help-19

[outer wadi]

I think that was the theorem?

let me go back and look

for each positive divisor k of n, the group <a> has one subgroup of order k, oh so does this mean that Z_p^i has a subgroup of order p

and p is a positive divisor of p^i

ohh

were you asking me

to give <a^(p^i/p)>

[echo ginkgo]

it's one way to see it, take a=1 here

you have <1> = Z_p^i, and the theorem tells you that the only subgroup of order p of <1> is <1^(p^i/p)> aka <p^(i-1)>

[outer wadi]

if we let a = 1

oh man I think I'm getting confused with regular exponents, why isn't 1^(p^i/p) = 1

oh oh

ok nvm I see

[echo ginkgo]

addition vs product notation yeah

[outer wadi]

I see I see

ok let me move to the next part of the proof now that I understand why G has a subgroup of order p

mmm second principle of mathematical induction

I hadn't really practiced with that before

[echo ginkgo]

they're assuming the proposition is true for the smaller group G/K

that's what makes it strong induction

[outer wadi]

yeahh I think ima need help with this one

[outer wadi]

hmm ok let me take a look again

by proposition your referring to the corollary that they are assuming to be true?

[echo ginkgo]

the actual thing you're proving right now

it wouldn't be induction otherwise

essentially this

you replace G by G/K in the statement of the prop, and that's what they're stating at the circled bottom

[short terrace]

I've never heard that referred to as the second principal of natural induction

[echo ginkgo]

weird name for strong induction I agree

what about the third one ? 👀

[short terrace]

I forget what the third one is, well-ordering of naturals?

[echo ginkgo]

well-ordered induction would be a good third principle ig

[short terrace]

I've been doing foundations stuff so induction genuinely scares me

[echo ginkgo]

👻 foundations stuff 👻

@outer wadi gtg soon so if you have other qs, it's right now

or xavier can take over maybe

[outer wadi]

Sorry I was moving locations I’m back

Thank you for the help platypus

currently looking up strong induction since ive never actually used it before

[odd edge]

@outer wadi Has your question been resolved?

[outer wadi]

mm so I looked in to strong induction

but im still not quite understanding how its being used here

why do we know G/K has a subgroup of the form H/K for H <= G

oh platypus said we replace G by G/K

mmmm

ok so we have G/K which a finite Abelian group of order n/p

and m/p divides n/p because m divides n

so we apply the hypothesis to this new group

[odd edge]

@outer wadi Has your question been resolved?

[outer wadi]

ok let me get back to this ive taken a long enough break

Can subgroup of G/K only be of the form H/K where H is a subgroup of G

[green elm]

do you know the correspondence theorem?

[outer wadi]

hmm maybe though they might call it something else in this book

is that the first isomorphism theorem?

or do you mean the fact that if there exists a bijection between two sets then there is a 1-1 correspondence?

if im then I dont think I know it

lets see

G/K has the form {aK, bK, cK,...} for a,b,c in G

so a subgroup of this group would defintely have the form {hK, h2K,..} for h in H

[green elm]

it's this theorem:

[outer wadi]

I should probably try and prove that ths would still be a group tho

[outer wadi]

is that the same as this?

[green elm]

no

[outer wadi]

hm not sure what canonical epimorphism is

[green elm]

what book is that, maybe i can locate it for you

[outer wadi]

its Contemporary Abstract Algebra by Gallian

what book is that by the way

I could use another book as a source

[green elm]

it's just the map that sends each g in G to the coset gK

[outer wadi]

oh ok

[green elm]

ah looks like gallian leaves it as an exercise and doesn't name the theorem:

that's in chapter 10

[outer wadi]

oh ok I see, yeah I dont think I tried that one

Itll probably be good to try it now tho

[green elm]

but anyway yea, that's the result you're looking for

[outer wadi]

before I try that though

[green elm]

basically the subgroup structure of G/K is "the same" as the structure of set of subgroups of G that contain K

[outer wadi]

I dont really understand how the strong induction is being used here, so say there is subgroup of the form H/K where H is a subgroup of G, then I get that |H/K| = m/p and from the theorem you gave we can see |H| = m but I dont see how strong induction is allowing this

[green elm]

(where K is any normal subgroup of G)

[outer wadi]

I haven't really used strong induction before so im still a bit confused about it

[green elm]

are you referring to this?

[outer wadi]

yeah I dont see how strong induction is completeing this proof or I guess I should say I dont get whats being inducted on?

idk I think im just confused in general

usually with induction you take n and then show it works n + 1

with a base case

[green elm]

oh, haha, he's referring to the same exercise i showed above, i bet (with a slightly different exercise number because different editions of the book)

you're inducting on n

assume the result holds for all groups of order less than n

and show it must hold for n

[outer wadi]

mmm I see

[outer wadi]

ok everything makes sense now after I prove this

[green elm]

nice

[outer wadi]

thank you for the help Bungo ill try proving this and hopefully I can do it without any help lol

[green elm]

yw

good luck!

