help-26

[novel solar]

because there's a 50% chance of having 1 boy and 1 girl or 1 girl and 1 boy

I thought the probability part

like the 1/2^whatever

would be stronger than the numerator

and it would approach 0 at larger numbers

like for 4

it'd be bbgg or ggbb

but the prob of either is 1/16 and 1/16*2 = 1/8

so the 4 case happens with prob 1/8

4 * 1/8 = 1/2

1 + 1/2 + ...

i thought it'd keep getting smaller

and converge to something

can you walk me through this

[hearty turret]

well equivalent it suffices to show that if X_n is a simple symmetric walk starting at 1

then setting N = first time we hit 0

E(N) = infinity

condition on your first step

so $\mathbb{E}[N] = \mathbb{E}[N \mid X_1 = 0]\mathbb{P}(X_1 = 0) + \mathbb{E}[N \mid X_1 = 2]\mathbb{P}(X_1 = 2)$

[thorny flame]

LY

[hearty turret]

notice that E[N | X_1 = 2] = 1 + 2E[N]

so we have E[N] = 1 + E[N]

which only holds if E[N] is infinity

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[novel solar]

one sec

i'm reading through

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[final swift]

how is e^x^1/2 differentiated?

[final swift]

nvm im dumb

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[green dock]

Is there a way not to use number of Euler in these proofs

[green dock]

nvm found it

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[rocky salmon]

4 points fall randomly into a circle, what is the probability that 4 point fall into a same semicircle

[rocky salmon]

I could solve that for 3 points it is 3/4, but could not solve for 4

[eternal spoke]

i found some youtube videos that can help you understand this problem

https://www.youtube.com/watch?v=vxF2qh7lXhc

https://www.youtube.com/watch?v=cTYFF4DPvhg

this is for the general case but you can apply it when n=4

hope it helps

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[rocky salmon]

TYSM!!!!

[rocky salmon]

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[silk tide]

I'm trying to conceptualize how sequences diverge or converge.

I know if a sequence does not settle near a value, then it is divergent, and when it does, it's convergent.

Here's a problem i've done for example. an = 5 as n approaches infinity, but when i put the equation into a graph, it doesn't look anything like that.

Can anyone help me try to understand this?

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[unique moon]

hi, need a little help with this

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[thick oyster]

so if im trying to find where a function is increasing or decreasing, i took the first derivative of the function and found the x values but what if an x value isnt in the domain?

[loud oasis]

what do you mean?

can you be more specific about your problem?

[thick oyster]

sure. the problem is f(x) = x^2 / (x^2 - 4)

i derived it to get -8x/(x^2 - 4)^2
and i got x = 0, 2, -2

but the domain is (-inf, -2) U (-2, 2) U (2, inf)

do i just ignore the 2 and -2?

and test values of 1 and -1 (around 0)?

[loud oasis]

so those are the 3 critical points of the function

[thick oyster]

ya thats what i mean

[loud oasis]

that means that the function is always increasing or always decreasing between those points

critical points don't have to be in the domain of the function

[thick oyster]

oh really

oke then :D thx

[loud oasis]

all we really care about is what happens between them

[thick oyster]

oke so ill test -3, -1, 1, and 3

thx

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[fossil moss]

I'm very much unsure how to even begin to solve 6 and 11
I was in hospital when the lectures for this part of the topic were on and I've too much to catch up on but this assignment is due really soon
Could someone please help?

[olive birch]

Try to solve problem 6 graphically

That is, on a cartesian plane label the range of time of Sam as an x-axis and the range of time of Jane as an y-axis

[fossil moss]

I'm trying rn

I'll get back to you

If I can do anything or no

I can't think of how the graph would look, cuz there's a probability Sam's there at 12 but not Jane

how can I have an x but no y unless its on the axis?

Or i draw y=x between 11:55 and 12:25?

Idk I suck at probability

[olive birch]

And then find the area

[fossil moss]

of?

Above y=x means jane is there first and below means sam is there first?

[olive birch]

Above $y=x$ means a point can be represented as $(x, x+\epsilon)$

[thorny flame]

timuko

[fossil moss]

I'm really sorry but that doesn't mean a thing to me at the moment

I absolutely cannot get a grasp on even basic probability

I don't understand it in a more intuitive way like say calculus or matricies

I can't see what to do

@olive birch can you pls solve the problem and show me how / why you did each step? If it doesn't make sense to me I'm just fucked for the rest of probability

[olive birch]

Here, point $(a,b)$ means that Sam showed up at $11:55 + a$ time and Jane showed up at $11:55 + b$ time

[thorny flame]

timuko

[olive birch]

Note that $10 \le b \le 20$

[fossil moss]

I have a diagram like that but I got the probability to be ½

[thorny flame]

timuko

[fossil moss]

I was being dumb

[olive birch]

So 6a answer is .5

[fossil moss]

Nah probably not

I was wrong

probability is 150/300.... wait it is ½?

