help-17

[sharp compass]

Oh I didn’t notice sorry!

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[white hearth]

Hey guys.
I'm having some difficulties starting a proof :
https://imgur.com/a/Gf4JDUO

My major difficulty is working with many conditionals like this.
Thanks in advance.

[peak matrix]

what's that C?

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[cunning slate]

Hi, this is applications of integration (with physics ish). does anyone mind helping?

[cunning slate]

.

[hushed pewter]

<@&268886789983436800>

[cunning slate]

oh yeah theres only 3 questions left in this unit finally :D

then ill be done with the whole unit

and not just the "sub" unit

[hushed pewter]

Are you familiar with W=mgh?

[cunning slate]

W = mass gravity height?

[hushed pewter]

yes

[cunning slate]

somewhat

[hushed pewter]

You need that

[cunning slate]

So then, its circular

So its a cylinder

[hushed pewter]

The general formula is $W=\int \vec{F}\cdot d\vec{s}$

[twin meteor]

SWR

[hushed pewter]

But, for your application, this will reduce to $W=g\int h\cdot dm$

[twin meteor]

SWR

[cunning slate]

Hm, interesting

so how do i get the values necessary to compute this?

g, h, and dm

[hushed pewter]

g, you should know

[cunning slate]

9.8

[hushed pewter]

dm will be a (constant) function of h

You use the volume of a cylinder formula, along with the density of water

[cunning slate]

1000 kg/m^3

[hushed pewter]

$dm=\rho \pi r^2 dh$

[twin meteor]

SWR

[hushed pewter]

Where $\rho$ is density, and $\pi r^2 dh$ is the differential cylindrical volume

[twin meteor]

SWR

[cunning slate]

dm = (1000)(pi)(6^2)dh

What would dh be?

height?

distance?

[hushed pewter]

dh is dh

[cunning slate]

oh

so dh is just the

its like dx

[hushed pewter]

You use dh in $W=g\int h dm$

[twin meteor]

SWR

[hushed pewter]

yes

[cunning slate]

Oh.

W = 9.8 * integral with integrand (1000)(pi)(6^2)(3-x)? upper limit 0 to 3

[hushed pewter]

Looks good

[cunning slate]

can you tell me if im right for this problem?

So first, i found the upper and lower limits, 0 to 8 which corresponds to the depth of the water

then, the gravity, 9.8 with the integral from 0 to 8

after, i found hdh, so

the area for a sphere is 4/3 pi r^2

h

the height is (16-x)

the radius is 8

so the ending of this would be

9.8(1000) (1000 is the density btw) times the integral of a upper limit 8 and lower limit 0 with an integrand of: 4/3pi (8^2) (16-x)

[hushed pewter]

How are you getting 1000 for the density?

[cunning slate]

.

wait

good point.

would the density be...

50?

[hushed pewter]

the weight is 50 (pounds)

[cunning slate]

.

so. um.

what do i do?

[cunning slate]

it says per cubic feet though ._.

[hushed pewter]

oh derp

I'm illiterate

u right, u right

[cunning slate]

when i put in the 50 though

i still got the answer wrong

what did i do :(

[hushed pewter]

show me what you have in full

[hushed pewter]

8² is the wrong radius

radius depends on x

[cunning slate]

x? so

4?

oh wait

since its half of a sphere

should i just cut the volume in half?

i could do 4/6pi instead of 4/3

[hushed pewter]

no

radius is a function of x, yes. But it is neither 4 nor x

Also, the 4pi/3 shouldn't be here at all

You're not doing volume of a sphere

[cunning slate]

oh, i thought that was the volume

oh

[hushed pewter]

You're integrating differential cylindrical volumes

[cunning slate]

what does that mean?

[hushed pewter]

For a given x, what is the radius of the horizontal slice of the sphere?

[cunning slate]

Im not sure,

[hushed pewter]

Use some trig and take some time to work it out

Here, your radius is CB

and CD is your x value

[cunning slate]

ok

[hushed pewter]

Not trig, pythag

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[limpid current]

video is uploading

[limpid current]

Hey everyone, I'm a game developer and trying to complete my ocean waves which use 8 various sizes and speeds of Gerstner waves.

The problem should be solved with this concept (timestamped - about 20 seconds):
https://www.youtube.com/watch?v=kGEqaX4Y4bQ&t=802s

You can see in the video i uploaded, i have the goal of making the red part sit ontop of the ocean while not moving in x and y coordinates. The white cube shows the result of "testing" that x and y location. The Gerstner wave is deforming x, y as well as z of the ocean plane so this is expected.

math here:

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[limpid current]

let me know if you need more info to answer, i've tried to say everything useful i could

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[quasi cloak]

.close

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[upbeat gyro]

.close

[tribal sapphire]

how do i solve this?

[tribal sapphire]

please ping me

[autumn stratus]

!status

[vocal sleet]

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

[tribal sapphire]

idk how to start it

well i do

[autumn stratus]

ok

[autumn stratus]

what have you tried

[tribal sapphire]

@autumn stratus im basically running in circles 😂

[sullen shoal]

Try to eliminate the x term in 1 and 2
Then solve 3 and thing u got
From there ull get y and z
Then ull get x

[undone basin]

how can you solve this?
if we take any values of x,y and z then you will get a,b and c respectively cause they are all different variables

[proven nebula]

im confused do we have to find the proof here?

how will it even have a proof

[sullen shoal]

they are asking to solve in terms of a b c prolly

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[smoky trench]

how do i prove that the set of all rational numbers is countable

[vast shale]

What's the definition of "countable"

Do you know?

