help-49

[minor heath]

at x = 0

but idk what else

[rare maple]

We know that till 0 the graph is identical to 2^x

And for less than 0

The function changes to 2^-x

[minor heath]

ooooooo

so it looks like

a parabola kinda?

[rare maple]

Yes

[waxen willow]

Just dont forget it should be pointy

[rare maple]

Now we can check values of c for which the graph of y=c intersects our g(x) graph twice

[minor heath]

ok

[waxen willow]

Like it should have a spike at x = 0. It shouldnt be smooth

[minor heath]

how

something like this?

[rare maple]

We know g(x)=c [linear function] and g(x)= our graphed function

So we can graph both of these

And see the number of points of intersection

[rare maple]

The intercepts are slightly off but it is fine

[minor heath]

theres two

how does one go from the graph concept

to the

find value of c

[rare maple]

I will start will linear functions

We consider y=3x+5

Where does this function equal 8

[minor heath]

1

x=1

[rare maple]

How would we find this graphically

[minor heath]

on the right?

[rare maple]

We need y=8

[minor heath]

oh yea

[rare maple]

And it also must satisfy the function

I will show it with a graph

[minor heath]

ok

[rare maple]

The blue line is our function

[minor heath]

yes

[rare maple]

and we need the function to equal 8

This the y coordinate of the function is 8

[minor heath]

ok

[rare maple]

So now how do we translate this idea to find the number of solutions for a given value of c

[minor heath]

y=c?

[rare maple]

We we graph y=c and y=g(x)

The points of intersection of these 2 will give us the solutions

[minor heath]

ok

but like idk the points

do i just make the y equal each other

[rare maple]

We don't need the points

[minor heath]

how do i find possible vlaues then

[rare maple]

We just need a rough sketch of how the graph would look like

[minor heath]

ok

[rare maple]

There can be infinitely many possibilities

[minor heath]

right

oh

is c just

a horizontal line

[rare maple]

Yes

[minor heath]

so it cant

be at the vertex

and such vertex is at

y = 1

so c >1

bro you had me lost there for a second

but now it makes

so much more sense ty

ahhh

i can probably do the rest of the quesiton then

ty

actually

it might help if you explain Qb and Qc

pls

[rare maple]

We first need the second function

[minor heath]

yes

idk what happens

to the pointy parabola

if it gets wrapped around another absolute

isnt it still the same graph?

[rare maple]

If y is greater than 0 yes

But we are subtracting d

[minor heath]

and if thats absolute

then it moves up?

[rare maple]

Which means our graph shifts below

I should probably clarify what what up implies here

[minor heath]

oh yeah sorry

um

is it the asymptote

[rare maple]

It will look something like a parabola

But the middle region is pushed above the x acis

[minor heath]

hasnt it always been above the x axis tho

the whole function

[rare maple]

,w graph y= |x2 - 5|

[grand pond]

Results provided by WolframAlpha

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Requested by triaengle

[minor heath]

oh what the heck

[rare maple]

Wait this isn't the complete thing

I will draw it

Something like this

Pointed at the top of course

[rare maple]

We first graph g(x)-d

What will that look like

[minor heath]

YOOOOO YOU GOT A CHALKBOARD

ohhhh

[rare maple]

I don't really like whiteboards

[minor heath]

you make senese now

yeah i get it

OOOO

then do you solve at x = o

x = 0 for y

oh wiat but we finding d

[rare maple]

First can you explain how we get this graph

[minor heath]

ok sortry

because it drops below the x axis

then you put the absolute around the function

so the part below the x

reflects

[rare maple]

Yeah now we can find d

[minor heath]

is d

just when x = 0?

[rare maple]

Sprry

I read that as y

[minor heath]

np

d = 4?

ooh

and for c

you just find a value of d

such that it is jsut

the original g(x)?

so its 1

thank you mystery math man

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[latent snow]

Help

<@&286206848099549185>

[vivid yoke]

!15m

[midnight plank]

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

[vivid yoke]

chill out dude

[turbid merlin]

what have you tried to start with?

[latent snow]

How to simplify the equation

[calm niche]

This easy bro

[latent snow]

I used cross multiplication.

[rare maple]

What did you get

[grand pond]

Couldn't find an attached image in the last 10 messages.

Sep 8, 2025, 11:40 AM

[latent snow]

Is this right

[rare maple]

There seems to be a calculation erro

[turbid merlin]

something's off here

also, imo, cross-multiplication isn't necessary here either

[rare maple]

Both terms in numerator cannot be negative

[latent snow]

What to do

[turbid merlin]

in the left fraction, you have a denom of x

[rare maple]

You can try to take the lcm

[turbid merlin]

on hindsight, i'll let you work that out unless you've seen it

but yes you can take the LCM of the two denominators

[turbid merlin]

that works, but you can simplify this

[latent snow]

How

[turbid merlin]

this is going to be a bit hard to spot

but technically in the numerator you have a common factor of sqrt(x) between the two terms

so factor that out

[turbid merlin]

wait what happened here

[latent snow]

Ha

[turbid merlin]

may i ask what you were trying to do here?

