of curl, my fingers wanna go the other way

let me know if im rwong

[pearl pond]

@sleek sandal Has your question been resolved?

[little elk]

Ifthe rotation is counter clockwise then the curl is towards you

Not sure where your arrow is pointing

Should be towards you

[pearl pond]

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[stoic seal]

i want help to plot this or get an intuition for it!

[pearl pond]

@stoic seal Has your question been resolved?

[viscid shale]

Do it numerically

[stoic seal]

fair

i kinda have a rough idea, i will try that thanks

.close

[pearl pond]

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[pearl pond]

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[sharp smelt]

could someone please help me with this

[sharp smelt]

so I get that I want E(kXbar)= \theta /2 I suppose?

[past perch]

i assume you find the expected value and then use that as the mean of the sample mean

[past perch]

idts i think you want k such that E(kXbar) = theta

[sharp smelt]

my issue is x depdns on theta

[past perch]

where Xbar would be theta upon 2

[sharp smelt]

and theta is unkown

[past perch]

we also know each X_i is uniform on (0 theta)

and we want kXbar to be an unbiased estimator of theta

ie E(kXbar) = theta

[sharp smelt]

mhm

and X bar would just be E(X_i) = \frac{1}{\theta} no?

[past perch]

no

Xbar would be E(X_i) = ||more formally|| \int_0^{\theta}{xf(x)} dx

[sharp smelt]

oops

theta /2 right

[past perch]

yeps

[sharp smelt]

so k theta /2 = theta so k=2

[past perch]

from what i understand yes

since E(Xbar) = theta/2, 2 makes the estimator hit theta on average

[sharp smelt]

okay, checks out with the key

thanks

.close

[pearl pond]

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[pearl pond]

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[tawny topaz]

Q30

[tawny topaz]

How do i do this

...

[hoary relic]

Try finding the slope ofcthe equation

Let x be the year and y be the salary

You have the equation y = mx+b where m is the annual incremnt and b his starting salary

[tawny topaz]

Wsit

I did a method

[tawny topaz]

Is this correct?

Different method tho

[proper nova]

,rccw

[hoary relic]

Oh i was talking about 28 not 30 lol sry

[tawny topaz]

What is mx + b

Im gonna do 28 soon

[hoary relic]

Linear equation

[tawny topaz]

But is this correct

[tawny topaz]

I didnt understand

[hoary relic]

I suppose so, havent checked

[hoary relic]

Well you can practice here https://en.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations

[tawny topaz]

The answer is 50 and 1300 right for 28

Is this correct @hoary relic

[hoary relic]

Yup.

[tawny topaz]

Ty

[hoary relic]

Anyways one thing that might help you when you get these point problems is point-slope form

Its for when you have 2 points

The general point-slope form is y-y_1 = m(x-x_1)

Where (x_1;y_1) is the coordinates of a point and m is the slope

For example in 28, find the slope first by (1800-1500)/(10-4) = 300/6 = 50 which is m
Then plug in the first point (50;1800) which gives y-1800 =50(x-10)

Simplifying into y= 50x + 1200

Now you see that m = 50 and b = 1200, which might be easier than what you did.

[pearl pond]

@tawny topaz Has your question been resolved?

[hoary relic]

@tawny topaz are you done? If yes please type .close

[pearl pond]

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[sharp galleon]

For fucks sake

How do i even start c

Bro

[rough forge]

Good luck

Did you try the hint?

[sharp galleon]

Brother

What hit

Hint

[rough forge]

Also I would try IBP

[sharp galleon]

???

Omgg

I know what to fo

Bro

Im not fucking seeing this

In the exam

In this time

[rough forge]

Are you okay

[sharp galleon]

You’re supposed tondo

Like

Speerate the sechs

So its like

^n-2 and ^2

[rough forge]

Is the a word limit per msg

[sharp galleon]

Aight bro

Real funny

Wait

Are you a girl or a boy

Don’t report me for calling you bro pls

I almost got banned for calling someone bro

🫩

[rough forge]

Maybe don't call people bro, real

[sharp galleon]

Bro

Sybau

Wait leme try

[toxic fractal]

maybe dont spam bs either

[sharp galleon]

Wydm

[rough forge]

the word limit per msg

[sharp galleon]

Ah

Star does my work make sense

What i said?

[rough forge]

no

[sharp galleon]

Why not

[rough forge]

I lost track

[sharp galleon]

If i turn it into this

Because part b has to be for something

And then i can use ibp

[rough forge]

but now you got the sech²(x) term

[steep saddle]

maybe try case when n is zero and two?

[sharp galleon]

What

[sharp galleon]

Yeah

Look at part b

And then

I do ibp

With like

Integrate the sech2tanh2

And differentiate sechn-2

Am i a genius or what

[rough forge]

Try, I don't see the vision yet

[sharp galleon]

Why?

Whats wrong w the vidion

Ay bro

How doni integrate sechx lol

Is it -sechxtanhx

@rough forge

Come back bro

🫩🍃

[buoyant cypress]

based on the hint iit would be good if you get rewrite as sech^n(x) tanx then ibp

[pearl pond]

@sharp galleon Has your question been resolved?

[pearl pond]

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[sharp galleon]

.reppen

[sharp galleon]

Why

[pearl pond]

@sharp galleon Has your question been resolved?

[pearl pond]

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[jolly parrot]

Renato

[broken timber]

Try thinking how a square can be divided into 4 smaller squares. Try dividing the smaller squares. For each dividing, how many new squares are obtained?

[stoic imp]

can you do a drawing

[pearl pond]

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[pearl pond]

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[left arch]

so far, i know TPs is 90
x = 30
and TSP = 29

[left arch]

how do i use those to find the measure of RST

[south ice]

what have you tried so far

[timid turret]

Can you explain how TPS is 29?

[left arch]

wrong way

fixed

[left arch]

90-29