help-38

[dusty sleet]

Is v a scalar or a vector?

[cobalt cloak]

Scalar?

oh shit

i misread oops

Av=lambdav

[dusty sleet]

Nice, letters don't matter actually, but it's common to use v for vectors

Ok so Av = λv

[cobalt cloak]

Yep

[dusty sleet]

Now, what's A²v ?

Think of how you can obtain A² in another way

[cobalt cloak]

uh

i am not sure

[dusty sleet]

What does A² mean?

[cobalt cloak]

Wait

Do we need to use characteristic equation?

[coral ember]

No

Whats the map associated yo A^n?

[dusty sleet]

What does 4² mean? @cobalt cloak

[cobalt cloak]

What is a map 😭 sorry

[cobalt cloak]

4*4

[dusty sleet]

That's it

So A² = A*A simply 😅

[cobalt cloak]

yep

ohh

ok yeah 😭

[coral ember]

A linear map or application whatever you call it

[dusty sleet]

So now, A²v = (A*A)v = ...

(hint: use associativity of matrix product)

[cobalt cloak]

A(A*v)

And Av = λv

[dusty sleet]

Perfect

Therefore you have A²v = A * λv
But λ is just a scalar, thus
A²v = λ * Av

Now, use again the fact that Av = λv and you get your result

[cobalt cloak]

ohhhh

A^2v = λ^2v

woa

h

i can use MI to prove this is true for all n?

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[dusty sleet]

What's MI?

[dusty sleet]

You can prove the statement is true for all n by using induction

[cobalt cloak]

mathematical induction

yeah

[dusty sleet]

Ah alright, that's correct yep

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[boreal shuttle]

$z^4 + 1 = 0$ wouldnt the solution just be $\sqrt{i}$

[solid kiln]

rajel

[stoic iron]

~~For one, that solution would be the fourth root of i.~~Secondly, there is more than one solution just like there is more than one solution to x² = 1. Here there are four solutions

My mistake your current solution is correct, try find the others

[boreal shuttle]

ok

[boreal shuttle]

i was just confused because in the video that im watching he didnt find my solution ( in the solutions)

[deft flame]

sqrt(i) is one of those solutions

try squaring sqrt(2)/2 *(1+i)

you'll get i

[boreal shuttle]

oh that makes sens thx

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[untold sedge]

i rlly don't get the second step

[brittle barn]

Rearrange to get $(x+1)^2=1-y^2$

[solid kiln]

Your LaTeX Helper

[brittle barn]

then take the square root of both sides.

So $|x+1|=\sqrt{1-y^2}$, i.e. $x+1=\pm\sqrt{1-y^2}$.

[solid kiln]

Your LaTeX Helper

[brittle barn]

Solving for $x$, $x=-1\pm\sqrt{1-y^2}$.

[solid kiln]

Your LaTeX Helper

[brittle barn]

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[untold sedge]

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[merry scaffold]

u needent do this to find x

just put the 1 on the otehr side and factorise it

[untold sedge]

i totally got it now

[merry scaffold]

oh i thought u want to find x..

[untold sedge]

k

yea, to find the possible value for x

that's the question

[merry scaffold]

then doiung this is way to long

[untold sedge]

yea, that's the only step that pop up in my mind

[brittle barn]

!xy

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[untold sedge]

thx

[brittle barn]

that typically gives you better results

[untold sedge]

yup

[unique pine]

damn complex numbers.. 11th? or revision? 😭

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[cobalt cloak]

If A and B are both transformational matrices, does AB=BA

[hasty yew]

what is a transformation matrix

do you mean transition matrix

https://en.wikipedia.org/wiki/Transformation_matrix

wikipedia seems to define it as the matrix representation of linear maps

if so

then your answer is no

matrix multiplication is not commutative in general

[vagrant marsh]

what is true though is |AB|=|BA|

[cobalt cloak]

oh i c

thx

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[outer cradle]

i get that you could just expand it

and then you integrate sinx ^2 +1 - 2sinx

but is there an easier way to do this

that method just feels too long

[kindred pier]

nope

[outer cradle]

omg

ok

great

[mild thunder]

sometimes the fastest way is just kinda brutal

calc 3 triple integrals be wilding like that too

[outer cradle]

yeah well

thanks i guess

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[crisp wyvern]

Hey. I'm wondering if it is possible to calculate the vector product in 4D. i want to calculate the Hypervolume of a slanted Tesseract this way. (i know its possible via the determinant and for a regular tesseract its just a⁴ but for a slanted it should be possible to do it with the vector product).

im not sure but I've seen (a x b) • (c x d) online. not sure if that is correct though

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[crisp wyvern]

<@&286206848099549185>

[fair bison]

the cross product only works for three-dimensional vectors

it is special to 3D specifically

you're best off just taking the determinant of the four vectors put together

[crisp wyvern]

hmm i figured. I heard it was also possible to use it in 7D though. how yould you calculate it there. is it possible to calculate a volume with 6 7D vectors using the vector product there?

also (might be a weird question) couldn't i define a form of special product for myself which works the different from the vectorproduct but gives the same solution for every input as the determinant does?

[fair bison]

that sounds like it would just be the determinant then? although, it'd have to be a product of four different 4D vectors

the reason that the cross product works in 3D is that every oriented area is also uniquely specified by a vector

this is because area is 2D, vectors are 1D, and 2 + 1 = 3

you can take a product of two vectors in any dimensional space to get an oriented area, but you will get as a result a bivector, not another vector

[crisp wyvern]

huh i see. will the bivecotor not help me to figure out the Hypervolume?