[smoky trench]

no i know what it means, but not the "mathematical" definition

[vast shale]

The mathematical definition is that, a set $A$ is infinitely countable iff there exists a bijective function $A \to \mbb{N}$

[twin meteor]

45

[smoky trench]

"infinitely countable"? arent these opposites?

[vast shale]

No

Q is infinite

Thus infinitely countable

There are sets, like the real and complex numbers, that are "bigger" than the natural and rational numbers

And are called infinitely uncountable

[smoky trench]

okayyy..

[vast shale]

I'm guessing this is your first time taking a discrete math class and you're learning about countability

I suggest reading about Cantor's diagonal argument if you're interested

In learning more

[smoky trench]

ive an exam tomorrow

[vast shale]

But back to the question

You need to prove that there exists a bijective function mapping the rationals to the naturals

[smoky trench]

and youre gonna guide me how to do this, right?

[vast shale]

Guide yes, but I won't spoon feed you the answer

And it won't be easy

#❓how-to-get-help

[smoky trench]

im ready

[vast shale]

Okay

What is a bijection?

[smoky trench]

a function that is injective and subjective both

[vast shale]

Yes

[smoky trench]

*surjective

[vast shale]

Yep

Okay

So you want to hit all elements in N and you want to hit them only once

[smoky trench]

yeah yeah

[vast shale]

One sec, I'll show you something

I want you to consider an infinite grid @smoky trench, this is something I wrote a paper on back in high school, and it's being graded right now, so I can't send it here but I can send it in dms

What we want to do is show that $\mbb{N} \cong \mbb{Q}^+$ first

[twin meteor]

45

[vast shale]

and I'll explain why

$\cong$ denotes that they're of the same size, or that a bijection exists

[twin meteor]

45

[smoky trench]

is that sign of congruence

this is high level maths tbh

[vast shale]

Yes, but it's used in different fields differently

e.g. isomorphism in Group Theory

Anyway

Here we will use it as a sign that two sets are of the same size

[smoky trench]

ok..

[vast shale]

so what you want to do now is

consider the set $\mbb{Q} = \left{\frac{p}{q} : p \in \mbb{Z}, q \in \mbb{N}, \gcd(p, q) = 1\left}$

[twin meteor]

45
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[vast shale]

Are you familiar with gcd?

Actually, we can even do without that

consider the set $\mbb{Q} = \left{\frac{p}{q} : p \in \mbb{Z}, q \in \mbb{N}\right}$

[smoky trench]

whats gcd

[vast shale]

Don't worry about it

[twin meteor]

45

[smoky trench]

q is not equal to 0

[vast shale]

Yes, that's why q is in N

[smoky trench]

ok ok

[vast shale]

I'm just thinking which proof I want to use here, since there's several

[smoky trench]

simples+easiest

[vast shale]

do you know what you're allowed to use? Tool-wise

[smoky trench]

of my exam

[vast shale]

yeah

what are you allowed to use

I have a neat proof for this but it's a bit technical

it's short though

[smoky trench]

so this is the syllabus

[vast shale]

and I don't know if you're allowed to take for granted that the countable union of countable sets is countable

[vast shale]

I think you're jumping ahead. Where does it suggest that you'll be asked to show that a set is countable?

This sounds like an introductory course in discrete math

[smoky trench]

its a pyq

[vast shale]

Okay

[smoky trench]

2nd semester

[vast shale]

?

If you state this, do you think they'll ask for a proof?

[smoky trench]

ofc

it came for 5 marks 2 years back

[vast shale]

Let's assume that the (countable) union of two countable sets is countable. We'll take this for granted. Then we can define a function $\phi : \mbb{Q} \to \mbb{Z} \times \mbb{N}$, where $\times$ denotes Cartesian product. Define this function $\phi$ to be such that for all $r = \frac{p}{q} \in \mbb{Q}$; $\phi\left(\frac{p}{q}\right) = (p, q)$ where $(p, q)$ is a tuple with $p \in \mbb{Z}$ and $q \in \mbb{N}$. We can prove that this is an injection trivially. We use the following result: Let $X$ and $Y$ be sets and $f : X \to Y$ an injection. Then $X$ is also countable. This shows that $\mbb{Q}$ is countable.

And p and q have no common factors^^

Otherwise we don't have an injection

[twin meteor]

45

[smoky trench]

ok

[vast shale]

Just memorize this tbh

You need to know two results

1. Countable union of countable sets is countable

2. If f : X \to Y is an injection and Y is countable, then X is countable

There's also a more "handwavy" way of showing this by constructing an infinite grid and showing that there's a direct bijection between N and Q^+, which is then extended to Q by showing that there's a bijection between Q^+ and Q^-

[smoky trench]

na this sounds hard

[vast shale]

It's not

But this method is quicker

[smoky trench]

i'll use this

[vast shale]

The only issue is

you're using two results not having proved them

Proofs can be found online tho

[smoky trench]

aight

thanks buddy

.close

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[obtuse zenith]

Solution to the implicit derivative of x^2 + y^3 = cosx + lny. One is mine, and the other is from my teacher's material. Is one better than the other? Or are both equally valid final answers?

