help-0

[white marsh]

k=1

[gray isle]

!nosols

[lone heart]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[normal walrus]

oh wait !

i forgot that constants become 0

[white marsh]

oh I am sorry

[normal walrus]

my bad

i forgot that constants like 5 become 0 when getting the derivative of a function

yeah i but i kept the 5 making me really confused

my brain is fried

ive been solving stuff since yesterday

ok thanks everyone

.close

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[cinder sundial]

how many real number "a", can we find?

above is the question from my book

[lone heart]

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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[cinder sundial]

does it makes sense

my sentence

[tardy tapir]

yes, it does

[modern sedge]

try applying FTC

(fundamental theorem of calculus)

that should help you to rewrite the integral

[cinder sundial]

i did, but i also notice that it is a odd function

[modern sedge]

what is odd function? The derivative? The f(x) itself?

I feel like none of those are odd

[cinder sundial]

f(x) is odd

[modern sedge]

why do you think?

[cinder sundial]

these area my cancel out.

[modern sedge]

you are integrating f derivative of x

not f itself

and that doesnt imply that f is odd

neither f' is odd

[cinder sundial]

god

i get tricked by the prime symbol

maybe im tired

i think i need a 5 min break

the question is temporily solved

[modern sedge]

alright, take a break and then try approaching it with FTC

[cinder sundial]

for sure

thank you

.close

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[simple saddle]

why fcp not useful here?

and combinations used?

[ionic jewel]

what is fcp

[merry depot]

It is useful here….

[simple saddle]

fundamental countig principle

[simple saddle]

how

[ionic jewel]

Yeah it's required(I think?) to solve the problem

[simple saddle]

its giving wrong result

[merry depot]

Count the number of ways to get freshman, and then the number of ways to get sophomores.
Fcp says multiply those together

[simple saddle]

7x6x5x4

5x4x3

but the answer is not correct here

[azure swan]

The order of picking the people is here irrelevant

[simple saddle]

ohh rightt right

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[tawdry oyster]

i am need help with this

[vapid shuttle]

!status

[lone heart]

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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[tawdry oyster]

!status

[lone heart]

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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[tawdry oyster]

I don't know where to begin

[sleek gyro]

why dont you start by writing down the first few elements of the sequence? maybe you notice something

[lone heart]

@tawdry oyster Has your question been resolved?

[thick prism]

There is a proof for this by contradiction that goes like this: Suppose xn+1n sin(xn) is periodic with period N. Then, since xn+1n sin(xn) = xn mod N, we have xn+1n=xn mod N. Since xn+1n=xn mod N, there exist integers m and h such that xn+1n\sin(xn)=xn+mn+h=x(n+m)+h. This contradicts the result that the sum of sines is not periodic.

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[humble hearth]

The question and my attempt

[tardy tapir]

it is correct

[humble hearth]

Oh thanks, so to get my f(x) I just need to let one side go to 0?

[tardy tapir]

what?

wdym by get f(x)

[vale wigeon]

your handwriting could be better... make those letters and numbers bigger and more distinguishable

also you should mention explicitly that the function (x-2)^2 - x/(x+2) is continuous on [2,3]

otherwise you haven't explained why the intermediate value thm applies

[humble hearth]

To get the function that let's me input x=2, x=3 in order to find the interval

[humble hearth]

Ahh okay gotcha, I've always had trouble with my handwriting lol

[humble hearth]

Ahhh I see, but does the same method always apply for all equations? Can I always get f(x) by minusing both sides by whatever that's there

[vale wigeon]

B tier wording

but yes

[humble hearth]

Sorry my English isn't that great lol

OK gotcha thanks 👍👍

Side question, how could I have worded that better? Just for future reference

@vale wigeon

[vale wigeon]

"Can I always turn my equation into f(x) = 0 by subtracting the RHS from both sides?"

probably

[lone heart]

@humble hearth Has your question been resolved?

[lone heart]

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[elfin vector]

Hi! Can someone help me with this probability question pls, I'm not sure if it's correct or not.

[real gazelle]

looks good! another way to do it is to note that there are 5C3 = (5*4*3)/(3*2*1) = 10 total possibilities for how to order them and only 2 of them result in A being ahead, so you get 2/10 = 1/5

[elfin vector]

Thx! Also, how did you know that there are 2 possibilities for A to be ahead (without writing out the possibles combinations that i did)?

also the 5C3 means that in five spots which 3 can be the A?

[real gazelle]

Oh I just wrote out the possibilities like you did

[real gazelle]

yup!

exactly

[elfin vector]

ah i see

thx for the help

[real gazelle]

no problem!

[elfin vector]

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[alpine sable]

Need help with algebra 1

[alpine sable]

i will send the question

what is continuous or discrete graph? I do not think I learn this in 7th grade

algebra 1 math 8th gradwe

[wind cloak]

If you can draw a graph without lifting your pen up, it's continuous

If it involves dots on the plane, it's discrete

[alpine sable]

oh wait i think my science teacher taught me this

6th grade

wait

so for 13

is it continuous

ok hold on let me write out the equations because the images are super pixelated

13. c = 6w
14. n = 24 - 2t
15. C = 37 - 1.5t

[wind cloak]

Didn't know that discontinuous functions are also considered discrete

[alpine sable]

ok i see, discrete is like that type of data but continuous has data that keeps on movig

[wind cloak]

indeed

no holes

[alpine sable]

well now i gotta solve these

for 13, c = 6w pretty simple model for a slope equation

its probably continuoos

but wait, what does a discrete graph look like as an equation?

[wind cloak]

The function would be piecewise

[ocean seal]

NEON

[wind cloak]

Something like this

[alpine sable]

I notice that discrete cannot be broken into fractions

[wind cloak]

wdym

[alpine sable]

hold on

[lethal belfry]

Wouldn't that be discontinuous, I don't think a discreet function is the same as a discontinuous function

[wind cloak]

I was giving an example for a piecewise function

mb should have kept it relevant

but their example clearly indicates the floor function as discrete which throws me off

[wind cloak]

I think we might be missing context

[alpine sable]

kind of

i think

maybe not

[pale kestrel]

thats honestly not a great explanation.