[fair bison]

it does, you can take the product of all the vectors together to get the (oriented) volume, but this is the same thing as taking a determinant

it's different from a cross product because the cross product outputs a vector

[fair bison]

I'm not too familiar with how the 7D cross product works unfortunately, but it is a bit weirder than the 3D cross product. There might be a formula to calculate volume of 7 7D vectors using it, but I don't know it.

[fair bison]

(this product is called the wedge product)

[crisp wyvern]

hmm i see. can you show me (or link a page) how i can calculate these?

[fair bison]

hmmm @frank folio do you know anything about the 7D cross product and using it to calculate volumes btw?

[crisp wyvern]

not only 7D. or wait. was the calculation with bivecotors only in 7D? i would need it in 4D

[fair bison]

Here's one but it might be a little abstract https://math.stackexchange.com/questions/2842911/how-to-perform-wedge-product#2842962

[fair bison]

No, bivectors work in any d, but again, the best way to calculate a volume is with the determinant

Is there a reason you want to avoid the determinant

[crisp wyvern]

i will look at it later. one other question. can you calculate other volumes in 4D or higher? i saw a video on how it is possible with integration for a sphere. also any ways to calculate surface area? (i couldn't find a lot on the Internet sadly)

[fair bison]

I can give you an example calculation with bivectors. If you have the two vectors (1,0,1) and (1,1,0), you can write them as e1 + e3 and e1 + e2. Then, you can take the wedge product as follows:
(e1+e3) wedge (e1+e2) = e1 wedge e1 + e1 wedge e2 + e3 wedge e1 + e3 wedge e2
= 0 + e1 wedge e2 + e3 wedge e1 - e2 wedge e3
= -1 e2 wedge e3 + 1 e3 wedge e1 + 1 e1 wedge e2.
This is a bivector. Note how this corresponds to the fact that the cross product of those two vectors is (-1, 1, 1).

(The way that you calculate with wedges is that anything wedge itself is zero, it distributes over addition just like regular multiplication, and when you flip the two arguments of a wedge product it flips sign just like with cross products)

[fair bison]

What do you mean by "other volumes"

As in, calculate a 2D area of a surface in 3D for example?

[crisp wyvern]

as in: how to calculate the Hypervolume of a 4D body

[fair bison]

Of any 4D body? Or a parallelogram

[frank folio]

You mean wedge product

The volume calculation aspect of the determinant is entirely due to properties of the wedge product

[crisp wyvern]

any but parallelogram would be optimal

[frank folio]

And is only valuable whenever you use a special case to make the determinant look just like the wedge product

I.e. whenever you look at a full n Dimensional volume

[fair bison]

To calculate the volume of a parallelogram, you can just use the determinant? I'm confused why you want to avoid the determinant

[frank folio]

I do not, the octonions don't make sense to me yet

[fair bison]

To calculate it for an arbitrary body, you would probably have to do some multivariable calculus

But the techniques are very standard and work for any number of dimensions

Basically you parameterize the body, take some derivatives, take the determinant, and then integrate it

You can look up some examples of calculating volume integrals or surface integrals on the Internet

But for parallelograms it is very easy

Now, if you wanted to calculate a 3D volume using 3 4D vectors for example, you would use the wedge product, like described above

[crisp wyvern]

i see. and for the parallelogram you would just use the determinant?

[fair bison]

Yes, if you have 4 4D vectors; as wraith said, that's the special case where the number of vectors matches the dimension

(And it's equal to what you would get with the wedge product)

[crisp wyvern]

i have to do a small assignment on teaching the step from 3D to 4D calculations. i wanted to start off with the determinant but give an alternative as well

(aqually not quit i wanted to start with the vectorproduct becaus this is how they explain it for 3D in the book. then work my way to 4D with it and give the determinant as an alternative. but now i will just show that in 3D the volume calculation with the vectorproduct is equal to the determinant and from there make my way into the 4th dimension

[fair bison]

Yeah the vector product (v1 × v2) · v3 will not generalize well

however if you view it as v1 ∧ v2 ∧ v3 instead, it will generalize to other dimensions

[frank folio]

The vector product has 1 generalization to N dimensions, and 2 generalizations in 7D. The cross product can be extended as an (n-1)-ary operation in N dimensions, by wedging n-1 vectors and taking a duality back to a vector. And in 7D as a product by utilizing the octonion product

The only usable one boils down to just use wedge/determinant

[crisp wyvern]

ahhh i see. ive seen that symbol aswell. is it for the wedge product?

[fair bison]

wraith do you have a decent resource introducing wedge products without much abstraction? I'm trying to find one; all the resources I find start with abstract definitions of multilinearity and the exterior algebra and so on which is probably unnecessary overhead here

[fair bison]

yup

[crisp wyvern]

(I don't know how well this would suit my assignment though, since its supposed to be made for school usage. i think Determinants are easier in this regard)

[fair bison]

wedge products are pretty easy to calculate

the thing with the determinant is that it's useful when the number of vectors matches the dimensionality

so for example you can't calculate the 3d volume of 3 4d vectors easily with it

or the 2d volume of 2 3d vectors

but it's perfect for calculating the 3d volume of 3 3d vectors, or the 4d volume of 4 4d vectors

[crisp wyvern]

hmm maybe i can try it. i have a pre-presentation on friday. i can ask about it there and if i have to use an alternative i will come back and ask again :D

[fair bison]

okay yea feel free to ping me

there's only really three things you need to know when calculating stuff with the wedge product

[crisp wyvern]

yeah, should be enough. i dont have to do a whole lot. just 4-8 pages

[frank folio]

Maybe https://www.mdeetaylor.com/wp-content/uploads/2018/02/GA_quick_course.pdf ?

