help-0

[alpine sable]

When you split it how do you evaluate -1/ln(1+x^2)

?

[long axle]

Wait ur not doing quotient rule right?

[wind cloak]

$\lim_{x \to 0} \frac{e^{x^2} \sin x}{x\ln (1 + x^2)} - \frac{1}{\ln (1 + x^2)}$

[long axle]

Just wanna make sure

[ocean seal]

NEONPerseus

[alpine sable]

Yeah, how would you evaluate the RHS

[wind cloak]

That's a problem for future you

[alpine sable]

Yh but I tried this

[alpine sable]

Wdym

[long axle]

Did u use quotient rule

[alpine sable]

No

[long axle]

Hmm

[alpine sable]

Why would i

[long axle]

Ohh u just did LH twice?

[wild trail]

did you miss an x in the denominator of second fraction?

[wind cloak]

$\lim_{x \to 0} \frac{e^{x^2}}{\ln ( 1 + x^2)} \cdot \lim_{x \to 0} \frac{\sin x}{x}- \lim_{x \to 0} \frac{1}{\ln (1 + x^2)}$

[alpine sable]

It cancels

[wind cloak]

No I cancelled it

[wild trail]

ah, okay

[ocean seal]

NEONPerseus

[alpine sable]

Yes but how do you evaluate the last one?

I have no idea

[wind cloak]

This becomes infinity minus infinity

Right

[alpine sable]

The answer is 5/6

[wind cloak]

Ofc it is

$\lim_{x \to 0} \frac{e^{x^2}}{\frac{1}{1 + x^2}} \cdot \lim_{x \to 0} \frac{\sin x}{x}- \lim_{x \to 0} \frac{1}{\ln (1 + x^2)}$

[ocean seal]

NEONPerseus

[wind cloak]

I'm gonna cry now

[alpine sable]

same

[wind cloak]

@wild trail how would we ever evaluate the limit on the the far right

Here is the future you

[wild trail]

lemme check if the limit even exists brb desmosing

[alpine sable]

bro this is in my high school homework 😭

why

[wind cloak]

:(

Whoosh

[alpine sable]

lol

[wind cloak]

Well it does exist

[wind cloak]

Yeah

Brute force L'Hospital

How did that work

[wild trail]

well, the answer is 5/6. I have no idea how to even reach that

[alpine sable]

its like

beyond my physical capabilities

.

[wild trail]

I have a guess that this will require using expansion

I'll give it one try

[alpine sable]

i tried that too

[wild trail]

what happened?

[alpine sable]

actually

wait

i'll try it again

[wind cloak]

Brute force Hospital didn't work

[wild trail]

GOT THE ANSWER

Had to do expansion

taylor expand everything to even 2 terms helps

[wind cloak]

Everything?

[wild trail]

-x will cancel with x and then take x^3 common from above and below

wdym?

[alpine sable]

yh i got that

but then what

[wild trail]

you should be left with 5/6 on top and 1 in bottom

[alpine sable]

(-x^3/3! + ...)/(x^3 + ...)

hmm

[wild trail]

that is wrong you'd have one more term of x^3

[alpine sable]

OH YEAH

[wild trail]

Should I post my gross solution

[alpine sable]

OH SHIT

I GOT IT

TY

[wild trail]

noice

[lone heart]

@alpine sable Has your question been resolved?

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[alpine sable]

hi im confused on how to prove that the endpoints are continuous for the fundamental theorem of calculus part 1

[alpine sable]

from the proofs that i can find online all that is proved is F(x) is differentiable and continuous on (a,b). it says also that you can do one sided limits to prove continuity for a and b

but im not sure how to do that

$\lim_{h\to 0^{+}} \frac{g(a+h) - g(a)}{h} = f(a)$

[ocean seal]

いろさん

[alpine sable]

$g'(a) = f(a)$

[ocean seal]

いろさん

[alpine sable]

i manage to get this but its not even true

because apaprently g is not differentiable at a or b

what the heeell

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

<@&286206848099549185>

[alpine sable]

halp

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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[umbral mortar]

so what I got for the negation is
∀x ∈R ∃y ∈R x + y ≤ 0, true?

im still iffy on how the logic works for if the statement is true or false

[noble sinew]

The negation is correct

[umbral mortar]

I think its actually false cause there are multiple numbers x or y could be where its still less than 0

but im unsure

nvm I think I got it

I'd just have to make ∃y a set number and plugin different numbers in for ∀x just to check

[noble sinew]

What

[umbral mortar]

no?

I have to find out if the statements are true also

or no

[noble sinew]

Okay so does there exists a real number x such that for all y we have x+y>0?

[umbral mortar]

no

[noble sinew]

Yes

[umbral mortar]

yes?

[noble sinew]

?

