help-43

[fathom basin]

It’s asking for positive solutions, so we can ignore any x<-1

[robust heart]

observe that tan x is i a increasing function on interval (0,pi/2) then tan pi/4 <= pi/4+1 and lim x -> pi/2 tan x>=x+1 so there has to be an intersection om that interval

[digital trench]

yea

[fathom basin]

y = x+1 is increasing over (-1,infinity), so the earliest intersection will be the smallest y

Do you need to formalize the proof, or just explain

[digital trench]

explain

just explai

[fathom basin]

👍

Still need help, or do you have it?

[digital trench]

i need help

i havent done much trig in my 11th so

[fathom basin]

Ok. So there’s an infinite number of solutions for the equation you’re given, because tan (x) is periodic.

[digital trench]

es

[fathom basin]

Oh my bad, I actually misread the problem a little. It’s asking for the smallest x value not smallest y

[digital trench]

oh

ye

[fathom basin]

So it’s just asking you for the first time it intersects when x>0

[digital trench]

yea..

[fathom basin]

Which happens to be in (pi/4,pi/2)

[digital trench]

oh

wait

yea

im dumb bro

[fathom basin]

Nah that’s just what these channels are for

You’re not dumb because you’re asking for help

[compact pewter]

@digital trench Has your question been resolved?

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[bleak venture]

Im confused

[bleak venture]

I did this

And got wrong answer apparently

[arctic portal]

i think you did -4+2=2 by accident c:

[bleak venture]

oh

also i need help checking this question

[chilly basalt]

Where did 8y - 4z = -2 come from?

[bleak venture]

look to the right

ontop

at the right

[chilly basalt]

Small hiccup

@bleak venture have you fixed it yet?

[bleak venture]

not yet

but i will

gimme one seoncd

*second

i fixed the -2 thing

it was supposed to be -12

i can just redo it later

thanks for the help though

.close

[compact pewter]

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[kind wind]

guys

[kind wind]

can anyone explain

circular permutations

[eager thunder]

uhh

[kind crane]

not really enough context

[eager thunder]

so essentially, we can deconstruct any permutation into a set of circles

[kind crane]

are you working on a specific problem

[kind wind]

no

im struggling with the theory

[eager thunder]

okay here lets think of a permutation this way

you have children 1 through n

[kind wind]

wat'

[eager thunder]

they then toss a ball to a person per a given permutation

so essentially, the circular permutation represents the sequence through which any ball goes through

[kind wind]

wai wait

ahmm

say with

5 elements

ABCD and e

E

[upper bane]

i recall you asking the same question earlier. what did you understand from that session?

[kind wind]

yea

i mean

[eager thunder]

ok sorry i gtg

[kind wind]

no one replied

[upper bane]

i am pretty sure a lot of people replied to your previous session. i just wanted to know how much you understood what they said

[compact panther]

I think about circular permutations this way, circular permutations are not very much different from linear permutations, circular permutation just doesn't have a reference point. So, generally in circular permutations u first select a object as reference then the other objects can be arranged the same way you do in linear permutations. Though i am not sure if u can think this way

[kind wind]

I think about circular permutations this way, circular permutations are not very much different from linear permutations, circular permutation just doesn't have a reference point. So, generally in circular permutations u first select a object as reference then the other objects can be arranged the same way you do in linear permutations. Though i am not sure if u can think this way

this

and the fact that many arrangements of abcde are taken as single just one arrangement

[upper bane]

did you just copy the explanation of the person above you lol

[kind wind]

because it was literally it

lmfao

[bleak flower]

[compact pewter]

@kind wind Has your question been resolved?

[compact pewter]

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[grim nimbus]

i feel like this is useless

[grim nimbus]

because i always end up solving the issue

or getting stuck so badly and overcomplicating it

but im gonna try this time

here we go

is this correct?

because 5^6th is 15,625

minus the 1 in a geometric series formula

which is 15,624

then the bottom is 5-1 or 1 - (positive)5 its the same thing

so that's 4

15,624/4 is 3,906

then i took that number, brought the 15,624 from the "sum of the first 6 terms" in the geometric series formula down

and divided it by the NEW NUMBER

3,906

to which that told me it was... 4.

so the first term is. 4?

15,624 before the = was S sub nth term

the max terms here is 6 and it already gave me the answer to that

all i did was plug in

[arctic portal]

yes, this seems right :)

[grim nimbus]

ok thank you

but then why is khan academy telling me i could've also done it this way?

is this problem commutative?

i meant it is

had to fix that 3 times sorry 😭

[crystal lintel]

It's just using the formula for the sum in a geometric sequence and plugging the values

but yours is as correct

[grim nimbus]

oh okay

[grim nimbus]

thanks

should i keep this open

[crystal lintel]

of course

[grim nimbus]

in case i have another issue

or

i'll close for now

[crystal lintel]

idk i just joined the surver lol

[grim nimbus]

oh

well thanks for the insight

.close

[compact pewter]

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[crystal lintel]

that was my first message

[grim nimbus]

😭

.reopen

[compact pewter]

✅

[grim nimbus]

im in need of another helper

alrighty

just double checking my answers

i think thats how it goes

cause they told me to add all 9 terms

but that's a big ass number

idk

i think geometric series are supposed to have big numbers though?

