help-17

[trim walrus]

Oh ye

Inequality change

When dividing by neg

I'm dumb

Thanks

.close

[vocal sleet]

Channel closed

Closed by @trim walrus

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.

[trim walrus]

Hi again

[trim walrus]

Well

I tried putting m=2 n =3
and I considered 2 cases 0 and 1
And got f(5) > 2

As I thought 2005 = 2000 + 5

I must be helpful

[urban edge]

f(even number)=0

Thats all i got

Wait no, hold on

f(9996+3)-f(9996)-f(3)=0

You shoild be able to find f(3)

[trim walrus]

I'm not sure how should I find f(9996)

[urban edge]

Do you wanna know how i got this?

[trim walrus]

Ofc yes

I'd very happy

[urban edge]

ok lol

first thing, rearrange to get
f(m+n)=f(m)+f(n)

n always equals 2

then you can do m=2 then 4 then 6

And so on

[trim walrus]

Alr so f(6+2)= f(6)

How exactly do we know that f(6) is even

??

[mild flower]

did you miss the "or 1" in the question?

[trim walrus]

And perhaps this is for the first case, f(m+n)=1+f(m)+f(n)

This is the second case

[urban edge]

OH

Crap

Apologies

[mild flower]

what could f(3) be though?

or maybe it's best to see what f(1) could be first

[trim walrus]

And how exactly, are we gona find f(1)

[mild flower]

we have a constraint

we might not be able to get it down to one value

[trim walrus]

m=3, n=-2

Smth like this?

[mild flower]

what are the properties of f that we know?

i count six listed in the description

[trim walrus]

n can't take neg

f(3-2) - f(3) -f(-2)

This would be invalid

Then

[mild flower]

correct, f isn't defined at -2

so property number 1 is that f has a domain of the natural numbers

what's the next property we have?

[trim walrus]

[mild flower]

we know its domain, it seems natural to figure out its range

or at least codomain / try to get a handle on its range even if we don't know the full extent of it

[trim walrus]

The only question is how now...

[mild flower]

well

the second phrase of the question would tell you

although it's phrased kind of oddly

i read that as saying the function takes on values that are non-negative integers

where "takes on" I read as "can output"

[trim walrus]

Alr

So our range starts from 0

[mild flower]

and is integers

[trim walrus]

Yep

[mild flower]

so now again what can f(1) be? we know it can be one of 0, 1, 2, 3, 4, 5, 6, ...

we do have four more properties of f that are given in the question

three of which are simple and one of which is fairly complicated to state

[trim walrus]

Three of which are f(2)=0, f(3) > 0 and f(9999)=3333

And the fourth one...

?

[mild flower]

the long one

[trim walrus]

And even the long one as two cases

[mild flower]

well, as two possibilitids

you could phrase that as f(m + n) - f(m) - f(n) < 2

but that's maybe less helpful

but you have everything you need to calculate f(1) or at least to constrain it to a handful of values

for example, is it possible that f(1) = 20?

[trim walrus]

Ig not

Or maybe it might be

Well m and n can take any integers but if we take a negative it won't be valid...
And only way to write one is 0+1 (in form of integrs >= 0)

[mild flower]

true, so f(1) can't be broken down into m and n

but that relationship applies to all values

[trim walrus]

Yes...

[mild flower]

the only useful piece of information we really have right now is f(2)

because even though f(9999) - f(9998) - f(1) = 0 or 1 we don't know f(9998)

[trim walrus]

Yep..

[mild flower]

think backwards

spoiler: ||m = n = 1 - set up the equation and you'll get one possible value for f(1)||

[trim walrus]

Considering 0;

f(1+1)-f(1)-f(1) =0
-2f(1)=0
f(1)=0

Considering 1;
f(1+1)-f(1)-f(1)=1
F(1)=-1/2 [invalid]

We got f(1)=0

[mild flower]

yep

cool! so now you should be able to fill in the rest of the table all the way up to 9999

[trim walrus]

Not sure what you mean by that...

[mild flower]

what's f(3)?

[trim walrus]

Well,
f(3) = 1

As 2+1

f(2+1)

We might find f(5) now

[mild flower]

how did you get that f(3) was 1? you're right but I'm not sure your reasoning is right

f(3) = f(2) + f(1) + (1 or 0)

[trim walrus]

Cuz

We reject 0

Says f(3) > 0

[mild flower]

ah ok yep

just making sure you had the right procedure

[trim walrus]

Alr

If I try finding f(5)
We again end up with 2 cases...
f(5) = 1 and f(5)=2

As f(3+2)-f(3)-f(2)=(0 or 1)
f(5) - 1 = (0 or 1)

[mild flower]

okay cool
why'd you skip over f(4)?

[trim walrus]

Oh ye

F

[mild flower]

i mentioned a table before; I'd definitely recommend one here

[trim walrus]

Table??

