help-49

[atomic grail]

So 0<x<1

[runic hamlet]

thats not how that works

x^2=9 has two solutions

not just x=3

[atomic grail]

If integer part is -1

Ah

[runic hamlet]

write x as [x]+{x} again

[atomic grail]

And i assum that integer part is -1?

[runic hamlet]

for now, yes

[atomic grail]

Wait a sec

What was the solution for a^2+a+1=0?

[runic hamlet]

quadratic formula

how did you get that equation

[atomic grail]

It gave me that but a=fractional part

And if i replace integer part

It gaves you that

[runic hamlet]

,rotate

[atomic grail]

But it s imposibble in real numbers

[runic hamlet]

-1-2=-3

[atomic grail]

Ah dang

I alwayd make this

Than fractional part is 3

False

Idk if i should replace them like this

[runic hamlet]

a^2+1=3a would be the correct equation

[atomic grail]

It gives {x}=3-sqrt5/2

I guess that s an x

A solution

This is the only solution

[x] can be just -1

[runic hamlet]

why

[atomic grail]

No

-1 or 1

Because if abs[x] is 2 or higher

The quadratic equation will give a solution that is not between 0 and 1

Withour them of course

[runic hamlet]

why does it give no solution between 0 and 1?

[atomic grail]

Cause i calculated

For abs[x]=2 it will give 1 and 2 or -1 and -2

If it s higher you can solve it just in C

[runic hamlet]

you made a mistake

[atomic grail]

Where

[runic hamlet]

well you didnt show your work so I dont know

[atomic grail]

No because I put a calculator to find out

The solutions

Why are you saying i made a mistake?

[runic hamlet]

because there is another solution

[atomic grail]

There are 2 solutions for x

Yes

I corrected myself when i said that [x] can be both 1 and -1

So 2 solutions

[runic hamlet]

we already discovered earlier that [x] needs to be negative

[atomic grail]

Yeah

Than

What s the big deal

[runic hamlet]

check [x]=-2 again

[atomic grail]

Ok

The roots are

2 and 1

[runic hamlet]

no

[atomic grail]

So the equation is a^2-3a+2=0

Where a is fractional part

[runic hamlet]

no

[atomic grail]

Shit

Is 6a

My dumbass forgot second grade

Wait a sec

It has one solution

It s like 0,35

[runic hamlet]

not quite

[atomic grail]

0,76

?

@runic hamlet

So [x] can be -1 or -2

[runic hamlet]

yes

[atomic grail]

Good

I understood

Thank you very much sir

.close

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[pallid wind]

fellas
HOW DO I EVEN START THIS HTING

[kindred dagger]

what method are you planning to use?

• shell
• washer
• disk

[pallid wind]

shell method

[kindred dagger]

ok. first, can you identify the thickness of each shell?

[pallid wind]

i'll be honest im blanking so hard rn

[kindred dagger]

i marked the thickness of one shell

can you identify the upper and lower part of this purple line?

[pallid wind]

wdym by identify

o hwait

so R2 and R1 are trhe excess bits?

idk how to describe it

[kindred dagger]

@pallid wind what given function touches the lower part of the purple line

[pallid wind]

2 x^1/4

[kindred dagger]

and what given function touches the upper part of the purple line

[pallid wind]

y=2

[kindred dagger]

so the thickness of the shell spans from $2 \sqrt[4]x$ to $2$

[grand pond]

King Leo

[kindred dagger]

so, in terms of x, what is the thickness of th eshell

[pallid wind]

uhh

2- 2 x^1/4

[kindred dagger]

✅

and what is the radius of each shell

[pallid wind]

wait

wouldnt you use the washer method

[kindred dagger]

shell or washer. either works

[pallid wind]

ah ok

[kindred dagger]

length of this red line

[pallid wind]

1?

idk

im throwinaim throwing whatever and see if shit sticks

[kindred dagger]

remember, the purple line is at some position x

so what is the left & right part of the red line

[pallid wind]

gothca

wouldnt that just be 2 x^1/4?
or am i thinking about this the wrong way

[kindred dagger]

remember, we know that the purple line is at some position x

so where is the left of the red line

[pallid wind]

x - 2 x^1/4?

idk

[kindred dagger]

the left part of the red line is just x

can you identify the right part of the red line?

pay attention to where the right of the red line is touching

[pallid wind]

ok i'm just lost now

is the red line being split up by 2x?

[kindred dagger]

no, because we're only paying attention to R2

[pallid wind]

ok

wait

ok yeah i dont know

[kindred dagger]

since we're rotating about the axis x = 1, the right part of your red line would be 1

gtg

[pallid wind]

oh

[midnight plank]

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[sly horizon]

is there anything i can do to simplify this or get a nicer form?

[fallow scarab]

don't think so

,w int sqrt(16-x^2)

[grand pond]

Results provided by WolframAlpha

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

Requested by rie.mann

[fallow scarab]

yea your boxed answer's right

[sly horizon]

alr thanks

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[small zenith]

hi, what is the point of inflexion?

[small zenith]

point(s)*

idk why i couldnt calculate (0,0)... instead, when i calculated f''(x), i found the point of inflexion as (-0.1, -0.259) which was weird af

[dawn dagger]

well it's not differentiable in (0,0)

[small zenith]

wdym?? dym explaining

[dawn dagger]

the slope is not definite

from the left you would expect something positive, from the right negative

[small zenith]

so it's not a point of inflexion?

what about (-0.1, -0.259)? why isnt it obvious on the graph?