[odd edge]

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[swift heron]

hi this is not exactly clicking for me

im not sure if i used the hint properly

I think the first paragraph is right, mbut im not sure what to do frrom here

[pseudo summit]

What type of math is that

[swift heron]

linear algebra

[pseudo summit]

What grade level

Oh it's calc 2

[swift heron]

🗿 calc?

[pseudo summit]

Ye

Calculus

[swift heron]

this isnt calculus

🤔

[pseudo summit]

Says it is

[swift heron]

wut

[faint pine]

Where does it say that linear algebra is calculus??

[pseudo summit]

Oh nvm

[swift heron]

🗿

anywaysssss this question is fucking me up 💔

[pseudo summit]

I saw vectors and thought of calc 😭

[swift heron]

u dont deal with vectors in calc 2 anyways, its mostly a calc 3 thing

[pseudo summit]

I just started integration

So I don't know much

[swift heron]

all good

[odd edge]

@swift heron Has your question been resolved?

[swift heron]

okay I think i solved it, but let me know if this makes sense

i feel like missing somethign

[odd edge]

@swift heron Has your question been resolved?

[odd edge]

@swift heron Has your question been resolved?

[sinful grove]

What’s ur def of “diagonalization”?
Since S = [I]_BE with E standard basis, then u directly get that [L_A]_BB = D

[odd edge]

@swift heron Has your question been resolved?

[odd edge]

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[slim vortex]

Hi

[slim vortex]

I need help with something

I'll give you a problem

And solve it

And tell me if the final number is negative or positive

Solve H

[errant gust]

are you asking if H is negative or positive

[slim vortex]

Yes when u calculate it all

[errant gust]

negative numbers raised to an odd power is negative

raised to an even power is positive

negative exponents will not generate negative numbers if the base is positive

[slim vortex]

Here is me solving it

[sick plaza]

could you explain the problem you're having? Do you just need us to check your work?

[slim vortex]

Check pls

Cause

Idk if the final answer should be -5292 or 5292

[sick plaza]

there are 3 negatives

one is squared

one is just there

and another is to the power of 2025

does that help

[slim vortex]

Ok I'll ask a question that'll help me

[slim vortex]

Tell me

If this is correct

The second photo

[sick plaza]

yes

[slim vortex]

Correct?

[sick plaza]

you get -1 * (-1)^-2025 * (-1)^2 = 1

that's the sign calculation

ignoring the size of the numbers

[slim vortex]

Wut

[sick plaza]

-1 from in front

[slim vortex]

Where did u get

[sick plaza]

(-1)^-2025 from below

[slim vortex]

Yes

[sick plaza]

and (-1)^2 from the (-35)^2

ignoring 35^2

just taking the -1

[slim vortex]

Kk

[sick plaza]

so the end result is positive

[slim vortex]

I'ma write it down

Then I can tell u what u can help me with more

Now

Here is the question I want to you to help me with

Number 5

It says

We need to prove that E =3^2m x 2² x5²

[sick plaza]

could you translate the entire question of 5?

[slim vortex]

Ok

E and F are numbers
F=2025
E=9^m+2 +9^m x 19

m is a natural number and m>4

Prove that

E= 3^2m x 2² x 5²

[sick plaza]

do you know how to split $a^{b + c}$?

[clever fjord]

Katharine

[slim vortex]

split powers?

[sick plaza]

yes

[slim vortex]

I did this at first

The 9 turned it into

3^2m+2

[sick plaza]

that's not correct

you're forgetting something

[slim vortex]

3^2(m+2)?

[sick plaza]

$9^{m + 2} = (3^{2})^{m+2}$

[clever fjord]

Katharine

[slim vortex]

Oh

Hmmm

Ok so when I do that

I have to always put

()

Ok understood

[tacit wasp]

Of course 😅

[sick plaza]

but i would go a different path first

[slim vortex]

() are my worst enemies

[sick plaza]

in the expression you want to prove

[tacit wasp]

Because m+2 is the exponent of the whole base (hence 3²)

[sick plaza]

E = 3^2m × 2^2 × 5^2 there is a 3^2m multiplied by other stuff

[slim vortex]

Yes

[sick plaza]

and 3^2m = 9^m

[slim vortex]

Yes

[sick plaza]

and you can see in the starting expression that there is $9^m \times k + 9^m \times l$

[clever fjord]

Katharine

[sick plaza]

if you split the powers

you start with $E = 9^{m + 2} + 9^m \times 19$

[clever fjord]

Katharine

[sick plaza]

and splitting the powers to get another 9^m alone

that make sense?