[mild flower]

of inputs to outputs

just to keep track of it

like

x | f(x)
--------
1 | 0
2 | 0
3 | 1
4 | ???
5 | 1-2

[trim walrus]

And also f(4) also has two possibilities l

f(2+2)- f(2)-f(2)= (0 or 1)
f(4)=(0,1)

And
f(3+1)-f(3)-f(1) = (0,1)
f(4)=(0,2)

Like 3 possibilities

0,1,2

[trim walrus]

Alr

[mild flower]

hmm. what happens if you take f(1+3) instead

[trim walrus]

f(3+2)-f(3)-f(2)=(0 or 1)
f(5) - 1 = (0 or 1)
f(5) = (1,2)


f(4+1) -f(4)-f(1)= (0,1)

Considering 3 case of f(4)

f(5) = (0,1), (1,2), (2,3)

[trim walrus]

Still f(1+3)-f(1)-f(3) = f(4)-1 = (0,1) => f(4)=(1,2)

[mild flower]

oh right it's symmetric duh

[trim walrus]

,rcw

[trim walrus]

I think we can see a pattern

After x=4

[mild flower]

how can f(5) = 3?

that doesn't work as f(2+3)

what i suspect will happen

is that you'll need to minimize these values to just barely satisfy that last constraint

[trim walrus]

f(4+1)

??

f(5)-f(4)=(0,1)

[mild flower]

it's gotta work for all of them

no matter what n and m are (as long as they're natural numbers), that equation has to be satisfied

[trim walrus]

Ye

[hard atlas]

you can determine f(4) exactly

[trim walrus]

Yes?

[hard atlas]

f(4)=f(2)+f(2)+(0,1) <= 1. and f(4)=f(3)+f(1)+(0,1) >= 1. so f(4)=1

next it might be better to skip f(5) and first determine f(6)

[trim walrus]

Like
f(4)=(0,1)

f(4)=(1,2)

Taking intersection

f(4)=1
Smth like this

Perhaps

[hard atlas]

yes that is what I just wrote down

[trim walrus]

Alr

[hard atlas]

yes

now f(5)

and then continue until the pattern emerges

[trim walrus]

The possibility of f(5) = (1,2)

I don't think we can find exact value of f(5)

As f(3+2) = 1 + (0,1)
f(4+1) = 1 + (0,1)

[hard atlas]

actually, f(7) first is better

[trim walrus]

Alr

[hard atlas]

urgh I'm making mistakes in my head. but anyway, you got the idea now. now its a bit of playing around with the conditions, playing them against each other

[trim walrus]

I think we see the pattern after 4

1 at 4
Then 1,2 at 5
Then 2 at 6
Then again 2, 3 at 7

Let me calc

8

Intersection is (2,3)

Again

My brain is hurting

Will solve this later

Thanks for the help tho

.close

[vocal sleet]

Channel closed

Closed by @trim walrus

Use .reopen if this was a mistake.

[vocal sleet]

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.

[old niche]

im confused about a midpoint calculator

[old niche]

calculation

am i tripping or is the midpoint calculated incorrectly

[vocal sleet]

@old niche Has your question been resolved?

[flat whale]

what do you think it should be?

[hardy rivet]

Simpson's rule.

[vocal sleet]

@old niche Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[vocal sleet]

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.

[vast shale]

What am I doing wrong for e) ii

?

[vocal sleet]

@vast shale Has your question been resolved?

[vast shale]

<@&286206848099549185>

[peak lark]

in the line where you rationalise the denominator, you are multiplying the numerator and denominator by (4-2sqrt(3)) right?

have another look at how you are multiplying out the denominator

[vast shale]

Ohhhh

I forgot the minus

[peak lark]

it actually took me a scary amount of time to spot that mistake lol

very easy to make

[vast shale]

Thank you

.close

[vocal sleet]

Channel closed

Closed by @mellow harbor

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.

[vast shale]

How do I do this?

[vocal sleet]

@vast shale Has your question been resolved?

[vast shale]

<@&286206848099549185>

[dusk mantle]

with ur hand

just jkin

[solid sage]

is this hsc seems familiar

[vast shale]

Yeah

😭

<@&286206848099549185>

[rigid kernel]

Man u annoying asf

[civic drift]

Why are you even here if you dont wanna help

[vast shale]

Jee thanks

I thought this server was made to help people

[rigid kernel]

What grade u on?

[vast shale]

Thanks for the help

.close

[vocal sleet]

Channel closed

Closed by @mellow harbor

Use .reopen if this was a mistake.

[vocal sleet]

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.

[fickle rose]

can someone explain this solution to me

i do not understand

[patent nymph]

What is the problem that this is a solution to?

[patent nymph]

what do you not understand?

[fickle rose]

why did the proof start with 4 x 5? shouldnt it start for 3x4 for the minimum?

and then the next inequality im confused about

i feel like im being dumb

maybe i should leave this

[vocal sleet]

@fickle rose Has your question been resolved?

[vocal sleet]

@fickle rose Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[vocal sleet]

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.

[viral copper]

$\lim_{n \to \infty} \sum_{k = 1}^n \left ( \frac{k}{n} \right )^n$

[twin meteor]

neonperseus

[viral copper]

I don't know where I'd begin on this

[blissful saffron]

uh oh

look who got caught

start by looking at partial sums

[viral copper]

shut up derp

[viral copper]

what's that

[blissful saffron]

basically

Define

$s_n = \sum_{k=1}^{n} \paren{\f{k}/{n}.}^n$

[viral copper]

Uh okay

And then what

[blissful saffron]

Wait does

[hard atlas]

yes

[viral copper]

Does what

[hard atlas]

your fears are real

[viral copper]

What fears

Derp what is happening

[blissful saffron]

[hard atlas]

I'm just guessing but maybe he just noticed the terms depend on n

[viral copper]

ah

So uh

What does that

Imply for your partial sum

[blissful saffron]

Okay I may have defined it wrongly then

But it still exists

[viral copper]

How do these sums help me solving the problem though

[blissful saffron]

just fix the partial sum definition

well

infinite series is just a sequence of partial sums

and you can batter it with a bunch of theorems

like monotone convergence theorem

and co

See if the thing converges

[viral copper]

It has a finite answer

[blissful saffron]

oh it converges

i guess partial sums are still useful tho

[viral copper]

It's e/(e - 1)