[fair bison]

1. it distributes over addition in the usual way, so (a + b) ∧ (c + d) = a∧c + a∧d + b∧c + b∧d
2. when you flip the arguments, it changes sign, so a∧b = -b∧a
3. the same thing wedge itself is zero, so a∧a = 0

so it's very similar to the cross product

but it works in all dimensions

[crisp wyvern]

okay, and for calculating surface areas? (surface volumes in 4D). any easy way to do it there? for a tesseract its 8L³ iirc and then for other bodys (like a 4D parallelogram?)

(or maybe even a 4D sphere, but i probably need integration there)

[fair bison]

yeah you'll need integration since it's a curved surface

[crisp wyvern]

makes sense

[fair bison]

it's a little annoying because you have to add up the areas of a bunch of faces

but calculating the area of one of the faces is pretty straightforward

actually I just checked, and there is a formula for the volume of such things using a determinant

(from the pdf wraith sent)

so if your face is determined by the vectors a1, ..., ak, then you can calculate the surface area (surface volume) of the face using this formula

and then just add up all the different faces and you're done

[crisp wyvern]

waittt. why is the determinant different from (a_1, a_2...)? also why the root?

[fair bison]

well let's say a1, a2, a3 are 4D vectors

and they represent a 3D face of your tesseract thing

then you can't take the determinant of (a1, a2, a3)

because it's a 4-by-3 matrix

you can only take the determinant of square matrices

[crisp wyvern]

nvm mb

[fair bison]

that's why you can't use that same formula for calculating 3D volumes in 4D (just like you can't use it for calculating 2D areas in 3D)

[crisp wyvern]

i thought you meant the volume of a 4D body. you meant the surface volume of a 4D body

or nD in that case

[fair bison]

yeah

[crisp wyvern]

okok. so hey, thats actually pretty easy. and i can try showing off how it works by using a tesseract and showing that it will also be equal to 8L³

[crisp wyvern]

just to be sure. • means the dot product, right? (i will look at the paper later)

[frank folio]

You might be able to use a properly ordered series of octonion products to determine area in up to 7D, now that I think about it

Er 7D cross products for up to 6D

[crisp wyvern]

ahhh i see. that makes sense

but i dont think thats what im aiming for, since it should still be easy for the kids

[frank folio]

Definitely not

[crisp wyvern]

and starting off with "so first we have to go into the 7th dimension"... hahaha xDDD

okay so thanks again for the help both of you! (im from germany so its hard to find good papers on the Internet but you helped me out a lot!). i will close this one and reopen it or open a new one if needed.

also, 4D is pretty cool. i wish they would teach some of it in school. every time i think/talk about these things i get excited and realize how cool maths is (lol).

[fair bison]

yea!

[crisp wyvern]

(or any higher dimension in that case)

[fair bison]

ah, viel Glück

[crisp wyvern]

:DDD Danke!

thanks again. Have a great day!

[fair bison]

bitte!

[crisp wyvern]

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[barren crescent]

how do I solve this?

[subtle leaf]

What do sin, cos, and tan mean as points on the unit circle?

[subtle leaf]

so what are the coordinates of the intersection on the unit circle

[barren crescent]

that i do not know

[subtle leaf]

where is the intersection point in that image?

[barren crescent]

cos 0, sin 0?

[subtle leaf]

theta, not zero, but yes

[barren crescent]

yeah i didnt have the symbol on hand

so cos is 0.559 and sin -0.829

does that make my coordinates?

[subtle leaf]

watch your sign, but yes

[barren crescent]

90 percent of the time the answer is stupid simple

thanks very much! so patient

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[bold prawn]

how do i do 51

well i got sqrt34/-5

but the answer is apparently sqrt34/5

and i was wondering how do i know whiere to p[ut the negative on

the 3 or 5

in the triangle

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[bold prawn]

<@&286206848099549185>

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[bold prawn]

bro

💀

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[maiden lynx]

how does cosx + sinx = 0 get to -sinx/cosx=1?

[burnt mulch]

$\cos x=-\sin x \implies \frac{-\sin x}{\cos x}=1$

[solid kiln]

Civil Service Pigeon

[maiden lynx]

ohhhhh

tysm

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[naive karma]

i need help like with solving algebra because i cant explain it step by step to my sister 🥲

[past river]

Do you have a specific problem in mind

[naive karma]

6/b-1 = 9/7 its basically solving proportions

im kinda having difficulty to recall binomials so yeah

i couldnt teach it to her 💀

[past river]

,, \frac {6}{b -1} = \frac 97

?

This?

[naive karma]

b-1 together

[past river]

You‘ll have to use parentheses for clarity then in the future (:

[solid kiln]

𝔧𝔪𝓣𝛾𝜑𝜽

[naive karma]

im so sorry

[past river]

Nono it‘s fine, just letting you know that notation can get confusing :)

[naive karma]

but yeah that's the one i cant recall

😭

[past river]

So, did you teach her that she‘s allowed to do operations (like adding, multiplying and such) on an equation, so long as she does it to both sides?

[naive karma]

yeah i believe distributive property is also involved right?

[past river]

Yeah, but the first thing you be getting rid of the left fraction, since we want b alone on its side

And there‘s two ways one could go about that, either multiply by (b-1) on both sides, or "flip" both sides

[naive karma]

i did 6 x 7 = 42

42 = 9b - 9

i dont know if im correct 💀

[past river]

Oh, you just "cross multiplied" right?

