help-10

[timid silo]

Thx guys

.close

[obtuse pebble]

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[humble dagger]

does anyone know how to solve this one?

[limpid oar]

what is cot?

[teal turret]

what have u tried @humble dagger

[humble dagger]

co tangent

[limpid oar]

which is?

[humble dagger]

we use the u substitution for y right ?

y=u^2

[limpid oar]

not necessarily, how could you express cot in some other way?

[humble dagger]

derivative of cot is -csc^2

the professor doesn’t cover material too well in class

[limpid oar]

how could express cot(x) in terms of sin(x) and cos(x)

[humble dagger]

cos(x)/sin(x)

[limpid oar]

what rule could you use to solve the derivate of that

[humble dagger]

ohh quotient rule?

lo d high high d lo /lo ^2

thank you!

[limpid oar]

are you given a formula sheet at all for exams?

[teal turret]

@limpid oar why would u even turn to sin and cos if they know this

no need for quotient rule

[limpid oar]

i somehow completely missed that lol, i asked for the formula sheet just in case they were already given derivate of cot and thought this excercise was to practice for rules

my bad @humble dagger ! may have given you a bit more extra work to do haha

[obtuse pebble]

@humble dagger Has your question been resolved?

[humble dagger]

oh okay yeah i was a bit stuck

we’re not provided a formula sheet ;_;

[limpid oar]

if you have the derivative of cot(x) = -cosc^2,
and you have cot(x)^2 what rule could you use to differentiate that

[humble dagger]

power rule?

[obtuse pebble]

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[timid silo]

We need to find the next missing number

There's a pattern in the sequence of number

[stable wyvern]

is this an arithmetic or geometric series?

[timid silo]

Dunno

We need to find it ourselves

[stable wyvern]

you can use a sereis to help you find the next term

or you can count

,w arithmetic series

[warm shale]

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[stable wyvern]

,geometric series

,w geomtric series

[warm shale]

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[timid silo]

I dunno what you're doing

[stable wyvern]

I meant geometric and aritrhmetic sequences** not series

sorry

[timid silo]

Oh

,w geometric sequence

[obtuse pebble]

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[charred elbow]

Heya, so i tried the integral of $sin^4(x)$ and was off by a bit, Answer from wolfram alpha is $$\frac{sin(4x)-8sin(2x)+12x}{32}$$

[warm shale]

Hemesfere

[lone oyster]

I would've used reduction formulae

Is there a reason you've done it like this?

[strong pike]

I think you lost a *2 in line 4 -> 5 but you shouldve just used product-to-sum or power reduction in the first place

[charred elbow]

Idk how to do it in any other way tbh

[charred elbow]

Ah yes, cuz its 2×e.. and e..×2

Ty

[lone oyster]

Reduction or p2s formulae

I would've thought you'd learn that well before this

[charred elbow]

Im learning integral techniques from yt videos so the teaching isn't exactly orthodox😭

[charred elbow]

Idk either of these rip

[lone oyster]

Ah I see I see, impressive work then

[lone oyster]

Would make your life easier I promise 🙏

[charred elbow]

I think it's a rlly nice substitution tho cuz i rlly dislike trig identities😭
But ig it wouldn't work for stuff like tan or sec cuz that's division

[charred elbow]

Ill look into them ty

[remote skiff]

You made your calculation more error prone by not structuring it like you would in the binomial theorem

Write down (e^ix - e^-ix)^4 and make sure your first step is such that you get the 1 4 6 4 1 coefficients in your next step before simplifying.

[obtuse pebble]

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[hybrid mortar]

so i don't get how i should find the domain of the graph

[woeful void]

What is the definition of domain?

[hybrid mortar]

its the input

[woeful void]

Right, what do you think the domain is then?

First, what are the smallest and largest x-values?

[hybrid mortar]

-8, and

9

[woeful void]

Yes. Are those inclusive?

[hybrid mortar]

no?

[woeful void]

Well, it is a black dot

Not an open circle

So it includes the point

[hybrid mortar]

oh

[woeful void]

Yeah.

Now, let's check for its continuity. Is it everywhere continuous?

On it's domain

Like are there any breaks?

Or discontinuities?

[hybrid mortar]

there aren't

[woeful void]

Yeah. So what do you think the domain would be?

[hybrid mortar]

im kinda new to this, so uhm i was wondering if you could explain?

[woeful void]

Right. We know that it is continuous, right?

[hybrid mortar]

yes

[woeful void]

on it's domain

And, the end points are inclusive.

So, we would use the notation [-8,9] to denote it's domain.

Note that there are two types of brackets

[] and ()

() is used when we are not including that point

For example, if it was (-8,9) then we are not including x=-8 and x=9.

