help-27

[alpine python]

because the leg is perpendicular so i guess it's the shortest path to the x-axis

[grand ledge]

ok

we also need to prove it this part using geomtry then?

[lusty sapphire]

First, is sin(x)<x (for x>0) something you can use right away, or are you concerned you need to prove it first?

[grand ledge]

We never learned it so I think I might need to prove it

[lusty sapphire]

okay sure. I have that all ready to show too, thankfully

[grand ledge]

btw all of this question is just 1 of 4 parts in one of 7 questions too. so they might not expect something this good. but im too invested 🙂

[lusty sapphire]

sign of a true mathematician

It's also why I am here, and put so much work into it

[grand ledge]

Im learning aeronautics engineering 😮

[lusty sapphire]

Okay, let us show that $\sin\theta<\theta$ for $0<\theta<\frac{\theta}2$

[woven radish]

SWR

[lusty sapphire]

yeah you definitely don't need all of this work then

but whatevs, we're here

Here's our geometric setup

[alpine python]

you don't actually need to find the right and left limits to argue that the limit doesn't exist

but yeah, since we're here

might as well see it through

[lusty sapphire]

We can look at this after then, I'd be interested to see your idea

[grand ledge]

me too

[grand ledge]

ok

[lusty sapphire]

We want to show that $\overline{CD}<\frown{CB}$. Any idea how we can do this?

[woven radish]

SWR

[lusty sapphire]

[grand ledge]

nope

geometry aint my thing

[alpine python]

what was it they say about the shortest distance between two points?

is that an axiom? probably

[grand ledge]

oh

a straight line

[alpine python]

yeah

[grand ledge]

and $\sin{\theta}^2 + \overline{DB}^2 <= \overline{CB}$

[woven radish]

Shachar

[alpine python]

in fact it equals CB

and <= theta

oh CB^2 i meant

[grand ledge]

right its a right triangle

so it's equalcs CB

and CB smalled than CB_arc

than sin theta for sure smaller than CB_arc which is theta

wow geometry is beatiful

[lusty sapphire]

yup

there you go

That was a lot of headache

but hey, we did it

I think you can do the rest now

It took a lot of working backwards to get to this little geometry problem

[alpine python]

u can use triangle inequality to say CD <= DB

[lusty sapphire]

what?

[alpine python]

wait i'm dumb

but we cheated because we dropped the 2 exponent

[lusty sapphire]

You $\textit{will}$ need to prove $\overline{CD}<\overline{CB}$, but I do not think you can do that with triangle inequality.

[woven radish]

SWR

[alpine python]

you're right

[grand ledge]

well damn @lusty sapphire thank you. no way it is what they intended for me to try and do, but it's beatiful

[alpine python]

oh sin = sqrt(sin^2) < sqrt(sin^2 + DB^2) = CB

[grand ledge]

axe what way you wanted to use?

im intriged

[alpine python]

i'm surprised you're still up for it

[grand ledge]

I can't not hear it now

[lusty sapphire]

Proving that $A<B$ implies $\sqrt{A}<\sqrt{B}$ is actually not so trivial

[woven radish]

SWR

[alpine python]

you need sqrt increasing

we're going deep

lol

[alpine python]

uh it starts in a similar way

but forget the upper bound

for $x\in (0,\sqrt{\pi}),\ \sin (x^2) \geq 0$ so $\frac{\sin (x^2)}{x}\geq 0$ and $\left\lfloor \frac{\sin (x^2)}{x}\right\rfloor \geq 0$

wait i'm dumb

[grand ledge]

x <= pi

[woven radish]

Axe

[grand ledge]

ok

[lusty sapphire]

ohhh. I see it

[alpine python]

for $x\in (-\sqrt{\pi},0),\ \sin(x^2) > 0$ so $\frac{\sin (x^2)}{x} < 0$ and $\left\lfloor \frac{\sin (x^2)}{x}\right\rfloor \leq -1$

[woven radish]

Axe

[grand ledge]

ok

then you limit both sides of the inequality?

[alpine python]

no

this is enough

[grand ledge]

what this gives you?

[alpine python]

this gives you:

$lim_{x\to 0} \left\lfloor \frac{\sin (x^2)}{x}\right\rfloor$ does not exist

[grand ledge]

based on what?

there is no two side limit here

[woven radish]

Axe

[alpine python]

um

intuitively, if the right limit is to exist, it must be >= 0, and if the left limit is to exist, it must be <= -1

[lusty sapphire]

This was something I was thinking about too, @alpine python. I get the logic of what you are trying to say right now. I think the nuance of what needs to be said next is that, if the right limit were to exist, then it must be ≥0. And if the right limit existed, then it must be ≤-1.

[lusty sapphire]

ah yes. Exactly

[alpine python]

yeah

[lusty sapphire]

Proving that, though, might be hard

[alpine python]

i guess u do need delta-eps after all for this method

[grand ledge]

so can't I just take the limit 0+ of the first inequality

[alpine python]

https://tenor.com/view/no-bugs-bunny-nope-gif-4095597578532971838

[grand ledge]

and the limit 0- of the second inequality

[alpine python]

well maybe

i'm not sure

[lusty sapphire]

We do not know yet if the limit exists

[grand ledge]

intuitivly

it ez

tho the question itself is ez intuitivlly

only hard to prove formally

[lusty sapphire]

Basically, we have $f(x)\ge0$ for $x\ge 0$, and $f(x)\le-1$ for $x<0$. Taking any limits here is too broad.