?

[olive birch]

Yes

For 6b, note that if the point lies below line $y=x-12$, the point can be represented as $(x, x-12-\epsilon)$ for $\epsilon > 0$. Because $x-(x-12-\epsilon) > 12$, it follows that Sam showed up more than 12 minutes after Jane arrived, so Jane will be already left.

[thorny flame]

timuko

[olive birch]

Thus, the chance that Same faces the angry Jane is the same as finding what fraction of area of a rectangle lies between lines $y=x$ and $y=x-12$

[thorny flame]

timuko

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[fossil moss]

🙏

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[fossil moss]

Wait @olive birch

Why x-12?

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[clever shard]

if a function has a (x-6) in the numerator and a (x-6) in the denominator, is the hole at x=6 or x=-6

[torpid sparrow]

Replace x with each value and evaluate

[true coral]

In general for (x-n) the hole will be at n

Logically this makes sense as (n-n) will become 0

Hence making a function discontinuity

[clever shard]

x-n?

[torpid sparrow]

Why whould dividing by -12 would make a whole?

[true coral]

In the case of x-6 it would be at x=6

This is because the denominator would be 0 which makes the function undefined

[clever shard]

o okt ty

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[limpid peak]

my teacher is confusing me help pls

[limpid peak]

finding reciproque or i think its inverse function in english

[hollow drum]

What step are you confused at?

[torpid shard]

its inverse function, what it basically tells you is to find the value of x in terms of y, instead of y in terms of x (which is usually how functions work)
once you solve, then switch y and x, and thats your inverse function}

[limpid peak]

why is the answer the same as the question

[torpid shard]

you do that by moving terms / factors to the other side

defining f(x) as y

also, not all functions can be inversed

[hollow drum]

Coincidence

[torpid shard]

i think his teacher did things wrong 💀

[hollow drum]

No it's right

[limpid peak]

why did he move the 2 to the left, just to move it back to the right 😭

[hollow drum]

You want to solve the equation for x

[limpid peak]

im hoping thats what it is

[hollow drum]

So the first step was the subtract 2

[hollow drum]

It's correct

[dim ruin]

It is

I believe he is confused at the moment x and y kind of switch places

[limpid peak]

i get that part, but why did he move the 2 from the right to the left, then move it back to the right

[hollow drum]

As I mentioned, you're setting the equation equal to x, so the first step was the subtract 2

[limpid peak]

i got this from the way i learned but im doing it wrong somehow, my friend got this aswell

[hollow drum]

That's where this came from

[limpid peak]

alr

[hollow drum]

It's the same

As your teacher

[dim ruin]

they move the 2 to LHS so he can then divide all of that. If they didn't, you would have to divide by only y and you would get
1 = 3/y(x-2) + 2/y

and that's annoying

[hollow drum]

3/(x-2) + 2 is the same as 3/(x - 2) + 2(x -2)/(x - 2)
Method of getting a common denominator to add fractions

[dim ruin]

Note that if you have a = b/c you can algebraicly turn that equation into c = b/a (multiplied c/a on both sides)

[hollow drum]

You pretty much simplified yours to one fraction, your teacher did not combine the two

[limpid peak]

which one is the inversed function

[opaque hearth]

I'm in the same class as him, im lowkey trippin 😭

but they're

different from my perspective

[hollow drum]

I literally showed how they are the same, you are adding them together using a common denominator

[torpid shard]

like, they are almost the same, its -2 instead of +2

@hollow drum can you fact check this?

[hollow drum]

You realize it's $\frac{3}{x - 2} + 2$

[thorny flame]

CaptainNova22

[hollow drum]

You did x + 2

[limpid peak]

ohhh ok im starting to get it

[hollow drum]

Both are equivalent

[limpid peak]

wait so was this alla coincedence or smth

usually when i find inverse functions they look way different yk\

[torpid shard]

oooh, thats why, mb mb

[hollow drum]

Your solution and the teacher's are the same

[limpid peak]

alr

[hollow drum]

That's what I said at the beginning

[hollow drum]

.

[torpid shard]

yeah, i simply thought that the symbol was a +, not a -

[limpid peak]

oh yeah 😭

also rq, how ab this one

[hollow drum]

Where are you confused?

[hollow drum]

@limpid peak

[limpid peak]

mb gimme a sec

are these the same

[hollow drum]

,rotate

[hollow drum]

Yes

If you're unsure, combine the terms that your teacher has, using a common denominator

[limpid peak]

alright thank you

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[raven ledge]

Bottom question @hollow drum

[raven ledge]

So i got that 24

remainder is -6, how do i solve for d?

since its the constant

[hollow drum]

You agree up to here, correct?