[crystal lintel]

you can also use the formula

[grim nimbus]

i know but its like

aghhhhhhh

😭

[crystal lintel]

i know, i know

that's math

[grim nimbus]

its like the quadratic formula it took me a while to get down

-b plusminus square root b squared - 4 * a * c

divided by 2 * a

i think thats it?

[crystal lintel]

yep

[grim nimbus]

and if its a negative square root number its square root that number * square root -1?

[crystal lintel]

see, you got it

[grim nimbus]

which is really just i?

so imaginary

yeah

its cause i was doing it over the span of like a few days tho

its fun

[crystal lintel]

correct

[grim nimbus]

alright ill try to get the formula down

S sub n = a * the r^n - 1 / r - 1?

[crystal lintel]

yes

[grim nimbus]

and i think the r's and 1's are commutative

could be wrong

[crystal lintel]

if you swap them both in the numerator and the denominator then yes

but i don't think you'll need to swap them

(im sorry in advance if my terminology is incorrect, i am still thinking in my native language)

although they shouldn't be that different

[grim nimbus]

oh

so when i do the sum of the first 9 terms

can i just use the common ratio and go up to the 9th exponent of it?

like 3^2

[crystal lintel]

yes that's it

[grim nimbus]

3^3

3^4

[crystal lintel]

yeah

exactly

[grim nimbus]

oh my god

that's easier but it seems harder

does that make sense

it LOOKS harder

but its not

[crystal lintel]

yeah yeah

[grim nimbus]

tysm bro

appreciate you so much

[crystal lintel]

of course

i am always happy to help

but if you see

you got it yourself

i just affirmed it

so congrats to you

[grim nimbus]

oh this is so niceee

alright ima close this ticket now

im actually happy

tysm

.close

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[quartz yoke]

tried assuming x^x = t and solving

but it goes too long + sm stuff isn't substitutable

[keen granite]

||NB: ln(ex)=ln(e)+ln(x)=1+ln(x)||

[quartz yoke]

did that

[keen granite]

then I think u=x^x is a very good substitution

[proper igloo]

Recall that $x^x=e^{x\log x}$.

[boreal girder]

EightyEightEpsilons

[quartz yoke]

hol up lemme comprehend

yeah

this is true

[proper igloo]

The derivative of $x^x$ is $x^x(1+ \log x)$

[quartz yoke]

O

H

[boreal girder]

EightyEightEpsilons

[quartz yoke]

WAI

IG OT IT

I GOT IT THX

[proper igloo]

Yay

[keen granite]

[proper igloo]

👏

[quartz yoke]

nah this lil dumb 😭 i was doing the differenciation wrong

.close

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[slender ferry]

the fractional part of x when k = 2,4,6,8

[slender ferry]

why is it 1/2,1/4,2/4,3/4...

[eager thunder]

uhh

i think we need context about the whole problem

[eager thunder]

oh uh sorry i got distracted

[eager thunder]

okay so they are simply calculating all points at which the value of the expression in terms of x changes

so if we keep on increasing x

given the nature of the floor function

the value of the function will remain the same, and then jump by an integer at certain values

or wait nvm

uhh

or yes actually

essentially the function can "jump" by an integer whenever one of the values being floored reaches an integer

thus the floor of 8x will increase by 1 when x is a multiple of 1/8

multiple of 1/6 for 6x

multiple of 1/4 for 4x

and multiple of 1/2 for 2x

[slender ferry]

so we are calculating all possible values of k{x}?

[eager thunder]

no we are looking at all possible values of {x} where the function goes up by at least 1

[slender ferry]

what does the first line of the second page mean by "how the expression cahnges"

[eager thunder]

the function is mainly constant, with the exception of jumping at certain values by an integer

the expression "changes" at these points

the values where it jumps, are when the functions being floored reach the next integer

for example, as x goes from 2.9999 to 3, the floor of x/3 goes from 0 to 1

[slender ferry]

wdym by jump

[eager thunder]

as in this

[slender ferry]

by "the integer part of x changes by n" do they mean that the new integer part is now floor(x+n)?

[eager thunder]

in they mean that floor(new x)=floor(old x)+n

integer part is floor

[slender ferry]

so 20xfloor(old) + 20n

ah

[slender ferry]

i still dont get it

floor(k)x{x}=1,2,3,4..... yes

and then we divide the whole expression by floor(k) to get 1/(floor(k),2/(floor(k)),3/(floor(k)),.....

but then when k=2,4,6,8

shouldnt we deal with the values seperately?

[slender ferry]

so if k=2 then the expression will be 1/2,2/2,3/2... and when its 4 the expression will be 1/4,2/4,3/4....

[compact pewter]

@slender ferry Has your question been resolved?

[slender ferry]

.close

[compact pewter]

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[slender ferry]

.reopen

[compact pewter]

✅

[slender ferry]

this is my question

[slender ferry]

{x} should be like this

[vivid breach]

Im thinking about trying something else for this

[slender ferry]

this solution uses the euler toitient function

[vivid breach]

for refernece the full list is
{x} = 1/2, 1/4, 2/4, 3/4, 1/6, 2/6, 3/6, 4/6, 5/6, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8

2{x} changes when x = 1/2
4{x} changes when x = 1/4, 2/4, 3/4
6{x} changes when x = 1/6, 2/6, 3/6, 4/6, 5/6
8{x} changes when x = 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8

[vivid breach]

this is not how you read the floor function

we're going to skip that you mentioned this and move on

[slender ferry]

its kxfloor({x})?