[rigid terrace]

Do like this buddy

[turbid merlin]

appreciate the help, but a quick reminder that normally we're not supposed to directly show solutions

[rigid terrace]

Oh sorry was unaware

[latent snow]

From where do you get(1-/-x

[turbid merlin]

but i'm not sure how else to explain to the helpee atm

[rigid terrace]

When we factorise root of x then x becomes root of x if you get what I mean

Like I take it out

And the x becomes it

Because root of x multiply root of x equals x

[latent snow]

Okkkk I get it 😂😂 it is easy

[latent snow]

It's took me 35 minutes to understand it

[turbid merlin]

well done

[latent snow]

Thx all for help

[zenith snow]

Np

!done

[midnight plank]

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[latent snow]

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[steel cedar]

I don't understand the part when he says "Clearly"

[steel cedar]

Why does that fact necessarily mean n is perpendicular to the plane?

[vivid yoke]

if n is normal vector then any vector r will have the same projection in direction of n

That's not true if n is not normal vector

[steel cedar]

Where r is any point on the plane?

[vivid yoke]

It's not a point it's a vector position, but yeah you know the idea

[steel cedar]

o yeahh

[vivid yoke]

If n is not normal vector

you can see r1 and r2 have different projection

[steel cedar]

That's actually rlly helpful

Thanks a lot!

.solved

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[grim vector]

i just can't manage to get the binomial coefficient

i know 2^2n is the amount of every possible walks of 2n length

in other words why is the amount of positive walks of length 2n is nC(2n)

[fallow scarab]

2n is the number of steps, and of those you choose n to go up because n number of them must also must go down to reach S_(2n) = 0

[grim vector]

but the positive walk doesn't need to have S_2n = 0 right ?

i might need to construct it

[fallow scarab]

i don't know what you're referring to. i'm talking about these two

[grim vector]

im referring to the second situation where S_i are positive

but i got your idea of n amount of +1 and n amount of -1 to go back to 0

i just find it too restrictive for what i need

cuz it don't imply S_(2n) = 0 neccessarly does it?

[fallow scarab]

maybe show the rest of the page.

[fallow scarab]

don't the paragraphs after (3.4) explain

[grim vector]

holy

if i get it correctly

so we have that its all > is 1/2 u_n and then we have that
1/2 u_n = 1/2 P(S_i >= 0) and so u_n = P(S_i >= 0) up to 2n-1

and then neccessarly till 2n cuz S_2n+1 is odd

thanks so much riemann i think im all good with this

[midnight plank]

@grim vector Has your question been resolved?

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[silent river]

so I was trying to develop law of cosines to simplify that and I'd reached on a formula on the right, can someone explain to me if its right my relations and the process?

[midnight plank]

@silent river Has your question been resolved?

[silent river]

<@&286206848099549185>

[fallow scarab]

you have a lot of variables without defining them. try drawing a right triangle and labeling it with your variables

[silent river]

i js want to know when I can use that formula, i tested on law of cosines and didnt work

[midnight plank]

@silent river Has your question been resolved?

[midnight plank]

@silent river Has your question been resolved?

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[last slate]

Ik this is a simultaneous equations question but im not sure where ive gone wrong

[last slate]

Thats my working

[fallow scarab]

where do you first two equations come from

oh are you using htis

[last slate]

Yeeee

[fallow scarab]

how did you get this eqn

[last slate]

I collected like terms from eq 1

[fallow scarab]

and where did 105 come from

[last slate]

18-75

180*

Ah shite

Its add isnt it

[fallow scarab]

-75 + 75 = 0 yes

[last slate]

So i add 75 to 180

Ohhhh i see Kk I’ve gotten the rivht answer

Tysm for the help @fallow scarab

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[fallow scarab]

<@&268886789983436800> spam

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[serene plover]

what do i do here? (literal equations)

[shell wigeon]

I'm assuming the question is "solve ... for a"

[serene plover]

yes

[shell wigeon]

You can multiply both sides of an equation by the same quantity

[serene plover]

oh yeah i noticed i didnt do the other side

so ku=a?

[shell wigeon]

Not quite

Can you show your work?

[serene plover]

ohh swap them

[shell wigeon]

Why did the u move?

[serene plover]

uuuh um the thing youre solving for goes first? but thats what my teacher does so idk

[tender dew]

$\frac{k}{a} (k) = \frac{k^2}{a}$

[shell wigeon]

Wait did you multiply both sides by k?

[grand pond]

Peter

[tender dew]

meanwhile you cancelled them

[serene plover]

yes to balance it out

and like cancel it out

[shell wigeon]

You don't cancel a numerator by multiplying by the same thing

[serene plover]

oh

[shell wigeon]

You cancel a denominator

[serene plover]

but the a is the denominator i need that

[shell wigeon]

Yes but you multiply on both sides

[serene plover]

yes i did that

[serene plover]

theres a (k) with the u on the other side

[shell wigeon]

Yes, I saw, but we're talking about a

[serene plover]

hm ok

[shell wigeon]

What happens if you multiply both sides by a? Just try

[serene plover]

OH SHI I CAN THEN DIVIDE AU

hidden 2 step equation..

a=k…. im so smart

[shell wigeon]

Hold up

Did you divide by u?