Your work isn't too clear so I rewrote it

[raven ledge]

Yes

[raven ledge]

Yea sorry ran out of space

[hollow drum]

So you know that the remainder is -6, meaning that when you subtract the last part, you should get -6, agreed?

[raven ledge]

Yea

[hollow drum]

So if you subtract the last part, what do you get?

In terms of d

[raven ledge]

Oh, d = 18

Makes sense thanks

[hollow drum]

Yep

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[neon iron]

[rapid lake]

lol what happened

[neon iron]

idk...

[agile harness]

g is not odd sir and h is not neither

[neon iron]

exactly bro....

[agile harness]

g would be odd if g(-x) = -g(x)

x^2 - 4x ≠ -x^2 - 4x

[neon iron]

preciate dat...

[agile harness]

gotchu bro

[neon iron]

and 4 is even because theres no x so its always 4

?

[agile harness]

no need to be condescending sir

[agile harness]

yep

h(-x) = 4

[neon iron]

how is that condescending

[agile harness]

…

[neon iron]

u gotta touch grass bro

chronically online

bro thinks people thankin him got attitude now

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[full dagger]

Im doing homework on chain rule and the problem is
f(x) = x*g(h(x))
g(4) = 2
g'(4) = 3
h(3)=4
h'(3)=-2
find f'(3)

[full dagger]

The answer key given by the teacher is this, but I thought that there would be a parenthesis before the problem is multiplied by negative 2 instead of just multiplying the 3(g(4))

because i get that product rule adds multiplication in I just thought it would deal with that rather than multiplying just one side of the product rule

[maiden cipher]

Too many parentheses can get a bit confusing.. have you tried leaving off the "(x)" from all your function labels and seeing if that makes more sense?

Honestly it looks like the only reason there are parentheses around the -2 is to isolate the negative sign along with the 2.. so it doesn't look like subtraction

[full dagger]

Yeah my problem is mainly that I thought there would be parenthesis around the term getting hit by the product rule

but there doesn't seem to be in the answer key, which my understanding of chain rule is that there should be based on past problems

[maiden cipher]

You can, but.. it's not mandatory, if your notation makes it clear that multiplication is taking place.. hence the use of the dots

Sorry.. what do you mean? What do you think it should look like?

[full dagger]

My belief is that instead of 2+3(3)(-2) near the end bit it should be (2+3(3))(-2)

because the term that becomes (2+3(3)) would be one term in the chain rule which is expanded by taking its derivative through product rule

[maiden cipher]

That splits things up incorrectly.. taking the product rule and part of the chain rule results while excluding other parts of the chain rule results

The + comes from your product rule, though

[full dagger]

So the product rule would completely split it up into 2 different terms?

[maiden cipher]

Yes

[full dagger]

Okay i was confused on that

thank you

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[autumn plaza]

can someone pls help how do i keep getting non real in calc

[smoky sparrow]

you can't use your GDC to find the derivative of an expression in an exam

the GDC only gives you a number

the derivative at a particular point

[autumn plaza]

whats gdc

[smoky sparrow]

graphing display calculator

[autumn plaza]

i think my calculator is broken

[smoky sparrow]

it's not

[autumn plaza]

his one does it but mine doesnt

for integration

wait i forgot the squared

[smoky sparrow]

you're trying to do something your calculator can't do

[autumn plaza]

brb

[smoky sparrow]

you have no bounds on your integral

[autumn plaza]

Oh i didnt

[smoky sparrow]

that's the issue

[autumn plaza]

how come their one does it?

[smoky sparrow]

they have computer algebra mode on

you can't use that mode in the exam

[autumn plaza]

r u sure? because my teacher does it too

[smoky sparrow]

you're doing HSC right

[autumn plaza]

WACE

[smoky sparrow]

oh right

https://senior-secondary.scsa.wa.edu.au/__data/assets/pdf_file/0009/1088910/Mathematics-Specialist-ATAR-Year-12-Syllabus-for-teaching-from-January-2024_pdf.PDF

okay I was wrong you can use CAS mode

[autumn plaza]

oh

how do i make it work

[smoky sparrow]

no idea

ask your teacher

[autumn plaza]

oh 😔😔

we on exam break im in the library by myself

i just finished my chem exam :D

[autumn plaza]

wait

do u think we're allowed assist mode in calculator

i just found assist mode

and it worked

[smoky sparrow]

you are allowed, my bad I forgot you were doing WACE

[autumn plaza]

wait

[autumn plaza]

how u know

does it say in the syllabus

[smoky sparrow]

you told me

yeah I checked your syllabus and indeed it says CAS allowed

It is assumed that candidates sitting this examination have a calculator with CAS capabilities for Section Two

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[autumn plaza]

how would i do iii?

