help-39

[misty trout]

due to issues with my helper

can a diffrent helper teach me chapter 3.6

oh wait ur not going?

[spiral coyote]

I hv an exam tmrw lol i gotta study but yeah hope u get the help u need.( enough i really should study)

[misty trout]

good luck, and really thxs, ur teaching meant so much to me

[misty trout]

(When someone is here pls ping me)

(If I don't respond just explain a stech, shrink, translation, and reflection and how to the question. Also explain it by using ur explantion in 1 problem in chapter 3.6)

I'm back

ohhh 15 min pass

<@&286206848099549185>

chapter 3.6

[prisma kernel]

What's your question again?

[misty trout]

A wondeful person help me with cahpter 3.5

but also need to study on her test

so left me to find another helper to teach me chap 3.6

[prisma kernel]

I guess transformation had an equation

Oh wait yeah, just use y = mx + c

[green flicker]

@misty trout For 3.6, increase c. For 3.5, equate y to 0.

[prisma kernel]

m is the slope, i.e. the angle of the line, and c is the y-intercept, so can find things accordingly

[misty trout]

explain?

step by step if possible

[prisma kernel]

Alright see

[misty trout]

And I just learn chapter 3.5

[prisma kernel]

Do you know y = mx + c

[misty trout]

no

but familiar with y=mx+b

[prisma kernel]

Oh yeah that's just the same thing, b is the y-intercept right?

[misty trout]

yes

b is y-intercept

Ok we have y=mx+b

whats next

[prisma kernel]

Alright, so you're given f(x) = x

It can be written as f(x) = 1*x + 0

So the y-intercept here is origin, and the slope is 1

[misty trout]

hmmm

f means

original line

and g means the new line?

[prisma kernel]

Yes

[misty trout]

ok we're givien this

so x

means right by 1 up by 1

if by itself

[prisma kernel]

Yes, 1 by 1, 2 by 2, and so on

[misty trout]

Just fowarding it

so

[prisma kernel]

And the slope of it is 1, because if we substitute it into y = mx + b, then we get y = 1*x + 0, hence the slope is 1 and the y-intercept is 0

[misty trout]

mhm I'm following

So

f(x) = 1*x + 0

also means

[prisma kernel]

OneRepublic, nice

But yeah, that's your graph

[misty trout]

mhm ok

but than

we are getten g(x)

or our new line

[prisma kernel]

Okay, so we need to convert f(x) into g(x) yes?

[misty trout]

I think the paper want us to do that

[prisma kernel]

Okay, now to convert it, be first multiply f(x) by 3

[misty trout]

ok 1st multiply f(x) by 3

[prisma kernel]

Which means stretching the graph up, such that f(1) = 3, f(2) = 6, and so on

[prisma kernel]

Like tilting it to be more steep at this rate

[misty trout]

mhm ok

[misty trout]

quick question where did u do this

what make you do this specifficaly

where in the equation

[prisma kernel]

Because g(x) = -3x - 2 yeah

And f(x) = x

[misty trout]

the -3x?

[prisma kernel]

Yes, we first multiply f(x) with 3 to get 3x, then we reflect it along the x-axis to get -3x, and then we pull it's y-intercept down to -2 to get -3x - 2

[misty trout]

ok

so translation

than reflection

than translation

[prisma kernel]

I mean not sure if you'll call multiplying it by 3 as translation, but I'm terrible with terms so up to you on this

[misty trout]

when multiplying by 3, what does that do?

[prisma kernel]

It basically "tilts" it

Like it still passes through origin, but you rotated it a bit counter-clockwise

[misty trout]

mhm close to this

[prisma kernel]

I mean yeah if it helps, just a lot more tilt

[misty trout]

ok

it tilts

than it reflects

[prisma kernel]

Like more than 25°

[misty trout]

ok

so first it tilts

almost 25 degrees

[prisma kernel]

Yes

[misty trout]

than it reflects over x-axis

[prisma kernel]

Yes

[misty trout]

than it translate down some units down(translate means move)

[prisma kernel]

Yes, it's y-intercept goes down by 2 units

[misty trout]

ok what equation r we doing again?

I want to see it and imigine it

[prisma kernel]

f(x) = x
g(x) = -3x - 2

Convert f(x) to g(x)

[misty trout]

ok

ic

so first we tilt

by multipying x by 3

which gets us g(x)=3x

[prisma kernel]

Yes

[misty trout]

than we flip it

g(x) = -3x

[prisma kernel]

Yes

[misty trout]

than we translate(move) down two units

to get

g(x)=-3x - 2!!!!

[prisma kernel]

Wait wait

Oh you scared me there

Nice!