[serene plover]

yes

[shell wigeon]

on both sides?

[serene plover]

OH SHI

a=ku

crisis averted

WAIT HOLD ON

THATS NOT RIGHT

MY A** IS NOT THINKING

[shell wigeon]

Oh I was also half-thinking

[serene plover]

a=k/u

[shell wigeon]

Yes, correct

[serene plover]

crisis averted 2x

btw is this correct

i meant this one

[sudden yacht]

No no

Do things on BOTH SIDES

[serene plover]

ohhh true

[sudden yacht]

You must never forget this

[serene plover]

so a =n+p/m

[mortal mirage]

dont forget parentheses c:

[serene plover]

hm uh ok

[mortal mirage]

the idea is correct though c:

[shell wigeon]

(or write the fraction correctly on paper)

[serene plover]

my teacher didnt do that tho to my memory

[mortal mirage]

your teacher probably used fractions

[serene plover]

like this

[mortal mirage]

yea that's correct

[shell wigeon]

In text when you write $n+p/m$ it means $n+\frac{p}{m}$, not $\frac{n+p}{m}$

[grand pond]

Nel

[mortal mirage]

but in single line typing, be sure to notate it as (n+p)/m

[serene plover]

ohh ok i did not know that

[shell wigeon]

Because we can't know that the slash is for both n and p

[sudden yacht]

Have you not been taught operations precedence? (Or PEMDAS)?

[shell wigeon]

When writing a fraction, it's obvious because there's a big horizontal bar

[serene plover]

i have been taught pemdas??? how did i get thru the earlier grades of school w/o it

[sudden yacht]

Well, the reason for this is simply pemdas 🤷‍♂️

I've asked you because people most of the times do not realise this, even if they have learnt it

[serene plover]

oh ok

[mortal mirage]

if you write it like this, according to pemdas, you do the division first, not the addition

[serene plover]

hold on is this correct

[shell wigeon]

It is, though we don't usually keep a negative sign in the denominator like that

[serene plover]

hm then how would you fix it

wait is it uh

flipping the values or something

[shell wigeon]

You can multiply both the numerator and denominator by -1

[mortal mirage]

you mean sign?

[serene plover]

i mean i guess i think

[mortal mirage]

ok, well how would you proceed with Nel's idea?

[serene plover]

like is it this

[shell wigeon]

Now instead of -g+b you can simply write b-g

[mortal mirage]

sometimes we write b-g just because though

[shell wigeon]

But both are correct

[serene plover]

oh true

what would i do here theres 2 x’es

[frigid patio]

cancel out

they cancel out

at least i think

i mgiht be dumb

[mortal mirage]

not yet c:

[serene plover]

hm yes but then there would only be g=-c

[mortal mirage]

first, get rid of the denominator

how would you do that?

[serene plover]

multiply?

[mortal mirage]

both sides by...

[frigid patio]

ohh i am dumb

[serene plover]

by c

[mortal mirage]

issok

[serene plover]

-c

[frigid patio]

well why?

[shell wigeon]

Just like you did there

[serene plover]

corrected it

ah

[frigid patio]

reverse order of operations

[mortal mirage]

what's in the denominator?

[serene plover]

x

[mortal mirage]

yep! so what do you multiply both sides by?

[serene plover]

ohh by x

typo

[mortal mirage]

yep!

so what would your new eq be?

[serene plover]

gx=x-c

and then add c

[mortal mirage]

hmm, i wouldnt

[shell wigeon]

Well you can do that, but you'll need to group the x's

[mortal mirage]

you wanna isolate x, no?

[serene plover]

oh true

so move it like

[shell wigeon]

If you don't know what to do, you can always try to get 0 on one side and everything else on the other

[serene plover]

so now g=2x-c?

[mortal mirage]

it's $g\cdot x$, not $g+x$

[grand pond]

Mathemusician

[mortal mirage]

so you can't move the x to the other side just yet

[serene plover]

oh ok

so divide i’ll do that

[mortal mirage]

heres a hint: move all terms w/ x to one side

[serene plover]

wa

[mortal mirage]

and all terms without to the other

[serene plover]

so x to gx?

[mortal mirage]

yes

[serene plover]

would it be g2x

i feel like thats wrong

[mortal mirage]

hmm, not quite.

move the x to the other side and write out the equation?

[mortal mirage]

not quite

what is gx - x?

[serene plover]

g… or not

[mortal mirage]

(if you factor out the x)

wait

[serene plover]

second guessing myself here

[mortal mirage]

i just gave you the answer

[mortal mirage]

when you move the x to the lhs, what does it become?

[serene plover]

uuh uuuh idk

[mortal mirage]

how would you move the x to the lhs?

[serene plover]

hhh we did something like this in our notes let me see

[shell wigeon]

You tried to subtract on the right and add on the left; you cannot do that

If you subtract x on the right, you subtract x on the left as well

[serene plover]

ohh ok

but then there would be no x

we’re solving for x

and it would end up g=-c

[shell wigeon]

There would still be x, because subtracting x from gx does not give 0

Nor does it give g

[serene plover]

OH YEAH

TRUTHNUKE

gx-x =-c?