[umbral mortar]

Y can be negative tho

all real numbers

[noble sinew]

I said yes your answer is correct

[umbral mortar]

oh

well ty

.close

[lone heart]

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[alpine sable]

okay so im confused because i know i should have root-9

shouldnt

[vague mirage]

My lord my eyes

You took -x^6 common

[alpine sable]

can you explain pls

[vague mirage]

Here you are taking out -x³

But what if x^3 is a positive integer

Then you are taking out a imaginary number

Ummmm

Well you know you should do this $\frac{-(9x⁶+1)}{x⁶}$

[ocean seal]

Arnab

[vague mirage]

Cause you can't take out x⁶ that way I think what if x is a positive real number

Maybe your answer will be right in both case

But this is a more appropriate way to do this

[alpine sable]

can i do this

[wild trail]

yep

[vague mirage]

Yes actually

Here it is ok

[wild trail]

you missed ^6 in the second step though

[vague mirage]

He skipped a step and wrote x³ directly I think

[wild trail]

in, sqrt{1 + 9x}

[vague mirage]

Oh yeah

Mistake in second step

[alpine sable]

well if i put x^3 into the root then it will be x^6 right

i divided whole root by x^3

then put x^6 into the root

[vague mirage]

$\frac{1+9x⁶}{x⁶}=\frac{1}{x⁶}+9≠\frac{1+9x}{x⁶}$

[ocean seal]

Arnab

[vague mirage]

You have written 9x in the second step

[alpine sable]

oh yea i didnt mean that

[vague mirage]

I know

[alpine sable]

im doing something wrong because i am getting -3

also it is -infinity

[wild trail]

is x going to infinity or - infinity?

[wild trail]

The denominator should be positive I guess

[alpine sable]

me too

[wild trail]

Well, I always create a new variable t=-x and substitute to avoid this sort of thing

[alpine sable]

how does that work

[vague mirage]

,w d³/dx³(√(1-9x⁶)/(5-x³))

[ocean seal]

Results provided by WolframAlpha

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

Requested by 𝓐𝓡𝓝𝓐𝓑 𝓟𝓐𝓛#6985

[vague mirage]

Fuck

[wild trail]

simple change of variable similar to integration

[alpine sable]

well i havnt gotten to integrals yet

supposed to be solving algebraically

[wild trail]

ohh

I mean it's basic algebraic manipulation

[alpine sable]

so why change all variables to -

because of - infinity?

[wild trail]

replace x with -t

everywhere

now, since x -> -infinity, t=-x -> infinity

[alpine sable]

okay ill try

i dont understand why the original way im doing it is wrong

this is showing that i should have had -sqrt1/x^6+9x^6/x^6

i dont understand why there is a negative in front of sqrt

i dont get it

hows -sqrtx^6=x^3

[vague mirage]

-√x⁶≠x³

-√2⁶≠2³

For example

I don't get it too

[alpine sable]

the negative in front of numerator comes from nowhere

i dont get this at all its apparently right and this program did everything the same as i did

except it put a negative before the numerator for no apparent reason

[vague mirage]

I don't get it

What did the positive sign

Do

Why replace the positive sign with negative

Like my ex replaced me with her boyfriend

🥲

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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[fresh parcel]

how many arrangements of checkers on a 5x5 board satisfy the condition that no two checkers share are adjacent?

[fresh parcel]

im wondering if theres a recursion for this

[honest geyser]

This problem can be solved using recursion, specifically using dynamic programming. Let's define a function $f(i, S)$ that represents the number of valid arrangements of checkers on a $5\times i$ board where the $i$th column has state $S$. Here, the state $S$ is an integer from 0 to 31 that represents the positions of the checkers in the $i$th column. Specifically, if the $j$th bit of $S$ is 1, then there is a checker in the $j$th row of the $i$th column.

We can define the function $f(i, S)$ recursively as follows:

$$f(i, S) = \sum_{T} f(i-1, T)$$

Here, the sum is taken over all states $T$ of the $(i-1)$th column such that no two checkers in $T$ are adjacent to any checkers in $S$. In other words, we can only add a checker to the $i$th column if it doesn't conflict with the checkers in the $(i-1)$th column.

We can calculate the base case of $f(1, S)$ directly since there are no previous columns to consider. Specifically, $f(1, S) = 1$ if there are no adjacent checkers in $S$, and $f(1, S) = 0$ otherwise.

Finally, the answer to the original problem is the sum of $f(5, S)$ over all states $S$ of the 5th column.

To implement this algorithm efficiently, we can use a 2D array $dp$ where $dp[i][S]$ stores the value of $f(i, S)$. We can fill in this array iteratively starting with the base case of $dp[1][S]$ and using the recurrence relation above to calculate $dp[i][S]$ for $i > 1$.

The time complexity of this algorithm is $O(32^5)$, which is reasonable since there are only 32 possible states for each column. However, we can optimize this algorithm further by using bit manipulation to check if two checkers are adjacent, which can be done in constant time.

[ocean seal]

Theodore

[honest geyser]

The software I used to figure this out was ExEmO if you're curious or wanting it.

[fresh parcel]

thanks, im going to read that solution

ahh i see

i was just thinking if theres a specific way to do this without iterating over all the columns

like a specific recursive formula

[honest geyser]

Hmm.

I'd just use the formula/calculation I found, it seems more logical and ethical but, if you really want the specific way, I'd go through some of the algorithms.

[fresh parcel]

is it even possible to do without coding

[honest geyser]

Not from my indication.

But it could be.

[fresh parcel]

ah rip

[honest geyser]

Indeed RIP.

If this answered your question though, use .close

[fresh parcel]

okay

thanks for your help!