[misty trout]

😄

but

how to we get

the answer now

Ik how it works

but whats the equation

to show (What we did) to get the answer

[misty trout]

Because it say describe the translation

so now we need to write an answer

[prisma kernel]

Even I'm not sure, I guess you just list down these steps to show that you actually know what you're doing

[misty trout]

mhm ok

[misty trout]

Ok part 2

oh wrong foward

[misty trout]

much better

ok now I need to know a strech

shrink

translation

and reflection

(strech and shrink is something vocab I NEED TO KNOW)

[prisma kernel]

I guess they're related with curves, although I'm not sure, they might be applicable here too

Maybe stretching is what that "tilt" was

[misty trout]

ok thats fine

can we work on a problem

im not undertstanding

[misty trout]

In question 22

[prisma kernel]

Okay so

[misty trout]

Says f(x+3)

[prisma kernel]

If f(x) = x, then what is f(x + 3)?

[misty trout]

let f(x) = 3x + 4

describe the transformation

[prisma kernel]

Yes yes, but let's start with this basic first

[misty trout]

Ok original line

@prisma kernel

R u good?

[prisma kernel]

Yes, I'm waiting for you to answer that question

[misty trout]

Oh mb mb dint see

[prisma kernel]

That's fine

[misty trout]

f(x+3) might be moving the unit?

thats where it lost me

the "( )"

[prisma kernel]

Yes, but you need to figure out what it does

Okay tell me

If f(x) = x, then what is f(1)?

[misty trout]

it means if it goes right by 1 goes up by 1?

[prisma kernel]

Yes that kind of a graph, but I want you think equation wise

[misty trout]

f(x) = x + 1

[prisma kernel]

No no, okay wait I'll tell you

f(x) = x

Then f(x + 3) is just (x + 3)

[misty trout]

the y intercept starts at 3?

if x is 0?

[prisma kernel]

Whatever is inside the bracket of f(), we replace x with that in our equation

[prisma kernel]

Yes

So if f(x) = 3x, then f(x + 3) becomes 3(x + 3)

You need to evaluate the equation first, after that thinking graphically would be very easy once you've simplified the equation

[misty trout]

hmmm somewhere here u lost me...

but ima guess

So we have h(x) = f(x+3)

[prisma kernel]

Ig that happened because we moved from graphical to equations for this question

[misty trout]

this means

when x is at 0

the y axis is at 3

:3

[prisma kernel]

No okay see

First of all, you have to get this clear

[misty trout]

mhm

[prisma kernel]

If f(x) = ax + b, or anything that has x, then f(w) = aw + b, you replace x in your equation with whatever is in that bracket

[misty trout]

Wait hold up it moves the unit

left

3 units

[prisma kernel]

First you have to get this equation clear, once you do that it'd all be fairly easy

[misty trout]

ok replace x with whatever is in bracket

[prisma kernel]

So if you have f(x) = 3x + 4, and they've said that h(x) = f(x + 3), then you just replace x with (x + 3) in your original equation, that becomes h(x) = 3(x + 3) + 4

[misty trout]

So we have h(x) = f(x+3) replace x is h(x+3) =

[prisma kernel]

Oh no no, you can't do that among functions, you do it when a function is equal to an equation

[misty trout]

wait original equation

[misty trout]

ohhh ok

ok we have

f(x) = 3x + 4

and

h(x) = f(x + 3)

[prisma kernel]

Yes

[misty trout]

take the bracket

we have (x + 3)

[prisma kernel]

Yes now first find f(x + 3)

Replace x with (x + 3) in f(x)

[misty trout]

f(x) = 3x + 4

so f(x) = 3(x + 3) + 4

MMMMMMMMMMMMM

[prisma kernel]

Okay i get what you did here

[misty trout]

I'm getting close

ur answer and my answer is closer

I'm getting there

[prisma kernel]

f(x + 3) I assume

[misty trout]

yeah

but wait

we replace it

[prisma kernel]

Okay, so you've got this, and that's just h(x)

[misty trout]

h(x)

ok

1 more time

[prisma kernel]

Sure

[misty trout]

ok we have
f(x) = 3x + 4
and
h(x) = f(x + 3)

we take the bracket

[prisma kernel]

Yes

[misty trout]

(x + 3)

go back to f(x)

f(x) = 3x + 4

replace x with bracket

[prisma kernel]

Yes

[misty trout]

h(x) = 3(x + 3) + 4

[prisma kernel]

Exactly

Now just simplify it

[misty trout]

Simplify it....

[prisma kernel]

Like open the bracket

3(x + 3) becomes?

[misty trout]

hmmm

do we divide from both sides?

[prisma kernel]

No no

Like 2x + 4 can be written as 2(x + 2)

So what can 3(x + 3) be written as

Or like 4(x + 3) can be written as 4x + 12

[misty trout]

3x + 9 😄

AHHHHH ok ok

[prisma kernel]

Yes, so rewrite h(x) after you've simplified 3(x + 3)

[misty trout]

h(x) = 3x + 9 + 4

and simplify that

is

h(x) = 3x + 13

[prisma kernel]

Yes

And originally f(x) was 3x + 4

So you basically just moved it's y-intercept up by 9 units

[misty trout]

hmmmm ok

So that means

blue is the original line

and green is the new line

and this was a vertical translation of postive 5

[prisma kernel]

Yes, so just moved up

[misty trout]

up by 5 right

[prisma kernel]

Won't it be called transition?