[obtuse zenith]

Acording to mathway, both are equal to the other, mathematically

Or at least that is what i interpreted from that lol

[heavy yoke]

you just multiply the top and bottom of the left one by y and you get the right

[obtuse zenith]

Hm true, didn't notice that

.close

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[hazy sage]

Please let me know how to solve this. I somehow got 67 but the correct answer was 8. Don't understand at all how

[peak shard]

what is $\frac{124}{2}$

[twin meteor]

tobi

[hazy sage]

??

yo i got it

[peak shard]

okay do you know what I mean with my question

[hazy sage]

thx

ye ye

i got it

[peak shard]

what is it

[hazy sage]

its 62 and 62 square root is like 7.9

closest square # is 8

i got it thx g

[peak shard]

calculator?

[hazy sage]

what?

[peak shard]

did you use a calculator for it?

[hazy sage]

yea

.close

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[olive jungle]

How can I solve this integral?

[hidden kelp]

,rotate cw

[hidden kelp]

My first thought is factoring the denominator

[olive jungle]

ok

after that

[hidden kelp]

What class is this for?

[olive jungle]

12th

[hidden kelp]

So, calc 2 ish?

I‘m not american

[olive jungle]

I didn't understand

[olive jungle]

Are you able to photograph your solution and send it?

[hidden kelp]

I‘m trying to know what techniques of integration you know, and therefore I can use with you

Ah, I think I see it now

So you have $\int\frac{1-\ln x}{(x+\ln x)^2}\dd x$ right?

[twin meteor]

ℑμΤ𝛄𝛗θ

[olive jungle]

I did this step, but after that I was not able to complete the solution

[hidden kelp]

,rotate ccw

[hidden kelp]

Right

So this form

Don‘t you think it looks like the quotient rule? $\dv{x} \frac{f}{g} = \frac{f^\prime g - fg^\prime}{g^2}$

[twin meteor]

ℑμΤ𝛄𝛗θ

[hidden kelp]

So integrate both sides, you get f/g is integral of the big fraction, and you can compare the terms to find f and g

[olive jungle]

Yes, but I did not find anything suitable for the specialized derivative

[hidden kelp]

Well, g is obviously $x+\ln x$, and $f\cdot g^\prime = \ln x$

[twin meteor]

ℑμΤ𝛄𝛗θ

[hidden kelp]

So you can find f

[olive jungle]

can u speak arabic?

[hidden kelp]

I cannot

[olive jungle]

Because I started not understanding

[hidden kelp]

I can ask if another helper speaks arabic in the helper channel if you want

[olive jungle]

Yes please

[hidden kelp]

I asked. Can‘t guarantee anyone will respond, but fingers crossed 🤞

[olive jungle]

If you can find someone, tag me

thank u

[olive jungle]

What does the letter g represent?

and f?

@hidden kelp ?

[bleak sapphire]

Arbitrary functions

Have you learned the quotient rule?

[olive jungle]

yes

but I did not find anything suitable for the specialized derivative

[bleak sapphire]

Yeah ok

I don't think it's obvious

But you have a differential equation for f

[hidden kelp]

Just diff g and use fg‘ no?

No ode required

[bleak sapphire]

No cause you might have cancellation

In the numerator

I gtg shower but I think the ODE should be pretty straightforward..

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[eager moon]

I don't get the "Whole work will be done in (3 × 5) = 15 days" part

Like

Why did 1/5 became 5 and where did the 3 came from?

[dark crag]

how much work does A do in 3 days?

1/20+1/20+1/20+1/30+1/60

that equals to 12/60

or better said

1/5th

to do all 5 of the 5ths

A needs 3x5

[eager moon]

Wait how did you konw that there are "five 5ths"

[dark crag]

well A needs to do 1 whole of work

or how you would say it

[eager moon]

ohhhhhh

[dark crag]

20/20

which is 1/1 or 5/5

[eager moon]

1/ (1/5)?

oh

i see

and then the 3?

[dark crag]

so A needs 3 days to do 1/5

how long will he need to do 5/5

[eager moon]

yeah

ohhhhhhh

wow

got it now

thanks

.close

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[dark crag]

no problem

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[bold locust]

Can anyone help me at b)? I already solved a) but I am stuck

[bold locust]

I feel like I dont get the task completely. Would I show that the long definition in b) with the "if and only if" implies, that the eigenvalues of BB^T are non-negative? Or how would you approach the question

<@&286206848099549185>

[mellow oyster]

observe that BB^T is positive semidefinite

this helps out with one direction of the implication

if A = BB^T and BB^T is PSD, then A is PSD

the other direction of the implication to show the existence of B is definitely a lot trickier

[bold locust]

Wait

Am I trying to prove the following two statements? \

1. If $A$ is symmetric and positive semi-definite then there exists a $B\in\mathbb{R}^{n\times n}$ such that $A=BB^T$ \
2. If there exists $B\in\mathbb{R}^{n\times n}$ such that $A=BB^T$ then $A$ is symmetric and positive semi-definite.

[twin meteor]

Mathmatix

[mellow oyster]

yes

that's how you approach if and only if proofs

show both directions

[bold locust]

And you basically showed the 2) direction correct?

[mellow oyster]

well kind of

i showed that if A = BB^T then A is PSD

since BB^T is PSD

to show that A is symmetric is also simple enough, just show that BB^T is symmetric

a matrix is symmetric if it equals its own transpose

so what's (BB^T)^T?

[bold locust]

If BB^T is symmetric then it follows BB^T = (BB^T)^T and so on

[mellow oyster]

well yeah so that's your proof for 2

[bold locust]

The only trouble I have right now for 2) is, if we are proving the direction 2) why can we use the fact that BB^T is symmetric? Because arent we trying to show that in the sentence:

If there exists $B\in\mathbb{R}^{n\times n}$ such that $A=BB^T$ then $A$ is symmetric and positive semi-definite.

