help-0

[urban bane]

7/stuff - 1/t?

how would I make them common though? I'd have to multiply with t*sqrt(49 + t) which would make the other fraction
1/(t^2 * sqrt(49 + t)

and I can't do that

[rustic coral]

What?

just get a common denominator

which is $t\sqrt{49+t}$

[ocean seal]

Civil Service Pigeon

[rustic coral]

the first fraction already has that denominator

now rewrite the second fraction to have the same denominator

[urban bane]

oh wait I see the mistake I think

multiplying to get common denominator doesn't double up the terms, you are just multiply to "add" a term so to speak

so both bottoms are t * sqrt(49 + t)

ok so that makes sense, although I don't know how to take it from here, most of the terms ar eunder the square root and I can't cancel them out can I?

[rustic coral]

$\frac{7-\sqrt{t+49}}{t\sqrt{t+49}}$

[ocean seal]

Civil Service Pigeon

[rustic coral]

I could've sworn you said it yourself above

use the conjugate

(note that multiplying by the conjugate is basically glorified difference of squares with more clutter, but I digress)

[urban bane]

all I said was I know that I could use a conjugate, but specifics and anything more beyond that I can't remember and am clueless about

its just something that can be used when dealing with square roots

[rustic coral]

tbf there's rlly one thing here that you can (meaningfully) take the conjugate of

[urban bane]

ok so I wind up with this... and but I'm not done because you can still plug in T and end up with the indeterminent form so nothing is really solved ultimately

so what are some of the things I can even do to wind up with a epxression that won't lead to an indeterminent answer

[rustic coral]

.

yk the definition of a conjugate, right

[urban bane]

I don't, like I said the extent of my knowledge of a conjugate is that its "something that can help with removing square roots"

I can try googling it

[rustic coral]

The conjugate of $a+\sqrt{b}$ is defined to be $a-\sqrt{b}$

[ocean seal]

Civil Service Pigeon

[rustic coral]

Oh I already said it so

tbf there's rlly one thing here that you can (meaningfully) take the conjugate of

So circling back to this, what can we (meaningfully) take the conjugate of

[urban bane]

staritng from here yeah?

[rustic coral]

yeah

[urban bane]

I mean the top, its the most similar to a + sqrt(b)

7 - sqrt(49 + t)

although its a minus

[rustic coral]

(and vice versa)

[urban bane]

but what do I just flip the sign? do I do another multiplication of the same fraction with that and somehow that cancels out?

[rustic coral]

.

meaning that the conjugate of $a-\sqrt{b}$ is $a+\sqrt{b}$

[ocean seal]

Civil Service Pigeon

[urban bane]

(7 + sqrt(49 + t))/(t*sqrt(49+t))

[rustic coral]

What?

You can't just magically change a sign in the original expression

The point is that you want to multiply the thing (in this case, the numerator) by its conjugate

[urban bane]

so then expresison then becomes as follows?

[rustic coral]

This is not equal to 1

meaning that again, you're fundamentally changing the original expression

[urban bane]

ok this isn't working

how would you solve it?

[rustic coral]

.

Recall that if you divide a quantity by itself, then the result is 1

take that as you will

[urban bane]

I'm actually stumped, idk if I just have a bad block but I can't apply that, yes I understand you have a quantity/quantity, if its the same they equal 1. but I'm still here so how does that work in practice? 7/t disappears? the sqrts disapear?

[rustic coral]

My point was that if you multiply the numerator be $7+\sqrt{t+49}$

[ocean seal]

Civil Service Pigeon

[rustic coral]

then you have to multiply the denominator by $7+\sqrt{t+49}$ as well

[ocean seal]

Civil Service Pigeon

[urban bane]

oh wait so the expression is wrong, so the section on the right needs to be the same top and bottom?

[rustic coral]

yes

[urban bane]

yikes that took longer than it should have, bloody hell

so i.e. you take the original expression

and to do the conjugation mess, you take the numerator, flip the sign, write denominator to be the same giving you a fraction that by itself = 1 because of division

[rustic coral]

sure

[urban bane]

so now just distribute the conjugate to the expression now? or can I cancel out/remove anything in here?

[rustic coral]

the goal is to make the numerator "nicer", so I would multiply out the numerator

Note that there is a ||difference of squares||

I would hold off on multiplying out the denominator tho

You'll see why, and this is a recurring pattern that you will observe

[rustic coral]

so I would multiply out the numerator

[urban bane]

7^2 + (sqrt(49 + t))^2 it becomes?

[rustic coral]

how did you get a + sign in between?

[urban bane]

no clue just figured I was combing = add

[rustic coral]

You can always just multiply it out properly

Note that there is a difference of squares

I also said this

[urban bane]

I don't know what those are

[rustic coral]

$a^2-b^2=(a-b)(a+b)$

[ocean seal]

Civil Service Pigeon

[rustic coral]

that's the difference of squares factorisation

[urban bane]

I'm lost on how that would apply here

(7-sqrt(49 + t)) * (7 + sqrt(49 + t)
(A) * (B)
so it evolves to?
(A)^2 - (B)^2

[rustic coral]

$a=7$ and $b=\sqrt{49+t}$

[ocean seal]

Civil Service Pigeon

[rustic coral]

[urban bane]

wha? within a smaller scope?