[slim vortex]

Yup

So we are left with?

[sick plaza]

so you get $E = 9^m \times 9^2 + 9^m \times 19$

[clever fjord]

Katharine

[slim vortex]

Ok

Then

[sick plaza]

which i think you know how to work with

[slim vortex]

W break em down?

[sick plaza]

you have a common term with the +

[slim vortex]

I'ma translate that n comeback

i don't understand English that well :3

Do you mean

That between +

We have

Same base

9

I can't solve if I can't understand 😭

[sick plaza]

this

[slim vortex]

Oh u mean that

So

9^m x9² + 9^m x 9^m

Is it correct?

Uhhmmm

Hello?

.close

[odd edge]

Channel closed

Closed by @slim vortex

Use .reopen if this was a mistake.

[sick plaza]

.reopen

[odd edge]

✅ Original question: #help-19 message

[sick plaza]

not correct @slim vortex

you added another 9^m

[slim vortex]

Wait wait

[sick plaza]

you have this

where 9^m is x

[slim vortex]

Yes

If a

Is 9²

What's b

[sick plaza]

19

[slim vortex]

Oh so not the other 9^m

[slim vortex]

.

[sick plaza]

$E = 9^2 \times 9^m + 19 \times 9^m$

[clever fjord]

Katharine

[sick plaza]

this way it's clearer

[slim vortex]

Ye mb

[sick plaza]

right so you apply that rule

[slim vortex]

I thought u meant 9^m (9²+9^m)

Ok

[waxen talon]

we are only here to help you with homework btw, we don't give answers directly

[slim vortex]

I alr solved that I just wanted to check I fo was fully correct

[slim vortex]

Ok I give I'm a certified waste of brain space but My brain froze 😭

Idk if the answer is Infront of my eyes

And I couldn't find it

Or I'm I just not able to get it

😭

Pls help

[sick plaza]

ok so applying the rule gives

[prisma spade]

Guys i need some one he understnd math i need him to help me how write the equation

[sick plaza]

$E = 9^m (9^2 + 19)$

[clever fjord]

Katharine

[sick plaza]

please open a new channel

[slim vortex]

Yes

I did that

But

The 19 makes me freeze

[sick plaza]

well now

add them up

in the brackets

[slim vortex]

(9^m x 9² + 9^m x 19)

[sick plaza]

this bit

[slim vortex]

Yes

Do I calculate em

[sick plaza]

yes what is the number you get

[slim vortex]

9² is 81 +19 is 100

[sick plaza]

yes

good

so now you have

$E = 9^m \times 100$

[clever fjord]

Katharine

[sick plaza]

do you know what factoring is

[slim vortex]

Breaking them down?

We break the

100 down

?

[sick plaza]

yes

[slim vortex]

Ok

5²* 2²

[sick plaza]

yes

[slim vortex]

And

[sick plaza]

good

[slim vortex]

9^m

Is 3^2m

[sick plaza]

exactly

yes

[slim vortex]

It was all in that 19 😭

I thought

I had to multiply it by 9^m

[sick plaza]

so you've turned $E = 9^{m + 2} + 9^m \times 19$ into $E = 3^{2m} \times 2^2 \times 5^2$

[clever fjord]

Katharine

[slim vortex]

Ok now

Pgcd

And ppcm

[sick plaza]

you should go back to it

and see if you understand all the steps

[slim vortex]

Ok

I'ma re say em

Without lookin

So

We separate the 9^m+2

Into

9² and 9¥m

9^m*

Then

We get commun term

Common*

9^m(9²+19

Which is 100 x 9^m

We break 100

To 5² and 2²

And simplify 9^m into 3^2m

[sick plaza]

that's correct

[slim vortex]

Now

Into pgcd and ppcm

Question says

With the value m calculate

Pgcd (E,F)

And Ppcm (E²,F)

[sick plaza]

i'm sorry i don't know what pgcd and ppcm are

[slim vortex]

Oh I forgot

French words

Wait I'ma search for those terms

In english

Oh it's gcd

And

[slim vortex]

Is this a term in maths

Greatest common thingy

[sick plaza]

greatest common divisor

[slim vortex]

Yes

It's pgcd

[sick plaza]

biggest number that is in both

makes up both

[slim vortex]

Back

Sorry my phone almost died

Anyways

We need to do gcd (E,F)

[sick plaza]

F = 2025

you need to break up 2025 too now

[slim vortex]

Ok

I need to break it down but how does need to look

[sick plaza]

just like you did with the 100

[slim vortex]

So

Break it into 5 ?