[blissful saffron]

oh e

hmm

[viral copper]

Which suspiciously looks like a geometric series

[blissful saffron]

yeah it reminds me of the

[hard atlas]

I'm wondering if maybe you can see this as a riemann sum

[viral copper]

But I didn't want to introduce bias by bringing in the answer

[viral copper]

yeah I did that in a previous problem I brought in here

It worked

[maiden iron]

the n exponent makes this not possible I think

but we can still try

[hard atlas]

yeah the exponent is also bothering me

[peak matrix]

$\frac{\sum_{k=1}^{n}k^{n}}{n^{n}}$

[twin meteor]

methisalwaysright

[peak matrix]

maybe this?

and limit to infinity

[viral copper]

Hmm

well

does that help

[hard atlas]

well you just run into a infty/infty situation

[peak matrix]

geometric series has closed form partial sum, so it might help

oh wait it's not geometric

damn

nvm ignore me

[blissful saffron]

how is this manipulation justified

[viral copper]

He just factored out n^n

[maiden iron]

1/n^n

[viral copper]

yeah

[blissful saffron]

oh ok nvm I thought you moved it out of the limit

lol

[viral copper]

haram

[blissful saffron]

what was the definition for e again

[viral copper]

$\lim_{n \to \infty} \left ( 1 + \frac 1n \right )^n$

[hard atlas]

putting the fraction on the left is cursed

[blissful saffron]

i agree

also

x instead of n

[twin meteor]

neonperseus

[blissful saffron]

thanks neon

[viral copper]

If we were to factor out n from here

$\lim_{n \to \infty} \frac{1}{n^n} \left ( n + 1 \right )^n$

[twin meteor]

neonperseus

[blissful saffron]

lim (n+1)^n/n^n

yep

[viral copper]

ooo

Tantalizing

[blissful saffron]

looking close actually

[blissful saffron]

this idea is now viable again

[viral copper]

Is it legal to use the binomial expansion on (1 + n)^n here

[blissful saffron]

yes

(why?)

[viral copper]

Even if the variable is in the exponent

Idk maybe it gives us something useful

[blissful saffron]

The answer is just due to how limits work lol

[viral copper]

Because if we had

nvm

Idt it would help

[blissful saffron]

Idk if that goes very far tho

[viral copper]

yeah

[blissful saffron]

Also gosh darn you is this your way of baiting me to do a limit

Open a help channel with it

[viral copper]

And now you're stuck with me

Helpers Honor

[blissful saffron]

truly I am

unless I decide to go to sleep

,ti

[twin meteor]

The current time for derp_z is 01:42 AM (AWST) on Sun, 02/07/2023.

[blissful saffron]

let's ask wolfie

[viral copper]

We know the answer

e/(e - 1)

[blissful saffron]

hmm

[wind kestrel]

it would be pretty difficult to find the limit using binomial expansion

[hard atlas]

WA doesnt know

[viral copper]

Man that's literally

The geometric

Series

[blissful saffron]

[viral copper]

For e, e^2, e^3...

[blissful saffron]

MathIsAlwaysRight may have a point lol

[viral copper]

Can each of these limits individually be written as some sort of power of e

Which then gets GP'd

[wind kestrel]

can you write out what you're trying to say lmao

[blissful saffron]

he already did

[viral copper]

Indeed

[blissful saffron]

[viral copper]

What's the formula for e^x again

[blissful saffron]

er

[viral copper]

$\lim_{t \to \infty} \left ( 1 + \frac xt \right )^t$

[twin meteor]

neonperseus

[viral copper]

Hmm

[blissful saffron]

yeah

[viral copper]

I think

[blissful saffron]

its tight

right

[viral copper]

$\lim_{t \to \infty} \sum_{k = 1}^{\infty} \left ( 1 + \frac kt \right )^t$

[twin meteor]

neonperseus

[viral copper]

THIS IS TANTALIZINGLY CLOSE

[hard atlas]

so, we have our sum (1/n)^n + (2/n)^n + ... + ((n-1)/n)^n + (n/n)^n. focusing on the second to last term, that's just (1-1/n)^n and converges to e^-1. the others from the right converge to e^-2, e^-3 and so on

[blissful saffron]

err the sum is to t right not to infty

[viral copper]

oh bruh

[hard atlas]

e/(e-1) is actually the GP for 1/e

[wind kestrel]

you can't just do that lol

[viral copper]

Does it matter

[viral copper]

yeah

[blissful saffron]

Ah yes 1/e^n

[viral copper]

I think what denascite said is right

[hard atlas]

yeah but its missing details

[blissful saffron]

the one which I said is not convergent

[hard atlas]

like lag said, you cant just do that

[viral copper]

Such as?

Hmm

[hard atlas]

with each new n there are new terms appearing. why is that fine

and if we kind of write this down where each row represents a certain value of n. we are then doing the limits by column. why is that allowed

[viral copper]

I don't get what you're saying

[hard atlas]

yeah not sure how to express what I mean without getting out paint

or typing a bunch

[viral copper]

it's alright

[blissful saffron]

does neon need to know analysis to justify these lol

[flat whale]

Why not

Neon do you know dominated convergence theorem

[viral copper]

nope

[blissful saffron]

That sounds like analysis lol

[viral copper]

sounds sexy

doesn't anymore

[wind kestrel]

because the number of terms is not constant

it depends on n

[viral copper]

n is very very big

[hard atlas]

we are roughly switching the order of the limits. we are first letting k->inf and then n->inf

[flat whale]

Nevermind what I said then

[blissful saffron]

lesbesgue dominated convergence theorem?