[slender ferry]

no sorry floor(k*{x})

[vivid breach]

kx?

whats the x doing in there

[slender ferry]

multiplication

[vivid breach]

does it occur to you that maybe having x represent multiplication and the letter x is a bad idea?

go use *

[slender ferry]

sorry

[vivid breach]

and also, you dont need to type the * in there since k {x} already means k times {x}

you can just type it like the problem, which is floor(k{x})

[slender ferry]

oh

[vivid breach]

so theyre listing all the numbers where this would change values

[vivid breach]

yea the floor() is missing from the explanation, I need to add that in
floor(2{x}) changes when x = 1/2
floor(4{x}) changes when x = 1/4, 2/4, 3/4
floor(6{x}) changes when x = 1/6, 2/6, 3/6, 4/6, 5/6
floor(8{x}) changes when x = 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8

[slender ferry]

cant x be any nummber?

except 1

[vivid breach]

lets say x is 0.00001

what would this evaluate to?

[slender ferry]

0

no

[vivid breach]

what about 0.00002?

[slender ferry]

wait

0

[vivid breach]

yea theyre both 0

you dont sound like you believe this for whatever reason

[slender ferry]

i initially didnt believe it

now i do

[vivid breach]

alr

so at some value of x, this will then change value

[vivid breach]

youll notice all thats been added is floors

so most of the time, a small change would just be rounded off

changes in value only happen at specific places

so what we can do is look at what we have then see when itd happen

for example 20 floor(x) would change value when x is an integer

(so {x} = 0)

we can ignore this for now

as before,
floor(2{x}) changes when x = 0, 1/2
floor(4{x}) changes when x = 0, 1/4, 2/4, 3/4
floor(6{x}) changes when x = 0, 1/6, 2/6, 3/6, 4/6, 5/6
floor(8{x}) changes when x = 0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8

so the expression changes value when
{x} = 0, 1/2, 1/4, 2/4, 3/4, 1/6, 2/6, 3/6, 4/6, 5/6, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8

again we ignore 0

what youre left with is that list of numbers that theyre mentioning, likely not intended to have the order matter

[compact pewter]

@slender ferry Has your question been resolved?

[vivid breach]

@slender ferry are you still stuck on something

[slender ferry]

i was afk for a while

[slender ferry]

dont they change when x is any value except for 1?

[vivid breach]

how so?

we didnt see a change when x moved from 0.00001 to 0.00002

[slender ferry]

floor(2{x}) changes when x is 1/8 aswell

floor 1/4 is 0 so 0

[vivid breach]

first of all, that was evaluated incorrectly

second of all, at x=0 is also 0

so thats literally not a change

[slender ferry]

so the only possible values are 2 and 0?

[vivid breach]

no

[slender ferry]

for floor(2{x})

[vivid breach]

human lets do this one step at a time

[slender ferry]

okay

[vivid breach]

youre going too far

lets first try to check something simpler

first, whats the range of {x}

[slender ferry]

0 to 1

not including 1

[vivid breach]

be more specific

[slender ferry]

inclusive of 0

[vivid breach]

so [0, 1)

[slender ferry]

yea

[vivid breach]

now what about the range of 2{x}?

[slender ferry]

if x < 0.5 then its 0 to 1 but not 1

if x = 0.5 then its 0

if x > 0.5 then its >1

[vivid breach]

whats 2{1.3}

[slender ferry]

2/3

no

3/5

[vivid breach]

yes

also, the range is presumably just the possible numbers that 2{x} can be

what youve given me is

not that

I see on the right its 0, or 0 to 1 but not 1, or > 1

[slender ferry]

0 to 2 but not 2?

[vivid breach]

[0, 2), yes

the range of {x} is [0, 1)

the range of 2{x} is [0, 2)

because its just 2 times [0, 1)

[slender ferry]

yea

[vivid breach]

now whats the range of floor(2{x})

[slender ferry]

0,1,

[vivid breach]

{0, 1}

yep

so floor(2{x}) can only ever be 0 or 1

now do you think, for most values of x, floor(2{x}) would change?

or does floor(2{x}) only change at specific values?

[slender ferry]

wdym by change

[vivid breach]

as in "not stay the same if x varies by a small bit"

@slender ferry alr lets try this

can you name the values of x where floor(2{x}) is 0?

[slender ferry]

specific?

[vivid breach]

that question was poorly worded, lets try something easier to say

[slender ferry]

oh

[vivid breach]

such as: can you name the values of x where floor(2{x}) is 0?

[slender ferry]

0 to 1 but not 1

[vivid breach]

are you sure?

[slender ferry]

0 to 0.5 but not 0.5

[vivid breach]

theres more values out there than just the numbers from 0 to 1

[slender ferry]

0 to 0.5 but not 0.5 right?

[vivid breach]

those numbers are correct at least

you need to name more numbers than that though

[0, 0.5) and what other numbers out there?

would x=1 work?

[slender ferry]

yes

any integer will work

[vivid breach]

so for what values of x does floor(2{x}) equal 0?

[slender ferry]

[0,0.5) and all positive integers including zero

[vivid breach]

...what about x = 1.1?

@slender ferry hello?

[slender ferry]

2

[vivid breach]

whats {1.1}?

[slender ferry]

1/10

[vivid breach]

2 {1.1} is?

[slender ferry]

1/5

[vivid breach]

floor(2 {1.1}) is?