[sick plaza]

into smaller numbers

so you can see which of them are in both

[slim vortex]

Oh ok*

I can break down 2025 into 5 x 405 or

25 x 81

[sick plaza]

break either of those down further

[slim vortex]

Ok

[sick plaza]

until you have some 2^k 3^l 5^m

[slim vortex]

25

Is 5²

81

Is 9²

9²

Is 3⁴

I need 2

[sick plaza]

not necessarily

[slim vortex]

Ok

[sick plaza]

k = 0 is also ok

[slim vortex]

So 5²

And 3⁴

[sick plaza]

so you get $3^4 5^2$

[clever fjord]

Katharine

[slim vortex]

Yes

[sick plaza]

so $E = 2^2 3^{2m} 5^2$ and $F = 3^4 5^2$

[clever fjord]

Katharine

[sick plaza]

m > 4

is what you said right

[slim vortex]

Yes

M>4

Or at least hopefully because

They used

Something that's looks like >

[sick plaza]

could you write the arabic down here?

the exact wording of that sentence

[slim vortex]

It says

حيث m عدد طبيعي
m>4

[sick plaza]

yeah ok

[sick plaza]

ok so m > 4

so m = 5 or m = 6 or m = 7 etc

[slim vortex]

Yes

Since it says calcul with the value of m

We need to find gcd without replacing m with a number

[sick plaza]

this means that E has multiple possible values

[slim vortex]

So it stays m

[slim vortex]

Yes

[sick plaza]

let's look at the smallest one

m = 5

what is E when m = 5

[slim vortex]

Let me see

[sick plaza]

just fill in m don't try to multiply

[slim vortex]

2² . 3^2.5 .5²

[sick plaza]

yes

good

so when you calculate 2*5

[slim vortex]

10

3¹⁰

[sick plaza]

you get $E = 2^2 3^{10} 5^2$

[clever fjord]

Katharine

[sick plaza]

what numbers are in both E and F?

that is the pgcd

[slim vortex]

3 n 5

We choose em both

And

[sick plaza]

3 and 5 but to what power

[slim vortex]

Take lowest power

[sick plaza]

yes

[slim vortex]

Right?

Ok nice

Sweet

[sick plaza]

so the pgcd is?

[slim vortex]

Let me calcul

3⁴ .5²

2025

[sick plaza]

so PGCD(E, F) = F

[slim vortex]

Nice

[sick plaza]

interesting result :D

[slim vortex]

Really inteesting

And sus

Don't u think

[sick plaza]

i think the person who made the exercises chose that

[slim vortex]

Hopefully 👀

Now

Ppcm

I couldn't find the term in english

But I know what to do

We do tleverything we just did

But with E² and F

And choose all the numbers we get

And the highest power for each one

[sick plaza]

important detail, you can use PGCD(E, F) = F

[slim vortex]

K

[sick plaza]

because this result means E = k * F

and so E^2 = m * F

[slim vortex]

Huh

[slim vortex]

Can u explain

[sick plaza]

because the pgcm is F

[slim vortex]

Ok

[sick plaza]

it means that both E and F can be written as some number multiplied by F

[slim vortex]

Oh

That's true

[sick plaza]

$E = 2^2 \times 3^6 \times F$

[clever fjord]

Katharine

[sick plaza]

and $F = 1 \times F$

[clever fjord]

Katharine

[sick plaza]

remember that we're taking m = 5

[slim vortex]

Yes

[sick plaza]

since we can do this

we can write E^2 as a multiple of F too

[slim vortex]

Oo

[sick plaza]

which means the result of PPCM(E^2, F)

is simple

can you think of what it might be

[slim vortex]

U said

E²

Is multiple F

And

[sick plaza]

since E is, so is E^2

[slim vortex]

Yes

So

Ppcm for E² and F is

Oh wait

Btw

I found the term in english

Least common multiple

Is ppcm E?

[sick plaza]

no

[slim vortex]

I jinxed it

[sick plaza]

can you write E = k * E^2?

[slim vortex]

Impossible

Oh wait

Helpy :3 cause with this idk how I get good grades in math 😭

My brain

[sick plaza]

like PGCD(E, F) = F

[slim vortex]

Freezes

Yes

That's ez

[sick plaza]

this one is also interesting

since you can write E^2 as a multiple of F

[slim vortex]

Yes

[sick plaza]

the ppcm (least common multiple)

is just E^2

since the smallest possible multiple of E^2 = 1 * E^2

that's what i was trying to hint at

[slim vortex]

So

It's litterally

This is way too sussy

First one was F

Sus but we let it pass

This one is also E^2

Is it correct ig

But sus

But ig

With that

U helped me understand

The hardest things

That I would've never understood

Tysm

Now I just need

To train

.close

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Closed by @slim vortex

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[odd edge]

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[loud hearth]

Alo
I need to find the root of equation $y=-x^2+x+0.75$ by converting it to $x=g(x); g(x)=x^2-0.75$
and then using fixed point iteration method
yet it is not converging
it only converges when i use 0.5 as an initial value

[clever fjord]

A brief description and guide on how to use me was sent to your DMs!
Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!