[pale kestrel]

And god this definition sucks

[alpine sable]

true

doesnt really tell me what it is

[pale kestrel]

bruh nah this is absolutely awful

that thing is not even a function

[wild umbra]

how can it have

[wind cloak]

lol

[wild umbra]

more than 1 values of y

[pale kestrel]

steep steps work but nah

obviously not what they drew

[alpine sable]

doesnt even start anywhere its literally just an arbritary point that keeps on staggering

[pale kestrel]

This is just some godawful notes

[alpine sable]

mb

sorry

anyway

[pale kestrel]

Ok use this:

Firstly, you know what a function is right?

[alpine sable]

yeah

[pale kestrel]

for every x, you have exactly 1 output y

[alpine sable]

its a input output thing

ye

[pale kestrel]

Right next

continuous means you can draw the graph

without taking your pen off the paper

[alpine sable]

Hm ok so like a slope equation

[pale kestrel]

so 1st one is continuous, 2nd isnt.

And 'discrete' is an awful name

I would simply say "not continuous"

or "non-continuous"

[alpine sable]

just say continuousn't

[pale kestrel]

If someone said 'discrete function' to me, I'd be thinking of something else

but yeah if they call it discrete in your class they call it discrete, but now you know 🙄

[alpine sable]

so discrete is not a slope equation

[pale kestrel]

f(x) = ax + b (linear function) will always be continuous

any linear function, any quadratic function - those are continuous.

Or any polynomial in fact

[alpine sable]

sorry if i sound dumb but what does a and b stand for in f(x) = ax + b

[pale kestrel]

any constants

sorry shoulda said

[alpine sable]

oh oh my bad

[pale kestrel]

f(x) = 3x - 2 is an example

[alpine sable]

yeah i understand

the video im watching is giving me the impression discrete data is data that cant be broken down into decimals because its giving me things like tickets and people which you cant split

[pale kestrel]

[alpine sable]

the video is not helping

[pale kestrel]

people usually use 'discrete' to mean something else than what this says

and that video you screenshotted look bad

[alpine sable]

idk why people said it was helpful

all i got from the video is that discrete is dots and discrete is not able to be split

It helps somewhat

[real gazelle]

Yeah, so just to give an example, the number of people is discrete because you can't have half of a person or something

The length of a table is continuous because you can have tables with any positive length (can be 1, 1.5, 1.24, 1.4546 meters, etc)

[pale kestrel]

You can still have decimals in a discrete data set

(but note this usage of the word 'discrete' has nothing to do with the function things you were seeing up there - thats continuity of functions)

[alpine sable]

isnt this just

a slope equation slipped around

[real gazelle]

Yeah, to give another example the amount of money you can have is discrete because it jumps from $1.25, $1.26, etc and you can't have like $1.2555 or anything

[real gazelle]

Do you mean slope-intercept form?

[alpine sable]

n = 24 - 2t

[alpine sable]

yeah

[real gazelle]

If so then yes it's basically just slope-intercept form

[alpine sable]

so the truck with 24 chairs

the 24 chairs is b

and 2t is the mx?

y = b - mx??/

hm??

[tacit arch]

What exactly is your question

[alpine sable]

this

[tacit arch]

I read it yes

What is your question

[alpine sable]

is n = 24 - 2t continuous

graph

[tacit arch]

Equations of lines are continuous yes

[alpine sable]

ok thank you

next question

my answer is just a bunch of metrics jargon and then I said something like, the water continuously decreases therefore its a continuous graph

is that a good answer

<@&286206848099549185>

[real gazelle]

wait

[tacit arch]

?

[real gazelle]

this is not continuous (in the sense of continuous vs discrete), no

because you only remove 2 chairs at a time

so t can only be 0, 1, 2, etc

it can't be something like 0.5

[alpine sable]

you remove 2 chairs per trip

[real gazelle]

so the graph should look like a bunch of dots

[real gazelle]

yeah, so the number of chairs is 24, then jumps to 22, then jumps to 20, etc

[alpine sable]

Can't see it

Is dat I or t

[real gazelle]

it never takes on any of the intermediate values like 23 or 23.5

t

[alpine sable]

wait

are the first 2 questions discrete then

oh wait no

14 is only

[real gazelle]

Yeah, just 14 is

for 13, you can have any number of pounds of walnuts

so it should be a continuous line, basically

and for 15, you can have any number of hours passed

[alpine sable]

so for 14 can I say that its discrete because its 2 chairs at a time and it cannot be any number of chairs

[real gazelle]

yeah

exactly

[alpine sable]

boom

dude i gotta get these right bro they counting my gpa in 8th grade

i think

because i got algbra 1 maths

i gotta hustle

at least my teacher gave me the homework friday dude i would be busted if it was by tomorrow

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

no

i mean yeah

ok goodbye

[lone heart]

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[alpine sable]

How did we manage to reduce it from n squared to just n when we extracted it from the root

[wind cloak]

$\sqrt{x^2} = x$ for all $x > 0$

[ocean seal]

NEON

[wind cloak]

This formatting of yours is ugly and misleading

[tacit arch]

$\eta \neq n$

[ocean seal]

riemann

[alpine sable]

.close

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[weak seal]

could this

[weak seal]

be equal to

1-P(X<8|Y=1)?

[real gazelle]

yup they are equal

[weak seal]

alright thank you

.close

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[placid ivy]

I was playing a game & wanted to figure out how much upgrade crystals it'd take to upgrade items on average. Each upgrade(+1) reduces the starting 100%(+0) success chance by 2% & failing downgrades the item once(-1). Max upgrades are +10 +15 +20 & +25 for the different rarities & each rarity costs different amounts of upgrade crystals.

[azure swan]

If the item is downgraded due to a failed upgrade, does the success probability increase or does it stay the same?

[placid ivy]

yeah it goes back up by 2% if it fails & decreases by 2% if it succeeds.

I've been trying to make a formula where I can just insert the upgrade crystal cost & the max upgrade possible for the item to get the average amount of upgrade crystals used but I can't figure it out.