[dawn dagger]

and well you also have this sharp twist

[dawn dagger]

no

[dawn dagger]

well if you compressed the y-axis you would see it better

[small zenith]

ah ok nice

[small zenith]

still cant, but thank you

[dawn dagger]

yea it's really subtle

seem to go from concave down to concave up

[midnight plank]

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[rustic saffron]

There are 7 kids on a bus with no bus driver. For each kid, there is 7 backpack, for each backpack, there are 7 big cats, and for each big cat there is 7 small cats.
This just really hurts my brain

[torn prawn]

whats the question asking...

[frigid gorge]

and what do you need help with

[rustic saffron]

Right I forgot how many cats are in the bus. I don’t know where to go with it

[torn prawn]

small or big?

[rustic saffron]

Both

[torn prawn]

uh

alr

[frigid gorge]

7x7x7

[rustic saffron]

343

[torn prawn]

*7

[rustic saffron]

2401

[torn prawn]

first is big

second is small

[rustic saffron]

Yes

[torn prawn]

cool

so u good now?

[rustic saffron]

Wait so is it literally just 2401 + 343!?!

I was overthinking that so much

[torn prawn]

lmao yea

[rustic saffron]

I think it’s time for me to sleep dude my performance be going 📉📉📉📉

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[lime olive]

I don't understand how to do it. Can you guys help me?

[dawn dagger]

there is a piece of information missing

[lime olive]

So I can't solve it?

[dawn dagger]

no, i am saying there is some context to f(x) you haven't shown

[lime olive]

There isn't any other pieces of information

Just this

Well, thanks then

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[fading vessel]

help

[dawn dagger]

Do you know the arc length formula

[fading vessel]

no

HELP

me

pls

<@&286206848099549185>

[rose trout]

Does this not ring any bell?

[wise depot]

@fading vessel , you should revise the line integral topic. I suspect you missed a class or two

[fading vessel]

no man

im completely lost

[rose trout]

To compute the length of the curve you need this formula. If you have not seen it at all you should try to find it in the course material you're following.

[rose trout]

It might not have looked exactly like this, but it should be similar.

[midnight plank]

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[midnight plank]

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[rustic tusk]

Can this be divided using polynomial long division?

[rustic tusk]

I'm getting a quotient of 1/y and a remainder of 3

[blissful trench]

seems fine to me

[rustic tusk]

I tried solving a differential equation by dividing y + 4 with y² + 4 like this but got it wrong

Where am I going wrong?

Sorry for the bad handwriting

[dull yoke]

@rustic tusk

integrate with this route

[rustic tusk]

Yeah I know it can be done this way

I just don't understand where I'm going wrong with the division method

[smoky walrus]

why's that y+1
(y^2 + 4)*1/y = y + 4/y

oh I see
you've mistaken y^2 + 4 as y^2 + y

you should probably change up how you write your 4 or y so that they don't look so similar

[dawn dagger]

best helper

[midnight plank]

@rustic tusk Has your question been resolved?

[rustic tusk]

Oh 💀

I see

In the middle right where I add them up to check, right?

So the division I did was incorrect?

With quotient 1/y and remainder 3

[midnight plank]

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[rustic tusk]

.reopen

[midnight plank]

✅

[waxen willow]

Typically if you have a polynomial P and Q then $\frac{P(x)}{Q(x)}$ if degree of P is smaller than degree of Q. There is nothing you can do

[grand pond]

casework

[waxen willow]

From just that you can see that aint no way you are getting rid of the linear term in the numerator (without also adding a $\frac{1}{y}$ term in it)

[grand pond]

casework

[rustic tusk]

Oh ok

I was under the impression you could divide this way

Thanks for the help

[smoky walrus]

it was referring to the division on the top right
I haven't read the other parts

[rustic tusk]

Oh ok

I got it nonetheless

Thanks

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[gritty hatch]

yo

[gritty hatch]

ok so

[grim vector]

Yo

[opal tiger]

yo

[gritty hatch]

i got this polynomial right

[buoyant yoke]

yo

[gritty hatch]

and i was using synthetic division and the rational roots test to solve it

so i got

(X - 1) + 1x^3 - 3x^2 - 5x - 15 = 0

after using synthetic division with the number 1

but the thing is

is that after doing that

i have been unable to solve the second part

and i noticed the root has to be something like 4.8 because 4.5 is too low and 5 is too high meaning it may not have a solvable root

[hard shard]

you need to factor as (x-r)(q(x)), not (x-r)+q(x)

[gritty hatch]

wdym

[hard shard]

you did x-1+the rest

instead of (x-1)*the rest

plus vs times

[gritty hatch]

huh? you multiply it vs itself

so you do 1 times 1 and then add the next number iirc

that is how synthetic division works yes?

and if thats true its only addition

since a value of 1 is just 1

[hard shard]

you get that part down i assume

but when you write the polynomial partially factored, you need to multiply (x-root)*the rest

[gritty hatch]

can you explain the terminology you are using e.g., (x-r)(q(x)) vs the other thing and what you mean by the last part

bc without the terminology its akin to saying nothing at all

[hard shard]

look at this message here

x-1 is the (x-r) part

1x^3-3x^2-5x-15 is the q(x) part

what you did was add them together when you should be multiplying them together

if you work out the math and combine like terms, you should get the original polynomial

[gritty hatch]

tysm

[hard shard]

when you add them, they dont add up to the original polynomial, but they do multiply together to get the original polynomial

[gritty hatch]

(X - 1) (1x^3 - 3x^2 - 5x - 15) = 0

alr here

[hard shard]

stop the parentheses at the equal sign

[gritty hatch]

however that doesnt appear to solve the question i had thats just a technical error

because that was still the correct answer

that did not influence my calculation in any way unless i desired to revert it

but AI still managed to get the inverse correct

even with my error

[hard shard]

do you mind showing me your synthetic division?