But [] means that we include x=-8 and x=9.

[hybrid mortar]

ohh, thank you

[woeful void]

np.

Please close the channel if you have no more questions!

Good luck!

@hybrid mortar

Please close the channel.

Thanks a lot!

Close it if you don't have any more questions 🙂

[hybrid mortar]

.close

[obtuse pebble]

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[hybrid mortar]

sorry

[obtuse pebble]

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[tender tusk]

what do u need help with

[true trail]

I am confused with angle

6i+8j
Cos theta and
-6i-8j cos theta are same?

[obtuse pebble]

@true trail Has your question been resolved?

[true trail]

@tender tusk

[tender tusk]

oh yo

aight notice how -6i-8j is in the third quadrant

$\cos(\theta)=-\frac{3}{5}$ as cos is negative in the third quadrant

[warm shale]

Galaxy

[tender tusk]

hence $\theta=\arccos(-\frac{3}{5})$

[warm shale]

Galaxy

[tender tusk]

.

[true trail]

Thank you

.close

[obtuse pebble]

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[plain nebula]

how do u solve for x

if y=0

[deep skiff]

is y = 3e^-x

?

if so then you have

3*e^-x = 0

clearly from there you have

e^-x = 0

now look at that carefully and think

[obtuse pebble]

@plain nebula Has your question been resolved?

[plain nebula]

ohh okay

thanks

.close

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[austere sleet]

!help

[obtuse pebble]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[austere sleet]

hi

help

[obtuse pebble]

@austere sleet Has your question been resolved?

[obtuse pebble]

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[solemn bough]

Hi i have the following pproblem which i have to describe as i dont know the name.
i want a formula which converts a slope degree to a speed multiplier.
the values look like this:
-45 = 2
0 = 1
45 = 0.5
it has to be linear
is this a linear equation?

[tardy tusk]

Well it's increasing

and actually the formula of linear is

mx+b

Where m = (y2-y1)/(x2-x1)

Where x2, y2 is poins that are more right

b=1 ig or put x and y and m to find

[solemn bough]

maybe what i need isnt a linear equation?
im very bad with higher math stuff tbh

[tardy tusk]

Me either

[solemn bough]

Its for my game project and mapping every value pair would be a real pain.
So im trying to find out if there is a methematical solution to my problem.
But it also seems like i completly missunderstood what linear equations are 😅

[tardy tusk]

I tried

[tardy tusk]

And you can write it as x/90 + 1

It is linear

[tardy tusk]

Just use this

To find

[solemn bough]

oh let me try this out

i tried x/90 + 1
well but now 45=1.5 and -45=0.5
but 45 should be 0.5 and -45 should be 2

[tardy tusk]

What?

[tardy tusk]

Here

You literally said

-45 is 0,5

0 is 1

45 is 2

Maybe just put -

Ok one sec

Hm

Strange

I tries -x/90 +1

And it's
-45 1,5
0 1
45 0,5

Okay

It's strange

I think it's not linear

Or

It's piecewise function

And it's the combination of two functions

[solemn bough]

youre right i messed up my first post

corrected that

[tardy tusk]

We can say that it's combination of linear functions

But when second function is less than 0

[solemn bough]

Oh wow thats strange.
How to handle something like this?

[tardy tusk]

Idk it's exactly like that

Like points are the same

But to write it as one function

I don't know

Don't have such a knowledge

[solemn bough]

i dont know if its even possible, maybe i should first check the input value and branch then

[tardy tusk]

It's possible if it's piecewise function

Actually we can express any shit graph as some function

[solemn bough]

i found something about unit step functions

Which you may use to find a function identical to
1 on an interval of the form (a,b) and 0 elsewhere:

i dont get that...
im a programmer

I think that the way im doing it, looks about right to me

the output of >= 0 seems fine, but < 0 looks wrong
output is at the top left its always angle followed by speed modifier

[obtuse pebble]

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[pine glacier]

how do i elimate t to make the function nice to graph x= cos(t)+1 , y=sin(t)-1

[opaque idol]

isolate t in each equation

[pine glacier]

im trying to sketch the vector function r(t) = (cos(t)+1)i + (sin(t)-1)j

[pine glacier]

ye but how do i do that without making it hard to graph

[fathom flicker]

x-1=cost

y+1=sint

So…

[pine glacier]

kinda looks like a circle?

would it be (x-1)^2 + (y+1)^2 = 1?

[wary badger]

yea you just square both sides and add the equations

[pine glacier]

got it

ty

[fathom flicker]

🔠

[lyric basalt]

What does this mean?