[rustic coral]

??

what is "smaller scope" supposed to mean

[urban bane]

so you mean within the actual expression

so its
(7^2 - sqrt(49 + t)^2) * (7^2 + sqrt(49 + t)^2)

[rustic coral]

what

$(7-\sqrt{49+t})(7+\sqrt{49+t})=7^2-(\sqrt{49+t})^2$

[ocean seal]

Civil Service Pigeon

[urban bane]

into one term? but this is still two different terms that are also squared

oh wait

ok did it on paper, I definetly have a block but I see the connection of it

but this is just the numerator

so the numerator now is 7^2 - sqrt(49 - t)^2

[urban bane]

but the denominator is still as such here

[rustic coral]

typo

49+t

also have you tried simplifying the numerator at all afterward

[urban bane]

7 = 49
sqrt(49 + t)^2 = ?

OH WAIT

sqrt(x)^2 = x?

[rustic coral]

you should rlly write (sqrt x)^2

b/c \text{sqrt(x)^2} could mean $(\sqrt x)^2$ or $\sqrt{x^2}$

[ocean seal]

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

[rustic coral]

the second of which is equal to |x|

[urban bane]

i'm used to programming where there are math method apis for sqrt(variable here)

so my bad

so

49 - (49 + t)
0 + t
t

so the numerator now is just t

[rustic coral]

check your signs

[urban bane]

the - needs to get distributed?

after 7?

[rustic coral]

yes

[urban bane]

geuss it's better to leave the parenthisis there for myself and not drop it

but wait now it becomes -t?

7^2 - sqrt(49 + t)^2
49 - sqrt(49 + t)^2
49 - 49 - t
0 - t
-t

ok

sanity checking there

so now becomes

-t/( (t*sqrt(49 + t)) * (7 + sqrt(49 + t) )

[rustic coral]

you there?

[urban bane]

am

[rustic coral]

did you

continue

[urban bane]

oh no, I'm just right here

I assume no difference of squares here, none of the terms really match

and it'd just be a distribution of the left side

so like

t*sqrt(49+t)(7 + sqrt(49 + t))```

[rustic coral]

did you ever try looking for cancellation

[urban bane]

im assuming it cancels out to 1? and the other T's are untouched because they are sheltered with the sqrt bracket

[rustic coral]

this is fine

[urban bane]

oh wait

I think I see the path

orr no I geuss no

is sqrt(49 + t) treated as one whole term? i.e. if distributed on the same term but on the right it would become effectively sqrt(49 + t)^2? but because of the power the bracket disapears and it's just 49 + t?

but the 7 will have the sqrt(49 + t) stuck to it

[rustic coral]

remember that you should always be checking to see if direct substitution is viable

[urban bane]

what would I even be substituting?

t? with that?

[rustic coral]

mate

[urban bane]

so now that just leads to
-1/(7 sqrt(49) + 49)

no more zero or indeterminant

but is that the final answer? I don't see how it could be simplified further

oh wait

7*7 = 49

pfff

gimme a sec

yikes

oh wait

signnns

or no its there

well ouch... 2 hours later and I atleast can see now how to get to -1/98 after putting you through the ringer

yikes, well thank you for putting up with a dumbass like me

[rustic coral]

oh yeah

[rustic coral]

the reason why I said this

is because you can do this in 30 seconds if you have a little intuition

[
\frac{7-\sqrt{t+49}}{t\sqrt{t+49}} = \frac{7-\sqrt{t+49}}{((t+49)-49)\sqrt{t+49}}
]

[
= \frac{7-\sqrt{t+49}}{(\sqrt{t+49}+7)(\sqrt{t+49}-7)\sqrt{t+49}}
]

[
= \frac{-1}{(\sqrt{t+49}+7)\sqrt{t+49}}
]

Substituting ( t=0 ) yields

[
\frac{-1}{(\sqrt{49}+7)\sqrt{49}} = \boxed{-\frac{1}{98}}.
]

[ocean seal]

Civil Service Pigeon

[urban bane]

well I lack intuition as you saw, my fundamentals are whack

ok

I just went through all of the steps you described and got to the same place on my own just to see if I could get there, and it looks like I did

now I can see the flow in this context so it makes sense

had a really bad block though mentally while trying to understand it, couldn't see the flow

[lone heart]

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[timber isle]

i need help solving this. i forgot the steps to get the answers to what they’re asking for on the right. i remember in class that one answer led you to the next one. i know how to factor but knowing where to put everything is confusing. i’m lost on where to go from here. i’m in algebra 2 and this is higher degree polynomial equations

[paper mango]

start by factoring

[timber isle]

oki

[paper mango]

ah it's more about stepwise answers

k well for the first you need to consider how many roots a polynomial of degree n can have

possible roots "p/q" refers to the rational root theorem

[timber isle]

is it how big the biggest power is, like 3 in -x^3

[paper mango]

for the third you factor as usual

[paper mango]

yes if x³ is the highest power, then the polynomial has a degree of 3

and for the max possible number of roots

consider that a polynomial of degree 1 can have up to one root

a polynomial of degree 2, a parabola, can have up to two roots

etc

[patent vale]

ts is the fundamental theorem of algebruh say its name🗿🗿

[paper mango]

you can also deduce this by knowing that a polynomial of degree n can have at max n linear factors

and therefore n roots

[timber isle]

wait do i factor it by doing this, like splitting it in half then factoring

i’m sorry i feel so stupid right now 😭

[paper mango]

dw

you found that 1 is a rational root

so you know that one of the factors is (x-1)

from here you can apply polynomial division

[timber isle]

i have -x^2(x-8) as the factored form so far but i don’t think it’s right because i wouldn’t be able to factor -x^2 from -19x-12 on the right

[paper mango]

hm no need to do so

you have (x-1) as correct factor

so polynomial division is the way

[timber isle]

would it be “number i factored out (x-1)”on the left and right

[paper mango]

didn't get that

are you unsure about the poly div or where are you stuck

[timber isle]

like the number i factored out on the blanks then (x-1)

[paper mango]

nono you don't need to split

[timber isle]

these are the notes i took, and i’m unsure about poly division and the part where im stuck

[paper mango]

yeah here they apply the polynomial division synthetically

[timber isle]

ohhhhh ok

[paper mango]

same approach needs to be done for your case -x³+8x²-19x+12

with the factor x-1

should I show the steps?