.close

[lone heart]

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[abstract nova]

how would i solve this type of question? do i take the antiderivative and find the value of it when t is 20 and 48 or do something else?

[quasi vector]

nvm

what does the antiderivative represent

does it not just represent L?

[abstract nova]

oh yeah it does

[quasi vector]

how does calculating L help

since the question asks about the rate of growth

[abstract nova]

couldnt i use L(20) and L(48) to plug in to the rate function

[quasi vector]

well

are you not given the rate function?

[abstract nova]

yeah

hm now that i think about it i dont think i could find the antiderivative of it since it has L in it

oh wait is it just a logic question

[quasi vector]

you could do it by separating the variables, but that may be above the pay grade

and

it isnt needed in this question

[abstract nova]

oh wait i just realized the 20 and 48 are the lengths already

so i already have the L

[quasi vector]

yes

[abstract nova]

so i would just plug those into the rate function and it would be greater at 20cm so it grows faster then

right?

[quasi vector]

yes

[abstract nova]

ok thanks i was overthinking it before

.close

[lone heart]

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[rocky abyss]

not math just C# coding

[alpine sable]

#old-network

[rocky abyss]

how to set the direction of something from code

what the fuck is network

[outer lark]

this is very vague

[rocky abyss]

oh yeah

so i create a object

now i want to change its direction

in unity

[outer lark]

coordinate systems are often specific to the library you choose to use

[rocky abyss]

but i wanna change its direction from the script

[outer lark]

I mean this is still pretty vague

[rocky abyss]

so you know

how

it can rotate

[alpine sable]

#old-network -> Computer Science
Google -> Library documentation

[rocky abyss]

so im new to C# and dont really know what things to use like transform and getaxis or whatveer

[rocky abyss]

i tried them b4 they shit

[outer lark]

right, thats more of a unity specific thingy

[rocky abyss]

yeah

[outer lark]

I can't say I have experience with unity but they might be able to help you in the unity disc

its fairly big

[rocky abyss]

im just tryna do a basic task of changing the rotating like 90 degrees just ti test

whats the unity disc

[outer lark]

if it is basic stuff it can mostly be handled by reading the docs

https://docs.unity3d.com/ScriptReference/Input.GetAxis.html

[rocky abyss]

wait what the hell is delta time

[outer lark]

the time inbetween the previous frame and the current frame ig?

making movement independent from framerate

you dont HAVE to have it

[rocky abyss]

okay pretty od

[outer lark]

no , its pretty useful actually

[rocky abyss]

yeah i come from the basic scratch thing

so all this shit is weird

[outer lark]

ah right

[rocky abyss]

so i use delta time to make my movement

different

from the last time the script ran

[outer lark]

I would probably go through the language you're gonna be spending most of your time in and make sure you know how it functions first :PP

[rocky abyss]

like the basic C# languege its self

[outer lark]

yes

[rocky abyss]

or the unity bit of the languege

oh i learnt the languege

[outer lark]

well unity 'is' c#

[rocky abyss]

yeahhh but they got a few other things

that come with it

[outer lark]

in c#

and you wont understand the docs if you dont know c#

w3schools is fairly ok for getting basics down

[rocky abyss]

ill check out this document

us it free

[outer lark]

https://www.w3schools.com/cs/index.php you can follow that all the way through

[rocky abyss]

thanks buddy

wow wow wait a minute

if you still there i have a new question

@outer lark

when it comes to

0.1

like

decimal

numbers'

you know how you gotta do all that dumb shit by putting int or float or

when it comes to flaot

[outer lark]

data casting?

[rocky abyss]

uhhhh

ye

[outer lark]

I mean you would have to read the docs for the specific function you are using

some require ints some take floats

all things movement would probably accept a float

[rocky abyss]

float Cheese = 0.9

um

why

do i have to put letter F

at the end

of my code

[outer lark]

[rocky abyss]

i mean after the 0.9

in unity

so if i just put down 0.9

it would red line

but if put 0.9f

it would be hapoy

[outer lark]

[rocky abyss]

ok nvm

[outer lark]

yeah you kind of plop a F for float or D for double L for long

many more

[rocky abyss]

oh

[outer lark]

but this is all just stuff you should try to learn through that w3schools website

[rocky abyss]

ok

[lone heart]

@rocky abyss Has your question been resolved?

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[flat mica]

Why is $-a^2-1$ and not $a^2+1$ as in the solution?

[ocean seal]

Kenshin H.

[lone heart]

@flat mica Has your question been resolved?

[lone heart]

@flat mica Has your question been resolved?

[flat mica]

Not yet

[lone heart]

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[ivory gull]

Hey I need help solving this question,

[ivory gull]

I don't know how to integrate e^-x

Someone suggested u sub

[tacit arch]

What's the question

[ivory gull]

I watched a whole video with u-sub and i sort of got it now, but I didn't solve an example like this from the video and I tried doing what I normally did but its not working out for me

I just need help with integrating 41. I can do the rest

[exotic arrow]

hello I also need help and this is really confusing me

[ivory gull]

Nvm I think I got it

.close

[lone heart]

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[alpine sable]

I need help

I put 0 but it said it was wrong

Is it not 0?

[waxen flame]

Is that a test?

[alpine sable]

No

Homework

[waxen flame]

What did you submit as your first answer?