[twin meteor]

Mathmatix

[bold locust]

A is symmetric? For me its sounds a bit like we use the fact we want to prove as argument

[mellow oyster]

you are using that

you want to conclude that A is symmetric and PSD

BB^T is symmetric

since A = BB^T, you can conclude A is symmetric

BB^T is PSD, since A = BB^T you can conlude A is PSD

this isn't circular reasoning

[bold locust]

But our premise is "there exists a B such that A=BB^T". The conclusion is "A is symmetric and psd". But in the premise we dont know if BB^T is symmetric and psd? We just know that A is the product of B and B^T or not?

[mellow oyster]

but you can prove that BB^T is symmetric and psd

[bold locust]

And how would you proof that its psd and symmetric without using the fact that A is psd and symmetric?

[mellow oyster]

(BB^T)^T = BB^T

so BB^T is symmetric

regardless of what we know about A

[mellow oyster]

and this is the proof for BB^T being PSD

[bold locust]

Ahhhhhhh

I think I missed that for any matrix X, XX^T is symmetric

.close

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[vocal sleet]

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[prisma stag]

Okay somethings been bugging me

[prisma stag]

so this is a true statement i think

but

this

e^ix/2ix integrated from 0 to inf is undefined

[sly sierra]

sure

[prisma stag]

because Ei(0i) is undefined

[sly sierra]

just shows that you can't decompose it that way and still integrate

another simpler example,
$$\int_0^\infty 0,dx = 0$$

[twin meteor]

Bungo

[sly sierra]

but $0 = x - x$

[twin meteor]

Bungo

[sly sierra]

and $\int_0^\infty x ,dx - \int_0^\infty x,dx$ has the same problem

[twin meteor]

Bungo

[prisma stag]

i see

[sly sierra]

"integral of sum equals sum of integrals" is only true if the integrals are all convergent and finite

[prisma stag]

got it

[vocal sleet]

@prisma stag Has your question been resolved?

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[vagrant sable]

What does f:R—->R mean

[vagrant sable]

In functions

(That is supposed to be an arrow btw)

[mellow oyster]

you know how functions have domains and codomains?

[vagrant sable]

Ues

[mellow oyster]

the thing to the left of the arrow is the domain, the thing to the right is the codomain

[vagrant sable]

Absolutely

[mellow oyster]

so that's read as a function from R to R, or a function that maps real numbers to real numbers

[vagrant sable]

So a function like that had domain R and codomain R

[mellow oyster]

yeah

[vagrant sable]

Mhm so

F:R(+)—->R means that the domain is positive real numbers and the codomain is all real numbers

And f:[-1,5]—->R means the domain is the interval and the codomain is R

(Just want to clarify)

[mellow oyster]

yes

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[blazing sail]

anybody good at drawing those word problem vector diagrams with planes and ships and shit 😭 I cannot draw a single one properly

[blazing sail]

this is what I draw but idk how to figure out the rest/enough angles inside the triangle to solve the problem

[vocal sleet]

@blazing sail Has your question been resolved?

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[desert glen]

How could I solve for x here? $2xe^{x^2}=0$

[twin meteor]

dingypine

[desert glen]

gotta find the critical points and this is my derivative

[desert glen]

devidie both sides by 2x

[tawny nacelle]

I see

[desert glen]

but im left with e^x^2=0

which is undefined

and im like nah that cant be right

[tawny nacelle]

okay

first things first, e^{x^2} is well defined!

it's a function

the other thing is that, I would definitely not recommend dividing both sides by 2x

[desert glen]

yeah i didnt know what else to do lol

[tawny nacelle]

to see why, suppose I have x(x + 2) = 0, and I divide both sides by x

then I am left with x + 2 = 0, so x = -2

which is correct, but clearly x = 0 solves our original equation too!

so we've deleted a solution

[tawny nacelle]

use the zero product law

[desert glen]

oh i see

so basically never divide a 0 by x

[desert glen]

whats that

[tawny nacelle]

if ab = 0, then either a = 0 or b = 0

just replace "a" with "2x", and "b" with "e^(x^2)" here

[desert glen]

(2x)(e^(x^2)) = 0

so i see one of the solutions is just 0, for 2x

e^(x^2)=0, take ln of both sides and thats undefined tho

so can i just ignore htat one?

[tawny nacelle]

correct

[tawny nacelle]

that's correct, but we don't need to invoke logarithms to know that

e^x is always positive, not matter what x is

[desert glen]

ohh

[tawny nacelle]

in general, e^(any real number) is positive

[desert glen]

okay i see

[tawny nacelle]

x^2 is certainly a real number if x is

[desert glen]

i see

[tawny nacelle]

so this tells us that e^(x^2) has no roots

so the only solution is x = 0

[desert glen]

i see

okay thank you

i appreciate it

[tawny nacelle]

[desert glen]

.close

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[tawny nacelle]

happy to help!

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[desert glen]

why do i need to use lhoptials rule here

[desert glen]

im not really sure why this is considered indeterminate

i know its indeterminate when its 0/0 or something, but not like this

[lavish geyser]

you don't have to but if you do, it makes the continutiy part really easy
else do the lhs, then rhs and then fx

[heavy yoke]

$0 \cdot \infty$ is indeterminate

[twin meteor]

cloud

[lavish geyser]

it takes little time to solve lhs, rhs, fx
but if you do it with lhopital then it is really fast

[desert glen]

alright i see, thank you guys

.close

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[mellow wyvern]

The answer is 1%, but I got 0.1/1.57 which is around 6%

[mellow wyvern]

what am I doing wrong?