[prisma kernel]

No no, 9

[misty trout]

yea

[misty trout]

up it went up by 9

HMMMMMMMM

Ok

So that what "( )" bracets mean

[misty trout]

Number 26

also looks a bit tricky

and we do that

after that I think I can do anything

in chapter 3.6

[prisma kernel]

Okay so h(x) = 3f(x)

[misty trout]

yes

[prisma kernel]

Now f(x) was 3x + 4

[prisma kernel]

So just substitute f(x) with 3x + 4 here

[misty trout]

ok we have f(x) = 3x+4

h(x) = 3f(x)

So we replace the bracket

so h(x) = 3(x)+4

but than

that makes another h(x) = 3(x)+4

:/

[prisma kernel]

No no see

Here, they haven't done anything INSIDE the bracket yeah

You replace x with that when they've done something inside f()

[misty trout]

ohhhhhhh

ic

os basicly

h(x) = 3(x)+4

is h(x) = 3x+4

@prisma kernel

[prisma kernel]

No no you don't do that here, you do that complicated inside the bracket stuff when they've given something other than x inside f(x)

[misty trout]

in that case

[prisma kernel]

If they haven't done anything INSIDE f(x), then you just treat f(x) as just another variable

[misty trout]

we leave it

h(x) = 3(x)+4

no changes

[prisma kernel]

No but h(x) was 3*f(x) yes

So it's basically 3(3x + 4)

[misty trout]

f(x) was x + 4

h(x) = 3(x)+4

so

here

[prisma kernel]

No no, f(x) was 3x + 4 ig

[misty trout]

ok we have f(x) = 3x+4
h(x) = 3f(x)

So

we need find the diffrence

what will you do?

[prisma kernel]

Okay let me help you with something

If h(x) is equal to f(k), where k is anything but x, then you do the stuff we did earlier (replace x with k and evaluate the equation)

But if there's no change and there's a difference between h(x) and f(x) outside f(x) (like h(x) = -f(x), h(x) = 2f(x)), then you can just treat f(x) as y and then solve the equation simply

[misty trout]

mhm ok

so we solve the equation simply

[prisma kernel]

Yes, here we do

[misty trout]

Does that means

we multiply it

by 3

or divide it

[prisma kernel]

Multiply

[misty trout]

Ok

[prisma kernel]

Like if there's no change inside f(), then it's just y

So y = 3x + 4

[misty trout]

f(x) = 3x+4
h(x) = 3f(x)

[prisma kernel]

And they've asked you h(x) = 3y

[misty trout]

y = 3x + 4

and we want h(x) = 3y

[misty trout]

first we subtract 4

...

right?

[prisma kernel]

No why

Shouldn't it be 3(3x + 4)

As y = 3x + 4

And it was 3y

[misty trout]

y = 3x + 4
and we want h(x) = 3y

so we multiply by 3

[prisma kernel]

Yes

[misty trout]

y=9x+4

....

or will it be

y = 3x + 12

[prisma kernel]

Yes

[midnight haven]

which

[misty trout]

ok

it will be

y = 3x + 12

[prisma kernel]

We're solving question 26 mate, the worksheets above there somewhere

[prisma kernel]

No, 9x + 12

[misty trout]

OHHHHH

[midnight haven]

send me

lemme see

[misty trout]

WE MULTIPLY THE WHOLE THING

[prisma kernel]

YES

[misty trout]

AHHHHH

My brain is braining

THAT MATCH MY GRAGH

IT'S CORRECT

9x + 12

Oh oh oh

ok ok ok

y = 3x + 4
h(x) = 3y

so

when when

the y intercept variable

is shown

multiply the whole thing

[prisma kernel]

Yes

[midnight haven]

h(x) = 3x + 4?

[misty trout]

How much time u got?

left

till u need to do something else

cause if u have time we can do question 27

to wrap things up

[prisma kernel]

Yeah sure, I'm here till you need me

[misty trout]

THis time

I do it myself

[midnight haven]

h(x) = 22

[misty trout]

and u tell me if it correct

f(x) = 3x + 4

and

we take bracet

6x

[prisma kernel]

Okay, so they've made some change INSIDE f() here, so you do the process we did earlier

[misty trout]

and we get

[prisma kernel]

Yes

[midnight haven]

3(6x) + 4

[misty trout]

f(x) = 3(6x) + 4

and than

we simplify

f(x) = 18x + 4

Is that correct?

[prisma kernel]

Perfect!

[misty trout]

No giving answers out, check rules

[misty trout]

😄

So....

[midnight haven]

im making as well

[misty trout]

I got a test tmr

on chapter 3.6

what do u thinl @prisma kernel

[midnight haven]

cant i?

[misty trout]

mhm well sure

[prisma kernel]

Well, something tells me you're gonna ace it

[misty trout]

well Thxs @prisma kernel

[prisma kernel]

Glad to help man

[misty trout]

And thxs @spiral coyote

Ima close it

.close

[pearl pond]

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[pearl pond]

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[cloud shuttle]

Is it true?

