help-33

[sullen ice]

in y

and x is a little bit less than -1.5

[pliant island]

Yah

Maybe 1.6 or 1.7?

[sullen ice]

let me find the actual answer with calc rq to see if its close

[pliant island]

Okie okie

[sullen ice]

it seems you expanded it out wrong btw

[pliant island]

😦

[sullen ice]

should be 4x

[sterile yarrow]

wait so what is the up and down cause we know the min and max

this got me confuzzled

[pliant island]

Up and down is concave, my teacher wants me to figure out what it is

I did this side perfectly fine but I didn’t find out the concave for #2

My teacher just straight up gave the answer

[sterile yarrow]

LMAO

hmm

[sullen ice]

the answer is 1.6667

[pliant island]

How’d you find out??

[sullen ice]

calc

[sterile yarrow]

calculator carry

[pliant island]

Gib formula

[sterile yarrow]

🥶

[sterile yarrow]

^^

[sullen ice]

let me send you a vid on derivatives to make it easy for u in the future lol

[pliant island]

Okie okie

[sullen ice]

dont watch it now u should focus on finishing

just write the point as 1.6

https://www.youtube.com/watch?v=aL15O6rS9z0&list=PLybg94GvOJ9ELZEe9s2NXTKr41Yedbw7M&index=6 just follow this video blindly, dont worry about why for now since all you're using it for is checking, if you're genuinely curious though watch the entire playlist

watch after ur done tho

[sullen ice]

watch until 5:05

https://www.youtube.com/watch?v=pvLj1s7SOtk&list=PLybg94GvOJ9ELZEe9s2NXTKr41Yedbw7M&index=13 then watch this

https://www.youtube.com/watch?v=15awMHeP1Yc then this for practice

can you do the same for B now

[pliant island]

B is just 1 for this one!

[sullen ice]

mhm!

other one is 1.667 but i doubt your teacher thinks you'll figure that out so just 1.6

we got there from just estimation

if you want to know how to get the exact answer watch all of those videos

[pliant island]

Can you explain 4x?

[sullen ice]

you forgot to put an x on the 4

when expandiong

[pliant island]

Cause I used ai to check and they said y is suppose to be 4 but I couldn’t understand cause the points on x are -4, -1, and zero

[sterile yarrow]

what kinda ai r u using

put me on

please

🙏

[pliant island]

Photomath 😛 and this other website hold on

[sullen ice]

uh its wrong it should be x^3+5x^2+4x

i mean ur literally just multiplying everything by x

inside the bracket

[pliant island]

😭😭

[sullen ice]

$=x(x+4)(x+1)$
$=x(x^2+x+4x+4)$
$=x^3+x^2+4x^2+4x$
$=x^3+5x^2+4x$

[elfin berry]

Joshii

[sullen ice]

yeah see

you forgot here too btw

[sterile yarrow]

joshii can u pretty please with a cherry on top help me with my geometry

[marsh citrus]

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[shy horizon]

im having some trouble with this quadratic problem, ive tried some fraction combinations that have a product of -25/12 but none match the sum. it feels like theres something im missing since this seems difficult to trial and error

[stoic saddle]

i mean you could just write down the quadratic in question from it could you not

[shy horizon]

i did

[stoic saddle]

if you know the quadratic formula or completing the square

[shy horizon]

i dont

[stoic saddle]

you can use those

rip

ok then alright

let the roots be a and b

you know a+b and you know ab

you also know the value of (a+b)^2 of course

and you might also know that (a ± b)^2 = a^2 ± 2ab + b^2

(do you?)

[shy horizon]

i do

[stoic saddle]

right

this means we can find the value of (a-b)^2

since ab is known and we can subtract 4 times it from (a+b)^2

so if we also specify that a is the higher root and b is the lower root, we can find the value of a-b from (a-b)^2 by taking the square root

[shy horizon]

one moment let me write this down

so (a + b)^2 is (-55/72)^2

[stoic saddle]

indeed

[shy horizon]

or (55/72)^2 sorry

[shy horizon]

im not following how this lets us figure out (a-b)^2

[stoic saddle]

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

=> (a-b)^2 = (a+b)^2 - 4ab

[shy horizon]

ahhh

so it would be (55/72)^2 - 25/3

or + 25/3

since youre subtracting a negative

[stoic saddle]

yup... might need some effort to clean it up.

72^2 admittedly isn't a very nice denominator by virtue of being so big.

[shy horizon]

mhm

[stoic saddle]

but it is what it is, i think. not much else we can do about the fractions we're given

[shy horizon]

i dont think it would make you do that though

[stoic saddle]

i mean this is the only alternative i can think of to guess and check lol

[shy horizon]

so what do we do with (a - b)^2

[stoic saddle]

we square-root it to get a-b

and once we know a+b and a-b, it is easy to get a and b themselves.