[flat whale]

Right

[flat whale]

What about the equivalent Monotone convergence theorem?

[viral copper]

no idea

[blissful saffron]

booo

[hard atlas]

$\int_{n=0}^{\infty} \int_{k=0}^n \left( \frac k n\right) ^n \dd k \dd n$ in counting measure or something

[blissful saffron]

chapter 2.3, understanding analysis

[viral copper]

omg integrals

[blissful saffron]

measure theory

[viral copper]

Wait that's not a nice integral

[flat whale]

I guess squeeze theorem is the last result

[viral copper]

yay I know what that is

[last creek]

double integrals

[hard atlas]

nah wait I messed up

[blissful saffron]

this can be solved with just MCT?

[flat whale]

Bound (1-k/n)^n from above by e^(-k)

[hard atlas]

that shows that it converges. but does it show that it converges to the correct thing?

[flat whale]

And below by something that approaches to e^(-k) you get from binomial expansion

[blissful saffron]

huh neat

[flat whale]

Takes a lot more work, but the same idea of showing each term converges to exp(-k)

Involves characteristic functions and integrals

[viral copper]

I'll probably just let this go for now

Especially if it calls for math beyond my level

[hard atlas]

do you have a specific proof in mind? involving char functions?

[viral copper]

I hope that was directed to riemann

[hard atlas]

yes

[flat whale]

Yea it's on MSE

[hard atlas]

can you share?

[flat whale]

https://math.stackexchange.com/questions/164074/how-to-evaluate-lim-limits-n-to-infty-sum-limits-k-1n-fracknn

[blissful saffron]

love how it ends with

[flat whale]

Characteristic being indicator, not the Fourier transform one

[blissful saffron]

"Can someone please help me?"

[viral copper]

Will you close the channels yourselves after you're done?

[hard atlas]

thanks

ah yes I first thought you meant the fourier ones

[flat whale]

Lol yea I realized that after I sent the link

[hard atlas]

but of course using the indicator ones like that is much smarter

[flat whale]

This was the answer I was describing
https://math.stackexchange.com/a/4546589

[hard atlas]

I thought somewhere along those lines but not enough to make it precise

been some time since I've done analysis like that

[flat whale]

Same. No way I can think of this in just a few minutes

I just Google fast and pretend like I understand

[hard atlas]

I also googled but didnt find it

[blissful saffron]

wow that is clever

[flat whale]

It's a good problem. Love that there are so many solutions

[hard atlas]

yeah its a nice one. and I think I would have gotten it with enough time, so I'm happy with that

[cobalt crypt]

weird that they didnt just apply MCT to the counting measure

[cobalt crypt]

your mentioning DCT gave it away

[flat whale]

Snow you might as well !nosols me

[cobalt crypt]

[hard atlas]

I mean whether you apply MCT or DCT to counting measure doesnt make a huge difference at that point

[cobalt crypt]

doing it with char funcs is funky notation

[hard atlas]

it removes the dependence on n for the upper bound which was one thing I was struggling with

[cobalt crypt]

[vocal sleet]

@viral copper Has your question been resolved?

[viral copper]

I don't know has it

[hard atlas]

well we are happy with the solution

are you?

[viral copper]

My satisfaction doesn't matter lol

I think it's beyond my level rn

[hard atlas]

well its your channel

[viral copper]

If you're satisfied I'm satisfied

[vocal sleet]

Channel closed

Closed by @viral copper

Use .reopen if this was a mistake.

[viral copper]

Thanks for your time though

[vocal sleet]

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.

[empty linden]

something isnt right. i thought only the value inside the bracket must be squared

[empty linden]

so shouldnt it be 2 sqroot(x), and not 4 sqroot(x)?

[hard atlas]

2 is also in the bracket?

$(2\sqrt x)^2 = (2 \sqrt x)(2 \sqrt x) = 2\cdot 2\cdot \sqrt x \cdot \sqrt x = 4 x$

[empty linden]

it wasnt in it at first

[twin meteor]

denascite

[empty linden]

but he put the bracket there after

oh, nvm

[hard atlas]

he is squaring both sides

everything on both sides

[empty linden]

if its an equation, we square the constant behind the sq root?

but in an expression, the constant remains unchanged right?

if there was no bracket, would we square the 2, or the x only?

[hard atlas]

$2 \sqrt x ^2 = 2 x$

[twin meteor]

denascite

[hard atlas]

if thats what you mean

[empty linden]

yeah but

i was just wondering if we should square the constant behind the square root only if its an equation

since an equation squares both sides, but an expression doesnt

[hard atlas]

this has nothing to do with whether its an equation or not

2 is either included in the bracket or not

[empty linden]

lets say we have csqroot axb. we wouldnt square the c

[empty linden]

hm

so if the question has a bracket, we square it, but if it doesnt have a bracket, we dont?

since a bracket applies to the whole thing

[hard atlas]

$2 \sqrt x ^2 = 2(\sqrt x)^2$

[twin meteor]

denascite

[hard atlas]

and 2 is not in the bracket

[empty linden]

yeah, so 2 sqroot(x)^2 is just 2sqrootx

since the square cancelled the square root

[hard atlas]

its 2x

[empty linden]

but in this example, he squared the constant behind it

which is why im confused

shouldnt this be 2x and not 4x?

[hard atlas]

he squared the whole thing

you cant just square part of the thing

[empty linden]

but in csqroot axb, we dont square the c

thats what i found in one of the answers

[hard atlas]

you dont square anything in there because you havent written down any square

[empty linden]

so it ended up being c sqroota sqrootb

so why do we sometimes square the constant, but sometimes not?

why do some examples say we squaare the constant behind the root, and sometimes not?