[cloud shuttle]

this too?

[eager jewel]

yes

[cloud shuttle]

So, its like a magnitude of A effective in the direction of B?

Isn't that given by |a|Cos (theta)

[fierce sky]

yeah, think of a sled, maximum work is done when you're pulling in the direction of the sled, as opposed to pulling 90° from the side in which case it won't budge

[tardy reef]

it'll be better if the statement gets rewritten as shine a light in a direction perpendicular to B, coz based on how you shine the light, you can change the length of the shadow, but yea, its good enough

[cloud shuttle]

what about this? |A| Cos( theta ) is already giving me the magnitude of A effective in the direction of B

[tardy reef]

well, yea

thats correct

but the statement is giving more of an intuitive interpretation of what the projection is just so that you can visualize what is meant by that

[cloud shuttle]

Then what's the difference between |B|.|A| Cos( theta ) and |A| Cos( theta )

[tardy reef]

that is the scaling

a dot product is more than simply projection of one vector on another

[cloud shuttle]

So,
projection says "A is this much effective in this direction"
dot product says "A is this much effective in this direction while also considering the already present effect of B"

right?

[tardy reef]

dot product says "A is this much effective in this direction while also considering the already present effect of B"
well, you could say that, technically, extending the projection definition. But its more realistic to think of it in the way it gets applied, so no one would use that

A more palatable interpretation would be A and B together are |A|.|B| strong, but now that they are facing in different directions with an angle theta between them, they are only |A|.|B| cos(theta) effective

But I'd recommend you go rigorously through your textbook/problems and not pick up intuition from other sources

intuition is something that you need to develop on your own based on your own understanding of the concepts, picking that up from elsewhere is simply useless

[cloud shuttle]

they just want me to know "How to solve it" not "What it really means?"

[cloud shuttle]

hmm, this force me to think like dot product is telling about the effectivness of both (together) in the direction of B

[cloud shuttle]

okay, let me postpone it to linear algebra course

[tardy reef]

well, a dot product is a scalar, so it has no direction, so talking about the directed effectiveness is pointless

in the direction of B
I'd remove this

[cloud shuttle]

combined directionless effectivness?

will it be correct?

really?

people were saying that combined effectivness would be |B| + |A| Cos( theta )

[tardy reef]

I'd suggest you look up the physical cases where dot products get applied

such as concept of work in mechanics and so on

[cloud shuttle]

okay!

thanks for your time!

.close

[pearl pond]

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[dreamy wharf]

I can show that (1) and (2) can't both be true

Now i have to show that not (1) implies (2) or not (2) implies (1)

Not (1) means that $Ax = b$ has no solution for $b>0$.
Due to Farkas' lemma there exists y such taht $A^T y(b) \leq 0$ and $b^t y(b) > 0$

How do I proceed?

[jolly parrot]

krumpler

[pearl pond]

@dreamy wharf Has your question been resolved?

[dreamy wharf]

nvm i found the proof online

[pearl pond]

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[bitter herald]

hi

[bitter herald]

testing

[bitter herald]

.close

[pearl pond]

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[weary ledge]

er what exactly are you testing?

[pearl pond]

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[velvet ruin]

What are you confused on this?

[meager scroll]

no idea how to continue

[velvet ruin]

Can you show me your current progress?

[meager scroll]

i have no progress

i liturately dont know what to do any trig sub doesnt work

and hyper trig functions dont either

[barren path]

thats kinda messy

[velvet ruin]

I believe this is a non-elementary integral

[barren path]

yeah

[velvet ruin]

Do you know how to express in hypergeometric function?

[meager scroll]

when we say non elementary what do we mean

[strong relic]

,w integrate sqrt(x^5/2 + 1)

[jolly parrot]

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Requested by ash_.a

[barren path]

it doesnt have closed form

[meager scroll]

whats F1

[meager scroll]

closed ?

[barren path]

try to use u-sub

[meager scroll]

$$
can i do some with this x_k = \exp!\left( i,\frac{2}{5}(\pi + 2\pi k) \right), \qquad k = 0,1,2,3,4
$$

[jolly parrot]

Taebek

[strong relic]

weird ahh hypergeometric function

[meager scroll]

is there no other way to express it

maybe with a sum ?

[strong relic]

not in elementary functions

[meager scroll]

what about with a sum

[meager scroll]

what functions do we call elementary

[ivory basin]

That's the wrong function

,w integrate sqrt(x^(5/2)+1)

[jolly parrot]

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Requested by xavier_dragon

[ivory basin]

Still no solution, but the correct way this time

[ivory basin]

Anything you're familiar with tbh

[meager scroll]

wdym

is e^ix a elementary function?

or like Log

wait no this one

[ivory basin]

,w integrate sqrt(x^3+1)

[jolly parrot]

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Requested by xavier_dragon

[ivory basin]

!original please

[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.