[azure swan]

Hmm you could model it as a Markov Chain on a straight graph

[placid ivy]

i have no idea what that is lol

[azure swan]

Wait I will try to sketch it

It looks like this

Each Circle is a state, which Describes the number of upgrades, you currently have

(n is the amount of upgrades you want to achieve)

[placid ivy]

ok

[azure swan]

And the arrows describe the probability, of how the states changes

Now we can form a system of equations, to get the expected value of the needed tries

Let's denote $E_k$ as the expected amount of steps to get to the state k

[ocean seal]

Jangler

[azure swan]

Then we get by rekursion
$$E_n = (1-(n-1)0.02) E_{n-1} + 1$$
$$E_{n-1} = (1-(n-2)0.02))E_{n-2} + 1 $$
$$E_0 = 0$$
$$E_k = (1-(k-1)(0.02))E_{k-1} + (k+1)0.02E_{k+1} + 1 $$
for $ 1 \leq k \leq n-1$. ($E_0$ is the start state).

[ocean seal]

Jangler

[azure swan]

Welp and this needs to be solved for E_n oof

[lone heart]

@placid ivy Has your question been resolved?

[lone heart]

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[alpine sable]

how can -cos(x) be the antiderivative of sin(x), therefore give the signed area of sin(x), but have the value 0 at x=1.5? The area of sin(x) from 0 to 1.5 isnt actually 0, is it?

[vale wigeon]

just because $-\cos(x)$ is an antiderivative of $\sin(x)$ does not specifically mean that $-\cos(x) = \int_0^x \sin(t) \dd{t}$. we write indefinite integrals with $+C$ for a reason.

also it's at pi/2 and not at 1.5 that you see the zero.

[ocean seal]

Ann

[alpine sable]

but that opposes everything I just learned

I thought that F(x) = integral from a to x of f(t) dt

[vale wigeon]

$\int_0^{\pi/2} \sin(x) \dd{x} = -\cos(t) \big|^{\pi/2}_0 = -\cos(\pi/2) - (-\cos(0))$

[ocean seal]

Ann

[vocal quiver]

blatant pi =/= 3 propaganda

[vale wigeon]

sure is, but if you want to calculate the antiderivative of sin and get specifically -cos(x) and NOT 1 - cos(x), then you will need a = pi/2 and not a = 0.

[vale wigeon]

are you serious or joking rn

if joking then like. please don't interrupt my genuine/serious explanation w/ a joke. i don't like it when people do that.

[vocal quiver]

Gotcha, sorry

[alpine sable]

I'm sorry, what does 1 - cos(x) have to do with this? I thought the anti derivative of sin(x) is -cos(x)?

[vale wigeon]

you realize there's no such thing as the antiderivative, right

it's defined only up to a constant shift

you need to grapple with that, it is impossible to progress otherwise

[alpine sable]

yeah, I get that

-cos(x) + C

[vale wigeon]

yeah

so you should be ok with the idea that -cos(x) and -cos(x)+1 are two functions that are both antiderivatives of sin(x)

[alpine sable]

yes

[alpine sable]

in what way does C matter when trying to calculate the integral?

[vale wigeon]

wym

for a DEFINITE integral it doesnt

(but you need to be clear about your bounds)

for an INDEFINITE integral it matters insofar as you actually describe the whole family of antiderivatives

[alpine sable]

when I put it into an integral calculator, I get this value

[alpine sable]

but when trying to do it with the method I just got to know

integral = F(b) - F(a), both a and b equal 0

but the integral itself doesn't equal zero

[tardy stag]

F(x) - F(a) = integral from a to x of f(t) dt

[vale wigeon]

doesn't match up with the screenshot tho

[alpine sable]

oh my god hold up

[vale wigeon]

@tardy stag do you mind taking this over

im gonna go crazy if i continue w thsi

[alpine sable]

my a in this scenario is -1

so it's actually 0 - -1 isn't it

and therefore it equals 1

[tardy stag]

ya

which is indeed the area below sint from 0 to pi/2

[alpine sable]

oh I had - F(a) missing in my function

[tardy stag]

yeah that's an important part

[alpine sable]

but they didn't include that here in the book

for some reason

[tardy stag]

oh that's wrong

unless they already established that F(a) = 0

[alpine sable]

but I suppose that for this function it doesn't matter

[tardy stag]

wtf that is so messed up

that is also wrong

[alpine sable]

but even then, what about 0 to a?

[tardy stag]

if you have that as a physical book i authorize you to add "- F(a)" to that diagram

[alpine sable]

how can the 14th edition of a book wrong lmao

I just have the png version tho

[tardy stag]

pls tell me you mean pdf

[alpine sable]

I mean pdf

lmao

[tardy stag]

all i can think of is that maybe there's some missing context, like they chose a particular antiderivative such that F(a) = 0

but that is not how it's usually defined at all

,tex .FTC1

[ocean seal]

Hayley

[tardy stag]

i should change that to "any antiderivative"

[alpine sable]

but this is correct, right?

[tardy stag]

yes that is correct

if you think about it, the F(a) drops out bc it's a constant

and you're left with F ' (x)

[alpine sable]

they have that later in the book, idk why the earlier definition is different

[tardy stag]

they call that part 2? that's weird

but whatever sure

[tardy stag]

actually can you go get equation (1) for me

[tardy stag]

okay that's better but still weird

like it's technically correct because F(a) = 0 (prove this!)
but that's a very confusing diagram

[alpine sable]

why exactly is F(a) = 0 true?

[alpine sable]

in this diagram, a and x seem to be similarly at the same height

https://media.discordapp.net/attachments/490557019623915520/1139994503764258947/image.png?width=388&height=390

[tardy stag]

that doesn't matter - F is the function that gives the area under the curve. The curve is f

[tardy stag]

well plug a into the equation for F and what do you get?

[alpine sable]

so F(a) is 0 when it starts at 0?

I mean the interval

when it's from a to b and a is 0

[tardy stag]

their F should really be called like $F_a(x)$ or something

[ocean seal]

Hayley

[tardy stag]

because they've defined $F_a(x) = \int_a^x f(t)\dd{t}$

[ocean seal]

Hayley

[tardy stag]

so if you take $F_a(a) = \int_a^a f(t)\dd{t}$ then that's equal to $0$

[ocean seal]

Hayley

[alpine sable]

I suppose I understand, that the derivative of F(x), at any given x in the interval a, b in the function f(t) equals the area from 0 to x. Am I correct here or am I mistaken?