[shell haven]

I think it will converge, you just have to do quite a lot of iterations

[loud hearth]

over 1000 and it still didnt tho

[shell haven]

I mean quite a lot is quite a lot

[clever fjord]

EthMiC_

[forest sky]

a sufficient condition for the fixed point iteration to converge is for $\abs{g'(x)} < 1$ for all $x$ between the initial value and the fixed point

[clever fjord]

κλαουντ ☁ (cloud)

[forest sky]

if that's not true you can't reasonably expect it to converge

[loud hearth]

are there any other rules i should know
like somewhere i read that it cant converge if |g'(x)| > 1, is this true

[quasi sparrow]

http://mathonline.wikidot.com/the-convergence-of-the-fixed-point-method

[loud hearth]

oh, i got beat to it

[quasi sparrow]

cloud summarized the theorem for you, the link expands to give the proof

[loud hearth]

thank u

i shall close now

,close

hmmm

[quasi sparrow]

.close is the command

[loud hearth]

.close

[odd edge]

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• Be polite and have a nice day!

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

[sinful goblet]

Save me dear god

[fair forum]

what do you understand or not understand about these questions?
and which one?

[sinful goblet]

I don’t understand any of them 🥹

[fair forum]

do you first understand what a causal relationship is?

[sinful goblet]

Nope

😔

[fair forum]

a causal relationship is where A causes B

and that is pretty much it

[sinful goblet]

I see

[fair forum]

for the first three questions, you are to determine if such a relationship exists, and if it does, which event causes/affects the other

[sinful goblet]

Ooh

What about 4

From A. to D.

[fair forum]

can you tell me what you understand about what these are? because I'm not sure if it's logical to have been handed these without being taught about any of them

(esp. part c)

[sinful goblet]

So what had happened was that i forgot my notebook and I didn’t get any of the notes for it 🥹. I put the notes on another paper and I lost that paper.

Yeah…

[fair forum]

and you don't remember a single thing you wrote on those notes?

do you at least have the lecture slides for this topic then?

[sinful goblet]

Horrible memory

[sinful goblet]

I don’t think so.

[fair forum]

honestly then I'd suggest you do a fresh relearning

[sinful goblet]

Okay

Ill try

Thanks for your help kind fellow ❤️

[fair forum]

no worries

[sinful goblet]

.close

[odd edge]

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[granite smelt]

thanks in advance

[odd edge]

Please don't occupy multiple help channels.

[wooden python]

close prev?

[wooden python]

also really unclear what's even being said here at all

do you have the original problem in writing/print somewhere

[granite smelt]

its in german

[shell haven]

If you could do so

[shell haven]

Can you attempt to translate

[granite smelt]

I need to define that I get all students that have one or more common favorite programm

its 2 b

[odd edge]

@granite smelt Has your question been resolved?

[clever fjord]

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

[glossy basin]

are you comfortable with english? Why did you write favourite "food" here when in the original question it mentions favourite programming language

You can use 'a1,a2,a3,a4 in R such that there exists b1,b2,b3,b4 in R for which...'

[granite smelt]

Didn't know the word for programming language so I just took food

[granite smelt]

I dont understand the theory or logic behind it. Is there a video that explains it or something?

[grim jewel]

hdllo

[granite smelt]

hello

[odd edge]

@granite smelt Has your question been resolved?

[odd edge]

@granite smelt Has your question been resolved?

[granite smelt]

.close

[odd edge]

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[smoky niche]

how do I even start with this

[amber schooner]

vertex form

[smoky niche]

ok so something lik...

[nocturne belfry]

usually it means completing the square

[smoky niche]

oh

[nocturne belfry]

you want to write $-3x^2 + x - 1= a(x-h)^2 + k$

[clever fjord]

jan Niku

[nocturne belfry]

completing the square is usually the best way to get there

[smoky niche]

ohhhhhh

sooo -3(x-h)+k

[fair forum]

whoa there. gotta find out your h and k first

[smoky niche]

oh

ok

[fair forum]

to do that, you will have to complete the square, as mentioned

[smoky niche]

uhhhhhh we cant really split 1x/2

I keep messing up completing the square...