[meager scroll]

so this F1 is what

[ivory basin]

Not an elementary function

[meager scroll]

i still dont get what we refer to as an elementary function

[ivory basin]

Trig, exp, log, poly

And any combination of these

[meager scroll]

like can we express this as integral function

or as an infinite Sum

.close

[pearl pond]

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[pearl pond]

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[pearl pond]

@stoic imp Has your question been resolved?

[pearl pond]

Channel closed

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[stoic imp]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[stoic imp]

,w absolute max and min (x-6)^2 - y^2 in (x-5)^2 + y^2 <= 25 and (x-2)^2 + y^2 >= 16

[tender egret]

This function will go from +inf to -inf

[jolly parrot]

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Requested by renatoxbz

[pearl pond]

@stoic imp Has your question been resolved?

[barren path]

didnt u want to solve using lagrange multip?

[pearl pond]

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[stoic imp]

why do I need to multiply the jacobian after changing from cartesian to cylindrical coordinates in triple integrals

[pearl pond]

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@stoic imp Has your question been resolved?

[rocky socket]

Think dimensional analysis then. The angle is dimensionless

dx dy and rho drho dphi at least have the same units

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[dreamy grail]

Hello, I need help my probability statistics, with simulations. A professor has 8 keys, trying to open a door. He tries one at a time at random from his pocket. All keys look the same and he does not put them back in his pocket after trying. Conduct simulation with random number table and conduct 20 trials.

1. For this part I had a number list, having 000-124 be success and 125-999 be not success. However, I am confused whether I am supposed to do just 20 individuals "numbers" or do the type of trial where I keep going until I hit a success. If I do it the second way, I run out of numbers to test, ending in 13 trials rather than the 20 required.

2. What is the expected number of tries needed for him to find the correct key?
   For this one, I assumed its just (1+2...+8)/8, as each number counts for a try, with that may possibilities, divided by the keys. However I do not know if this is the case because Im not sure if Im supposed to use my simulation data.

3. What is the probability it will take more than 4 tries to find the right key?
   I would think this is 1-(P(x<_3) with (8 0)+... (8 3) being subtracted from the 1.

[chrome plank]

remember, your list of numbers is only correct when there are 8 keys in his pocket.
when he tries one, he doesn't put it back.

[dreamy grail]

Oh, so one trial would consist of 8 numbers, then noting down which number by order he managed to get the right key?

[pearl pond]

@dreamy grail Has your question been resolved?

[dreamy grail]

<@&286206848099549185>

[pearl pond]

@dreamy grail Has your question been resolved?

[dreamy grail]

<@&286206848099549185>

[green flicker]

May I ask what software you are using?

If you are using MATLAB, try linear algebra or permutations and combinations.

Instead of asking us, why not ask your peers or professors?

[dreamy grail]

This is for statistics on paper, and I also do not know what any of that is. This due tomorrow and I do not have contact with my teacher.

[pearl pond]

@dreamy grail Has your question been resolved?

[pearl pond]

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[latent quail]

Alright, latexing

I'm not confident but I assume the question is merely asking me to prove $n^2+1 \ge n+1$\

[jolly parrot]

Acknowledged Scumbag

[brave sluice]

i'm not so sure about that

[latent quail]

I mean if we have a sequence of distinct real numbers

we can always find a subsequence of it which yields a either strictly increasing or decreasing sequence if ordered properly, right?

[brave sluice]

i don't think you can reorder it

we should check the definition of subsequence

In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements.

[latent quail]

hmm

yeah ig that's the flaw of this question

I mean, if that's what it intended, there should be no guarantee

[brave sluice]

idk it seems plausible to me

[latent quail]

yeah I can find an example that does not fulfill it

[brave sluice]

oh nice

what is it?

[latent quail]

n = 2 sequence {7, 6, 8, 5, 9}
you can not find any sub sequence that is either strictly increasing or decreasing with the length of 3

[brave sluice]

7, 6, 5

[latent quail]

hm

yeah forgot that something can be omitted

[crystal dew]

I seem to recall this being a theorem

maybe look up ||erdos-szekeres|| (not sure if this is too direct)

[latent quail]

Alright, will continue later

Ty all

.close

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[exotic gale]

i needa help on this question

[exotic gale]

What is the area, in square units, of the region enclosed by |x-3| + |y+5| = 8?

do we just set it to 0?

[latent quail]

better visualizing it prior to doing anything

[exotic gale]

yeah im trying to graph it on desmos and see

[brave sluice]

you could break it into four cases, depending on the signs of x-3 and y+5

[exotic gale]

pls help i dont understand

[brave sluice]

for example, when x-3 >= 0 and y+5 < 0, graph the line (x-3) - (y+5) = 8

then graph the other three cases

[pearl pond]

@exotic gale Has your question been resolved?