[mortal mirage]

yes!

[serene plover]

so then i divide both sides by g to remove the g

like this?

[mortal mirage]

not quite

[shell wigeon]

Not yet

That would leave x - x/g on the left

[serene plover]

ah

hmmmmmm

hmmmmmmmmmm

then what would i do

[mortal mirage]

can you factor something out of both terms?

[serene plover]

uhhhh hm

[shell wigeon]

Do you know how to factor?

[serene plover]

i forgot what that means mb

hold on i need my smart pfp for this

[shell wigeon]

If I give you 2 meters + 3 meters, can you tell me the result?

[serene plover]

5 meters

[shell wigeon]

Nice

Do you know what a coulomb is?

[serene plover]

no…….

[shell wigeon]

That's great, don't worry about it

A coulomb is a unit, just like meters, but meters are for distances and coulombs are for something else

We can write like 4C to say "four coulombs"

Can you tell me what 5C + 7C would be?

[serene plover]

12 coulombs?

[shell wigeon]

Exactly

Now, say I make some new unit and call it x

And I measured the quantities f and g, in x

Can you tell me what fx + gx would be?

[serene plover]

uum fgx?

[shell wigeon]

No, think about what you did with 5 and 7 to get 12

[serene plover]

i added them

the numbers so i got that

[shell wigeon]

Right, so why did you multiply f and g just now?

[serene plover]

oooh i just realised that

uuuuuuuh f+g+x???

[shell wigeon]

No

[serene plover]

i knew that

[shell wigeon]

Can you say step by step in details what you did to go from 5C + 7C to 12C

[serene plover]

5+7 = 12, and they were both like terms so thats why the c’s were together

wait was it f+g+2x

[shell wigeon]

Did you mean to write 5+7 = 12?

[serene plover]

YES YES MB

[shell wigeon]

Ok let's forget about x for a moment and continue with coulombs instead

Say I measured 6C somewhere and there's some unknown quantity K, also in coulombs, nearby

And I want to add them so I can find K by doing more measuring in both places

How would I add 6C and KC?

[serene plover]

hmmmm

would you move the c?

[shell wigeon]

The C is a unit of measurement

Would you prefer I write "how would I add 6 coulombs and K coulombs" ?

[serene plover]

hmmm

hold on i need to take a brain break

10 mins

[shell wigeon]

Fair enough

[serene plover]

back

@shell wigeon

wait maybe you could write out the answer and explain why its like that

maybe i’d understand more like that

and then i’d solve another example

[shell wigeon]

I'd rather you figure it out, then you'll remember

[serene plover]

oh ok

[shell wigeon]

We can go back to meters if you want

[serene plover]

yeah

[shell wigeon]

Alright

I have a measuring stick that's 1 meter in length, and I need to measure a table's side length

I put my measuring stick on it against a corner, and leave a mark at the end, then I try measuring the rest

It's a very precise measuring stick, so I get a lot of decimals

It just so happens that when I measure the rest of the length, I get the number pi to 20 decimals

You know pi, right?

[serene plover]

hmmmm

yes

3.14…..

[shell wigeon]

Ok, so in total the length of the table's side is 1 + pi

[serene plover]

yep

[shell wigeon]

But that's in meters, right?

[serene plover]

yes

[shell wigeon]

I measured 1 meter, then pi meters

And I added them up to... ?

[serene plover]

4.14…. ?

[shell wigeon]

No, I don't want an approximation

[serene plover]

oh ok mb

[shell wigeon]

And I want to keep the unit, meters, in the answer

So 1 meter + pi meters = ?

[serene plover]

1+pi meters?

[shell wigeon]

Yes!

[shell wigeon]

So, back to this

6 coulombs + K coulombs = ?

[serene plover]

6+k coulombs?

[shell wigeon]

Great

Now I don't want to write out coulombs every time, I'd rather just write "C"
So 6C + KC = ?

[serene plover]

yeah so 6+k c

[shell wigeon]

Alright but be careful with how you write it exactly

If you write "6+k coulombs", it's obvious that the 6+k go together

[serene plover]

yes

[shell wigeon]

If you write "6+k c", it looks like 6, and then k coulombs

That doesn't make much sense

[serene plover]

true

[shell wigeon]

How do you group them?

[serene plover]

just spaced it out since i wasnt sure

(6+k)c?

[shell wigeon]

Exactly

Now, remember the new unit I invented, "x" ?

[serene plover]

yes

[shell wigeon]

I said I measured f x and g x, and I want to, surprise, add them

[serene plover]

so (f+g)x

[shell wigeon]

Yes!

[serene plover]

gah how did i not see that

[shell wigeon]

.

So what do you do here?

[serene plover]

hmm so (g) = -c?