[shy horizon]

alright

what i got is a = 15/8, b = -10/9

[stoic saddle]

let's check

,w 15/8 - 10/9

[elfin berry]

Results provided by WolframAlpha

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Requested by ann_dec

[stoic saddle]

,w 15/8 * 10/9

[elfin berry]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by ann_dec

[stoic saddle]

checks out

[shy horizon]

nice!

thx for the help

.close

[marsh citrus]

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[tender mantle]

Hints

[gilded cape]

do you know how to compute dy/dx

[tender mantle]

y/x = tant

dy/dx = tant

[grand halo]

i thin you have to find dy/dt

and dx/dt

[tender mantle]

I see

Let me do it and tell you

[brave spire]

Find dy/dx (by using dy/dt and dx/dt) and ur done with the numerator

[marsh citrus]

@tender mantle Has your question been resolved?

[tender mantle]

It is getting messy

[stark trail]

@tender mantle

https://tenor.com/view/war-is-coming-cat-of-war-cat-cats-gif-10315471693685787999

[marsh citrus]

@tender mantle Has your question been resolved?

[tender mantle]

<@&286206848099549185>

[fathom ridge]

Waaaaaaait a minute

Isn't that just the radius of curvature

Why don't you just find $\frac{v^2}{a_{\perp}}$

[elfin berry]

jan Nejon

[fathom ridge]

where v and a are the velocities and perpendicular accelerations at that point

[tender mantle]

What is v and a in the equation

[fathom ridge]

I said so already

[tender mantle]

How to find V and A but?@fathom ridge

I meant by x and y equations

[rigid sequoia]

Try x²+y²=a²cos2t

And find dt/dx from the given x(t), you'll need it

[tender mantle]

??

[tender mantle]

I understand this

[rigid sequoia]

I think I made a mistake in solving

But x²+y²= a²cos2t should help

[tender mantle]

I guess i am not able to understand this question

What should I do now?

[tender mantle]

From this

Differentiation

[rigid sequoia]

Yes differentiate it

[tender mantle]

,w dx/dt of x=a cost √(cos2t)

[elfin berry]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by arjunn5589

[tender mantle]

I did it here too

[rigid sequoia]

Yea the numberator will become sin3t I accidentally did Sint

[tender mantle]

a will be cancelled out

I'm going to close it as no one is looking at the work i have done. No more wasting my time

<@&286206848099549185>

.close

[marsh citrus]

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[still temple]

Hey I’m trying to find the value of cos 2 degree with a triangle

[still temple]

So what’s the length’s of opposite and adjacent side’s?

[umbral rover]

mind showing the question?

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

How can a cancel out, when's Theres a in all options 😭

I got -cos(2t+t)/sin(2t+t)

[marsh citrus]

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[still temple]

There’s no any question

[tender mantle]

take common

[still temple]

I just thinked it about it when I woke up

[marsh citrus]

@still temple Has your question been resolved?

[marsh citrus]

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[opal verge]

x^x = 256
Solve for x
The answer can be found easily with some guesswork (x=4), but is there any way to solve this mathematically?

[opal verge]

My math teacher says no but im skeptical

[fathom ridge]

sure but it needs special functions

[quaint elm]

what does "mathematically" mean?

[fathom ridge]

Analytically I presume

[opal verge]

without guessing

[stoic saddle]

honestly your math teacher is right

[opal verge]

what kind of special functions?

[quaint elm]

probably the lambert W function

which is the inverse of $xe^x$

[elfin berry]

jan Hayley

[fathom ridge]

I don't understand it well enough to explain it to you though

All I know is how to use it

[quaint elm]

there are techniques to solve that numerically though

and get a "solution" like 4.0000112315923

[tawdry thunder]

thats actually an interesting question

[tawdry thunder]

and youll go really far being skeptical like that

[fathom ridge]

well y = x^x increases and y = 256 is a straight line which means 4 the only solution

[quaint elm]

increases on x > 1 at least

[fathom ridge]

yeah

Mb

[quaint elm]

and below that i think it caps at e or something silly

[fathom ridge]

1/e right

[quaint elm]

sure

[fathom ridge]

The derivative would be x^x(1 + ln x)

[amber escarp]

hey i wanted to ask that how do you know which eqation of motion to use when solving physics problems. also i am in 9th

[marsh citrus]

@opal verge Has your question been resolved?

[opal verge]

based on my understanding, the lambert w equation uses the form
xe^x = y
how would I manipulate x^x = y into something similar to xe^x = y?

[quaint elm]

good question]

[opal verge]

i have also found a stackoverflow post but i cant wrap my head around how the answerer got this result

https://math.stackexchange.com/questions/54210/is-xx-y-solvable-for-x

[marsh citrus]

@opal verge Has your question been resolved?

[opal verge]

.close

[marsh citrus]

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[marsh citrus]

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[tropic forge]

#1 is the general formula for a mod b, how do you manipulate #2 to #3 based on #1? If you need more info, I can post the whole question.

[marsh citrus]

@tropic forge Has your question been resolved?

[marsh citrus]

@tropic forge Has your question been resolved?

[marsh citrus]

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[nocturne burrow]

would it be correct to say: each noncommutative group has a nontrivial commutative subgroup?

[nocturne burrow]

due to sylow saying that there is a subgroup H of G with order H=p^k if order G is mp^k

[marsh citrus]

@nocturne burrow Has your question been resolved?

[nocturne burrow]

<@&286206848099549185>

[marsh citrus]

@nocturne burrow Has your question been resolved?

[devout mauve]

you dont need sylow. you can just take any element and consider the subgroup generated by that element

also p-groups dont have to be abelian

[marsh citrus]

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[wooden grotto]

Could someone help me with this problem?

