help-26

[lusty roost]

That's not correct

1/(1/x + 1/y) isn't x+y

^I think that might be the error

[solar musk]

it is, isn't it

1/1/x = x

[lusty roost]

no

[solar musk]

1/1/y = y

[lusty roost]

yes

[solar musk]

reciprocal of 1/x is x

[lusty roost]

But adding them is different

1/x + 1/y = x+y/x*y

[solar musk]

oh right

so then what

ugh im so confused rn

[lusty roost]

this is very close

[solar musk]

we do need the system of equations to solve obviously

[lusty roost]

not this time

[solar musk]

we just need to isolate one

[solar musk]

what

[lusty roost]

just x, right?

[solar musk]

ye

[lusty roost]

divide both sides by (m+n)

[solar musk]

which leads us to?

[lusty roost]

1/x+1/y = a+b/m+n I think

[solar musk]

hmm

[lusty roost]

bc (m+n)(1/x + 1/y) = (m+n)/x + (m+n)/y

[solar musk]

yes

[lusty roost]

Now, subtract both sides by 1/y

[solar musk]

hmm

simiplifying that now

[lusty roost]

and divide 1 by both sides

[solar musk]

so m+n/a+b -y

ig

[lusty roost]

Remember?

Weird looking equation, ik lol

[solar musk]

ye if only a senior could come and see lol

im in 8th grade

[lusty roost]

I just finished 11th

[solar musk]

country?

[lusty roost]

Australia

[solar musk]

I'm from India

[lusty roost]

cool!

[solar musk]

thing is

this question was discussed

like 5 months ago

but i forgot the answer

can't even verify 😦

[lusty roost]

x = 1/((a+b)/(m+n) - 1/y)

Should be that

don't think you can simplify (too much) more than that

You can though

[solar musk]

a lil bit but not more than that yes

[lusty roost]

If I did it right

should be that

I hope

[solar musk]

we both are using wolfram lolll

[lusty roost]

I didn't

[solar musk]

then what is that?

[lusty roost]

I used desmos to render that

did the algebra by hand

[solar musk]

i tried putting the main expression gave that, then put 1/x + 1/y = a+b/m+n gave that so im troubleshooting more to check if its correct

[solar musk]

ohh

nice

[lusty roost]

thx

Very useful for that

Worlfram alpha is pretty cool

The creator of it is a literal genius, too!

Wrote a genuine paper on quantum field theory at the age of 15

[solar musk]

i didn't know that

[lusty roost]

Astonishingly bright man.

[solar musk]

damn

[lusty roost]

Here's his first ever paper I talked about: https://content.wolfram.com/sw-publications/2020/07/hadronic-electrons.pdf

[solar musk]

at 1975??

[lusty roost]

yes

[solar musk]

without the internet just thru books

damn that's hard

[lusty roost]

Can barely understand a word of it lol

[solar musk]

well gotta study for other exam topics so bye, we had a nice chat

[solar musk]

fr

.close

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[reef copper]

Translated, the two sets A and B are non-empty. If (the relation above), then which of these options is untrue. Why (B) ?

[native notch]

It's because if A is a subset of B and B is a subset of A both sets have to be equal

A intersection B and A union B will also be equal to A = B

Since they are non empty their intersection cannot equal phi

[reef copper]

Shouldn’t “⊆” be used instead? If they are equal

[native notch]

I guess

But there is no other way this is possible

[reef copper]

Well yes that what I thought

Thanks then

[native notch]

[reef copper]

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[cunning talon]

How to calculate CPI?

[full dagger]

Consumer price index?

[cunning talon]

Yes.

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@cunning talon Has your question been resolved?

[cunning talon]

<@&286206848099549185>

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@cunning talon Has your question been resolved?

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@cunning talon Has your question been resolved?

[sweet shard]

https://www.wikihow.com/Calculate-CPI

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[quick mantle]

A company has 15 employees divided into three departments: Research (6 employees), Development (5 employees), and Testing (4 employees). The company needs to form specialized project teams under the following conditions:

• Each team must consist of 5 employees with at least 2 employees from Research and at least 1
employee each from Development and Testing.

• Within each team, the employees are assigned distinct roles (e.g., Lead, Analyst, Tester, Devel-
oper, and Coordinator) (Hint: the arrangement of the employees matters)

If you need to form 8 different project teams under the conditions specified above, how many
different arrangements are possible?

[timber crystal]

where are you stuck?

[nova vector]

so what is about the 8 different project teams

if its just an example number then i may have a solution to get all combinations

[quick mantle]

I am just getting the wrong answer currently. I know that there are 1,450 ways to select the team members satisfying the requirements for a single team and that there are 120 ways to assign the unique roles to the 5 team members.

I multiplied both of those to get 174000

and then did 174000^8 to get all combinations

but apparently that's wrong

so that's where I am stuck as of now

[quick mantle]

Sorry I don't understand the question

[nova vector]

how dis you get your numbers

[quick mantle]

Which numbers?

the 1450 and 120?

[nova vector]

1450

yes

[timber crystal]

How did you get 1450

They already chose 4 of your 5 members

[nova vector]

the number is much bigger when i calculated it

[timber crystal]

This means you'll choose 1 out of 3 picks to full up the spot

So 3 combinations

[timber crystal]

?

[nova vector]

no if you dont choose the same guy for the 2 first spots it counts as seperate combination

[quick mantle]

So basically I just did C(6,2) * C(4,2) * C(5,1)

[nova vector]

no

you need to use arrangements

[quick mantle]

I know the 1450 is correct tho

like it's verified

[timber crystal]

That's for the 2nd rule

We're doing step by step

[nova vector]

ok

[quick mantle]

Anyways for the 1450 I basically just selected 2 from research, then selected 1 from dev, then selected 2 from test

and then multiplied

[nova vector]

yes but its missing

you can select RRRDT
or RRDDTT
orRRDTT

[timber crystal]

But the question doesn't say that there's researcher 1, r2, r3, etc...