[shell wigeon]

No, where did the x's go

[serene plover]

hmmm

im not so sure there

[shell wigeon]

Ok hold on

Let's go back to coulombs for a second

[serene plover]

yes ok

[shell wigeon]

I measured two new quantities, p and q, in coulombs

[serene plover]

yes

[shell wigeon]

p is a total, and q is one part of it, and I want the rest

So I want to subtract q from p, right?

[serene plover]

yes

[shell wigeon]

So if I write it all out, I need pC - qC

What would that be?

[serene plover]

(p-q)C

[shell wigeon]

Nice

Now I realize that q is just 1

That part I measured is just 1 coulomb

[serene plover]

mhm

[shell wigeon]

So now I'll just write pC - 1C

Just making sure, what do you think that would be?

[serene plover]

(p-1)C

[shell wigeon]

No

[serene plover]

SHI WHONG ONE

i meant -

[shell wigeon]

Alright, yes

Now usually when we use some unit like coulombs or meters and we have a value of 1, we always write that 1 in front

But when manipulating variables, we don't

[serene plover]

but what about standard form

[shell wigeon]

Wdym?

[serene plover]

you put the integers last

[shell wigeon]

No no no I'm talking about something else

[serene plover]

ohok

[shell wigeon]

If I have 1 meter, I write 1m

If I have 1 coulomb, I write 1C

[serene plover]

(-1p)C?

[shell wigeon]

?

No I'm no longer talking about the (p-1)C

That was correct

I'm talking about why I wrote pC - 1C instead of pC - C

If I have a variable y in an equation and it's just 1y instead of 2y or -3y or ey, then I just write y, not 1y

Do you follow?

[serene plover]

hm yeah

oh yeah ik what youre talking about

[shell wigeon]

Yeah?

[serene plover]

thine common sense theres an invisible 1 knew that

[shell wigeon]

Ok great

Can you rewrite gx - x = -c with that in mind?

[serene plover]

hold on which one would be 1

[shell wigeon]

Pretty much anything

Multiplying a value by 1 doesn't change it

[serene plover]

1g1x-1x=-1c

[shell wigeon]

Yeah, that's fine

Maybe don't write it in between two variables though, like in "g1x"

That doesn't read well

You already have a 1 at the very front

So then how do you go further?

[serene plover]

hmmmmmmm

[shell wigeon]

You literally just did it with pC - 1C

[serene plover]

oh true i think

wait hold on this isnt working give me 3 mins

[midnight plank]

@serene plover Has your question been resolved?

[serene plover]

i dont wanna work on this anymore its too much so

[midnight plank]

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[midnight plank]

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[still flax]

hello, I need some help for a, ii) she needs five or fewer attempt to pass. for a, i) my answer was 0.01536

[still flax]

this is geometric distribution

[drowsy ferry]

How far have you got?

[still flax]

I think the formula I need to use is Pr=P(1-p)^r-1 so do I got to do this 6 times and add them?

[still flax]

I've only done i) and not sure how to start ii)

[drowsy ferry]

You can write "five or fewer attempts to pass" as "does not fail the first 5 times".

Then the answer will be P(5 or fewer attempts to pass) as 1 - P(first 5 attempts fail) given suitable assumptions.

[still flax]

is it 1 - because im finding the opposite of what I want then finding the probability i'm left with?

[drowsy ferry]

Yes

[still flax]

I see thank you

[drowsy ferry]

A hidden assumption that I'm using is that she doesn't give up before 5 attempts.

[still flax]

Could another assumption be "the test conditions stay the same"?

[drowsy ferry]

Yes

[still flax]

okay thank you 👍

[drowsy ferry]

Also that the probabilities are independent.

[turbid merlin]

another assumption you could make here would be that Genevieve doesn't tilt after failing once (may affect probability too)

[still flax]

does this mean that each test probability doesn't effect another?

[turbid merlin]

which is kinda tied to this point actually

[still flax]

ohh your right

[turbid merlin]

if a fail decreases the chances of subsequent passes, then the probabilities are not independent

[still flax]

okay thanks guys 🙏

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[midnight plank]

@tidal turret Has your question been resolved?

[potent veldt]

In the green room in a garden it is known that the following is true:

a) if everybody knows Juan, then everybody knows Camila.(we can suppose it is because they always walk together)

b) if Everybody knows Juan, then everybody knowing Camila implies everybody knowing Gonzalo

The question is: Is it true that if everybody knows Juan then everybody knows Gonzalo? Resolve using Propositional Logic

[last arch]

Well I guess yea

Unless I'm missing something

[potent veldt]

bet

Anyhow I'm assuming you're supposed to invoke the laws of logic in your explanation?

So what do you think the answer is and what rules do you know?

!occupied

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[thin shore]

I don't even know where to start this question

i'm assuming it must be a linear graph

[vocal moon]

its quadratic clearly, so yeah indeed it should be linear.

[turbid merlin]

it is

[pearl hull]

have you found the original function?