[gritty hatch]

alright

15+1 | 16-52 | -36 + 20 | -16 + 16 | 0 (negative one is a root)

the | denotes the next operation, e.g., 15+1 moving on would be 16-52

https://cdn.discordapp.com/attachments/1018703621841494076/1342179855139082311/image.png?ex=67b8b1e5&is=67b76065&hm=514ce5ddf35ab469ceb8784e9740cebce3c6ce2dea565d9b3f8d8d754c36ee55&

[hard shard]

so how did you get your coefficients for q(x)

[gritty hatch]

?

im just gonna do it again

for clarity purposes

[hard shard]

how did you get 1x^3+...

[gritty hatch]

well

X^3 = 1x^3

they are the same

so you simplify to just 1 since we are only doing the coefficients

but here

15 + 1 - 52 + 20 + 16
trying one as a root (it will all be addition bc anything times 1 is just itself)

15+1 = 16 | 16-52 = -36 | -36 + 20 = -16 | -16 + 16 = 0```

so as you can see i was correct

the root is 1

[hard shard]

is this how you do the synthetic division on your paper?

[gritty hatch]

uhhhh i dont use paper

i type

[buoyant yoke]

Use paper.

[gritty hatch]

uhhhh no?

like can you give me a actual reason to do so?

that would take me a lot of time and waste a lot of resources

also on the topic of AI even if it did hallucinate i could probably tell i mean its not like i cant see the steps and tell if they atleast appear very rational (and the video shows the answer so it literally cannot be wrong especially not overtime)

[hard shard]

but it would save you a lot of time and resources struggling to learn this

[gritty hatch]

plus its scientifically shown IIRC people can multiple times faster using it so

[gritty hatch]

? how?

the only like

issue i have rn is just

i dont format very well

but i dont think that will be a problem for long

[hard shard]

thats a big issue

[gritty hatch]

this is very clear in what it means though do you not understand this?

[hard shard]

i do, but its barring you from using one of the most helpful parts of synthetic division

[gritty hatch]

the only reason i would have a hard time understanding it isnt because i dont understand it as much as it is im new to the conversion so moving that into a full equation is like

a bit annoying but thats just bc i havent done it before

[gritty hatch]

which is?

[hard shard]

i would suggest you do it by the book to start

[gritty hatch]

dude can you just answer the question i had its not that complicated

if you want ill just use AI here wait

cuz AI = paper (it has formatting)

[hard shard]

the results of the addition give you the coefficients

[gritty hatch]

(and dont say it will hallucinate i already know what the right answer is)

[hard shard]

if you dont want to use paper, you can also use any sketching/art program

[gritty hatch]

alright so you are basically saying i made an error where i didnt write the coefficients/answers afterwords to rebuild the equation as I was doing it yes?

[hard shard]

i am saying that you should perform synthetic division the way it was presented to you to perform the factoring properly

[gritty hatch]

Ok.

wait

confused as hell

what do i even do? The change of format is confusing.

Aura

anddd hes gone

great

i knew i shouldve just done this myself

and he went without a single moment actually mentioning any type of error i made

or what i couldve done better besides drawing it erm

and bonk if youre reading this you cant argue i didnt explain properly cuz it doesnt get simpler than heres the completed equation how do i use synthetic division to solve this or are there no solutions

that simple

.close

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[sinful topaz]

patience 🙏

[gritty hatch]

hi

can you help?

i have big confusion

[sinful topaz]

nah cant, im busy, but if you want to you can reopen this channel and wait for someone to help

[gritty hatch]

alr ty

[buoyant yoke]

hard disagree

[gritty hatch]

back it up

[buoyant yoke]

if you follow everything from the video to the tee, there should be no confusion

[gritty hatch]

i think i actually know what the error was

but you are definitely wrong

if im right about that

basically the only way i can even be wrong is if I misinterpreted something

[buoyant yoke]

you probably did

[gritty hatch]

you saying im wrong if i missed something despite me providing the original source is basically faulting me for making a mistake

which is the most natural thing ever in a school context

if a teacher cannot spot a simple error given the original source or atleast show what the correct sequence is

they Are at fault

unless you can substantiate a logical explanation that would supersede my current logical explanation for why that is true

as i provided clearly the method and original equation meaning it is impossible for me to be wrong

[dawn dagger]

we are not your teachers

[gritty hatch]

because all a teacher would have to do to correct any and all errors is follow the simple sequence provided and show the correct path (funnily enough exactly what you said meaning you agree with me they are wrong)

(im not arguing semantics but you are as a teacher is any individual who teaches another X info unless you mean not me specefically incase alright)

[midnight plank]

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[lone hound]

I have an exam tmrw its late so pls help fast if u can

[lone hound]

I just need help with C

[dusty portal]

What do you think the question means?