So it wouldn't matter the order

[quick mantle]

mb i forgot to tlak abt the other cases

[timber crystal]

[nova vector]

it would

[quick mantle]

theres two other cases as well

[timber crystal]

Nope

Where does it say you need to select 1,2,3,4,5 chronologically in order

[quick mantle]

I basically did all 3 cases and then added to get 1450(600 + 400 + 450)

[nova vector]

then why did it tell you you have 6 employees instead of 2

[timber crystal]

?? They're all the same employees they're not labelled differently

[nova vector]

oh wait le me do the math to check this

[timber crystal]

Think of it as colours

[quick mantle]

suree

ok wait so can we go back to the original q

😭

[timber crystal]

... What are they

[quick mantle]

C(6,3) * C(5,1) * C(4,1)

[timber crystal]

you didn't give the full question

[quick mantle]

and C(6,2) * C(5,2) * C(4,1)

[timber crystal]

No I mean the cases themselves not the calculations

[quick mantle]

oh

[timber crystal]

!xy

[topaz sinew]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[quick mantle]

ok so the three cases are

c1: 2 research, 2 test, 1 dev
c2: 3 research, 1 test, 1 dev
c3: 2 research, 1 test, 2 dev

[wary tulip]

[quick mantle]

Ok one sec

[timber crystal]

I thought by cases you meant rules? I misunderstood ig

[quick mantle]

This is exactly it

and then there are diff parts to the q

a. How many ways are there to select the team members to satisfy the departmental requirements
for a single team?

b. For each selected team, in how many distinct ways can the employees be assigned to the 5 roles,
considering each role is unique?

c. What is the minimum number of project teams that must be formed to ensure at least one
employee is included in at least 4 teams?

d. If you need to form 8 different project teams under the conditions specified above, how many
different arrangements are possible?

a, b, c are all correct

d is what i need help with

a is 1450, b is 120, c is 10

[nova vector]

ok the first 2 are right let me check the last

[quick mantle]

👍

[nova vector]

the second is just 5 factorial

[quick mantle]

yep

[nova vector]

i dont understand the third cant just that employee never gets selected in the first place or am i misunderstanding

c

[quick mantle]

i think all positions need to be filled

but anyways

from here on out

how would I go abt doing d. correctly

[timber crystal]

Why is the hint at the very bottom of the 3rd paragraph, this could easily be misunderstood as a hint FOR THAT PARAGRAPH, ie 5! which is ALSO the arrangement of employees

[nova vector]

are you sure of a result

[quick mantle]

Ah yeah idk why they put it there

[quick mantle]

for c?

yes

a, b, c are all correct

[timber crystal]

What

That's just 4 not 10

Unless it means ALL employees

This whole question is just weird wording

Hold on lemme look at d

[quick mantle]

well there are 15 employees 3 teams per employee makes it 45 positions

each team has 5 employees so all pos would be how many number of teams * 5 and since we need 1 in >=4 teams it basically has to be greater than 45/5

so greater than 9

[timber crystal]

That's just too huge of a number, (1450*120)⁸

[quick mantle]

and that would be 10

[quick mantle]

yeah

[timber crystal]

So yeah the question has bad wording

[quick mantle]

yeah lol

[nova vector]

pretty sure a is wrong

[quick mantle]

it is nottt

lol

[nova vector]

C(6,2) * C(5,1) * C(4,1)

look at this

this is how we will choose 4 emplyers

and then we can choose the third one from the remaining

[quick mantle]

ur talking abt c?

[nova vector]

no a

[timber crystal]

6!/4!x5x4x11

[nova vector]

so its 15X5X4

the multiply it by the last persan

[quick mantle]

how would I approach d tho?

[nova vector]

which is c(total -used,1)

your approach is correct

[quick mantle]

hmm

[nova vector]

but the value of a is wrong

so let me continue the a value should be 15X5X4Xc(15-4,1)

15X5X4X11

3300

[timber crystal]

6!/4! For 2 researchers, 5 for having 5 developers, 4 for having 4 testers, last one is just 11 due to 15-4

[nova vector]

then for d you have to do what you did before

3300*120

all to the power of 8

try that

[quick mantle]

sure

[timber crystal]

15?

No it's 6!/4! ie 30

6x5

[quick mantle]

396000^8 tho is too huge of a number

[timber crystal]

You used c

[nova vector]

its less than that cuz here you are calculating the change between the 2 first researchers

[timber crystal]

You need p

[nova vector]

which we added after

[quick mantle]

3300*120 = 396000

[timber crystal]

you switch the positions of the 2 researchers thus x2

[nova vector]

no no in 6*5 that case is calculated

for example

1.2.3.4.5.6

[timber crystal]

exactly that's why we use 30

[nova vector]

no let me explain to you

so if you do 6*5

its like making R,R a soldid part of all combinations

its correct

but the question doesnt want that

[timber crystal]

it's just colors I understand

[nova vector]

it wants How many ways are there to select the team members to satisfy the departmental requirements
for a single team?

so it just wants to select 2 random R

[timber crystal]

I understand now it's just colors lol

[quick mantle]

this wouldnt work

team has 5 members no

[nova vector]

ye we choose 4 then add the last

then mix between them

[quick mantle]

whats wrong w these cases:

[nova vector]

you can do it like that as well it should give similar result

[quick mantle]

thats how i did it and got 1450

[timber crystal]

They're correct

[quick mantle]

C(6,2) times C(5,1) times C(4,2) = 450
C(6,2) times C(5,2) times C(4,1) = 600
C(6,3) times C(5,1) times C(4,1) = 400

450 + 600 + 400 = 1450

[timber crystal]

Each department has r1,r2,r3,...; d1,d2,...; t1,t2,..