[hard atlas]

this is not about 2 being a constant or not

2sqrtx is one thing

you are squaring the entire thing

because you square both sides of the equality

[empty linden]

what if its not an equation, but an expression?

[hard atlas]

an expression includes the square already. so either the 2 is included in the bracket or its not

[empty linden]

what if its not

[hard atlas]

then you dont square it

[empty linden]

but in this example, we didnt square the c

well, the brackets in lns arent necessary but

[hard atlas]

well why should we square anything there

there is no ^2 in sight

[empty linden]

i had to expand it

to get rid of the sqroot

which is how the answer turned out to be ln c + ln a/2 + lnb/2

[hard atlas]

and?

[empty linden]

ok so basically what i got from this is

[hard atlas]

you didnt square anything

[empty linden]

if bracket, then square the whole thing

if no bracket, then square whats inside the root

thats it right?

[hard atlas]

square the things inside the bracket

[vocal sleet]

@empty linden Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

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.

[zenith zephyr]

how do this?

why it wrong

[paper depot]

cos^2(8x)...

tho it's also sus how exactly you got that expression

[zenith zephyr]

so its this?

are you postivie on that

dont sell bro

[paper depot]

... i'm a woman, don't call me bro

[zenith zephyr]

bro is gender neutral term

[paper depot]

your experience is not universal.

you haven't done what is asked for, was my point.

[zenith zephyr]

wdym?

[paper depot]

rewrite the expression in terms of first powers of the cosines of multiple angles

you've got a second power in there still

[zenith zephyr]

does this seem right

[paper depot]

dunno let's check

[vast shale]

Siiii

[paper depot]

,w simplify 4sin^2(4x) - (3-4cos(4x)-cos(8x))

[twin meteor]

Results provided by WolframAlpha

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

Requested by ann_dec#0

[paper depot]

nope

[zenith zephyr]

thats not what i put

[paper depot]

i had WA simplify 2 * (original expression) - 2 * (your answer)

if your answer was correct it would have come out as 0

[zenith zephyr]

,w simplify 1/2[3-4cos(4x)-cos(8x)

[twin meteor]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by nythzn#0

[zenith zephyr]

bro you sold so hard

you sold smh

smh smh smh

you dont deserve to have the helpful role

youre a bot

gg

[vast shale]

https://cdn.discordapp.com/emojis/1094139869959299082.webp?size=48&quality=lossless

[paper depot]

i don't know what "sold" means, but also i told you not to call me bro

you chose to double down on it

[vast shale]

What does selling mean

[paper depot]

and now accusing me of being a "bot"

[pallid zenith]

what is selling

[hasty pulsar]

||It's used in the context of a multiplayer online team-based battle game where you give the enemies kills (by dying from them), and your allies will say that you're selling, because you're giving them (free) kills or points.||

[flat whale]

Ann your sales numbers are through the roof

[hasty pulsar]

It seems POV has raged and left the server, channel may get closed.

[vocal sleet]

@zenith zephyr Has your question been resolved?

[paper depot]

???

[hasty pulsar]

What they meant with selling

[paper depot]

how's that apply here

what did i sell

[flat whale]

Just gamer rage talk

Nothing logical about it

.close

[vocal sleet]

Channel closed

Closed by @flat whale

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.

[pallid zenith]

whatd you try

[boreal star]

What's the domain of arccos?

[rotund hornet]

-1,1

[boreal star]

So for what domain of x can sin take values in [-1,1]?

[rotund hornet]

Wdym?

[boreal star]

What are the possible values of x such that sin x lies in [-1,1]?

[rotund hornet]

Isn’t it jus -1 and 1?idk

[boreal star]

Why? What are the possible values x can take, not sin x

It should be R, right?

[rotund hornet]

R?

@boreal star

[boreal star]

(-infty, infty)

[rotund hornet]

Ohhhh

I though you meant what values can x in sinx take between -1 and 1

My bad

[boreal star]

Yeah so the required domain is R then, right?

[rotund hornet]

Yeah

[boreal star]

Now for the range, sin x takes all the values in R, so the range of 2arccos(sin x) should be twice of arccos, right?

[rotund hornet]

Because of the 2?

[boreal star]

Yes

[rotund hornet]

Yeah and the range of arccos is 0,pi

[boreal star]

Yes

[rotund hornet]

So then what?

[boreal star]

Yeah so the range is [0,2pi]

[rotund hornet]

So you multiplied them?

[boreal star]

Yes

[rotund hornet]

That’s also what you did here with -1,1?

[boreal star]

No

That was domain

[rotund hornet]

Why can you multiply for range but not domain?

[boreal star]

What is domain?

[rotund hornet]

The possible x values

[boreal star]

Basically the domain of a function is the set of values the input of the function can take so that the function returns a value in the codomain.

Here, the domain is the set of values of x such that the function gives real values. If I multiply the function by any real number (but 0), it does not affect the domain, as I can still take the same values of x and get a real output from the function

On the other hand, range is the set of values that the function returns from the argument. So if I'm multiplying a real number to the function, it changes the range of the function.

[rotund hornet]

Ohhhhhhhh

Understood

[rotund hornet]

So yk how sinx range is -1,1 do we multiply 0 and 2pi by it?

[boreal star]

No. See we need the range of y=2arccos(sin x), the range of sin x is [-1,1], so y ranges from 2arccos(-1) to 2arccos(1), right? Now since the domain of arccos is [-1,1], [arccos(-1),arccos(1)] is the range of arccos, which is [0,pi], so 2arccos(-1) to 2arccos(1) would be [0,2pi]

[rotund hornet]

So you did multiply it But it just gives you the same thing?