[wooden grotto]

For part a, to prove that 1 can be written as a linear combination of cd and k, I need to use the fact that gcd(c,k)=1 and gcd(d,k)=1, right?

[marsh citrus]

@wooden grotto Has your question been resolved?

[wooden grotto]

<@&286206848099549185>

[marsh citrus]

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[regal owl]

uhm

[regal owl]

is anyone there?

Put a reaction if anyone is here?

so i can put my problem

and not waste time

[main idol]

Not how this works

[main idol]

Read this

[regal owl]

okay

since your here

can you help me with this

[regal owl]

This is an explanation

this is the problem

<@&286206848099549185>

how would I do this problem?

[jaunty creek]

I’m not familiar with this math. Sorry lol

I can only really help with trig and pre calc 🙃

But I’m assuming the answer is Cx based on the example above

[regal owl]

oh.

[jaunty creek]

Y =Cx

[regal owl]

btw me too

idk what this math is

There's like ton of these

like these

[regal owl]

let me see.

[regal owl]

Yeah it is correct.

Thanks

it's kinda not difficult but also difficult

as the same time

thanks

.close

[marsh citrus]

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[dull bay]

I have a uniquely annoying problem.

Given f(x) = x^n + x, where n is any number greater than 1, what is the inverse function of f(x) for any arbitrary value of n?

[young owl]

it doesn't necessarily exist

x²+x is not one to one for example

[dull bay]

The inverse of that exists

[young owl]

if you say so

[dull bay]

(1 +/- sqrt(1+4x))/2

If we say the above equation is g(x), we prove g(f(x))=x (with extraneous solutions), and f(g(x))=x, therefore they are inverses

@young owl the quadratic formula is your friend

[young owl]

yeah ok bye

[marsh citrus]

@dull bay Has your question been resolved?

[marsh citrus]

@dull bay Has your question been resolved?

[marsh citrus]

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[chrome mirage]

!show

[marsh citrus]

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

[steep nexus]

I didnt do much

the answer is suppose to be 256 but I have no clue how to even get there

nvm I was looking at the wrong answers the whole time

.close

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[hazy lion]

what'd you try?

[civic skiff]

i think it is 5

[whole sleet]

What's f⁻¹(1)?

[hazy lion]

hmm by what reasoning

[civic skiff]

5

[whole sleet]

Then what's f⁻¹(5)?

[civic skiff]

2

[whole sleet]

Perfect. So,
f⁻¹(f⁻¹(1))
= f⁻¹(5)
= 2

[civic skiff]

so that is the answer correct

[whole sleet]

Well, does the logic make sense? You never know, I could be wrong

[civic skiff]

yes it does

i was just over thinkimg

[whole sleet]

All good haha

[civic skiff]

per usall hahahah

thank you for your time and help

.close

[marsh citrus]

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[tender mantle]

There are m arithmetic series and every series first element is 1 with difference 1,2,3...

[tender mantle]

What will be the sum of nth terms?

[marsh citrus]

@tender mantle Has your question been resolved?

[tacit vale]

(n+1)n/2

do you know how to get this?

are you expected to derive it

[marsh citrus]

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[tender mantle]

.reopen

[marsh citrus]

✅

[marsh citrus]

@tender mantle Has your question been resolved?

[tender mantle]

bro it is not sum of arithematics series formula

we do not have one series there are many

[tacit vale]

ah i see

[tacit vale]

you want the sum of the nth term of each series?

[tender mantle]

yes

[tacit vale]

@tender mantle does this make sense?

[tender mantle]

i did not understand the third term

why added and minus?

[tacit vale]

(n - 1) ?

its from the first line

nth term formula

[tender mantle]

can you write in words what you did in third term?

[tacit vale]

which third term,

?

[tender mantle]

thanks

let me read it carefully

[tacit vale]

no problem

[tender mantle]

wow

you did it so simply

.close

[marsh citrus]

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[tender mantle]

.reopen

[marsh citrus]

✅

[tender mantle]

But answer is given C

[tender mantle]

.close

[marsh citrus]

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[worthy flume]

Yo

[worthy flume]

Doing function formulas/evalutions

How would i go about solving this

[karmic crest]

hey

whats ur question?

[worthy flume]

One sec

https://i.imgur.com/LmuyAkG.png

@karmic crest

[compact crescent]

emm

[tropic forge]

so g is a function right?

[worthy flume]

yeah

[tropic forge]

looks like g(x) = cube root of x

[worthy flume]

correct

[tropic forge]

so what's the issue?

[restive sun]

are you trying to evaluate g at 64/27

and are asking for help about simplifying the right side

[worthy flume]

How to evaluate it

ik how to do it with whole numbers just not fractions

[restive sun]

you can apply the root to the numerator and denominator separately

$\sqrt[3]{\frac{64}{27}}=\frac{\sqrt[3]{64}}{\sqrt[3]{27}}$

[elfin berry]

Desync

[worthy flume]

ooh gotcha

[tropic forge]

dam how do u type latex that fast

[worthy flume]

Now my teacher doesnt alllow calculators how would i do cube root on paper?