[pine glacier]

its the first 4 letters of the english alphabet

[lyric basalt]

but why

what does it imply @pine glacier

[fathom flicker]

Always be clearin doubts

Means like question resolved

[pine glacier]

i be doubting

[teal turret]

https://tenor.com/view/rememeber-the-titans-leave-no-doubt-run-it-up-gif-15050615

[fathom flicker]

We all be

[teal turret]

https://tenor.com/view/zucchini-fnaf-minecraft-zucchini-fnaf-minecraft-fnaf-minecraft-gif-2064541954878560303

JIMMY

@pine glacier u have any more questions?

[pine glacier]

not atm

[lyric basalt]

Ty

[obtuse pebble]

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[obtuse pebble]

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[short orchid]

Can someone help me with this?

[true trail]

Multiple channels?

[vestal thorn]

the other one is technically closed

[short orchid]

Sorry I’m new to this

[crude comet]

what is the maximum value of sin(k)?

[short orchid]

honestly no clue

3?

Since its peak is (0,3)

[crude comet]

so it is possible for sin(k) to = 3?

can you think of a number k that makes sin(k) = 3?

,w graph sin(x)

[warm shale]

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[crude comet]

the maximum is 1

likewise, the maximum value of sin(pi/14(x+8)) is 1

[short orchid]

is it

just + 3?

[crude comet]

wdym + 3

[short orchid]

Wait sorry

[crude comet]

the maximum value of

sin (anything) is 1

[short orchid]

Right so it can’t equal 3 right for any real number

[crude comet]

yes

if you want it to be 3

you must multiply sin(x) by a number

if the maximum value (or amplitude) of sin (x) is 1, then multiplying sin(x) by what will make the amplitude 3

[short orchid]

You multiply sin x by 3 to make it 3sin(x)

[crude comet]

yes correct

,w graph 3sin(x)

[warm shale]

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[crude comet]

notice, if you have 3sin(kx), the frequency of the wave will depend on k

every time kx increase by 2pi is one cycle, right?

[short orchid]

Right

[crude comet]

ok

so lets say k = 50

a small value of x will be required to make kx increase by 2pi

so the frequency will be higher

,w graph 3sin(50x)

[warm shale]

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[crude comet]

,w graph (3sin(50x)) 0<x<2pi

[warm shale]

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[crude comet]

the frequency is very high here

compared to

,w graph 3sin(x) 0<x<2pi

[warm shale]

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[crude comet]

likewise, we can shift the wave left or right

do you know how to shift a function left or right?

lets say there is a function f(x)

do you know how to shift f(x) left or right?

[short orchid]

it’s

f(x-a value) for right

[crude comet]

yes

correct

[short orchid]

And left it’s f(x+a) for left

[crude comet]

correct

so the general equation for a wave is given by

$A\sin(bx + c)+k$

A denotes the amplitude

well

one sec

[warm shale]

hrdxpqurcxktdbanql

[crude comet]

A denots the amplitude

b represents the angular frequency (not frequency, that is different), c is the shift left or right, and k is the shift up or down

From the picture it is clear A = 3

Is the function shifted at all left or right?

[short orchid]

Left

[crude comet]

ok how far left

[short orchid]

I believe it’s 8 which is why I did x+8

[crude comet]

,w graph sin(x+8), -8<x<0

[warm shale]

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[crude comet]

notice, if it is shifted left 8, then your sin function starts at 0 at x = -8

[short orchid]

so the value is incorrect?

[crude comet]

yes its incorrect

,w graph sin(x)

[warm shale]

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[crude comet]

we know that a cycle of sin(x) starts at a point where y = 0 and sin is increasing

as seen in the image

so where does the graph = 0 and increase here?

[short orchid]

There’s no point where the graph equals 0

it’s like 2.75 when it crosses over the y axis

[crude comet]

sorry

i mean

where does the y value of the graph = 0 and increase?

[short orchid]

I want to say at -11.5

[crude comet]

ok very good

so we know c =?

[short orchid]

-11.5

[crude comet]

is your function shifted left or right?

[short orchid]

Left

[crude comet]

ok

so is c positive or negative

[short orchid]

Negative

-11.5 like we said

[crude comet]

,w graph f(x-20) = (x-20)^2 , 0<x<30

[warm shale]

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[crude comet]

is thsi function shifted left or right?

[short orchid]

Right ✅

[crude comet]

ok

its shifted right 20 units correct?

[short orchid]

Correct

[crude comet]

ok

so we know if we have a function f(x), then f(x-20) will be 20 units right

right?

[short orchid]

yes

[crude comet]

ok

so if our function is sin(x-11.5), will it be shifted left or right

[short orchid]

11.5 to the right

[crude comet]

ok

so if c = -11.5

then

is this function shifted left or right

[short orchid]

Right

[crude comet]

ok

[short orchid]

let’s go bro

we’re getting somewhere

I’m hyped

[crude comet]

but we know the graph in your homework is shifted left 11.5

not right

so is c=-11.5?