[lone hound]

What do you mean by that

It means

if u draw a circle with the points A B and C on the vertices

think of it as a circum circle

where would the point D lie

outside the circle, on it or inside

[dusty portal]

Yeah. Would point D be on the circle?

[lone hound]

if I knew the answer to that i wouldn't be here right now

[dusty portal]

Do you know what cyclic quadrilateral are?

[lone hound]

yes

a quadrilateral with all four vertices passing through a circle

[dusty portal]

Great. Is ABCD a cyclic quadrilateral or no?

[lone hound]

it is only possible if the sum of the opposite angles are 180

[dusty portal]

What are some properties?

[lone hound]

in this case no it ain't cyclic

[dusty portal]

And that means?

[lone hound]

thats the only property ik 😭

[dusty portal]

It’s the only one we need.

[lone hound]

it doesn't lie on the circle

[dusty portal]

Okay.

Now, let’s analyze the angles of ABCD.

Heres a hint: if angle B is greater than angle D, then D is inside the circle; if not, it is outside.

[lone hound]

right

so here B is clearly greater

So its inside the circle?

@dusty portal

[dusty portal]

Yes.

[lone hound]

BRUH

chat GPT told me its outside

[dusty portal]

!nogpt

[midnight plank]

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).

[lone hound]

alright

I see

so thats it

if the opposite angle is bigger then its inside otherwise its outside

[gritty hatch]

you should enable search + reasoning to use O3 Mini which is the best available free tier model more likely to get these things correct but remember to check if the actual logical steps are correct and run it by multiple questions to see if the method it provided actually works.

[lone hound]

ok

[gritty hatch]

alrr

https://media.discordapp.net/attachments/1015759484708208792/1045835008012714017/a_3a6633f55a6525e045c618e50b97782a12.gif

[lone hound]

wait

whats search + reasoning

[gritty hatch]

basically if your using chatgpt already (i didnt tell him to use it dont warn me) at the bottem left under the like text bar the buttons will be visible it basically makes the AI reason about its responses (if you still dont see it google regarding the buttons its pretty neat)

[dusty portal]

[lone hound]

thx

that helps

🥰

.close

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[lucid dew]

Consider positive integers a, b, c, d such that ab = cd, which of the following is a possible value of a+b+c+d?
A 97
B 101
C 301
D 401
E None of the above

Question: how should I start thinking about this? I notice that all the answer choices are prime except 301. how can I use that?

[lucid dew]

sorry typo

ab = cd

[tawny pumice]

That's the same

[silent parcel]

Great observation

Make a = cd/b

And factorise M = a + b + c + d by substituting

Then proof M is composite by contradiction

[lucid dew]

should I substitute more than one variable?

actually nvm I figured it out!!

you can factorize M after you substitute

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[obtuse totem]

I dont understand why after carrying, you get 1+2^-51

i think when you round you are adding 1 so it accounts for the 1 + portion before 3x 2^-53

idk

[midnight plank]

@obtuse totem Has your question been resolved?

[tawny pumice]

For 2^-52+2^-53?

[obtuse totem]

yea

[midnight plank]

@obtuse totem Has your question been resolved?

[obtuse totem]

ok ill prob ask again later

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[viral dagger]

how many integers n exist such that $n^4+2n^3+3n^2+4n+5$ is a square

[grand pond]

skissue.in.a.teacup

[viral dagger]

honestly for these questions normally i try to either mod chase or factor but both doesent seem to work

[lethal path]

you can bound it!

you need to bound it between two consecutive squares

[lyric charm]

consider: ||what is n^4+2n^3+3n^2**+2n+1**?||

[lethal path]

^

[viral dagger]

(n^2+n+1)^2

oh so basically you want to prove that $(n^2+n+1)^2=n^4+2n^3+3n^2+2n+1<n^4+2n^3+3n^2+4n+5<n^4+2n^3+5n^2+4n+4=(n^2+n+2)^2$?

[grand pond]

skissue.in.a.teacup

[lethal path]

yeah like the only cases are when that equals (n^2 + n + 1)^2 or (n^2 + n + 2)^2

otherwise it's in between and hence cannot be a perfect square

[viral dagger]

$0<2n+4<2n^2-1$

[grand pond]

skissue.in.a.teacup

[feral sedge]

this inequality does not fail to hold for finitely many n

[lethal path]

why you gotta be so mean (confusing)

yeah ofc there's finitely many

[viral dagger]

why cant it be like (n^2+n)^2 or like (n^2+n+3)^2

[lethal path]

oh wait it's integers sorry

[feral sedge]

no i'm saying the inequality is false in general

[lethal path]

hmmm like it's not strictly bounded for negative ints

yeah should be <= methinks

but then it just doesn't work otherwise somehow, gives complex n

[feral sedge]

you can just decrement the squares for the negative case

works out the same way

[viral dagger]

for negs cant you use (n^2-n+1) and (n^2-n+2)

[lethal path]

no because the term is 2n^3, not -2n^3

[viral dagger]

so for this, it holds for all pos int n

[viral dagger]

i mean like sub n=-m, m>0, m^4-2m^3+3m^2-4m+5

[lethal path]

ahhhhhh okay I see, exactly

nice strat

[feral sedge]

i fail to see why that should work

if this inequality fails to hold true as written, then certainly it won't be true with a substitution x -> -x

[viral dagger]

$m^4-2m^3+3m^2-2m+1<m^4-2m^3+3m^2-4m+5<m^4-2m^3+5m^2-4m+4$

[grand pond]

skissue.in.a.teacup

[viral dagger]

$0<-2m+4<2m^2-1$

[grand pond]

skissue.in.a.teacup

[viral dagger]

ok yea doesent work

[feral sedge]

so what do you think should work?