[vocal sleet]

@mellow wyvern Has your question been resolved?

[dark kiln]

0.01

between 1.565 and 1.575 difference of 0.01

you;re not given a number like 0.4 but a number like 0.32

[mellow wyvern]

isn't the uncertainty like 1.57+-0.1m^2

oh

oh ok that makes sense

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[olive jungle]

How can I complete this integration?

[karmic forge]

Try doing cotx=cosx/sinx

And then sinx=t

[olive jungle]

It's not possible

[karmic forge]

It works try it

[olive jungle]

ok

[olive jungle]

Here you mean by assumption؟

[karmic forge]

Ya

We get cosxdx=dt

Since sinx=t
Cosx=√(1-t^2)

[olive jungle]

Wait, I want to think about it

[fast path]

I think you need to lower cos^5 first

[orchid wren]

Haha

i love to see this integral and will probably give you one of the best methods to solve odd powers of tan and cot ,

Only three steps doesnt matter how long the problem is

[olive jungle]

I hope

[orchid wren]

So here you will consider ln cosecx = z

Bruh keep trust in me

[orchid wren]

Do this

So cotx dx = -z dz

Now cosecx = e^z

Or cosec²x = e^2z
or 1 +cot²x = e^2z

[olive jungle]

You mean to make an assumption?

[orchid wren]

Or cot²x = e^2z - 1

[orchid wren]

i mean substitution

[olive jungle]

yes

[orchid wren]

So our integral will become int (e^2z - 1)² and if you know binomial expansion

Then expand it next step

And write answer on the last one

As the easiest integration is of e^ax giving e^ax/a

@olive jungle Got it ?

[olive jungle]

thank u

close.

Question, is this method possible for all odd exponents in cot and tan? @orchid wren

[orchid wren]

Ofc

[olive jungle]

Sorry for bothering you

[orchid wren]

Np bro

[olive jungle]

close.

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[sweet flower]

!status

[vocal sleet]

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

[fervent wasp]

Square both sides

[smoky trench]

done. now

[fervent wasp]

You’re done?

[smoky trench]

x^2 +y^2 -2xy <= |x|^2 + |y|^2 + 2|x||y|

[smoky trench]

is this it?

just language now, right? that absolute value of x is greater than x and blah blah blah?

[mellow oyster]

might be useful to notice that |x|^2 and x^2 are the same thing, you could simplify this inequality

[fervent wasp]

You can simply remove the x^2 and y^2 from both sides

[smoky trench]

okay
-2xy <= 2|x||y|

[mellow oyster]

might as well divide by 2

[smoky trench]

-xy <= |x||y|

[mellow oyster]

now it's just a matter of casework for the possible signs of x and y

when x and y are both nonnegative, |x| = x and |y| = y

repeat this for x negative and y positive, x and y negative, and y positive and x negative

[smoky trench]

aight thanks buddy

can i ask next question here?

[fervent wasp]

Sure, but I suggest you open a new channel

[mellow oyster]

alsi

also

i just realized

you don't have to do the casework

it's much simpler

|x| is either +/- x

[mellow oyster]

so divide both sides of this by |x|

then x / |x| is either 1 or -1

same thing when you divide both sides by |y|

so the right side becomes 1

and the left side becomes -1 * +/- 1 * +/- 1

which is always either 1 or -1

so your inequality just becomes 1 >= +/- 1

which is always true

[smoky trench]

yeah this is much shorter

appreciate it!

imma open a new channel now.

.close

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[bold kayak]

can someone help me rq for my math exam

that happens very soon

for question 3

[bold kayak]

how did i create this venn diagram.

[golden yew]

Let's say that the number of people in the intersection is x

Then what kind of equation do you think we can create

[bold kayak]

what

oh vm

nvm

i understand it now

.close

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[smoky trench]

prove that every equation of the nth degree has n roots and no more

[sly sierra]

using what theorems?

this is essentially the fundamental theorem of algebra

[smoky trench]

correct

[vocal sleet]

@smoky trench Has your question been resolved?

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[wet cipher]

Give Alpha or Beeta =7

Given

<@&286206848099549185>

.close

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[wraith path]

hi，guys I'm new to here.😁
I heard from other students, that there is lot of masters that will help with my question.can you guys help? thank you so much!!!😆

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[onyx marsh]

part 2

[onyx marsh]

i tried u=(1-t^2)^(n+2) and dv=1 for F_{n+1}(x)

$F_{n+1}(x)=\int_0^x\frac{\dd t}{(1+t^2)^{n+1}}=\frac{t}{(1+t^2)^{n+1}}|_0^x-\int_0^x-(n+1)\frac{2t}{(t^2+1)^{n+2}}\dd t=\frac{t}{(1+t^2)^{n+1}}|_0^x+(n+1)\int_0^x\frac{2t}{(t^2+1)^{n+2}}\dd t$

[twin meteor]

pirateking0723

[onyx marsh]

oh wait i will continue maybe that will lead to something

[onyx marsh]

$=\frac{x}{(1+x^2)^{n+1}}+(n+1)\frac{(x^2+1)^{-n-1}}{-n-1}-(n+1)\frac{1}{-n-1}=\frac{x-1}{(1+x^2)^{n+1}}+1=(x-1)\frac{\dd F_n(x)}{\dd x}+1$

[twin meteor]

pirateking0723

[onyx marsh]

is this correct

<@&286206848099549185>

[craggy rain]

hmm

is correct

[craggy rain]

yes

[onyx marsh]

tysm

.close

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[onyx marsh]