[fresh musk]

nono this is wrong

[timber crystal]

Every single person is considered different within each department

[fresh musk]

mela mnayek

[timber crystal]

You need to choose 2 researchers out of 6

So you pick 1, then 2 and divide by 2 to remove order

6*5/2

Then choose 1 developer

There's 5, so you can pick 1 only which give 5

Choose 1 tester
There's only 4 so you pick one which gives 4 possibilities

You took 4 of the 15

Now you can choose ANY of the rest of the employees so 11

(6*5/2)x(5)x(4)x(11)

[quick mantle]

right

[nova vector]

do you have a way to verify the answers

like is it an online test or smthing

[quick mantle]

a, b, c are verified correct

its an autograder

[nova vector]

you sure a was 1450

[quick mantle]

yes

100% lol

[nova vector]

did you try the other number i gave you *120 to the power of 8

[quick mantle]

let me see

3300 * 120?

[nova vector]

ye

[quick mantle]

its wrong 😦

[nova vector]

do you lose points if you try wrong answer

[quick mantle]

nope nope

i can resubmit unlimited times

until it says correct

[nova vector]

can you try 1800*120 to the power of 8

[quick mantle]

also wrong 😦

[nova vector]

oh sorry not 1800 its 2000

[quick mantle]

still wrong :((

[nova vector]

doesnt make sense

[timber crystal]

Send an email to your teacher

It's wrong

[quick mantle]

hahahaha i wish

:((

[timber crystal]

ask your friends what they got

[quick mantle]

lol none of them have done it

or the ones who have they're stuck

since its due tonight in like 4ish hours

they'll probably do it last minute

[wary tulip]

what are you guys working on?

[quick mantle]

.

part d

[wary tulip]

ok i can see what i get but it’ll be a few mins before i can start

[nova vector]

hey try 1450*120

just that

[quick mantle]

no problem!

[quick mantle]

sure one sec

[nova vector]

i think this might be it

[quick mantle]

it is not wfewhjfbe

[nova vector]

oh wait the teams are distinct

so its 1450X120 factorial /(1450X120-8) factorial try that

so its 1450X120 X1450*120-1X1450...

[quick mantle]

Hmmm

Ok lemme see

[nova vector]

i can calculate it if you want

[hazy lynx]

it's https://sourcely.co/PNRHitX

apologies if notation is incorrect in the query

regardless its a ridiculously large number

[nova vector]

yes thats what i just said

[hazy lynx]

oh apologies gng

thought u wanted the raw answer

[nova vector]

oh no its a big number not practical to get it

but thx anyways

[wary tulip]

If you need to form 8 different project teams under the conditions specified above, how many
different arrangements are possible?
what are the conditions specified above?

[nova vector]

@quick mantle did it work

[nova vector]

i will send them

[wary tulip]

but you only have 15 employees? are people allowed to be assigned to multiple teams

you need 8 teams and they have 5 employees each?

[nova vector]

the same employee can be in anoter team

[wary tulip]

weird but ok

[quick mantle]

I’m
Checking

Sorry we had a fire alarm haha

[nova vector]

lol

[quick mantle]

this is like the 10th time someone in our apt overcooked chicken

😭

[nova vector]

hhhh

[quick mantle]

840087000000000000000000000000000000000000 didn't work

[wary tulip]

if you have researcher 1 and researcher 2 in a team, are these the same?

1. assigning researcher 1 to lead and researcher 2 to analyst
2. assigning researcher 1 to analyst and researcher 2 to lead

or are the researchers distinguished from each other

[quick mantle]

the roles are distinguished from each other

[wary tulip]

but what about the people within a department

[quick mantle]

assigning 1 to "Lead" and 2 to "Analyst" is different compared to 1 to "Analyst" and 2 to "Lead."

people within the same department are distinguished from each other

[wary tulip]

ok

[nova vector]

8.40087×10^41

[quick mantle]

does not work :((

[wary tulip]

hm

[wary tulip]

i don't think this will work but try 840086679969274908724650056048477891040000

and 20835483134158603887020090675805503250

[wary tulip]

this is also 174000 choose 8 btw

[quick mantle]

yeah neither worked

😦

[wary tulip]

what the heck

[quick mantle]

😭😭😭

[wary tulip]

well i'm not sure i understand the question then. maybe something to do with "under the conditions specified above"

[quick mantle]

A company has 15 employees divided into three departments: Research (6 employees), Development (5 employees), and Testing (4 employees). The company needs to form specialized project teams under the following conditions:

• Each team must consist of 5 employees with at least 2 employees from Research and at least 1
employee each from Development and Testing.

• Within each team, the employees are assigned distinct roles (e.g., Lead, Analyst, Tester, Devel-
oper, and Coordinator) (Hint: the arrangement of the employees matters)

these are the conditions.

[nova vector]

pretty sure there is a mistake or input pattern you have to follow

[wary tulip]

writing "under the conditions specified above" on part d of a question is bad practice

[quick mantle]

hmmmmmmmm yeah

i thought the 174,000 number is correct
but since it's 8 different teams, we cant have repeats
so could it be $174,000 * (174,000-1) * (174,000-2) \cdots$

[thorny flame]

ani

[topaz sinew]

@quick mantle Has your question been resolved?

[topaz sinew]

@quick mantle Has your question been resolved?