I believe, that's what they are trying to tell me here

[tardy stag]

F(x) is defined to be the area under the curve of f(t) from t=a to t=x. (0 and b don't really enter into this at all)

the derivative of F(x), at any x, is equal to the function f(t) evaluated at t=x, which is to say f(x)

honestly if you just watch the 3b1b "essence of calculus" video it explains this concept far better than i can over text

[alpine sable]

I watched part 8 and 9 twice which exactly goes over this right here, but is the proof of f(x) being the derivative of F(x) really just the fact, that when you extend the interval a bit, and let the width run against 0, that the formula that describes the rectangle that is created (marked red in the image above), is the same as the difference quotient?

[tardy stag]

yes, exactly you're adding a little tiny rectangle to the area the height of which is f(x)

[alpine sable]

was this figured out coincidentally, or did Newton or Leibniz really think that, when adding an extra rectangle and letting the width run to zero, that the formula created turns out to be the difference quotient

[tardy stag]

honestly that's a great question and i don't know

it's not, like, that big of a jump to make once you already have the idea of an infinitesimal

[alpine sable]

well I suppose, that if they already knew how to use the h running towards 0 when calculating derivatives, and wanted to find out the derivative of the imaginary function F(x), whose value gives the area of f(x), they just used the same method, just not with difference in slope, but with the difference in area.

or is my thinking wrong

[tardy stag]

they may have even started with integrals tbh, derivatives feel more fundamental to us, but i wonder if "what's the accumulation" was a more natural question

[alpine sable]

yeah according to google it seems to be integration

crazy how differentiation and integration connect tho

[tardy stag]

wild, isn't it? i remember looking at the formulas for area and circumference of a circle and then suddenly going "hey wait a minute"

[alpine sable]

when imagining the original function f(t) as v(t) and the integral as s(T), like 3b1b explained, it really becomes intuitive

thanks so much @tardy stag for your immense effort

[tardy stag]

of course! Ann helped a lot too and you ask good questions

[alpine sable]

.close

[lone heart]

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[tidal bane]

im trying to scale the chebyshev polynomial such that its on the interval [0,t], this is my first time doing this sort of thing so I was hoping I could get a second pair of eyes to make sure that what im doing is correct. I'll provide both an image and the latex I wrote out:

\subsubsection{Scaled Chebyshev Polynomials}
We will also consider scaling the Chebyshev polynomials to be orthogonal on the interval $[0, t]$. To scale this to the interval $[0,t]$, we substitute $x' = \frac{x}{t}$:

\begin{align}
    \omega(t,x) &= \frac{1}{\sqrt{1-(\frac{x}{t})^2}} \\
    &= \frac{1}{\sqrt{1-(\frac{x^{2}}{t^{2}})}} \\
    &= \frac{1}{\sqrt{\frac{1}{t^{2}}(t^{2} - x^{2})}} \\
    &= \frac{1}{\frac{1}{t}\sqrt{(t^{2} - x^{2})}} \\
    &=\frac{t}{\sqrt{t^2 - x^2}}
\end{align}

The orthonormal polynomials are then:

\begin{align}
    p_n(t,x) = \frac{1}{\sqrt{\int_0^t \omega(t,x) dx}} T_n(\frac{x}{t})
\end{align}

Evaluating the integral:

\begin{align}
    \int_0^t \omega(t,x) dx = \int_0^t \frac{t}{\sqrt{t^2 - x^2}} dx = \pi t
\end{align}

Therefore, the properly scaled orthonormal Chebyshev polynomials on $[0,t]$ are:

\begin{align}
    p_n(t,x) = \sqrt{\frac{1}{\pi t}} T_n(\frac{x}{t})
\end{align}

Satisfying orthonormality on $[0,t]$:

\begin{align}
    \int_0^t p_n(t,x) p_m(t,x) \frac{1}{\sqrt{t^2 - x^2}} dx = \delta_{nm}
\end{align}

[ocean seal]

BeeGass

[smoky tangle]

Bruh I was typing :/

[tidal bane]

sorry

I found a couple of typing errors and added a bit:

\subsubsection{Scaled Chebyshev Polynomials}
We will also consider scaling the Chebyshev polynomials to be orthogonal on the interval $[0, t]$. To scale this to the interval $[0,t]$, we substitute $x' = \frac{x}{t}$:

\begin{align}
    \omega(t,x) &= \frac{1}{\sqrt{1-(\frac{x}{t})^2}} \\
    &= \frac{1}{\sqrt{1-(\frac{x^{2}}{t^{2}})}} \\
    &= \frac{1}{\sqrt{\frac{1}{t^{2}}(t^{2} - x^{2})}} \\
    &= \frac{1}{\frac{1}{t}\sqrt{(t^{2} - x^{2})}} \\
    &=\frac{t}{\sqrt{t^2 - x^2}}
\end{align}

The orthonormal polynomials are then:

\begin{align}
    p_n(t,x) = \frac{1}{\sqrt{\int_0^t \omega(t,x) dx}} T_n(\frac{x}{t})
\end{align}

Evaluating the integral:

\begin{align}
    \int_0^t \omega(t,x) dx = \int_0^t \frac{t}{\sqrt{t^2 - x^2}} dx = \pi t
\end{align}

Therefore, the properly scaled orthonormal Chebyshev polynomials on $[0,t]$ are:

\begin{align}
    p_n(t,x) = \sqrt{\frac{1}{\pi t}} T_n(\frac{x}{t})
\end{align}

These satisfy the orthonormality condition:

\begin{align}
    \int_0^t p_n(t,x) p_m(t,x) \omega(t,x) dx = \delta_{nm}
\end{align}

Where the weight function is:

\begin{align}
    \omega(t,x) = \frac{t}{\sqrt{t^2 - x^2}}
\end{align}

The orthogonal polynomial basis is:

\begin{align}
    {p_0(t,x), p_1(t,x), p_2(t,x), \ldots}
\end{align}

[ocean seal]

BeeGass

[tidal bane]

ok just being thorough here:

\subsubsection{Scaled Chebyshev Polynomials}
We will also consider scaling the Chebyshev polynomials to be orthogonal on the interval $[0, t]$. To scale this to the interval $[0,t]$, we substitute $x' = \frac{x}{t}$:

\begin{align}
    \omega(t,x) &= \frac{1}{\sqrt{1-(\frac{x}{t})^2}} \\
    &= \frac{1}{\sqrt{1-(\frac{x^{2}}{t^{2}})}} \\
    &= \frac{1}{\sqrt{\frac{1}{t^{2}}(t^{2} - x^{2})}} \\
    &= \frac{1}{\frac{1}{t}\sqrt{(t^{2} - x^{2})}} \\
    &=\frac{t}{\sqrt{t^2 - x^2}}
\end{align}

The orthonormal polynomials are then:

\begin{align}
    p_n(t,x) = \frac{1}{\sqrt{\int_0^t \omega(t,x) dx}} T_n(\frac{x}{t})
\end{align}

Evaluating the integral:

\begin{align}
    \int_0^t \omega(t,x) dx = \int_0^t \frac{t}{\sqrt{t^2 - x^2}} dx = \pi t
\end{align}

Therefore, the properly scaled orthonormal Chebyshev polynomials on $[0,t]$ are:

\begin{align}
    p_n(t,x) = \sqrt{\frac{1}{\pi t}} T_n(\frac{x}{t})
\end{align}

These satisfy the orthonormality condition:

\begin{align}
    \int_0^t p_n(t,x) p_m(t,x) \omega(t,x) dx = \delta_{nm}
\end{align}

Where the weight function is:

\begin{align}
    \omega(t,x) = \frac{t}{\sqrt{t^2 - x^2}}
\end{align}

The orthogonal polynomial basis is:

\begin{align}
    {p_0(t,x), p_1(t,x), p_2(t,x), \ldots}
\end{align}

where:

\begin{align}
    p_0(t,x) &= \sqrt{\frac{1}{\pi t}} T_0(\frac{x}{t}) \\
    p_0(t,x) &= \sqrt{\frac{1}{\pi t}} \cdot 1 \\
    p_0(t,x) &= \frac{1}{\sqrt{\pi t}}
\end{align}

[ocean seal]

BeeGass

[lone heart]

@tidal bane Has your question been resolved?

[tidal bane]

no

[lone heart]

@tidal bane Has your question been resolved?

[tidal bane]

No

[lone heart]

@tidal bane Has your question been resolved?

[tidal bane]

No

[naive crystal]

MInd considering advance math SECTION..

[twin nimbus]

@tidal bane for clarity, you're attempting to move the chebyshev polynomials from the domain $[-1, 1]$ onto the domain $[0, t]$ while preserving their orthonormality with respect to the standard inner product for $T_n$: $\langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \frac{\dd{x}}{\sqrt{1 - x^2}}$ in an analogous fashion with respect to the new variable $x'$.

Is this correct?

[ocean seal]

OmnipotentEntity

[twin nimbus]

The reason I'm asking for clarity, is your work seems to assume we are going from [0, 1] rather than [-1, 1]

There's also the possibility that I'm both misremembering my previous work with and misinterpreting the references that I'm currently using on Chebyshev polynomials.

[lone heart]

@tidal bane Has your question been resolved?

[tidal bane]

yes I believe so; just to be clear I am reading this paper where in section B.1.1 Properties of Legendre Polynomials they are also doing this. Im just trying to make it happen for chebyshev

https://arxiv.org/pdf/2008.07669.pdf

[tidal bane]

oh I see, yeah that is an error. I very much want to translate from [-1, 1] to [0, t]

[tidal bane]

no I think you might be correct

[tidal bane]

how do I do this?

[tidal bane]

not yet

[tacit arch]

#get-advanced-access but I'd be more specific with your question first

[tacit arch]

Maybe try this in your own first

[real gazelle]

Hey! I'll take a look

@tidal bane what exactly is your question? it seems like you already posted a formula for how to scale the chebyshev polynomials, no?

is there some part of the formula that you don't understand, or something which isn't addressed?

I'm just kinda confused

[lone heart]

@tidal bane Has your question been resolved?

[lone heart]

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[glossy jolt]

why did 1/(4/3)p turn into -3/4p

[vale wigeon]

1/(4/3) = 3/4

more generally 1/(a/b) = b/a

[glossy jolt]

why is it negative then

[vale wigeon]

wym?

the minus sign was just sitting there untouched

[glossy jolt]

oh right

thanks

.close

[lone heart]

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[alpine sable]

how

[alpine sable]

i dont understand how this is correct

i get that {A} subsets B

I'm not sure why A doesn't and where they got {{A}}

you verify it by yourself tho, i mean i dont have any proof, but if we make subsets of B we can cleary see that A isnt subset of B

[alpine sable]

you dont got it? you dont got the part

{{A}} is subset of B

?

[tawny condor]

A is a subset of B if and only if, every element of A is an element of B

All we know, is that B contains A and {A}. Those are SETS.

B contains only 2 things, which are A and {A}

{A} is a subset of B, because every element of {A} is in B. {A} is a set with just the element A, and we know that A is in B, so that holds.

{{A}} is a subset of B, because again, every element of {{A}} (which is just {A}) is in B

[alpine sable]

so why exactly is A not a subset of B?

[tawny condor]

Do you know what elements A holds?

You don't

[alpine sable]

no

[tawny condor]

And you for sure don't know whether B has those elements aswell.

[wanton pebble]

cause b has only two elements

[tawny condor]

You just know that A, as a set, is an ELEMENT of B. It's not a subset of it

[wanton pebble]

{A} and {{A}}

[alpine sable]

A isn't subset of B

A is Element of B

[tawny condor]

No, the elements are A and {A}

[alpine sable]

Set containing the elements of it are its subset

its element arent its subset

[alpine sable]

so where is A?

is A not defined as an element?

it's a set?

but are they saying A = B

[alpine sable]

A is a set which is as whole considered as element for B

[tawny condor]

It is an element. It's also a set. But it's not a subset of B because B doesn't have every element A has (in fact, you don't even know what A has)

[alpine sable]

so then how do you know that it doesn't have every element of A in it

[tawny condor]

Because you just don't.

B = {A, {A}}

If A had the element 1 in it,

does B also have 1?

[alpine sable]

i think i understand where you're trying to get at but I dont know why A is serving as a set and an element

are they different

[tawny condor]

B is a set, and it can hold whatever you want. In this case, it has an element that is ALSO a set.