[short orchid]

oh dear

hmm

wouldn’t it be positive 11.5 then..

[crude comet]

yes

correct

so c = 11.5

sorry

[short orchid]

all good

[crude comet]

is the graph shifted up or down at all?

[short orchid]

yes I mean it goes down to -3 and up to 3

[crude comet]

yes

youre right

let me ask this

,w graph sin(x)

[warm shale]

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[crude comet]

is this graph shifted up or down?

ie would it be written as sin(x) + k?

or is it sin(x) + 0

[short orchid]

sin(x) + 0…

[crude comet]

yes

obviously i wrote it lol

[short orchid]

😂

[crude comet]

consider the case where sin is shifted up or down

,w graph sin(x) + 4

[warm shale]

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[crude comet]

does your graph look like that?

[short orchid]

Wouldn’t it be + 3?

or am I tripping

[crude comet]

ok

lets see what +3 looks like

,w graph 3sin(x+11.5) + 3, -30<x<0

[warm shale]

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[short orchid]

Uh oh

Doesn’t look like mine

[crude comet]

it doesnt

note, the amplitude is the maximum the wave can increase

+3 adds 3 to the maximum and subtracts 3 from the minimum

we know the amplitude here is 3

so if it was shifted up 3, then the graph would touch y = 6, right?

[short orchid]

yes

[crude comet]

ok

in the graph, the amplitude is 3 and the max it touches is y = 3, so the shift up or down is ___

[short orchid]

0

[crude comet]

correct

so we know

A = 3, c = 11.5, and k = 0

all we're missing is b

note

the equation is now

$f(x)=3\sin(bx+11.5) +0$

[warm shale]

hrdxpqurcxktdbanql

[crude comet]

b is the angular frequency of the wave

it determines how quickly the part on the inside increases by 2pi or one cycle, correct?

[short orchid]

yes

Correct

[tacit vale]

THERES A PARADE INSIDE MY CITY YEAHHHH

[crude comet]

consider the case:

[short orchid]

😂

[crude comet]

,w graph 3sin(50x+10) 0<x<2pi

[short orchid]

@tacit vale focus brother

[warm shale]

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[crude comet]

,w graph 3sin(1x) 0<x<2pi

oops

[warm shale]

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[crude comet]

clearly, what is multiplied by x determines the frequency of the wave, or how quickly the wave goes through one cycle, right?

[short orchid]

yessir

[crude comet]

ok

so how long does it take for this wave to go through one cycle?

aka

[short orchid]

I see 4.5 to 11.5

So 7

[crude comet]

how long does it take for a y value to repeat twice**

[short orchid]

-7 technically

[crude comet]

you right

but

consider ,w graph sinx

,w graph sinx

[warm shale]

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[crude comet]

how long does one cycle take?

[short orchid]

1 pi

[crude comet]

ie how long before the graph repeats itself??

are you sure?

so after pi, the graph repeats?

[short orchid]

it’s 2pi

[crude comet]

,w graph sinx, 0<x<pi

yes

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[crude comet]

thats correct

so how long does a cycle take here

[short orchid]

14

[crude comet]

ok

we know

for any waveform

say cos(n)

every time n increases by 2pi, the cycle repeats right?

[short orchid]

yes

[crude comet]

ok

so every time kx increase by 2pi, the cycle will repeat, right?

[short orchid]

yes right

[crude comet]

ok

so we know

every time x increases by 14, the cycle repeats

[short orchid]

yessir

[crude comet]

ok

so every k(14), the cycle repeats, right?

[short orchid]

indeed

[crude comet]

so this occurs when 14k=2pi

can you solve for k?

[short orchid]

k should be pi/7

[crude comet]

ok

so we know b = pi/7

what is the equation?

f(x) = ??

[short orchid]

3sin(pi/7x + 11.5) + 0

[crude comet]

ok correct

[short orchid]

no way

We did it?

[crude comet]

yes

it should be correct

lets see

,w graph 3sin(pi/7x+11.5) ,-20<x<0

[warm shale]

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[short orchid]

YOOOOO

[crude comet]

it is incorrect

are you sure the function begins at -11.5?

[short orchid]

It begins at 4.5..

aha

[crude comet]

ok

so

,w graph 3sin(pi/7x+4.5) ,-20<x<0

[warm shale]

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[crude comet]

oh wait

[short orchid]

do I dare say it starts at 2.75?

[crude comet]

yes

2.75?

you sure?