[restive sun]

it's half of my job

[restive sun]

64 and 27 are cube numbers

so if you memorise a few, you can spot them

but apart from that, just cube a few small numbers

[tropic forge]

It's mostly about intuition, cubes: 1, 8, 27, 64, 125

[worthy flume]

Oh i get it now ty

[tropic forge]

or you can take the prime factorization of the number

[worthy flume]

lemme test this

So the answer is this:

https://i.imgur.com/ZpOxuAj.png

@restive sun that look right?

[restive sun]

yes

[worthy flume]

Cool tyvm both of yall

.close

[marsh citrus]

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[vernal shard]

is this true

[vernal shard]

i can't find a good counter example but it doesn't make sense to be true

[fathom ridge]

On the left you have the derivative at a

just pick something like f(x) = sin x and a = 0

[vernal shard]

oh i c

i tried like x^2 and it ended up being true

but ye makes sense

[fathom ridge]

yeah that's because the derivative of x^2 is 2x

So both would give 0

You can try just x though that should work

Or even some constant function

[vernal shard]

ohhh ye true

[marsh citrus]

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[lost stratus]

how would I get the reduced row echelon form of this is an easy way?

I cant find a way to do it

I see no pattern

[restive sun]

subtract (multiples of) row 1 from 2 and 3

you can eliminate most of row 2 immediately by doing this, and get rid of the real component of most of the row 3 entries as well

[lost stratus]

but then you are left with -3 in row 2 for example

[restive sun]

so divide it by -3

and that's a pivot you can use later

[lost stratus]

aight

[restive sun]

you can't do it all in one step

[lost stratus]

not sure what to do with row 3 and 4 now

just turn them into 0? Idk

[restive sun]

what do you have right now

[lost stratus]

I have:

1 i 1 0
0 0 1 0
2i -2 -2-i 0
i -1 -1 0

[restive sun]

subtract 2*row 4 from row 3

then the next steps looks like it should be multiplying row 4 by i, then subtracting row 1 and 3 from it

or at least something similar to that

[lost stratus]

alright

so right now I have:

1 i 0 0
0 0 1 0
0 0 -i 0
-1 -i -i 0

[restive sun]

add row 1 and subtract row 3 from row 4

that eliminates row 4 to all zeros

divide row 3 by -i then subtract row 2 from row 3

that eliminates row 3 to all zeros

[lost stratus]

Alright, thanks!

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[frosty tusk]

help me ,how to find eigen vector when all rows and columns are propotional..

[odd crest]

what do you mean?

[marsh citrus]

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[marsh citrus]

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[crisp kraken]

I'm watching a video on MIT, they propose a problem and go with an explanation on how to solve it.
It seems to me they "guess" the solution out of nowhere. How on earth would I be able to solve this?

[stoic saddle]

two ways

[crisp kraken]

I'd like to know where to start to think about this

im listening

[stoic saddle]

1. recognize the whole thing as a riemann sum and calculate the integral it corresponds to (which is not uniquely deteremined but there are ways to reduce your headache for it)
2. just work out the sum in the honest way, expanding stuff out and eventually using the formula for the sum of the first n squares

and also remembering that n is a constant and so 1/n multipliers etc. can be factored out.

[crisp kraken]

they go with n1, so i don't know yet how you managed to recognize it
2 is i think what i've tried to do. gosh I wish i didnt delete m attempt. give me a sec iìll try to show where i got to

[stoic saddle]

"they go with n1"?

[crisp kraken]

yes

[stoic saddle]

what is n1

[crisp kraken]

your first solution

[stoic saddle]

oh

ah.

[vernal forge]

number 1

[crisp kraken]

ok so in my first attempt i've not factored out the 2/n and i got to something like this
a is the number changing
a/n^3 - 2/n
or
(a-2n^2)/n^3

so i have a changing but i cannot find a rule for it. the rest is constant

then i've tried to take the limit of n but i was confused about what was going on.

if i expand to the last term , factoring out 2/n
i get
(2(n-1)/n)^2 -1

if i take the limit for n to infinity i get 2 -1

the meaning of this to me is: the larger the n the better approximates some function.

but i might be wrong, because i think about taylor series.

but also i'm really confused in taking the limit of just this part and not the 2/n factored out

[marsh citrus]

@crisp kraken Has your question been resolved?

[crisp kraken]

i didnt know about the formula for the sum of the first n squares. never heard of it

[crisp kraken]

ok so i got here:
2/n*( 4/n Sum(i^2) + Sum(-1) )

so i use the formula for the sum of the first n squares for the first sum using i=n-1

the last sum is = -(n-1)

so i get this:

2/n*(4/n*( (n^2-n)(2n-1)/6 ) -(n-1)

would this makes sense?

🤨

is this my function? and should i take the limit now?

[marsh citrus]

@crisp kraken Has your question been resolved?