@timber isle

[timber isle]

yes please

[paper mango]

k start with the coefficients of your polynomial

-x³+8x²-19x+12

has coefficients -1, 8, -19, 12

in order

and then put the root found in the next row at the start

you can also put it in the first row, doesn't matter

by that I mean you can also write it like this

[timber isle]

the 1 that im dividing by, is it negative or positive?

[paper mango]

sry meant root, not constant

since you found the root 1

you can put the 1 at the start

if you had found the root -1, you'd put -1 there

[timber isle]

oki

[paper mango]

Now the first resulting value you get by simply copying the first value:

then go diagonally

and multiply with the value at the start, which is 1

so (-1) * 1 = -1

then you add the second column together to get the second result

8 + (-1) = 7

then the steps repeat, go diagonally and multiply with 1

[timber isle]

did i do it right

[paper mango]

yeah

[timber isle]

YAYYYYYYYYY

[paper mango]

nice :]

and you read your result this way:

[timber isle]

wait wut do i do now

[paper mango]

since it's a division

you can have a remainder

if you already know that (x-1) is a root and you divide by it, then the remainder is always just 0

because (x-1) is a factor

but if you instead for instance divide by (x-14913), then you'll get a remainder

what the result means is that you divided -x³+8x²-19x+12 into (x-1) * (-x²+7x-12)

which is the crucial part

after that you need to figure out the roots of -x²+7x-12

e.g. with pq / quadratic formula

I'll leave some hints here as I have to go

||Applying the quadratic formula should yield that 3 and 4 are roots of -x²+7x-12, meaning -x²+7x-12 = -(x-3) * (x-4)||

[timber isle]

in my notes i divided by the number in the function that equals 0, was i supposed to plug in all of the possible roots then find what equals 0 then divide

[timber isle]

and awww 😦 you’re so nice

[paper mango]

sure first try all rational roots

in this case you'll find 3 and 4 work

so you have two more factors

[timber isle]

oki

[paper mango]

at which point you wouldn't even need to apply polynomial division

since you already know all roots

1, 3, 4

and thereby the factors

(x-1), (x-3), (x-4)

but since a polynomial of degree 3 can only have 3 factors

-x³+8x²-19x+12 = -(x-1)(x-3)(x-4)

you directly get the result

however this doesn't always apply

since some roots can be irrational

[timber isle]

f(3)=0, i plugged it in

[paper mango]

at which point you'd need approaches like the pq formula

to find the irrational roots

nywys gl with the rest! 🦩

if more guidance is needed just ask for help here again and another wanderer will come c:

[timber isle]

awww goodnight 🫶 and oki

thank you for your help

[paper mango]

np

[lone heart]

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[lone heart]

@timber isle Has your question been resolved?

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[prime nebula]

The question at a is this

[lone heart]

@prime nebula Has your question been resolved?

[prime nebula]

<@&286206848099549185>

<@&286206848099549185>

[astral flicker]

Try to explain what that problem is saying in your own words

1. What are the properties of convex quadrilaterals

[prime nebula]

All the lines that connects 2 points from the qudrilateral is in the qudrilateral too

[astral flicker]

Correct

Now what does {O} = AC intersection BD mean?

[prime nebula]

Well they are the diagonals

And O is the intersection

[astral flicker]

So O is the intersection pt

Right

[prime nebula]

So i tried to write that thing like a×OP + b×OA + c×OB=0

Where those are vectors OA OB OC and 0

And wrote a b c like there with segments

But idk what should I do after

I tried some Stewart theorem to write OP somehow but it doesn t really help me

@astral flicker

[astral flicker]

How would you parse OA.OB.CD.OP

Here OP is a vector

What are OA OB CD

[prime nebula]

What means parse?

[astral flicker]

I mean what sense do you make of it

[prime nebula]

OA OB CD are segments

And they re product is a scalar

[astral flicker]

Segment lengths ?

[prime nebula]

Yes

They re like a scalar for OP to me

[astral flicker]

So we have three vectors

Multiplied by some scalars

OP, OA, and OB

[prime nebula]

Yes

And now what should I do

@astral flicker

[prime nebula]

Bro @astral flicker

[lone heart]

@prime nebula Has your question been resolved?

[lone heart]

@prime nebula Has your question been resolved?

[astral flicker]

Sorry

Was in class

[lone heart]

@prime nebula Has your question been resolved?

[lone heart]

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[deft plover]

I was doing IBP and then tried to split -1/(x+x³) into two fractions but i think i messed it up cuz i got -(1/x + -x/(1+x²)) but i dont think that is right?

[vale wigeon]

,w simplify 1/x - x/(1+x^2)

[ocean seal]

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[vale wigeon]

why do you think it isn't right

[deft plover]

hmmm

cuz my integral diverges ;-;

and it converges apperently

[ivory fern]

The original integral converges but after splitting you obtain two integrals that diverge

[deft plover]

aaah

that makes sense

ye cuz the other one will be -1/2 ln(1+x²)

[ivory fern]

You could integrate from 1 to M instead, then find the limit when M goes to infinity and finally justify why that is the same number as the original integral

[deft plover]

Aaah i might be able to do that

[ivory fern]

If you are not familiar with measure theory and Lebesgue integration you can probably skip the last part though

[deft plover]

i might know something i can do

wait please ill show u hopefully it works

i saw a mistake wait please ;-;

i forgot the -1/2

[ivory fern]

Yes

[deft plover]

is the answer pi/4 + 1/2 ln(2)?