,w graph sinx

[warm shale]

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[crude comet]

letme try again

[short orchid]

I’m stumped

[crude comet]

,w graph 3sin(pi/7x+4.5) ,-20<x<0

[warm shale]

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[crude comet]

thats weird

let me check on desmos

[short orchid]

oh no

[crude comet]

one second

[short orchid]

No worries

[crude comet]

wait i think i know

we can try to solve it a different way

we have

$y=3\sin(bx+4.5)$

[warm shale]

hrdxpqurcxktdbanql

[crude comet]

what is the value of the graph at -15?

[short orchid]

(15,3)

[crude comet]

ok

[short orchid]

-15,3

[crude comet]

its 3

so

we know when y = 3, x = -15

can you plug it in here and solve for b?

[short orchid]

yes

[crude comet]

ok

[short orchid]

3 = 3sin(-15b + 4.5)?

[crude comet]

I think i told you incorrectly how to find b initially

yes

[short orchid]

b is long integer though

if I’m not mistaken

[crude comet]

h wait

no we had it right

its just the graph on wolfram

the x axis is in terms of pi

,w graph 3sin(pi/7x+4.5) ,-20<x<0

[warm shale]

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[short orchid]

3sin(pi/7x + 11.5) + 0

^?

[crude comet]

it still looks wrong tho

,w graph 3sin(pi/7x+4.5pi) ,-20<x<0

[warm shale]

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[crude comet]

I see

[short orchid]

AH

[crude comet]

that makes sense

to shift it one unit left, you have to do f(x+pi)

since if x = -pi then sin(0) = 0 ????

doesnt make sense

let me try another grapher

[short orchid]

Alright

[crude comet]

<@&286206848099549185> i need help

,w graph sin(x-3pi)

[warm shale]

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[crude comet]

,w graph sin(x-pi)

[warm shale]

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[crude comet]

,w graph sin(pi*x-pi)

[warm shale]

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[short orchid]

oh dear

[crude comet]

i just googled it

[short orchid]

what was the result

[crude comet]

the phase shift depends on

,w graph 3sin(pi/7x+2pi/4.5) ,-20<x<0

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[crude comet]

1 sec

I see

finding the phase shift isnt so simple

ϕ=(xpeak−P/4) for the sine graph

$c=x_{peak}-period/4$

[warm shale]

hrdxpqurcxktdbanql

[crude comet]

the period is 14, that is how long it takes

where do the peaks occur?

at what x value

@short orchid

[obtuse pebble]

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[ripe hazel]

im not in physics i dont get it

[teal turret]

u dont need to be in physics

it gives u a formula

v = sqrt(2gh)

u know g and v

just solve for h

[ripe hazel]

oh ok

[obtuse pebble]

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[static pawn]

.

[static pawn]

how do i ge tthe derivative of this

[obtuse pebble]

@static pawn Has your question been resolved?

[worn yoke]

product rule

[obtuse pebble]

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[burnt anvil]

i need help

[burnt anvil]

we have two functions

f(x) = -x^2 + 5x

g(x) = -2x + 15

the distance A depends on the value of x

determine A as a function of x

[tranquil beacon]

You would just need to take the difference of the two functions

so g - f

[burnt anvil]

why

oh

A = g - f

[tranquil beacon]

yes. Just the standard distance formula

[burnt anvil]

oh damn alright

thx for the help

.close

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[shell oar]

how is this thing a vector???

[pseudo swift]

M is a matrix

e_0 is a vector

[shell oar]

yh

[shell oar]

does a matrix times a vector always give a vectir

vector

[pseudo swift]

yeah?

[shell oar]

ok think my mind just went blank

thanks

.close

[obtuse pebble]

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[pseudo swift]

.reopen

[obtuse pebble]

✅

[pseudo swift]

it's been a while you've been asking about that thing, no ? @shell oar

[shell oar]

wdym

[pseudo swift]

idk I remember seeing that exact picture a few weeks ago

maybe it wasn't from you

[shell oar]

i think i asked this before yh

a month or so ago

just doing some revision

and forgot something

[pseudo swift]

ah ok fine fine

alright I'll reclose then

.close

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[whole jewel]

Hi, I'm trying to collect the likes terms x^2 and 16x^2/9. I'm not sure how to combine them, some help would be really appreciated!

[bronze nova]

are u trying to add the terms?

[whole jewel]

yeah

[bronze nova]

u factor out x^2

u get x^2(1+16/9)

=x^2(25/9)

since x^2 is the like term, u can factor it out

[whole jewel]

oh awesome thanks

[whole jewel]

why do you add a 1 in the brackets?

[bronze nova]

x^2+16x^2/9 = x^2 (1+16/9)

when u factor out x^2 from x^2, u r left with 1

cuz 1*x^2 = x^2

[whole jewel]

oh ok thanks

[whole jewel]

and how did you end up with 25/9

[bronze nova]

1 + 16/9 = 25/9

[whole jewel]

bedmas?