[woven radish]

SWR

[vagrant lichen]

the way you guys talk is so fun

[grand ledge]

wym

[vagrant lichen]

idk you js sound so smart rn

[grand ledge]

they are smart

[vagrant lichen]

ikr

[devout snow]

@grand ledge Has your question been resolved?

[lusty sapphire]

@grand ledge @alpine python

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[lusty sapphire]

.bye bye everybody. It was fun

[devout snow]

Available help channel!

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[spring oasis]

How many three-digit numbers are divisible by $13$?

[woven radish]

938c2cc0dcc05f2b68c4287040cfcf71

[fast mist]

how many 3 digit numbers are divisible by 2

[spring oasis]

,calc 999/2

[woven radish]

Result:

499.5

[solid osprey]

how many numbers less than 1000 are divisible by 13?

[spring oasis]

like 499 I guess

[wicked turtle]

2 doesn't have three digits

[fast mist]

what is a 3 digit number

[spring oasis]

100 to 999

[fast mist]

so why 999 / 2

[spring oasis]

idk, I had a brain fart

[fast mist]

🙃

so how many divisible by 2?

[solid osprey]

the subtract by the number of numbers less than 100 that are divisible by 13

[spring oasis]

,calc 1000/13

[woven radish]

Result:

76.923076923077

[spring oasis]

,calc 76*13

[woven radish]

Result:

988

[spring oasis]

,calc 100/13

[woven radish]

Result:

7.6923076923077

[spring oasis]

,calc 7*13

[woven radish]

Result:

91

[spring oasis]

69

[fast mist]

🎉

[spring oasis]

okay, but it wasn't easy

,calc 76-7

[woven radish]

Result:

69

[fast mist]

another way (maybe if you have a calculator): the first multiple of 13 is 104 = 13 * 8. the last is 13 * 76 = 988. so there are 76 - 8 + 1

[spring oasis]

why +1

[fast mist]

how many numbers integers are there between 1 and 10 (inclusive)

[spring oasis]

10-1+1

[fast mist]

why +1

[spring oasis]

idk

you said inclusive

[fast mist]

right

so that's why +1

i want to include every number between 8 and 76, endpoints included

[spring oasis]

yeah I guess

.closs

.close

[devout snow]

Channel closed

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[devout snow]

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[fair juniper]

Given two numbers $x$ and $m$ where $gcd(x,m)=1$, let $n$ be the smallest positive integer such that $x^n \equiv 1 (\mod m)$. Show that $n | \phi(m)$.

[woven radish]

LocalLunatic

[lament kraken]

n is the order of x mod m

Ah

[fair juniper]

I know by FLT, x^phi(m) is congruent to 1 mod m, but i kinda forgot how to do the rest

yes, n is the order

[lament kraken]

I seem to remember that it was done by contradiction

[fair juniper]

oh i think i get it

[lament kraken]

Yay i actually forgot it lol

[fair juniper]

if x != m, then n>1

[lament kraken]

Ok

[fair juniper]

Assume by contradiction n does not divide phi(m)

[lament kraken]

Ok

[fair juniper]

then x^nk = 1 for all k

[lament kraken]

(u dont have to help me but ty)

Ok

[fair juniper]

im tryna work it out too

[lament kraken]

(i can check the proof ig)

[fair juniper]

then we do euclid algo

m = nq+r

[lament kraken]

Uh huh

[fair juniper]

so we pick k=q

correction: phi(m) = nq+r

[lament kraken]

Yes

[fair juniper]

then we get x^nq = 1 but also x^nq+r = 1

[lament kraken]

Ah

[fair juniper]

so 1*x^r = 1

r < n which contradicts the statement that n is the smallest

[lament kraken]

Yes

[fair juniper]

heh nice

[lament kraken]

Nice

[fair juniper]

thanks for keeping up

.close

[devout snow]

Channel closed

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[devout snow]

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[heavy monolith]

I got up to here

But the answer is wrong apparently

Can someone point out where I messed up?

[fair juniper]

"their centres form a square"

[heavy monolith]

So would it be 40-the side of the square?

[fair juniper]

you start with 4 spheres on the table placed in a square formation

then you get a fifth one and put it on top

[devout snow]

@heavy monolith Has your question been resolved?

[eager gyro]

Their centers form a square, square side will be 2times radius so 20cm

If you still have not solved yet i would like to solve if and drop pic😇

[heavy monolith]

Yes sure

I don’t mind

[heavy monolith]

You can send

[devout snow]

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[heavy monolith]

I solved it though

[devout snow]

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[heavy monolith]

But I’m happy to see your diagram if you still want to send it

[heavy monolith]

Yo

[eager gyro]

I solved just now too lol

[heavy monolith]

The answer is 34.14

[eager gyro]

Its easy tbh

[heavy monolith]

Ye

I just didn’t plan it accordingly

I always miss something when reading the question

Js gotta do more of them

[eager gyro]

Answer is 20+10sqrt2

Yeah appr 34.14

[tribal vessel]

Proof that 1+1=2

[heavy monolith]

Yo Sandro can you help me with this question

[eager gyro]

Will try ofc

Idk hollow inverted cone

Idk english words

[heavy monolith]

It’s like a cone upside down without the base

[eager gyro]

Oh ok

This? Well it is not supposed to be touching base of cone

Oh wait

With this numbers it will be something like this

Ice cream lol

Is answer 1?