[viral dagger]

wait hold on what if you use (m^2+m)^2 as that forces the whole thing to be (m^2+m+1)^2

[feral sedge]

getting somewhere

[viral dagger]

$$m^4-2m^3+m^2<m^4-2m^3+3m^2-4m+5<m^4-2m^3+5m^2-4m+4$$
$$0<2m^2-4m+5<4m^2-4m+4$$

[grand pond]

skissue.in.a.teacup

[viral dagger]

0<2m^2-4m+5 det is <0, so it always holds

0<2m^2-1 holds for all pos int m (so all neg n)

so we can force it to be equal to (m^2+m+1)^2

m^4-2m^3+3m^2-2m+1=m^4-2m^3+3m^2-4m+5
2m-4=0
m=2

err but m=2 should hold...

[feral sedge]

what about it?

oh that message was sent a few seconds ago

carry on then, continue your work

[viral dagger]

2m^3+3m-2 doesent have rational roots

[radiant roost]

$n^4+2n^3+3n^2+4n+5=(n^2+n+2)^2-2n^2+1<(n^2+n+2)^2$ for $n<0$

[viral dagger]

but m=2 <=> n=-2 is a square

[grand pond]

Axe

[radiant roost]

does this help

[viral dagger]

its basically this

[feral sedge]

i'm not sure where exactly your contradiction is arising but

you can indeed verify that

$3^2 = (2^2 - 2 + 1)^2 = 2^4 - 2 \cdot 2^3 + 3 \cdot 2^2 - 4 \cdot 2 + 5 = 9$

[grand pond]

Mqnic_

[feral sedge]

so

good enough

[viral dagger]

oh wait

also i missed n=0 which breaks the pos int inequality but that doesent hold so doesent matter

ok ty

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[viral dagger]

if $x_1,x_2,x_3,x_4,x_5$ are roots of $x^5-3x^4+2x^3-6x^2+7x+3$, find $|\prod^{5}_{i=1}(x_i^3-4x_i^2+x_i+6)|$

[grand pond]

skissue.in.a.teacup

[lethal path]

,w factor x^5 - 3x^4 + 2x^3 - 6x^2 + 7x + 3

what the-

[viral dagger]

x^3-4x^2+x+6=(x+1)(x-2)(x-3)

[lethal path]

ahhhhhh

yeah this is bullshit, expecting you to recognise

[waxen willow]

RRT

[viral dagger]

i really feel like i have to do something with the ^5 polynomial so that it has a root of either -1,2,3 but i dont know how to apply it

[waxen willow]

Oo i see

Write the polynomial like
$$(x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)$$

[grand pond]

casework

[lethal path]

ah wait it's just $(x_1 + 1)(x_2 + 1) \cdots (x_5 + 1)$

multiplied by $(x_1 - 2)(x_2 - 2) \cdots (x_5 - 2)$

multiplied by $(x_1 - 3)(x_2 - 3) \cdots (x_5 - 3)$

[grand pond]

south

[lyric charm]

^

[lethal path]

yeah then you can just collect the coefficients after considering each row

[viral dagger]

oh

[lyric charm]

well it's more like you dont need to expand anything

[lethal path]

yeah I just couldn't figure out the initial step haha

[viral dagger]

if the polynomial is f(x) the product is just |f(1)f(-2)f(-3)| then?

[lyric charm]

just recognize each row as the value of the quintic at -1, 2 and 3 resp

[lethal path]

yeah I mean that

[lyric charm]

are we not making any sign errors

[lethal path]

like you take 1, -3, 2, -6, 7, 3, and sub them in the right places for the multiplications

[waxen willow]

Its all in the absolute values

[lethal path]

and then the others go -2, (-2)^2, (-2)^3 ..... (-2)^5

[viral dagger]

yea the 2 hints complement eachother really well

thanks guys

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[twilit field]

$x = t \cos(t);y=t\sin(t)$
\
$x^2+y^2 \implies t^2\cos^2(t)+ t^2 \sin^2(t) = t^2$

[grand pond]

What a wonderful world it is !

[twilit field]

and $z^2=t^2$

[grand pond]

What a wonderful world it is !

[twilit field]

which completes our proof

tq

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[atomic grail]

How do I solve this

[lyric charm]

do the brackets stand for floor?

@atomic grail

[atomic grail]

It s integer part

[opal briar]

use sum in ap formula

[lyric charm]

ok that's floor then

[lyric charm]

are you sure about this?

the roots don't form an AP.

[opal briar]

n/2 times (2a + (n-1)d)

[lyric charm]

sqrt(1), sqrt(2), ..., sqrt(2025) is not an AP.

the AP sum formula does not work.

[opal briar]

Ohh

nvm mb

I am mistaken

[lyric charm]

@atomic grail the x can be pulled out from the floor function

so the equation is $x + \floor{\sum_{k=1}^{2025} \sqrt{k}} = 2025$

[grand pond]

ann.in.a.teacup

[lyric charm]

and the real challenge is to figure out how to work out the floor of that sum of roots

[atomic grail]

I didn t work with this floor function

[opal briar]

either way u can pull x out

[atomic grail]

I mean by floor you mean integer part?