[alpine sable]

0

For both of them

[waxen flame]

Hmm, that's what I would have answered as well. 🤔

[alpine sable]

But it says it’s wrong

[waxen flame]

You may want to "ask your teacher" why it is wrong. This wouldn't be the first time online math HW was wrong.

[alpine sable]

Okay

I asked

[wary stream]

Click practice another and see if it gives you the answers for the practice question so you can see what the answer should look like

[alpine sable]

It said it was right when I put 0

For “practice another”

[waxen flame]

My money is on the website is wrong.

[alpine sable]

Most likely

It worked

I put the 0’s again

And it said it was correct now

[waxen flame]

😖

[alpine sable]

[wary stream]

What if you were typing in o's instead of 0's?

[alpine sable]

I’m pretty sure they were 0’s

There’s 5 steps for this question

[waxen flame]

You can keep asking questions. This is your channel.

[waxen flame]

What does the zero-factor theorem state?

[alpine sable]

A B?

[tacit arch]

You need to read the whole theorem

[alpine sable]

a b c d?

[waxen flame]

Intuitively, I would refer back to Step 1.

$\underbrace{(x-5)}{0} \underbrace{(x-9)}{0} = ?$

[ocean seal]

Kookiemon

[tacit arch]

Screenshot and upload the full statement

[alpine sable]

Wdym

[tacit arch]

.

[alpine sable]

Idk

[wary stream]

Literally given here

[tacit arch]

Read and understand that screenshot

[waxen flame]

Or at least let it marinate for a bit until you understand what it means. 🧑‍🍳

[alpine sable]

The one on top would be 0

[waxen flame]

What would be 0?

[alpine sable]

(x-5)(x-9)

[waxen flame]

So (x-5)(x-9) = what?

[alpine sable]

=0

[waxen flame]

Correct.

[alpine sable]

Now I have to use the foil method

[waxen flame]

Yes.

[alpine sable]

Mmmm

Would it be the equation again

(x-5)(x-9)

[waxen flame]

You need to use FOIL to expand it.

It wants the answer in ax^2 + bx +c = 0 form.

[alpine sable]

I got it

x^2-14x+45=0

Ty

.close

[lone heart]

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[winter wing]

#help-39 message

[winter wing]

@vague mirage could you please explain?

[acoustic patrol]

u guys are ass

jk

[lone heart]

@winter wing Has your question been resolved?

[lone heart]

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[rotund grove]

I was supposed to get 1/6…. Anyone know what I did wrong?

[lone heart]

@rotund grove Has your question been resolved?

[harsh swallow]

Did you remember than your integration was for u^-3 and u^-4 so that the term you're dividing by to get the antiderivative is -2 and -3?

[lone heart]

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[fluid karma]

Need help Asap

[fierce prairie]

find the derivative

plug in 2 for x

[keen mason]

Remember 4/x = 4x^-1

[lone heart]

@fluid karma Has your question been resolved?

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[reef flame]

Need help on 3 and 4

I'm pretty sure I got 3 right but I'm not sure what Lc=b and Ux=c would look like

[lone heart]

@reef flame Has your question been resolved?

[reef flame]

<@&286206848099549185>

I just don't get how I would find L

is L my first matrix that I solved for in problem 3?

and then U is what it multiplies to get my original matrix A?

[lone heart]

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[alpine sable]

I throw five coins, what is the probability that I will get more heads than tails?

[alpine sable]

for this, is throwing five coins equivalent to throwing one coin 5 times?

is this a correct and efficient method?

because I certainly wouldn't list out:

HHHHH
HHHHT
HHHTT
HHTTT

etc

[tacit arch]

Use binomial theorem

[alpine sable]

I'm not meant to

[tacit arch]

Then what are you supposed to use

[alpine sable]

because this question is for 15-16 year olds in my country, and they dont learn binomial at that level

just basic knowledge of probability

[alpine sable]

so i was thinking of this:

when you throw one coin 5 times, you can get these possibilities

5H
4H 1T
3H 2T
2H 3T
1H 4T
5T

[tacit arch]

That doesn't really answer my question

[alpine sable]

addition rules, sample space diagrams for combined events etc

[tacit arch]

Find the probabilities of those outcomes

[alpine sable]

is that important? because there are 3/6 possibilities where heads will be greater than tails

so 1/2

is that a right way of thinking

[tacit arch]

If it wasn't a fair coin then your logic would be wrong

[alpine sable]

ah i see, i want to know why it would be wrong if the coin wasn't fair

[tacit arch]

.

[alpine sable]

they are all 1/32

right?

oh wait no

top and bottom are 1/32

1/32
5/32
10/32
10/32
5/32
1/32?

but i just wanna know why my logic is wrong if it wasnt a fair coin

@crisp iron

[tacit arch]

Right for a fair coin

[tacit arch]

Do the same calculations for P(H) = not 1/2. Say something simple like 3/4 or 2/5

[alpine sable]

243/1024
405/1024
270/1024
90/1024
15/1024
1/1024

@tacit arch

[tacit arch]

Now calculate P(H) / P(T)

[alpine sable]

3/4 / 1/4?

3?