[ivory fern]

I don't know

[deft plover]

aaah ill show my work okay?

[ivory fern]

Yes

[deft plover]

i did skip some notation stuff cuz didnt want u to wait for too long

[ivory fern]

It is correct
Note that you could have done
$$\int_2^{a}\dfrac{du}{u} = [\log u]_2^a$$
and then you don't have to simplify the $\sqrt{1+a^2}$

[ocean seal]

d

[deft plover]

smarttt

that would have saved time

but wait how am I allowed to at the limits then tho?

oooh wait i get it

[ivory fern]

Also you can't just do $[\log u]1^\infty = \lim{a\to\infty} \log a$

[ocean seal]

d

[deft plover]

true but wanted to do it fast

[ivory fern]

Ok

[deft plover]

this is really helpful tho

[ivory fern]

So do you know how to justify it?

[deft plover]

Justify what exactly?

[ivory fern]

Why you can do the limit thing

Again, are you familiar with Lebesgue integration?

[deft plover]

No

but i do get why it is allowed tho

Wait let me try to write it down please

Hmm i think i did something wrong :/

wait my answer is wrong it should be pi/4 + 1/4 ln(4)

[ivory fern]

Ohh I finally see what you did wrong

[ivory fern]

So this is why you can't do this

You have to integrate from 1 to a from the beginning

So that when you do the u substitution you get the sqrt(1+a^2) in the integration limit

[ivory fern]

It's correct because log(4)/4 = log(2)/2

[deft plover]

aaah okay luckily

[deft plover]

could you maybe type / write it out for me

i dont really see what you mean

[ivory fern]

$$\int_1^{a}\dfrac{x}{1+x^2}dx=\int_2^{\sqrt{1+a^2}}\dfrac{1}{u}du$$

[ocean seal]

d

[deft plover]

aaaah

okay

smart

and then it works?

[ivory fern]

It should

[deft plover]

imma try

shouldnt it be 1+a²?

instead of the root?

[ivory fern]

So what you should do is
$$\int_1^{+\infty}\dfrac{1}{x+x^3}dx=\lim_{a\to+\infty}\int_1^{a}\dfrac{1}{x+x^3}dx$$
before anything else and then simplify, until you can find the limit by substituting

[ocean seal]

d

[ivory fern]

Yes it should

[deft plover]

alright perfect

yes it works

thank you very much!

imma close this now have a good night or day!

.close

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[ivory fern]

You too!

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[fickle wraith]

Not a single progress

[forest marsh]

Multiply by conjugate

[gritty dune]

try rationalization

[fickle wraith]

Sqrt x is gon be the same as x

Since x is 0

[gritty dune]

the numerator

[fickle wraith]

Is the conjugate the - after the sqrt x?

Sorry im not familiar with deciding conjugate

[forest marsh]

A + B conjugate quantity is A-B

So

[fickle wraith]

Thanks

[forest marsh]

Here

How can you use it

[fickle wraith]

Yeah i did that hold on

Yeah but like are you sure this is correct it seems like a very tedious step

[marsh hamlet]

You should get: $\lim_{x\to 0}\frac{10\sqrt{x}}{\sqrt{x}(\sqrt{4+5\sqrt{x}}+\sqrt{4-5\sqrt{x}})}$ and thus should subsitute 0 in again to get the desired result.

[ocean seal]

mathisfun

[marsh hamlet]

And make sure to cancel the sqrt(x)'s in the numerator and denominator where applicable.

[gray isle]

note that the use of conjugates is an application of
the factorisation of a difference of two squares
(a-b)(a+b) = a^2 - b^2
all that tedious expansion was unnecessary and overcomplicates things

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[onyx oyster]

∙

[rich escarp]

do you know how RSA encryption scheme works?

@onyx oyster

[onyx oyster]

i have no idea

[rich escarp]

why not look it up

[onyx oyster]

i might i probably forgot

[rich escarp]

then come back, if you still dont see where primes pop up

[onyx oyster]

my memory's are fry

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[silver agate]

Does anyone know how can i solve the second question here ,it basically says find the the expression of f(x) while G(x,y) verifies cauchy-riemann's conditions. When i tried to solve i tried using cauchy-riemann conditions i didn't get really an answer that seems appropriate as i found f(x) = 0

[lone heart]

@silver agate Has your question been resolved?

[tiny fable]

can I see your work? what Equations did you get from the cauchy riemann conditions?

[fast lichen]

i cant find nontrivial f either

[silver agate]

Okei give me one moment pls

[silver agate]

Here sorry for taking a bit long

Idk where to go on after reaching the system

[tiny fable]

okay well that is correct... from what i can tell yeah, it seems like the only solution is 0

can you find specific conditions for which you can find a correct f?

or does it have to work for all R?

0 is the only one that works for all R, but there are other solutions that work in only specific regions in R^2

[silver agate]

Welp i tried this solution in my exam and i think my professor doesn't agree that is correct since he gave me a 0 lol

[silver agate]

I tried diffrential equation in the first one to find f(x) but didn't find anything that makes sense

[silver agate]

Hmm

[fast lichen]

it does say f is from R to R, no?

[fast lichen]

you can consider y neq 0 in the 2nd equation

[wary fable]

whats the question?