[opal briar]

right

[lyric charm]

it is the same as your []

[atomic grail]

Ah ok then

[lyric charm]

better notation for the same

[atomic grail]

Ok ok

[opal briar]

if its normal bracket u can pull it out

[lyric charm]

it's so that you do not have to scream "[.] REPRESENTS GIF" every single fucking time

[lyric charm]

no, those [] are not just normal brackets.

[opal briar]

oh ok

mb again

ill just listen

[atomic grail]

Well that s my problem

[lyric charm]

have you worked with integration before

[atomic grail]

No, i m class 9

[lyric charm]

wow ok

wonder how you are meant to do this then...

[opal briar]

ask openai

one sec

[lyric charm]

!nogpt

[midnight plank]

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).

[lyric charm]

!nogpt

[midnight plank]

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).

[atomic grail]

It s on 9th grade math olympiad question

[lyric charm]

(is your keyboard's apostrophe key broken?)

[opal briar]

oh mb

[atomic grail]

I tried to write smth like this

[lyric charm]

i dont think this will help much unfortunately.

err wait hold on

are we sure x belongs to N

and not to Z

because this sum of roots is bigger than 2025 all by itself

[atomic grail]

It s natural

It shouldn t be

[lyric charm]

you are SURE that x is meant to be natural?

[atomic grail]

Yes

[lyric charm]

absolutely 100% certain? could swear your life on it?

[atomic grail]

Yes

[lyric charm]

well then there's no value of x that works.

[atomic grail]

Nah let me do the calculus

[lyric charm]

the sum of roots is definitely bigger than 2025, and even bigger than 2026, all by itself.

you can do this without calculating it.

notice that there's 2025 terms in the sum, and all except the very first are bigger than 1.

[opal briar]

oh shit yeah

so x has to be negative

dam

[atomic grail]

Nah the question is mistaken

It sure is like this

[lyric charm]

.

[atomic grail]

I asked my math teacher for clarification

I have one more question tho @lyric charm

What if it was, let s say, sqrt of numbers till 200 or smth like that

I mean smth more smaller so it could work

How would i do it

[lyric charm]

i honestly don't know how to do it bc my only idea has to do with integration

but we don't have access to that

[atomic grail]

Can you tell me how?

I know what the integral is

I worked at physics with it

[lyric charm]

well my idea was that $\sum_{k=1}^{2025} \sqrt{k} \approx \int_0^{2025} \sqrt{x} \dd{x}$ but then there's the question of how far off this approximation is

[grand pond]

ann.in.a.teacup

[lyric charm]

(and having tested it with desmos, it is quite far...)

[atomic grail]

So the function is f(x)=sqrtx

[runic hamlet]

have you worked with integration before?
no
I know what an integral is, I worked at physics with it
dude

[lyric charm]

ok yeah idk which time you lied and which time you told the truth

[atomic grail]

Sorry man I just wanted to know if i could make it with 9th grade math

[runic hamlet]

you can

[atomic grail]

How

[runic hamlet]

because of this

[atomic grail]

Yeah ok that s wrong question in my book

But what if the last sqrt is sqrt of 200

Or 100

[runic hamlet]

just because the answer is "there is no solution" that does not make the question wrong

[atomic grail]

Ok i know this but still

[runic hamlet]

you can probably also prove the relevant bound by induction

but from where you are supposed to get the bound? no clue

[atomic grail]

Nah man probably just integration solves this

Ok I understood thanks for your time

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[twilit field]

I was thinking We know that $f_{xy} = f_{yx}$ by Clairaut's theorm

[grand pond]

What a wonderful world it is !

[twilit field]

So $f_{xy} = f_{yx} =g$

[grand pond]

What a wonderful world it is !

[twilit field]

It then follows that $f_{xyy} = f_{yxy}= g_{y}$

[grand pond]

What a wonderful world it is !

[twilit field]

Is this fine?

[modern sapphire]

yea, that sounds good

[twilit field]

I feel like the next bit is going to be messy

[modern sapphire]

nahh, next bit is also clever clairaut

[twilit field]

hmm

I see we have $f_{y(xy)} = f_{y(yx)}$

[grand pond]

What a wonderful world it is !

[modern sapphire]

mhm

thats good

[twilit field]

No, I don't know this

I'm trying to split this

[subtle blaze]

Let g = f_y

[twilit field]

oooh

[subtle blaze]

What do the derivatives of f tell you about g

[twilit field]

$g_{xy} = g_{yx}$

[grand pond]

What a wonderful world it is !

[subtle blaze]

Why

[twilit field]

oh

clairaut's

[subtle blaze]

What does it require

[twilit field]

That the 2nd derivative is continuous in an open disc around the point

[subtle blaze]

Do you have that

[twilit field]

yes

[subtle blaze]

From where

[twilit field]

f's third derivative is continuou

[subtle blaze]

Which means what for g

[twilit field]

g's twice differentiable

[subtle blaze]

Not only that

Its second derivative is continuous

[subtle blaze]

This condition

Its this that allows for clairnaut’s theorem

[twilit field]

Got it

thanks

thanks

[sinful topaz]

Wait, if g is defined as the second partials of f, then if we know that f has continuous 3rd partials, don’t we only know that g has continuous first partials?