[agile harness]

if A and B are independent then P(A) * P(B) = P(A and B)

[autumn plaza]

i wrote that done

down

[agile harness]

yea and what else should you use

[autumn plaza]

i have probability of A

[agile harness]

you were given P(A or B)

and P(A)

[autumn plaza]

yea

[agile harness]

$P(A \cap B) = 0.2P(B)$

$0.68 = 0.2 + P(B) - 0.2P(B)$

[thorny flame]

knief

[agile harness]

from P(A or B)

[autumn plaza]

what

what is -0.2P(B)

[agile harness]

^

[autumn plaza]

wait what is this

why?

[agile harness]

P(A) = 0.2

[autumn plaza]

yes

now what

[agile harness]

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

P(A or B) = 0.68

P(A) = 0.2

we determined this

[autumn plaza]

can u use the U and n symbols instead pls?

[agile harness]

since they were independently

[autumn plaza]

this is kinda confusing me

[agile harness]

$\cap$

[thorny flame]

knief

knief

knief

[agile harness]

better?

@autumn plaza

[autumn plaza]

yea

[agile harness]

does this make sense now

[autumn plaza]

hmm i started doing this

can this work as well?

bc if AUB = 0.68 P(B) has to equal 0.68 - P(A) -P(AnB)?

[agile harness]

yea i know how you got it

[autumn plaza]

does that also work

or am i overcomplicating

[agile harness]

shouldn’t it be + P(AnB)

i think my way is much simpler thouyh

[autumn plaza]

should it? idk im bad at probability

why is it +?

[agile harness]

from here, $P(B) = P(A\cup B) - P(A) + P(A\cap B)$

[thorny flame]

knief

[agile harness]

@autumn plaza

[autumn plaza]

hi

[autumn plaza]

oh what? how come?

[agile harness]

hi

[autumn plaza]

i thought probability of only B we subtract P(AnB)?

too

[agile harness]

^

just isolate P(B)

[autumn plaza]

im trying to visualise with venn diagrams tho

if we have P(AuB) then its all of them then to get P(B) we subtract the middle and P(A)?

[agile harness]

no that’s B not A?

if we have P(AuB) then subtract P(A) we’d get rid of the intersection P(AnB)

so we have to add that back

[autumn plaza]

oh is probability of B also P(AnB)?

[agile harness]

yes because P(AnB) is in B

[autumn plaza]

ohhh

ok makes sense

[agile harness]

so it would be 0.48 + x

which also works

but i think my way is much quicker

[autumn plaza]

probability of A also includes P(AnB) so wouldnt there be 2 P(AnB)?

am i dumb..

[agile harness]

what

maybe you should draw it

[autumn plaza]

ok brb

[agile harness]

P(B) is just the B circle, we don’t care if it also happens to be in A

[autumn plaza]

this is P(A) right?

[agile harness]

ok please erase the black shit though

[autumn plaza]

what do u mean

[agile harness]

it makes the image hard to look at

[autumn plaza]

huh how?

[agile harness]

i’ll draw it then

[autumn plaza]

oh

oka

[agile harness]

,rotate

[agile harness]

if we just did AuB - A we’d exclude the green region

which is still part of B

so we have to add that back

to get all of B

[autumn plaza]

ohhh

omg

ok that makes sense

[agile harness]

good

[autumn plaza]

i have trouble visualising things ):

[agile harness]

no worries

[autumn plaza]

how would we write only B?

[agile harness]

the purple?

[autumn plaza]

yes

[agile harness]

B-A or BnA^c

A^c is A complement

[autumn plaza]

ohh ok

and if we want to include the sample space too we do BuA^c?

i mean the outside

[agile harness]

well no, because B or A^c would include the intersection of A and B

[autumn plaza]

oh

[agile harness]

and generally we assume the sample space is the same as AuB tbh

i just draw it in because it’s standard

the outside region is A^c n B^c though if you were curious

[autumn plaza]

o

how did u learn probability?

was it from teacher

bcoz its my weakest topic

[agile harness]

nah

[autumn plaza]

yt?

[agile harness]

i self studied ap stats

but i also had some exposure in algebra 2

and i’m currently taking a second course in statistics rn for college

[autumn plaza]

oh

american?

[agile harness]

yea

it’s late

ik

3 am

[autumn plaza]

Lol its 3 pm for me

[agile harness]

oh fire

[autumn plaza]

im on the car going back home rn

in

[agile harness]

australia?

[autumn plaza]

yea

[agile harness]

nice accent

new zealand accent is probably my favorite though

https://tenor.com/view/korg-ginormous-spaceship-marvel-guardians-of-the-galaxy-wsbn-gif-21930517

[autumn plaza]

yea its fire we have a lot of kiwis in australia

[agile harness]

never had one tbh

you ever see a kangaroo

[autumn plaza]

no

...