[slender ferry]

0

[vivid breach]

so looking at the numbers between 1 and 2,

[slender ferry]

ohh

[vivid breach]

which of these numbers also make floor(2{x}) equal 0?

[slender ferry]

1 to 1.5 without 1.5

am i wrong

[vivid breach]

thats correct, [1, 1.5)

a thing you can think of is that {x} only knows the decimal part of x

so since floor(2{x}) has its first step be doing {x},

floor(2{x}) cant tell apart 1.1 from 0.1

or 2.1 from 0.1

or -0.9 from 0.1 (since -0.9 + 1 = 0.1)

[slender ferry]

so there are infinite possibilities

[slender ferry]

then?

[vivid breach]

using this, try coming up with all the numbers for which floor(2{x}) is 0

so far we have [0, 0.5) U [1, 1.5)

{x} is a function that repeats for +1 to x and for -1 to x

[slender ferry]

[a,a.5) where x belongs to whole numbers

[vivid breach]

you couldve just said [n, n + 0.5)

but yes thats the idea

so for $[n,n+0.5)$ for some integer $n$, the value is 0

[boreal girder]

mtt

[vivid breach]

now since the value can only ever be 0 or 1,

for which values is floor(2{x}) equal to 1?

[slender ferry]

(n+0.5, n+1) for some integer n

[vivid breach]

thats not correct, try saying that again

think about what kind of interval youre describing before you say this

[slender ferry]

is this correct?

[vivid breach]

[n + 0.5, n + 1)

[slender ferry]

but then we'll get one

[vivid breach]

???

isnt 1 what we want?

[slender ferry]

oh nvm

continue

[vivid breach]

[n + 0.5, n + 1) is correct

[slender ferry]

yes

[vivid breach]

alr, now lets notice what happens with these intervals

any time {x} is 0, thats the integers, right

[slender ferry]

yea

[vivid breach]

now youll notice that when {x} = 0, floor(2{x}) changes value, right

before, the value was 1
after the value was 0

[slender ferry]

yes

[vivid breach]

now when {x} = 0.5, floor(2{x}) also changes value
from 0 to 1
right

[slender ferry]

yea

[vivid breach]

so floor(2{x}) only changes at when {x} = 0 or when {x} = 0.5

it doesnt change at any other value of x

do you see how that happens?

[slender ferry]

yes

ohh

OHHH

I GOT IT

TY

.close

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[hazy obsidian]

Let $$B = { x\in \mathbb{R} \mid \log_2(-x^2+7x-10) + 3 \sqrt{\cos(\pi \sqrt{x^2+7}) -1} = 1 }$$
Find the sum of all of the members in $B$

[boreal girder]

Copter

[hazy obsidian]

well we get $$\pi\sqrt{x^2+7} -1 = 2\pi n$$

for integer n, because cos(...) <= 1, but the square root has to be >=0 which implies cos(…) >=1

is this reasoning right? thats all im asking

[boreal girder]

Copter

[hazy obsidian]

and so the big sqrt(cos) term is 0

[boreal nimbus]

you are not supposed to subtract 1 there

but, sure, yes, the idea is that the 3sqrt() term vanishes everywhere when defined

[hazy obsidian]

oops

i see, thanks!

this showed up in exam

.close

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[polar peak]

Hello. Is my Maths answer for this Past Paper correct? Do i get 5/5 marks?

[polar peak]

It is a Past paper ( Exam past paper ) for the Mathematics course i am currently undergoing.

Is my answer correct? 5/5 mark

Minus the estimation.

I didnt do that yet.

[sullen gorge]

First we deposited 36 euros

Then we had to pay 15 euros every month, for 12 months

That is 180 euros

So 216 euros is the total cost of the payment plan

Now how much is an 8% less price of 240 euros because we have to compare it with the payment plan to see whether the payment plan saves the @polar peak more or less

Oh yes pounds, mb

,w 0.08 × 240

[boreal girder]

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[sullen gorge]

,w 240 - 19.2

[boreal girder]

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[sullen gorge]

220.8 > 216
So we end up saving more using the payment plan

Yes

But in the conclusion

You haven't stated whether you are making more profit or loss

And that is the question

So no full marks

You said "no"

But it does save 8 percent, and even more

The answer is "yes"

Idk, a wrong answer would get you maybe 2 marks for step marking or 1.5

The reasoning you made is flawed, better luck next time

You said 5 marks*

Not 6

Depends on how you were taught

If the amount has to be exact then you are correct

You get all 5 marks

It's actually more better that way, we need not to talk more than needed or it causes problems

Imagine someone asking you whether I will be able to save 8% on this 20 pounds on this sweatshirt, and you go like YES, you even save 6 dollars more

And the other person is like: But who asked?

Please be polite

!done

[compact pewter]

If you are done with this channel, please mark your problem as solved by typing .close

[sullen gorge]

Question: Giving 5 marks on such a basic question make me wonder which class are you in? We learn this type of percentages in class 5th. So are you a 5th grader by chance?

[foggy jungle]

!nogpt

[compact pewter]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[covert crater]

this looks like gcse

are you doing gcse?

[foggy jungle]

you DID receive an answer, @polar peak: your answer will NOT receive full points, because you did NOT answer the question

[covert crater]

you'd get like 3 ish marks for that

[wanton ferry]

why are you so obsessed with this? it doesnt matter.