[twilit field]

we're discussisng part 2

[sinful topaz]

Oh g is defined by the first partial

Fair enough

[twilit field]

This now

[
\pdv{u}{x_i} = a_i e^{\sum_{i=1}^{n} a_i x_i}
]

[
\pdv^[2]{u}{x_i} = a_i^2 e^{\sum_{i=1}^{n} a_i x_i}
]

[grand pond]

What a wonderful world it is !
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[twilit field]

[
\pdv{u}{x_i} = a_i e^{\sum_{i=1}^{n} a_i x_i}
]

[
\pdv[2]{u}{x_i} = a_i^2 e^{\sum_{i=1}^{n} a_i x_i}
]

[grand pond]

What a wonderful world it is !

[twilit field]

Adding these up, as the summation of a_i ^2 is 1, this is u

.close

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[zealous schooner]

!status

[midnight plank]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[twilit field]

1

like consider $e^{x+y}$

[grand pond]

What a wonderful world it is !

[twilit field]

It has infinitely many partial derivatives

[zealous schooner]

of n'th order?

[twilit field]

nth order as in $\pdv[n]{f}{x}$

[grand pond]

What a wonderful world it is !

[twilit field]

right

[zealous schooner]

no

[modern sapphire]

theres more than 1 nth order derivatives

[zealous schooner]

that is one example of an n'th order derivative

[twilit field]

oh

[modern sapphire]

f_xx, f_xy, f_yy are all 2nd order

[twilit field]

2 nth order partial derivatives then

oops

no

[sinful topaz]

$\pdv{f}{x}{y}$ would be another example

[twilit field]

2^n

[grand pond]

eugene_krabs_has_cake

[zealous schooner]

why?

[modern sapphire]

use clairauts to reduce this number

Actually, binomial thm

[sinful topaz]

That’s probably for part b no?

[modern sapphire]

oh yea

[zealous schooner]

I think you are supposed to use that for part b

[twilit field]

$2^n$ nth oder partial derivatves. Each time we derive , we have two choices, derive either with respect to x or with respect to y

[twilit field]

2^{n-1} then

[grand pond]

What a wonderful world it is !

[twilit field]

(b) would be 2^{n-1}

right

[zealous schooner]

no

why would it be 2^(n-1)?

[twilit field]

Half the derivatives will be the same no

[zealous schooner]

why half?

[modern sapphire]

same as what?

Are you saying half are same as eachother and other half are all distinct?

[twilit field]

like f_{xy} = f_{yx}

[twilit field]

no

half the derivatives are unique

wait, no

[zealous schooner]

what about f_{xyx} and f_{xxy} and f_{yxx}

[twilit field]

they are equal

assuming the nth derivative is continuous

[zealous schooner]

so your claim that exactly two n'th order partial derivatives are equal is false

[modern sapphire]

||use binomial theorem as I suggested earlier||

[zealous schooner]

do we really need it here?

I think simple counting just works

[modern sapphire]

It does provide the idea is what i mean

[zealous schooner]

I suppose

[twilit field]

oh

(n-1)!?

[zealous schooner]

factorial...?

[twilit field]

how?

[sinful topaz]

Doesn’t that just grow larger than 2^n?

[twilit field]

yes

[sinful topaz]

You can’t have more distinct partials than number of partials

[modern sapphire]

for two variables, the nth partial derivative is a string with k x's and n-k y's

[twilit field]

yes

[modern sapphire]

so each of the nth derivative is gonna appear as a string similar to (x+y)^n

you know there are gonna be 2^n derivatives

[twilit field]

I don't see how that helps

yes

[modern sapphire]

and binomial thm tells you exactly which ones are identical

(your statement that half are identical is wrong, which you can see from the binomial expansion)

[twilit field]

hmm

So what will the binom expression be

I'm really confused

[modern sapphire]

(x+y)^n = nC0 x^n + nC1 x^(n-1) y ... + nCn y^n

that means there are a total of nC0 nth partial derivatives with value identically equal to d^n/dx^n

and so on

[grand pond]

kheerii

[zealous schooner]

If you assume commutativity then by counting the new total number of terms you get the answer for part (b)

Do you see why?

[zealous schooner]

By this same logic you can also do part (c) relatively easily

[midnight plank]

@twilit field Has your question been resolved?

[twilit field]

$^nC_0$?

[grand pond]

What a wonderful world it is !

[twilit field]

Sorry for dipping, have a viva in a while

.close

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[modern shard]

well

i thought

assuming an AP

but pretty sure that wouldn't work

i mean would be diff

[lyric charm]

you haven't said anything so far

[modern shard]

lemme show

[lyric charm]

brb ordering food but i will say you probably just have to introduce variables for the first term and common diff and write out the equations

[modern shard]

thats what i did

nvm kill me

i made mistake in signs

.close

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[lyric shore]

first one

i got 15

[lethal path]

is that Ukrainian btw?

[lyric shore]

yes

[lethal path]

omg slava ukraini

[lyric shore]

героям слава

[lyric shore]

how come you recognized it?

[lethal path]

I know that Russian doesn't have i

[lyric shore]

i see

[lethal path]

,w integrate (6x + 14 - 8/x^2) from -2 to -1

[grand pond]

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[lethal path]

ah something must have happened

some sign error

[lyric shore]

Ah I forgot to divide by -1 at the start

Yep got 1 now

dk if or what i did wrong here

[lethal path]

ah, -1,5 = -27/18

hmmmmm seems all correct

,w integrate (3/x^2 - x/9) from 1 to 3

[grand pond]

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[lethal path]

exactly

[lyric shore]

so i got the right answer here?