[bronze nova]

yes

[whole jewel]

sorry still confused how you got the 25

[humble valve]

1 = 9/9 so then 9/9 + 16/9 you just add the top part straight across which is 25/9

[whole jewel]

oh right thanks

.close

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[twilit dove]

I have the answer

remainder is 0

quotient is the anser

but what do I fill out for f(x)=

[tame narwhal]

if the remainder is 0, it's just a factorization

so f(x) = g(x) * q(x)

[twilit dove]

what if it wasn't 0

x^2-2x-4 is my answer

(x-3)(x+1) or something like that would be factored

not factorable

,rotate

[tame narwhal]

if the remainder wasn't 0, I think it'd be f(x) = g(x) * q(x) + r(x)

[twilit dove]

g and r being?

[tame narwhal]

you're given g. r would be the remainder

[twilit dove]

okay I see

maybe it's factored form like this?

[twilit dove]

would I do the multiplication or just leave it as an equation

[tame narwhal]

if you do the multiplication you just get f back

[twilit dove]

I see

[obtuse pebble]

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[vague violet]

is this answer right?

[obtuse pebble]

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

[vague violet]

/close

.close

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[languid cloud]

Whom needs help

[obtuse pebble]

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[obtuse pebble]

Please don't occupy multiple help channels.

[vague violet]

this one hella confusing

cause isnt it top minus bottom?

[tame narwhal]

<@&268886789983436800> troll

[vague violet]

here lemme write it out

@tame narwhal this is what i thought

[tame narwhal]

that's what I'd put

[vague violet]

ok

isnt this one B

@alpine verge thats right?

[alpine verge]

yes

[vague violet]

ok thank you

[alpine verge]

yw!

[vague violet]

this is right too?

sorry im just making sure

[tame narwhal]

yeah, looks right

you can always check by taking the derivative of your answer choice

[vague violet]

yea thats what i thinking

preciate it

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[cobalt aspen]

Can someone help me graph this?

[shell scarab]

make it into a sine wave first

[cobalt aspen]

.close

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[lilac kayak]

This is a question from the book Putnam and Beyond. I can't seem to wrap my head around the solution they provided. I know you are supposed to prove this by contradiction and understand the assumptions they made. But I don't get how S1 ∪ S2 is greater than 1+8/9 and the following logic. Can somebody explain their solution?

[obtuse pebble]

@lilac kayak Has your question been resolved?

[obtuse pebble]

@lilac kayak Has your question been resolved?

[lilac kayak]

<@&286206848099549185>

[obtuse pebble]

@lilac kayak Has your question been resolved?

[stoic yacht]

assume that the statement is false, that is that no two surfaces overlap their area by >= 1/9

then if you take any two surfaces and combine their areas with each surface having area 1, then assuming the surfaces perfectly overlap this combined area is 1, and if they don't overlap at all then their combined area is 2

now their combined area decreased the more that they overlap

let's say they overlap by exactly 1/9, then their combined area has to be 2 - 1/9

right?

[lilac kayak]

makes sense

[stoic yacht]

okay that's where the 1 + 8/9 comes from for two surfaces, and since their overlap is supposedly less than 1/9, their combined area > 1 + 8/9

now you take the third surface, and let's assume it overlaps by exactly 1/9 with each of the two other surfaces, then their combined area must be greater than the 1 + 8/9 of the two surfaces plus another 7/9 (the area 1 from the third surface minus the 1/9 overlap with each of the two other surfaces)

which means that their combined area of the three is at least 1 + 8/9 + 7/9

all they've done is repeat this reasoning and use the worst case scenario that each surface has just marginally under 1/9 overlap with each of the other surfaces

then they get the contradiction, and i think you got it from there

[lilac kayak]

ahh okay

the contradiction is that the combined area turns out to be over 5

right?

[stoic yacht]

yes

and the problem tells you it's exactly 5

[lilac kayak]

yes got it

thanks mate

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[graceful ridge]

does this converge or diverge?

[bronze nova]

what do u think

[graceful ridge]

Because I just stumbled on a way to make it converge on my own, idk if its valid though, say y = e^1 + e^2 + e^3 +... or y = e (1 + e^1 + e^2 + e^3 + ...) but wait wait that's just y = e (1+y), so y = e/1-e, LMAO Hilarious result

[bronze nova]

u cant manipulate divergent series like that

[tacit leaf]

Let's use the graph epower x

[graceful ridge]

it's the same kind of result as 1 + 2 + 3 + 4 + ... = -1/12

[tacit leaf]

Integration will help

[graceful ridge]

umm it follows the laws of algebra, what's the problem with it? It's just a very weird result though

[tacit leaf]

How can you do that

[graceful ridge]

Apparently it's called Borel regularization

[tacit leaf]

Their is no multiplication

[graceful ridge]

I just looked it up

[graceful rapids]

and a series that converges to infinity = -1/12 is obviously nonsensical
in fact it's a good example of why manipulating infinite series in general is nonsensical

[crude comet]

your reasoning seems fine

[tacit leaf]

Leting a = e^1+ e^2.....