@tacit arch

oh

is it because

my logic is based off the fact that

there's an even distribution?

but whereas with an unfair coin it's not even

but how come I can't use my logic to figure out the probability of me getting at least 1 head, when I flip 1 coin 5 times?

5H
4H 1T
3H 2T
2H 3T
1H 4T
5T

my logic would be 5/6 being the probability but the actual answer would be 31/32

<@&286206848099549185>

or my logic is completely flawed and it only worked for the first question because of pure coincidence

because for it to work every time, the probabilities of 5H, 4H 1T has to all be the same

which isn't possible, even for a fair coin

there's always a distribution/bell curve

================================================================
so how would I tackle this question without binomial theorem?

I throw five coins, what is the probability that I will get more heads than tails?

<@&286206848099549185>

[stray estuary]

@alpine sable

[alpine sable]

hey

[stray estuary]

wait is this unfair coin

[alpine sable]

no no, fair coin

[stray estuary]

oh okay

so yes your logic would be flawed

[alpine sable]

ye, so now I want to figure out a way to find an answer for this q without using binomial theorem

because it's in a textbook used by students where binomial theorem isnt taught to them yet

[stray estuary]

ah i see

okay so

5 coins?

[alpine sable]

ye

[stray estuary]

hmm

okay so no binomial

can you not just draw a tree diagram for this?

[alpine sable]

i could, but its before the tree diagram chapter 😅

[stray estuary]

ahh fuck

LOL

[alpine sable]

so not using tree diagrams either

[stray estuary]

uhhh

is it not going to be just 1/2?

without binomial theorem, cant you reason that you only either have more heads or more tails?

and nothing in between?

[alpine sable]

can u elaborate further?

[stray estuary]

well

when you throw 5 coins

what are the two possible outcomes?

[alpine sable]

heads or tails

[stray estuary]

exactly

each is half as likely

since its an odd number of coins, then you either only have more heads or more tails

so its just 1/2

[alpine sable]

wait

i need to digest this

😂

[stray estuary]

yeye nw

[alpine sable]

what if you had 6 coins

6H
5H 1T
4H 2T
3H 3T
2H 4T
1H 5T
6H

[stray estuary]

you would have to subtract the probability of getting 3 heads and tails

[alpine sable]

ohhh

[stray estuary]

right yeah

[alpine sable]

wait but why

because we're only concerned with

more H > T

[stray estuary]

well for 6 coins, you can have H = T

but for 5 coins, its only either H > T or H < T

[alpine sable]

but we've still got 3 scenarios where it's more H than T no?

[stray estuary]

thats for an even number of coins

so for an odd number of coins, we only have two possibilities, # H > # T or # H < # T

for an even number of coins, we have three though

the two i said above and # H = # T

[alpine sable]

im just tryna understand why you have to subtract it

i must be missing something or its just 4am

[stray estuary]

well its not good its 4 am LMFAO

but anyways, why wouldn't we subtract it?

does H = T count as H > T?

[alpine sable]

it doesn't

[stray estuary]

therefore we subtract it

LOL

it doesnt count so we have to subtract it

[alpine sable]

but if it doesnt count then can't we just ignore it?

[stray estuary]

no, because its still a possibility to occur

[alpine sable]

ohhhhhhhhhhhhhhhhh yeah it's still part of the entire system so to figure out the probability of #H > #T that H = T is gonna be part of the denominator right?

[stray estuary]

right

[alpine sable]

but then that isnt subtracting its just

taking it into account

[stray estuary]

well i was starting from 1

but yes

thats fine

[alpine sable]

wdym?

1 coin?

[stray estuary]

well

okay

let me apply this with smaller amounts of coins

lets take 1 coin

we either have heads or tails right?

so

$1 = \frac{P(H > T)}{2} + \frac{P(H < T)}{2}$

[ocean seal]

blanket

[stray estuary]

hopefully its easy to see that the stuff on the numerator will both be 1

so 1/2 plus 1/2 equals 1

right?

[alpine sable]

ye

im with you

[stray estuary]

okay cool

so

lets do two coints

we either have less or more or equal right

so

$1 = \frac{P(H > T)}{4} + \frac{P(H < T)}{4} + \frac{P(H = T)}{4}$

[ocean seal]

blanket

[stray estuary]

wat

[alpine sable]

bruh 💀

[stray estuary]

but does this make sense? @alpine sable

[alpine sable]

yes that makes sense

[stray estuary]

okay cool

hopefully its also simple to see that

[alpine sable]

wait

[vapid steppe]

Wrong channel sorry

[stray estuary]

$P(H > T) = P(H < T)$

[ocean seal]

blanket

[stray estuary]

(for fair coins)

so we can rewrite this as

$1 = 2P(H > T) + P(H = T)$ for even coins

[ocean seal]

blanket

[stray estuary]

so

$\frac{1 - P(H = T)}{2} = P(H > T)$

[ocean seal]

blanket

[alpine sable]

wait are we not missing smth

[stray estuary]

what are we missing?