[toxic lichen]

that or maybe you know how to graph absolute value functions, then you can get away with doing a bit less case work

[pearl pond]

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[sharp smelt]

Define $\psi: G \to H \ \phi: G \to G \ \alpha: H \to G$.
\Then define $\Omega_{ \phi} = \alpha \circ \phi \circ \psi$

[jolly parrot]

wai

[sharp smelt]

This feels off as an isomprhic mapping b/w aut(G) and aut(H)

[tropic saddle]

draw diagram

[sharp smelt]

wdym

oh that

[sharp smelt]

Define $\psi: G \to H\ \phi: G \to G; \alpha : H \to H$. Then define $\Omega_{ \phi} : Aut(G) \to Aut(H) $ as $\alpha \circ \psi \circ \phi$

[jolly parrot]

wai

[sharp smelt]

where alpha and psi are fixed

[tropic saddle]

was your response to "draw diagram" really just repeating your other message rn?

[sharp smelt]

SHould I send the diagram

[tropic saddle]

well that would be good

what is alpha

[sharp smelt]

.

[tropic saddle]

no

what does it mean that H and G are iso

intuitively

[sharp smelt]

there's an iso b/w them

[tropic saddle]

intuitively

[sharp smelt]

or they are similar in some sense

[tropic saddle]

they are the same thing, just with different names

imagine a boardgame in english and german

same game

different boards

they are iso

your isomorphism is a translation between them

so lets have an isomorphism phi:H->G

aka a "translation" from H to G

you know automorphisms of G

how can you use these and your translation to get automorphism of H

[sharp smelt]

I first translate, get an image in G and then translate back to H

[tropic saddle]

yes

so what is that in formulas

[sharp smelt]

let g be some automorphism

phi^{-1} o g o phi

[tropic saddle]

yes

[sharp smelt]

oh I see, that's much simpler

[tropic saddle]

what is the diagram

[tropic saddle]

better

but make a square

two Hs and two Gs

[sharp smelt]

is this not enough?

[tropic saddle]

you dont get back to the "same" H

you get back to alpha(H)

so its good to write a second H

same way you wrote two Gs

[sharp smelt]

something like this?

[tropic saddle]

yes

you can now also put in alpha:H->H

so you can either get H->H by going along alpha or going along the other arrows

thats the basic idea

usually its draw with the two Hs on the first line and G on the bottom

[sharp smelt]

nioted

[tropic saddle]

that diagram occurs all the time

eg also for basis change in LA

[tropic saddle]

this is a very fundamental idea

translate, do stuff, translate back

[sharp smelt]

Noted

thanks

.close

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[stoic imp]

how to solve for x in here cos^2(x) = sin^2(x)?

[fierce sky]

move everything to one side and use a double angle identity

[stoic imp]

is there a simpler way with out double angle identity

[fierce sky]

you could solve tan²(x) = 1

[stoic imp]

im cooked

[eager jewel]

yes just do this..

one case tan x = -1 other case tan x = +1

and then use general solution

for trignometric equations

[stoic imp]

what

[eager jewel]

yes now use general solution

[stoic imp]

can i just use arctan

[reef kestrel]

honestly the double angle identity is probably still easier

even if it’s more to remember

[eager jewel]

alr u can use that as well

[stoic imp]

what is that

[reef kestrel]

periodicity

[stoic imp]

how do i use it

[eager jewel]

tan x = 1 does not have 1 single solution..look at the graph of y=tan x

[stoic imp]

i don't know how to graph it

[eager jewel]

u have tan theta = 1 = tan (pi/4) so ur alpha is pi/4..now u can vary the values of n

its fine u dont need to

[stoic imp]

u used arctan?

[eager jewel]

no

[stoic imp]

,w arctan(1)

[eager jewel]

oh u mean like that

[stoic imp]

im cooked

can we use double angle

[eager jewel]

yes

[reef kestrel]

lol yes

[stoic imp]

how

[reef kestrel]

$\cos(2x)=\cos^2(x)-\sin^2(x)$

oops wait

[stoic imp]

but

[jolly parrot]

plantsyavi

[stoic imp]

what about double angle for sin

[reef kestrel]

$\sin(2x)=2\sin(x)\cos(x)$

[jolly parrot]

plantsyavi

[reef kestrel]

this one is less useful here

[reef kestrel]

if you work with this you’ll get this

e.g. take (1-sin(2x)^2)=cos^2(2x)

[reef kestrel]

but eventually you willl reach this

or at least that is the goal

[stoic imp]

when is cosine 0

[reef kestrel]

look at the unit circle

[stoic imp]

no

pi/2

by sohcahtoa

ok Thanks

.olved

.close

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[sharp smelt]

just want to be sure., we want upto isomorphism, right

So for instance as <(2,0)> is iso to <(0,2)> we won't count them as distinct elements

[cursive wraith]

we're not asking up to isomorphism here

[sharp smelt]

I see

well, so each element is 1 or 2

Atleast one element has to be 1 or 2

so 3^n-1?