[boreal star]

It does not give me the same thing. If the range of y is [a,b], then what should be the range of 2y?

[rotund hornet]

2a,2b

[rotund hornet]

No I mean the 0,2pi

Multiplying it by -1,1 gives the same thing

[boreal star]

Why would you multiply it with -1 or 1?

[rotund hornet]

Wait nvmmm

I get it

Thank you so much

.close

[vocal sleet]

Channel closed

Closed by @rotund hornet

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.

[hearty delta]

Hi, I need help with this problem, not the answer jsut an explanation to what it is asking:

[hearty delta]

is the boundary just the highest/lowest c values on the map or?

[vocal sleet]

@hearty delta Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[vocal sleet]

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.

[vocal sleet]

@elfin moon Has your question been resolved?

[blissful saffron]

open a complex analysis textbook

[elfin moon]

.close

[vocal sleet]

Channel closed

Closed by @elfin moon

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.

[trim walrus]

How do you get started with

[paper depot]

so A = ds(4444^4444), B = ds(A), and we're asked for ds(B), where ds stands for digit-sum.

[trim walrus]

Yes

[paper depot]

in other words we're asked for ds(ds(ds(4444^4444)))

about how many digits do you expect 4444^4444 to have?

[trim walrus]

I mean like how am I suppose ans that

They are finite

But idk how many

[paper depot]

well, what happens to the digit length of a number when, say, you square it?

roughly

[trim walrus]

They double there self

But there are expectations perhaps like 2²=4, 3³= 9??

[paper depot]

did you mean "expectations" or did you fall victim to autocorrect

[trim walrus]

Exception

[paper depot]

ok you fell victim to autocorrect

anyway, your wording of "They double there self" is clunky, but yes. the digit length approximately doubles.

n^2 is certainly not more than twice as long as n itself.

what do you think would happen to a number's length if you raised it to the 4444th power?

[trim walrus]

Well, like if you square then the lengths double then might be that if you raise it to 4444th Power the length might be x4444

[paper depot]

yes

so about how many digits would you expect 4444^4444 to have?

[trim walrus]

Ummm to have 4444 digits

Roughly

[paper depot]

are you sure

or are you not reading the things you yourself are saying

raising to the 4444th power multiplies the length of the number (roughly!) by 4444,
but what is the length of 4444 itself? certainly not 1!

[trim walrus]

Alr I get it

4×4444

[paper depot]

yes

and this is a rough estimate, but it'll be enough for us.

[trim walrus]

but we only know the length not the digits ....

[paper depot]

our gigantic power has, let's say, at most 18,000 digits.

[trim walrus]

Alr

[paper depot]

and while we do not know exactly what those digits are,

this info still lets us put an upper bound on ds(4444^4444).

do you see how or should i prompt you further?

[trim walrus]

Ahhhh.....NO

I don't see it

Guide me ....

[paper depot]

what's the biggest that one individual digit can be?

[trim walrus]

9

[paper depot]

so with a number 18000 digits long, at most how big can its digit sum be?

[trim walrus]

9×18000

[paper depot]

exactly

[trim walrus]

That's it??

[paper depot]

no, that's not yet it.

but we know A = ds(4444^4444) ≤ 162000.

this means A is at most a 6-digit number starting with a one.

that severely limits what ds(A) could be !

[trim walrus]

How can you claim this?? I don't get it...

[paper depot]

A = ds(4444^4444) ≤ 162000.

A ≤ 162000

do you understand this or not

[trim walrus]

Yes I certainly do, but how does that and that relate??

[paper depot]

if A ≤ 162000, can A be longer than 6 digits?

[trim walrus]

Yep, I'm dumb

Alr

So, ds(4444^4444) can't be longer than

6

[merry python]

IMO 1975 Problem 4

[trim walrus]

Pw dpp problem number 2

[paper depot]

ds(4444^4444) can't be longer than 6 digits

[trim walrus]

Alr

[paper depot]

so what's the biggest that ds(ds(4444^4444)) could be, given this?

[trim walrus]

Welll, as ds(4444^4444) can't be greater than 6, the atmost level the max number will be 999999 ds(999999)....

Ap 9×6 = 56

[paper depot]

9*6 is not 56

[trim walrus]

54

F

[paper depot]

also we can do a bit better than that, since not all six-digit numbers are available as values for A

but i suppose that actually doesn't matter.

so B ≤ 54 now

[merry python]

You can keep the 1 in the front
Since the ds is no more than 162000
so instead of 999999 find ds of 199999

[paper depot]

as ds(4444^4444) can't be greater than 6
this is incorrect as written btw

[trim walrus]

Why is that so??

[paper depot]

A ≤ 162000, so if A is six digits long then its first digit has to be 1

[trim walrus]

Hmm

Alr

ds(199999) = 1+9×5 = 46

B <= 46

[paper depot]

yeah, ok

[trim walrus]

We need to find ds(B)

[paper depot]

now find an upper bound for ds(B)

[merry python]

apply the same thing again
First digit will be 4 and two digit number

[paper depot]

we won't be done even after finding it, however.

[trim walrus]

So B =< 42, it can't be greater than 2 (length) so at peak it has to be 42 again??

[merry python]

how 42?

the length must be two and first digit must be 4

so the max you can have is 49

find its ds

[paper depot]

39, actually.

no number between 40 and 42 can beat 39's digit sum

[merry python]

well that doesn't make a difference

[trim walrus]

49

[merry python]

okay so what is its digit sum

[trim walrus]

13

[merry python]

So we're now done with the first part
Now are you aware of modulo?