@heavy monolith Ig you already did it yourself but still (sorry for ping)

[devout snow]

@heavy monolith Has your question been resolved?

[devout snow]

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[native stone]

good morning guys, i need help with this exercise

[native stone]

the translation in english is just: ,,find out F(x),,

[ancient sluice]

Ok

[native stone]

looks like a very easy exercise but i cant figure out what to do

[native stone]

oh hi xor

[ancient sluice]

1/cos⁴ x = sec⁴x = (1+tan²x)sec²x

Now it can be visualised

[native stone]

ok

[ancient sluice]

Else you can just consider tanx = z and write an extra step

[native stone]

so tg'x time 1/cos^2x

so actually its (tan'x)^2

so the derivative of tanx to the second power

[ancient sluice]

Huh ?

[native stone]

we dont use sec around here, but sec^2x = derivative of tan x

if im not mistaken

and we use the derivative not sec

[ancient sluice]

int sec²x = tgx and int tan²xsec²x = tg³x/3

int = integration

[native stone]

yep yep

[ancient sluice]

Yeayyy !

[native stone]

the second one i dont know

but anyway

[ancient sluice]

Nah

[ancient sluice]

Use a sub , tanx = z

sec²x = dz

So we get int z²

That gives z³/3

[native stone]

ah okk

[ancient sluice]

You understand now ?

[native stone]

got any solution thats without sec tho?

[native stone]

yes i do

it doesnt seem that complicated

[ancient sluice]

Yeah tgx + tg³x/3 is the soln

[native stone]

interesting

lets try to figure out a way to write the fraction

so that we get something like what you got

but without the need of sec

if you want ofc, ur solution is correct as well, but i just wanted something different

oh wait

i can just write the nominator as sin^2 x + cos^2 x and it comes out easily

sorry for the bother

i got it

.close

[devout snow]

Channel closed

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[devout snow]

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• Show your work, and if possible, explain where you are stuck.
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[native stone]

i thought that this is just an exercise with a usual cos alfa substitution

but it doesnt work here

any other clues on how to solve it?

[fair juniper]

you can use a u sub

[grand siren]

well it's multiple choice so you could just differentiate all of the answers and see which one matches...

[fair juniper]

lmao yeah

[native stone]

oh yea i could

i dont wanna do it that way tho

feels like cheating

so like

[grand siren]

also these don't look like particularly fun expressions to differentiate so it might take a while

[native stone]

yea

there also choices until f

so thats a bit too many

[grand siren]

but anyway yeah there is a useful u sub here

[native stone]

so what the substitution?

[grand siren]

u = x^2 + 1

or at least i think that works

[native stone]

you think that its useful?

sure let me try

[grand siren]

because then the square root just becomes sqrt(u) which is a lot nicer

[native stone]

i hope you can read my writing

but as you can see

while it does look close to one of the multiple choices, its not exact them

[native stone]

are you still here?

[grand siren]

i am now

i think you lost a factor of 1/2

[native stone]

oh yea you are right

[grand siren]

and the other thing is that you need to replace the u with x^2+1, because you want an answer in terms of x

[native stone]

yea thats fine

[grand siren]

also you need a +C
after that i think it should be identical to one of the answers

[native stone]

i can see it without chaning it back to

x

[grand siren]

oh fair enough

[native stone]

yea, im just lazy

thanks a lot tho

.close

[devout snow]

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[devout snow]

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[lost laurel]

Let $0\leq x_1 \leq y_1$.Define $x_{n+1} = \sqrt{x_n y_n}$ and $y_{n+1} = \frac{x_n+y_n}{2}$. Show that lim $x_n$ and $y_n$ both exist and are equal.

[woven radish]

math_rocks

[lost laurel]

so here can I just assume x_1 and y_1 to be arbitrary constants

[fair juniper]

ooh the AGM

[lost laurel]

yeah

hmm

[lament kraken]

Intresting ngl

[lost laurel]

yeah, it is. But also looks hard

$x_{n+1}= \sqrt{x_n(2y_{n+1}-x_n})$

would this help in any way

[woven radish]

math_rocks

[lost laurel]

Hmm

$x_{n+1} \leq y_{n+1}$ makes sense yeah

[woven radish]

math_rocks

[lost laurel]

hmm, and $y_n \geq y_{n+1}$ makes sense too

[woven radish]

math_rocks

[lost laurel]

but how does that imply their limits must be equal

okay, so monotone convergence theorm

hmm

Let em think a bit more

thanks

.close

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[fierce girder]

Can anyone help me understand how to calculate the surface area of a sphere? I heard aristotle did it by realizing that a cylinder of base radius 'r' and the height '2r' can completely enclose a sphere of radius 'r' and the shadows of the sphere fall exactly on the cylinder so the surface area of the cylinder must equal the surface area of a sphere and thats how he found that it was 4πr².