[cursive wraith]

definitely not that many

[sharp smelt]

well, why not

[cursive wraith]

I'm almost sure you're overcounting just subgroups generated by 1 element

[sharp smelt]

oops, I counted the total number of elements of order 3

[cursive wraith]

you do know that <(2,0)> is the same as <(1,0)> for example right?

we're close to getting the number of subgroups generated by 1 element though

(even though it's not exactly what we're looking for)

[sharp smelt]

well, Z_3+ SZ_3 has 2 subgroups of order 3

[cursive wraith]

really?

and by that you mean $\bZ_3 \oplus \bZ_3$ right

[jolly parrot]

Raphaelisius Maximus MMIII

[sharp smelt]

yes

[cursive wraith]

you didn't find all subgroups of order 3 then

[sharp smelt]

(1,0),(0,1),(1,1) are the genertaors I suppose

so 3

[cursive wraith]

still not

you're missing one

[sharp smelt]

hmm

surely not (1,2)

[cursive wraith]

yes surely

[sharp smelt]

huh

[cursive wraith]

well is <(1,2)> any of the previous subgroups of order 3 you created?

[sharp smelt]

no

fair enough

[cursive wraith]

basically

you found there are 3^2 - 1 = 8 non zero elements

this overcounts how many subgroups generated by 1 element there are

by how much? by a factor of 2

since each of these subgroups has 2 possible generators

so, (3^2 - 1)/2 = 4 subgroups of order 3

[sharp smelt]

noted

[cursive wraith]

now, we need subgroups of order 1 and 9

{0} and Z3+Z3

so 1 + 4 + 1 = 6

[sharp smelt]

got it

thanks!

[chrome plank]

{<(1,0)>, <(0,1)>, <(1,1)>, <(1,2)>}

[sharp smelt]

.close

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[frosty grail]

i’m given matrix A as shown

and i’m trying to find its determinant using cofactor method

so i chose to “eliminate” row 2

apparently the determinant is supposed to be -2 based on the answer key

but i’m not getting that

where am i going wrong

[feral leaf]

Det of a 2 x 2 is AD - BC

You did AD + BC

[frosty grail]

ohhh

thank you!

[pearl pond]

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[loud viper]

Pls help with this question

[acoustic compass]

Did you try using the identities

[loud viper]

I wrote pi/18=theta and got a very complex expression which I couldn't simplify in terms of theta

[acoustic compass]

which function?

Trigonometric function I mean

[loud viper]

Tbh I didn't find anything useful

Nor in f(x) nor in g(x)

[acoustic compass]

What are the options? Maybe you could use the series expansion formula

[loud viper]

It's an integer type

[acoustic compass]

Hmmm

[loud viper]

Answer is 17

[pearl pond]

@loud viper Has your question been resolved?

[loud viper]

<@&286206848099549185> pls help

[loud viper]

.

I simplified to this

[pearl pond]

@loud viper Has your question been resolved?

[pearl pond]

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[blissful brook]

how do i do part b

[plush bramble]

use part a) and the initial condition

[blissful brook]

x = 0, t = 0?

[reef kestrel]

Yup

[blissful brook]

how does this help me

f(t) = k/3 - ke^(-3t)/3

subbing in those just gives 0 = k/3 - k/3

[plush bramble]

then your answer from part a looks wrong

[blissful brook]

nah

its right

[plush bramble]

!show

[pearl pond]

Show your work, and if possible, explain where you are stuck.

[plush bramble]

where did you use this?

[blissful brook]

i didnt

do i jst set f(t) to 7

[plush bramble]

well there you go

do you know what "long term" means?

[blissful brook]

yeah

[plush bramble]

what does it mean

[blissful brook]

as t -> inf

[plush bramble]

yes

[blissful brook]

but how do i represent that

lemme try

well

as t-> inf, e^-3t -> 0

so 7 = k/3 - k/3

this makes no sense tho

7 = 0 lol

[plush bramble]

correct here

[blissful brook]

hmm

nvm

im tripping

7 = k/3 so k = 21

.close

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[mental depot]

Where did i go wrong?😭

[tulip ore]

you solved for C incorrectly

[mental depot]

Im so dumb

I substituted t and then didnt remove it wtf😭😭

Mb

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[tulip ore]

I dont know what you mean by that

when you substitute t, the C is still there

you solved for C incorrectly, it should be 1/2 (ln 5 - ln 4)

[mental depot]

Ye i calculated it wrong

I assumed t=x^2 then i didnt square the values anyways i got it

[tulip ore]

damn I didnt notice that

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[full basin]

Could somebody please explain why the radius of the yellow segment is the same as OE?

[twin portal]

What are the initial conditions of the figure?

Because it's clearly written that AO = r = OB = r = OE = r.

So the radius AO of the yellow segment is r = OE.
It's not like everyone has done the exact same problem and knows what's going on - the initial conditions / the problem statement.

[full basin]

yes sorry, here it is

we arent given that AO=r, so I was wondering how the given solution came to that conculsion

[toxic lichen]

it's a regular 7-gon that we start with

[full basin]

Oh my gosh youre so right

thank you

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[twin portal]

Np.

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[naive marlin]

. close

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.reopen

[pearl pond]

✅ Original question: #help-39 message

[naive marlin]

what are some good eample of isomorphic rings ?