[silver agate]

Yeah it is

[fast lichen]

and put f'(x)=-1/2f(x)

[forest marsh]

||Il y a pas mal de fautes dans les questions quand même||

[silver agate]

où

[forest marsh]

"Est ce F verifier"

[silver agate]

Ah

[forest marsh]

"G verifier les conditions"

[silver agate]

juste une erreur d'impression je pense

[forest marsh]

Oui, heureusement que ça ne change rien aux maths derrière

[silver agate]

Well i think I'll just move on to something else this took way too much time i really appreciate the help guys thx a lot.

.close

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[dapper mason]

Is this right?

[oak chasm]

,rccw

[dapper mason]

Mb

[oak chasm]

Looks good.

[dapper mason]

Aight 👍

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[alpine sable]

how do we find minimum of this function

idk what even this post used to do it

[minor trout]

choped sqrt

[alpine sable]

lmao how u know

[left socket]

minimum is always derivative in physics

[alpine sable]

its not necessaryt in this case

[left socket]

so try that maybe

idk

[minor trout]

thin line at top + diagonal at side + context

[alpine sable]

i see lmao

Cauchy-Shrawrtz inequality they used

[minor trout]

infinity no?

,calc (3-500)^2+(0-(500^2))^2

[ocean seal]

Result:

6.2500247009e+10

[minor trout]

yeah maximum is infinity

[alpine sable]

but i asked for minimum

[minor trout]

oh

oops

whatever makes the second term lowest as x increases exponentially

so 0 in this case

nvm

one squared is one so it'd be that

[alpine sable]

0?

[lone heart]

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[sleek fog]

I need help in 3)b)

[high rapids]

,rotate

[sleek fog]

In case you need help from other questions

[high rapids]

So they asked you to integrate

By parts

[sleek fog]

This what i get and idk what to do

[sleek fog]

Yeah which i did

Any idea to continue ?

Since i just came to a dead end

[marsh hamlet]

@sleek fog This is just an iterated formula, like using w_n+w_{n-2}=I_n

What is I_n in terms of w_n?

[sleek fog]

You will find it in here 3)

[marsh hamlet]

I'm asking you.

Wait a minute. I_n as you defined in the English portion is not the same as the french portion. Which one really is it?

[sleek fog]

In is just sequence

In = wn +w(n-2)

[marsh hamlet]

Look at how you defined it.

[sleek fog]

I am confused

Like wdym to defind it
yeah ig i just going to all night with it

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[alpine sable]

so im doing this problem let me show what i have done

and looking forwward to if i did any

mistake

if these two are right then i think i did everything correct

[violet jetty]

yup

[alpine sable]

can i ask for one more following this problem

just one is this looking fine

[violet jetty]

yes

[alpine sable]

Cool! i think i got the idea now

thank you !

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[crystal stone]

i am struggling witht he problem ∫1/(sqrt(x-x^(3/2))) my initial idea is to use the idenity that the intigral of 1/sqrt(a^2-u^2) = arcsin(u/a) where i have a as sqrt(x) and u as (x^(3/4)) then get a final answer of arcsin(x^1/4) but this is not the right answer from what i see

[tight pier]

yea because a is a constant

[crystal stone]

a has to be a constant?

[tight pier]

yes

[crystal stone]

thanks my teacher explained it could be anything so long as it was real

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[tight pier]

hmm let's see

okay

[tight pier]

yea a constant that is a real number

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[lone heart]

Please don't occupy multiple help channels.

[final crescent]

4 b

Why is y/(1/3)

Not y(1/3)

Because the question said compress vertically

Is for y right

Not for x

[wispy plaza]

y/(1/3) would mean y times 3

[final crescent]

Yeah

That’s the answer key said

[wispy plaza]

In a sense you can think of it as each unit of Y having three times more power

[final crescent]

But is compress right?

[wispy plaza]

Therefore you need 1/3 of the y to reach the same effect

[final crescent]

Ohhhhhh

[wispy plaza]

So multiplying compresses

And division decompresses

[final crescent]

I think I get because is compress by 1/3

[wispy plaza]

When n>1

[final crescent]

So is actually expanding

Right

Because is less than 1

If it said compress vertically by 3 is like 1/3

Like word trick

[wispy plaza]

Yes, it’s like each y unit has “more power”

[final crescent]

Can you explain question 5

5 c

Wait

I am still kinda confuse

For question 4c

Is like bigger than 1 right why does it still need to divide by 4/3?

[final crescent]

Please help

I don’t get it by do you need to divide , like y/4/3 for question 4 c

You can’t just do y(4/3 right?

U only divide when is X right?

[wispy plaza]

I’m a bit confused by your question

[final crescent]

Like for question 4c

The answer key said is y/(4/3)

[wispy plaza]

I think of it this way, we need 36 requirement to get our result. Each y and x produces a certain amount of requirement the is equal to the constant multiplier

[final crescent]

But why though I though u only do that to x, for y you just do y(4/3)

[wispy plaza]

The higher the multiplier to x or y, the less of x or y we need to reach the requirement, and it gets squished

And the converse for a smaller multiplier

[final crescent]

Emmmmmmm

I don’t get it sry

Why y/(3/4)

That’s the answer key

[final crescent]

But why though shouldn’t we just answer what the questions want ?

The question want to expand the y

So we times by 4/3 for y

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@final crescent Has your question been resolved?

[lone heart]

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[native cloud]

Hello

[native cloud]

Is this an existing notation?

$\int _0^{2\pi ^+}:f\left(x\right)dx$

[ocean seal]

Chaewon

[tiny fable]

I've never seen it

[summer dirge]

what does that notation mean?