[vocal moon]

Observe that its inverted

[pearl hull]

just diff it and sketch the new function

[vocal moon]

so the leading term is negative

[vocal moon]

The graph isnt labelled

[thin shore]

in these types of questions you aren't given an original function

[vocal moon]

Yeah figures

[sudden yacht]

But you can find it

[vocal moon]

In that case

[pearl hull]

Set the coordinates, find the function, diff it, and sketch the new one

[sudden yacht]

(only because the units are said to be the same)

[vocal moon]

Too much hassle ngl it'll just be a linear function which attains zero where the quadratic attains its maximum

and the slope is -ve thats all

[thin shore]

I think i'll just mark like stationary points

or something

[pearl hull]

It's safe and difficult to yields any mistake. Better for beginners.

[vocal moon]

Ig so

[thin shore]

and then I assume the graph has to be above the x axis where it's increasing

[pearl hull]

Yes

[vocal moon]

mhm

[thin shore]

does this point have any significance?

as another stationary point

[vocal moon]

Its the only stationary point

[pearl hull]

it's the maxima of the function

[vocal moon]

and here f'=0

yes

[sudden yacht]

Not there...

[thin shore]

so how do I draw this line

hm

[sudden yacht]

The derivative is 0 at that x

0 means on the x- axis

[thin shore]

hang on

this is what the answer looks like

[pearl hull]

hmm

Let's begin with some elementry approach

-# They're gone, so I gotta bring out my method

[pearl hull]

Consider every square on the graph to be 1 unit^2, can you find the function for me?

[vocal moon]

damn

[thin shore]

.close

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[jade magnet]

im not sure what to do

i thought maybe i should just expand them out

and then subtract g(x) from f(x)

to get f(x) - g(x) = 0

but i realised that doesnt seem like the simplest solution

surely theres a shortcut

[jade magnet]

and then you can probably do the derivative and find when the min and max points are 0

again though

it seems difficult

[midnight plank]

@jade magnet Has your question been resolved?

[wary epoch]

Weird that I found that q=1 and r=2. I might do something wrong.

I reckon that $x^2(x-1)^2(x-2)+p(x-q)^2(x-r)^2=0$ implies that $x^2(x-1)^2(x-2)=0$ and $(x-q)^2(x-r)^2=0$ separately

[grand pond]

Xwtek

[jade magnet]

hmm

that would make sense actually

[jade magnet]

hmm

ill try that

i think im gonna discuss this with my math teacher tbh

.solved

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[slate ferry]

if the diameter of a circle grows to twice its size, what happens to the radius?

[slate ferry]

GROWS TO FOUR TIMES ITS SIZE RIGHT???

IM NOT TRIPPING

[pearl hull]

Have you heard of that pizza meme?

[slate ferry]

no 😭 ?

2r ---> (twice its size) ---> 4r

no????

[pearl hull]

pi r^2

[slate ferry]

whats with that?

[pearl hull]

r --> x2
size --> x4

[pearl hull]

that's how you find the area of a circle

[slate ferry]

yes, but i dont see how thats relevant

[slate ferry]

r here is radius... right

[pearl hull]

wait, you mean the relation between diameter and the radius?

diameter = 2 * radius

diameter * 2 = 2 * radius * 2

[slate ferry]

so if the diameter grows to twice its size... radius also grows to twi-

oh. my.

okay i failed 2nd great math

[pearl hull]

I don't quite get your question honestly

[slate ferry]

its lowkey out of context

[pearl hull]

I thought you were asking about radius and the area

[slate ferry]

no

well

im studying electromagnetisim and- theres proportion

yeah i okay i understand now

ty ig

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[pearl hull]

have a good one

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[slate ferry]

if a=b/c^2 , and im comparing between the (a) of 4 different things... like i wanna know which thing has biggest a for example and im given b and c. to give the most accurate comparsion, knowing that none of these variables can be negative,

can we ignore the square on the c? like just evaluate b/c and see which one of the 4 has it the biggest.

[pearl hull]

[runic hamlet]

no

[slate ferry]

why?

[runic hamlet]

frankly I dont wanna cook up counterexamples

[shell wigeon]

2/3^2 < 1/2^2 but 2/3 > 1/2

[slate ferry]

ok ig makes sense

what if the variables cant be any number between 0 and 1

[shell wigeon]

?

4/9^2 < 2/5^2 but 4/9 > 2/5

You won't find a criterion

[slate ferry]

alright understood

and p can be ignored if a=b/(p c^2) (p is a constant)

and p can also be ignored if a=b/(pc)^2 as long as p is a constant

(these are questions)

nevermind! proved it in two seconds (im dumb)

ty people

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[fair kayak]

guys, for 23b, what do they mean 'these numbers are not independent of the previous numbers'? do they mean repetition or something

[lyric charm]

yes, they do mean repetition. you've got digits taken from the same source twice

[fair kayak]

i seee, okay thanks!

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[barren wharf]

Find the fraction closest to 3/4 but not exactly 3/4 of the form a/b where (a,b) = 1 and b <= 15.

a and b are integers.
Consider that a/b > 3/4.
Hence, 4a - 3b is an integer.
It cannot be 0 because that would mean 3/4 = a/b.
But since a/b > 3/4, 4a - 3b must be a positive integer.
Let 4a - 3b = n
Then 4(a/b) - 3 = n/b
Let a/b = R
4(R) - 3 = n/b
Hence 4R = 3 + n/b
R = 3/4 + n/(4b)
Now we shall minimize n/4b
This is minimized trivially at n = 1 and b = 15
Let's check if this works.
If b = 15 and n = 1, then a = (3b+n)/4 = (46)/4 = 11.5
This isn't an integer.
Hence this solution is exiled.
We now think to get a fraction that is at the same distance from 3/4 as 11.5/15 or even less.