I tried to find the volume of a cone, sphere by using integration and I was successful. But I have tried to compute the surface areas of sphere and cone by intergration but failed. I do not care about cones much but I wish to know how to compute the surface area of a sphere by using integration with one variable.

[devout snow]

Please don't occupy multiple help channels.

[fierce girder]

<@&286206848099549185>

[crude wasp]

<@&268886789983436800> this person seems to be spamming a few channels with russian that's unrelated? (can't speak russian but google translate says it's spam)

(see hlounge)

[merry patio]

first of all we want to know how to draw a circle in 2d

and half of it should be enough so we can rotate it around either the x or y axis

[merry patio]

what is the equation for something that looks like a circle?

[fierce girder]

$\sqrt{1-x^2}$

[woven radish]

bagelguy3

[fierce girder]

@merry patio

[merry patio]

we can make it generic if you want

and say r^2 instead of 1

wait hold on lemme check if i rember correctly how to do it

[fierce girder]

ok

thats fine

[woven radish]

bagelguy3

[fierce girder]

$\f(x) = sqrt{r^2-x^2}$

[woven radish]

bagelguy3
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fierce girder]

$\ff(x) = sqrt{r^2-x^2}$

[woven radish]

bagelguy3
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fierce girder]

${f(x) = sqrt1-x^2}$

[woven radish]