[crisp kraken]

not solved but i have no more time today, i'll ask another day thanks

[marsh citrus]

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[hazy lion]

Show that $\frac{1}{\Gamma (x)} \int _0 ^\infty \frac{t^{x-1}}{e^t-1} \dd t \sim (x-1)^{-1}$ as $x\to \infty$

[elfin berry]

jan Niku

[hazy lion]

I think this problem is easier than it looks and that I'm making it

prof gave a hint to split the range of integration into 0 to 1 and 1 to infty

i don't really see whats supposed to happen

This fact is stated on the wiki page, without proof

[blazing pulsar]

I don't really want to try this problem tbh but I imagine there is a trick involving the beta distribution due to that hint

[hazy lion]

its far from that involved

the bits and pieces i remember from the hint were that you should use the taylor expansion of e^t

something something happens

[blazing pulsar]

I don't think that's a very involved bar to just use some established fact to control the convergence on 0 to 1, then you can use whatever methods like a taylor expansion on the rest

[hazy lion]

you end up with $t^{x-1} t^{-1} + O(t^{-2})$

[elfin berry]

jan Niku

[hazy lion]

or something?

[hazy lion]

i mean one of his problems wouldnt rely on this

we have watsons lemma

but it doesnt seem helpful

we cant just apply the maclaurin series for e^t on the entire range of integration can we

like you have something something something

the entire contribution comes from $\int_0^1 \frac{t^{x-2}}{e^t-1}\dd t$

[elfin berry]

jan Niku

[hazy lion]

taylor expand $\int_0^1 \frac{ t^{x-2}}{t+\frac12t^2+\dots} \dd t$

[elfin berry]

jan Niku

[hazy lion]

argue that t is the dominant contribution here in the denominator?

so $\int _0 ^1 t^{x-2} \dd t \sim \frac{1}{x-1}$

how far off am i

[main idol]

looks like the right direction but the last step is dubious?

[hazy lion]

yea

[elfin berry]

jan Niku

[hazy lion]

should be this

also i have no idea how to go about making this even halfway rigorous

[main idol]

i think you should factor the numerator out and factor t from the denom and use

[hazy lion]

what exactly happens with the other integral

[main idol]

(1 + t/2 + o(t^2) )^(-1) = 1 - t/2 + ...

[hazy lion]

factor the numerator out? how do you mean?

[main idol]

$\frac{t^{x-2}}{t + ... } = \frac{t^{x-2}}{t} \cdot \frac{1}{1+ (...)/t}$

[elfin berry]

riemann

[hazy lion]

right

so the lower integral can look like $\int _0 ^1 \frac{t^{x-2}}{t} \sum_0 ^{\infty} \dots$

[elfin berry]

jan Niku

[hazy lion]

oh wait something happened

shouldnt be x-2

we get to $\sum _0 ^\infty \frac{(-1)^n}{n!} \int _0 ^1 t^{x-2+n} \dd t$

[elfin berry]

jan Niku

[stoic saddle]

did someone ping me here

[hazy lion]

i dont thtink so

[marsh citrus]

@hazy lion Has your question been resolved?

[marsh citrus]

@hazy lion Has your question been resolved?

[main idol]

Did you do this integral and sum?

[marsh citrus]

@hazy lion Has your question been resolved?

[hazy lion]

sorry got an appt ill be back

[marsh citrus]

@hazy lion Has your question been resolved?

[hazy lion]

oh, so $\sum _0 ^\infty \frac{(-1)^n}{n!(x-1+n)}$

[elfin berry]

jan Niku

[hazy lion]

but what about the other integral?

the expansion isnt good there is it?

I guess okay so weve now got it split

$\sum _{n=0}^\infty \frac{(-1)^n}{n!(x-1+n)} + \int _1 ^\infty \frac{t^{x-2}}{e^t-1} \dd t$

[elfin berry]

jan Niku

[hazy lion]

my question: should it be obvious the second integral is asymptotically negligible against the first term of that sum? how do you show it?

my professor would usually say integrate by parts

actually now I'm doubting, since we still have the 1/gamma factor out front, this doesnt even show the dominant behavior correctly, does it

[main idol]

Yea the second integral looks hard yo

[hazy lion]

its the second hardest problem on a set of 9

[hazy lion]

oh, its not integrable

its what the incomplete zeta function

if that even exists

i think the goal is argue its neglgible not to integrate it

but i dont think the expansion holds weight for t that large so im not sure how to argue

[main idol]

,w plot t^10/(e^t-1) for 1<t<20

[elfin berry]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by rie.mann

[hazy lion]

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[still temple]

I had questions about the rational functions. If I have a hole in a function, is it excluded from both the domain and range?

[main idol]

What do you mean range

[still temple]

domain and range

[main idol]

Give an example function

[still temple]

Okay. (x+5)/(x^2+x-20)

If the hole is (-5,-1/9)

Then -5 is excluded from the domain and -1/9 is excluded from the range, right?

[main idol]

At x=-5 only. You have to check y can't be -1/9 at other points too

So no, not necessarily

[still temple]

Oh

So does that point not get excluded, but just the x value?

[main idol]

Only the domain yes

[still temple]

Oh ok

Thank you.

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

Yes

[marsh citrus]

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[viscid sky]

For 2nd Question ,

First Catogarize the repetations of letters

[rancid charm]

2T 4O 2B 2E 1R 1N right

[viscid sky]

3T**

[rancid charm]

ah yeah sorry 3t 4o 2b 2e 1r 1n

[viscid sky]

So we have total 13 letters

so arrangment of 13 letters = ?

[rancid charm]

13 factorial?