.reopen

[vocal sleet]

✅

[onyx marsh]

but this leads to some problems

for part 3

$I_1=\lim_{x\to\infty}F_1(x)=\lim_{x\to\infty}(x-1)(\frac{\dd F_0(x)}{\dd x}+1)=\lim_{x\to\infty}\frac{x-1}{x^2+1}+1=1$

[twin meteor]

pirateking0723

[onyx marsh]

$I_0=\frac{\pi}{2}$ so using 3)a) $I_1=\frac{2(0)+1}{2(0+1)}\frac{\pi}{2}=\frac{\pi}{4}\neq 1$

[twin meteor]

pirateking0723

[onyx marsh]

wait i have a mistake here the integral that i used is F_n(x)

not F_{n+1}(x)

[onyx marsh]

all of this is wrong

i messed up

$F_n(x)=\int_0^x\frac{\dd t}{(t^2+1)^{n+1}}=\frac{t}{(t^2+1)^{n+1}}-\int_0^x-(n+1)\frac{2t^2}{(t^2+1)^{n+2}}\dd t=\frac{t}{(t^2+1)^{n+1}}+(n+1)\int_0^x\frac{2t^2}{(t^2+1)^{n+2}}\dd t$

[twin meteor]

pirateking0723

[onyx marsh]

is this correct

if yes then what to do from here

i dont see anything that can be done from here

<@&286206848099549185>

<@&286206848099549185>

[vocal sleet]

@onyx marsh Has your question been resolved?

[onyx marsh]

<@&286206848099549185>

anyone willing to help I am not finding a way to solve part 2

<@&286206848099549185>

[vast shale]

ive just taken a course on this. my best suggestion is that you should try subbing, or try the A B C method. not sure if it the method A B C is its actual name (ps i should probably remove my helper tag as i am now struggling with the same thing you are)

[onyx marsh]

by ABC do you mean decomposing a rational function into a sum of rational functions??

[vast shale]

yes

[onyx marsh]

<@&286206848099549185>

[vast shale]

exactly thats what i thought about

[onyx marsh]

but how would that help

i will have to sum like n+2 rational functions

[vast shale]

thats true

[onyx marsh]

and the terms that i am summing will be products of many factors too

so i dont think thats the way to go

<@&286206848099549185>

[viral plover]

t^2 = (t^2 + 1) - 1

[viral plover]

for this ^

[onyx marsh]

but i will have the dF_n/dx in the expression too right

[viral plover]

yh

[onyx marsh]

also here for the first term which is not the integral t should be replaced with x

[viral plover]

but i would just leave it explicit rather than writing as dF/dx

[viral plover]

yh

[onyx marsh]

so something like $F_n(x)=\frac{x}{(x^2+1)^{n+1}}+(n+1)F_n-2F_{n+1}\implies F_{n+1}=nF_n+\frac{x}{(x^2+1)^{n+1}}$?

[onyx marsh]

why is that

[viral plover]

because in the next part youre going to take the limit as x -> infinity

[viral plover]

this is not quite right

your coefficients infront of Fn and F(n+1) are wrong

[twin meteor]

pirateking0723

[viral plover]

i mean in the first equation sorry

here

[onyx marsh]

oh yes the coeff of F_n should be 2(n+1)

but the coeff of F_{n+1} is correct

[viral plover]

no

[onyx marsh]

right

[viral plover]

this is right

but the coef of F(n+1) should be the same

[onyx marsh]

ohh yes you are right

but with different signs

the coeff of F_{n+1} is -2(n+1)

the coeff of F_n is 2(n+1)

[viral plover]

yes

[onyx marsh]

$F_n(x)=\frac{x}{(x^2+1)^{n+1}}+2(n+1)F_n-2(n+1)F_{n+1}\implies F_{n+1}=\frac{2n+1}{2n+2}F_n+\frac{x}{(2n+2)(x^2+1)^{n+1}}$

[viral plover]

very nearly

this coeff is wrong

[onyx marsh]

2n+1

not -1

[viral plover]

yh

[twin meteor]

pirateking0723

[onyx marsh]

now as x goes to inf F_{n+1} goes to kF_n where k is the above coeff

which proves 3a

[viral plover]

yes

[onyx marsh]

but with the lim for kF_n too

[onyx marsh]

is there another way to figure part 2 without this trick

[viral plover]

i mean probably

but that is the definitely the intended and easiest way i would say

[onyx marsh]

yes that was neat

[viral plover]

also a very common trick (adding 0 i.e. adding and subtracting something in order to allow your to rewrite the expression in a more favourable way)

you also see this with multiplication

multiplying by 1, for example when you rationalise the denominator

[onyx marsh]

yes it didnt come to my mind this time although ive seen it alot before

[viral plover]

$$\frac{1}{\sqrt{3} + 1} = \frac{1}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \dots$$
is the same principle

[twin meteor]

Acman

[onyx marsh]

yes but with multiplication instead of addition

[viral plover]

yeah it can be tricky to spot sometimes

yeah

[onyx marsh]

for 3b $I_0$ clearly satisfies what we are trying to prove, now suppose that $I_k=\frac{(2k)!}{4^k(k!)^2}$ where $k\in\mathbb{N}$ then using 3a $I_{k+1}=\frac{2k+1}{2(k+1)}I_k=\frac{(2k+1)(2k)!\pi}{4(k+1)4^k(k!)^2}=\frac{(2k+2)!}{4(k+1)^24^k(k!)^2}\frac{\pi}{2}=\frac{(2(k+1))!}{4^{k+1}((k+1)!)^2}\frac{\pi}{2}$

[twin meteor]

pirateking0723

[onyx marsh]

and this proves it

now i still have 3c

let me think about it for a sec

ok i figured it out

[viral plover]

nice

[onyx marsh]

oh wait i was doing smething wrong

ill rethink

[viral plover]

ill leave a small hint ||its easier to use the result from 3a than 3b||

[onyx marsh]

i am trying to reach something like (2n)!<2^n(n!)

i didnt see the hint yet and i will try more before seeing it

[viral plover]

do you mean not to have the square on the RHS ?