[snow tiger]

Maybe you aren’t supposed to consider the roles for part d

So drop the 120 from the calculation

[snow tiger]

Or do this but subtract 120 each time instead of 1

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[wide sentinel]

if the line is perpendicular to xy plane then is it just the z axis

[sullen tundra]

not necessarily

the z-axis is perpendicular to the XY plane, but there's infinite lines that are also perpendicular to the XY plane

[wide sentinel]

i see

[agile harness]

the z axis is where x and y are zero but you could however visualize a line that is perpendicular to the xy plane without x and y being zero, you should go to desmos 3d if you need help visualizing this

or just lay your hand flat, you could put a pencil perpendicular to your hand but it could be shifted along your hand and still be perpendicular

[wide sentinel]

for this question is it (3,2,-1) + lambad(0,0,1)

vector equation

[agile harness]

yep

lambad

[wide sentinel]

ok thanks

lambda

lol

.close

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[steady sail]

help

[steady sail]

i found total number of 3 digit terms that are divisible with 3 and 2 respectively, and subtracted it with total number of 3 digit terms divisible by 6, in order to get rid of the overlaped terms

then i just checked total number of terms divisible by 7, and subtracted it from the value we got above

[lavish dagger]

why get rid of the multiples of 6? is the word "either" implying exclusive or?

[snow brook]

thats what im wonderin

wondering*

[pearl fog]

i belive they tried inclusion-exclusion

[steady sail]

lets take an example, the number 12 is gonna eb repeated in 2 as well as 3 ryt, since its a multiple of both, but we cant take it twice can we? so in order to get rid of one of the 12's, we gotta subtract all the multiples of 6

[snow brook]

oh oka

okay*

[lavish dagger]

oh yh sure

[steady sail]

maths is annoying 😭

[lavish dagger]

nah its great, anyway ig its just 900/2 + 900/3 + floor(900/7) - 900/6?

[steady sail]

what;s floor?

@lavish dagger

[lavish dagger]

10/3 = 3.333...
floor(10/3) = 3

[steady sail]

oh no

thats not the ans

[pearl fog]

you should add the floor of the ammount where its not divisible by 2 or 3 but is by 7

[steady sail]

i think we need sm jee students to join in 🫡

do u wanna know the ans?

[lavish dagger]

or wait yeah idk why i added divided by 7

900/2 + 900/3 - floor(900/7) - 900/6?

i mean, that doesnt help actually

[pearl fog]

900/2+900/3-900/6-floor(900/7)+floor(900/21) i belive

[lavish dagger]

divisble by 21 means its divisible by 7

[steady sail]

bruh 💀 what's wrong with what i did though

[steady sail]

.

[lavish dagger]

its the same thing ?

[pearl fog]

whats the answer

[steady sail]

514

[pearl fog]

really?

[steady sail]

yup

i got 470

[pearl fog]

oh right

wait tf? i got 513 lol

[lavish dagger]

is it 901?

no that doesnt

make sense

wait yes it does

[steady sail]

guys just one thing, could we solve it without floor? we don;t recognise floored answers in jee

if possible please

[lavish dagger]

100 + 10 = 110, but there are 11 numbers in that range (100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110)
so, 100 + 900 = 1000, but there are 901 numbers in that range

so its 901 ig

[pearl fog]

well 900/2+900/3-900/6 only counts those that count the first two conditions, if you subtract by floor(900/7) then you subtract ALL numbers from 100 to 900 that divides 7, you should only divide numbers that satisfies the first 2 conditions and divides 7

[steady sail]

wai but

so see

we got first condition ryt?

[pearl fog]

yes

[steady sail]

900/2 + 900/3 - 900/6?

okay so for 2nd condition

lets just use AP

a = 105, d = 7, last term = 994?

994 = 105 + (n-1)7, where n is the number of the last term, which is 994

u get n = 128, hence number of 3 digit terms that are divisible by 7 are 128

isnt that 3rd condition?

[pearl fog]

lets take a smaller example, say numbers less than 25
by brute force we can see there are {2,3,4,6,8,9,10,12,15,16,18,20,22,24} (14)

lets try the non bruteforce way
24/2+24/3-24/6=16
{2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24} (this has 16)
we have to subtraxt 14 and 21 here, if you just subtract by 24/7 you will subtract by ALL numbers divisible by u that are less than 25, so {7,14,21} (3), if you subtract 3 from 16 you will get 14 which is incorrect, as you shouldnt subtract the 7, we can counteract this by adding the numbers that are not divisible by 2,3, but are divisibly by 7, which in this case would be roughly 24/21 which is 1, so we add 1 to 13 to get 14 (correct)

problem is for finding why you divide by 21 you have to be careful

[steady sail]

ohh

makes sense makes sense

ty got it

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[jagged wigeon]

What sense does it make that when we talk about continuity we imply that we are talking about continuity in the domain of the function and not about continuity of the function in R?

[jagged wigeon]

I mean, i've always talked about continuity of the function in R (high-school) but now (university) the book prefers to talk about the continuity of the function in its domain.
I mean, what sense does it make to study the function only in its domain?

I mean, wouldn't it be more right to say that the sentence "study the continuity of the function f" implies to study it in R and not in f's domain?

[main shale]

what would it mean for a function to be continuous or not at a point outside its domain?

the entire definition of continuity at any given point depends on local behaviour of the function near that point

if the function doesn't exist there you can't say anything

[jagged wigeon]

Let's take this example:
{ x if x < -1
f: { 0 if x = 0
{ x if x > +1

The book says that the function is continuous in 0 cause 0 is an isolated point of the function and for that reason the function is continuous in that point...

[main shale]

in the epsilon-delta definition, you can just take delta to be 1/2

then your vertical change is always 0<epsilon since you're fixed at that point
or if you prefer sequential continuity, the only sequences that converge to that point are the eventually constant sequences there

functions are always continuous at isolated points because of that

[jagged wigeon]

But when the exercise asks me to study the continuity of the function (without specifying the field) i understand that i have to study it in R, so the function is NOT continuous in R.
But if it asks me to study it in its domain, then f is continuous in its domain and i don't understand why in 0.
I mean, how can i say that the funciton is continuous in 0 if i cannot study it nerby 0?