[viscid sky]

yup

now in this arrangement we have counted the arrangements of shuffling of similar letter so what will we do?

[rancid charm]

ah take it away idk ? 😭

[viscid sky]

We divide them

[rancid charm]

13! over 3! 4! 2! 2! 1! 1!

[rancid charm]

like that?

[viscid sky]

Yea 13!/ 3! 4! 2! 2!

Just solve it u will get the ans

[rancid charm]

I got B thank I understood that

[viscid sky]

idk for these

[rancid charm]

ah alr

no worries

[viscid sky]

Ping helpers after 15 mins if anyone doesn't show up here

[rancid charm]

who are helpers sorry 😭

[viscid sky]

its a role

[rancid charm]

ah ok

will do thank u

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[open kayak]

Prove or give a counterexample: If $u$ does not have any zeros, then $f = u \circ v$ also does not have any zeros. \[5pt]
We can do this with contraposition: If $f$ has a zero, then $u$ must also have a zero, namely $v(x)$. \ Could we also prove it directly?

[marsh peak]

I guess we could simply say that u(x) =/= 0 for all x, u(v(x)) =/= 0 for all x, f(x) =/= 0 for all x

[open kayak]

I guess we would need to justify the u(v(x)) =/= 0 part more

[marsh peak]

u(t) =/= for all t in Dom(u) and v(x) is in Dom(u) for all x in Dom(v) -> u(v(x)) =/= 0 for all x in Dom(v)

How about this?

[open kayak]

Hm, I guess that should work, thanks

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[muted solar]

Hi

[muted solar]

this is the derivative of a function f

how would we draw the function f

[leaden trout]

The derivative of f is basically the local rate of change of f.

So what does it mean for f if f' is positive/negative?

Or if it's 0?

[fathom ridge]

try modelling those equations and integrate

[marsh citrus]

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[marsh citrus]

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[burnt abyss]

I need to prove that a function with degree 4 has a maximum of 4 zeros

[burnt abyss]

@stoic saddle

Sorry for being afk

Can you still help?

[cerulean pollen]

ooo I remember doing this a long time ago and there was a really good khan academy video on it

let me see in the case I can get it 1 sec

[burnt abyss]

Thanks

[cerulean pollen]

I think the idea is just that if you were to factor something with a degree of 4, the maximum number of zeros can be four because you have an (x) in each factor let me get a screenshot

like if you had a function with a degree of four you can't factor this any more

the unit on khan academy is Algebra 2, unit 5, lesson 1

[burnt abyss]

Ok

[cerulean pollen]

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-zeros/v/polynomial-zeros-introduction

I'm not sure this unit like explicitly will mention degrees but I think you could see the pattern and explain it

[burnt abyss]

thanks

what about this:

prove that a function of degree 3 has a minimum of 1 zero

[cerulean pollen]

well so what you are saying is why does a 3rd degree polynomial HAVE to cross the x axis

[burnt abyss]

yeah

[cerulean pollen]

let's just take the parent function, x^3 okay

you can input literally any x value into that function right

[burnt abyss]

👍

[cerulean pollen]

so because 3 is an odd number right any negative x value will produce a negative result

and any positive x value will produce a positive result

[burnt abyss]

ok

[cerulean pollen]

because you have an infinite number of x values no matter where you move around the function eventually it HAS to cross the x axis right

[burnt abyss]

is this an actual proof though

[cerulean pollen]

sorry

[burnt abyss]

ok thanks

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[desert dirge]

hm?

[minor vessel]

How

[minor vessel]

Find x in R

[desert dirge]

youre solving the above for an inequality for x?

[minor vessel]

Yre

Yes

[desert dirge]

you sure?

[minor vessel]

Yes

Wdym

[desert dirge]

in that inequality x could be anything and itll hold

[minor vessel]

U tried working it ?

Yes

Thats what I found too

[desert dirge]

then x is just any real number

[minor vessel]

Well then what am doing is right

I got anothor one

[desert dirge]

hm, kind of hinges on whether |x|-1<0 or |x|-1>0
if its |x|-1<0 then |x-2|+x^2+1>0
if its |x|-1>0 then |x-2|+x^2+1<0

[minor vessel]

Hmm

Okay

Ty

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[cerulean bloom]

I do not understand the simplifying in the denominator

[cerulean bloom]

how 1-(10/x)^2 turns into x^2-10^2 over x^2

[marsh citrus]

@cerulean bloom Has your question been resolved?

[buoyant turret]

you got to first expand (10/x)^2

for the next step you need to think about how to combine 1 + a/b

[cerulean bloom]

I am so lost

@buoyant turret this is what I see

[buoyant turret]

okay lets forget about this part first and focus on 1 - a/b

how would you combine it?

[cerulean bloom]

lcd

[buoyant turret]

you can only combine it when both denominators are equal

so how would you make 1 to have the denominator b

[cerulean bloom]

multiply both sides by x/x?

well they're not sides..

[buoyant turret]

err not quite

[cerulean bloom]

well 1 is already 1 so can i just make it x/x?

[buoyant turret]

you would want to express 1 as b/b

yes

thats the idea

but you need to expand (10/x)^2 before you can proceed

[cerulean bloom]

can I just do 10^2/x^2?