[trim walrus]

Yes but

Wait

[paper depot]

yes? is there something wrong?

[trim walrus]

B <= 42, so it as to be a smaller number less than 42 and should have two digits so how's 42 not correct??

[paper depot]

are you asking why B ≤ 42 doesn't mean ds(B) ≤ ds(42)?

[trim walrus]

I mean it is 2 digit long

[merry python]

digit sum of 39 or 49 is greater than digit sum of 42

[trim walrus]

Like here, I'm confused here again how's the first digit being one??

[paper depot]

can A be 2##### or 3##### or 4##### etc.?

[trim walrus]

No, but A can be 161999??

[merry python]

at most 162000

[paper depot]

it can. and?

[trim walrus]

Yes

And be 162000 at its peak

It can't go beyond that

But here B <= 42, how's 49 coming up??

[paper depot]

we're looking at the biggest digit sum possible among the numbers 1, 2, 3, ..., 42

and saying that none of them have digit sum greater than 4 (max value of tens digit) + 9 (max value of units digit)

[merry python]

well just so you say we can consider 39 which is less than 42

[trim walrus]

Yes

[paper depot]

do you wish to argue AGAINST any part of my argument?

yes or no

[trim walrus]

No

Alr we'll continue

So we get 39 as per I understand

[paper depot]

"we get 39" is vague

we get that ds(B) ≤ 12.

[merry python]

.

[trim walrus]

Alr

Yes

[paper depot]

now consider

for ANY natural number n, it is true that ds(n) ≡ n (mod 9)

do you understand this or not

[trim walrus]

I understand that 9 | ds(n) - n

[paper depot]

should i read this as "I understand that 9 | ds(n) - n, but not ds(n) ≡ n (mod 9)"?

yes or no

[trim walrus]

No

I understand both

I understand what you're saying

It is true

[paper depot]

why not just say "yes"

saves effort for both you and me

[trim walrus]

Alr alr

[paper depot]

anyway, applying this three times, we get that ds(B) has the same remainder mod 9 as that big power we started with, 4444^4444

are you able to find the remainder of 4444^4444 mod 9?

[trim walrus]

0

[merry python]

No

[trim walrus]

1

Alr

[merry python]

Are you just guessing

[paper depot]

[trim walrus]

Bruh, you just gave the himt

[paper depot]

i mean there are finitely many options

basudev, you should give straight answers to yes/no questions that people ask you

right now you AVOIDED that

and instead started guessing

i find that extremely annoying

please do not do this ever again

[merry python]

first of all find remainder of 4444 mod 9

[trim walrus]

Alr

9×493+smth= 4444
smth = 4444-9×493

7??

[merry python]

well 7 is right

Now try to find remainder of what power of 7 mod 9 is one

[trim walrus]

Leaves a remainder of 0, so

??

[paper depot]

WHAT leaves a remainder of 0?

[trim walrus]

Fuck shake dude 9-7= 2

2^x = 1

X=0

0000000

Alr I got it

[merry python]

[paper depot]

i think you in fact did not get it.

[paper depot]

7^n is never 0 mod 9 btw

[merry python]

oh shi-

rem 1 is what I meant

Okay I'm out. Better let Ann help

[paper depot]

i'm having trouble helping actually

[trim walrus]

Alr I'll figure it out

Myself

Thanks for the help

.close

[vocal sleet]

Channel closed

Closed by @trim walrus

Use .reopen if this was a mistake.

[vocal sleet]

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.

[mint iris]

Hello, can anyone help out with this question (e) . I tried using the pq formula, but could not get to a suitsble answer

[mint iris]

.help

[vocal sleet]

Commands:
clopen: .close, .reopen, .solved, .unsolved
consensus: .poll
factoids: .tag
help: .help

Type .help <command name> for more info on a command.

[outer warren]

pq formula is overkill for that,
still show us how you tried to apply that

[mint iris]

.I can show

everything above Retry

Is what I have done

[outer warren]

you're using the wrong p value

p is the coefficient of x,
which is 14, not 7

[mint iris]

I get it

I will try and inform if i stuck

There is the answer I think

I have got it right, thanks for helping out

.close

[vocal sleet]

Channel closed

Closed by @mint iris

Use .reopen if this was a mistake.

[vocal sleet]

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.

[celest kite]

Im stuck in this question please help.

[celest kite]

I try writing a system of equations then use a matrix that represents the system and perform row operations until the matrix is in echeleon form. However, I seem to do something wrong in the process

[vast shale]

-96 = a(-3)^3 +b(-3)^2 +c(-3) +d -2 = a(-1)^3 + b(-1)^2 +c(-1) +d 4=a(1)^3 + b(1)^2 +c(1) +d 19=a(2)^3 +b(2)^2 +c(2)+d then you solve this system of 4 equations to find the values of a,b,c and d

oh show me how you did it

[celest kite]

Too many steps but. I started with this

Not a neat handwriting sorry

[vast shale]

don't worry

[celest kite]

Then I interchange rows 1 and 3

From the matrix

Then I did operations to perform row operations that make 0 appear below in column 1

R2=r1+r2

Beginning with that

[karmic imp]

If you noticed, you can simplify R2 and R3 first, it would make things much more easier

If you added the two, you just have b and d left

[celest kite]

Simplify? Would there only be R1 and R4 if I add R2 and R3? So it be only two equations

But what about the -2 and 4 on the other side

[karmic imp]

I'm saying the left side would be much easier to simplify since you have zeros

[celest kite]

Is there a way to show me because I got lost? Do you mean add them up to have 0=2?