Now taking e common

So

[bronze nova]

assigning finite values to a divergent series is already nonsensical

[tacit leaf]

e(e^0+e^1+e^2...)

[crude comet]

e^1 + e^2 + e^3 = e (1 + e^1 + e^2 + e^3 + ...) = e (1 + e (1 + e^1 + e^2 + e^3 + ...)) = e (1 + e (1 + e^1 + e^2 + e^3))

lol

[tacit leaf]

Or we can integrate e from 1 to infinity

As the area of the graph is the answer

[timid silo]

The value keeps increasing...so duh

Diverging

[molten grove]

yes it's true for any finite stopping point, it's literally just a geometric series with common ratio e. But this isn't a finite series you're asking about, it has no stopping point

[graceful ridge]

I just did some math, and it's still the same result that way too

[molten grove]

and if you're familiar with the fact that geometric series diverges for common ratio geq 1

[tacit leaf]

E power infinite is infinite so answer is diverging

[tacit leaf]

The answer is diverging

[graceful ridge]

What I wanna know is what mistake I made here that led to this nonsensical result.

[limber pelican]

Since e^x is a function you can use the integral test

[molten grove]

you assigned the series to the value of y, doing so assumed the series converges in the first place which it does not

[limber pelican]

You will see that it diverges once you integrate it

[crude comet]

y = e(1+e(1+e(1+e......)

[graceful ridge]

now divide by e both sides, you get y/e = 1 + y

[molten grove]

please read this sir @graceful ridge

[crude comet]

y/e = 1+e(1+e(1+e(1+e...

[molten grove]

my brain cells please

[crude comet]

y/e - 1 = e(1+e(1+e).....

[graceful ridge]

oh okay, so just avoid assigning values to geometric series when they're known to be divergent?

[molten grove]

Yes

[crude comet]

((((((y/e-1)/e-1)/e-1)/e-1)/e-1)/e-1...) = y

[molten grove]

you've assigned a finite value to a series diverging to infinity

And then are doing infinity / infinity

When you think you are doing finite / finite

the manipulation you are doing is how you get a closed form for a geometric series in general, but it requires that the geometric series converges a priori

and that holds when the common ratio r satisfies |r| < 1

which e certainly does not

[graceful ridge]

Wolfram Alpha says it's a regularized result, and it's the same result that we got by assigning the series to y, does this mean although our method led to the answer, it's a wrong way of assigning value to a divergent series or is our method also valid for a "regularized result"?

[molten grove]

Sorry I am unfamiliar with borel summation extension so I don't have a good answer

intuitively it'll be that you're getting the value that makes the "most sense" if everything were to work nicely

but Ive not been exposed to the idea formally so that's the best I can say

[graceful ridge]

"In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series."

"In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization."

So, this kind of summation should only be applied in specialized mathematical contexts?

[molten grove]

yes

[graceful ridge]

ok thanks

.close

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[sacred merlin]

how do i prove that cos(1/n) is monotonic

[sacred merlin]

wtf does monotonic even mean

[molten grove]

Monotone means it's either increasing decreasing

[keen verge]

It's not

Wait nvm were working in N

[sacred merlin]

i gotta do that shi

[molten grove]

You've chosen the wrong bn

[keen verge]

I think b_n = cos(1/x) is right

[molten grove]

o I see I kept reading cos(n) for some reason

[sacred merlin]

cos(1/n) diverges by nth term

[molten grove]

my mistake

[sacred merlin]

lol ur good

[keen verge]

Prove that it is monotonic by showing that cos(1/n) < cos(1/(n+1))

For all n ofc

[sacred merlin]

thank u ❤️

[keen verge]

Np

[sacred merlin]

goat

more help please 😭

idek what to set as an or bn

[keen verge]

The second sum is bounded by pi^2/6 or smth

[sacred merlin]

wtf

[keen verge]

Ikr

Actually stupid af

[sacred merlin]

how ???

where tf does pi come from

[keen verge]

If you take it to infinity it becomes a famous problem called the Basel problem. There's heaps of ways to solve it

None of which I know of the top of my head sorry

[sacred merlin]

lol ur good

[keen verge]