[fair forum]

maybe show an attempt and helpers can see what went wrong

[smoky niche]

ok

[fair forum]

expand this and see if you get back the original

[smoky niche]

ok

[fair forum]

(hint: you won't. the -2 squared gives you a 4, and multiplying that by -3 gives you -12, so subtracting 1 from it gives you -13, which is not the same as the original constant term of -1)

ntm you kinda jumped a few steps here

[smoky niche]

ahhhhhhhh

oh no

ok ok hmmmmmmmmmmmm

Im lost I keep messing up

@fair forum I may need some help

[fair forum]

consider factoring a -3 out of the expression first

(yes this will create fractions, but it's fine)

[smoky niche]

ah

[smoky niche]

so like this?

[fair forum]

what is happening here

[smoky niche]

I-

I dont know

[fair forum]

I asked to factor a -3 from the whole expression

while you did that correctly for the x^2, I'm not sure what happened to the other terms

don't do any completing the square yet. just factor

[smoky niche]

Ok

Um so....-3(x+-2)²?

[fair forum]

no, don't do any completing the square yet!

just faithfully factor a -3, and only a -3 out from all three terms

(this will create two fractions, one for the coefficient of x, and one in the constant term. that is fine)

[smoky niche]

Ok so (1-3?)

[fair forum]

sorry, what do you mean by that?

[smoky niche]

Well isnt x² the a?

[fair forum]

ax^2, yes

but I don't get your point still

[smoky niche]

So then a is 1?

Right?

[fair forum]

a was -3, but if you're talking about post-factoring, then yes a = 1

that's why we want to factor out a -3 in the first place; to turn a into 1 so we don't have to deal with it when actually completing the square

and like I said, you factored out -3 correctly from the x^2 term, just not the x term and the constant term

[smoky niche]

I thought you toss c away

[fair forum]

😅

there's a reason I told you to factor first, complete the square afterwards

don't mash these two steps together

[smoky niche]

Oh

I think I keep mixing when to use completeing the square, factoring and quadratic equations

[fair forum]

that's a fair point, and maybe sometime later we will get into that

but here I explicitly mentioned factoring

and only factoring

we'll get into the actual completing the square part later

[smoky niche]

Oooohh

[fair forum]

ps: don't overthink the factoring part

[smoky niche]

Oh

Ok hmmmmmmmmmmmmm

[fair forum]

if you want to use the shortcut method, I suppose you can

after all, this method came from completing the square to begin with

[smoky niche]

I just wanna understand

ok sooooo

Im just... not making progress

https://tenor.com/view/shinji-ikari-gif-25054017

[fair forum]

have you done the factoring first?

(assuming we are still on the same question)

if not, show what you have tried, where you are stuck, and perhaps what you did not understand about what it means to "factor a -3 out of the whole expression"

[smoky niche]

So shove it to the other side

[fair forum]

no

suppose I have an expression 2x + 4

I want to factor a 2 out of this entire expression

then I would just write it as 2(x + 2)

(I assume you understand what factoring itself means, just not how it applies here. if you don't understand what factoring itself means, then please say so)

[fair forum]

this is what I'm looking for in the original problem - to factor out the -3

no shoving to the other side required (and in fact there kind of is no other side to shove stuff into anyway. we're working with only the expression given here)

[short terrace]

Here's an easier way to complete the square

Let's say you have ax² + bx + c

And you want to write it as a(x+h)² + k

[short terrace]

You can expand this and then equate coefficients

[fair forum]

I... didn't think of that way. yeah, that works too

my bad

[short terrace]

No worries no worries

[odd edge]

@smoky niche Has your question been resolved?

[smoky niche]

But HOW do I get there

[short terrace]

Well you have -3x² + x - 1

Now you want to write this as a(x+h)² + k

Expand this

Also Mika feel free to chime in anytime

This was your channel originally, so I'm happy to leave it to you

[odd edge]

@smoky niche Has your question been resolved?

[smoky niche]

Ok

Thank you

[odd edge]

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[vernal yacht]

Same, I'd like my work checked

[vernal yacht]

a) 1
b) 1
c) 2
d) 3

[fair forum]

correct

[vernal yacht]

Cartesian product question:
ik I've asked this a couple weeks ago but I'd still like to fortify my understanding about the reasonings.

[vernal yacht]

A x B x C simply means the Cartesian product of three sets

Therefore, every subsets should contain 3 elements

nevertheless (A x B) x C means we get the Cartesian product of A and B in the first place. Therefore, we'll get subsets containing 2 elements respectively as a result.

[fair forum]

the second part of the explanation sounds incomplete

you are right that A x B x C is the Cartesian product of three sets and will thus result in ordered triples

[vernal yacht]

nevertheless (A x B) x C means we get the Cartesian product of A and B in the first place. Therefore, we'll get subsets containing 2 elements respectively as a result.
is it here?

[fair forum]

yes, that part sounds incomplete. you correctly mentioned that the final product will consist of ordered pairs instead of ordered triples, but you should probably also say something about the nature of the first element in each pair

(because now the first element in each pair comes from A x B)

[vernal yacht]

I see, thanks.