[agile harness]

unfortunate

[autumn plaza]

oh wait yes in a zoo

[agile harness]

fake aussie

[autumn plaza]

wait was it

idk i forgot

[agile harness]

you probably live in a city then

not in the outback

is that what you call it

[autumn plaza]

yea or country

[agile harness]

#serious-discussion message

who wouldn’t want to pet them

they’d probably kill you though

they’re not very friendly i hear

i’ve seen videos of them stealing peoples dogs and shit

they’re huge

and they can fight

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[fresh adder]

Do I use a^x formula or take log on both side? ( I need to find deravative)

[little pine]

is that $7^{x cosec x}$?

[fresh adder]

Yes

[thorny flame]

Sepdron

[little pine]

you can do either

[fresh adder]

Ik but what will be better to do? I have like 500 questions total homework and if I get 5 wrong I will have to do it all again

And if I do the way the teacher doesn't want me to then he will consider it as incorrect

So what's better?

[little pine]

daym, 500? that's a lot
taking ln on both sides is probably easier

[little pine]

have you got teached both methods?

[fresh adder]

Both ways are easy but which would be the safer bet?

[fresh adder]

Yeah

[little pine]

is your teacher currently teaching one of them?

[fresh adder]

Nah currently implicit derivatives just got finished

Logarithmic trigonometric and inverse trigonometric were done before that

[little pine]

actually ig you're doing product rule anyway,
so doing it directly is probably safer because it has less steps

[fresh adder]

First the a^x then uv rule yeah

[little pine]

then except if they have it on the question, it should be ok to do either no?

[fresh adder]

I think so but the teacher is very strict so just to be safe

Thankss

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[fresh adder]

How to do log base 10 sinx

Log base b a is log a/ log b then how to derive 1/log10

It's jus 0?

.reopen

[topaz sinew]

✅

[little pine]

what are you trying to do with it?

[fresh adder]

Y'

Difference wrt x

It's 1/log10

Not logW

[little pine]

ah, just treat the 1/log 10 as a constant then

[fresh adder]

Ohh

Genus

Cause log of constant is co stant

Wow

[fresh adder]

Derivative of( 1+sinx)/cosx taken by quotient rule and taken by written 1/cosx as secx are different which one is correct or are both correct

[little pine]

what're the results?

[fresh adder]

I did with quotient rule and got (1+sinx)/cos²x

This is with taking 1/cosx as secx

[little pine]

how'd you get that?

[little pine]

,rcw

[fresh adder]

?

[little pine]

yea nvm, it seems correct

oh yea both of them is identical

[little pine]

you can transform that answer to yours

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[static juniper]

Let z1 and z2 be two complex numbers, such that 𝑧1 + i𝑧2 = 0 and arg (z1, z2) = 𝜋. Then find arg (z1)

[static juniper]

how would we do this

[fair thorn]

arg(z1, z2)??

how can an argument take in two inputs

[static juniper]

im confused about it too

[fair thorn]

you mean arg(z1) - arg(z2)?

[static juniper]

i dont know man in our assignment the question was given like that

does it mean the angle between z1 and z2?

[buoyant horizon]

do you refer to this maybe

atan2

[static juniper]

explain

[fair thorn]

???

still doesn't take in two inputs

[fair thorn]

just fancy definition of arg

[static juniper]

.

[buoyant horizon]

could be

[static juniper]

oh but they didnt explain why its relevant

[fair thorn]

no it's just the piecewise definition of arg

since normal arctan doesn't quite hold up

you don't need to worry abt that tho

[static juniper]

so what do i do?

leave the question?

[buoyant horizon]

yes

i guess

[static juniper]

@fair thorn

does the two inputs possibly mean the angle between the complex numbers?

like arg(z1) - arg(z2) i suppose

that would give the angle between them no?

i dont know man

[topaz sinew]

@static juniper Has your question been resolved?

[fair thorn]

i assume?

see the problem is

through only a bit of rearranging

you get

[thorny flame]

ren

[fair thorn]

which means that they have an argument of pi/2 between them

the issue is finding out what arg(z1, z2) means

@static juniper lmk if you get any updates

[static juniper]

ill ask my teacher about it

thanks

,close

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[uneven ginkgo]

how do I do b

[opal vault]

law of total events, P(A) = P(B)P(A|B) + P(notB)P(A|notB)...

[uneven ginkgo]

how do u know

[opal vault]

i told you law of total events

[uneven ginkgo]

theres a law?

i have to memorise that?

[opal vault]

P(A) = P(A and B) + P(A and notB)

[opal vault]

do we agree with that?

[uneven ginkgo]

let me try draw

[opal vault]

(A and B), (A and notB) are disjoint events

so sum of probabilities = probability of union

union = A

[uneven ginkgo]

this?

[uneven ginkgo]

i think i got this part

what now?

[opal vault]

ok so

[opal vault]

now we use definition of conditional probability

[opal vault]

does that still make sense?