[jagged wigeon]

Can you explain it better please?

[main shale]

are you using the epsilon-delta definition of continuity

[jagged wigeon]

Not yet, i'm stuck with the definition of continuity given by that equation: continuity if lim(x->x0) f(x) = f(x0)

[main shale]

then what's your formal definition of a limit

[jagged wigeon]

The limit exists only if the right and left limit of that point x0 are the same

but if i can't study the limit of f(x) in that point (0), then how can i say that the function is continuous in that point?

[main shale]

I meant more along the lines of $\forall\varepsilon>0:\exists\delta>0:\forall x\in X: 0<|x-c|<\delta\rightarrow|f(x)-L|<\varepsilon$

[thorny flame]

Desync

[main shale]

for a function f:X->R

[jagged wigeon]

Yeah

[main shale]

so take delta = 1/2

[jagged wigeon]

Ok

[main shale]

if |x-c| < delta = 1/2 then x necessarily is c (= 0 in this example)

since there's only that single point in the domain of f in that interval

so |f(x)-f(c)| = |f(c)-f(c)| = |0-0| = 0 < epsilon

for any epsilon

[jagged wigeon]

Wait

I don't get it.
You saying that, taken an epsilon = 1/2 what happens?
That interval has no values but 0

[main shale]

so x must be 0

[jagged wigeon]

Yeah, i suppose

[main shale]

oh, or rather

sorry, sec

[jagged wigeon]

BUT in the definition of limit, x must not be equal to x0 or not?

[main shale]

yeah, it's slightly different from normal epsilon delta definition of continuity (in that, you just take x=x0)

you can't have x=x0, so in this case the implication is vacuous

so it still holds

[jagged wigeon]

Yeah, i know but what i am saying is: if studying continuity of a function means checking if the equation $lim_{x->x_0}f(x) = f(x0)$ is true for every x of the field (domain or R), how can i study continuity (using this equation) if i can't study the limit cause there are no sufficient exes nerby x0 to know what f(x0) points to?

[thorny flame]

Quantum

[main shale]

you need to verify that $0<|x-c|<\delta\rightarrow|f(x)-L|<\varepsilon$

[thorny flame]

Desync

[main shale]

if the left is false the implication is vacuously true

there's nothing to check

there are no valid x's

[jagged wigeon]

i can't verify it cause the I(L) has no values in it but L itself

[main shale]

do you know what a vacuous truth is

[jagged wigeon]

means that if x ∈ I(c) then f(x) ∈ I(L)?
But in the example i gave this means that i have to verify that if x ∈ I(0) then f(x) ∈ I(0)?
But what does it mean?
I mean, both intervals contain just 1 value

[jagged wigeon]

no

[main shale]

if you have an implication P -> Q, then the implication is true if P is false

you are trying to prove an implication (if x is such that 0<|x-c|<d, then...), but in this case P (that is, 0<|x-c|<d) cannot be satisfied, so the implication is vacuously true

[jagged wigeon]

i still don't understand...
Why are we demonstrating it by using intervals?
I am just wondering what does it mean studying a limit that cannot be studied?

I'm stuck there

At that equation: "f is continuous if lim x->x0 f(x) = f(x0)

[main shale]

ignore the limit interpretation and just look at the definition $\forall\varepsilon>0:\exists\delta>0:\forall x\in X: 0<|x-x_0|<\delta\rightarrow|f(x)-f(x_0)|<\varepsilon$

[jagged wigeon]

Ok

[thorny flame]

Desync

[main shale]

f is continuous if, given some e>0, we can come up with a d>0 such for all x, the implication ( 0<|x-x0|<d -> |f(x)-f(x0)|<e ) is true

do you agree

[jagged wigeon]

Means that if x ∈ I(x0) then f(x) ∈ I(f(x0))?

[main shale]

what's your I(...) notation

[jagged wigeon]

Interval

I(x0) = (x0 - epsilon, x0 + epsilon)

[main shale]

then no, because you disallow x being x0

patching that is beyond the point

[main shale]

do you agree that this is a correct characterisation of continuity

[jagged wigeon]

Yes, this is the definition of continuity

[main shale]

then lets take d=1/2

what x (in the domain of f) satisfy 0<|x-x0|<d?

[jagged wigeon]

x0?

[main shale]

no, because |x-x0| = 0

and we need 0<|x-x0|

[jagged wigeon]

Then no x

[main shale]

yes, good

so the implication is

[jagged wigeon]

false

[main shale]

False -> |f(x)-f(x0)|<e

which is a true implication

[jagged wigeon]

Wait

You saying that taken d = 1/2, there is no value of the interval but x0?

[main shale]

yes, in an interval of radius 1/2 around x_0, there is only x_0

[jagged wigeon]

Ok, yea

[main shale]

but we disallow x=x_0 in the definition of a limit

[jagged wigeon]

Ok

[main shale]

so the set of possible xs is empty

[jagged wigeon]

Yes, so we can't study the limit of x0

[jagged wigeon]

.

if we can't study the limit in x0, then we can't check if the limit is equal to f(x0), so we can't determinate if f is continuous in x0

[main shale]

do you understand that we reached this point False -> |f(x)-f(x0)|<e

[jagged wigeon]

Yes

[main shale]

do you understand that False -> anything is a true statement

[jagged wigeon]

no

Oh

I think i got it, yeah but it makes no sense

No, i don't understand it

Like: if apples are blue, then something is true

[main shale]

yes

[jagged wigeon]

makes no sense

[main shale]

but apples aren't blue, so that statement isn't telling you anything

it doesn't say that the consequent is true

[jagged wigeon]

Yes

indeed

[main shale]

for a more mathematical example, do you agree that $$\forall x\in\mathbb{Z}, x>1\rightarrow x^2>2$$?