.solved

[odd edge]

Channel closed

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[odd edge]

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[eager tinsel]

Ummm, just when I though I had no problems with this I'm back at it. I am trying to do $(f \times g)'(2)$, to which the result gave me $-14$. Which by the way is half the solution. $f'(2) = 2$, so, $f'(x) = x$, $g(x) = (3 - x)^2$, so $g'(x) = -3x^2 + 18x - 27$, so the result is $(2)(-3(2)^2+18(2)-27) = -14$, but it happens not to be the case.

[clever fjord]

ℕ∈RD ALERT: Gonçalo Gonçalves

[eager tinsel]

Which now that I check is actually -6

The solution is -7

[wooden python]

uhh hold up

can you show the problem as given in the book? rn there's tons of context missing

@eager tinsel

[eager tinsel]

ok

38.3

[wooden python]

ok so

f'(2)=2, so f'(x)=x
this is what we call jumping to conclusions.

another big thing:

(f*g)'(2) is NOT the same thing as f'(2)*g'(2)

also g(x) is (3-x)^3 so you made a typo there.

[eager tinsel]

oops, guess I did

[wooden python]

more importantly though, you need to learn and apply the product rule for derivatives.

[eager tinsel]

I did... but then I learned

from another guy

but maybe I learned wrong

about

wait no

[wooden python]

what other guy

where and when

[eager tinsel]

the product rule is (f \times g)'(x) = f'(x) \times g(x) + g'(x) \times f(x)

[eager tinsel]

just now

[wooden python]

that is correct

[eager tinsel]

I don't remember the name

[eager tinsel]

yeah, so I applied that

like this:

[wooden python]

i recommend btw that you should write out in several lines, one thing per line:

[eager tinsel]

alright. I'll just send my resolution here, it's line b

[wooden python]

• f(2) = ...
• f'(2) = ...
• g(x) = ...
• g(2) = ...
• g'(x) = ...
• g'(2) = ...
• f(2)g'(2) = ...
• f'(2)g(2) = ...
• f(2)g'(2) + f'(2)g(2) = ...

rn what you did is find g'(x) in a long and painful way

also your laptop camera has poor resolution or the lighting is poor or both

[eager tinsel]

umm..

probably lighting

[eager tinsel]

how should I find it then?

should I just have kept

[wooden python]

chain rule imo

[eager tinsel]

I read something about that

I just read it, how does that help, there isn't a function inside a function

shouldn't the chain rule be for that?

[wooden python]

yes there is

3-x inside the cube

[eager tinsel]

hmm

so we create a new function with a cube

[wooden python]

you don't create anything new

[eager tinsel]

then we nest it inside of it?

oh

ok

then what?

[wooden python]

no, you understand that (3-x)^3 already IS the composition of g1(x) = x^3 and g2(x)=3-x in that order

[eager tinsel]

I'm not in university, I'm in 11th grade, maybe it's because of the language but I don't really understand, I know that it's annoying.

[wooden python]

i don't speak portuguese so we're gonna have trouble with that i think

[eager tinsel]

ok

so

hold it a minute, I'll see if chat gpt helps simplify what you just said

[wooden python]

!nogpt

[odd edge]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[eager tinsel]

oh

ok

plan B

ohhh

ok

so

g1(x) is an element of the function

and g2(x) is another element

you nest them, and you apply the chain rule, is that it

?

[wooden python]

mm

close enough

[eager tinsel]

ok

uh

ok

so I did it again

with the chain method

here it is

allr

what do you think

I see that, the result is different from the other painful method I used

OH

NO WAI>T

it's -27 + 18x + 3x^2

it's the exact same

forget it

anyways, now we do (f \times g)'(x)

of 2 actually

[eager tinsel]

btw, why is this not the same

just for my information

[wooden python]

ok so you would actually do better to keep g'(x) as -3(3-x)^2

easier this way unless you like pain

[eager tinsel]

umm

ok?

[wooden python]

well you know the derivative of x^2 is 2x, yes?

[eager tinsel]

yea

[wooden python]

but if you tried to split it up as x*x first

you would find yourself calculating 1*1

which is just 1, and nowhere near what it should be

[eager tinsel]

umm, yea, I guess that makes sense

anyways, maybe it's not the right moment to say this

but, at x = 2

f(x) has of image 2, just saying

[wooden python]

what

i don't know what you mean by this

f(x) has of image 2
this seems to be a translation issue

[eager tinsel]

so f'(2) = 2

you see that point A on the x axis

intersecting at 2

A(2, 3)

that's it's cordenates, just wanted to say it in case it changes things

[wooden python]

yes

[wooden python]

coordinates

3 is f(2), and it is one of the pieces you need

[eager tinsel]

yea I figured

ok I have questions (again). So, for line a, if you see on the exercise, I substituted f(x), by t(x), by accident, and it worked?