[native cloud]

Um

Just a thought I had

[crimson dawn]

I mean what would it mean though

[native cloud]

$\lim _{x\to 2^+}\left(\int _0^x:f\left(x\right)dx\right)$

[ocean seal]

Chaewon

[native cloud]

What about this one?

[crimson dawn]

So this is just

[crimson dawn]

This shortened

[summer dirge]

your bound and integrand can't be functions of the same variable

[crimson dawn]

Oh yea

[summer dirge]

should be f(t) dt

[alpine sable]

heyyyy

[native cloud]

Oh yeah right

[alpine sable]

im new here

[native cloud]

#❓how-to-get-help

This is my channel

[alpine sable]

i am currently stuyding AI coding

[native cloud]

Leave

[alpine sable]

ohhhh

[native cloud]

.close

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[glacial jetty]

I have a combinatorics related question. u_0=1, u_1=2 and u_n = 6u_{n-1} - 8u_{n-2} + 3n
Need to figure out the simplified formula. We had an exercise last week where we had a similar exercise but this one feels a lot harder. w esolved that by trying with c2^n and going from there. here i dont know what to start trying with even

[glacial jetty]

oh we had an example like that not an exercise but point still gets across

<@&286206848099549185>

[lone heart]

@glacial jetty Has your question been resolved?

[glacial jetty]

here is maybe a bit better pic of the exercise

ofcouse translated etc but point

So i think i need to figure out some recursions so that: $u_n=a_n+b_n$. and go from there. but i dont know what i should try. For example in the example we had i believe $C2^n$ was used etc etc but now i cant figure it out

[ocean seal]

Olav

[glacial jetty]

<@&286206848099549185>

[lone heart]

@glacial jetty Has your question been resolved?

[dire gazelle]

i did it with matrices

using jordan normal form

but can you post the example they're talking about?

[glacial jetty]

translation according to chat GPT

@dire gazelle

[dire gazelle]

👀

interesting

[glacial jetty]

how so?

[dire gazelle]

what if you try a_n=An+B?

i don't know how to solve for b_n

[glacial jetty]

i am quite lost too here

[dire gazelle]

well i got a(n)=n+10/3

and n+10/3 does appear in the answer

[glacial jetty]

those are the first enough many u_n:s

[dire gazelle]

you can solve it like this

[glacial jetty]

amm i suck with matrixes. what is the final formula?

[dire gazelle]

[dire gazelle]

do you have any more explanation for this process?

how to solve for b_n

[glacial jetty]

sorry finnish

[dire gazelle]

oh

i see

[glacial jetty]

sorry finnish

[dire gazelle]

2 and 4 are roots of x^2-6x+8

[dire gazelle]

we aren't supposed to give away answers

[lone heart]

@glacial jetty Has your question been resolved?

[glacial jetty]

but you actually got a formula that worked and gave you the correct numbers?

a_n=An+b
b_n=6(b_{n-1})-8(b_{n-2}+3n?

[glacial jetty]

doesnt work 😦

been there before. term nro is on a b is the one calculated by that formula and c is with the recursive 😦

[dire gazelle]

B and C match

[glacial jetty]

no they dont

d is just c-b

[dire gazelle]

oh

probably a rounding error

[glacial jetty]

it works for like 28 first terms but breaks then

you think?

ok if thats the issue im pissed

cause i was here well 4 and a half hours ago or so

xD

[dire gazelle]

well you can check if the formula satisfies the recurrence

$n+\frac{10}{3}-\frac{7}{2}2^{n}+\frac{7}{6}4^{n}=6\left(n-1+\frac{10}{3}-\frac{7}{2}2^{\left(n-1\right)}+\frac{7}{6}4^{\left(n-1\right)}\right)-8\left(n-2+\frac{10}{3}-\frac{7}{2}2^{\left(n-2\right)}+\frac{7}{6}4^{\left(n-2\right)}\right)+3n$

[ocean seal]

Axe

[dire gazelle]

otherwise you can write a program using a data type that can handle more precision

[glacial jetty]

jeh ill go with it and see if i get the points even though im allready late xD

[glacial jetty]

@dire gazelle thank you.... im heading to bed now. im slightly annoyed since i got to that formula at like 3-4 am in the morning and decided that nah it cant be that..... welll lets hope its early enough that it counts! thanks again!

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[sharp hollow]

how do u rotate shapes

[north rover]

!xy

[lone heart]

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.

[sharp hollow]

oh aight

[north rover]

Imagine the top part is on the left

[sharp hollow]

okay

[sharp hollow]

my problem is that i cant find the center of rotation

[sharp hollow]

and i need help doing vectors :((

[north rover]

Assume the center of rotation is the origin if not specificed

[sharp hollow]

oh

[sharp hollow]

thank u very much

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[deft plover]

Find the convergence radius and interval

[deft plover]

so -1<R<-7 but for the interval, do I solve the sum (3^n (-3)^n)/sqrt(n) ?

and also for positive 3?

[wary fable]

what did you try?

[deft plover]

alternating series test wont work but I am wondering if I even have to do all this

cuz this is from an exercise exam and I feel like this would take way too long

[wary fable]

where the fuck did a x+4 term go to?

[deft plover]

that is just -3 or 3?

for the interval?

[wary fable]

where is it in the expression of an+1/an

[deft plover]

huh ;-;

for the limit or?

[wary fable]

(x+4)^(n+1) / (x+4)^n

is ?

[deft plover]

(x+4)

[wary fable]

where is it?

[deft plover]

we were told

doing lim |cn/cn+1 | for convergence radius is just the same as lim |a_n+1 / an|

[wary fable]

what is c_n?