[karmic imp]

I never said 0 = 2

I said you can add up R2 and R3

[dreamy viper]

yes^

0s so you add less in columns 1 and 3

[karmic imp]

If you noticed how in R2 and R3, the a and c terms have equal coefficients but are opposite

So they would cancel out

[dreamy viper]

then interchange R3 and R1 if you haven't

[vocal sleet]

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.

[vast shale]

welp

I'm stupid

[karmic imp]

Do you understand the process of Gaussian Elimination, in general?

You want to end up with a upper triangular matrix

[celest kite]

Not really I got started. I can follow steps given but understanding not really. I just started learning echelon form

[karmic imp]

Since you are new to echelon, I suggest this video
https://www.youtube.com/watch?v=eDb6iugi6Uk&ab_channel=TheOrganicChemistryTutor
And I suggest working with smaller matrices at first, to get the hang of it before going into a 4x4 and bigger as it takes more algebra

Here's a 4x4 example
https://www.youtube.com/watch?v=4VAXv6yRULU&ab_channel=TheOrganicChemistryTutor

[dreamy viper]

I thought it was just called reducing the matrix to echelon form didnt know it had a name lol

[karmic imp]

A lot in math has a name

[celest kite]

Thanks!
I think for the problem I have to interchange row 1 and 3. Then R2=r1+r2 next R3=27r1+r3 but if there is an easier way...

[karmic imp]

Overall, you have slightly bigger numbers so it's going to take time

[celest kite]

Yeah... It's just too many steps that Im doing at some point I eventually make a mistake

[karmic imp]

Just take your time with the math

[vocal sleet]

@celest kite Has your question been resolved?

[celest kite]

I finally got it for R3=27r1+r3 I incorrectly did 27(1)-96 instead of 27(4)-96 thats I was off.

[karmic imp]

Fyi you should react to the bot's message so the channel doesn't close

[karmic imp]

Can you post your full work?

[karmic imp]

React meaning hit the check if you are done with the channel or x meaning you need it open still

[vocal sleet]

Channel closed

Closed by @celest kite

Use .reopen if this was a mistake.

[karmic imp]

.reopen

[vocal sleet]

✅

[celest kite]

Sure i would have to write it neatly though

Its chicken scratch

right now

[karmic imp]

Were you done with the channel? Because the check means you are done and it closes the channel

[celest kite]

Yeah im done I got 3x^3-2x^2+3

[karmic imp]

If you got the answer, then what did you mean by

I finally got it for R3=27r1+r3 I incorrectly did 27(1)-96 instead of 27(4)-96 thats I was off.

[celest kite]

I was explaining why I wasn't getting it correctly I was doing the row operations and accidentally put a 1 instead of a 4

[karmic imp]

Alright since you're done with the channel, I'll close it

.close

[vocal sleet]

Channel closed

Closed by @karmic imp

Use .reopen if this was a mistake.

[karmic imp]

Btw no need to because I thought you were saying you did math wrong with this statement

I finally got it for R3=27r1+r3 I incorrectly did 27(1)-96 instead of 27(4)-96 thats I was off.

If you want you can dm the work, still

[vocal sleet]

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.

[warm atlas]

Hello, I might just be doing this wrong but the answers for angle A don't match up (which is underlined in red). I have looked over it but I can't seem to find what I've gotten wrong. It could also be that the answer for BC was wrong.. (Calculator was used btw)

[vocal sleet]

@warm atlas Has your question been resolved?

[warm atlas]

<@&286206848099549185>

[wraith python]

You added a 2 in front of sin(2A) when you shouldn't have.

[warm atlas]

Ah damn, I see. Tyvm.

.close

[vocal sleet]

Channel closed

Closed by @warm atlas

Use .reopen if this was a mistake.

[vocal sleet]

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.

[spring gate]

i understand everything here

exept where that 1/4 came from?

am i being dumb and just missing something?

[urban edge]

Substitute du/4 for dx

[last creek]

[vast shale]

^^^

[spring gate]

um... ok but why does dx = 1/4?

[last creek]

dx = du / 4

and not 1/4

[last creek]

u have to substitute du / 4 in place of dx

[spring gate]

but both are in the same line?

int (4x-3)^3 dx

[spring gate]

where are they getting du = 1 then?

[last creek]

where are they getting du = 1 ?

[last creek]

[spring gate]

im really not following this at all

[last creek]

that's int u^3 du/4

u can take the constant term out of the integral

[spring gate]

what constant?

[last creek]

1/4

1/4 is a constant value

[spring gate]

so the 1/4 comes from the (4x-3)

?

its taken from there

right?

[last creek]

read each line thoroughly
they are explaining everything

if u substitute something u have to find a relation as well

as in this example we substituted 3x+4 as u

but can't solve $\int u^5 dx$

[spring gate]

substitution seems so pointless

[twin meteor]

_kapp_

[last creek]

so we need to find the relation between du and dx

[spring gate]

(ax-c)^b = 1/a*(ax-c)^b+1/b+1

is that formula correct in terms of integration

[last creek]

2 /
??

[spring gate]

2/ what

[last creek]

can u write in latex or on a notebook and send me a picture

Usually the method of integration by substitution is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. Doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function.

[spring gate]

ignore the qudratic at the top

[last creek]

seems correct to me

but as i said above

substitution in general is better than memorising so many results

[spring gate]

ok so

all im hearing is that its a better meathod for advanced stuff

[last creek]

yes

[spring gate]

but rn its just overcomplicating simple formula

[spring gate]

why does "du/dx" = 4

d(4x-3) / dx