[thorny flame]

Desync

[jagged wigeon]

Yes?

[main shale]

but what about x=0?

then the implication reads 0>1 -> 0^2>2

False -> False

[jagged wigeon]

Yeah

[main shale]

and similarly, x=-2 yields -2>1 -> (-2)^2>2

False -> True

[jagged wigeon]

Then False implies something that can be true or false, we can't know it

[main shale]

but the implication itself is true

the forall is like an infinite AND over your set

[jagged wigeon]

But against hypothesis: x > 1

I don't get how that should be helpfull with my questions xD

[main shale]

when I say the implication, I mean the statement P -> Q

not Q

or P

if P is false, then P -> Q is true

why it's helpful is because we reduced the definition down to False -> ...

which is true

which is what we wanted to show

[jagged wigeon]

What does it mean that an implication is true?
I mean, i can say that P or Q is true but not the 'implication'

OHH

wait

you saying that the implication 'False implies something that can be true or false' is true?

[main shale]

false -> Q for any statement Q is true

[jagged wigeon]

Yes

[jagged wigeon]

"false -> Q for any statement Q, the implication is true" or "false -> Q for any statement Q is true"?

[main shale]

I don't know what you're trying to distinguish there

[jagged wigeon]

What's true? Q or the implication?

[main shale]

the implication

[jagged wigeon]

Yes, it is

And?

[main shale]

f is continuous if, given some e>0, we can come up with a d>0 such for all x, the implication ( 0<|x-x0|<d -> |f(x)-f(x0)|<e ) is true

we had this

we want to prove that the implication 0<|x-x0|<d -> |f(x)-f(x0)|<e is true

we saw that this reduces to false -> |f(x)-f(x0)|<e

because there are no valid xs

[main shale]

this is an implication of this form

which is true

[main shale]

so we have satisfied this definition, as we have proved the implication true

[jagged wigeon]

And what does it mean that the implication is true?
I mean, what have we come up with by doing so?

[main shale]

I think this is going to take more time than I have free

I recommend looking into stuff on symbolic logic, and possibly other definitions of continuity to see how they connect to the limit definition

[jagged wigeon]

Alr, alr.
Thank you for your time btw

[main shale]

nw, gl

[jagged wigeon]

thx : )

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[knotty heron]

The k over n

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[knotty heron]

Confused on this

I have no idea what the 4 over n means

[rigid ivy]

!msgdel

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[knotty heron]

My bad

.close

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[still bough]

is this 2x = 90

divide it by 2 which is 45?

[exotic remnant]

not enough info

[topaz sinew]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[tame ingot]

if the triangle is inscribed in the circle, then angle x is half the arc length of the opposite chord

if the side opposite x is 90 degrees (out of 360) then yes x is 45

[exotic remnant]

not enough info given to know

[tame ingot]

well yeah i dont know how mayo drew it

just saying that he can approach it that way

[still bough]

the triangle is an isosceles triangle so i assumed 2x and 90

[exotic remnant]

if it's an isosceles triangle then yes

[tame ingot]

it has to both be an isosceles triangle, and the longest side has to be the diameter of the circle

otherwise you're not guaranteed an angle of 45

[sharp vigil]

Is the red line a diameter?

[still bough]

yess

[sharp vigil]

Then it's fine

X is 45°

[sharp vigil]

Considering red is diameter

And both those blue sides are equal

[still bough]

got it thank u

.close

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[topaz sinew]

@serene tangle Has your question been resolved?

[buoyant horizon]

I would try to use the work-energy theorem from the moment the block is released

Till it stops again after compression

hmm i guess you were using the theorem from somewhere in the middle of the trajectory

[buoyant horizon]

in this case the initial and final velocities would be 0

yes

and because the ball momentarily stops

we can use WET till that moment

so you just equate the total work done to 0

I mean...

the work done is equal

i'm not sure if the forces are

If you apply the theorem like I said it comes out like this

$mg(h_0+x)-\frac{1}{2}kx^2=0$

[thorny flame]

@buoyant horizon

[buoyant horizon]

x=19cm, m=700g, k=400N/m

side note, if you want to determine the work done by a spring you need to integrate the spring force since the force varies with the displacement

that is

$W_\text{spring}=\int\limits_{x_0}^{x}-kx\dd{x}=-\frac{1}{2}k\left(x^2-x_0^2\right)$

[thorny flame]

@buoyant horizon

[buoyant horizon]

x_0 is the initial compression/extension

x is the final

here x_0 is 0 since the spring was in its natural state

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[ruby heron]

Idk how to do 7a or 7b

[ruby heron]

I assume the ratio test is used for both

But if so, the algebra got messy for me

[neon venture]

can you show the calculations you did ?

[ruby heron]

You mean the one line of work i did and said yeah im screwed?

Sure!

[neon venture]

yea

[neon venture]

$\frac{n+n!}{(n+1) + (n+1)!}$

[thorny flame]

Herels

[neon venture]

🤔

maybe we can try this :

$\frac{n+n!}{(n+1)! (\frac{1}{n!} + 1)}$

[ruby heron]

Uh

[thorny flame]

Herels

[ruby heron]

Howd u do that

The + in the bottom is misin

Missin

[neon venture]

factoring by (n+1)! on the denominator

[ruby heron]

Ok lemme see one sec

Ok i see it

But now what

Do it to the numerator?

Eh idk what thatd do

[neon venture]

also :
$\frac{n!(\frac{1}{(n-1)!} + 1)}{(n+1)! \frac{1}{n!} + 1)}$

[thorny flame]

Herels

[ruby heron]

Yeah im lost

How is n!/(n-1)! = n

[neon venture]

cuz n! = n(n-1)!