[wary fable]

no

[deft plover]

huh

[wary fable]

do an+1 / an stop overcomplicating stuff

[deft plover]

cn is 3^n/sqrt(n)?

what do I do now?

<1?

or <|1|?

[onyx swallow]

you forgot an absolute value around 3(x+4) when pulling it out

[deft plover]

Aaaah smart!

So radius is (-1/3,1/3)
-4-1/3 < x < -4+1/3

[onyx swallow]

not quite. ratio test tell you -4-1/3 < x < -4+1/3 then it is convergent

how about when x>-4+1/3

and when x<-4-1/3

and when x=-4+1/3 and x=-4-1/3

[deft plover]

aaah so now

plug in -1/3 and 1/3 ?

[onyx swallow]

right

[deft plover]

then figure out if it will still be convergent right?

perfect imma do that rq

[onyx swallow]

this is divergent by ratio test again

[onyx swallow]

same for this

[deft plover]

so for x=1/3 it will be divergent

[onyx swallow]

shouldn't it be-4-1/3 and -4+1/3

[deft plover]

ye true

but i plugged it in for x+4

for x= -4-1/3 will be divergent because alternating series test and for -4+1/3 it will be divergent because of p-series?

How do I know for outside the this radius? so x<-4-1/3 and x>-4+1/3 cuz I dont really get what u mean with ratio testing it, you mean plugging it in and do the ratio test with that?

[onyx swallow]

can you tell me how you did the alternating series test?

[deft plover]

sure, bn>0
1/sqrt(x) , x>= 1
bn+1 < bn, f'(x) = -1/2 * 1/(xsqrt(x)) which will be <0 for x>=1
Limit x-> inf 1/sqrt(x) = 0

[onyx swallow]

its part of ratio test

when 3(x+4)>1 then the series diverges

[deft plover]

aaah

So that is why we must check it for the -4-1/3 and -4+1/3 becaues it gives L=1 and then the test fails

damn this was really helpful

[onyx swallow]

why f'(x)?

[deft plover]

that way I can proof that bn+1 < bn?

[onyx swallow]

oh ok, I think you can just use the fact that $\sqrt {n+1}>\sqrt n$

[ocean seal]

qwertytrewq

[onyx swallow]

so their reciprocal will flip > into a <

[deft plover]

;-;

that works way better ye

[onyx swallow]

and also this, by alternating series test, says its convergent?

[deft plover]

divergent

wait no

convergent

[onyx swallow]

yeah

[deft plover]

omg i must focus

[onyx swallow]

you got the test but forgot the conclusion

[deft plover]

true ;-;

it is just so muchhh is so little amount of time

alright anyways

thank you very much

this was really helpful

[onyx swallow]

np

[deft plover]

imma write the extra stuff down and close it then

have a good day/night

@onyx swallow sorry one last question

so how do I write my conclusion now

R = [-4-1/3,-4+1/3) ?

[onyx swallow]

im pretty sure yeah

[deft plover]

thank you

[onyx swallow]

np

[deft plover]

.close

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[covert brook]

does anyone have any tips on how to proceed? i know this has to do with hensel's lemma

[covert brook]

to apply hensel's we have to ensure that f'(a) = d * a^(d-1) =/= 0 mod p

[lone heart]

@covert brook Has your question been resolved?

[covert brook]

so looking at a^(d-1)
we know a is an element of Z/p^n Z
we know d | d-1 and therefore d is not divisble by p
just looking at a1, we can conclude that a^(d-1) is also not zero
so since f'(a1) =/= 0 mod p we can conclude its roots can be lifted ?

mm i think i get it thank u !!

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[ionic maple]

yo guys i'm trying to make blackjack in minecraft. So far so good.
but I want a blackjack version that only has card values 1-10(for simlpicity and my mental health's sake) and I was wondering if anyone knew the rules of a 1-10 decked version of blackjack
like what would be the target value?
without soft cards pls... it's hard to make it in minecraft

[ionic maple]

yeah ik it's not related to math but I don't have any friends so... I'm here i guess

It doesn't have to be perfectly balanced tho...
anything up to like 5% house win is accepted

(I don't have suits or anything like that. I just have values from 1-10)

[broken pivot]

anyone knew the rules of a 1-10 decked version of blackjack
cant you just google this 👀

[ionic maple]

there isnt

[broken pivot]

Oh are you asking for custom rules that will give extra 5% win rate to the house?

[ionic maple]

yeah

[broken pivot]

Idk then, but consider programming a simulation

[violet jetty]

Are you trying to prescribe exactly 5% or do you just want "a small house edge"?

[broken pivot]

I assume 55% house, 45% player

(or 52.5, 47.5)

[violet jetty]

I mean does it have to be exactly 5% or is the goal to just maintain the house's edge?

[ionic maple]

a small house edge

bigger than normal

[broken pivot]

Didnt read the question, mb

[broken pivot]

So do you just want the smallest advantage thats still an advantage?

[violet jetty]

Then stnadard rules should be fine. Aces and face cards both benefit the player, removing them should move the house edge from ~0.5% to some number that's slightly higher.

[ionic maple]

nah

around 5% is fine

i wanna build a casino in mc

[broken pivot]

Btw in your mc blackjack, are you making Ace be 1/10

Or is Ace just 1

[ionic maple]

just 1

i said no soft cards

those are hard to code

[violet jetty]

Im guessing no doubling or splitting?