[covert crater]

maybe 2 if the examiner is strict

but there's usually a mark scheme for it

[wanton ferry]

you keep asking "how many points do i get" correcting your answer is more important than 2 points.

[foggy jungle]

(box the "no" please)

[covert crater]

okay now you're just spamming

[wanton ferry]

lets invite everybody who asked to a party! i can afford that.

[foggy jungle]

...

[covert crater]

don't use gpt, it's unreliable

[foggy jungle]

!nogpt

[compact pewter]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[sullen gorge]

My argument is that you still saved 8 pounds even if it's more than 8 pounds

[wanton ferry]

gpt is flattering you, its in its nature.

[foggy jungle]

...for what it's worth, saying "no, it saves 10%" and "yes, you pay no more than 92%" are both correct answers

unfortunately, I see calculations but not justifications 😭

[compact pewter]

@polar peak Has your question been resolved?

[upper bane]

is GPT marking your actual work?

because if it's not and you're inclined to believe it more than humans checking your work, then are you asking us to validate GPT or are you actually asking for your work to be checked?

[snow finch]

.close

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[languid gate]

hi guys is there any way to simplify this further? it looks rlly weird. the answer is supposed to simplify to a=4bcos(2x)cos(x)

[covert crater]

I'm guessing it involves using the trig identity for sin(3x) and cos(3x)?

[languid gate]

welll we haven’t covered triple angle identities (yet?) and I feel like if I go down that road it’s gonna turn out more complicated 😭

[worn loom]

sin(2x + x)

[languid gate]

yea I tried that it didn’t work out D:

[covert crater]

you can derive them from using the trig addition formula from sin(2x+x) and cos(2x+x)

[languid gate]

oh in that case I did that

[covert crater]

could you show it?

the working out

[languid gate]

ok mb it’s like really cramped but here’s what I did im not sure if it’s right 🧍‍♀️

[covert crater]

looks right but I can't really see a way to simplify it further

[languid gate]

yea same here

i think the approach i took with this question might be wrong I’ll js try again

ty

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[ebon wharf]

how ts work?

[ebon wharf]

how does rate of change stuffs work

[strange ermine]

First, start by naming the variables

For instance, let's call the width with x

[ebon wharf]

ye

[strange ermine]

Now, what's the length?

[ebon wharf]

3x

A = 3x*x

A = 3x^2

[strange ermine]

Awesome

So now we need to find dA/dx

[ebon wharf]

yeah what does that mean

sorry

i havent done rate of change yet

[strange ermine]

Or A'(x) in other words

[ebon wharf]

ohh

i understand it more like that

[strange ermine]

It's just the derivative

[ebon wharf]

its area with respect to width?

same thing?

[strange ermine]

Huh?

[ebon wharf]

oh

no

i know what derivative is

ok just lets continue

🥀

[strange ermine]

A'(x) will be 6x, right?

[ebon wharf]

ye

[strange ermine]

Wait sorry

[ebon wharf]

o

alg

idk what went wrong but alg

😭

[strange ermine]

No no my bad

Forget what I said

[ebon wharf]

ok

[strange ermine]

Time is not involved here

[ebon wharf]

ok

we know A = 3x^2

[strange ermine]

Because it's rate of change with respect to width

[strange ermine]

.

[ebon wharf]

yeee

was this right then?

[ebon wharf]

ye making sense

nvm it was wrong nvm what i said

but yeah A'(x) = 6x

[strange ermine]

Yeah I didn't understand what you were referring to sorry

[ebon wharf]

o ok

[strange ermine]

So we need the value of x

[ebon wharf]

we put A'(x) as 75 or 0

or something else

[strange ermine]

Nope, 75 is the area, and not the derivative of the area

[ebon wharf]

oh true

[strange ermine]

So 3x² = 75 and we can solve for x

[ebon wharf]

x=5

do we do A'(x) when x=5 then?

A'(5) = 30

[covert crater]

yeah that's correct

[ebon wharf]

so its like

rate of change is same as like

slope at a point x?

but in this its rate of change at specific width?

[ebon wharf]

or gradient

[covert crater]

yeah you're looking at how the area changes with respect to the change of the width

[ebon wharf]

and then if im finding the gradient at a point its how y changes wrt x?

[covert crater]

it's the same principle basically

[ebon wharf]

ic

man theres so many different ways to describe this derivative stuff

atleast i get how to do the questions

🥀

[strange ermine]

The gradient at some point is exactly the derivative of your function

[ebon wharf]

ye i get that

[strange ermine]

Actually, that's why derivates are used a lot 💪

[ebon wharf]

icic

well ill wrap my head around it slowly

thanks a bunch guys

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[floral wadi]

Can someone double check please

[compact pewter]

@floral wadi Has your question been resolved?

[magic wren]

I think your domain for number 4 should be the union of two open intervals

same for the range

but other than that everything looks good

You can also use Desmos to check your answers yourself

[compact pewter]

@floral wadi Has your question been resolved?

[upper bane]

!occupied

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[upper bane]

oh

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[plain marlin]

if the equation of line are written with constant terms c1 and c2 positive then the equation of the bisectors of the angle containing the origin is obtained by using positive sign in equation of bisectors

[plain marlin]

can someone help me understand why this is the case

this thing bassically

like if upon putting a coordinate into equations of 2 lines it yields the same sign at the end what does that mean ?

[compact pewter]

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[mental jacinth]

I tried to use formula A^-1 = 1/detA * objA and tried to obtain the degree of the determinant, can someone please help ?