Zero divisors ?

[pearl pond]

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[naive marlin]

.reopen

stupid ass bot

[river hazel]

first statement is correct yeah

i forgot what was the second question?

[naive marlin]

ignore that one

[river hazel]

rings of integers modulo n

[naive marlin]

other ones ?

[river hazel]

uhhh like trivial examples or just in general?

[naive marlin]

both

[river hazel]

i mean an obvious example is Z and Z[x]

for constant polynomials

[pearl pond]

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[dapper sorrel]

can i have some help with this i forgot how to do it?

[toxic lichen]

gonna need to bust out your protractor. and also draw a North line from both cities bc for some reason you aren't given any...

[dapper sorrel]

its like just out the picture north is up

then what and wehre does the north line go across

[toxic lichen]

you need to draw a north line starting from A and also one starting from B

[dapper sorrel]

okay

[pearl pond]

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[naive marlin]

elrenato

[stoic imp]

can we use parametrization for this?

[naive marlin]

mui no pablo espagnol

bye

[median swan]

You probably can do it, but it seems better to just use Lagrange multipliers

[midnight haven]

.tr

$tr

$translate$

[jolly parrot]

░▒▓█PIE█▓▒░

[midnight haven]

🤦🏻‍♂️

[pearl pond]

@stoic imp Has your question been resolved?

[stoic imp]

are you here @median swan ?

[median swan]

Why did you consider y=5 here? You already got the values for y when x=0, and y=5 doesn't seem to solve the system of equations. In the second image I'm sending, you substituted the values in the opposite way, you found that y=11/2. In the third one, you've passed the equation wrong, it should be a 2 not a 6 in the last line. In the last image, why is that implication true?

[stoic imp]

yes, yes, and wdym?

[median swan]

I mean that you need to correct some mistakes

Basically you ignored the points $\left(\pm\frac{3\sqrt{11}}{2},\frac{11}{2}\right)$ that were solutions to the system, and you have some minor mistakes across the solution that I've marked

[jolly parrot]

KonoEmllikDa

[stoic imp]

in the scale of 1 to 10 how ready I'm for the exam?

@median swan

[median swan]

I'm not quite sure, I don't know what else you have to solve. But if it's just lagrange multipliers I would say you are at least at a 7, you need to have more caution when solving the equations. Don't take this value too serious I'm in no position to grade you

[stoic imp]

i will fix my shit and ping you, alright?

[median swan]

sure

[pearl pond]

@stoic imp Has your question been resolved?

[pearl pond]

@stoic imp Has your question been resolved?

[stoic imp]

.close

[pearl pond]

Channel closed

Closed by @stoic imp

Use .reopen if this was a mistake.

[pearl pond]

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[carmine juniper]

{} is a set?

[carmine juniper]

Right

But it’s “empty set”

[steady vapor]

👏

[carmine juniper]

It is still a set

If it’s empty

[split void]

it is a set

[cinder flower]

that is indeed the empty set

[ivory swallow]

The empty set is both empty and set

[carmine juniper]

A ø {B}

“A doesn’t belong in B set?”

[split void]

but $\phi$ is the empty set

[jolly parrot]

Annie Maqionde

[cinder flower]

doesn’t parse

[rigid mist]

oh hey slayla

[cinder flower]

did you mean $\not\in$

[jolly parrot]

slayla

[cinder flower]

hi

[cinder flower]

also if it’s that, what you said still isn’t right

[rigid mist]

do you still remember me?

[cinder flower]

how could i forget

[carmine juniper]

Let’s say

Set A

1,2,3

Set B

1,2,3,4,5,6,7,8,9,10

It’s then

A e {B}

[cinder flower]

no

$A\subseteq B$ is true

[sick lantern]

hi

[jolly parrot]

slayla

[cinder flower]

{B} is a set with one element in it (even if B is a set)

[rough forge]

oh u r back

[pearl pond]

@carmine juniper Has your question been resolved?

[cinder flower]

indeed

[carmine juniper]

.close

[pearl pond]

Channel closed

Closed by @carmine juniper

Use .reopen if this was a mistake.

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Remember:
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• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
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[sharp smelt]

Show that any two groups of prime order isomorphic.

[sharp smelt]

This feels sus

like U(7) isn't isomorphic to U(11)

[cursive wraith]

Two groups of same prime order

[pulsar flax]

Hint: ||There is a simple group of order p for every prime p||

[sharp smelt]

A group of prime order is always cyclic , any two cyclci groups of same cardinalit are isomo

[sharp smelt]

my courses didn't do simple groups 😭

[pulsar flax]

I meant simple as in the word "simple"

not the term

[cursive wraith]

As in easy

[safe marten]

directions cosines and ratios and confusing

[pulsar flax]

!occupied

[cursive wraith]

They meant ||Z_p||

[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).

[sharp smelt]

thanks

.close

[pearl pond]

Channel closed

Closed by @sharp smelt

Use .reopen if this was a mistake.

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Remember:
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[carmine juniper]

Why don’t we write

R c {R}?