[naive karma]

yu

p

[past river]

That‘s the "shortcut" if you want to think of it that way

[past river]

Well yeah, once you have this, standard addition, multiplication and whatnot to get b alone

[naive karma]

but im not sure if its applicable so yeah i pretty much forgot binomials and all that

[past river]

Well, for any a,b,c,d $\frac ab = \frac cd \ \implies bc = ad$

[solid kiln]

𝔧𝔪𝓣𝛾𝜑𝜽

[past river]

That‘s "cross multiplying"

a,b,c,d can be numbers or expressions or whatever

[naive karma]

i sort of added like +9 to both sides (42 and 9) in the equation which left me to like 51/9 = 9b/9 💀 but i think that's when im wrong

[naive karma]

let me save this

[past river]

Hm, no you‘re doing fine, just 9b/9 simplifies some more

[naive karma]

yezz i think it would be b alone

but 51 isnt divisible by 9

that's where im stuck

[past river]

,calc 51/9

[solid kiln]

Result:

5.6666666666667

[past river]

It‘s still a number

Just not one that has a "pretty" form

[naive karma]

I guess, but im not sure if her teacher would allow that in her homework but if i tell her to divide 51/9 and it results 5.6 right would it suffice

[past river]

If she‘s allowed a calculator, approximating as 5.6 is probably fine. If not, leave it as is

[naive karma]

51/9 = 9b/9 could work as it is?

[past river]

Well, 9b/9 = b, so b =51/9 (which simplifies to 17/3 btw). And the goal was to find b right?

[naive karma]

yeasss the goal was to find b

[past river]

Then you got your answer

[naive karma]

tysm

ill js tell her to find the gcf in the simplifying part

i mean

the factors*

how do i close the channel again?

😭

[past river]

With .close

[naive karma]

thank you again!! i appreciate the help

🫶 🙂

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[echo linden]

Hi, can anyone help me with this? Thanks

[echo linden]

Question 4

[kindred pier]

,rccw

[kindred pier]

!status'

[trim jolt]

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

[vagrant kraken]

hmmm

lets see

[echo linden]

2

[vagrant kraken]

i am positive i have seen a kind of question like this, let me recall

well

i do have a complex way of doing this question

[echo linden]

So I drew a diagram, I think you’ll have to use pythag somewhere

[vagrant kraken]

do you know how to solve a quadratic function system?

[echo linden]

I don’t think I’ll have to use it tho

The unit is based off pythag and surds

[vagrant kraken]

that is the point.... i have no idea how to solve it without using quadratic function system, but I can provide you with an answer

[echo linden]

Do you just sub in the values into the equation?

[vagrant kraken]

?

anyways

[echo linden]

Is there anyone else that would know how to do it without quadratics

[vagrant kraken]

hmmm i am not quite sure, my answer went to be 2sqrt(74-20sqrt(6))

i prob made a mistake in my calcs

[echo linden]

It is

I’m just not sure where I have to use pythag

.coose

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[outer cradle]

what do i take as u

i think 2x right

but then how do I integrate tan

or do i expand tan2x into its identity

[sudden mist]

tan u = sin u / cos u

[marsh forum]

i'd start with 2x=u

[outer cradle]

okay

so tanu

[outer cradle]

ok

[marsh forum]

no

[outer cradle]

sin u / cos u

[marsh forum]

tan(u)/2

[outer cradle]

yeah yeah

then on?

by parts?

[sudden mist]

Yeah well he was then asking how to integrate tan, so I think a coefficient wasnt necessary to include for the hint

[sudden mist]

You can make another substitution

(If you want to)

With t = cos u

[outer cradle]

oh

i didnt know you could do that

like two subs

but idk why not

it makes sense

[sudden mist]

Nothing is stopping you, but you have remember you made these

[outer cradle]

and sub back later

[sudden mist]

But specifically in this case it’s sort of not needed since you can just compare the form to the integral of f’(x)/f(x)

[outer cradle]

why dont i just then directly make it sin(2x)/cos(2x) adn then cos(2x) = u

[sudden mist]

Yeah sure

[outer cradle]

i dont follow

[sudden mist]

It’s just a shorthand rule but eh if you haven’t seen it before you’re not missing anything. It’s just a u-sub for a specific form

[outer cradle]

hmm

alright

anyways i got -1/2ln(cos2x) + c

think thats right

thanks for the help

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[lethal lodge]

please step by step

i have first changed it to the form ((a-1) + bi) / ((a+1)-bi) and multiplied by the complex conjugate

which is (a+1)+bi

[fleet bear]

That should work fine once you've multiplied it out

A slightly easier way to get there would be leaving it all in z form

The conjugate of $z^* + 1$ is $z + 1$

[solid kiln]

StrangeQuarkAL

[fleet bear]

Multiplying the numerator and denominator by $z + 1$

$$ \frac{z - 1}{z^* + 1} \cdot \frac{z + 1}{z + 1} = \frac{z^2 + 1}{|z + 1|^2} $$

[solid kiln]

StrangeQuarkAL

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[lethal lodge]

Thanks guys

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[lethal lodge]

Multiplying it worked

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[wraith hinge]

Find the mistake:

[livid thunder]

using a ring light

[wraith hinge]

Mistake?

[livid thunder]

yeah thats it

[wraith hinge]

Good bc i can control the intensity of ligh?

[wraith hinge]

Is it right?

@livid thunder HEELLOUUU

[nova spire]

why do you think there is a mistake?

[wraith hinge]

Video didnt factor -(x-4)

[nova spire]

?

[wraith hinge]

In thevid its left without my last step

In the vid they didnt cansel these and didnt add the -1

[nova spire]

which vid?