[onyx marsh]

yes

[onyx marsh]

is this wrong

[viral plover]

yes

(2n)! = (2n) (2n-2) (2n-4) ... (4) (2) (2n-1) (2n-3) ... (3) (1) = 2(n) 2(n-1) 2(n-2) ... 2(2) 2(1) (2n-1) (2n-3) ... (3) (1) = 2^n n! (2n-1) (2n-3) ... (3) (1)

so (2n)! is much bigger than 2^n n!

[onyx marsh]

oh yes

i can rewrite (2n-1)(2n-3)...(3)(1) as (2n-1)(2(n-1)-1)(2(n-2)-1)...(3)(1)=2^n(n!)-c<2^n(n!)

so (2n)!<4^n(n!)^2

and $\frac{(2n)!}{4^n(n!)^2}>0$ so $\lim_{n\to\infty}0<\lim_{n\to\infty}\frac{(2n)!}{4^n(n!)^2}<\lim_{n\to\infty}1\implies 0<\lim_{n\to\infty}\frac{(2n)!}{4^n(n!)^2}\frac{\pi}{2}<\frac{\pi}{2}$ so $I_n$ converges

[twin meteor]

pirateking0723

[onyx marsh]

$\lim_{n\to\infty}I_{n+1}=\lim_{n\to\infty}I_n$ but that doesnt prove convergence

[twin meteor]

pirateking0723

[viral plover]

i was going more for I(n+1) < I(n) and I(n) > 0 for all n

[onyx marsh]

is this correct

[viral plover]

hm im not convinced tbh

[slim patrol]

k

[onyx marsh]

how does that help

[onyx marsh]

what didnt convince you

maybe i can convince you unless i did something wrong

[viral plover]

what have you actually done on convergence

like what have you been taught to do for questions where you need to justify the convergence

[viral plover]

if you have done monotone convergence theorem then that is suffficient to imply convergence

[onyx marsh]

to show that as n goes to inf I_n goes to a constant ?

[onyx marsh]

no

we didnt do stuff like this just the def of convergence

[viral plover]

diff of convergence ?

oh def ?

yh ok

[onyx marsh]

yea mb

[onyx marsh]

ive studied these alone but i didnt use them in a very long time so i am starting to forget them

but even if i still know them quite well i cant use them in the school rn

because its not something that we have covered in class

[viral plover]

yeah thats the issue with questions like this, you dont actually have the machinery to do what its asking

so you have to be very hand wavey

[onyx marsh]

and here this is probably insufficient i mean maybe the sequence isnt approaching a number

although is it bounded

[viral plover]

here's how i would do it then

[onyx marsh]

just like the lim of sinx as x goes to inf for example

although sinx is bounded but it doesnt converge

[onyx marsh]

so this doesnt prove anything probably

[viral plover]

yeah

you havent really shown that it converges

youve said if it converges this is the range it would converge in

[onyx marsh]

so how would you do it

[viral plover]

i think you just have to write something vague like '(2n+1)/(2(n+1)) = (1 - 1/(2(n+1)) and so the ratio of successive term is getting closer and closer to 1, meaning the change in size of I(n) gets slower as n increases' or smth like this

i dont think there is a satisfying way to answer that question given the content youre allowed to use

[onyx marsh]

hmm but also that doesnt prove it because it depends on the convergence of I_n in the first place

so we are letting -I_n/(2n+1) go to 0 as n goes to inf although we dont know anything about I_n as n goes to inf

[viral plover]

well we know that 0 < I(n) < pi/2

so it is true that I(n)/(2(n+1)) -> 0

it just also isnt a real proof of convergence

but neither is that what theyre asking for or what is achievable

they just want to you to say something along these lines to justify it

[onyx marsh]

ohh ok

[viral plover]

i think best to ask your teacher about it, they will have a much clearer idea on what is an expected response to that type of question

[onyx marsh]

yes you are right about this

tysm for your help and sorry if i wasted your time

have a nice day/night

.close

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[viral plover]

yw and not at all :)

[viral plover]

you too

[vocal sleet]

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[violet compass]

Someone help in differentiation here

😭

[scenic ravine]

!status

[vocal sleet]

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

[violet compass]

Between 1 and 2 ig

[scenic ravine]

essentially $\frac{4 \pi r^3}{3} \rho=k$

[twin meteor]

ƒ(Why am. I here)=I don't know

[scenic ravine]

makes sense ?

[violet compass]

This was the only question for which i had to see solution and i still dont get it

[violet compass]

Yes

So the

[scenic ravine]

differentiate this please

[violet compass]

Differentiation would be

Zero

[scenic ravine]

no, differentiate the LHS

[violet compass]

Is thus correct

[scenic ravine]

I think so yeah

so now what do you have

[violet compass]

Equals to zero

?