[ionic maple]

maybe doubling

splitting no

[broken pivot]

Doubling is just doubling the bet, and then hitting right

[ionic maple]

yeh

[violet jetty]

yeah, and no more hits after

[ionic maple]

and only one double

yeah

[broken pivot]

_ _

[ionic maple]

yeah that could work... if i wasn't so dumb

[broken pivot]

So double = double the bet, hit, stand

[violet jetty]

You probably just wanna monte carlo it yeah

But like back of the napkin i think standard rules where aces are 1 and no face cards probably makes the house edge around 10%?

[broken pivot]

#old-network message perhaps this server can help you

[ionic maple]

alr thx

[violet jetty]

This actually sounds interesting, I might script it sometime

[ionic maple]

please do.

thx

[ionic maple]

alr

that's.... fine?

i want to make a casino in minecraft with blackjack and roulette

and 777

[violet jetty]

ill dm you if/when I do

[lone heart]

@ionic maple Has your question been resolved?

[lone heart]

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[coarse elm]

Ok so I already did this question, but I forgot what I did because im confused by my own steps lol

why is f(-2) = 1???

[tacit arch]

this part of the function

must equal this part of the function

for f to be continuous at x=-2

[coarse elm]

ya i got that

but how is f(-2) = 1

i think im just slow

like what

isnt it m(-2) + b = u dont konw

oh

ya i still dont get it

same with 4

[tacit arch]

what's this when x=-2

[coarse elm]

why did i plug it in for the last function

[coarse elm]

ya but that function is x < -2

ohh

[tacit arch]

for $f$ to be continuous at $x=-2$, $\lim_{x \ra -2^-} f(x)=lim_{x \ra -2^+} f(x)$

[ocean seal]

riemann

[coarse elm]

ohh ya

OHH

ya nevermind i did just forget

ya i remember it now LOL

thank you

.close

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[strange schooner]

Well, first things first

[marsh hamlet]

For which question?

[tribal lion]

She's right ✅️

[strange schooner]

The rounding criterion is arbitrary, so I assume your class has agreed to round up from 0.5 and up from there?

[marsh hamlet]

The correct answer is yes, since 7.350 would indeed round up to 7.4.

[strange schooner]

Fairly sure it's false

[marsh hamlet]

I mean, arbitrarily, if you chose 7.3499, what would you get?

[strange schooner]

That statement is equivalent to: 7.349 is the number that immediately precedes 7.35

Which is quite obviously false, just choose 7.34900000001

This number is greater than the one stated by the exercise and clearly would round down

[tribal lion]

She is incorrect

✅️

[strange schooner]

Let's think this through

Let's assume that what she says is right

Then, clearly, any number greater than 7.349 would round up, at least, to 7.4, right?

[tribal lion]

Gang, you could argue both points but she's wrong

[strange schooner]

Yeah, the answer is that the statement is false, I'm trying to argue that

Do you see how both statements are equivalent?

[tribal lion]

If in the question it said there was a limit of 2 decimal places then she right but there isn't

Or was it 3

Same thing

[strange schooner]

That's the rounding

The number of digits on N is not bounded

[tribal lion]

To be fair it does say significant figures

You could hardly call .0000001 significant

[strange schooner]

For the rounding, no relation to the choice of N

No

[strange schooner]

You're confusing rounding with N

You see what she's saying is:“ any number greater than 7.349 will surely round to something equal or greater than 7.4“

[tribal lion]

7.349 gang over here

[strange schooner]

I am not

Alright, let's say that ε is a real number we choose, be it great or small

[tribal lion]

The significant figures is still lurking man

She's right

She is so right

[strange schooner]

What the girl's saying is that 7.349+ε rounded is greater or equal than 7.35

[tribal lion]

Because you have to round to 2 significant figures

[strange schooner]

You're getting the notion wrong

You're comparing N before the rounding

The thing about the significant figures is, to put it bluntly, quite irrelevant for this

[strange schooner]

The only thing that's doing is guarantee that you can only round to numbers of the form a.x

[tribal lion]

But she right and wrong

[strange schooner]

The statement is false

[tribal lion]

It is not true that a value higher than 3.349 would round up but it is true that 3.35 would round up to 7.4

[strange schooner]

You can find a counterexample to the claim, which means the statement is wrong

[tribal lion]

So shhhhht

[strange schooner]

Yeah, well, the statement is saying two things:“N cannot be greater than 7.349“ and “7.349 rounds to 7.35“, if either one is false, the resulting statement is false

[tribal lion]

But there is still truth in it

[strange schooner]

This follows from elementary propositional logic

[strange schooner]

That statement is of the form “A and B are true“, trivially, of either one are false, the whole statement will be false

[tribal lion]

She's obviously wrong and that's a fact but technically speaking half of her theory is correct

[strange schooner]

Doesn't matter

[tribal lion]

https://tenor.com/view/blackbeard-blackbeard-writing-taking-notes-writing-gif-17583038544116987898

[strange schooner]

If I said:“pi is an irrational number and all snakes can fly“

Would I be right?

Said all

[tribal lion]

All snakes you dumb ahh 🤣

My bad, that might be offensive, sorry

[strange schooner]

Would you rather I write it out in terms of logic?

[tribal lion]

Snakes are a type of reptile that can be found all over the world. They are known for their long, narrow and limbless bodies and overlapping scales. Snakes can vary in their size and weight, depending on the species. Some are huge, hulking predators measuring many meters in length.

[strange schooner]

It was just an example, regardless, my point is that finding a single snake that can't fly makes my statement false

[tribal lion]

Also , yes there is a specie of snake that can fly

[strange schooner]

Doesn't matter, if you can find a single counterexample, the statement crumbles

[tribal lion]

They glide really, can't everything glide?

So all snaked can fly

Hold up