[wraith hinge]

I reviewd my answear on internet

[nova spire]

also if they didn't "add" the -1 (since 4-x = -(x-4)) that means they got the sign wrong

and btw the limit you got is correct

[crisp bronze]

Looks good to me but you don't have to keep writing lim_x->4 when you plug 4 in for x

[nova spire]

also that

when you managed to plug x = 4 you don't keep the "lim" part

[wraith hinge]

They just didnt factor with minus . They left it like thaz

Big eq

[nova spire]

in any case (4-x)/(x-4) = -1

whether you did it with the extra step (4-x)/(x-4) = -(x-4)/(x-4) = -1 or not

if they don't simplify (4-x)/(x-4) into -1 then they're wrong, otherwise you are both correct

[wraith hinge]

Okay I think its a fake alarm by my caffeine intoxicated brain.

Thanks btw!

.close ahhhhhhhhhhh

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[wraith hinge]

Hello! My question is: What is the probability that Christmas (on a randomly chosen year) falls on a Monday, Tuesday, Wednesday, &c.? Accounting for the leap years and the 400-year repeating cycle, I wrote a program to model this situation and my answers were:

Sunday 58/400
Monday 56/400
Tuesday 58/400
Wednesday 57/400
Thursday 57/400
Friday 58/400
Saturday 56/400

I verified my answers with my teacher, and they're correct, but I need to solve the problem on pen and paper, without empirically going through every case in a 400-year period. What would be a simple/subtle answer to this? Also, I need to prove most of the assumptions I make in my solution (such as the Gregorian calendar repeating every 400 years).

I found a solution online, but I can't really make much sense of it: https://people.missouristate.edu/lesreid/Sol04_0809.html

Thank you!

[wraith hinge]

<@&286206848099549185>

[lethal lodge]

what an interesting question

[wraith hinge]

True : D I love questions like these, and teachers who ask them

[lethal lodge]

You created an algorithm?

[junior pawn]

hi

[lethal lodge]

Hi

[junior pawn]

I can't agree with your solution

as you mentioned

in the case of 400 year cycle

there are 146.1097 days

and it is 20871 weeks exactly

so probability in Monday is just 1/7

[lethal lodge]

1/7?

If over 400 years Christmas falls on a Monday 56 times then would be fair to say it’s 56/400? Although I don’t know how to achieve such an answer with math

Or how to prove said assumption

[junior pawn]

just as the problem, it is the probability not just a exact value

[lethal lodge]

That’s true

[wraith hinge]

It is an exact value

[lethal lodge]

Can someone ping me when this is answered

[wraith hinge]

Because of how the calendar is arranged, some days are more common than others

[junior pawn]

400 year is the cycle and there are exactly 20871 weeks during 400 years

so the probability of Monday is one of the seven days - a week

1/7

[wraith hinge]

Wait igtg for a sec

I'll get back to you in fifteen mins or so

Until then <@&286206848099549185> can someone help?

[wraith hinge]

tf-

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[wraith hinge]

<@&286206848099549185>

[icy moat]

Hi, I can try to help

[wraith hinge]

Hey! Please do : ) Thank you so much

[crisp bronze]

Kind of similar to the doomsday algorithm ngl, I'll try to do this too

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@wraith hinge Has your question been resolved?

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@wraith hinge Has your question been resolved?

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[brittle walrus]

If I have an implicit function, given by:

[brittle walrus]

And I have shown that the inverse value theorem applies near (0,1)

I am asked to calculate

I have differentiated and gotten the following expression for dy/dx:

dy/dx = (2e^x + y)/(1 + 2y)

Plugging in (0,1), I get the right answer

What I'm confused about is what is actually happening, since y is supposed to be expressed as y(x) and thus only have an x input

and so how do I justify also tossing in y = 1 into the differential expression at x = 0

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[orchid owl]

can i get help with slope please?

[pine trellis]

Please post your question.

[orchid owl]

Thing here I don't really know much abt it

[split chasm]

wheres the question about slope

[orchid owl]

Oh the question is on the bottom

asking if its a slope and how do i know

[split chasm]

i don't see the question about slope

i see something about function but not slope

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[wraith hinge]

i was looking at these conic section parabola and its defined on the vertex (0,0)

but what if vertex change?

how is that gonna affect

the equation

my book has no mention on this

[unique minnow]

You can always translate it to (h,k) with transformations x -> x-h and y -> y-k

[wraith hinge]

oh yes

[unique minnow]

This will yield the more common vertex form (x-h)^2 + 4pk = 4py

With the constants written in a more weird way but the points remains

[wraith hinge]

so i need to change y to y + k and x to x + h

that distance y should change to y + h ?

[unique minnow]

Yeah well in this the main difference is the coordinates of the focus. It will be at F(h, k+p)

[wraith hinge]

this makes sense let me derive equation from that then

[unique minnow]

Similarly the directix will be a y=k - p

[wraith hinge]

$\sqrt{(x-h)^{2} + (y - (k+p))^{2}} = | y + p + k |$

[solid kiln]

Gary

[wraith hinge]

@unique minnow right

[unique minnow]

Yeah I'd try to see if that works

[wraith hinge]

on solving its giving
(x-h)^2 = 4y(k+p)

[unique minnow]

Yep that's what we want!

Hmm wait

Yeah ok I think that works

[wraith hinge]

we can do same the for horizontal

parabola

too right

[unique minnow]

Yeah well again if you have any formula, you can always translate it to (h,k) with the transformations I gave you earlier

[wraith hinge]

Cool

really hate book didn't gave general equations

lol

but now i know

[unique minnow]

I think there is something slightly wrong here, but it might have been in the setup with the directix and stuff

[wraith hinge]

i will check it later on

[unique minnow]

Because the vertex is fixed at y=0 if you plot it

But the process shouldn't be too far off 🙂

[wraith hinge]

thank you!