bagelguy3

$\{f(x) = sqrt1-x^2}$
```Compilation error:```! Extra }, or forgotten $.
l.49 $\{f(x) = sqrt1-x^2}
                         $
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}{/usr/l```

[fierce girder]

$$f(x) = sqrt{r^2-x^2}$

[woven radish]

bagelguy3

[fierce girder]

$f(x) = sqrt{r^2-x^2}$

[woven radish]

bagelguy3

[fierce girder]

$\sqrt(r^2-x^2)$

[woven radish]

bagelguy3

[fierce girder]

$f(x) = /sqrt{r^2 - x^2}$

[woven radish]

bagelguy3

[fierce girder]

wtf

[devout snow]

@fierce girder Has your question been resolved?

[merry patio]

alr sorry

back

@fierce girder

,w graph sqrt(1-x^2)

[woven radish]

Results provided by WolframAlpha

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

Requested by _rysel_

[merry patio]

u there?

[devout snow]

Channel closed

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[fierce girder]

I am now

.reopen

[devout snow]

✅

[fierce girder]

@merry patio

for some reason discord notification sound isn't working

[merry patio]

oh ok

so basically lets imagine the 3d volume

[fierce girder]

ryse

send me a DM

lemme confirm that the sound isn't working

[merry patio]

we want to take these small "cylinders" and add them up all together

[fierce girder]

yes

waoh

[merry patio]

or circumferences

[fierce girder]

dam

[merry patio]

so basically we know the radius is just the function

root of r^2-x^2

but we need to know this length

[fierce girder]

I broke them into two hemisphere and decided to calculate just the top hemisphere and i tried doing from bottom to the top

[fierce girder]

yes\

[merry patio]

that length if we zoom in

[fierce girder]

is that dθ

[merry patio]

is just a right triangle

[fierce girder]

right

[merry patio]

and to find it we use pythagora

[fierce girder]

so thats dl?

[merry patio]

yes

dL

[fierce girder]

i SEE

isn't that what u use to calculate arclength of a curve?

[merry patio]

so dL = root( dy^2 + dx^2 )

yes exactly

same formula

[fierce girder]

sqrt(1 + f'(x)^2)

[merry patio]

we can group the dx^2 and put it outside the root

and yes

[fierce girder]

yes

so

I see

[merry patio]

we get root(1+f') dx

[merry patio]

so our formula will be the y length and the dL but rewriting it in terms of x

we need to apply the rotation

[fierce girder]

$/∫ sqrt{1 + f'(x)^2} dx$

[woven radish]

bagelguy3
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fierce girder]

BRUH

WHY

why doesn't it work where do I go wrong

[merry patio]

well its not even really a rotation

its just 2piy dL

integrate it

[fierce girder]

dont we get the square?

[merry patio]

square? there are no suqares

[fierce girder]

i meant

[merry patio]

you are just adding up every circumference

[fierce girder]

(f')²

[merry patio]

ih yes

[fierce girder]

how can we get f'

[merry patio]

i forgot

yes

our dy/dx

[fierce girder]

yes

[merry patio]

since our y = root(r^2 - x^2)

[fierce girder]

since we are using a² + b² = c²

[merry patio]

you can derive this to get dy/dx

[fierce girder]

dy = y' dx

[merry patio]

what is your doubt

[fierce girder]

how to calculate the surface area of a sphere with integration

not the circumference

[merry patio]

we add all these circumferences together

with an integral

[fierce girder]

uhm

so look

[merry patio]

from the left to the right point

[fierce girder]

assume

we take just one hemisphere

just the top hemisphere

[merry patio]

ye

[fierce girder]

cut it into the southern hemisphere and northern hemisphere

ignore the south one

just delete

imagine the north

[merry patio]

ye

[fierce girder]

now the circum ference from the equator (middle line) to the north pole (top) would be rθ where θ = π/2 so it would be πr/2
now, rotating that would give us πr/2 * 2π = π²r.

multiplying that by 2 to also include the area of the southern hemisphere gives us 2π²r

but we know that the answer is 4πr²

so where did I go wrong

[fierce girder]

that integral is the key and you gave me everything aside the integral

[merry patio]

not sure what you did there circumference from equator to north pole?

[merry patio]

yeah you integrate everything at the end, same thing as summing all the circumferences

[fierce girder]

i uh

dont understand

[fierce girder]

could u be more elaborate

[merry patio]

we have this image

[fierce girder]

yes

[merry patio]

we want to find the generic area of one circumference

and then add them all up together

from -r to r

or you can do from 0 to r and double it

same thing

we know the circumference is 2pir

problem is we also have that dL

which we found previously

[merry patio]

and basically you integrate this now

we know that our "radius"

is jsut our y of the equation

its our "height"

from the x axis

we can also find dy/dx which is just the derivative

from there you just integrate

[fierce girder]

whats written with purple color

[merry patio]

its a y

or f(x)

[fierce girder]

i see

so its basically

[merry patio]

which is our radius

[fierce girder]

2 ∫(0,r) f(x) * sqrt( 1 + f'(x)²) dx

[merry patio]

exactly

missing the pi

but yes

[fierce girder]

so

[merry patio]

and also the 2

so it would be 4pi isntead of the 2 outside

[fierce girder]

∫(-r,r) y dl

[merry patio]

you need the 2pi

we are not rotating anything

[fierce girder]

right so

[merry patio]

we are just adding all the circumferences up

we know the circumference is just 2pi * y * dL

[fierce girder]

4π ∫(0,r) y dl

[merry patio]

yes

exactly

if you start solving it

you will get to the right answer

our y is sqrt(r^2 - x^2)

then you can find dy/dx

and you just plug in everything

[fierce girder]

i see

thank y

ou

[merry patio]

np

[fierce girder]

I also have another question if u dont mind

[merry patio]

do you wanna see the process

sure

[fierce girder]

I understand derivatives

but why does doing the opposite of derivative and pluging in the limits give you the area under the curve between those limits

basically why does integration work?

I understand the infinite summation and I understand the process of making it into infinite rectangles and finding the area and stuff

but I dont see any rectangles when I do integration

just find the anti derivatives and plug in the limits

[merry patio]

im honestly not quite sure about where the calculation comes from too tbh

i think i forgot it

[fierce girder]