[ruby heron]

Ok that makes sense

[fervent ferry]

Better way where you don't drown in algebra: find a simpler series such that $a_n > \frac{1}{n+n!}$ that you know converges.

[thorny flame]

LooseEthics

[ruby heron]

Like 1/n!?

[fervent ferry]

Exactly

[ruby heron]

Which is bigger than the a_n

So i can do a root test with 1/n!

And if that converges

So does a_n

[fervent ferry]

yeah

[ruby heron]

Yeah that makes more sense to me thank you

[neon venture]

yea i think thats better

[ruby heron]

I respect the hell outta this though

U guys are awesome at math lmao

But what about 7b?

Idek how to go about that one at all

Never had a series like it in any hw before

[fervent ferry]

You can split that into $\sum{1/n}$ and $\sum{1/n!}$

[thorny flame]

LooseEthics

[ruby heron]

ok

now how would that work

like what if one part converges and the other diverges

[neon venture]

yes

[fervent ferry]

If both converge, it converges. If one diverges, it diverges. If both diverge, it gets complicated.

[ruby heron]

...

tf u mean it gets complicated

[fervent ferry]

You could end up with an indeterminate

Like if one diverges to infinity and the other to -infinity

[ruby heron]

bruh

[fervent ferry]

But that's not the case here

[ruby heron]

idr being taught this at all lmao

[ruby heron]

this part specifically

now i gotta go lookin for it

[fervent ferry]

If both converge then you're basically just adding two numbers.

[ruby heron]

but they dont

one diverges

just lookin at it

its just a matter if the other does

which im testin for in 7a

[fervent ferry]

A divergent series is either infinite or oscillates

[ruby heron]

maybe...maybe u should be my professor

[fervent ferry]

I did some tutoring back in the day

[ruby heron]

im curious what some of your guys' math backgrounds are tbh

do u have a degree?

if so, what is it, if u dont mind me askin

[fervent ferry]

just a computer science bachelor

[ruby heron]

"just"

ok

do u enjoy math so u chill in here to help?

[fervent ferry]

more or less

occasionally open up the server to see if anyone has an interesting problem they need help with

[ruby heron]

huh

ive enjoyed calc 1 and 2 for the most part

few exceptions here n there

probably the most fun ive ever had in classes, but some of the most ive struggled at the same time

like now

[fervent ferry]

calc is nice for the most part

[ruby heron]

im tryin to go for a biochem degree

i think calc 2 is where i stop

"math" wise

a part of me doesnt want to stop

but then washer and shell method, related rates, optimization, infinite series, all make me want to stop lol

[fervent ferry]

yeah, calc and maybe linear algebra is about all you'd need there I think

complex analysis is where the pain starts lmao

[ruby heron]

feels like pain rn

Lemme try to work out this number 7a and 7b

see if i can come back with an answer

@fervent ferry

So, off of what u said, 7b diverges i think

But idk what test its based off of

[fervent ferry]

1/n is the harmonic series

[ruby heron]

Yeah i get that

[fervent ferry]

I think you're supposed to integrate it and find out it's log(n)

Which goes to infinity

[ruby heron]

Ahhhh yes

My weakest test

The integral tesr

Test*

Truth be told i keep forgettin about it

Wait a damn minute

I have no idea how to integrate factorials

[fervent ferry]

You don't need that

But if you want to, look up the gamma function

[ruby heron]

Nah im focused on this

Wdym i dont need it

The integration of factorials or the integral test?

[fervent ferry]

integration of factorials

since you already know 1/n! converges

[ruby heron]

So why am i doing the integral test

I was told thats like my last resort lmao

[fervent ferry]

for 1/n

[ruby heron]

But p-series works for 1/n

[fervent ferry]

I don't remember what that is

[ruby heron]

I respect that

Bro knows about gamma functions and how to integrate factorials

But doesnt know the pseries test

Hell yeah

Lemme see if i can find it in my notes

Its like uh

1/n^(p)

If p=< 1 it diverges

If p>1 it converges

I know theres a command in here to rotate that, but idr what it is

[fervent ferry]

Wait, isn't p-series just for geometric series?

[ruby heron]

No

Geometric series is

1/5^n

For example

Not 1/n^p

[fervent ferry]

right

got it

[ruby heron]

Look at me tutoring the tutor

Nah im playin

But yeah thats what the p-series test is

Its like

1/n^(1.000000001)

It converges

1/n^(2/3)

[fervent ferry]

ya, I remember now

[ruby heron]

Diverges

Yup

So do i still need the integral test?

[fervent ferry]

nope

[ruby heron]

So what test confirms that the entire series diverges?

Dude i legit cant find a single example in my notes thats like it

[fervent ferry]

Summation is distributive, $\sum{a+b} = \sum{a} + \sum{b}$

[thorny flame]

LooseEthics

[fervent ferry]

So if a one of those converges and the other doesn't, you're just adding a number to a divergent series

[ruby heron]

Huh

Ok

So should i just write diverges?

[fervent ferry]

that's what I'd do

[ruby heron]

No test

It just does

Ill ask my classmates what theyve done

If theyre even up at this time

[fervent ferry]

Write what 1/n and 1/n! do separately, then that together they diverge

Or I suppose you could do the same thing as in 7a

1/n + 1/n! > 1/n

1/n diverges

therefore 1/n + 1/n! does too

[ruby heron]

Cant do that

UNLESS

I show it via limit comparison

But then im dealin with

[fervent ferry]

Why not?