Now consider a/b < 3/4
Hence 3b - 4a = n = a positive integer

3 - 4(a/b) = n/b
R = 3/4 - n/(4b)

We get a ratio extremely close to 3/4 when n is minimized and b is maximized.
Take n = 1 and b = 15.
This gives a = 11.

a/b = 11/15

Any fraction in the first case is > 11.5/15
3/4 = 11.25/15
Hence any fraction in the first case is at a greater distance than (11.5/15) - 11.25/15 = 0.25/15 from 3/4
11/15 is only 0.25/15 away from being 3/4.
And this fraction is the one closest to 3/4 than any other fraction in the second case.

Hence the closest fraction satisfying the requirements of the question is 11/15.

[barren wharf]

Is this solution correct?

[midnight plank]

@barren wharf Has your question been resolved?

[barren wharf]

<@&286206848099549185>

[steep wolf]

yeop

@barren wharf

power rangers are here what is your problem

https://tenor.com/view/ekoi-ekoi-dancekid-dancekid-gif-10799637110820905144

[ruby depot]

What you can do is if you can't get it exactly (which the whole point is to not do it) use the biggest b find the a such that it is equal (it can it self be rational) then roll the a as you wish(maybe to the closest or to the next integer)

so just 3/4 = (45/4) / 15 which is approximately 11/15

[civic lynx]

since b has a largest value of 15, 1/15 is the smallest possible distance you can move away from 3/4

[ruby depot]

Yes.

[civic lynx]

your solution works, though, since you derive those equations at the end and then take 1/15

[ruby depot]

Yes, do you have any ohter questions related to this?

[barren wharf]

thank you

[barren wharf]

nope

.close

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[shell wigeon]

That's not true in general

[civic lynx]

how so?

[shell wigeon]

If the constraint was b <= 18 the answer wouldn't be some k/18

Plus you phrased it wrong, 1/15 is not the smallest possible distance you can move away from 3/4; it's 3/4-11/15 = 1/60

[civic lynx]

oh you're absolutely right - disregard what i said then @barren wharf

[barren wharf]

yes

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[jaunty vortex]

pls guys 🙏

[barren wharf]

1

? = 1

[jaunty vortex]

acutally

question like that

[timid drum]

what is $a_n$?

[grand pond]

mersenne primes

[jaunty vortex]

idk

[timid drum]

theres no definition provided

[jaunty vortex]

my teacher said like this

[timid drum]

yeah u cant simplify it then

[jaunty vortex]

and i wrote

[jaunty vortex]

yea i tried but

turkish highschools 👎

[timid drum]

brah

[jaunty vortex]

😔

$n_a$

yes

[midnight plank]

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[twin kernel]

Is this correct graph

[queen ermine]

seems good to me

wait

holup

did you maybe mix up x and y for a?

[lyric charm]

uh yeah the tail is actually at (1,-2) there

[twin kernel]

Oh yeah

But it’s basically plot the points in the parentheses and then move it 3 right and then down 1 right. And draw an arrow to it from parentheses point

[queen ermine]

ye just fix the mix up

[buoyant linden]

are these vectors?

what's this notation called

[queen ermine]

frankly first time seeing this notation, but people like to write vectors in very weird ways

vectors are just this thing that can be anything if you really want them to be and they can look like anything, like they make up the universe or smth

[paper inlet]

angle bracket notation

[queen ermine]

every number is a vector if you want it to be

[buoyant linden]

interesting

thanks

[silk snow]

mnk

2+2

[twin kernel]

Is this correct

[twin kernel]

I think component form

[queen ermine]

draw larger diagrams

for your own sake please

use the whole page if you need to

trust me

[twin kernel]

Is it right tho 😭 it’s cus I got 17 problems to do

[queen ermine]

I can't tell if it is

your teacher can't either

practice more clear drawing, trust me, it'll help you in the longrun

no need for everything to be cramped

[twin kernel]

What about this

I already drew it so I’m not tryna erase everything

That counts for b) right

[midnight plank]

@twin kernel Has your question been resolved?

[midnight plank]

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[spiral rock]

Integral

[twilit field]

$E(X) = \int_{1}^{\infty} \frac{2}{x^2}dx$

[grand pond]

wai

[hard shard]

you multiply by for E

[lyric charm]

he did

[hard shard]

please dont delete it, just own your mistake

[lyric charm]

oh

bruh

[spiral rock]

Its adds no value...

[twilit field]

I think it's 2?