P(A) = P(B)P(A|B) + P(notB)P(A|notB)

[uneven ginkgo]

how come we added all of this stuff?

[opal vault]

wdym

what's the definition of P(A|B) for example?

[uneven ginkgo]

probability of a given that event b has already occured?

[opal vault]

no, the math definition

[uneven ginkgo]

is it that

[opal vault]

yes

so if P(A|B) = P(A and B)/P(B)

that means P(A and B) = ...

yes even though the question wasn't directed towards you 😅

[uneven ginkgo]

...

[uneven ginkgo]

yea what he said bc u multiply

[opal vault]

yes

P(A and B) = P(B)P(A|B)

[opal vault]

can you also see how the second term was replaced by P(notB)P(A|notB) now?

[uneven ginkgo]

how come the sum of those 2 gives us the p(a)

[opal vault]

I thought we agreed on this

[uneven ginkgo]

oh

thats so much

to think abvout

):

[uneven ginkgo]

ok I have this now what

[opal vault]

almost

but you know what

[uneven ginkgo]

oh..

no lets change it

it doesnt make sense

i change olk

[opal vault]

yeye you can

[uneven ginkgo]

ok:D

[opal vault]

it's still written wrong on the other line wait

it's either P(A and B)

[uneven ginkgo]

oh

i changed it at the bottom

[opal vault]

or P(A|B)P(B)

[uneven ginkgo]

ok now wat

@opal vault \

[opal vault]

you're still writing A and B instead of A|B

be careful

so

now there are a few unknowns

let's name p = P(B)

and let's try to express every unknown probability here in terms of p

[opal vault]

first of all and easiest, how can we write P(notB)?

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[fresh adder]

What is the difference between log base a sin x and the second one

[wooden osprey]

$\log_{\sin x}^a$

this looks incomplete. what is it taking the log of?

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[fresh adder]

That's the whole question

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[gleaming thunder]

int lnx can be computed by IBP

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[zealous fable]

,rotate

[barren marten]

Area for 360deg circle is pir^2
Then area for thetadeg circle is (pir^2)*(theta/360)

Angle = arc length/radius

Use these two formulas to calculate the answers.

[topaz sinew]

@zealous fable Has your question been resolved?

[zealous fable]

@barren marten

how do i illustrate the circle?

[barren marten]

Maybe something like this:

[zealous fable]

@barren marten

on 6 how can i calculate diameter?

2pi r^2

right

should i just change the formula with that

[static juniper]

radius will be equal to 11

diameter = 2 * radius

so you can use the same formula

[barren marten]

Yeah

[static juniper]

theta/360 (pi * r * r)

[barren marten]

Or you could use
(theta/360)*(pi x d/2 x d/2)

[zealous fable]

@barren marten

for e.arch length

what formula do i need to use?

@apex

@static juniper

[static juniper]

theta/360 (2pi r)

@zealous fable

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@zealous fable Has your question been resolved?

[zealous fable]

how about if the central angle is missing

@static juniper

[static juniper]

send the question

[zealous fable]

on the last one

[static juniper]

they have given you the arc length

do you know radian system of angles @zealous fable

[zealous fable]

nope

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[near laurel]

Hello, I was re going over differentiation and was just looking for a little bit of checking

Would this not be Dv/dt?

Oh wait, that’s the volume

[strong viper]

do you mean the quantity 80pi cm3s-1?

if so, yes

[near laurel]

Oh sorry, I edited out the question on the second picture

[strong viper]

yeah, that seems right, option 2

[barren marten]

@zealous fable
Use Angle = arc length/ radius
You'll get angle in radian
After that multiply by 180/pi to convert it into degrees.

[near laurel]

.close

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[torn shoal]

I looked at this and had no idea what to do

[whole geode]

That's an equilateral triangle + 3 circle segments

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

[torn shoal]

what should i do?

[whole geode]

Draw the equilateral triangle. And then you have the three circular segments, you can calculate the area of the triangle and the segment.

each of the three segments are the same area.

[torn shoal]

Im sorry but I dont know what that means

[whole geode]

It would be helpful if you can let me know what you aren't understanding, in that case.

[torn shoal]

The "equilateral triangle" and "circular segments"

[whole geode]

equilateral triangle is a triangle that has all three of its sides the same length.

[torn shoal]

oh ok

[whole geode]

it has 60 degree angles

and you can use trigonometry to find the area based on a side length

"circular segment" I actually linked the wikipedia page for.

so please give it a read.

[torn shoal]

Oh, i get it now, so i basically have draw the triangle in the shaded area

[whole geode]

Basically

[mystic kraken]

.close

[whole geode]

You can calculate it as "area of triangle" + 3 * "circular segment"

or you can calculate it as 3 * "circular sector" - 2 * "area of triangle"

you might find the latter more convenient.