[lethal path]

yep!

[lyric shore]

and i got a really ugly answer in the last one

not sure if thats right

[tawny pumice]

What is the og question

[lyric shore]

.

find the area of the figures

this is for the thrid one

im still doing the last

[tawny pumice]

Is this calc 1 or 2

[lyric shore]

dont know, does that matter much?

[tawny pumice]

Yes 😔

[lyric shore]

this is integrals

[tawny pumice]

Yeah I know

[lyric shore]

i do not know what system you're using

[tawny pumice]

US idk

[lyric shore]

this is from my math/algebra class in grade 11 Ukraine

why does it matter which theme it is btw?)

[tawny pumice]

Oh bc I'm in calc one is I didn't know if I should be able to do it

I could if you need help though

[lyric shore]

i just need to check if my answers are correct

i dont need a solving step by step

[tawny pumice]

Okayy

Do you have an answer key or no

[lyric shore]

no

[tawny pumice]

Weird

[lyric shore]

not sure how common that is in the US

[tawny pumice]

I'll try it and let you know in two hours once I'm out of school

[tawny pumice]

We always get one

[lyric shore]

if so then just DM me if you got the same answers

[tawny pumice]

Okay I'll add you rn then

[lyric shore]

i sent you a message request

.close

[midnight plank]

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[midnight plank]

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[safe onyx]

how can I put this in a TI 84 calculator?

[safe onyx]

How do I do it in summation form?

If I put the f(x) as a Y1

[gaunt imp]

I assume you're trying to compute part of a riemann sum for f(x)?

[safe onyx]

I want to be able to put the mid point, trapezoid etc

Instead of doing it by hand

[gaunt imp]

I don't get how trapezoids have to do with an integral though?

[safe onyx]

It can be more accurate than the left/right side method

[gaunt imp]

I have to admit, never heard of that

[safe onyx]

are you familiar with integrals?

[gaunt imp]

Well, I'd say I am(though I obviously don't know all there is to know about them)

Though as I said, never covered those three things, probably because they seem to be approximations

[safe onyx]

these are just different was of approximating area under the curve
midpoint and trapezoid are mentioned in calc one, but calc two spends more time on them in general

[gaunt imp]

Alright, the second one seems like just an alternative formulation of the classical rectangles under the curve approach

The first one is much more of an approximation, and the third one's the one I'm least familiar with

[safe onyx]

yes they are similar

I'm just trying to know the right way do put the f(xi) ... in ti 84

[gaunt imp]

If your question was how to compute them with a calculator, though, I think you'll have to just write the terms out manually and then input them on a calculator

[safe onyx]

$\sum^{10}_{n=1} \left ( \frac{\frac{b-a}{n}}{2} ( ??? ) \right)$

this is what I have

[grand pond]

smeagol

[safe onyx]

there we go

I am not sure how to use the f(x0) + f(x1) ...

[gaunt imp]

? Are the interrogation signs intentional?

[safe onyx]

I am not sure what to put there

to have it be the same as

[gaunt imp]

Well, I have to admit I'm not too skilled at that, I never did cover approximation methods so

I think someone else would be better suited to help you, if no one shows up after a while you can just ping helpers

[safe onyx]

ok

thank you

[safe onyx]

<@&286206848099549185> how can I use ti 84 for these integration area formulas?

yes there are. Please read #❓how-to-get-help

[midnight plank]

@safe onyx Has your question been resolved?

[summer jolt]

Friends I am new here

Will you all like to be my friend

[safe onyx]

If you would like to chat please use #discussion

[safe onyx]

I got it using
Delta X * SUM (Y1( Xi))

.close

[midnight plank]

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[swift monolith]

Hi do I go about solving this problem?

[swift monolith]

so far i am here

For the positon time graphdo i use the displacement of the following shapes i made

which corellates to the position

?

[midnight plank]

@swift monolith Has your question been resolved?

[opal briar]

Area under speed time graph gives displacement no?

Hence speed would be gradient

So to plot u would just have to find where at what times speed changes

[opal briar]

also, use a ruler bro

makes it easier for ur teacher to mark, and for you to see where u went wronf if u do go wrong somewhere

[midnight plank]

@swift monolith Has your question been resolved?

[rustic garden]

i think his work is clear enough, i dont think a ruler will significantly improve anything nor do i think it needs it

[opal briar]

Never hurts to use a ruler though

[midnight plank]

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[tacit rose]

<@&268886789983436800>

Spam scam

[grim vector]

<@&268886789983436800>

[lavish venture]

they’re sleeping

[grim vector]

,w hours in usa

[grand pond]

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[hard shard]

,w pst

[grand pond]

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[grim vector]

Ty

[midnight plank]

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[grim vector]

Lets go stEEam card with two e

[lavish venture]

i just got 50 bucks bro

[opal briar]

NO WAY

[hard shard]

steeamn

[opal briar]

my turn

Nooo

eher did it gooo

[midnight plank]

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[safe onyx]

I just realized I mixed up the equation 🥺

V final ^2 = V initial ^2 + 2 a Delta x

0 = (311 * 1000 / 60^2) ^2 + 2x * .81

x = - 4610 m/s^2