[hard shard]

youre learning

thats value

[twilit field]

As for $E(X^2)$. It diverges

[grand pond]

wai

[spiral rock]

Whats your question

[twilit field]

Wanted this verified

[spiral rock]

Well yeah you are right

[twilit field]

thanks

So $Var(X) = E(X-µ_X)$

right

[grand pond]

wai

[twilit field]

so $1=E(Y-1)$

[grand pond]

wai

[twilit field]

And $0.04=E(X-7)$

[grand pond]

wai

[spiral rock]

This is not the definition of variance

[twilit field]

so $1=E(aX+(b-1))$ and $0.04=E(X-7)$

[grand pond]

wai

[twilit field]

oops

right

[lyric charm]

E(X - mu_X) is always zero

[twilit field]

The whole thing is squared

[spiral rock]

The inside is squared

[twilit field]

$0.04= E((X-7)^2)$
\
$1= E((aX+(b-1))^2$

[grand pond]

wai

[twilit field]

So now what

[lyric charm]

this was kind of a bad idea imo

you should instead be trying to think about how mean and variance are affected by linear transformations

[twilit field]

hmm?

Wait

I forgot E[x] os the mean

💀

$E[X]=7;E[X^2] =0.04$
\
$E[Y]=a[X]+b$
\
$Var[Y]= E[Y^2]-(E[Y])^2$

[grand pond]

wai

[twilit field]

Is this correct

.close

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[quaint shoal]

How do I do this limit?

[quaint shoal]

I tried expanding but that gets me nowhere

[zenith snow]

Can u show ur work

Notice that 2³=8

U can use a³-b³ here

[quaint shoal]

no way I didn’t see that😭😭

[zenith snow]

Happens happens

It's fine

[quaint shoal]

this is a dumb question but how do I do difference of cubes when u have something like this

I’ve only ever done it with single numbers like 27+x^3

not (x+3)^3 -2^3

[zenith snow]

Consider x+3 as a

Un these case u would want to get rid of the denominator

Cuz that's what makes it undefined

So u need to use the formula (a-b)(a²+ab+b²)

[quaint shoal]

oh okay so like this?

[zenith snow]

Yep right track

[quaint shoal]

okay now I’ve got it

just gotta do a ton of practice problems I feel like for these

[zenith snow]

Honestly u didnt need to expand and all uk

U could js sub as it is

[quaint shoal]

and just pluggged in -1?

[zenith snow]

Yes after cancelling out the (x+1)

[quaint shoal]

yeah that’d make life a bit easier

I’ll keep that in mind for next time

[zenith snow]

Yea gd work btw

[quaint shoal]

thanksss I feel like I’m getting the ideas now

[zenith snow]

Yes it's fun once u get it

Np

[quaint shoal]

im also struggling a bit with this

I’m assuming I did something illegal somewhere?

[zenith snow]

U did

See (a+b)/x²=(a/x²)+(b/x²)

Not (a/x) +(b/x)

The denominator remains same

Both should be x²

And 2ndly try taking sinx common and then proceed

[quaint shoal]

hmm what would I do after this?

I’m not quite sure how to simplify that

[zenith snow]

Wait a minute

Lemme write it down

Ignore the top portion

[midnight plank]

@quaint shoal Has your question been resolved?

[midnight plank]

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[tidal turret]

let a and b be in R, Prove that for all n in N, ... = ... . Deduce the formula of the geometric series forall a not equal to 1. ...

[full moon]

LHospital:

[crude compass]

This post explains it well. Ping me if you want explanations: https://math.stackexchange.com/a/4255635

They use r instead of a.

[midnight plank]

@tidal turret Has your question been resolved?

[crude compass]

Actually, I prefer the answer just bellow: https://math.stackexchange.com/a/4255639, because it is more general.

[tidal turret]

how did this happen?

[crude compass]

What part?

[tidal turret]

like from start to bottom

[grand pond]

Sam

[crude compass]

Tell me if any part needs more details.

[midnight plank]

@tidal turret Has your question been resolved?

[tidal turret]

how does this relate to my problem?

[crude compass]

Don't you need to find the geometric series formula?

[tidal turret]

yes

[subtle blaze]

Do you ever use your brain and think for yourself or do you just hope people spoon feed you

[tidal turret]

I just started university

but yeah, sorry about that

[subtle blaze]

Literally every time you ask a question it’s “I don’t understand anything start from the beginning”

[tidal turret]

is true though

[subtle blaze]

I refuse to believe that if you tried you understand absolutely nothing

[subtle blaze]

A question like this is so open ended

You need to try to ask more specific questions

[tidal turret]

the problem is like I dont understand whats the link between the derivation of the geometric series with my exercise?

[subtle blaze]

Look at this

[tidal turret]

in my exercise they give me a different sum

[subtle blaze]

You’re soooo general with your questions

[tidal turret]

I mean as far as I am aware

I need to work with the given proposition they gave me

and somehow along the way, re derive the geometric series formula

[crude compass]

Right, I missed that.

[tidal turret]

I mean, is good that you sent this, is just that I still need to prove this using induction

[subtle blaze]

Well have you tried anything

[tidal turret]

yes

I proved the base case, when n = 1

[subtle blaze]

Okay?

[subtle blaze]

All you do is post the question and sit there