[alpine sable]

#H > #T will be 1
#H < #T will be 1
#H = #T will be 1

but that gives us 3/4 in total

[stray estuary]

no

there are two occurrences where H = T

1 H 1 T

1 T 1 H

[alpine sable]

ohhh 1 T 1 H

yh

[stray estuary]

yeah

[alpine sable]

lemme re-read

[alpine sable]

i see what you did here but how this will benefit us

[stray estuary]

its not supposed to

you should recognize that $P(H = T) = 0$ for even # of coins

[ocean seal]

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

[stray estuary]

so divide by 2

you should see 1/2

[alpine sable]

bro im so sorry maybe im just slow af rn but

0?

oh shjit

they're equally as likely with even coins but not 50%

when it comes to P(#H >#T) and P(#H < #T)

im with you

the sample space is

divided into three for an even number of coins

instead of two for odd

[alpine sable]

there's a symmetry

[alpine sable]

but not here

[stray estuary]

right

[alpine sable]

oh yeah, so for even coins

to find the probability for the more heads occurring than tails, we subtract the probability that there's the same amount from 1, then we can divide it by 2 because the two other sample spaces are equal

i get it

thank u sm bro

i can go sleep

[alpine sable]

its amazing to me how you explained it in this fashion

[alpine sable]

ahh u meant subtract from 1 (overall probability)

maybe my brain is no longer in deep sleep and im understanding things now

LOL

[alpine sable]

i think you meant for odd here

it all clicks

thanks man im so grateful

I wish you well

[lone heart]

@alpine sable Has your question been resolved?

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[alpine sable]

hello

[alpine sable]

im trying to estimate instantaneous velocity

i keep getting it wrong

but im pretty sure im doing it correct

[west bone]

is there a specific problem you are having trouble with?

hello?

.close

[lone heart]

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Use .reopen if this was a mistake.

[alpine sable]

dam

.reopen

[lone heart]

✅

[alpine sable]

yes

[west bone]

oh ok sorry

[alpine sable]

so i took avg from 2-3

and avg 3-4

added them and divided by 2 and it is wrong

[west bone]

hmm

<@&286206848099549185>

[alpine sable]

her is my math

[west bone]

how did you get 27.1

[gusty gorge]

hmm I think the issue here is that you're taking the average velocity over the interval [2, 4] and using that to estimate the instantaneous velocity

[alpine sable]

23.1-50.2/-1

[gusty gorge]

but the thing is that this thing is clearly accelerating, so that will be an underestimate

[alpine sable]

okay so what shall i do

avg 1-3 and 3-6?

[gusty gorge]

no that will be even worse

first, you should note that averaging the average velocities between 1 and 3 and 3 and 5 is just computing the average velocity from t=1 to t=5

well actually no

but

weighted average whatever

[alpine sable]

noted

[slow totem]

hey you with the red dress on
I gotta find a way to take it off
I got a lot of love and it's growing strong
When I see you with your red dress on

[gusty gorge]

in general, there isn't a way to get a single answer from it

like unless you know the full characteristics of the motorcycle's motion

it turns out this motorcycle can't be at a constant acceleration

but you can try fitting a quadratic or a cubic to it

and then figuring out the slope at the given point

[alpine sable]

so i must graph and solve with tangent line

like it asked me to but i assumed i could estimate close enough like this

[gusty gorge]

well the graphing technique doesn't work well either

it's the same problem

you don't have points near to 3

so how are you going to draw a tangent line?

a cubic fits pretty nicely to it though

[alpine sable]

not following anymore

also what did you use to graph that?

[gusty gorge]

https://stats.blue/Stats_Suite/polynomial_regression_calculator.html

[alpine sable]

nice

so uh

how i do dis?

[alpine sable]

well it wouldnt be a graph of a tangent

i would use a tangent line to narrow in on the instantaneous velocity

[gusty gorge]

you can't draw a tangent line without already having the points nearby

like not sure what you mean by "use a tangent line to narrow in on the instantaneous velocity" is supposed to mean

[alpine sable]

me either

ive just seen ppl do things like this to find the instantaneous rate of change

i dont really know about it

so do you know of a way i can find this?

[tacit arch]

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:average-rate-of-change/v/introduction-to-average-rate-of-change

[alpine sable]

but

i already found all of the averages.

i need to go to sleep

.close

[lone heart]

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[gusty gorge]

okay I have a question about some applied probability that I've been struggling with for a little:
Let X_i be distributed according to a mixture distribution: p proportion of the time, it's sampled according to Uniform(a, b), 1-p it is always 0

What is the distribution of X_1 + X_2 + ... + X_n? Clearly, we can compute the expectation and variance of X_i and use the normal approximation for large n, but is there an exact thing?

Further, the specific case I'm interested in is p = 1/3 or 1/4, a = 468, b = 1169, n = 5. I've calculated Var = 812696, E = 1364.16 for the total. Is a normal approximation good enough? I'm too lazy to simulate

[cold leaf]

dam

[gusty gorge]

I feel like I'm missing something simple here, but that's what happens when you slack off in class

[cold leaf]

imma be hosnet ive never seen a problem like that

[stray estuary]

wait so each of the x_i have mixture distributions?

[lone heart]

@gusty gorge Has your question been resolved?

[prime badge]

well suppose n is 2, then p² time it's 2 unfiorms, and (1-p)² time it's 0 and the rest it's one uniform?

is that wrong?

[lone heart]

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[tropic oasis]

how would i start on this?

[tropic oasis]

i guess graphing but

should i consider f(x) = 2xe^(-x)?