[mental jacinth]

<@&286206848099549185>

[shell tendon]

I'm guessing we're assuming A is invertible?
If so I think the best way might be to multiply A with the inverse and look what happens in the position (i,i) for an arbitrary i and equalise it with 1
Also what happens on the position (k,j) and equalise it with 0
Since we know q_{ij} is a polynomial of degree two write it as ax²+bx+c

[old mica]

A is an nxn matrix, and each entry is of degree 2, so det(A) can't have degree greater than 2n

[mental jacinth]

yeah it is sure it is but what obout objA ?

[old mica]

furthermore, each entry of the adjugate matrix to A is the determinant of an (n - 1)x(n - 1) submatrix of A

which can't be of degree greater than 2(n - 1)

so each entry of A^-1 is looking like degree 2(n - 1)/2n

[mental jacinth]

Could you explain it bit more 😢
you mean while calculating coefficients we will endup with (n-1) x (n-1) matrices and multiplyed by aij right ?

but since aij is also already 2. degree polinomial than how it could be 2(n-1) ?

thanks for the help btw < 3

[old mica]

each entry of the adjugate matrix is the determinant of an (n - 1) by (n - 1) submatrix of A by definition

[mental jacinth]

oww my bad I was also multiplying it with aij (like we did while calculating determinants)

well this proofs I need coffe

[old mica]

the determinant of an (n - 1) by (n - 1) matrix whose entries are degree 2 polynomials is a massive sum and product of quadratics with (n - 1) terms and (n - 1) quadratic factors in each term

[mental jacinth]

so our A^-1 will be rational ?

[old mica]

so the degree of the final polynomial of the submatrices is capped at 2(n - 1)

so each entry of the adjugate is a polynomial with degree max 2(n - 1)

divide that by the 1/det A factor, and each entry of A^-1 is of degree 2(n - 1)/2n

so yes, rational

[mental jacinth]

< 3

[old mica]

this is really just an exercise in book-keeping degrees of polynomials under products and sums

[mental jacinth]

🙁

[old mica]

why sad

[mental jacinth]

because I couldn't solved it

mehhh

[old mica]

it happens

maybe I should've let you do more of the problem

you had the right idea though

it all just came down to counting degrees

[mental jacinth]

anyway

[old mica]

and using definitions

[mental jacinth]

huge thanks for your help better than my friends

have a great day !

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[old mica]

have a great day too!!

[compact pewter]

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[young raft]

i need help understanding this part

[young raft]

turing machines can be simulated if

• sigma eventually becomes constant
• twice differentiable
• nonzero derivative on open interval

so why exactly does sigma=arctan not work like they say there?

or by "eventually constant" they mean derivative goes to zero on both sides AND $\lim_{x\to -\infty} \sigma(x) = \lim_{x\to +\infty} \sigma(x)$?

[boreal girder]

artemetra

[slate wyvern]

I feel like this is probably just "eventually constant, twice differentiable, non-zero derivative on open interval" ⇒ simulate-able, but there are others that can be as well? I'm very much not well versed in this area though, so I wouldn't rely on this answer

[young raft]

no i am not well versed as well xd

[slate wyvern]

What's this from?

[young raft]

but it just says

arctan and other [...] functions do not satify these hypothesis

[young raft]

https://www.sciencedirect.com/science/article/pii/S0005109800000509
this is the paper but i doubt context is of relevance here

[gusty elm]

sup

OH sorry wrong chat

[young raft]

it's alright haha

[young raft]

i really do think it's just that but it's weird wording nonetheless

[young raft]

page 15 btw

[slate wyvern]

Doesn't look like I can get access since I'm not currently in any academic area

[young raft]

i can dm you the pdf

[slate wyvern]

Is that pirating? 🙃

[young raft]

no i do have academic access

ah well who cares

[slate wyvern]

No, I mean, is sharing it considered pirating it to me

[rigid perch]

koiran 1996 for reference
https://www.sciencedirect.com/science/article/pii/S0304397596000886

[slate wyvern]

I would just wait for someone else, I really don't think I can help lol

[young raft]

thank you, was gonna check that out later

[rigid perch]

their example function is
0 if x <= 0, x if 0 <= x <= 1, and 1 if x >= 1. so it is a constant function for x >= some constant

[lost jackal]

Is reading the previous pages necessary for the part you are asking, cause I think that's a lot

[rigid perch]

which is presumably what they mean by eventually constant

[young raft]

aha

alright

so it's equal to a constant past some point

makes sense

[young raft]

thank you!

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[trail arch]

the time to failure of a unit is modeled by an exponential distribution with a variance of 40,000^-1 hours, the probability that a product requiring 3 identical and independent units connected in a standby configuration will survive 100 hour is?

[trail arch]

do i use gamma distribution for this one?

[shy current]

yep

[trail arch]

okay in the solution it said that the variance=1/limda^2
but in the book it says that the variance=r/limda^2

is there any reason why he used 1 instead of 3?

[shy current]

hmm?

the variance should be just 1/lambda^2

sum of n exponentially distributed random variables (that are independent) with parameter lambda will yield a Gamma distribution with params shape=n and rate=lambda

[trail arch]

why so

[shy current]

it's a well known property

[trail arch]

okay youre saying that the variance is 1/lambda^2 right?

but in the book it says r/lambda^2

[shy current]

I don't know what book you have nor the context of what they are claiming

[trail arch]

statistical quality control

[shy current]

they might be talking about the mean and not the sum of the variables

[trail arch]

okay appreciate it

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[compact pewter]

Show your work, and if possible, explain where you are stuck.