u knew at a point?

[merry patio]

dunno

i think idk

[fierce girder]

alright

i gotta go study now

cya

also thx for ur help

[merry patio]

bai

[fierce girder]

bai

.close

[devout snow]

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[devout snow]

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[restive river]

Here, 3x is simplified to 3

I get why

because x^0 = 1

3x if x = 1 is 3.

but then why is there an x next to the 14 in the bottom row?

or even the 2 in the middle row with the 2x.

shouldn't it just be -7 * 2 for the 2nd row and -14 for the bottom row?

[solar goblet]

this is not how power rule works

,tex .diff rules

[solar goblet]

refer to the first two rules

[restive river]

Ye I still can't make sense of it sorry

is it because the 3x has not exponent on the x and the -7x did?

ah because the power rule is only applicable when x != 0.

I think that's right, correct me if I'm wrong please.

.close

[devout snow]

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[solar goblet]

the value of x doesn't matter

[restive river]

.reopen

[devout snow]

✅

[restive river]

That's why the -7x^2 didn't have the x ommited though correct?

or what do you mean please?

[solar goblet]

the general rule of thumb when differentiating x^n is that you multiply the result with n, and then knock down one power in the exponent

i'd say the condition n != 0 is unnecessary here, it still holds for n = 0

the reason why the derivative of 3x is 3 is because 3x is actually 3x^1

so you would get $(3\times1)x^{1 - 1}$

[solar goblet]

which is 3

[merry patio]

would recommend using definition of derivative for the first few times so you can see it in action

[solar goblet]

same deal with -7x^2

[merry patio]

its not good to apply without knowing why

[solar goblet]

or, just prove the power rule

[merry patio]

that is a little harder but yes

[restive river]

I don't know what you mean exactly?

[restive river]

but then the exponent of 2 gets subtracted by 1 and the x because x^1

so shouldn't it just be -2

[solar goblet]

x^1 = x

what are you saying

[restive river]

oh wait

I think I got you.

[merry patio]

you should revise what a derivative is before applying its rules

[restive river]

give me a sec just gonna process

[merry patio]

they are just shortcuts in the end to do this

[solar goblet]

it just requires binomial theorem

[merry patio]

ye

[olive snow]

Also its h going to zero, little typo

[merry patio]

ye idk why the image uses a instead of h

[solar goblet]

[olive snow]

Both are useful

[merry patio]

other one should be c

[olive snow]

Depends situation

[merry patio]

well same thing

[restive river]

Just read this 3 more times

and it makes sense now

thank you so much

.close

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[restive river]

I thought I knew what a derivative is but I don't know how this applies here, thanks for replying anyway though, you don't have to of course but if you find a sec to relate it to the specific example, I'd appreciate it 👍

[merry patio]

basically the power rule and evey other rule is just a shortcut

you are technically applying the definition of derivative but mathematicians are lazy and find patterns

imo you should use the definition instead for the first few times instead of the shortcut to see what is going on

and for the ones with power rule just try applying the first one

[restive river]

I mean, I much rather, gonna just think about how to apply the definition of a derivative here.

[merry patio]

lemme find a better image this one is bad

[restive mauve]

also known as "first principles of derivatives" if u wanna do some research on it

[restive river]

thanks!

[restive mauve]

the image that ryse js deleted*

[restive mauve]

yeahh^^

[merry patio]

you had for example 7x^2

applying definition you get (7(x+h)^2 - 7x^2)/h

if you just solve the limit

you will find find out that its just 14x

mathematicians noticed a pattern with the power

and they just said x^n ---> n*x^(n-1)

[restive river]

This helps a ton, I'll do that moving forward just because it means I don't have to remember the rules and I can instead actually make sense of the solution.

Thank you!

[merry patio]

no problem!

[merry patio]

you should still remember the shortcuts otherwise you will take ages to differentiate lol

but yeah if you happen to forget you can always use the definition

[restive river]

https://www.youtube.com/watch?v=9VVcuFksx0A

Found a nice little video about it just before you clarified earlier

[merry patio]

yes thats good

proofs are harder with harder functions

[devout snow]

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[oblique rover]

How to show that $\lim_{x\to\infty} \sin x$ deosn't exist using delta epsilon?. I know that i need to show
[\neg(\exists L\in\bar{\mathbb{R}}\forall\varepsilon>0\exists\delta>0\forall x\in\mathcal{P}{\delta}(\infty):f(x)\in\mathcal{U}{\varepsilon}(L))\iff]
[\iff \forall L\in\bar{\mathbb{R}}\exists\varepsilon>0\forall\delta>0\exists x\in\mathcal{P}{\delta}(\infty):f(x)\notin\mathcal{U}{\varepsilon}(L))]
But how should i do it?

[woven radish]

Slowaq

[dire forge]

What is P_delta(infinity)?

[woven radish]

Slowaq

[dire forge]

does R with the bar above it mean the extended real numbers?

[oblique rover]

yes

[dire forge]

okay

suppose L is either infinity or -infinity, what have you tried so far?

[oblique rover]

so i tried putting epsilon equal to 0,5

with this there will always be infinitely many ones or -ones for example for x=pi/2+kpi

or hm

[dire forge]

yes

that's a good start

[oblique rover]

how should i continue?

[dire forge]

you'll want to verify that your choice actually works, so first what if L is infinity? then we choose epsilon = 0.5 and we fix some number delta too, you need to choose a value of x in P_delta(infinity) so that f(x) is not in U_epsilon(L)

what value of x do you choose?

[oblique rover]

for example x=pi/2+2kpi for big enough k?

[dire forge]

yeah what value of k?

well

what will sin x be equal to in that case, first

[oblique rover]

so that pi/2+2kpi is bigger than delta

-1

[dire forge]

you sure?

check again

[oblique rover]

ah sorry 1

[dire forge]

yeah

then, is sin(x) contained within U_epsilon(L)?

for that choice of x

[oblique rover]

for L being infty or -infty definitely not

[dire forge]