[wooden python]

it's not an accident

f and t TOUCH at that specific pt

[eager tinsel]

ohhhh

yeah

you're right

I completely forgot about that

I was a genious

alright.

ok

it gave 12

it's supposed to give -7

probably a calculation error

I'll send what I did

here

[wooden python]

this is hard to read

as in actually i cannot read anything at all, it's too faint

[eager tinsel]

ok

I'll take it with the phone cam

allr, can you see?

[wooden python]

i can see now yes

and i can see that you did it rather incorrectly

and didn't follow my instructions

these two formally are not dependent on one another but if you HAD followed my instructions there would be less chance of screw up

[eager tinsel]

so, I didn't follow your instructions at all and I did it incorrectly

alright

let me go back and see what you said

[wooden python]

here

[eager tinsel]

you're mentioning this?

[wooden python]

yes

[eager tinsel]

oh

ok

f(2) = 2

opps

no

forge

f'(2)= 2

f(2) = 2 * 2 - 1

g(2) = (3-2)^3 = 1

[wooden python]

paper

[eager tinsel]

ok

ok yea... now it's correct...

thanks @wooden python

.close

[odd edge]

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[twin vigil]

How to check the graph according to the given situation

Once the ball is given a velocity it will increase and when it touches The table horizontal surface its velocity will be slightly reduced

[willow zealot]

break the question up

first it rolls down a ramp

then across a horizontal tabletop

then it falls to the ground

in the first part, what do you think will happen to its acceleration

[twin vigil]

increase

no

[lone solar]

B

[wooden python]

!nosols

[odd edge]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[twin vigil]

say any magnitude of velocity is given then for table top it will be constant

[wooden python]

dont give out answers regardless if right or wrong @lone solar

[lone solar]

Oh.

Ok my bad.

[wooden python]

but you are looking for graph of accel not velocity.

[lone solar]

Can I explain?

[twin vigil]

Then acceleration must be 0

[wooden python]

accel is 0 while the ball rolls along the table yeah

[twin vigil]

From velocity looking in change in velocity is good idea

[wooden python]

in this case your answer is wrong btw

[lone solar]

Oh wait then what's the answer?

[wooden python]

broken wording,
but you can reason about acceleration directly actually

[twin vigil]

Answer is A

[wooden python]

like when the ball falls off obviously it is in freefall and its acceleration is g

when the ball rolls down the ramp its acceleration will be less

[twin vigil]

Then how the lines in the graph A shows this situation

[wooden python]

the trajectory of the ball consists of 3 parts

1. roll down ramp
2. roll along table
3. fall

within each part acceleration is constant and it changes abruptly in both transitions

agree or disagree?

[twin vigil]

Ahree

Agree*

Okay

.close

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[keen scroll]

ok so i was working through this problem

i got two verisons

i learnt that if u get a z shape like thius the angles are equal

but then i also get this

2*32+x=180?

[nimble blaze]

yes

[keen scroll]

oh

Wat

bruh

LOL

i was correct

yo

goat

ty

.close

[odd edge]

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[odd edge]

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[inland estuary]

sos @wanton bison

[wanton bison]

affen alarm

[inland estuary]

Zieh die Stäbe raus, doch pass gut auf!??!?!?!??!

[wanton bison]

haha

[inland estuary]

joaa xd uhmmm bin grad bei einer anderen prüfung am machen

[wanton bison]

schick

[inland estuary]

wollen wir erst von gerstern anschließen oder diese hier beenden?

[wanton bison]

wie du magst

[inland estuary]

weil das ist nur eine die eig einfach ist ich weiß nur net wie ich die Grade baue

erstmal die dann hab ich die erste Abiprüfung fertig

[inland estuary]

so bei e

c und d hab Ich hinbekommen glaub ich

aber bei e) weiß ich nicht wie ich die grade für die Untere stange baue

Also wir haben Sk aber wir haben nur S zu k=1 und nicht zu k=0.5 wie das hier gewollt ist

Die Obere stange haben wir ja müssen nur für k= 0,5 einmal machen

aber kein plan wie ich die untere Stange hinbekomme

[wanton bison]

Wenn du die untere Stange bestimmen möchtest, dann brauchst du für eine Gerade zwei Punkte prinzipiell

[inland estuary]

Ja?

[wanton bison]

Die untere hat ja G_k und S_k

[inland estuary]

Stützpunkt und Richtungsvektor also zwei dings da

[inland estuary]

yep

[wanton bison]

Ah okay das sind keine festen Punkte

[inland estuary]

nope

Wir haben S für k=1 vorgegeben aber wir brauchen S ja für k=0,5