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[unique minnow]

Yeah it should be |y - k + p| on the right! that's what was missing

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[wraith hinge]

yeppp

[wraith hinge]

(x-h)^2 = 4p(y-k)

im getting now

[unique minnow]

Perfect then that works as intended

[wraith hinge]

the directrix is y = k - p cuz we shifted the graph now the origin at (h, k ) and directrix is same distance below as the focus is high from vertex so -p

[unique minnow]

Indeed.

[wraith hinge]

yay Cool thank you so much i know so much more now!

[unique minnow]

It's always insightful to try and derive things yourself, it was a good idea you had 🙂

[wraith hinge]

yep

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[vernal fern]

If a sinusoidal graph has sequential maximum points at (3, 4) and (5, 4) and sequential minimum points at (4, 1) and (6, 1), what are the amplitude, midline, and period of the graph?

[vernal fern]

I think the amplitude. is 1.5, the midline is y= 2.5 and the period is 2

does that sound right

[kindred pier]

you wrote midline twice

[vernal fern]

oh

whoops

midline is y= 2.5 and period is 2

[kindred pier]

ok

looks good

[vernal fern]

woot

thanks

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[barren crescent]

how do i solve this?

what i know or think i know

sine value is y

actually thats for unit circle maybe i know nothing

[cosmic meadow]

In which quadrants are sin positive?

[barren crescent]

is this the thing you are talking about?

[dry nebula]

yeah this is wht u should use

[barren crescent]

just 2

[dry nebula]

theres acc another quadratns where its positive

[barren crescent]

but does the negative tangent value do anything?

oh??

[dry nebula]

look at all of the descriptions carefully

[barren crescent]

oh

yeah i can't read

so it could be in quadrant 2 or 1

[dry nebula]

yeah

but only one of those has a negative tan

[barren crescent]

?

[barren crescent]

only one has a negative tan, you mean quadrant 1?

<@&286206848099549185>

[white iron]

No

It cannot be the first one

Since all trig functions are positive in that quadrant

Also since only sine and cosec is positive in second quadrant, it must be the answer

As it naturally concludes that tangent would be negative at it.

@barren crescent did you get it?

[barren crescent]

sine and tangent are both positive in the first quadrant, sine is positive in the second quadrant while tangent is negative in the second quadrant, it can only be the second quadrant

[white iron]

Yep.

[vernal fern]

nice!!

@barren crescent this is why you should listen to me

[whole granite]

you are correct

[barren crescent]

not my fault you can't explain things well

thank you!

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[ashen plover]

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/cc-8th-approximating-irrational-numbers/v/comparing-values-with-calculator

[ashen plover]

why does he use a arbitrary number?

[stoic iron]

you linked a 4 minute video youre gonna have to be more specific

[ashen plover]

in the vid he picks an arbirtrary number to compare the two values

but why he's doing that for what reason?

[hallow kite]

timestamp would be useful

[ashen plover]

you could just compare the two number without the arbitrary number

[stoic iron]

he picks 0.3 because he knows its bigger than 22.9% and its considerably easier to square 0.3 than it is to square 22.9%

[stoic iron]

he does literally go on to do this right after

[ashen plover]

i know

but why is the arbitrary number required to compare the two values?

[stoic iron]

.

its easier to compare 0.3 and sqrt(0.45) than it is to compare 22.9% and sqrt(0.45)

if you know that 0.3 < sqrt(0.45) then you know that 22.9% is less than sqrt(0.45)

[ashen plover]

couldnt we just conver the percentage into a decimal?

[stoic iron]

sure but it is still easier to square 0.3 than it is to square 0.229

[ashen plover]

ok so instead of square rooting 22.9 he uses 30% because its easier and then he can compare 30% and 0.45 and then use that to compare against 22.9%?

[stoic iron]

instead of squaring 22.9%

but yes

[ashen plover]

yea squaring

sorry

ok makes sense thanks

[stoic iron]

he gets that 22.9% < 0.3 < sqrt(0.45) which tells him that 22.9% < sqrt(0.45)

[ashen plover]

ya i get it now. thanks

see you

how do i mark as solved

[stoic iron]

you can type .close 🙂

[ashen plover]

.close

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[analog pier]

study the continuity of the function f

[analog pier]

!status

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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

[analog pier]

1

i know that the limit is equal to fn+1/fn but idk what to do from here

i thought about expanding 1-x but that does not seem to help

[clear cloud]

Is fn equals to zero somewhere ?

[analog pier]

no

[clear cloud]

Are there any other problem such as dividing by zero in continuity ? (Idk)

[analog pier]

i dont think so

since f is the value of a limit

it is probably always continuous

except if it is infinity maybe

[vagrant prism]

consider the case when n is even and when n is odd nvm

[analog pier]

what about x

this what i should get

[vagrant prism]

not quite..

[analog pier]

no this is the answer

idk why they used 1/2

[vagrant prism]

ah i see

[vagrant prism]

wait what is the task?

study the continuity or find the function

or both

[analog pier]

both

[vagrant prism]

note that for every n, the derivative of this function is 0 at 0.5

try seeing that for yourself

[analog pier]

i did the derivative but how can you calculate it

you mean the original function right?

nvm

i found the solution

thanks anyways

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[slow locust]

dont know if yall do physics but can anyone confirm if my answer with a 35 degree triangle is correct? (starred part)

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[slow locust]

my answer for the original problem is correct, i just want to get an idea of how much the triangle angle affects the final answer.

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[slow locust]

.close

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[thorny wadi]

I need to use differential thing

Is this correct what i did?

[winged hinge]

What r u trying to do

[simple warren]

are trying to make f continuous?

[thorny wadi]

ehh

Differential quotients

[winged hinge]

wot

[thorny wadi]

what

okay wait let me make it cleaner

[winged hinge]

R u trying to find the derivative at x = 0?