Nah, just watch too many math youtube vids

[sacred merlin]

bro idk why my teachers giving this to us im in bc

this does not seem like bc

or maybe im j stupid

[keen verge]

I mean, you could just prove that the second sum is bounded by 478837378 by induction if you want too

U don't have to know the exact limit

[sacred merlin]

f it im j saying basel problem and hope that man doesn't mark me off

[weary reef]

Iirc, the inner sum is polylog_0 of 1

[keen verge]

No, say that it converges bc it's a p series where the power is bigger than 1 I reckon

[weary reef]

My bad, it's Li_2

[sacred merlin]

im ngl that sounds like a whole bunch of yap to me

idk what that is 💀

[weary reef]

Hmm, maybe you don't need it for this problem

Oh, I'm stupid nvm

The inner sum is finite

[sacred merlin]

yeah I think I'm trying to prove that it's convergent

[keen verge]

I reckon just use comparison test

[sacred merlin]

a_n is convergent through comparison test and geometric series

b_n is bounded and monotonic

so i think thats enough proof to say it converges

[keen verge]

Valid

[sacred merlin]

.close

[obtuse pebble]

Channel closed

Closed by @sacred merlin

Use .reopen if this was a mistake.

[obtuse pebble]

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[winter surge]

i missed a few lessons and have no idea how to approach b)

[keen verge]

Okay so depends on what direction u wanna go

I think forwards is easiest

But yeah, so you know that the equation for a circle is x^2+y^2=r right?

[winter surge]

yeah

[keen verge]

So just translate that so that P and Q lie on it ig idk

Alternatively

U can factorise that as (x + 5)^2 + (y + 45/2)^2 - 2213/4

[winter surge]

thats what i was originally thinkinf but no method seems clear from there to me

that works actually

ill try that

but i cant start with the thing im trying to show, right

[keen verge]

Nah, I reckon it's cool. Just show how you factorise that and your good

They give it to you, you just has to show that it holds

[winter surge]

ahhh that makes sense

[keen verge]

Alternatively, you could derive it yourself

[winter surge]

yeah i felt like it was implying i had to derive it

[keen verge]

But no one wants to derive anything they don't have to

[limber pelican]

1)Find middle point of PQ with Pythagorean and divide C by 2

[winter surge]

pq isnt necessarily along the diameter

[limber pelican]

Dosnt matter since 2 points will always make a line

[winter surge]

ah

also wdym divide c by 2

[limber pelican]

You know how to find the middle point coordinates in PQ?

You can check how x and y varies to go from P to Q

Then you divide the x and y variation you found

[winter surge]

oh i have the equation of a line

[limber pelican]

That’s gonna give you the middle point coordinates

Your comfortable with vectors?

You can than find a vector perpendicular to PQ

[winter surge]

yeah

i have a perpendicular bisector of the line

[limber pelican]

Then you can just start your vector from the middle point you found of PQ

That’s gonna give you two points to work with in order to find the equation

[winter surge]

aaaaalright

i dont see how this will work

[limber pelican]

At least that’s how I would approche it

You find the middle point of the line PQ

then find a perpendicular vector of PQ (vector product=0 when it’s perpendicular)

[winter surge]

i do have the vector

go on

[limber pelican]

Find the (a,b) of the vector you found

Then you add it to the middle point of PQ

You see where I’m going?

You will have two points of the line you want to find the equation off

[winter surge]

i can see what your goal is, but i dont necessarily see why that would work

the midpoint of PQ doesnt necessarily lie on the circle

[limber pelican]

Bisector of a line segment means the middle of the line

[weary reef]

I think what they are saying, you find the perpendicular bisector of PQ. That has to be a diameter

[winter surge]

ohhhh

i didnt think about that

[weary reef]

I think that's why the first question asked it

But yeah, that should be sufficient to solve it

[winter surge]

ok im getting it now, thanks

.close

[obtuse pebble]

Channel closed

Closed by @winter surge

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[obtuse pebble]

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• Be polite and have a nice day!

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[timid silo]

Y coordinate

Yes

[obtuse pebble]

Channel closed

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[timid silo]

For 0.5 radians

I forgot what we do

Is it 0.5/2pi

Any help

[crimson wave]

hmm

[timid silo]

Omd wait

Is it

Errrr

0.5x^2/2

Right

[crimson wave]

i need to calculate 1 min

[timid silo]

Theta = 0.5 right

[crimson wave]

right

[timid silo]

So

To find area = 2xy + ((x)^2 theta)/2

[crimson wave]

oh shit

i dont have this lesson

[timid silo]

To find perimeter

[crimson wave]

talk to @ruby sequoia

[timid silo]

We do

[timid silo]

Ok

[ruby sequoia]

ahem

[timid silo]

I’ll just type

So