why?

[oblique rover]

because $\mathcal{U}_{\varepsilon}(\infty)=(\varepsilon,\infty)=(\frac{1}{2},\infty)\nsupseteq \sin x$ since $\sin x \in [-1,1]$

[woven radish]

Slowaq

[dire forge]

but we said for that x, sin x is 1, right?

we're only looking at one specific value of x

is 1 in (1/2, infty)?

[oblique rover]

hm yes

[dire forge]

yeah, so we need to pick a different value of x if L is infinity and we're choosing epsilon to be 1/2

[oblique rover]

so then what ybout x=3pi/2+2kpi

then it works

[dire forge]

yeah good

that settles the case where L is infinity, what if L=-infinity?

if we choose epsilon to be 1/2, what would be a good value of x to choose

[oblique rover]

then ill use the first case

x=pi/2+2kpi

[dire forge]

yup good

that settles it for L being infinite

now what if L is finite

let's say epsilon is 1/2

how do we choose x?

[oblique rover]

hm now it depends on L doesnt it

for example if L is 0 then well use the same x=pi/2+2kpi as in previous

if L is [1;-0,5) then we use x=3pi/2+2kpi

similar for -1

[dire forge]

yeah, what are all the different cases for L you want to consider?

obviously we can't list them all out one by one

[oblique rover]

and if its ouside of [-1,1] then x from reals

[oblique rover]

if its ouside of [-1,1], if its [1;0) and if its [0,-1]

[dire forge]

okay, so you said if L is not in [-1,1] then x is from the reals, can you elaborate on that

[oblique rover]

because for any x from reals sinx wont be fully contained in U_epsilon(L)

ahh no

or yes

[dire forge]

well choose a number for x

remember x has to be a specific number in P_delta(infinity)

[oblique rover]

hm so then it just needs to be number greater then delta?

[dire forge]

okay which one?

[oblique rover]

tbh i kind of got lost wdym which one

[dire forge]

what value are you choosing for x

you need to pick a specific number

because the statement we're trying to prove is "there exists x such that ..."

[oblique rover]

ah so for example x=kpi?

[dire forge]

yeah good, why does that work

[oblique rover]

because its in distance of 1 from 1 and -1 thus i can not be in 0.5 neighbourhood of numbers outside of [-1,1]

[devout snow]

@oblique rover Has your question been resolved?

[devout snow]

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[devout snow]

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[tender reef]

Im looking for a good simple video i have this chapter that looks easy but also easier to get wrong i tried to look on youtube but all are very complex what im looking for exactly is explanation for Basic concepts of set theory

[tender reef]

Its a simple sheet but i would love explanation so i can study and pratice

[devout snow]

@tender reef Has your question been resolved?

[tender reef]

<@&286206848099549185>

[frosty portal]

so wht do u need help with

@tender reef

[devout snow]

@tender reef Has your question been resolved?

[frosty portal]

@tender reef

[tender reef]

I would like source explanation on basic cocepts of set and theory

Alot of videos in youtube are all complex videos if possible i only want to understand the basics like it shown in the sheet i sent

[supple knot]

https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/01%3A_Tools_for_Analysis/1.01%3A_Basic_Concepts_of_Set_Theory

[frosty portal]

so first we need to get clear on wht is a "set"

[devout snow]

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[rugged jewel]

How is it x(1+... and not x(1-... here?

[rugged jewel]

Probs something very basic but i dont see it

[carmine burrow]

yeah basically it's because you suppose x<0 here, so sqrt(x^2+4x) = -x sqrt (etc) for x small enough

[rugged jewel]

ah okay

|x| becomes -x instead of x

makes sense ig

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[stiff spire]

Help

[unreal thunder]

@stiff spire

[stiff spire]

Yes

@unreal thunder

[unreal thunder]

Which values are plotted on the horizontal axis ?

[stiff spire]

WTf is a horizontal axis 😭😭

[unreal thunder]

x- axis

—- (horizontal)

[stiff spire]

Idk what ur sbt styll

[unreal thunder]

| (vertical)

[stiff spire]

Ohhhh

Im a bit dumb like dat

Thsnks

[stiff spire]

A and B

[unreal thunder]

Do you see time should be plotted in horizontal axis?

[stiff spire]

Yhyh

I know A is 20, my man told me it 😄

[devout snow]

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[devout snow]

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[stiff spire]

nope

[devout snow]

@stiff spire Has your question been resolved?

[stiff spire]

Nope

[devout snow]

@stiff spire Has your question been resolved?

[stiff spire]

NO

[jade oak]

The scale is going to be linear, so just count how many squares away B is from 0, then multiply by 20 (since you know A is 1 square away and represents 20 seconds)

big dark squares

not the smaller lighter squares, if you count those instead each one represents 4 seconds (by dividing 20 by 5 since there are 5 small squares in each big square)

[stiff spire]

Its not 200 tho

wait what.

[granite shuttle]

not with that attitude!!

[stiff spire]

huh

[devout snow]

@stiff spire Has your question been resolved?

[devout snow]

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[rigid ore]

hello

[rigid ore]

so i have a question

First some quick vocab:

Giant Mantis Tame Chance: 11%

1 Elixir Of Influence = 5%

What is the exact probability of failing 2 times in a row the taming of a giant mantis with only 3 elixers of influence to help?

To solve this, you need to add 5 times the number of elixirs to the tame chance of the animal, then subtract it from 100, and divide it by 100. Then, attach an exponent to the number (the exponent is the failed tames in a row) and multiply it by 100 for your final result. (You may need to round to the tenths place) (THIS IS FROM THE WIKI)

(and this is for a game)

This question is multiple choice so here are the possible answers

A) 0%

B) 94.9

C) 35.2

D) 54.8

[winter torrent]

it sounds like you have a set of directions

have you tried following them?

[rigid ore]

yep but i got a number way off ;-;

[winter torrent]

okay, can you show what you did?

[rigid ore]

i dont have a picture i just typed the numbers down and kinda erased it ;c

like i did the math in my head and put the number down

and once i was done with it i put another number