What's a tho

[thorny wadi]

a is 1

[winged hinge]

ok

[thorny wadi]

okay here

is it better?

[winged hinge]

no there's a few mistakes

[thorny wadi]

okay

where

[winged hinge]

sin(1/0) isn't 0

It's not even defined

[thorny wadi]

oh okay

how do i fix it

defined?

[winged hinge]

(e^x - 1)/x isn't 1 either

[thorny wadi]

if i use

lhospital on it

it is

thats why I seperated the structure

or cant i do that

cus i need to figure out the limit

or not?

[winged hinge]

o u could do that

[thorny wadi]

so how do i fix my

result

i dont have solution thats why idk if it is right or not

[simple warren]

so you want to do

$\lim_{x \to 0} \frac{f(x) - f(0)}{x-0}$

[solid kiln]

vehnil

[thorny wadi]

Yes

[simple warren]

which becomes

$\lim_{x \to 0} \frac{x\sin(1/x) +e^x - 1}{x}$

[solid kiln]

vehnil

[thorny wadi]

Yes

thats right

But idk if I simplified it right

[simple warren]

okay so theres a couple things you should do

[thorny wadi]

yes

[tender vigil]

venjy

[simple warren]

...

first you notice $x\sin(\frac{1}{x})$ looks familiar

[solid kiln]

vehnil

[thorny wadi]

doesnt look familiar

never saw it

[simple warren]

but most important you have a oscilating discontinuity

sin(1/x) as x approaches infinity is undefined

[thorny wadi]

oscilating discontinuity what does that mean

[simple warren]

go on desmos

and plot sin(1/x)

see what happens

[thorny wadi]

?

[simple warren]

1/x as you get closer and closer to zero is larger and larger values

[thorny wadi]

Yess

closer to zero

[simple warren]

sin of larger and larger values is going to oscilate between -1 and 1

and in fact the limit of sin(1/x) as x approaches 0 is undefined

you can't calculate it

[thorny wadi]

Then what can I do

[simple warren]

but we don't have just sin(1/x) we have xsin(1/x)

I don't think we can substitute just yet

but we can do whats called a change of variables

we will say that u = 1/x

so

[winged hinge]

that's a fools errand

piece wise function xsinx, 0 at x = 0 isn't differentiable

So this isn't either

[simple warren]

?

[worldly comet]

hes not taking the derivative

[simple warren]

the limit of xsin(1/x) does exist

[winged hinge]

If u say u = 1/x u just find that limit sin(u) as u goes to infinity doesn't exist either

[simple warren]

as x approaches 0

[winged hinge]

But not sin(1/x)

[simple warren]

it doesn't need to

[winged hinge]

u have xsin(1/x)/x

[worldly comet]

the x term turns into 1/u

[simple warren]

okay let me just finish

anyways

[winged hinge]

If u don't believe me

https://math.stackexchange.com/questions/3704186/question-about-x-sin1-x-at-x-0

[simple warren]

its a limit

[winged hinge]

bruh

Yes derivatives are defined as limits

[simple warren]

I am not evaluating xsin(1/x) at x=0

im taking the limit

the limit will end up being 0

which won't matter

[winged hinge]

He's asking to evaluate

[simple warren]

because theres another term

[winged hinge]

The differential quotient at x = 0

I. E. The serivative at x = 0

[thorny wadi]

yes

[winged hinge]

But it doesn't exist

[simple warren]

please look at the difference quotient and tell me if the numerator is solely xsin(1/x)

[winged hinge]

That doesn't matter

[thorny wadi]

What if I change the a ?

[winged hinge]

if this were differentiable at x = 0, then so would the function in the stackexchange question be

[thorny wadi]

This is the full task

[winged hinge]

Since e^x is clearly differentiable

And derivatives are linear

Oic

[simple warren]

Okay

[winged hinge]

B just wants you to write out the expression in the limit in the derivative definition

And c is asking you if it's differentiable?

[thorny wadi]

Yes

[simple warren]

So anyways

[winged hinge]

ok well the answer to C is no

[thorny wadi]

why?

cus of the sin(1/x)

?

[winged hinge]

yeah

[thorny wadi]

Ah okay

What is the

xsin (1/x) wasnt there

Then it would be 1 right?

then a = 1?

[simple warren]

the point isn't to give him the answer

[winged hinge]

a = 1 even with sin(1/x) if you want the function to be continuous

[thorny wadi]

what if i want it to be differentiable

cus i dont really understand it

[winged hinge]

the limit of xsin(1/x) as x goes to 0 is 0, but the limit of xsin(1/x)/x as x goes to 0 doesn't exist

[thorny wadi]

basically we have an a = 1 to be continuous

[winged hinge]

If it were differentiable a would still be 1 cus differentiability implies continuity

[thorny wadi]

ah okay

thank you

[winged hinge]

np

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[fair bison]

(Please ping Helpers only after 15 minutes)

You might want to give some more context though, like where did these numbers come from?

Yeah sure, a pic of the original question would be good

[fair bison]

Oh the fact that it's supposed to go together in pairs changes things a lot

I'm not really sure what the answer is though, sorry

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@wraith hinge Has your question been resolved?

[wraith hinge]

+4, *10 it looks like

14+4 = 18

18*10 = 180

8+4 = 12

18 + 4 = 22

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[quiet turret]

can someone help me differentiate $\frac{\sqrt{3}a^2}{2} + \frac{4299.48}{\sqrt{3}a}$

[solid kiln]

pixel

[quiet turret]

should i rationalise denominator first

[grim sparrow]

you don't have to

what have you tried so far?