[scenic ravine]

good

carry on

[scenic ravine]

wait

[scenic ravine]

this is wrong

it should be

[violet compass]

Yea

I was stuck there only

[scenic ravine]

$\frac{dr}{dt}$

[twin meteor]

ƒ(Why am. I here)=I don't know

[violet compass]

Begun and got stuck

[scenic ravine]

as you differentating wrt time

[violet compass]

Ohh

[violet compass]

What to do of cube

[scenic ravine]

so 4πr^2 dr/dt

[scenic ravine]

let me write the entire thing out in TeX

wait a minute

$\frac{4 \pi r^2 dr\rho}{dt}+\frac{4 \pi r^3 d(\rho)}{3dt}$

[twin meteor]

ƒ(Why am. I here)=I don't know

[violet compass]

But why is it not dr^3/dt

[scenic ravine]

I differntiated that using the chain rule

[violet compass]

What method is this ig my teacher didnt teach me

[scenic ravine]

do you know the chain rule

[violet compass]

Yes

I do

No

Whats implicit

Differentiation

[scenic ravine]

Not required here

used the wrong term

my bad

[scenic ravine]

can you do it from here?

[violet compass]

no

i only knew i have to differentiate

dunno further

[scenic ravine]

ok, should I help ?

$4\pi r^2\rho\left(\frac{dr}{dt}\right)+\frac{4\pi r^3}{3}\left(\frac{d\left(\rho\right)}{dt}\right)=0$

[twin meteor]

ƒ(Why am. I here)=I don't know

[violet compass]

yes

[scenic ravine]

$4\pi r^2\rho\left(\frac{dr}{dt}\right)=-\frac{4\pi r^3}{3}\left(\frac{d\left(\rho\right)}{dt}\right)$

[twin meteor]

ƒ(Why am. I here)=I don't know

[scenic ravine]

yes?

$\left(\frac{dr}{dt}\right)=-\frac{r}{3}\left(\frac{d\left(\rho\right)}{\rho\ dt}\right)$

[twin meteor]

ƒ(Why am. I here)=I don't know

[scenic ravine]

makes sense until here?

[violet compass]

yes

[scenic ravine]

now what?

[violet compass]

now integrate or wha

[scenic ravine]

no

read the question again

[violet compass]

velocity

[scenic ravine]

and what is constant ?

[violet compass]

f/a

[scenic ravine]

wait

[violet compass]

but will have to differentiate accereleration

[violet compass]

its mass no?

so f/a

[scenic ravine]

$\frac{d\left(\rho\right)}{\rho\ dt}=k\ ;\ k\in\mathbb{R}$

[twin meteor]

ƒ(Why am. I here)=I don't know

[scenic ravine]

yes?

[violet compass]

yes

[scenic ravine]

now can you do it?

it reduces to ||v=-kr/3|| Spoilers!

[violet compass]

😭 im clueless

[scenic ravine]

do you follow until here?

[violet compass]

yes

[scenic ravine]

and this?

[scenic ravine]

if so, simplify this for me

[violet compass]

how

i only know basic differentiation

this is very new to me

ty for helping ill ask my teacher

[scenic ravine]

I'll help

*help

[violet compass]

you wont be able to on text

[scenic ravine]

okay

[violet compass]

cause this is very new to me

[scenic ravine]

I think you've almost got it

[scenic ravine]

what is velocity in terms of derivatives

dv/dt, right?

[violet compass]

wait isnt it

differentiation of acceleration

[scenic ravine]

I mean dr/dt

[scenic ravine]

no

[violet compass]

oh yea

dv/dt is acceleration

[scenic ravine]

yes

now what is velcoity

[violet compass]

dx/dt

[scenic ravine]

good

and here x =?

[violet compass]

x is r?

[scenic ravine]

yes

[violet compass]

damn

[scenic ravine]

so can you do it now?

[violet compass]

i do get it but im def having problem while differentiating

everytime i get stuck in physics its cause of complicated calculation 😭

[scenic ravine]

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1a/a/chain-rule-review

use thus

*this

[violet compass]

what is "inner" in our problem

[scenic ravine]

here, nothing, but I thought you may want to brush up on derivatives

I could be wrong though, I'm pretty bad at maths

But worse at other subjects

[violet compass]

how does chain rule apply here then

[scenic ravine]

used when we differentiate rho wrt t

and r wrt to t

[violet compass]

crazy but i get nothing

anyways

ty for help

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[odd olive]

,w that is the solution of ×^5-x^4+1=0

[twin meteor]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by lanallana

[odd olive]

Xpower5-Xpower4+1

[peak matrix]

,w x^5 - x^4 + 1

[twin meteor]

Results provided by WolframAlpha

Click here to refine your query online
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Requested by methisalwaysright

[odd olive]

Thanks!

[peak matrix]

np

you can use #bots or #latex-testing for this btw

or even DMing texit

[vocal sleet]

@odd olive Has your question been resolved?

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[sharp comet]

is this congruent

or no

would it be sas ?

because it can't be HL

because we don't know the hypotenuse side length

and for HL you need to know that

so it has to be congruent through sas right?

ok

thanks

[fervent wasp]

🙂

[sharp comet]

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[frail dew]

Brain can not compute.

f_y = 3(x) - 3x

That's what I got from plugging equation 5 into 4.

Where did they get y^4 = y?

[solid nacelle]

i am pretty sure that when you substitute the equation 5 in 4 they get y^4 but im pretty dumb

[frail dew]

yeah but how?
x = y^2
sub that into 3y^2 = 3x
you get 3x = 3x

[glossy maple]

Can you show the rest of the question?

If you sub x = y^2 into (4) you will definitely get 0

Because that's what x = y^2 is a solution of

Maybe they meant to sub it into the original function or something