[buoyant turret]

yeap

[cerulean bloom]

oh I see i see

[buoyant turret]

so then you can apply this concept to 1 - 10^2/x^2

[cerulean bloom]

this hw really leaves out all the juicy bits huh

[buoyant turret]

haha as long as you encounter this once you shd be good forever

[cerulean bloom]

Yeah I'm hoping so lol. I have an exam on monday and am just trying to get all these lil specifics down

gotts work on the foundational stuff more

it's always those things that mess me up

[buoyant turret]

good luck man u got this

[cerulean bloom]

THANK YOU

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[humble stream]

Given that $\overline{AB}$ is on the perpendicular bisector of $\overline{XY}$. Is $\overline{XY}$ then on the perpendicular bisector of $\overline{AB}$?

[elfin berry]

Monkagoras

[marsh citrus]

@humble stream Has your question been resolved?

[marsh citrus]

@humble stream Has your question been resolved?

[humble stream]

<@&286206848099549185>

[neat bay]

well you know XY is a line

so draw a line and do a perpendicular bisector of it

[humble stream]

Oh, I see

In this case the statement is false, right?

so because this is a counter example then the statement is not true?

or have I done something wrong?

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[thorny falcon]

could someone help me solve this im completely lost

[marsh citrus]

Please don't occupy multiple help channels.

[thorny falcon]

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✅

[desert dirge]

well, first we must establish what concave up and concave down mean

what do you know

[thorny falcon]

ik how to interpret concavity, but when working with sin and cosine i get confused bc i have to now think about it in terms of the unit circle

[desert dirge]

ah okay, so what is f''(x)

[thorny falcon]

should be 4cos(2x)

[desert dirge]

alrighty, in the interval [0,pi] when would that be positive or negative
you can ignore the 4, as for finding x
you can find values for 2x in the domain [0,2pi] then divide by 2

so basically when is cos(u) positive or negative on u in [0,2pi]
and u=2x

thats what you need to look at

[thorny falcon]

wait why is it 0,2pi and not 0,pi

[desert dirge]

im finding values for 2x, not x

so i just expanded the domain for ease

if you divide the bounds by 2 after, itll be fine

you could just look where cos(2x) >0 or <0 directly if you want on [0,pi]

up to you

[thorny falcon]

i think im confused bc id just assume the positive values for cosine would just be the 1 and 4th quadrant and the negative ones would be the 2nd and 3rd quadrant

[desert dirge]

cos(2x) will be negative for 2x in the intervals [pi/2,3pi/2] and positive for 2x in the intervals [0,pi/2] and [3pi/2, 2pi]

its like looking at a regular cos graph at this point

if you divide the bounds of the intervals by 2, youll get the intervals for x rather than 2x

[desert dirge]

i probably should have explained this better,
if 0<=x<=pi then 0<=2x<=2pi

is what i meant

[thorny falcon]

ik this is alot to ask but is there a chance you could maybe draw the steps past finding the derivative on snipping tool

bc im kinda lost rn lol

[desert dirge]

sure

one moment

diagrams always help

[thorny falcon]

thankyou so much

this helped alot

[desert dirge]

no problem

[thorny falcon]

I couldnt visualize it at first but when you graph it like that it makes sense

i see what you meant by multiplying 1/2

[marsh citrus]

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[lyric bay]

Hello

[lyric bay]

Can anyone help me with this problem

[marsh citrus]

@lyric bay Has your question been resolved?

[marsh citrus]

@lyric bay Has your question been resolved?

[lyric bay]

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[covert basin]

How would I solve this SAT problem without calculus

[late geode]

discriminant

use that to determine when the resultant equation after substitution/elimination of y only has one solution

[covert basin]

thank you

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[static kernel]

is this upper or lower triangular?

[static kernel]

when x is 0 its lower triangular, when x is 1 its not triangular

so im a bit confused

(the question)

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[pulsar flume]

how does one even get started with a problem like this?

[gilded cape]

recall the definition of a normal subgroup

[pulsar flume]

the left and right cosets for a subgroup in a group are equal

[gilded cape]

that’s one equivalent definition

alternatively, a subgroup that is closed under conjugation by any element of the containing group

this is a more useful condition for this question

[pulsar flume]

okay, is that the one thats like gng^(-1) is an element of N for n in a subgroup N and g in a group G?

[gilded cape]

yes

[pulsar flume]

okay then!

so do i start by taking two normal subgroups of G, like M and N, and then we are considering $M \cap N$?

[elfin berry]

goobybalooby

[gilded cape]

yes

[pulsar flume]

okay, and we want to show that is a normal subgroup too, and you said that second equivalent definition is more useful in doing that?

[gilded cape]

yes

so let x be in M n N

[pulsar flume]

okay

do we need to show M n N is a subgroup separately from showing that it is normal?

[gilded cape]

show that gxg^(-1) is in M n N

where g is an arbitrary element of G

[gilded cape]

yes, unless you’ve shown this before

you can try it and ask if you need help with that

[pulsar flume]

okay ill give it a shot

[marsh citrus]

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[torpid ivy]

not sure how to draw the diagram

[wet holly]

draw a perpendicular from the center to a side

[torpid ivy]

so like this

@wet holly

[wet holly]

Yeah

[torpid ivy]

so what’s the total area im trying to find

is it just the triangle

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[pulsar latch]

Pls help

[desert dirge]

you know of arctan?