But that doesn't matter.

It can be whatever.

[alpine sable]

so B is holding A

[tawny condor]

Yes

[alpine sable]

So if A = {3,2,1} then B = {3,2,1}??

[tawny condor]

No no no

A = {1, 2, 3}

Then,

[alpine sable]

{{A}} you forgot that one

[tawny condor]

B = { {1, 2, 3}, {{1, 2, 3}} }

[alpine sable]

but does it matter

[tawny condor]

Of course it does

1 isn't in B

[alpine sable]

if they have the same values wouldn't you just ignore it

like

[tawny condor]

2 and 3 also aren't.

[alpine sable]

A = {1,2,3,3} = {1,2,3}

[tawny condor]

You don't see 1 as an element in B by itself.

[tawny condor]

That is true, but we are not repeating anything

Look, let's take a simpler example

B = {A}

[alpine sable]

yes, the way you put bracket does matter and you kinda expressed a subset of B instead of complete B

[tawny condor]

B is a set containing the element A.

[alpine sable]

ok

[tawny condor]

Now, if A = {1, 2, 3}

Then B is a set, containing the set {1, 2, 3}

It has a single element, and that element is A SET.

It's NOT 1, 2, or 3.

B doesn't have 1, 2 and 3 as elements.

It has the SET {1, 2, 3}

Ok?

[alpine sable]

ok

I see

[tawny condor]

So again, A is not a subset of B in this case.

For A to be a subset of B, you need every element in A to be in B.

You need 1 to be in B, and 2 and 3. But that's not the case.

We only have a single element in B, which is a set. We don't even have any numbers in B.

[alpine sable]

and that element is A?

[tawny condor]

Yes.

[alpine sable]

so the element is A and not {1,2,3}

[tawny condor]

No, it's A which is {1, 2, 3}

B has an element which is {1, 2, 3}

It doesn't have 1, 2 or 3 as numbers.

It has a set which has those numbers

[alpine sable]

ok i see

[tawny condor]

Let's look at B = {A, {A}}

[alpine sable]

so that's why it doesn't subset A?

[tawny condor]

Yes.

[alpine sable]

ok

[alpine sable]

okay

[tawny condor]

In this case, B has 2 elements.

One is A, the other is {A}

Both are sets. A is a set, for example {1, 2, 3}

{A} is also a set, but it's the set which has a single element (that element is A)

I'll say this again: {A} is an element of B. This element is a set with a single element, which is A. It's a set inside a set.

[alpine sable]

yeah makes sense

[tawny condor]

For A to be a subset of B, we would need 1, 2 and 3 to be elements in B.

But they aren't. The only elements are A, and {A}.

Those elements aren't even numbers. Of course they can't be 1, 2 or 3.

If you wanted A to be a subset of B, you can add those elements:
B = { 1, 2, 3, A, {A} }

In this case, A is a subset of B. (A = {1, 2, 3})

[alpine sable]

yeah

but it is not like that?

[tawny condor]

Yeah it's not

I just made an example where it would

[alpine sable]

okay

wait so

if A = {1,2,3}

Why wouldn't B = {1,2,3, {1,2,3}} ?

[wanton pebble]

cause it is given

{1,2,3} ≠ 1,2,3

[alpine sable]

so if you say A = {1,2,3}

and B = {A}

B is not directly {1,2,3}?

B is just A?

[wanton pebble]

no

{A} = {{1,2,3}}

[alpine sable]

ahhhh

so {{1,2,3}} not equal to {1,2,3}?

so A is not equal to {A}

and {A} is not equal to {{A}}

[wanton pebble]

yeah cause {1,2,3} have three elements in it

but {{1,2,3}} is have just one

[alpine sable]

okay i understand

and they say it is a subset of {A} and {{A}} because they're inside the set

?

[wanton pebble]

what is subset of {A}

[alpine sable]

mb

{A} and {{A}} are subsets of B due to that reason?

because they're inside of B?

[wanton pebble]

yes

[alpine sable]

ohhhhhhhhhhhhhhhhhhhh

i see

okay, then, thanks

.close

[lone heart]

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[alpine sable]

whar

ok

[lone heart]

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[mossy siren]

4 a i Expand (4 + x) ^5 as far as the term in x^3.
ii Hence find the coefficient of x3 in the expansion of (3 − x) (4 + x) ^5. I just need help with part ii

[wanton pebble]

multiply 1st part with (3-x)

[mossy siren]

what do u mean

[vale wigeon]

have you done part i?

[lone heart]

@mossy siren Has your question been resolved?

[mossy siren]

yes

[wanton pebble]

so can you multiply 3-x in solution you got in part 1

[lone heart]

@mossy siren Has your question been resolved?

[upbeat hornet]

what did you get for part i?

[lone heart]

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[thick monolith]

so in the above problem the red line is 6/x, the blue line is 6/x^2 and the green line is y=1

Im supposed to figure out the grey area

so firstly I set up an integral to describe it

I figure I can calculate the area between x=1 and x=6 thats between the red and blue line

and than substract from it the area between the green line and blue line between x=sqrt6 and x=6

hold on ill post my maths

so my solution comes out to roughly 3.65 which is twice the area that the teacher calculated, roughly 1.8

teacher used a different method too, which is fine if my method is wrong

but I want to know why my method is wrong

im more suspicious of the first integral i wrote than the second

Okay so I just checked the solution provided by the book and its the same as my solution, but the teachers solution is different

this is the solution the teacher made

Ohhhhhhhhh hold up nevermind all this

I just saw that I didnt look at the full teacher solution, its the same as books

my bad

.close

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[wide trout]

guys is there a formula for finding the area of a triangle by the coordinates of his points?

[wise dirge]

like on a plane?

[wide trout]

yup

[wise dirge]

well it should be the same

[wide trout]

what?

[winter light]

Heron's formula

But I bet there are other methods as well, which I don't know actually

[wide trout]

you're saying that I should get the lengths by the coordintes and then use heron's

[winter light]

Yes, at least that's one way

[wise dirge]

you can use the law of areas or whatever that was called, $\frac{1}{2}ab*sin(C)$

[winter light]

But using vectors I believe there are easier/quicker formulas

But then you must figure out sinC only with coordinates, right?