[ruby heron]

Lim n-> inf (1/n + 1/n!)/(1/n)

[ruby heron]

The conditions of direct comparison

[fervent ferry]

the inequality holds for all n>1

[ruby heron]

Its not that

Wait

Actually

Nvm

It should work

Ur fine this was my fault

But the actual act of just sayin 1/n + 1/n! > 1/n bothers me for some reason lmao

It makes sense

[fervent ferry]

Nah, that's on me

[ruby heron]

No ur right

If the equality is correct

[fervent ferry]

Going on about distributivity when there's such a trivial way to do this

[ruby heron]

Nah ur way makes sense lmao

Just using direct comparison for such an ugly expression

Its just odd, yknow what i mean?

I see ! And my brain is wired to think ratio test

And again, ive never seen a problem like this, aside from a special limit like (1 + 1/n)^n

Like thats the closest thing ive seen to somethin lookin like that

Ill put both ways how about that lmao

Then ask my professor on tuesday

[fervent ferry]

sounds good

factorials are always kind of awkward to work with

but you get used to it

[ruby heron]

Yeah

Whats (n+3)!/(n+2)!?

Outta curiosity

Like if u simplified it, if possible

[fervent ferry]

n+3

all the terms below that just cancel out

[ruby heron]

I see

What about the reciprocal

(N+2)!/(n+3)!

[fervent ferry]

1/(n+3)

Factorial is just a product

$\frac{(n+2)!}{(n+3)!} = \frac{(n+2)(n+1)(n)...(2)(1)}{(n+3)(n+2)(n+1)(n)...(2)(1)}$

[thorny flame]

LooseEthics

[ruby heron]

Ah

So they just cancel

Ok what about

[fervent ferry]

yea

[ruby heron]

(N+2)!/(n+4)!

Is that 1/(n+3)(n+4)?

[fervent ferry]

correct

[ruby heron]

Ok what about

(N+2)!/(5n+4)!?

[fervent ferry]

Can't really simplify that.

[ruby heron]

Cuz of the 5n?

[fervent ferry]

Since the relative number of terms changes

yeah

[ruby heron]

I see

Yeah ive done a lot of calc now

And algebra

Think its time for a small break then bed

Thank you loose for your help and time

[fervent ferry]

yeah np

[ruby heron]

I appreciate it greatly

If it wasnt for this server id be so screwed

Someone here taught me uh

Was it alternating series?

No limit comparison and direct comparison

And it strengthened my knowledge of p series n geometric

So now im able to do most of these or at least give a valiant effort

So yeah, thank you

[fervent ferry]

ye, this place is pretty good for advice

[ruby heron]

Yeah

[fervent ferry]

better than quora and gpt at least

[ruby heron]

A decent amount of u too

[ruby heron]

Snapchat has a myAI thing

Thought it was stupid tbh

[fervent ferry]

yeah, ai can't do math

[ruby heron]

But when i needed help with math i didnt know who to go to

So i tried it

It actually works sometimes

Like i can take a picture of a problem and say "show whether number 21 converges or diverges"

And theyd actually do it

Steps n everythin

Off a picture

[fervent ferry]

yeah, but that's because it has these basic examples in the training set

[ruby heron]

Fair

[fervent ferry]

any slightly complicated problem and it doesn't know

even worse, it thinks it knows and will try to convince of the wrong result

[ruby heron]

Yeah i double check it

One time i sent it a series and it got the denominator wrong

But when i had no one else to turn to or need a really quick and simple answer

Or just to maybe double check a small thing

Its nice

[fervent ferry]

Try wolframalpha then

[ruby heron]

Ive heard of it

[fervent ferry]

It won't give you steps unless you pay them, but it can solve pretty much anything that is solvable

anyway, gotta go now

[ruby heron]

Ok

[fervent ferry]

best of luck

[ruby heron]

Thank you again

👍

.close

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• Show your work, and if possible, explain where you are stuck.
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[wraith pine]

hey quick discussion me and my friends are having
what is the correct ans

shift this function 5 units up, 3 units to the left and horizontal dilation of 1/2 . f(x) = x^3

either was

f(x) = (2x+6)^3 + 5
OR
f(x) = (2x+3)^3 + 5

[torpid sparrow]

That depends when you do dilation

[wraith pine]

What would you say it is?

[torpid sparrow]

How do you do dilation by 1/2 of (x+3)^3?

[torpid sparrow]

Oh you asked what not why

[wraith pine]

yeah

[torpid sparrow]

I would say, if we follow the orders strictly as stated, it should be the second

[wraith pine]

Alright thank you

.close

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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.

[placid flame]

$\int{ln{x+sqrt{x²+1}}}

[placid flame]

How to use those signs

[pearl fog]

$\int \ln(x+\sqrt{x^2+1}) dx$

[thorny flame]

Skissue ping4response

[pearl fog]

like this?

[placid flame]

Thanksvhow to solve it

Yeah exactly right

[acoustic tangle]

Thinking of ibp

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@placid flame Has your question been resolved?

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[placid flame]

Thanks

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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.

[supple shard]

I need help with this math question
I have one million light bulbs in a line, each one has a switch. When you switch the first one all the light bulbs turn on, when you switch the second 2,4,6,8… turn off. Then we switch the third and 3,6,9,12… turn on/off depending on its state. We continue to do that until we reach the millionth. How many light bulbs are on in the end?
I seem to have a problem approaching this question as I have never encountered anything like it. I don’t want to use AI cuz I know they make many mistakes and they’re not reliable
Your help will be appreciated

[long stirrup]

the bulb is on at the end if its number has odd amount of divisors

3 stays off because 3 is divisible by 3 and 1

9 stays on because it's divisible by 1,3,9

so you need to predict if it's going to be odd or even for any number

it's not very obvious

[odd flower]

for this

u can find out the number of lightbulbs

with an odd or even amt of factors

those with an odd amt of factors

stay on

[long stirrup]

also on this server people don't like when you answer questions, it's not like stack exchange

[odd flower]

wit

wot