(as 3 * a 60 degree circular sector is half a circle)

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

[torn shoal]

I dont understand how to calculate the segments because like if I would translate to my language it doesn't show and theres like words I never heard before being used in the wikipedia page

[whole geode]

what language?

[torn shoal]

swedish

[whole geode]

https://sv.wikipedia.org/wiki/Segment_(geometri)

https://upload.wikimedia.org/wikipedia/commons/4/4b/Cirkel-11.png

https://sv.wikipedia.org/wiki/Cirkelsektor

hope this helps

[torn shoal]

these equations looks kinda complex for 9th graders like me in my opinion

wait i think i found it

A = L.r /2

I dont know if its the same in the picture after drawing the triangle though

.close

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[torn shoal]

.reopen

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✅

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@torn shoal Has your question been resolved?

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[lilac folio]

i've attempted to prove this inductively, but i dont know enough about the gamma function complete this, or so i think. what am i doing wrong?

im aware of the other path, but id like to try to complete it inductively. if its a lost cause, please tell me so

[sweet shard]

\gamma seems unnecessary

[lilac folio]

it comes from Gamma(z + 1) = zGamma(z). made it easier to reason

[sweet shard]

In "case k=n+1" you should be differentiating the nth derivative result

[lilac folio]

oh okay

[sweet shard]

finding the nth derivative of this shouldn't be too hard

[lilac folio]

i actually had that earlier, but i couldnt figure out how to get the denominator to be right

if i even did it right

[sweet shard]

You need a formula for $\frac{d^n}{dt^n} (1-\beta t)^{-\alpha}$ first

[thorny flame]

riemann

[sweet shard]

E(Y^n) doesn't actually come into the calculations at all until you evaluate at t=0

[lilac folio]

ic, let me fix it

[lilac folio]

it should look like this, correct?

though, i dont exactly get how the denominator contains Gamma(\alpha). still stuck on this step

[sweet shard]

$\Gamma(\alpha + n) = (\alpha + n - 1) \cdot (\alpha + n - 2) \cdot \ldots \cdot 1$

[thorny flame]

riemann

[sweet shard]

your nth derivative stops at $\alpha$, not 1

[thorny flame]

riemann

[sweet shard]

also you should clean up your negative signs before moving on

in this step

[topaz sinew]

@lilac folio Has your question been resolved?

[lilac folio]

am i missing something still?

[topaz sinew]

@lilac folio Has your question been resolved?

[sweet shard]

You're still getting the definition of Gamma wrong

[sweet shard]

.

[lilac folio]

not trying being willfully ignorant, but i dont get it
isnt (a + n - 1)...(a+1)a...(1) the same as (a+n-1)(a+n-2)...?
it generates the same terms at the end of the day no?
in the case of the derivative, we would see a, (a+1) etc?

[sweet shard]

.

[sweet shard]

now it's just wrong

you take n derivatives, not alpha + n - 1 derivatives

do yourself a favor and just set alpha = 7, n = 3

see if you can take 3 derivatives

[lilac folio]

i see what you mean now

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[west scarab]

I am grade 12 doing adv functions, have a test on trig tomorrow and im currently doing a practice test

[west scarab]

Got stuck on this question

[visual monolith]

,rotate left

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[west scarab]

<@&286206848099549185>

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[finite mural]

Hi there How would you proof this ? Im havint troubles with it

[wheat kindle]

Identity
Inverse
Associativity
Commutativity

What have you managed so far?

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[jovial girder]

My instructions are to find the work done by the force field as it moves a particle along the curve C, in the given orientation.

this is 1(h)

<@&286206848099549185> in some dire need of help

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[rigid prawn]

why is the total number of upnos 2^9 and the total number of downnos 2^10?

Can somebody pls explain this to me

thanks

[whole geode]

@rigid prawn consider an upno

This number can only have up to 9 digits right?

Because digits are not allowed to repeat.

123456789 is the largest upno

[fading owl]

i did the exact same question today

omg

i was looking at aops too

[whole geode]

If we consider that each digit can only appear once

[fading owl]

and i got the same qn wrong

[whole geode]

And the order cannot be changed

[fading owl]

note that the digits cant be the same

[whole geode]

Then the only difference between any two given upnos are whether or not any given digit appears

[fading owl]

beacause the next digit is strictly less than the first digit

[fading owl]

yeah

[whole geode]

There are 9 digits that can appear, and each has two possibilities, so that's 2^9

[fading owl]

2023 amc 10b / 23

@rigid prawn u here?

[whole geode]

I guess they dropped the question and ran

It's a shame, interesting question too

[fading owl]

cant believe i just did that paper as practise for amc 10/12

[fading owl]

yes

i got it wrong js now 😭

reviewing answers rn 💀

got 112.5 exactly distinction 💀

Hey @rigid prawn

[topaz sinew]

@rigid prawn Has your question been resolved?