[paper raptor]

I would suggest using tan=sin/cos

[pulsar latch]

The button on the calculator with the -1?

[desert dirge]

yeah

[pulsar latch]

Idk how to apply it in this q

[desert dirge]

first things first is to do tan^-1(sqrt(3))

this will give you one of the x values

sorry, theta

[pulsar latch]

Oh, so pi/3?

[desert dirge]

youll wanna be in degrees here but yeah thats one of them

next is the fact tan is a repeating function every 180 deg

[pulsar latch]

Would it be two solutions, 60, 90+60 ?

[desert dirge]

right idea, wrong execution

180, not 90

[pulsar latch]

got it! Thank you!

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[craggy stump]

i was wondering what equation would fit best for this wave

[wary condor]

looks like the square of a sine or cosine function

[craggy stump]

i thought of it as the sine wave with the bottom cut off

[wary condor]

, w (sin(x))^2

[elfin berry]

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[wary condor]

, w graph (sin(x))^2

[elfin berry]

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[craggy stump]

oh thanks

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[craggy stump]

.reopen

[marsh citrus]

✅

[craggy stump]

wait

thats just a sine wave moved up

i was asking for an equation that doesn't have the bottom half or i guess an amplitude involved

like this

so uhh

<@&286206848099549185>??

[keen stump]

sin²x

[craggy stump]

yeah i tried that

[keen stump]

or maybe | sinx |

[craggy stump]

sin^2x is just a sine function moved up

[keen stump]

absolute value

[craggy stump]

but the image shown looks much more like a hop/bounce than a wave

[craggy stump]

hm

ok i see

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[heady junco]

i had a question about linear algerba, so if a system has a trivial solution, does it mean that its necessarily homogeneous?
or are there like

[heady junco]

other tips of solutions to it

[young owl]

if by trivial solution you mean all (0, 0, ...)

yes it was homogeneous

when you have a system of linear eq, you can rewrite is as Ax = b

where A have the coeff of the equations, x your vector of unknowns, and b the constant coeff

if A is invertible, you have a unique solution, that's essentially what row reduction/other solving algo etc do

and if A is not invertible, then either you have infinite solutions, because of free variables, either you don't have solutions at all

bc some of the equations contradict each other

and your question about homogeneity: if x = 0 vector is solution, Ax = 0 so b was 0, ie the system was homogeneous

[heady junco]

ahhhhh i seee thanks for the help!, that makes a lot more sense,, i think i just gotta put more thought into it toooo

but thanks a lottt!

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[untold umbra]

Sorry I sent this question earlier

But can someone please explain what they did on the fifth line of the solutions

How did they take the modulus of the thing?

[quartz plank]

here?

[marsh citrus]

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[earnest hound]

how do i solve this? idk where to start

[wary condor]

Take the derivative of p(x)

[earnest hound]

how..

[wary condor]

Mhmm. Do you know the product rule?

Chain rule? Etc

[earnest hound]

you want me to find the derivative of p(x) = (x-2)f(x)+xg(x) ?

[wary condor]

Yes

Wrt x

[earnest hound]

how can i find the derivative of that knowing the rules?

[wary condor]

Use the product rule

https://en.m.wikipedia.org/wiki/Product_rule

[earnest hound]

is it going to be really long?

because i got a long answer

and idk if its right

[wary condor]

It might be

[earnest hound]

p'(x) = f(x) + xg(x) + (x-2) (f'(x) + g(x) + xg'(x))

[wary condor]

What did you get?

Ah

Something went wrong

You should take the derivative of terms separately

And generally, you probably need a lot more practice with derivatives

[earnest hound]

which terms?

[wary condor]

Terms are separated by a plus or minus

Product rule applies to a ... Product

I recommend rereading your chapter on derivative rules

[earnest hound]

okay, i'll try

thanks

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[worthy flume]

Hi

[worthy flume]

Just wondering how to get started here is all

https://i.imgur.com/I2mnFyh.png

[cunning basin]

Domain is the set of values x such that H(x) is defined

[worthy flume]

Yeah so how would i go about solving the problem?

[cunning basin]

are there any value of x such that -7x+4 is not defined?

[worthy flume]

im not sure tbh

[cunning basin]

so

given any number x, you can always calculate -7x+4

[worthy flume]

I seee

So its all real numbers?

[cunning basin]

correct

[worthy flume]

Then what about something harder like this?

https://i.imgur.com/WdI0fKX.png

Wouldnt it also be all real numbers or no

[cunning basin]

to find the domain it's better to think about what can't be calculated

common examples would be square root of negative numbers and in this case denominator of 0

[worthy flume]

Why would the denominator be 0

Wouldnt it just be 0+9 making it 9?

[cunning basin]

I'm not saying the denominator is 0

I'm saying when denominator is 0 h(x) is not defined

[worthy flume]

i see

[vital kraken]

when would the denominator be 0

[worthy flume]

-9

figured it out on my own

ty guys

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[muted narwhal]

does anyone have a clue on how to do this

[muted narwhal]

what I did was sub 1 into x and then solved it then

[chrome mirage]

go ahead

[marsh citrus]

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