[quiet turret]

$\sqrt{3}a + \frac{4299.48}{\sqrt{3}}$

[solid kiln]

pixel

[quiet turret]

would this be the answer

[split chasm]

no

[grim sparrow]

the second term looks wrong

how did you get that?

[split chasm]

what rule are you applying when differentiating the second term

[wraith hinge]

why 4299.48

jeez

making it harder to write

[quiet turret]

$\sqrt{3}a - \frac{4299.48}{\sqrt{3}a^2}$

[solid kiln]

pixel

[quiet turret]

that should be it

[split chasm]

yes

[quiet turret]

thank you

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[fresh sphinx]

Q: find an eqn of a plane containing the intersection of ppanes 3x-2y+4z=5, 2x+4y-z=7 and passing through (2,1,2)

[fresh sphinx]

Ik i can solve it by finding the line of intersection of the 2 planes, then getting an arbitrary pt anf getting the second direction vector to get the plane im looking for, but the algebra looks awful and i was wondering if there is a better way

[verbal stream]

I think cross products could help here

[fresh sphinx]

How would that work? Do u mean to find the direction of the line of plane intersection by cross product of the 2 normals of the planes

[verbal stream]

yes

[fresh sphinx]

Yeah, but thatvwould still be quite tedious. My friend mentioned there is some way to solve by considering like a family of lines with the intersection

But idk how that works

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[turbid gazelle]

anyone good at probability i dont really get the explanation

[vagrant marsh]

lets say that people have a choice between four different toppings:
chocolate chips, cherries, almonds, or walnuts

anybody who wants chocolate chops will also want a chocolate cake

but if its friday it will be a chocolate cake

so then we can list the possibilities like this (day, flavor,topping)
(M,C,Choc chips)
(T,C,Choc chips)
(W,C,Choc chips
(Th,C,Choc chips)
(F,C,Choc chips)
(F,C,Cherries)
(F,C,Almonds)
(F,C,Walnuts)

now ive adjusted the cases so each case is equally likely

if you notice of all cases with chocolate, half of them occur on a Friday

oh wait what

oh i misread the problem

my bad

ok so the real explanation

we know that each birthday is independent

so there is a 20% chance that a birthday falls on a Friday

we can imagine 100 people in the office

20 people will have chocolatecake on a friday

we know 25 people will have chocolatw cake on their birthdays

so 20/25 people having chocolate cake will have it on a friday

in other words, 4 in 5

@turbid gazelle kinda whiffed the first explanation, hope this makes sense

[turbid gazelle]

all the employees have chocolate cake on their birthdays

[vagrant marsh]

no, 25% of people will have chocolate cake on their birthdays

[turbid gazelle]

oh shi mb all of them have cake but its not always choc

Ok that makes sense

[vagrant marsh]

yeah

[turbid gazelle]

TY

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[pulsar spade]

Hi

[kindred pier]

Did you have a question?

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@pulsar spade Has your question been resolved?

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[vague stirrup]

I searched up online that C^infty is the set of all functions that are infinitely smooth and have a nth derivative for all n.
But, how can a sequence of complex numbers be a C^infty function? I can't prove the statement until I can see how a set of sequences can possibly be the subset of a set of functions?

[left oriole]

i suspect that's not what C^infty means in this context

if i had to guess, it's supposed to mean the set of all complex sequences

[vague stirrup]

Oh, that's funny

How do you interpret such notation?

What does the infinity represent?

[left oriole]

it's not very good notation but i assume it's intended to be suggestive of C^n where n is finite

[hallow kite]

https://en.wikipedia.org/wiki/Sequence_space

might be what this is trying to define

[left oriole]

$\mathbb C^{\mathbb N}$ would be more correct

[solid kiln]

Bungo

[vague stirrup]

Awesome

Thanks you guys!

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[radiant crow]

i'm tired how is ADE = 90 - angle a? H is orthocenter

[radiant crow]

ah i get it

wait no i don't

uh

AEDB is cyclic so A + 90 + ADE = 180, ADE = 90 - A

nvm i get it

thanks guys

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[marsh vessel]

For a given positive integer m, you have a round table of 2m seats, and m couples attending this party (all 2m people are distinct). Given an odd integer
1 < n < m, you want to seat everyone such that the two people in each couple
are exactly n seats apart (as in there are n−1 seats between them). How many
distinct ways are there to achieve this?

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@marsh vessel Has your question been resolved?

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@marsh vessel Has your question been resolved?

[marsh vessel]

<@&286206848099549185>

[marsh vessel]

should the answers be 0 <@&286206848099549185>

[dull temple]

why do you think it's 0?

[marsh vessel]

I'm not sure I used catalan numbers formula

[wraith hinge]

Consider the arithmetic sequence 2 ,12 , 22 ,....... . .
a) What is the algebraic form of this sequence ?
anyone

[marsh vessel]

this is arithmetic series

AP

[wraith hinge]

not ap

[marsh vessel]

why is that relevant here tho

[wraith hinge]

its maths

[marsh vessel]

this is my help channel you can create your own

[wraith hinge]

i wanted help tho

oo

can you help

[marsh vessel]

i think it's AP

please ask your question on your own channel

[marsh vessel]

catalan number

is it correct?

im sending the question again

For a given positive integer m, you have a round table of 2m seats, and m couples attending this party (all 2m people are distinct). Given an odd integer
1 < n < m, you want to seat everyone such that the two people in each couple
are exactly n seats apart (as in there are n−1 seats between them). How many
distinct ways are there to achieve this?

<@&286206848099549185>

[dull temple]

i will say it seems to be possible for 6 people and n = 3

[marsh vessel]

not possible

i tried 6

check it out

oh wait sorry

[marsh vessel]

i think that's correct