[ocean seal]

紅卫兵

[minor needle]

it is

[wide trout]

?

[wide trout]

really?!!!

that's awesome

[winter light]

Yes, it comes from the concept of determinant of a matrix

Since it's the modulus of a cross product

[winter light]

But maybe this is too advanced for you

[minor needle]

I guess nobody knows it by heart so

just formula for specific cases

[wide trout]

aha that's abit advanced for me

[minor needle]

but I remember we used to have a list of formulas on the mature exams, including this one

[winter light]

Yeah indeed, but the formula is very easy so I think that you can use it with no problems

[wide trout]

yup

[minor needle]

yes, plug values and calculate

that's it

[wide trout]

thx guys

.close

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[azure tapir]

hi

[azure tapir]

can i get some help with the chain rule caluslus

[rocky grove]

Hey

[azure tapir]

/\frac{d}{dx}\left(\frac{3}{\sqrt{\left(1-4x\right)}}\right)/

\frac{d}{dx}\left(\frac{3}{\sqrt{\left(1-4x\right)}}\right)\

[rocky grove]

$$\frac{d}{dx}\left(\frac{3}{\sqrt{\left(1-4x\right)}}\right)$$

[winter light]

$\frac{d}{dx}\left(\frac{3}{\sqrt{\left(1-4x\right)}}\right)$

[ocean seal]

VulcanOne

Alberto Z.

[azure tapir]

i dont know how to apply the chain rule

[rocky grove]

Well

Metaphorically speaking, you can treat the function like an onion

[winter light]

Instead of / ... \ you need $ ... $ or $$ ... $$

[rocky grove]

Keep peeling the outer layers till you reach the center piece

[azure tapir]

thx

[winter light]

Yw

[ocean seal]

VulcanOne

[rocky grove]

[azure tapir]

$f(x) = 3x^-1/2$

[ocean seal]

Sunset

VulcanOne

[rocky grove]

Yeah

What's now the derivative?

[azure tapir]

$f(x)' = 1.5x^-1 1/2$

[ocean seal]

Sunset

[rocky grove]

Hm

[azure tapir]

how do you do that power thing

[rocky grove]

$3^{2-1 + \ln(7x^{e^{x^2}})}$

[ocean seal]

VulcanOne

[rocky grove]

You use { }

[azure tapir]

okey

[rocky grove]

Anyways

[azure tapir]

$f(x)' = 1.5x^{1/2$}

[ocean seal]

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

[rocky grove]

Oops

The dollar signs should be the last thing

[azure tapir]

yah

[rocky grove]

$f'(x) = 1.5x^{\frac 12}$

[ocean seal]

VulcanOne

[rocky grove]

But yeah you forgot a negative sign

And you forgot to subtract 1 from the power

[azure tapir]

$f'(x) = -1.5x^{\frac 12}$

[ocean seal]

Sunset

[rocky grove]

-1/2 -1 = ?

[azure tapir]

i'm brain dead

[rocky grove]

No don't worry

You're getting there

[azure tapir]

$f'(x) = -1.5x^{-\frac 12}$

[ocean seal]

Sunset

[azure tapir]

$f'(x) = -1.5x^{-1\frac 12}$

[ocean seal]

Sunset

[rocky grove]

Yep

[azure tapir]

lets go

[rocky grove]

-1.5 = -3/2

Now this counts as the first layer

The actual function's derivative looks the same but you just replace what's inside the square root

$f'(x) = -1.5(1-4x)^{-\frac 32} \cdot \text{something}$

[ocean seal]

VulcanOne

[rocky grove]

Now

What is the derivative of $f(x) = 1-4x$?

[ocean seal]

VulcanOne

[azure tapir]

-4

[rocky grove]

Yep

That's the second layer

And looks like it is the last onion layer

So all in all

The derivative of the function is

[azure tapir]

$f'(x) = -1.5(1-4x)^{-\frac 32} \cdot \text{something}(-4)$

[ocean seal]

Sunset

[rocky grove]

Yeah but the something is -4

[azure tapir]

yah i just copied an pace

[rocky grove]

Lol

$f'(x) = 3 \cdot -\frac 12 (1-4x)^{-\frac 32} \cdot (-4)$

[ocean seal]

VulcanOne

[rocky grove]

That's the whole derivative

[azure tapir]

nice

[rocky grove]

Simplify and you get your answer :)

[azure tapir]

i got another one

[rocky grove]

Alright

[azure tapir]

is like e{2x{2}}

$ e{2x{2}}$

[rocky grove]

You mean $f(x) = e^{2x^2}$?

[azure tapir]

$e{2x{2}}$

[ocean seal]

VulcanOne

[azure tapir]

yah

that

[rocky grove]

Okay

Same stuff

Replace 2x^2 with k

You get $e^k$

[ocean seal]

VulcanOne

[azure tapir]

okey

is this the first layer

[rocky grove]

What's the derivative of e^k with respect to k?

[rocky grove]

Yep

[azure tapir]

ek

[rocky grove]

e^k right?

[azure tapir]

yah

[rocky grove]

Itself

So we remove k and we put 2x^2 back

So we know e^{2x^2} is part of the derivative

Now let's focus on the second layer

2x^2

What's the derivative?

[azure tapir]

is f(x)' = (e^2x^2)(4x)

[rocky grove]

Yep you caught onto the pattern

[azure tapir]

$(e{2x}{2})(4x)$

[ocean seal]

Sunset

[azure tapir]

https://tenor.com/view/capoo-bugcat-cute-swole-strong-girl-gif-26565854

[rocky grove]

$f'(x) = e^{2x^2} (4x)$

[ocean seal]

VulcanOne

[azure tapir]

i was ment to write that

[rocky grove]

Btw #latex-help and #latex-testing will help you learn how to use LaTeX :)

[azure tapir]

okey thx

[rocky grove]

[azure tapir]

i really appricate the help

i might need help later on but i will close for now

/close

\close

how do i close

[rocky grove]

To close you type .close

[azure tapir]

.close

[lone heart]

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[azure tapir]

thanks

[rocky grove]

[azure tapir]

.close

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[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

<@&286206848099549185>

[uncut torrent]

idk

nice name tho

huge fan

[alpine sable]

thanks

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@alpine sable Has your question been resolved?