[rigid prawn]

sorry yall i had to do something

this helped a ton

thanks

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[leaden hatch]

where did the y_0 and x_0 go at the bottom?

[fickle mulch]

they equal zero so they just simply didn't write it

they could say y-0 but thats the same as y

[leaden hatch]

but it says x_0 not equal to 0

[fickle mulch]

o

idk 😭

[leaden hatch]

😓

[little pine]

expand out the RHS
they'll cancel

[leaden hatch]

😮

that's cool

thanks

.close

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[toxic hill]

i understand that the second derivative of g(x) is the slope the first derivative of g(x)

[agile harness]

did you get part a

[toxic hill]

which is why i believe that g''(x) is negative between -3 and 0 and positive between 0 and 3 what im confused about is what to say when the slope is 0

when it comes to part a im also a little confused

[agile harness]

ok then let’s start at part a

[toxic hill]

i know g(x) is increasing when its slope is positive and decreasing when its slope is negative

but what if the slope is 0

is it increasing or decreasing?

(-inf,3) its 0 and (3,inf) slope is 0 as well

[agile harness]

neither

[agile harness]

where are you getting this

[toxic hill]

(-inf,-3)

im looking at the graph

[toxic hill]

yeha it begins decreasing at -3 doesn't it?

fuck mb i just realized

[agile harness]

this is the graph of g’, if g is decreasing then g’ is negative, if g is increasing then g’ is positive

[toxic hill]

this is the graph of the derivative wherever it is negative the slope is negative

[agile harness]

from the graph, g’ seems to never be zero

[toxic hill]

my fault

[agile harness]

it asymptotically approaches zero

never crosses

[toxic hill]

it would be decreasing everywhere

[agile harness]

yep

except 0

[toxic hill]

ya

[agile harness]

because it’s not defined at 0 of course

[toxic hill]

yeah

mb for some reason i didnt clock i was looking at the graph of the derivative

[agile harness]

all good

so what is your answer for part a

[toxic hill]

decreasing (-inf,0)U(0,inf)

[agile harness]

yep

now for part b

[toxic hill]

part b the slope of the actual graph matters right

[agile harness]

g’’ is positive means g’ is increasing, g’’ is negative means g’ is decreasing

[toxic hill]

g''(x) = slope of g'(x)

concave down (-inf,0) and concave up (0,inf)

[agile harness]

yep

[toxic hill]

alr cool

thank you

[agile harness]

you’re welcome

[toxic hill]

.close

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[loud oasis]

c is a quantity which depends on the shape and cross-sectional area of the object and the density of the fluid (air)

it can be experimentally determined, yes

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[wispy mango]

can someone help me figure out how to go about solving this analytically? if there is a way?

or would it be more worth while to solve it numerically?

import numpy as np
from scipy.integrate import nquad

# Define the function to integrate
def integrand(z, y, x, b):
    return 1 / ((b - x)**2 + y**2 + z**2)

# Set limits of integration
a = 1  # Replace with your value of a
b = 2  # Replace with your value of b
limits = [[-a, a], [-a, a], [-a, a]]  # limits for x, y, z

# Perform the triple integral
result, error = nquad(integrand, limits, args=(b,))

print("Result of the integral:", result)
print("Estimated error:", error)

I got this from chatgpt (God bless ai)

<@&286206848099549185>

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[mortal heart]

The gradient of the curve y = x3 + ax2 + bx + 7 at the point (1, 5) is −5. Find the value of a and the value of b.

[true coral]

create 2 simul equations and solve

[mortal heart]

guys why am i wrong?

[true coral]

substitute in (x,y) and (x, dy/dx)

[true coral]

you went from -5 to 5 it looks like

on the lines that are highlighted

in your sol

[mortal heart]

ohhhhh ure right

i'll fix it

[mortal heart]

thankyouuu

i got ittt

.close

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[mortal heart]

thanks again

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[glad sierra]

Can you check if this is correct?

[glad sierra]

I’m trying to find the 2nth derivative of (x^2-1)^n

[odd pagoda]

you should write $u^{(2n)}$ instead of $u^{2n}$

[thorny flame]

Denascite

[odd pagoda]

but checks out

[glad sierra]

yes

I really mean, u^(2n)

My reasoning was after the first term

differentiating x^(2n-2) , 2n-2 times will give me (2n-2)!

But I still need to do two more differentiation to do it 2n times

ie the 2nth derivative is 0

Same applies to all other terms

Sounds correct ?

This is what my professor did,

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[wintry urchin]

This is my table. How can I use Gomory's cutting plane algorithm to find an integer solution? I know by inspection that the answer is x = 2, but how do I show that using the method?

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[wild zephyr]

Is it correct??

Idk how to do it

<@&286206848099549185>

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[fair thorn]

please don't ping us immediately

[wild zephyr]

Ohh ok sorry

How to do it?? Factor each polynomial completely 😦