[fierce prairie]

yeah

[tropic oasis]

rotate on x-axis

meaning we integrate by x

i think i see a washers method?

limit from 0 to 4?

[fierce prairie]

well, isn’t the question just asking for area?

[tropic oasis]

oh

[fierce prairie]

or do you also need to rotate around x axis

[tropic oasis]

i misread

i thought rotate on x-axis but it just said on the x-axis

[fierce prairie]

yeah lmao

[tropic oasis]

how would i go as for finding the area again?

[fierce prairie]

just integral from 0 to 4

as the integral is the area under the curve

[tropic oasis]

so graphing was kind of pointless? since i dont need a specific method?

[fierce prairie]

yeah pretty much

ig they just wanted you to visualize

[tropic oasis]

.close

[lone heart]

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[vapid shuttle]

,, \frac{i}{i-1}

[ocean seal]

AustinU

[vapid shuttle]

What is the easiest way to simplify this, without multiplying by the conjugate

nvm , it is multiplying by negative i

.close

[lone heart]

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[vapid shuttle]

.reopen

[lone heart]

✅

[vapid shuttle]

How can I find the argument of either

or

[ocean seal]

AustinU

AustinU

[naive valley]

well the second one is real

[vapid shuttle]

The second one is the magnitude of the first

[naive valley]

so its argument should hopefully be obvious

[molten pivot]

i/(i-1) = (i-1 + 1)/(i-1) = (i-1)/(i-1) + 1/(i-1) = 1 + 1/(i-1)

[naive valley]

for the first one, try expressing 1+i in polar coordinates

[vapid shuttle]

I get sqrt(2) * e^(i * arctan(1))

[naive valley]

and arctan(1) is what angle?

[vapid shuttle]

pi/4 ?

[naive valley]

yep

so what can you say about the argument of 1/(1+i)

[vapid shuttle]

4/pi ?

[naive valley]

no, it doesn't work that way...

do you know the notation e^(i theta) ?

[vapid shuttle]

I can represent it using euler's formula if that is what you are asking

[naive valley]

yep

what is 1 / e^(i theta) ?

it's e to the what?

[vapid shuttle]

Sorry, I am not very good at the complex manipulation stuff

Is it just the negative of the angle then

the argument then being -pi/4

[naive valley]

yep!

exactly

[vapid shuttle]

Okay nice

So

I basically have the imaginary one as the complex gain of my function at a certain point

and when I say what that complex gain actually is I take the magnitude

which is where the 1/sqrt2 comes from

Then I am trying to find the phase-lag, which is the negative argument of the complex gain

[naive valley]

ok

[vapid shuttle]

but do I use 1/sqrt2, or the imaginary part for finding that

[naive valley]

so negative of -pi/4 ?

[vapid shuttle]

which do I take the argument of

[naive valley]

1/sqrt2 is the magnitude

[vapid shuttle]

So the negative argument of that?

[naive valley]

you've already lost the phase information when you compute the magnitude

[vapid shuttle]

oh

[naive valley]

for phase you have to go back to the original complex number

[vapid shuttle]

Gotcha, so yes -pi/4

[naive valley]

yea

[vapid shuttle]

Okay thanks for talking this through with me

[naive valley]

or pi/4 if you need negative of that

[vapid shuttle]

mhm

thank you

[naive valley]

sure

cheers

[vapid shuttle]

For part a, could x=cos(t)-t be a solution?

and then b, 2(cos(t)-t)

But I need help with solving part c. It looks like I should use something to do with time-invariance, but I am not sure how to begin

any tips?

[gusty gorge]

you should be able to just shift the solution

something something LTI systems

[vapid shuttle]

that is what I am thinking

but

[gusty gorge]

also not sure about b

[vapid shuttle]

I am confused about the notation

[gusty gorge]

what notation

[vapid shuttle]

because it doesn't really give me any initial values?

how does q(t-1) tell us that we are shifting the initial values

[gusty gorge]

why do you need initial values

it's shifting the solution x

[vapid shuttle]

I've only ever done the time-invariance with explicit problems and initial values so I am just a bit confused on how to apply it to one like this where I don't know the function q(t) or the p(D)

[gusty gorge]

you don't have to though

if you know that a particular function satisfies the differential equation

[vapid shuttle]

cost and t both do

[gusty gorge]

then if you consider the modified one with q(t-1)

then the shifted functions will satisfy that one

[vapid shuttle]

so cos(t-1) or cos(t)-1 and t-1?