[alpine sable]

sure

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[buoyant stone]

Hello! How to compare 13^9 and 2023 choose 4 without a calculator?

[vale wigeon]

well i'd probably try to estimate first

2023 * 2022 * 2021 * 2020/24 should be something to the tune of single-digit trillions

uhhhh

can we get an idea of how big 13^9 is this way

[mystic vortex]

Does anyone know what log_10 of 13 is off the top of their head?

[buoyant stone]

I doubt it

[buoyant stone]

13^9 has 11 digits and 2023 choose 4 has 12 digits

[mystic vortex]

It's gotta be like 1.2 or smth

[vale wigeon]

how do you know this without a calculator?

[buoyant stone]

I used a calculator to find that but I’m trying to prove it without a calculator

[old heath]

can you not binomial this

for number of digits

[molten pivot]

Consider that 10^9 = 1 000 000 000

[old heath]

ok we can see this by powers of 10

the number of digit atleast

if they were same this would be harder

if you multiply by a 2 digit number

1 factor of 10 is getting added

mostly u get 1 extra digit

if you square the digit goes to

2k-1

if youre before a certain number

for 13

13^2 is 169

its square will have 5 digit

binomial better

the significant contribution to number of digits

will be first 2

so 10^9 +3 10^8 +9 10^7 + 27 10^6 ...+243 10^4

nah doesnt work

[buoyant stone]

Yeah I don’t think it does

[old heath]

yeah if i do this

only get 10digit

not 12

[buoyant stone]

What I tried to do was to say 2023 choose 4 > 2000x2000x500x300

[old heath]

why binomial not work tho

[buoyant stone]

how to do it with binomial

[old heath]

well one thing is certain that 2023c4 is 11 digit

this can be done with what i was saying

[old heath]

if split 13^9 as

10+3

the significant contribution

is from the

first 3 and last 3

tried doing it from here

3^9=27^3

well this has no contri aswell

[buoyant stone]

No actually 2023c4 has 12 digits I think

This is the number 695798479355

Idk how to prove that 13^9 has less than 12 digits tho

So this is where I am so far

13^9<16^9 and 16^9=2^36

[buoyant stone]

This means that 2023c4>6x10^11

[old heath]

uhm

[buoyant stone]

So we can compare 6x10^11 and 2^36

[old heath]

2023 x 2022 x 2021 x 2020

yeah

its 12

13^9 though

can we find least number whose power 9 is 12 digits?

[buoyant stone]

17

[old heath]

16^9

2^26

36

512

^4

has to be a better method tbh

oh.

ohh

we can factorize

2023c4

and compare factors

[buoyant stone]

Ohh that’s smart

[old heath]

approximate the factors

the entire 4! gets cancelled

[buoyant stone]

These are the factors

[old heath]

its 505x 2021 x 927 x 2023

take 169 from each and approximate ig

[buoyant stone]

I think it’s 337 and not 927

[old heath]

wait yeah my bad

[buoyant stone]

Cause it’s 2022/2/3

[old heath]

yeah

yeah the q is done

2023 c 4 is

approxing

13^8

x 3 x2 x12x 12

which is more than

13^9

[prime badge]

we kinda moved on from the original question but
2023 × 2022 × 2021 × 2020/24
is at least 40×50×40×50×40×50×40×2
20×25×20×25×20×25×20×(2^8)
20×25×20×25×20×25×20×16×16

[old heath]

we were comparing that only

[buoyant stone]

yeah that’s also smart

This is what I got

it looks good, thanks!

.close

[lone heart]

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[alpine sable]

let f(n,a) be the sum from k=0 to n of (-1)^k*k^a*(n choose k)

[alpine sable]

this came up during my attempt to solve a problem about the number of labeled trees on n vertices

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@alpine sable Has your question been resolved?

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✅

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@alpine sable Has your question been resolved?

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@alpine sable Has your question been resolved?

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[raw ginkgo]

how would I solve this

[vale wigeon]

do you know how to find the variance of a random variable in general?

[rocky grove]

@raw ginkgo

[raw ginkgo]

yah Ive had alot of questions recently

on this topic thanks for the help time and time again doe

[rocky grove]

Do you still need help?

[raw ginkgo]

yah

[rocky grove]

Okay do you have any formula for Variance?

[vale wigeon]

ok thats great and all but also you have not answered my question

but i'll take your "yeah" at the start as meaning yes

in which case i have to ask: what is troubling you for this one in particular

[raw ginkgo]

variance is s^2=1/n-1 (xi-x-)

wait lemme get a screenshot

[rocky grove]

Alright

[vale wigeon]

that's sample variance ⚠️

is this the only formula that was mentioned?

[raw ginkgo]

ye

[vale wigeon]

so $\mathrm{Var}(X) = E[ (X - \mu)^2]$ was never spoken of?

[ocean seal]

Ann

[vale wigeon]

that is odd.

[raw ginkgo]

wait no it was

nvm

[vale wigeon]

yeah

well apply it then

they give you mu for free

[raw ginkgo]

so the value of e is 20.59?

[vale wigeon]

what's e?

[raw ginkgo]

E

[rocky grove]

μ*

[vale wigeon]

E is not a number

E is the symbol for expectation

or expected value

[rocky grove]

<@&268886789983436800>

[raw ginkgo]

so variance=(x-20.59)^2

[rocky grove]

Hm

[raw ginkgo]

im so confused

.close

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[muted breach]

how to slove it?

[young finch]

start by factoring the denominators

there will probably be a common root too

[muted breach]

how to do it? like do the same denominator?

[young finch]

brother

factor the quadratics in the denominator

[muted breach]

I didnt get it

[winter light]

x² - 8x + 7 = (x - a)(x - b)

You have to find a and b from this

And similarly for the other two denominators

[muted breach]

how can I do it

[winter light]

Solve a trivial quadratic equations

[muted breach]

its like impossible

[winter light]

Not at all

Do you know how to solve a quadratic equation?? @muted breach

[muted breach]

ohhh yeah I think

like

X1 X2

[winter light]

Yes, my a and b were your x1 and x2, sorry for the different notation

[muted breach]

ohhh

get it I think'

[winter light]

But it's quicker if you use sum - product decomposition (if you know it of course, otherwise ignore)

[muted breach]

7 and 1