[compact cairn]

for the first one it seems g(x) is a jump meaning it is discontinuous and therefore DNE so the addition of both of them = DNE right

for the second one I see f(x) does converge on a point at 2 but there is no dot I think that means the limit still exists not sure but that would mean it is equal to 0 right then also g(x) does not get approached from both sides at that point so it is DNE making two also DNE right?

then for three again same situation there’s no dot but it is being approached from both sides at that point right but g(x) is only being approached from one side so it also DNE idk

for four same situation right idk

for five it seems from the left side f(x) is equal to -1 right?
so 3+f(x) would be like 2 idk

[rotund sphinx]

1. Fine
2. f(2) = 0. I think the graph of g just got cut off at x=2 tho ngl.
3. What do you mean by "g is being approached from one side"? g is defined both to the left and to the right of x=0.
4. See above
5. You want the limit of sqrt(3+f(x)), not 3+f(x).

[compact cairn]

maybe I dont fully understand how approaching works because idk what g(x) for 3 and 4 is

if you could explain that

[rotund sphinx]

This is what it looks like as x approaches 0 for g

from the left and the right, you can see that you approach what looks to be around 2.7 from both directions

same logic for 4

[compact cairn]

oh ok yea no I did the earlier but for some reason decided it was wrong

[rotund sphinx]

rip

[compact cairn]

so they are both zero in that case

because f(x) is 0

or what

[rotund sphinx]

Yes both question 3 and question 4 are zero

as $\lim_{x \to 0} f(x)=0$ and $\lim_{x \to 0} g(x) \neq 0$

[boreal girder]

Civil Service Pigeon

[compact cairn]

ok thank you bro got it

[rotund sphinx]

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[inland nacelle]

as a geo main this trivial geom is slightly annyoing

[inland nacelle]

diagram with lines added in

It is natural to construct right angles after given many equal lengths

But i find it hard to continue

Help is greatly appreciated, if you can please break it down and give me some hints thanks

[compact pewter]

@inland nacelle Has your question been resolved?

[inland nacelle]

<@&286206848099549185>

[compact pewter]

@inland nacelle Has your question been resolved?

[molten badger]

BAC=3x and ABN=2x? So BAC:NBA=3:2?

Ngl this problem looks more complicated than it actually is

[inland nacelle]

oops

stupidity

i change later

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[inland nacelle]

mb i gtg

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[molten badger]

Find x to maximize alpha

I used tan sum identity and differential to find max here

But that's awful and look like a mess

So I want to know if there is any other way to do it

[heavy basalt]

X=540/11?

[molten badger]

No

It's almost 71

Idk the exact here since I used calculator to solve for x

It only shows decimal

[heavy basalt]

Is those 2 triangles similar?

[molten badger]

No

That's all we have, AM and BN are perpendicular to MN

[heavy basalt]

Do u have the key?

The bot is not working

[grand lark]

Hmm, ya

[molten badger]

I'm not gonna read all this...

So coord bash

[grand lark]

Easy

[molten badger]

Where P and Q come from

You didn't mention it here

[kind viper]

!nosols

[compact pewter]

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.

[kind viper]

and probably:

!nogpt

[compact pewter]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[molten badger]

Ngl reading this kinda hurt my eye

[grand lark]

I'll try to explain it another way

[molten badger]

I hope this's faster than the tan way i used

[grand lark]

We have two poles: the one on the left is 60 m high (from to ) and the one on the right is 105 m high (from to ).

Between them is a "V" cable made up of two wires: one goes from to a point in the middle (), the other goes from to one point in the middle (), the other goes from to another point in the middle ().

Between them, there is a 15-meter horizontal stretch.

These two wires intersect at a point down there, forming an angle.

For each value of , you can calculate where and connect to and , and find the point where the wires cross.

With the coordinates of , we can calculate the angle using trigonometry (dot product of the vectors and ).

Doing this for various values of , we see where the angle reaches its maximum.

After doing this and calculating we arrive at

$x \approx 71 \quad\text{e}\quad \alpha_{\max} \approx 81{,}16^\cir$

[boreal girder]

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

[grand lark]

Ok?

[molten badger]

Not yet,hold on

[molten badger]

Forr each value of...?

[grand lark]

For each value of x

Sorry I'm having trouble with translation

Wait for someone to come help

[molten badger]

Yeah I think you missing some variable and point while translate

[compact pewter]

@molten badger Has your question been resolved?

[molten badger]

Your approach is to find vector AC and BC and use dot product?

Holy, that gives me cos(alpha)^2= f(x)/g(x) with f(x) and g(x) are both 4 degree polynomials

Yeah I'm not gonna do that

[compact pewter]

@molten badger Has your question been resolved?

[molten badger]

I will close this channel so we can have no occupied channel

.close

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• Show your work, and if possible, explain where you are stuck.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[compact storm]

I disagree

[subtle helm]

https://tenor.com/view/ace-attorney-phoenix-wright-objection-capcom-gif-20965907

[kind viper]

well then you can take it up with the <@&268886789983436800>

[compact pewter]

@compact storm Has your question been resolved?

Channel closed

Closed by @compact storm

Use .reopen if this was a mistake.