i actually had a similar question for this

ill send that and the answer

[winter torrent]

how about you go ahead and redo the problem and keep track of what you're doing

[rigid ore]

k

brb

.close

[devout snow]

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[rigid ore]

.reopen

[devout snow]

✅

[rigid ore]

ok

i think i might have an answer

is it 54.8

does that seem reasonable

ill just close it ig

.close

[devout snow]

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[winter torrent]

seems good to me

you have a 1/4 chance of catching it each time

so around 1/2 in two tries

[devout snow]

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[pale prawn]

how do i do this?

[safe jasper]

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

[pale prawn]

i dont understand

[olive snow]

Remark : 4x² = (2x)² and 9 = 3²

[devout snow]

@pale prawn Has your question been resolved?

[pale prawn]

oh okay thank you

[devout snow]

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[novel violet]

y = 4sin3x + 2

Find the smallest zero in the interval 2pi <= x <= 3pi

[novel violet]

I know that x = 7pi/6 and 11pi/6 between 0 and 2pi, but I dont know what to do next

And I must know how to do it algebraically

I know the new modified domain is 6pi to 9pi

But how do I make my current solutions into the new domain

PING IF HELPING

[idle phoenix]

Zero you mean y = 0?

[novel violet]

Yes

[idle phoenix]

If that so, 4sin3x + 2 = 0 implies that sin3x = -1/2
So, 3x needs to be 7pi/6 + 2k . pi or 11pi/6 + 2k . pi, where k is an integer, so you have to find k such that 3x is between 6pi and 9pi

Let 7pi/6 + 2k . pi >= 6pi, so 7/6 + 2k >= 6 -> 2k >= 29/6 -> k >= 29/12

You have to do that for 7pi/6 + 2k . pi <= 9pi

And do the same for 11pi/6 + 2k . pi

[novel violet]

Thank you 🙏

[devout snow]

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[late ruin]

can i get some help on smth

[late ruin]

wait

its

4

?

[supple knot]

,calc (4-1)^2

[woven radish]

Result:

9

[supple knot]

No

[late ruin]

well

uh

hold on i think i got it

5

or

-3

[supple knot]

Yea those are both right

[late ruin]

okay thanks

.close

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[devout snow]

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[peak blade]

Guys what do I do next

[peak blade]

I got the values of x

but I have a confusion of the signs

[supple knot]

set the left side equal to 0 and use quadratic formula or factor or complete the square to find the roots

quadratics/parabolas have a very specific pattern to being above the x-axis

[devout snow]

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[frozen stump]

for part a, do I integrate as usual and then sub in two cases where n--> inf and n--> 0?

[supple knot]

Drawing a picture might help

[frozen stump]

iv never had to deal with upper and lower rectangles before, so im not quite sure what it really is

After some research i assume this is what you mean

How do I implement this as an actual formula tho?

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@frozen stump Has your question been resolved?

[frozen stump]

<@&286206848099549185>

[outer ember]

Consider integrating 1/2^n and 1/2^(n+1)

[supple knot]

Use the bounds from your own integral

It's just one rectangle for upper and one rectangle for lower

[frozen stump]

Is this alright?

[frozen stump]

how do I "hence" for part b?

[outer ember]

You can expect this integral would larger than n/2

[frozen stump]

?

[outer ember]

Integrate from 1~2, 2~4, 4~8,....2^(n-1)~2^n. Each part would larger than 1/2.

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[frozen stump]

.close

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[vivid estuary]

What is the roll of nCr(1,1)

[sand quarry]

you mean "role"?

[vivid estuary]

ye lol

[sand quarry]

If your question is why they wrote it this way then it is just to illustrate the point they're trying to get across because 1C1 = 1. This probability model is what's called a hypergeometric distribution. Basically, the hypergeometric distribution describes the probability of drawing a specific number of successes in a series of draws from a finite population without replacement, which is what the question is describing i suppose

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[vestal bay]

I found that <CAF is 50 using alternate segment theorem and <ACD=50 alternating angles

[vestal bay]

I don't know what to do next to find <DCB

[tall stirrup]

Alternate interior angles

CD is bisecting <ACB

so, I think you have your answer

[vestal bay]

I don't know what <ACB is

[tall stirrup]

No need of it

since CD bisect ACB

<BCD = <ACD

[vestal bay]

Oh okay

thankss

.close

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[potent onyx]

hi

[potent onyx]

I want help in AP

here is the problem

can anyone tell me the solution + explain this

<@&286206848099549185>

hi @sullen island

[sullen island]

yeah you want to use the formula for the sum of an AP certainly

that formatting is so cursed tho

holy

[potent onyx]

same reaction

i use all the formula of AP given in my book

and my brain stopped braining

Sn = n/2[2a + (n-1)d]

this one ?

or the other one

[sullen island]

that should do it yes, a is the start and d is the difference right

[potent onyx]

can u help me with one more?

[sullen island]

ok

[potent onyx]

yes

can u help me with both?

or i have to wait for 4 jan for the answer key

[sullen island]

well one at a time but sure

[potent onyx]

ans for 2nd one is 3

[sullen island]

so you ok with the first one now?

[potent onyx]

no

[sullen island]

ok

[potent onyx]

help me with 1st one

it is okay if you dont help with 2nd one

cus I got the solution on google for 2nd one

[sullen island]

well ok what's 2+5+8+... using that formula then ?

[potent onyx]

if we use that formula

we need N

or Sn

if Sn is 23

[sullen island]

well n is unknown as of now

that's what we're trying to find remember

so just compute it for any n

[potent onyx]

lets go for 1

[sullen island]

keep the n, just plug in a and d

[potent onyx]

ok

Sn = n/2[1+3n]

[sullen island]

you sure a is 1 ?

[potent onyx]

no

i solved in my copy

a is 2

d is 3

[sullen island]

yeah ok

well 2a is 4 then

not 1

[potent onyx]

but

i solved that already

4-3

[sullen island]

ok nevermind can't read

sure