[gusty gorge]

because (Df)(t-1) = Df shifted by 1

[vapid shuttle]

so you are saying something like this right

If we have

P(D)x=q(t)

and we know its solutions

solving P(D)x=q(t-1)

equals

solving P(D)x=q(t) - q(1)

[gusty gorge]

the solutions to P(D)x = q(t-1) are the solutions to P(D)x = q(t), but shifted to the right by 1

[gusty gorge]

no

like a concrete example here

if you consider the differential equation d / dt of x = 2t

clearly, the general solutions are x(t) = t^2 + C

but since differentiation is a linear time-invariant operator, you know that the solutions of d/ dt of x = 2(t-1) are simply x(t) = (t-1)^2 + C

this holds for things for polynomials of D

[vapid shuttle]

oh okay

that does make sense

[gusty gorge]

because D(x shifted by 1) = (Dx) shifted by 1

and in general, any power of D will be like that

[vapid shuttle]

so if we had the normally solution being x=cos(t) then our shifted will be cos(t-1)

[gusty gorge]

yes

[vapid shuttle]

Thanks for that

I'm bad with the notation still, it all always throws me for a loop to see new stuff

[gusty gorge]

you can think of D as an operator that converts a function into another function

and p(D) is just some operator like (D^2 + D)

[vapid shuttle]

I understand what the differential operator notation is

it was mainly the q(t) side

[gusty gorge]

q(t) is just some function of t

could be anything

[vapid shuttle]

well yah

sorry, I guess I just meant to say that since I've been used to only like explicit questions, seeing one like p(D)x=q(t) without actual functions or knowing what P(D) is just sort of intimidates me I guess, because I feel a bit silly that I needed help with them now that I have them complete.

Thank you for your help

[gusty gorge]

I guess it's sort of trying to motivate some things like the Laplace transform

basically it ends up not really mattering what q(t) is, since there are general techniques

[lone heart]

@vapid shuttle Has your question been resolved?

[lone heart]

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[mellow sphinx]

hi

[mellow sphinx]

i need help

These questions use simple interest-buying on terms

<@&286206848099549185>

[modern bluff]

ok

[grave loom]

shut up

ugly

im trying to figure out the most efficent way to build a nuclear reactor

[modern bluff]

whyy meean

[grave loom]

im here to build nuclear reactors

not do money

[mellow sphinx]

ok

I just want some help thanks

I=PRN

[wheat onyx]

Which one

[mellow sphinx]

Both questions

but 9 and 10

would be best

[modern bluff]

iidk

[gusty gorge]

I know this isn't relevant at all

but please don't borrow money to go on an overseas holiday

especially not at 9% annually

brooke is actually a doofus

[north hemlock]

jesus...

this chat is so bad what am I reading 😭

[wheat onyx]

Its just a simple interest problem

[modern bluff]

<@&286206848099549185>

can hhelp

myy cousinn

nneeeedds help

[gusty gorge]

simple interest is just I =PRT

so figure out how much interest is paid

[modern bluff]

hiisss homework

[gusty gorge]

figure out how much was paid in total

[wheat onyx]

17% is count as for 4 years or annual?

[gusty gorge]

jesus this homework is just encouraging awful spending habits

[modern bluff]

3 15 35 63

1.find tthe nth terms

[gusty gorge]

why not borrow money to make an investment or buy a house

[mellow sphinx]

over 4 years

[gusty gorge]

why does it have to be stuff like getting a pool

[gusty gorge]

#❓how-to-get-help

[wheat onyx]

I don't know the meaning of 3% commission charged

[gusty gorge]

they basically tack an extra 3% onto the loan amount

[wheat onyx]

Total 21600 dollars over 208 weeks means 103 per week

[lone heart]

@mellow sphinx Has your question been resolved?

[modern bluff]

/?

?

!

1!??

!?

[rose sigil]

???

if you have a question you should read #❓how-to-get-help and post it in an open channel 🙂

[lone heart]

@mellow sphinx Has your question been resolved?

[lone heart]

Channel closed

Closed due to timeout

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Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
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[sinful citrus]

I need help

[sinful citrus]

How to solve

[brisk dawn]

using sin

[sinful citrus]

Sin = 67 /15?

[brisk dawn]

a) sin67 = a / 15

then find a

[sinful citrus]

I still don’t understand-

[vapid shuttle]

Tren do you know SOHCAHTOA?

[sinful citrus]

Yes

I know

[vapid shuttle]

So you know that sine of an angle equals the side length opposite to it divided by the side length of the hypotonuse?

sine = opposite / hypotonuse

right?

[sinful citrus]

Right

[vapid shuttle]

So for that first triangle

you have an angle, 67 degrees, and you know the side length opposite to it = "a" and the hypotonuse is 15cm

so what equation can you set up

[sinful citrus]

Sin 67= a/15

[vapid shuttle]

Exactly

and then can't you solve that for a? to find the side length

I'd suggest the same method for the second triangle as well 😉

[sinful citrus]

Oh just like that , thank you 🙏🏻 i can understand now

[vapid shuttle]

perfect, no problem

[sinful citrus]

.close

[lone heart]

Channel closed

Closed by @sinful citrus

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[alpine sable]

How do I do iii?

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

<@&286206848099549185>

[hard mountain]

Have you tried calculating $$ \frac{z^7 +1}{z+1}$$ ?

[ocean seal]

OOKK

[alpine sable]

I was thinking to do that but I don’t know where to go from there

[vapid shuttle]

do you know how to do the polynomial long division

[alpine sable]

Yes but no way I’m doing that

[vapid shuttle]

lol

it might be the only way!

[alpine sable]

I think the solutions have a different way

[vapid shuttle]

honestly, it doesn't look too bad

[alpine sable]

Want to see?

[vapid shuttle]

sure

[alpine sable]

I don’t understand why they multiplied it instead of dividing

[vapid shuttle]

They are showing that that will be the result of the long division

[alpine sable]

Ohhhhh