[compact storm]

Yep

[compact pewter]

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.

[sullen gorge]

1 + 1 - 2sin²x = 1. Solve for x.

[sullen gorge]

Use calculator if needed

1 - 2sin²x = 0

Sin²x = 1/2

[subtle helm]

ye

so what is the answer

[sullen gorge]

Let's see

Sinx = ± sqroot (1/2)

x = arcsin sqroot (1/2) or -arcsin sqroot (1/2)

[subtle helm]

[ \sin(\pi - x) = \sin(x)]

[sullen gorge]

How does this help then?

[subtle helm]

find this first

[sullen gorge]

Solving straight ahead with arcsin we get -pi/4 and pi/4

[subtle helm]

thats not all the answers

^

[sullen gorge]

OK so we can say

Pi - (-pi/4) for 1st quadrant

[subtle helm]

pi/2?

[sullen gorge]

180 - (-pi/4)

[subtle helm]

pi + pi/4, ye

[sullen gorge]

True

Then we group them together with a bracket and add +2kpi?

Or is there something to the periodicity that I haven't known of?

Like, (pi + pi/4, ..., ..., ....) + 2kpi?

[subtle helm]

think of the unit circle

[sullen gorge]

Tangent = 1/hyp?

[subtle helm]

the answer we've discovered so far are pi/4, 3pi/4, 5pi/4, 7pi/4, ...

[sullen gorge]

Yes

[subtle helm]

what do u notice

[sullen gorge]

We have both negative and positive values so the quadrants don't matter

[subtle helm]

yup

[sullen gorge]

So should I just +2kpi?

[subtle helm]

nah

[sullen gorge]

So I leave the answer at this?

[subtle helm]

nope

[subtle helm]

whats the difference btw each of the valueshere

[sullen gorge]

Pi/2

Oh

So (pi/2)k

[dire fiber]

-sinx right

[boreal girder]

k

[subtle helm]

oopsie

i meant sin(pi -x)

[sullen gorge]

.close

[compact pewter]

Channel closed

Closed by @sullen gorge

Use .reopen if this was a mistake.

[compact pewter]

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.

[main sail]

i need help with this question pls

[molten badger]

What have you tried?

I'm not so good at number theory but I will try my best here

[main sail]

uhh i tried the simons factoring thing

i cant rlly see anything i can do

[molten badger]

$n(x+y)=xy$

[boreal girder]

Alexis_Fx

[molten badger]

So

$nx+ny-xy=0$

[boreal girder]

Alexis_Fx

[molten badger]

$x(n-y)+ny=0$

[boreal girder]

Alexis_Fx

[main sail]

oh ok

[molten badger]

$x=\frac{ny}{y-n}$

[boreal girder]

Alexis_Fx

[molten badger]

Keep in mind x is positive integer, what can you get from here

[main sail]

ok

so we have to find the number of solutions for this where n = 10?

[molten badger]

I guess so

[main sail]

how do i do that?

is there some kind of ai in this server?

[molten badger]

No

[molten badger]

$x=n+\frac{n}{y-n}$

[boreal girder]

Alexis_Fx

[molten badger]

Agree?

[main sail]

yeah

[molten badger]

So x is integer and n is integer

That make n/y-n also has to be an integer

[molten badger]

And when n/y-n is an integer?

[molten badger]

Do you understand what i said here?

[main sail]

yeah

thats true

[molten badger]

Answer this, this is the key

[main sail]

x is integer?

[molten badger]

No

When 6/x be integer?

[main sail]

when x is a factor of 6

[molten badger]

Yes

[molten badger]

And here?

[main sail]

when n-y is a factor of n

[molten badger]

Yes

Now plug n=10

[main sail]

factors of 10 are 1, 2, 5, 10

so y = 11, 12, 15, 20

yeah?

[molten badger]

Yeah

With each y we only have 1 unique x so

Yeah

Now you know how to do the rest

[main sail]

oh thank u

so the answer for g(2000) = factors of 2000?

number of factors of 2000?

[molten badger]

$2000 = 2^4.5^3$

[boreal girder]

Alexis_Fx

[main sail]

k

[molten badger]

Do you have any idea to count it in mind?

[main sail]

uhh

its like the number of combinations

idk how to count it well though

[molten badger]

Yeah this's combinatorial problem

$2^4\cdot 5^3$

[boreal girder]

Alexis_Fx

[molten badger]

Now look at this, every divisor of 2000 has to include some power of 2 and 5 only

This can't be divisor of 2000 right? $2^2.5^2.3^2$

[boreal girder]

Alexis_Fx

[main sail]

yeah

[molten badger]

Since we have 3^2 here

[main sail]

yeah

[molten badger]

But this's $2^2.5^2$ a divisor of 2000

[boreal girder]

Alexis_Fx

[molten badger]

See any pattern here?

[main sail]

OHH ok

[molten badger]

Notice anything?

[main sail]

brb

[compact pewter]

@main sail Has your question been resolved?

Channel closed

Closed by @main sail

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.

[mossy latch]

hey guys can anyone help me out with this problem please im very lost:

[dusky kindle]

Well theres a very famous 10 digit number 3816547290

its the only 10 digit

with that property

[hushed magnet]

clearly the problem is about finding that number and showing it is unique

[dusky kindle]

yes

so it is not very hard to show that it is unique

so the 5th digit must be 5 and the 10th digit must be 0

and you go through using the same technique and slow youll reach the number